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Then, for all m \\in Z and all i \\in \\0,\\l…","labels":["prop:translationInvariance"],"detail_key":"p0"},{"id":"n4","layer":"informal","project":"p0","title":"def:glide","kind":"proof","summary":"We prove a stronger statement, called the glide symmetry of frieze patterns. First, consider th…","labels":["def:glide"],"detail_key":"p0"},{"id":"n5","layer":"informal","project":"p0","title":"cor:imageFinite","kind":"corollary","summary":"Let f be a nowhere-zero F-valued pattern of height n. Then, \\rm Im(f) := \\f (i,m) : i \\in \\1,\\l…","labels":["cor:imageFinite"],"detail_key":"p0"},{"id":"n6","layer":"informal","project":"p0","title":"Consider the finite set D = \\(i,m) : i \\in \\1,\\ldots, n\\, m \\in \\0,\\ldots, n\\\\. By Propos…","kind":"proof","summary":"Consider the finite set D = \\(i,m) : i \\in \\1,\\ldots, n\\, m \\in \\0,\\ldots, n\\\\. By Proposition…","labels":[],"detail_key":"p0"},{"id":"n7","layer":"informal","project":"p0","title":"def:flute","kind":"definition","summary":"A sequence (a_k), indexed by N^* and consisting of positive integers is called pandean if a_1 =…","labels":["def:flute"],"detail_key":"p0"},{"id":"n8","layer":"informal","project":"p0","title":"l:nFluteNonEmpty","kind":"lemma","summary":"For any positive integer n, the set Flute(n) is non-empty.","labels":["l:nFluteNonEmpty"],"detail_key":"p0"},{"id":"n9","layer":"informal","project":"p0","title":"It is clear that the constant sequence consisting entirely of ones is pandean, and such a…","kind":"proof","summary":"It is clear that the constant sequence consisting entirely of ones is pandean, and such a pande…","labels":[],"detail_key":"p0"},{"id":"n10","layer":"informal","project":"p0","title":"l:FibFlute","kind":"lemma","summary":"1) If n is odd, the n-tuple \\[ (F_2,F_4, F_6, \\ldots, F_n-1, F_n, F_n-2, F_n-4, \\ldots, F_5, F_…","labels":["l:FibFlute"],"detail_key":"p0"},{"id":"n11","layer":"informal","project":"p0","title":"These are a tedious but straightforward calculation.","kind":"proof","summary":"These are a tedious but straightforward calculation.","labels":[],"detail_key":"p0"},{"id":"n12","layer":"informal","project":"p0","title":"lem:FluteReduction","kind":"lemma","summary":"In a flute (a_1, \\ldots, a_n), one of the following two statements holds. 1) a_2 = 1 or a_n-1 =…","labels":["lem:FluteReduction"],"detail_key":"p0"},{"id":"n13","layer":"informal","project":"p0","title":"Suppose that 1) does not hold. In particular, a_2 - a_1 > 0 and a_n - a_n-1 < 0. We prove…","kind":"proof","summary":"Suppose that 1) does not hold. In particular, a_2 - a_1 > 0 and a_n - a_n-1 < 0. We prove that…","labels":[],"detail_key":"p0"},{"id":"n14","layer":"informal","project":"p0","title":"prop:FluteBounded","kind":"proposition","summary":"Fix a positive integer n, and let (a_1, \\ldots, a_n) \\in \\rm Flute(n). For any i \\in \\1,\\ldots,…","labels":["prop:FluteBounded"],"detail_key":"p0"},{"id":"n15","layer":"informal","project":"p0","title":"eq:ai","kind":"proof","summary":"Consider the statement \\[ P_n: \\textIf (a_1, \\ldots, a_n) \\in \\rm Flute(n) \\text, then a_i \\leq…","labels":["eq:ai","eq:ajs"],"detail_key":"p0"},{"id":"n16","layer":"informal","project":"p0","title":"def:arith_fp","kind":"definition","summary":"A Q-valued pattern of height n is said to be an arithmetic frieze pattern if it takes values in…","labels":["def:arith_fp"],"detail_key":"p0"},{"id":"n17","layer":"informal","project":"p0","title":"prop:friezeIffFlute","kind":"proposition","summary":"1) Let f be an arithmetic frieze pattern of height n. For all m \\in Z, the n-tuple \\[ (f (1,m),…","labels":["prop:friezeIffFlute"],"detail_key":"p0"},{"id":"n18","layer":"informal","project":"p0","title":"1) Note that we have f(1,0) = f(n,0) = 1 by definition. Moreover, f is arithmetic and so…","kind":"proof","summary":"1) Note that we have f(1,0) = f(n,0) = 1 by definition. Moreover, f is arithmetic and so the fi…","labels":[],"detail_key":"p0"},{"id":"n19","layer":"informal","project":"p0","title":"c:arithFriezePatSetNonEmpty","kind":"corollary","summary":"Fix a positive integer n. The set Frieze(n) is non-empty.","labels":["c:arithFriezePatSetNonEmpty"],"detail_key":"p0"},{"id":"n20","layer":"informal","project":"p0","title":"The proof of Lemma \\refl:nFluteNonEmpty showed that (1,1,\\ldots, 1) is a flute. The claim…","kind":"proof","summary":"The proof of Lemma \\refl:nFluteNonEmpty showed that (1,1,\\ldots, 1) is a flute. The claim then…","labels":[],"detail_key":"p0"},{"id":"n21","layer":"informal","project":"p0","title":"mainTheorem","kind":"theorem","summary":"For all n \\geq 1, we have \\[ u_n = F_n. \\]","labels":["mainTheorem"],"detail_key":"p0"},{"id":"n22","layer":"informal","project":"p0","title":"By Proposition \\refprop:friezeIffFlute, every entry of an arithmetic frieze pattern of he…","kind":"proof","summary":"By Proposition \\refprop:friezeIffFlute, every entry of an arithmetic frieze pattern of height n…","labels":[],"detail_key":"p0"},{"id":"n23","layer":"formal","project":"p0","title":"imageFinite","kind":"theorem","summary":"∀ (F : Type u_1) [inst : Field F] (f : Prod Nat Nat → F) (n : Nat) [inst : nzPattern_n F f n],…","labels":[],"detail_key":"p0","name":"imageFinite","module":"FriezePatterns.chapter1"},{"id":"n24","layer":"formal","project":"p0","title":"pattern_n","kind":"inductive","summary":"(F : Type u_1) → [inst : Field F] → (Prod Nat Nat → F) → Nat → Prop","labels":[],"detail_key":"p0","name":"pattern_n","module":"FriezePatterns.chapter1"},{"id":"n25","layer":"formal","project":"p0","title":"pattern_nContinuant1","kind":"theorem","summary":"∀ (F : Type u_1) [inst : Field F] (f : Prod Nat Nat → F) (n : Nat) [inst_1 : nzPattern_n F f n]…","labels":[],"detail_key":"p0","name":"pattern_nContinuant1","module":"FriezePatterns.chapter1"},{"id":"n26","layer":"formal","project":"p0","title":"pattern_nContinuant2","kind":"theorem","summary":"∀ (F : Type u_1) [inst : Field F] (f : Prod Nat Nat → F) (n : Nat) [inst_1 : nzPattern_n F f n]…","labels":[],"detail_key":"p0","name":"pattern_nContinuant2","module":"FriezePatterns.chapter1"},{"id":"n27","layer":"formal","project":"p0","title":"translationInvariance","kind":"theorem","summary":"∀ (F : Type u_1) [inst : Field F] (f : Prod Nat Nat → F) (n : Nat) [inst : nzPattern_n F f n] (…","labels":[],"detail_key":"p0","name":"translationInvariance","module":"FriezePatterns.chapter1"},{"id":"n28","layer":"formal","project":"p0","title":"FluteBounded","kind":"theorem","summary":"∀ (n : Nat), GT.gt n 0 → ∀ (f : flute n) (i : Nat), LE.le i (HSub.hSub n 1) → LE.le (f.a i) (Na…","labels":[],"detail_key":"p0","name":"FluteBounded","module":"FriezePatterns.chapter2"},{"id":"n29","layer":"formal","project":"p0","title":"FluteReduction","kind":"theorem","summary":"∀ (n : Nat) (f : flute n), Or (Or (Eq (f.a 1) 1) (Eq (f.a (HSub.hSub n 2)) 1)) (Exists fun i =>…","labels":[],"detail_key":"p0","name":"FluteReduction","module":"FriezePatterns.chapter2"},{"id":"n30","layer":"formal","project":"p0","title":"csteFlute","kind":"def","summary":"(n : Nat) → Inhabited (flute n)","labels":[],"detail_key":"p0","name":"csteFlute","module":"FriezePatterns.chapter2"},{"id":"n31","layer":"formal","project":"p0","title":"fib_flute_even","kind":"def","summary":"(k : Nat) → flute (HAdd.hAdd (HMul.hMul 2 k) 2)","labels":[],"detail_key":"p0","name":"fib_flute_even","module":"FriezePatterns.chapter2"},{"id":"n32","layer":"formal","project":"p0","title":"fib_flute_odd","kind":"def","summary":"(k : Nat) → flute (HAdd.hAdd (HMul.hMul 2 k) 1)","labels":[],"detail_key":"p0","name":"fib_flute_odd","module":"FriezePatterns.chapter2"},{"id":"n33","layer":"formal","project":"p0","title":"flute","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p0","name":"flute","module":"FriezePatterns.chapter2"},{"id":"n34","layer":"formal","project":"p0","title":"fluteSetNonEmpty","kind":"theorem","summary":"∀ (n : Nat), Nonempty ↑(fluteSet n)","labels":[],"detail_key":"p0","name":"fluteSetNonEmpty","module":"FriezePatterns.chapter2"},{"id":"n35","layer":"formal","project":"p0","title":"arithFriezePatSetNonEmpty","kind":"theorem","summary":"∀ n : Nat, Ne n 0 → (arithFriezePatSet n).Nonempty","labels":[],"detail_key":"p0","name":"arithFriezePatSetNonEmpty","module":"FriezePatterns.chapter3"},{"id":"n36","layer":"formal","project":"p0","title":"arith_fp","kind":"inductive","summary":"(Prod Nat Nat → Rat) → Nat → Prop","labels":[],"detail_key":"p0","name":"arith_fp","module":"FriezePatterns.chapter3"},{"id":"n37","layer":"formal","project":"p0","title":"fluteToFrieze","kind":"def","summary":"∀ n : Nat (g : flute n), Ne n 0 → arith_fp (frieze_f g) n","labels":[],"detail_key":"p0","name":"fluteToFrieze","module":"FriezePatterns.chapter3"},{"id":"n38","layer":"formal","project":"p0","title":"friezeToFlute","kind":"def","summary":"(f : Prod Nat Nat → Rat) → (n : Nat) → Nat → LE.le 2 n → [inst : arith_fp f n] → flute n","labels":[],"detail_key":"p0","name":"friezeToFlute","module":"FriezePatterns.chapter3"},{"id":"n39","layer":"formal","project":"p0","title":"frieze_f","kind":"def","summary":"n : Nat → flute n → Prod Nat Nat → Rat","labels":[],"detail_key":"p0","name":"frieze_f","module":"FriezePatterns.chapter3"},{"id":"n40","layer":"formal","project":"p0","title":"main1","kind":"theorem","summary":"∀ (n : Nat), Ne n 0 → ∀ (f : Prod Nat Nat → Rat), arith_fp f n → ∀ (a : Prod Nat Nat), LE.le (f…","labels":[],"detail_key":"p0","name":"main1","module":"FriezePatterns.chapter3"},{"id":"n41","layer":"formal","project":"p0","title":"main2","kind":"theorem","summary":"∀ (n : Nat), Ne n 0 → Exists fun f => Exists fun hf => Exists fun a => Eq (f a) ↑(Nat.fib n)","labels":[],"detail_key":"p0","name":"main2","module":"FriezePatterns.chapter3"},{"id":"n42","layer":"formal","project":"p0","title":"main3","kind":"theorem","summary":"∀ (n : Nat), Ne n 0 → Exists fun g => Exists fun x => Exists fun b => And (∀ (f : Prod Nat Nat…","labels":[],"detail_key":"p0","name":"main3","module":"FriezePatterns.chapter3"},{"id":"n43","layer":"informal","project":"p1","title":"f^* and f_*","kind":"definition","summary":"[f^* and f_*] For every continuous Function f : X \\rightarrow Y between topological Spaces, the…","labels":["def:f_star"],"detail_key":"p1"},{"id":"n44","layer":"informal","project":"p1","title":"f^* \\dashv f_*","kind":"lemma","summary":"[f^* \\dashv f_*] f^* is the right adjoint to f_*","labels":["lem:f_star_adj"],"detail_key":"p1"},{"id":"n45","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n46","layer":"informal","project":"p1","title":"Embedding","kind":"lemma","summary":"[Embedding] (Leroy Lemme 1) The following arguments are equivalent: \\item f^* is surjective \\it…","labels":["lem:embedding"],"detail_key":"p1"},{"id":"n47","layer":"informal","project":"p1","title":"This follows from the triangular identities.","kind":"proof","summary":"This follows from the triangular identities.","labels":[],"detail_key":"p1"},{"id":"n48","layer":"informal","project":"p1","title":"Embedding","kind":"definition","summary":"[Embedding] An embedding is a morphism that satisfies the conditions of \\reflem:embedding","labels":["def:embedding"],"detail_key":"p1"},{"id":"n49","layer":"informal","project":"p1","title":"Nucleus","kind":"definition","summary":"[Nucleus] \\mathlibok A nucleus is a map e : O(E) \\rightarrow O(E) with the following three prop…","labels":["def:nucleus"],"detail_key":"p1"},{"id":"n50","layer":"informal","project":"p1","title":"Nucleus","kind":"lemma","summary":"[Nucleus] (Leroy Lemme 3) Let e : O(E) \\rightarrow O(E) be monotonic. The following are equival…","labels":["lem:nucleus"],"detail_key":"p1"},{"id":"n51","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n52","layer":"informal","project":"p1","title":"Nucleus Partial Order","kind":"definition","summary":"[Nucleus Partial Order] \\mathlibok For two nuclei e and f on O(E), we say that e \\le f if e(U)…","labels":["def:nucleus_partial_order"],"detail_key":"p1"},{"id":"n53","layer":"informal","project":"p1","title":"Nucleus Intersection","kind":"lemma","summary":"[Nucleus Intersection] \\mathlibok For a set S of nuclei, the intersection \\bigcap S can be comp…","labels":["lem:nucleus_intersection"],"detail_key":"p1"},{"id":"n54","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n55","layer":"informal","project":"p1","title":"Sublocal","kind":"definition","summary":"[Sublocal] (Leroy CH 3) A sublocal Y \\subset X is defined by a nucleus e_Y: O(X) \\rightarrow O(…","labels":["def:sublocal"],"detail_key":"p1"},{"id":"n56","layer":"informal","project":"p1","title":"Sublocal Inclusion","kind":"definition","summary":"[Sublocal Inclusion] (Stimmt das?)(Leroy Ch 3) X \\subset Y if e_Y(u) \\le e_X(u) for all u. This…","labels":["def:sublocal_inclusion"],"detail_key":"p1"},{"id":"n57","layer":"informal","project":"p1","title":"Union of Sublocals","kind":"definition","summary":"[Union of Sublocals] (Leroy CH 1.4) Let (X_i)_i be a family of sublocals of E and (e_i)_i the c…","labels":["def:sublocal_union"],"detail_key":"p1"},{"id":"n58","layer":"informal","project":"p1","title":"Union of Sublocals","kind":"lemma","summary":"[Union of Sublocals] (Leroy CH 4) Let X_i be a family of subframes of E and e_i be the correspo…","labels":["lem:sublocal_union"],"detail_key":"p1"},{"id":"n59","layer":"informal","project":"p1","title":"The properties of the nucleus (idempotent, increasing, preserving intersection) can be ve…","kind":"proof","summary":"The properties of the nucleus (idempotent, increasing, preserving intersection) can be verified…","labels":[],"detail_key":"p1"},{"id":"n60","layer":"informal","project":"p1","title":"Intersection of Sublocals","kind":"definition","summary":"[Intersection of Sublocals] Let (X_i)_i be a family of sublocal of E and (e_i)_i the correspond…","labels":["def:sublocal_intersection"],"detail_key":"p1"},{"id":"n61","layer":"informal","project":"p1","title":"Nucleus Complete Lattice","kind":"lemma","summary":"[Nucleus Complete Lattice] \\mathlibok The Nuclei (and therefore the sublocals) form a complete…","labels":["lem:nucleus_complete_lattice"],"detail_key":"p1"},{"id":"n62","layer":"informal","project":"p1","title":"One can prove that the Nuclei are closed under arbitrary intersections by unfolding the d…","kind":"proof","summary":"One can prove that the Nuclei are closed under arbitrary intersections by unfolding the definit…","labels":[],"detail_key":"p1"},{"id":"n63","layer":"informal","project":"p1","title":"Complete Heyting Algebra","kind":"proposition","summary":"[Complete Heyting Algebra] \\mathlibok A complete Lattice is a Frame if and only if it as a Heyt…","labels":["prop:complete_heyting_algebra"],"detail_key":"p1"},{"id":"n64","layer":"informal","project":"p1","title":"(Source Johnstone:) The Heyting implication is right adjoint to the infimum. This means t…","kind":"proof","summary":"(Source Johnstone:) The Heyting implication is right adjoint to the infimum. This means that th…","labels":[],"detail_key":"p1"},{"id":"n65","layer":"informal","project":"p1","title":"Nucleus Heyting Algebra","kind":"lemma","summary":"[Nucleus Heyting Algebra] \\mathlibok The Nuclei form a Heyting Algebra.","labels":["lem:nucleus_heyting_algebra"],"detail_key":"p1"},{"id":"n66","layer":"informal","project":"p1","title":"Quelle Johnstone","kind":"proof","summary":"Quelle Johnstone","labels":[],"detail_key":"p1"},{"id":"n67","layer":"informal","project":"p1","title":"Nucleus Frame","kind":"lemma","summary":"[Nucleus Frame] \\mathlibok The Nuclei form a frame.","labels":["lem:nucleus_frame"],"detail_key":"p1"},{"id":"n68","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n69","layer":"informal","project":"p1","title":"e_U","kind":"definition","summary":"[e_U] Let E be a space with U, H \\in O(E). We donote by e_U the largest W \\in O(E) such that W…","labels":["def:e_U"],"detail_key":"p1"},{"id":"n70","layer":"informal","project":"p1","title":"e_U is a nucleus","kind":"lemma","summary":"[e_U is a nucleus] The map e_U is a nucleus.","labels":["lem:e_U_nucleus"],"detail_key":"p1"},{"id":"n71","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n72","layer":"informal","project":"p1","title":"Open sublocal","kind":"definition","summary":"[Open sublocal] For any U \\in O(E), the sublocal [U] is called an open sublocal of E.","labels":["def:open_sublocal"],"detail_key":"p1"},{"id":"n73","layer":"informal","project":"p1","title":"(6,7) Open Sublocal Properties","kind":"lemma","summary":"[(6,7) Open Sublocal Properties] (Leroy Lemma 6,7) \\item For all subspaces X of E and any U \\in…","labels":["lem:sublocal_properties"],"detail_key":"p1"},{"id":"n74","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n75","layer":"informal","project":"p1","title":"Complement","kind":"definition","summary":"[Complement] The complement of an open sublocal U of X is the sublocal X \\setminus U. (Leroy p.…","labels":["def:complement"],"detail_key":"p1"},{"id":"n76","layer":"informal","project":"p1","title":"Complement Injective","kind":"lemma","summary":"[Complement Injective] The complement is injective.","labels":["lem:complement_injective"],"detail_key":"p1"},{"id":"n77","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n78","layer":"informal","project":"p1","title":"Closed Sublocal","kind":"definition","summary":"[Closed Sublocal] A sublocal X of E is called closed if X = E \\setminus U for some open subloca…","labels":["def:closed_sublocal"],"detail_key":"p1"},{"id":"n79","layer":"informal","project":"p1","title":"Intersection of Closed Sublocals","kind":"lemma","summary":"[Intersection of Closed Sublocals] For any family X_i of closed sublocals of E, the intersectio…","labels":["lem:closed_intersection"],"detail_key":"p1"},{"id":"n80","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n81","layer":"informal","project":"p1","title":"(1.8) Properties of Complements","kind":"lemma","summary":"[(1.8) Properties of Complements] For any open sublocal V of E and any sublocal X of E, we have…","labels":["lem:properties_of_complements"],"detail_key":"p1"},{"id":"n82","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n83","layer":"informal","project":"p1","title":"(1.8bis) Properties of Complements Part 2","kind":"lemma","summary":"[(1.8bis) Properties of Complements Part 2] For any open sublocal V of E and any sublocal X of…","labels":["lem:properties_of_complements_part_2"],"detail_key":"p1"},{"id":"n84","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n85","layer":"informal","project":"p1","title":"Further Topology","kind":"definition","summary":"[Further Topology] \\item Int X is the largest open sublocal contained in X \\item Ext X is the l…","labels":["def:further_topology"],"detail_key":"p1"},{"id":"n86","layer":"informal","project":"p1","title":"Properties of Further Topology","kind":"lemma","summary":"[Properties of Further Topology] \\item \\barX = E \\setminus Ext(X) \\item \\partial X = E \\setminu…","labels":["lem:properties_of_further_topology"],"detail_key":"p1"},{"id":"n87","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n88","layer":"informal","project":"p1","title":"Measure on Locales","kind":"definition","summary":"[Measure on Locales] A measure on a local X is a map \\mu : O(X) \\to [0,\\infty) such that: \\item…","labels":["def:measure_on_locals"],"detail_key":"p1"},{"id":"n89","layer":"informal","project":"p1","title":"Caratheodory","kind":"definition","summary":"[Caratheodory] For any measure \\mu on a local X, the caratheodory extension is: \\[\\mu(A) = \\inf…","labels":["def:caratheodory"],"detail_key":"p1"},{"id":"n90","layer":"informal","project":"p1","title":"Proptery 0 (Commutes with sup)","kind":"lemma","summary":"[Proptery 0 (Commutes with sup)] (Leroy lemme 3.1) The caratheodory extension of a measure on a…","labels":["lem:commutes_with_sup"],"detail_key":"p1"},{"id":"n91","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n92","layer":"informal","project":"p1","title":"Caratheodory Extensions are monotonic","kind":"lemma","summary":"[Caratheodory Extensions are monotonic] The caratheodory extension is monotonic i.e. \\[A \\le B…","labels":["lem:monotonic"],"detail_key":"p1"},{"id":"n93","layer":"informal","project":"p1","title":"This is a direct consequence of the definition of the caratheodory extension.","kind":"proof","summary":"This is a direct consequence of the definition of the caratheodory extension.","labels":[],"detail_key":"p1"},{"id":"n94","layer":"informal","project":"p1","title":"Subadditivity","kind":"lemma","summary":"[Subadditivity] The Caratheodory extension is subaddtive: \\mu(A \\cup B) \\le \\mu(A) + \\mu(B)","labels":["lem:caratheodory_subaddtive"],"detail_key":"p1"},{"id":"n95","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n96","layer":"informal","project":"p1","title":"Regular Local","kind":"definition","summary":"[Regular Local] A local is regular, if for all open sublocals U of E, the open sublocals V such…","labels":["def:regular_local"],"detail_key":"p1"},{"id":"n97","layer":"informal","project":"p1","title":"Neighborhood","kind":"definition","summary":"[Neighborhood] A neighborhood of a sublocal A of X is an open sublocal V of X such that A \\le V.","labels":["def:neighborhood"],"detail_key":"p1"},{"id":"n98","layer":"informal","project":"p1","title":"Regularity of Sublocals","kind":"lemma","summary":"[Regularity of Sublocals] (Leroy lemme 3.2) In a regular local, any sublocal is regular, meanin…","labels":["lem:regularity_of_sublocals"],"detail_key":"p1"},{"id":"n99","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n100","layer":"informal","project":"p1","title":"Measure add compl eq top","kind":"lemma","summary":"[Measure add compl eq top] (Leroy Lemme 3.3) For any open sublocal U of a local X, the caratheo…","labels":["lem:measure_add_compl_eq_top"],"detail_key":"p1"},{"id":"n101","layer":"informal","project":"p1","title":"Siehe Leroy","kind":"proof","summary":"Siehe Leroy","labels":[],"detail_key":"p1"},{"id":"n102","layer":"informal","project":"p1","title":"Restriction","kind":"lemma","summary":"[Restriction] The Restriction of a Measure to any open Sublocal is a Measure.","labels":["lem:restriction"],"detail_key":"p1"},{"id":"n103","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n104","layer":"informal","project":"p1","title":"Property 2","kind":"lemma","summary":"[Property 2] (Leroy Lemm 3.4) For any open sublocal U and any sublocal A of a local E, the cara…","labels":["lem:restrict_add_compl_eq_top"],"detail_key":"p1"},{"id":"n105","layer":"informal","project":"p1","title":"Siehe Leroy","kind":"proof","summary":"Siehe Leroy","labels":[],"detail_key":"p1"},{"id":"n106","layer":"informal","project":"p1","title":"Property 3","kind":"lemma","summary":"[Property 3] (Leroy Lemm 3.5) For a increasing family V_\\alpha of open sublocals of E and any s…","labels":["lem:restrict_preserves_sSup"],"detail_key":"p1"},{"id":"n107","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n108","layer":"informal","project":"p1","title":"Restriction to a Sublocale","kind":"lemma","summary":"[Restriction to a Sublocale] Let A be a sublocale of E with the embedding i : A \\rightarrow E.…","labels":["lem:restriction_to_sublocale"],"detail_key":"p1"},{"id":"n109","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n110","layer":"informal","project":"p1","title":"strictly additve","kind":"proposition","summary":"[strictly additve] (Leroy theorem 3.3.1) For any measure on a local X, the caratheodory extensi…","labels":["prop:strictly_additive"],"detail_key":"p1"},{"id":"n111","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n112","layer":"informal","project":"p1","title":"reductive","kind":"proposition","summary":"[reductive] (Proposition 3.3.1) For any measure on a local X, the caratheodory extension is red…","labels":["prop:reductive"],"detail_key":"p1"},{"id":"n113","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n114","layer":"informal","project":"p1","title":"Commutes with inf opens","kind":"lemma","summary":"[Commutes with inf opens] (Leroy Lemme 3.6) For any measure on a local X and a decreasing famil…","labels":["lem:commutes_with_inf_opens"],"detail_key":"p1"},{"id":"n115","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n116","layer":"informal","project":"p1","title":"Commutes with inf","kind":"proposition","summary":"[Commutes with inf] (Leroy lemme 3.7 et principal) For any measure on a local X, the caratheodo…","labels":["prop:commutes_with_inf"],"detail_key":"p1"},{"id":"n117","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n118","layer":"informal","project":"p1","title":"Main Theorem (very important)","kind":"theorem","summary":"[Main Theorem (very important)] For any measure on a local X, the caratheodory extension is \\it…","labels":["thm:main"],"detail_key":"p1"},{"id":"n119","layer":"informal","project":"p1","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p1"},{"id":"n120","layer":"informal","project":"p1","title":"(1.10) Intersection of Open and Closed Sublocals","kind":"lemma","summary":"[(1.10) Intersection of Open and Closed Sublocals] For any U \\in O(E), and sublocal X of E we h…","labels":["lem:open_closed_intersection"],"detail_key":"p1"},{"id":"n121","layer":"informal","project":"p1","title":"Regular Top to regular local","kind":"lemma","summary":"[Regular Top to regular local] Any regular topological space induces a regular local.","labels":["lem:regular_top_to_regular_local"],"detail_key":"p1"},{"id":"n122","layer":"informal","project":"p1","title":"Opens","kind":"lemma","summary":"[Opens] (Leroy V.1 Remarque 2) The Open subsets of any good enough topological space correspond…","labels":["lem:opens_correspond"],"detail_key":"p1"},{"id":"n123","layer":"informal","project":"p1","title":"Subset Sublocal","kind":"lemma","summary":"[Subset Sublocal] (leroy V.1 Remarque 3) Any subset X of a good enough topological space E indu…","labels":["lem:subset_sublocal"],"detail_key":"p1"},{"id":"n124","layer":"informal","project":"p1","title":"Good enough topological space","kind":"definition","summary":"[Good enough topological space] blackbox to mathlib????","labels":["def:good_enough_topological_space"],"detail_key":"p1"},{"id":"n125","layer":"informal","project":"p1","title":"Subset to sublocal Part 1","kind":"lemma","summary":"[Subset to sublocal Part 1] (Leroy Proposition 5.1.1) For two subspaces X and Y of E and an ope…","labels":["lem:subset_to_sublocal_part_1"],"detail_key":"p1"},{"id":"n126","layer":"informal","project":"p1","title":"Subset to sublocal Part 2","kind":"lemma","summary":"[Subset to sublocal Part 2] (Leroy Proposition 5.1.2, 5.1.3) For an open subspace U of E and a…","labels":["lem:subset_to_sublocal_part_2"],"detail_key":"p1"},{"id":"n127","layer":"informal","project":"p1","title":"Part 3","kind":"lemma","summary":"[Part 3] For any subspaces X of E, we have: \\item \\[Ext[X] = [Ext X]\\] \\item \\[\\bar[X] = [\\barX…","labels":["lem:subspaces_part_3"],"detail_key":"p1"},{"id":"n128","layer":"informal","project":"p1","title":"Subset to sublocal preserves structure","kind":"proposition","summary":"[Subset to sublocal preserves structure] For two subspaces X and Y of E and an open subspaces U…","labels":["prop:subset_to_sublocal_structure"],"detail_key":"p1"},{"id":"n129","layer":"informal","project":"p1","title":"Measure top to loc","kind":"theorem","summary":"[Measure top to loc] Any measure on a good enough topological space X induces a measure on the…","labels":["thm:measure_top_to_loc"],"detail_key":"p1"},{"id":"n130","layer":"formal","project":"p1","title":"Sublocale.Neighbourhood","kind":"def","summary":"X : Type u → [inst : Order.Frame X] → Sublocale X → Set (Sublocale X)","labels":[],"detail_key":"p1","name":"Sublocale.Neighbourhood","module":"Leroy.Further_Topology"},{"id":"n131","layer":"formal","project":"p1","title":"Sublocale.closure","kind":"def","summary":"E : Type u → [e_frm : Order.Frame E] → Sublocale E → Closed E","labels":[],"detail_key":"p1","name":"Sublocale.closure","module":"Leroy.Further_Topology"},{"id":"n132","layer":"formal","project":"p1","title":"sup_compl_eq_top_iff","kind":"theorem","summary":"∀ E : Type u [e_frm : Order.Frame E] x : Sublocale E u : Open E, Iff (LE.le u.toSublocale x) (E…","labels":[],"detail_key":"p1","name":"sup_compl_eq_top_iff","module":"Leroy.Further_Topology"},{"id":"n133","layer":"formal","project":"p1","title":"Measure","kind":"inductive","summary":"X : Type u_1 → [h : Order.Frame X] → Type u_1","labels":[],"detail_key":"p1","name":"Measure","module":"Leroy.Measure.Basic"},{"id":"n134","layer":"formal","project":"p1","title":"Measure.caratheodory","kind":"def","summary":"X : Type u_1 → [h : Order.Frame X] → m : Measure → Sublocale X → NNReal","labels":[],"detail_key":"p1","name":"Measure.caratheodory","module":"Leroy.Measure.Basic"},{"id":"n135","layer":"formal","project":"p1","title":"Measure.caratheodory.mono","kind":"theorem","summary":"∀ E : Type u_3 [e_frm : Order.Frame E] m : Measure A B : Sublocale E, LE.le A B → LE.le (Measur…","labels":[],"detail_key":"p1","name":"Measure.caratheodory.mono","module":"Leroy.Measure.Basic"},{"id":"n136","layer":"formal","project":"p1","title":"Measure.caratheodory.subadditive","kind":"theorem","summary":"∀ E : Type u_3 [e_frm : Order.Frame E] m : Measure (a b : Sublocale E), LE.le (Measure.caratheo…","labels":[],"detail_key":"p1","name":"Measure.caratheodory.subadditive","module":"Leroy.Measure.Basic"},{"id":"n137","layer":"formal","project":"p1","title":"Measure.caratheodordy.preserves_iInf","kind":"theorem","summary":"∀ E : Type u_2 [inst : Order.Frame E] [Fact (regular E)] m : Measure ι : Type u_3 [Nonempty ι]…","labels":[],"detail_key":"p1","name":"Measure.caratheodordy.preserves_iInf","module":"Leroy.Measure.Reduction"},{"id":"n138","layer":"formal","project":"p1","title":"Measure.caratheodory.strictly_additive","kind":"theorem","summary":"∀ E : Type u_2 [inst : Order.Frame E] [Fact (regular E)] m : Measure (A B : Sublocale E), Eq (M…","labels":[],"detail_key":"p1","name":"Measure.caratheodory.strictly_additive","module":"Leroy.Measure.Reduction"},{"id":"n139","layer":"formal","project":"p1","title":"Measure.add_complement","kind":"theorem","summary":"∀ E : Type u_4 [e_frm : Order.Frame E] [e_regular : Fact (regular E)] m : Measure (U : Open E),…","labels":[],"detail_key":"p1","name":"Measure.add_complement","module":"Leroy.Measure.Regular"},{"id":"n140","layer":"formal","project":"p1","title":"Measure.add_complement_inf","kind":"theorem","summary":"∀ E : Type u_4 [e_frm : Order.Frame E] [e_regular : Fact (regular E)] m : Measure (u : Open E)…","labels":[],"detail_key":"p1","name":"Measure.add_complement_inf","module":"Leroy.Measure.Regular"},{"id":"n141","layer":"formal","project":"p1","title":"Measure.inf_filtered","kind":"theorem","summary":"∀ E : Type u_4 [e_frm : Order.Frame E] [e_regular : Fact (regular E)] m : Measure (A : Sublocal…","labels":[],"detail_key":"p1","name":"Measure.inf_filtered","module":"Leroy.Measure.Regular"},{"id":"n142","layer":"formal","project":"p1","title":"Sublocale.intersection_Open_Neighbourhhood","kind":"theorem","summary":"∀ E : Type u_4 [e_frm : Order.Frame E] [e_regular : Fact (regular E)] (a : Sublocale E), Eq a (…","labels":[],"detail_key":"p1","name":"Sublocale.intersection_Open_Neighbourhhood","module":"Leroy.Measure.Regular"},{"id":"n143","layer":"formal","project":"p1","title":"regular","kind":"def","summary":"(E : Type u_4) → [Order.Frame E] → Prop","labels":[],"detail_key":"p1","name":"regular","module":"Leroy.Measure.Regular"},{"id":"n144","layer":"formal","project":"p1","title":"Measure.restrict_sublocale_measure","kind":"def","summary":"E' : Type u_1 → [inst : Order.Frame E'] → (A : Sublocale E') → Measure → [Fact (regular E')] →…","labels":[],"detail_key":"p1","name":"Measure.restrict_sublocale_measure","module":"Leroy.Measure.Restrict"},{"id":"n145","layer":"formal","project":"p1","title":"Nucleus.frameHom","kind":"def","summary":"E : Type u_3 → [inst : Order.Frame E] → (n : Nucleus E) → FrameHom E ↑(Image n)","labels":[],"detail_key":"p1","name":"Nucleus.frameHom","module":"Leroy.Nucleus_Image"},{"id":"n146","layer":"formal","project":"p1","title":"Nucleus.gc","kind":"theorem","summary":"∀ E : Type u_3 [inst : Order.Frame E] (n : Nucleus E), GaloisConnection (⇑n.frameHom) (f_untens…","labels":[],"detail_key":"p1","name":"Nucleus.gc","module":"Leroy.Nucleus_Image"},{"id":"n147","layer":"formal","project":"p1","title":"image_frame","kind":"def","summary":"E : Type u_3 → [inst : Order.Frame E] → (n : Nucleus E) → Order.Frame ↑(Image n)","labels":[],"detail_key":"p1","name":"image_frame","module":"Leroy.Nucleus_Image"},{"id":"n148","layer":"formal","project":"p1","title":"Closed","kind":"inductive","summary":"(E : Type u_2) → [Order.Frame E] → Type u_2","labels":[],"detail_key":"p1","name":"Closed","module":"Leroy.Sublocale"},{"id":"n149","layer":"formal","project":"p1","title":"Closed.instInfSet","kind":"def","summary":"E : Type u_1 → [inst : Order.Frame E] → InfSet (Closed E)","labels":[],"detail_key":"p1","name":"Closed.instInfSet","module":"Leroy.Sublocale"},{"id":"n150","layer":"formal","project":"p1","title":"Open","kind":"inductive","summary":"(E : Type u_2) → [Order.Frame E] → Type u_2","labels":[],"detail_key":"p1","name":"Open","module":"Leroy.Sublocale"},{"id":"n151","layer":"formal","project":"p1","title":"Open.preserves_inf","kind":"theorem","summary":"∀ E : Type u_1 [inst : Order.Frame E] (U V : Open E), Eq (min U V).toSublocale (min U.toSubloca…","labels":[],"detail_key":"p1","name":"Open.preserves_inf","module":"Leroy.Sublocale"},{"id":"n152","layer":"formal","project":"p1","title":"Open.toSublocale","kind":"def","summary":"E : Type u_1 → [inst : Order.Frame E] → Open E → Sublocale E","labels":[],"detail_key":"p1","name":"Open.toSublocale","module":"Leroy.Sublocale"},{"id":"n153","layer":"formal","project":"p1","title":"Open.toSublocale_injective","kind":"theorem","summary":"∀ E : Type u_1 [inst : Order.Frame E], Function.Injective Open.toSublocale","labels":[],"detail_key":"p1","name":"Open.toSublocale_injective","module":"Leroy.Sublocale"},{"id":"n154","layer":"formal","project":"p1","title":"Sublocale.inf_apply","kind":"theorem","summary":"∀ E : Type u_1 [inst : Order.Frame E] (u v : Sublocale E) (x : E), Eq ((min u v) x) (iInf fun j…","labels":[],"detail_key":"p1","name":"Sublocale.inf_apply","module":"Leroy.Sublocale"},{"id":"n155","layer":"formal","project":"p1","title":"Sublocale.le_iff","kind":"theorem","summary":"∀ E : Type u_1 [inst : Order.Frame E] (u v : Sublocale E), Iff (LE.le u v) (∀ (i : E), LE.le (v…","labels":[],"detail_key":"p1","name":"Sublocale.le_iff","module":"Leroy.Sublocale"},{"id":"n156","layer":"formal","project":"p1","title":"Sublocale.sSup_apply","kind":"theorem","summary":"∀ E : Type u_1 [inst : Order.Frame E] (s : Set (Sublocale E)) (x : E), Eq ((sSup s) x) (iInf fu…","labels":[],"detail_key":"p1","name":"Sublocale.sSup_apply","module":"Leroy.Sublocale"},{"id":"n157","layer":"formal","project":"p1","title":"complement","kind":"def","summary":"E : Type u_1 → [inst : Order.Frame E] → Open E → Sublocale E","labels":[],"detail_key":"p1","name":"complement","module":"Leroy.Sublocale"},{"id":"n158","layer":"informal","project":"p2","title":"def:konk_rot","kind":"definition","summary":"Unsere Rotationsmatrizen sind:","labels":["def:konk_rot"],"detail_key":"p2"},{"id":"n159","layer":"informal","project":"p2","title":"Invertierbarkeit von A und B","kind":"lemma","summary":"[Invertierbarkeit von A und B] Es gilt det A\\neq 0 und det B\\neq 0 und damit sind A und B inver…","labels":["lemma:a_b_invertierbar"],"detail_key":"p2"},{"id":"n160","layer":"informal","project":"p2","title":"Folgt durch Nachrrechnen.","kind":"proof","summary":"Folgt durch Nachrrechnen.","labels":[],"detail_key":"p2"},{"id":"n161","layer":"informal","project":"p2","title":"def:konk_rot_erzeugt","kind":"definition","summary":"G bezeichnet die von A und B erzeugte Untergruppe.","labels":["def:konk_rot_erzeugt"],"detail_key":"p2"},{"id":"n162","layer":"informal","project":"p2","title":"lem:adjugate_fin_three","kind":"lemma","summary":"Die adjungierte 3x3 Matrix kann in einer bestimmten Form dargestellt werden... \\mathlibok","labels":["lem:adjugate_fin_three"],"detail_key":"p2"},{"id":"n163","layer":"informal","project":"p2","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p2"},{"id":"n164","layer":"informal","project":"p2","title":"Konkrete darstellung der Drehungen","kind":"lemma","summary":"[Konkrete darstellung der Drehungen] Wenn \\rho : R³\\RightarrowR³ ein Ausdruck in G der Länge n…","labels":["lem:darst_von_rot_res"],"detail_key":"p2"},{"id":"n165","layer":"informal","project":"p2","title":"Diese Behauptung folgt aus den Erzeugermatrizen und durch konkretes Multiplizieren eines…","kind":"proof","summary":"Diese Behauptung folgt aus den Erzeugermatrizen und durch konkretes Multiplizieren eines reduzi…","labels":[],"detail_key":"p2"},{"id":"n166","layer":"informal","project":"p2","title":"def:freie_grp","kind":"definition","summary":"Eine freie Gruppe G ist eine Gruppe, in welcher zwei Wörter auf einer spezifischen Erzeugermeng…","labels":["def:freie_grp"],"detail_key":"p2"},{"id":"n167","layer":"informal","project":"p2","title":"thm:freie_grp_an_rot","kind":"theorem","summary":"Die von unseren konkreten Rotationen aus \\refdef:konk_rot erzeugte Untergruppe G ist eine freie…","labels":["thm:freie_grp_an_rot"],"detail_key":"p2"},{"id":"n168","layer":"informal","project":"p2","title":"chillig.","kind":"proof","summary":"chillig.","labels":[],"detail_key":"p2"},{"id":"n169","layer":"informal","project":"p2","title":"Einheitskugel ohne Mittelpunkt","kind":"definition","summary":"[Einheitskugel ohne Mittelpunkt] Sei L=\\(x,y,z):x²+y²+z²\\leq1\\ die Einheitskugel. Wir definiere…","labels":["def:kugel_ohne_mittelpunkt"],"detail_key":"p2"},{"id":"n170","layer":"informal","project":"p2","title":"Orbit","kind":"definition","summary":"[Orbit] Zwei Punkte a und b gehören zum selben Orbit, genau dann, wenn ein \\rho in G existiert,…","labels":["def:orbit"],"detail_key":"p2"},{"id":"n171","layer":"informal","project":"p2","title":"Abzählbarkeit aller Orbits","kind":"lemma","summary":"[Abzählbarkeit aller Orbits] Die Menge aller Orbits ist abzählbar.","labels":["lemma:all_orbits_countable"],"detail_key":"p2"},{"id":"n172","layer":"informal","project":"p2","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p2"},{"id":"n173","layer":"informal","project":"p2","title":"repräsentative Punkte","kind":"lemma","summary":"[repräsentative Punkte] Wir können uns aus jedem Orbit einen repräsentativen Punkt auswählen.","labels":["theorem:rep_punkte"],"detail_key":"p2"},{"id":"n174","layer":"informal","project":"p2","title":"Dies folgt direkt mit dem Auswahlaxiom.","kind":"proof","summary":"Dies folgt direkt mit dem Auswahlaxiom.","labels":[],"detail_key":"p2"},{"id":"n175","layer":"informal","project":"p2","title":"Menge aller Repräsentanten","kind":"definition","summary":"[Menge aller Repräsentanten] M ist die Menge aller ausgwählten repräsentativen Punkte.","labels":["def:menge_rep_punkte"],"detail_key":"p2"},{"id":"n176","layer":"informal","project":"p2","title":"Fixpunkte","kind":"definition","summary":"[Fixpunkte] Sei Y eine Menge und f:Y\\rightarrow Y eine Funktion. Dann heißt ein Punkt y\\in Y Fi…","labels":["def:fixpunkte"],"detail_key":"p2"},{"id":"n177","layer":"informal","project":"p2","title":"Menge aller Fixpunkte","kind":"definition","summary":"[Menge aller Fixpunkte] Bezeichne mit D die Menge aller Punkte in L', welche Fixpunkte der Rota…","labels":["def:menge_fixpunkte"],"detail_key":"p2"},{"id":"n178","layer":"informal","project":"p2","title":"Abzählbarkeit von G","kind":"lemma","summary":"[Abzählbarkeit von G] G ist abzählbar.","labels":["lemma:G_abzaehlbar"],"detail_key":"p2"},{"id":"n179","layer":"informal","project":"p2","title":"Steht noch aus.","kind":"proof","summary":"Steht noch aus.","labels":[],"detail_key":"p2"},{"id":"n180","layer":"informal","project":"p2","title":"Genau eine Rotationsachse","kind":"definition","summary":"[Genau eine Rotationsachse] Jede Rotation in G hat genau eine Rotationsachse.","labels":["lemma:eine_rot_achse"],"detail_key":"p2"},{"id":"n181","layer":"informal","project":"p2","title":"Abzählbarkeit Rotationsachsen","kind":"lemma","summary":"[Abzählbarkeit Rotationsachsen] Die Rotationsachsen liegen auf abzählbar vielen Linien.","labels":["lemma:abz_rot_achsen"],"detail_key":"p2"},{"id":"n182","layer":"informal","project":"p2","title":"Steht noch aus.","kind":"proof","summary":"Steht noch aus.","labels":[],"detail_key":"p2"},{"id":"n183","layer":"informal","project":"p2","title":"Vereinigung X","kind":"definition","summary":"[Vereinigung X] X=\\bigcup\\limits_k=1^\\inftyA^-kM. X ist also die Menge aller Elemente von M, we…","labels":["def:vereinigung_x"],"detail_key":"p2"},{"id":"n184","layer":"informal","project":"p2","title":"Zerlegung in Mengen","kind":"definition","summary":"[Zerlegung in Mengen] P_1=S(A)M\\cup M\\cup X \\\\ P_2=S(A^-1)M\\backslash X \\\\ P_3=S(B)M \\\\ P_4=S(B…","labels":["def:zerlegung_L_D"],"detail_key":"p2"},{"id":"n185","layer":"informal","project":"p2","title":"Vereinigung der Zerlegung","kind":"lemma","summary":"[Vereinigung der Zerlegung] L'\\backslash D=P_1\\cup P_2\\cup P_3\\cup P_4","labels":["lemma:vereinigung_zerlegung"],"detail_key":"p2"},{"id":"n186","layer":"informal","project":"p2","title":"Steht noch aus.","kind":"proof","summary":"Steht noch aus.","labels":[],"detail_key":"p2"},{"id":"n187","layer":"informal","project":"p2","title":"Drehung zerlegte Mengen","kind":"lemma","summary":"[Drehung zerlegte Mengen] AP_2=P_2\\cup P_3\\cup P_4 BP_4=P_1\\cup P_2\\cup P_4","labels":["lemma:rot_zerlegte_mengen"],"detail_key":"p2"},{"id":"n188","layer":"informal","project":"p2","title":"Steht noch aus.","kind":"proof","summary":"Steht noch aus.","labels":[],"detail_key":"p2"},{"id":"n189","layer":"informal","project":"p2","title":"Verdopplung L' \\\\D","kind":"lemma","summary":"[Verdopplung L' \\\\D] L'\\backslash D=P_1\\cup AP_2 L'\\backslash D=P_3\\cup BP_4","labels":["lemma:verdopplung_L_D"],"detail_key":"p2"},{"id":"n190","layer":"informal","project":"p2","title":"Steht noch aus.","kind":"proof","summary":"Steht noch aus.","labels":[],"detail_key":"p2"},{"id":"n191","layer":"informal","project":"p2","title":"Abzählbarkeit der repräsentativen Punkte","kind":"lemma","summary":"[Abzählbarkeit der repräsentativen Punkte] Die Menge der Repräsentativen Punkte ist abzählbar.","labels":["lemma:abz_menge_rep_punkte"],"detail_key":"p2"},{"id":"n192","layer":"informal","project":"p2","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p2"},{"id":"n193","layer":"informal","project":"p2","title":"Äquidekomponierbar","kind":"definition","summary":"[Äquidekomponierbar] Zwei Mengen C und D heißen äquidekomponierbar, wenn C in endlich viele Tei…","labels":["def:aequidekomponierbar"],"detail_key":"p2"},{"id":"n194","layer":"informal","project":"p2","title":"Äquidekomponierbarkeit von L'\\\\D und L'","kind":"lemma","summary":"[Äquidekomponierbarkeit von L'\\\\D und L'] L'\\backslash D und L' sind Äquidekomponierbar","labels":["lem:aequidekomponierbarkeit"],"detail_key":"p2"},{"id":"n195","layer":"informal","project":"p2","title":"Da die Punkte in D auf abzählbar vielen Achsen liegen, kann man eine Linie l durch den Ur…","kind":"proof","summary":"Da die Punkte in D auf abzählbar vielen Achsen liegen, kann man eine Linie l durch den Ursprung…","labels":[],"detail_key":"p2"},{"id":"n196","layer":"informal","project":"p2","title":"Pi und Wurzel 2 haben kein gemeinsames vielfaches","kind":"lemma","summary":"[Pi und Wurzel 2 haben kein gemeinsames vielfaches] \\nexists p,q\\in Z \\sqrt2\\cdot \\fracpq=\\pi","labels":["lemma:ncm_pi_sqrt_2"],"detail_key":"p2"},{"id":"n197","layer":"informal","project":"p2","title":"Angenommen es gäbe p,q\\in Z mit \\sqrt2\\cdot \\fracpq=\\pi. Nach Definition von \\cos^-1 auf…","kind":"proof","summary":"Angenommen es gäbe p,q\\in Z mit \\sqrt2\\cdot \\fracpq=\\pi. Nach Definition von \\cos^-1 auf dem In…","labels":[],"detail_key":"p2"},{"id":"n198","layer":"informal","project":"p2","title":"Äquidekomponierbarkeit Kreis","kind":"lemma","summary":"[Äquidekomponierbarkeit Kreis] Ein Kreis ist äquidekomponierbar mit einem Kreis ohne einen best…","labels":["lemma:aequi_kreis"],"detail_key":"p2"},{"id":"n199","layer":"informal","project":"p2","title":"Wir kümmern uns um den Einheitskreis S¹=\\(x,y):x²+y²=1\\ ohne (1,0). Wir verwenden den Ein…","kind":"proof","summary":"Wir kümmern uns um den Einheitskreis S¹=\\(x,y):x²+y²=1\\ ohne (1,0). Wir verwenden den Einheitsk…","labels":[],"detail_key":"p2"},{"id":"n200","layer":"informal","project":"p2","title":"Äquidekomponierbarkeit Subset","kind":"lemma","summary":"[Äquidekomponierbarkeit Subset]","labels":["lemma:equi_subset"],"detail_key":"p2"},{"id":"n201","layer":"informal","project":"p2","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p2"},{"id":"n202","layer":"informal","project":"p2","title":"Äquidekomponierbarkeit Kugel","kind":"theorem","summary":"[Äquidekomponierbarkeit Kugel] Eine Kugel ohne ihren Mittelpunkt ist äquidekomponierbar mit der…","labels":["theorem:aequi_kugel"],"detail_key":"p2"},{"id":"n203","layer":"informal","project":"p2","title":"Die Konstruktion eines Kreises im Inneren der Kugel, welcher den Mittelpunkt der Kugel be…","kind":"proof","summary":"Die Konstruktion eines Kreises im Inneren der Kugel, welcher den Mittelpunkt der Kugel beinhalt…","labels":[],"detail_key":"p2"},{"id":"n204","layer":"informal","project":"p2","title":"Banach-Tarski","kind":"theorem","summary":"[Banach-Tarski] Eine Kugel ist äquidekomponierbar mit zwei Kopien ihrer selbst.","labels":["thm:BanachTarski"],"detail_key":"p2"},{"id":"n205","layer":"informal","project":"p2","title":"Im zweiten Unterkapitel haben wir gezeigt, dass die Kugel ohne ihren Mittelpunkt und den…","kind":"proof","summary":"Im zweiten Unterkapitel haben wir gezeigt, dass die Kugel ohne ihren Mittelpunkt und den Punkte…","labels":[],"detail_key":"p2"},{"id":"n206","layer":"formal","project":"p2","title":"P₁","kind":"def","summary":"","labels":[],"detail_key":"p2","name":"P₁","module":"banach_tarski.Decomposition"},{"id":"n207","layer":"formal","project":"p2","title":"X","kind":"def","summary":"","labels":[],"detail_key":"p2","name":"X","module":"banach_tarski.Decomposition"},{"id":"n208","layer":"formal","project":"p2","title":"rot_A_P₂","kind":"lemma","summary":"","labels":[],"detail_key":"p2","name":"rot_A_P₂","module":"banach_tarski.Decomposition"},{"id":"n209","layer":"formal","project":"p2","title":"union_parts","kind":"lemma","summary":"","labels":[],"detail_key":"p2","name":"union_parts","module":"banach_tarski.Decomposition"},{"id":"n210","layer":"formal","project":"p2","title":"D","kind":"def","summary":"Set ↑L'","labels":[],"detail_key":"p2","name":"D","module":"banach_tarski.Definitions"},{"id":"n211","layer":"formal","project":"p2","title":"L","kind":"def","summary":"Set r_3","labels":[],"detail_key":"p2","name":"L","module":"banach_tarski.Definitions"},{"id":"n212","layer":"formal","project":"p2","title":"rotationsAchse","kind":"def","summary":"Matrix.GeneralLinearGroup (Fin 3) Real → Set r_3","labels":[],"detail_key":"p2","name":"rotationsAchse","module":"banach_tarski.Definitions"},{"id":"n213","layer":"formal","project":"p2","title":"equi_kugel","kind":"theorem","summary":"","labels":[],"detail_key":"p2","name":"equi_kugel","module":"banach_tarski.Equidecomposable.Equi_Ball"},{"id":"n214","layer":"formal","project":"p2","title":"pi_sqrt_two","kind":"theorem","summary":"(Exists fun x => Eq Real.pi (HMul.hMul (↑x) sq_2)) → Eq false true","labels":[],"detail_key":"p2","name":"pi_sqrt_two","module":"banach_tarski.Equidecomposable.Rotations"},{"id":"n215","layer":"formal","project":"p2","title":"g_countable","kind":"theorem","summary":"Function.Injective map_G_to_Nat","labels":[],"detail_key":"p2","name":"g_countable","module":"banach_tarski.G_countable"},{"id":"n216","layer":"formal","project":"p2","title":"M","kind":"def","summary":"","labels":[],"detail_key":"p2","name":"M","module":"banach_tarski.Orbit"},{"id":"n217","layer":"formal","project":"p2","title":"M_countable","kind":"lemma","summary":"","labels":[],"detail_key":"p2","name":"M_countable","module":"banach_tarski.Orbit"},{"id":"n218","layer":"formal","project":"p2","title":"all_orbits_countable","kind":"lemma","summary":"","labels":[],"detail_key":"p2","name":"all_orbits_countable","module":"banach_tarski.Orbit"},{"id":"n219","layer":"formal","project":"p2","title":"same_orbit","kind":"def","summary":"","labels":[],"detail_key":"p2","name":"same_orbit","module":"banach_tarski.Orbit"},{"id":"n220","layer":"informal","project":"p3","title":"product_lower_bound","kind":"lemma","summary":"Let \\chi be a Dirichlet character modulo~N. Then for all \\varepsilon > 0, we have \\label eqn:pr…","labels":["product_lower_bound","eqn:product"],"detail_key":"p3"},{"id":"n221","layer":"informal","project":"p3","title":"This follows from a trigonometric inequality.","kind":"proof","summary":"This follows from a trigonometric inequality.","labels":[],"detail_key":"p3"},{"id":"n222","layer":"informal","project":"p3","title":"non_quadratic","kind":"lemma","summary":"Let t \\in R and let \\chi be a Dirichlet character. If t \\ne 0 or \\chi^2 \\ne 1, then \\[ L(\\chi,…","labels":["non_quadratic"],"detail_key":"p3"},{"id":"n223","layer":"informal","project":"p3","title":"Assume that L(\\chi, 1 + it) = 0. Then the (at least) quadruple zero of~L(\\chi, s)^4 at 1…","kind":"proof","summary":"Assume that L(\\chi, 1 + it) = 0. Then the (at least) quadruple zero of~L(\\chi, s)^4 at 1 + it w…","labels":[],"detail_key":"p3"},{"id":"n224","layer":"informal","project":"p3","title":"def:bad_char","kind":"definition","summary":"A \\emphbad character is an R-valued (hence quadratic) Dirichlet character such that L(\\chi, 1)…","labels":["def:bad_char"],"detail_key":"p3"},{"id":"n225","layer":"informal","project":"p3","title":"def:bad_char_F","kind":"definition","summary":"Define F \\colon C \\to C by \\[ F(s) = \\zeta(s) L(\\chi, s) & \\textif s \\ne 1 \\\\ L'(\\chi, 1) & \\te…","labels":["def:bad_char_F"],"detail_key":"p3"},{"id":"n226","layer":"informal","project":"p3","title":"F_entire","kind":"lemma","summary":"If \\chi is a bad character, then F is an entire function.","labels":["F_entire"],"detail_key":"p3"},{"id":"n227","layer":"informal","project":"p3","title":"This is easy for s \\ne 1 since we know analyticity of both factors. To prove analyticity…","kind":"proof","summary":"This is easy for s \\ne 1 since we know analyticity of both factors. To prove analyticity at s =…","labels":[],"detail_key":"p3"},{"id":"n228","layer":"informal","project":"p3","title":"zero_of_F","kind":"lemma","summary":"We have F(-2) = 0.","labels":["zero_of_F"],"detail_key":"p3"},{"id":"n229","layer":"informal","project":"p3","title":"Follows from the trivial zeroes of Riemann zeta.","kind":"proof","summary":"Follows from the trivial zeroes of Riemann zeta.","labels":[],"detail_key":"p3"},{"id":"n230","layer":"informal","project":"p3","title":"F_Euler_product","kind":"lemma","summary":"For \\Re(s) > 1, F(s) is equal to the L-series of a real-valued arithmetic function e defined as…","labels":["F_Euler_product"],"detail_key":"p3"},{"id":"n231","layer":"informal","project":"p3","title":"We have Euler products for both L(\\chi, s) and \\zeta(s).","kind":"proof","summary":"We have Euler products for both L(\\chi, s) and \\zeta(s).","labels":[],"detail_key":"p3"},{"id":"n232","layer":"informal","project":"p3","title":"nonneg_coeffs","kind":"lemma","summary":"The weakly multiplicative function e(n) whose Euler product is E takes non-negative real values.","labels":["nonneg_coeffs"],"detail_key":"p3"},{"id":"n233","layer":"informal","project":"p3","title":"It suffices to show this for prime powers. We have e(p^k) = (k + 1) if \\chi(p) = 1, e(p^k…","kind":"proof","summary":"It suffices to show this for prime powers. We have e(p^k) = (k + 1) if \\chi(p) = 1, e(p^k) = 1…","labels":[],"detail_key":"p3"},{"id":"n234","layer":"informal","project":"p3","title":"positivity_from_derivs","kind":"lemma","summary":"An entire function f whose iterated derivatives at s are all real with alternating signs (excep…","labels":["positivity_from_derivs"],"detail_key":"p3"},{"id":"n235","layer":"informal","project":"p3","title":"This follows by considering the power series expansion at zero.","kind":"proof","summary":"This follows by considering the power series expansion at zero.","labels":[],"detail_key":"p3"},{"id":"n236","layer":"informal","project":"p3","title":"derivs_from_coeffs","kind":"lemma","summary":"If a \\colon N\\to C is an arithmetic function with a(1) > 0 and a(n) \\ge 0 for all n \\ge 2 and t…","labels":["derivs_from_coeffs"],"detail_key":"p3"},{"id":"n237","layer":"informal","project":"p3","title":"The mth derivative of~f at~x is given by \\[ f^(m)(x) = \\sum_n=1^\\infty (-\\log n)^m a(n) n…","kind":"proof","summary":"The mth derivative of~f at~x is given by \\[ f^(m)(x) = \\sum_n=1^\\infty (-\\log n)^m a(n) n^-x =…","labels":[],"detail_key":"p3"},{"id":"n238","layer":"informal","project":"p3","title":"quadratic_char_nonvanishing","kind":"lemma","summary":"If \\chi is a nontrivial quadratic Dirichlet character, then L(\\chi, 1) \\ne 0.","labels":["quadratic_char_nonvanishing"],"detail_key":"p3"},{"id":"n239","layer":"informal","project":"p3","title":"Assume that L(\\chi, 1) = 0, so \\chi is a bad character. By Lemma~\\refF_entire, we then kn…","kind":"proof","summary":"Assume that L(\\chi, 1) = 0, so \\chi is a bad character. By Lemma~\\refF_entire, we then know tha…","labels":[],"detail_key":"p3"},{"id":"n240","layer":"informal","project":"p3","title":"dirichlet_char_nonvanishing","kind":"theorem","summary":"If \\chi is a Dirichlet character and t is a real number such that t \\ne 0 or \\chi is nontrivial…","labels":["dirichlet_char_nonvanishing"],"detail_key":"p3"},{"id":"n241","layer":"informal","project":"p3","title":"If \\chi is not a quadratic character or t \\ne 0, then the claim is Lemma~\\refnon_quadrati…","kind":"proof","summary":"If \\chi is not a quadratic character or t \\ne 0, then the claim is Lemma~\\refnon_quadratic. If…","labels":[],"detail_key":"p3"},{"id":"n242","layer":"formal","project":"p3","title":"mainTheorem_general","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] χ : DirichletCharacter Complex N t : Real, Or (Ne (HPow.hPow χ 2) 1…","labels":[],"detail_key":"p3","name":"mainTheorem_general","module":"Project.EasyCase"},{"id":"n243","layer":"formal","project":"p3","title":"ArithmeticFunction.iteratedDeriv_LSeries_alternating","kind":"theorem","summary":"∀ (a : ArithmeticFunction Complex), (∀ (n : Nat), LE.le 0 (a n)) → ∀ x : Real, LT.lt (LSeries.a…","labels":[],"detail_key":"p3","name":"ArithmeticFunction.iteratedDeriv_LSeries_alternating","module":"Project.EulerProducts.Auxiliary"},{"id":"n244","layer":"formal","project":"p3","title":"Complex.apply_le_of_iteratedDeriv_alternating","kind":"theorem","summary":"∀ f : Complex → Complex s : Complex, Differentiable Complex f → (∀ (n : Nat), Ne n 0 → LE.le 0…","labels":[],"detail_key":"p3","name":"Complex.apply_le_of_iteratedDeriv_alternating","module":"Project.EulerProducts.Auxiliary"},{"id":"n245","layer":"formal","project":"p3","title":"norm_dirichlet_product_ge_one","kind":"theorem","summary":"∀ N : Nat (χ : DirichletCharacter Complex N) x : Real, LT.lt 0 x → ∀ (y : Real), GE.ge (norm (H…","labels":[],"detail_key":"p3","name":"norm_dirichlet_product_ge_one","module":"Project.EulerProducts.PNT"},{"id":"n246","layer":"formal","project":"p3","title":"mainTheorem_quadratic","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] χ : DirichletCharacter Complex N, Eq (HPow.hPow χ 2) 1 → Ne χ 1 → N…","labels":[],"detail_key":"p3","name":"mainTheorem_quadratic","module":"Project.MainTheorem"},{"id":"n247","layer":"formal","project":"p3","title":"ourMainTheorem","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (χ : DirichletCharacter Complex N) (t : Real), Or (Ne χ 1) (Ne t 0)…","labels":[],"detail_key":"p3","name":"ourMainTheorem","module":"Project.MainTheorem"},{"id":"n248","layer":"formal","project":"p3","title":"BadChar","kind":"inductive","summary":"(N : Nat) → [inst : NeZero N] → Type","labels":[],"detail_key":"p3","name":"BadChar","module":"Project.PropertiesF"},{"id":"n249","layer":"formal","project":"p3","title":"BadChar.F","kind":"def","summary":"N : Nat → [inst : NeZero N] → BadChar N → Complex → Complex","labels":[],"detail_key":"p3","name":"BadChar.F","module":"Project.PropertiesF"},{"id":"n250","layer":"formal","project":"p3","title":"BadChar.F_differentiable","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (B : BadChar N), Differentiable Complex B.F","labels":[],"detail_key":"p3","name":"BadChar.F_differentiable","module":"Project.PropertiesF"},{"id":"n251","layer":"formal","project":"p3","title":"BadChar.F_eq_LSeries","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (B : BadChar N) s : Complex, LT.lt 1 s.re → Eq (B.F s) (LSeries (⇑B…","labels":[],"detail_key":"p3","name":"BadChar.F_eq_LSeries","module":"Project.PropertiesF"},{"id":"n252","layer":"formal","project":"p3","title":"BadChar.F_neg_two","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (B : BadChar N), Eq (B.F (-2)) 0","labels":[],"detail_key":"p3","name":"BadChar.F_neg_two","module":"Project.PropertiesF"},{"id":"n253","layer":"formal","project":"p3","title":"BadChar.e_nonneg","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (B : BadChar N) (n : Nat), LE.le 0 (B.e n)","labels":[],"detail_key":"p3","name":"BadChar.e_nonneg","module":"Project.PropertiesF"},{"id":"n254","layer":"informal","project":"p4","title":"def:empiricalDistribution","kind":"definition","summary":"\\notready Let S be a multiset of elements in X with finite cardinality. Then the associated emp…","labels":["def:empiricalDistribution"],"detail_key":"p4"},{"id":"n255","layer":"informal","project":"p4","title":"def:semicirclePDFReal","kind":"definition","summary":"The function sc : R \\times R_\\geq 0 \\times R \\rightarrow R defined by \\[ sc(\\mu,v,x) = \\frac12π…","labels":["def:semicirclePDFReal"],"detail_key":"p4"},{"id":"n256","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_def","kind":"lemma","summary":"Given a mean \\mu \\in R and a variance v \\in R_\\geq 0, the pdf sc : R \\rightarrow R of the semic…","labels":["lem:semicirclePDFReal_def"],"detail_key":"p4"},{"id":"n257","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n258","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_zero_var","kind":"lemma","summary":"If the variance v is given to be zero, then the pdf of the semicircle distribution is the funct…","labels":["lem:semicirclePDFReal_zero_var"],"detail_key":"p4"},{"id":"n259","layer":"informal","project":"p4","title":"By Definition \\refdef:semicirclePDFReal, the square root of a nonpositive number is defin…","kind":"proof","summary":"By Definition \\refdef:semicirclePDFReal, the square root of a nonpositive number is defined to…","labels":[],"detail_key":"p4"},{"id":"n260","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_nonneg","kind":"lemma","summary":"The pdf of the semicircle distribution is always nonnegative.","labels":["lem:semicirclePDFReal_nonneg"],"detail_key":"p4"},{"id":"n261","layer":"informal","project":"p4","title":"By Definition \\refdef:semicirclePDFReal, the square root of a nonpositive number is defin…","kind":"proof","summary":"By Definition \\refdef:semicirclePDFReal, the square root of a nonpositive number is defined to…","labels":[],"detail_key":"p4"},{"id":"n262","layer":"informal","project":"p4","title":"lem:measurable_semicirclePDFReal","kind":"lemma","summary":"Given a mean \\mu \\in R and a variance v \\in R_\\geq 0, the pdf sc : R \\rightarrow R of the semic…","labels":["lem:measurable_semicirclePDFReal"],"detail_key":"p4"},{"id":"n263","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n264","layer":"informal","project":"p4","title":"lem:stronglyMeasurable_semicirclePDFReal","kind":"lemma","summary":"Given a mean \\mu \\in R and a variance v \\in R_\\geq 0, the pdf sc : R \\rightarrow R of the semic…","labels":["lem:stronglyMeasurable_semicirclePDFReal"],"detail_key":"p4"},{"id":"n265","layer":"informal","project":"p4","title":"By Lemma \\reflem:measurable_semicirclePDFReal, we know the pdf sc with fixed mean \\mu and…","kind":"proof","summary":"By Lemma \\reflem:measurable_semicirclePDFReal, we know the pdf sc with fixed mean \\mu and varia…","labels":[],"detail_key":"p4"},{"id":"n266","layer":"informal","project":"p4","title":"lem:integrable_semicirclePDFReal","kind":"lemma","summary":"Given a mean \\mu \\in R and a variance v \\in R_\\geq 0, the pdf sc : R \\rightarrow R of the semic…","labels":["lem:integrable_semicirclePDFReal"],"detail_key":"p4"},{"id":"n267","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n268","layer":"informal","project":"p4","title":"lem:lintegral_semicirclePDFReal_eq_one","kind":"lemma","summary":"Given a mean \\mu \\in R and a nonzero variance v \\in R_> 0, the lower Lebesgue integral of the p…","labels":["lem:lintegral_semicirclePDFReal_eq_one"],"detail_key":"p4"},{"id":"n269","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n270","layer":"informal","project":"p4","title":"lem:integral_semicirclePDFReal_eq_one","kind":"lemma","summary":"Given a mean \\mu \\in R and a nonzero variance v \\in R_> 0, the integral of the pdf sc : R \\righ…","labels":["lem:integral_semicirclePDFReal_eq_one"],"detail_key":"p4"},{"id":"n271","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n272","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_sub","kind":"lemma","summary":"For any pdf sc : R \\rightarrow R of the semicircle distribution, the following relation is sati…","labels":["lem:semicirclePDFReal_sub"],"detail_key":"p4"},{"id":"n273","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_add","kind":"lemma","summary":"For any pdf sc : R \\rightarrow R of the semicircle distribution, the following relation is sati…","labels":["lem:semicirclePDFReal_add"],"detail_key":"p4"},{"id":"n274","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_inv_mul","kind":"lemma","summary":"For any pdf sc : R \\rightarrow R of the semicircle distribution, the following relation is sati…","labels":["lem:semicirclePDFReal_inv_mul"],"detail_key":"p4"},{"id":"n275","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n276","layer":"informal","project":"p4","title":"lem:semicirclePDFReal_mul","kind":"lemma","summary":"For any pdf sc : R \\rightarrow R of the semicircle distribution, the following relation is sati…","labels":["lem:semicirclePDFReal_mul"],"detail_key":"p4"},{"id":"n277","layer":"informal","project":"p4","title":"def:semicirclePDF","kind":"definition","summary":"Let f : R \\times R_\\geq 0 \\times R \\to R denote the real-valued semicircle density defined in D…","labels":["def:semicirclePDF"],"detail_key":"p4"},{"id":"n278","layer":"informal","project":"p4","title":"lem:semicirclePDF_def","kind":"lemma","summary":"For all \\mu \\in R , v \\in R_\\geq 0, the extended pdf. \\barsc : R \\to [0,\\infty] satisfies: \\bar…","labels":["lem:semicirclePDF_def"],"detail_key":"p4"},{"id":"n279","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n280","layer":"informal","project":"p4","title":"lem:semicirclePDF_zero_var","kind":"lemma","summary":"If the variance v is zero, then the extended pdf. is identically zero: \\forall x \\in R, \\quad \\…","labels":["lem:semicirclePDF_zero_var"],"detail_key":"p4"},{"id":"n281","layer":"informal","project":"p4","title":"This follows immediately from the definition of \\barsc as h(sc(\\mu,0,x)), and the fact th…","kind":"proof","summary":"This follows immediately from the definition of \\barsc as h(sc(\\mu,0,x)), and the fact that sc(…","labels":[],"detail_key":"p4"},{"id":"n282","layer":"informal","project":"p4","title":"lem:toReal_semicirclePDF","kind":"lemma","summary":"Let \\mu \\in R , v \\in R_\\ge 0, and x \\in R . Then the real value recovered from the extended se…","labels":["lem:toReal_semicirclePDF"],"detail_key":"p4"},{"id":"n283","layer":"informal","project":"p4","title":"Since sc(\\mu, v, x) \\ge 0, we have h(sc(\\mu, v, x)) = sc(\\mu, v, x), and thus \\[ \\barsc(\\…","kind":"proof","summary":"Since sc(\\mu, v, x) \\ge 0, we have h(sc(\\mu, v, x)) = sc(\\mu, v, x), and thus \\[ \\barsc(\\mu, v,…","labels":[],"detail_key":"p4"},{"id":"n284","layer":"informal","project":"p4","title":"lem:semicirclePDF_nonneg","kind":"lemma","summary":"If v > 0, then for all \\mu, x \\in R, the extended pdf is nonnegative: \\[ 0 \\le \\barsc(\\mu,v,x).…","labels":["lem:semicirclePDF_nonneg"],"detail_key":"p4"},{"id":"n285","layer":"informal","project":"p4","title":"This is immediate from the definition of \\barsc as h(sc(\\mu,v,x)) and the nonnegativity o…","kind":"proof","summary":"This is immediate from the definition of \\barsc as h(sc(\\mu,v,x)) and the nonnegativity of sc (…","labels":[],"detail_key":"p4"},{"id":"n286","layer":"informal","project":"p4","title":"lem:semicirclePDF_finite","kind":"lemma","summary":"For all \\mu, x \\in R, and v \\in R_\\ge 0, we have: \\barsc(\\mu,v,x) < \\infty.","labels":["lem:semicirclePDF_finite"],"detail_key":"p4"},{"id":"n287","layer":"informal","project":"p4","title":"Since sc(\\mu,v,x) \\in R_\\ge 0, we have \\barsc(\\mu,v,x) = h(sc(\\mu,v,x)) < \\infty.","kind":"proof","summary":"Since sc(\\mu,v,x) \\in R_\\ge 0, we have \\barsc(\\mu,v,x) = h(sc(\\mu,v,x)) < \\infty.","labels":[],"detail_key":"p4"},{"id":"n288","layer":"informal","project":"p4","title":"lem:semicirclePDF_ne_top","kind":"lemma","summary":"For all \\( \\mu, x \\in R \\), and \\( v \\in R_\\ge 0 \\), the extended pdf is finite: \\[ \\barsc(\\mu,…","labels":["lem:semicirclePDF_ne_top"],"detail_key":"p4"},{"id":"n289","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n290","layer":"informal","project":"p4","title":"lem:support_semicirclePDF","kind":"lemma","summary":"Let \\( \\mu \\in R \\) and \\( v \\in R_> 0 \\). Then the support of the extended pdf is \\[ supp(\\bar…","labels":["lem:support_semicirclePDF"],"detail_key":"p4"},{"id":"n291","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n292","layer":"informal","project":"p4","title":"lem:measurable_semicirclePDF","kind":"lemma","summary":"The function \\( x \\mapsto \\barsc(\\mu,v,x) \\) is measurable for all \\( \\mu \\in R \\), \\( v \\in R_…","labels":["lem:measurable_semicirclePDF"],"detail_key":"p4"},{"id":"n293","layer":"informal","project":"p4","title":"Since h is measurable, and h is a measurable map \\( R_\\ge 0 \\to \\overlineR_\\ge 0 \\), thei…","kind":"proof","summary":"Since h is measurable, and h is a measurable map \\( R_\\ge 0 \\to \\overlineR_\\ge 0 \\), their comp…","labels":[],"detail_key":"p4"},{"id":"n294","layer":"informal","project":"p4","title":"lem:lintegral_semicirclePDF_eq_one","kind":"lemma","summary":"If v > 0, then the total integral of \\barsc with respect to Lebesgue measure is 1: \\[ \\int_R \\b…","labels":["lem:lintegral_semicirclePDF_eq_one"],"detail_key":"p4"},{"id":"n295","layer":"informal","project":"p4","title":"This follows from the equality: \\[ \\int_R h(sc(\\mu,v,x)) \\, dx = h( \\left( \\int_R sc(\\mu,…","kind":"proof","summary":"This follows from the equality: \\[ \\int_R h(sc(\\mu,v,x)) \\, dx = h( \\left( \\int_R sc(\\mu,v,x) \\…","labels":[],"detail_key":"p4"},{"id":"n296","layer":"informal","project":"p4","title":"def:semicircleReal","kind":"definition","summary":"The semicircle distribution with mean \\mu and variance v, denoted \\sigma(\\mu, v), is the Dirac…","labels":["def:semicircleReal"],"detail_key":"p4"},{"id":"n297","layer":"informal","project":"p4","title":"lem:semicircleReal_of_var_ne_zero","kind":"lemma","summary":"If v \\neq 0, then the definition the semicircle distribution is defined as the measure with den…","labels":["lem:semicircleReal_of_var_ne_zero"],"detail_key":"p4"},{"id":"n298","layer":"informal","project":"p4","title":"Follows directly from definition of semicircle distribution.","kind":"proof","summary":"Follows directly from definition of semicircle distribution.","labels":[],"detail_key":"p4"},{"id":"n299","layer":"informal","project":"p4","title":"lem:semicircleReal_zero_var","kind":"lemma","summary":"If the variance is 0, then the semicircle distribution is exactly the Dirac measure at \\mu.","labels":["lem:semicircleReal_zero_var"],"detail_key":"p4"},{"id":"n300","layer":"informal","project":"p4","title":"Follows directly from definition of semicircle distribution.","kind":"proof","summary":"Follows directly from definition of semicircle distribution.","labels":[],"detail_key":"p4"},{"id":"n301","layer":"informal","project":"p4","title":"lem:instIsProbabilityMeasuresemicircleReal","kind":"lemma","summary":"For all \\mu \\in R and v \\in R_\\ge 0, semicircleReal is a probability measure.","labels":["lem:instIsProbabilityMeasuresemicircleReal"],"detail_key":"p4"},{"id":"n302","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n303","layer":"informal","project":"p4","title":"lem:noAtoms_semicircleReal","kind":"lemma","summary":"If v > 0, then the semicircle distribution has no atoms.","labels":["lem:noAtoms_semicircleReal"],"detail_key":"p4"},{"id":"n304","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n305","layer":"informal","project":"p4","title":"lem:semicircleReal_apply","kind":"lemma","summary":"For a semicircle measure with mean \\mu and variance v > 0, the measure of any measurable set s…","labels":["lem:semicircleReal_apply"],"detail_key":"p4"},{"id":"n306","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n307","layer":"informal","project":"p4","title":"lem:semicircleReal_apply_eq_integral","kind":"lemma","summary":"For any mean \\mu \\in R, any variance v > 0, and any measurable set s of real numbers, the semic…","labels":["lem:semicircleReal_apply_eq_integral"],"detail_key":"p4"},{"id":"n308","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n309","layer":"informal","project":"p4","title":"lem:semicircleReal_absolutelyContinuous","kind":"lemma","summary":"For a semicircle distribution with mean \\mu and variance v > 0, the measure \\sigma(\\mu, v) is a…","labels":["lem:semicircleReal_absolutelyContinuous"],"detail_key":"p4"},{"id":"n310","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n311","layer":"informal","project":"p4","title":"lem:rnDeriv_semicircleReal","kind":"lemma","summary":"The Radon–Nikodym derivative of the semicircle measure \\sigma(\\mu, v) with respect to the Lebes…","labels":["lem:rnDeriv_semicircleReal"],"detail_key":"p4"},{"id":"n312","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n313","layer":"informal","project":"p4","title":"lem:integral_semicircleReal_eq_integral_smul","kind":"lemma","summary":"Let f : R \\to E be a function where E is a normed vector space over R. For the semicircle distr…","labels":["lem:integral_semicircleReal_eq_integral_smul"],"detail_key":"p4"},{"id":"n314","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n315","layer":"informal","project":"p4","title":"lem:semicircleReal_map_add_const","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, then for any constant y…","labels":["lem:semicircleReal_map_add_const"],"detail_key":"p4"},{"id":"n316","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n317","layer":"informal","project":"p4","title":"lem:semicircleReal_map_const_add","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, then for any constant y…","labels":["lem:semicircleReal_map_const_add"],"detail_key":"p4"},{"id":"n318","layer":"informal","project":"p4","title":"Obvious from commutativity between x + y and y + x.","kind":"proof","summary":"Obvious from commutativity between x + y and y + x.","labels":[],"detail_key":"p4"},{"id":"n319","layer":"informal","project":"p4","title":"lem:semicircleReal_map_const_mul","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, then for any constant c…","labels":["lem:semicircleReal_map_const_mul"],"detail_key":"p4"},{"id":"n320","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n321","layer":"informal","project":"p4","title":"lem:semicircleReal_map_mul_const","kind":"lemma","summary":"Given semicircular measure \\sigma with mean \\mu and variance v, then for any constant c \\in R,…","labels":["lem:semicircleReal_map_mul_const"],"detail_key":"p4"},{"id":"n322","layer":"informal","project":"p4","title":"Use commutativity between Xc and cX.","kind":"proof","summary":"Use commutativity between Xc and cX.","labels":[],"detail_key":"p4"},{"id":"n323","layer":"informal","project":"p4","title":"lem:semicircleReal_map_neg","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, the pushforward of \\sig…","labels":["lem:semicircleReal_map_neg"],"detail_key":"p4"},{"id":"n324","layer":"informal","project":"p4","title":"Special case of the multiplication by constant map with constant being -1.","kind":"proof","summary":"Special case of the multiplication by constant map with constant being -1.","labels":[],"detail_key":"p4"},{"id":"n325","layer":"informal","project":"p4","title":"lem:semicircleReal_map_sub_const","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, then for any constant y…","labels":["lem:semicircleReal_map_sub_const"],"detail_key":"p4"},{"id":"n326","layer":"informal","project":"p4","title":"Use the map by addition of constant and substitute constant for its -1 multiple.","kind":"proof","summary":"Use the map by addition of constant and substitute constant for its -1 multiple.","labels":[],"detail_key":"p4"},{"id":"n327","layer":"informal","project":"p4","title":"lem:semicircleReal_map_const_sub","kind":"lemma","summary":"Given semicircular measure \\sigma(\\mu, v) with mean \\mu and variance v, then for any constant y…","labels":["lem:semicircleReal_map_const_sub"],"detail_key":"p4"},{"id":"n328","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n329","layer":"informal","project":"p4","title":"lem:semicircleReal_add_const","kind":"lemma","summary":"Given a real random variable X \\sim \\sigma(\\mu, v) then for a constant y \\in R, X + y \\sim \\sig…","labels":["lem:semicircleReal_add_const"],"detail_key":"p4"},{"id":"n330","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n331","layer":"informal","project":"p4","title":"lem:semicircleReal_const_add","kind":"lemma","summary":"Given a real random variable X \\sim \\sigma(\\mu, v) then for a constant y \\in R, y + X \\sim \\sig…","labels":["lem:semicircleReal_const_add"],"detail_key":"p4"},{"id":"n332","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n333","layer":"informal","project":"p4","title":"lem:semicircleReal_const_mul","kind":"lemma","summary":"Given a real random variable X \\sim \\sigma(\\mu, v), then for a constant c \\in R, cX \\sim \\sigma…","labels":["lem:semicircleReal_const_mul"],"detail_key":"p4"},{"id":"n334","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n335","layer":"informal","project":"p4","title":"lem:semicircleReal_mul_const","kind":"lemma","summary":"Given a real random variable X \\sim \\sigma(\\mu, v), then for a constant c \\in R, Xc \\sim \\sigma…","labels":["lem:semicircleReal_mul_const"],"detail_key":"p4"},{"id":"n336","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n337","layer":"informal","project":"p4","title":"lem:integral_id_semicircleReal","kind":"lemma","summary":"If X \\sim \\sigma(\\mu, v), then its expectation E[X] = \\int x d \\sigma(\\mu, v) = \\mu","labels":["lem:integral_id_semicircleReal"],"detail_key":"p4"},{"id":"n338","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n339","layer":"informal","project":"p4","title":"lem:variance_fun_id_semicircleReal","kind":"lemma","summary":"If X \\sim \\sigma(\\mu, v), then its variance Var(X) = v","labels":["lem:variance_fun_id_semicircleReal"],"detail_key":"p4"},{"id":"n340","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n341","layer":"informal","project":"p4","title":"lem:variance_id_semicircleReal","kind":"lemma","summary":"The variance of a real semicircle distribution with parameter (\\mu, v) is its variance paramete…","labels":["lem:variance_id_semicircleReal"],"detail_key":"p4"},{"id":"n342","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n343","layer":"informal","project":"p4","title":"lem:memLp_id_semicircleReal'","kind":"lemma","summary":"All the moments of a real semicircle distribution are finite. That is, the identity is in L_p f…","labels":["lem:memLp_id_semicircleReal'"],"detail_key":"p4"},{"id":"n344","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n345","layer":"informal","project":"p4","title":"lem:centralMoment_two_mul_semicircleReal","kind":"lemma","summary":"E[(X - \\mu)^2n] = v^n C_n","labels":["lem:centralMoment_two_mul_semicircleReal"],"detail_key":"p4"},{"id":"n346","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n347","layer":"informal","project":"p4","title":"lem:centralMoment_fun_two_mul_semicircleReal","kind":"lemma","summary":"E[(X - \\mu)^2n] = v^n C_n","labels":["lem:centralMoment_fun_two_mul_semicircleReal"],"detail_key":"p4"},{"id":"n348","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n349","layer":"informal","project":"p4","title":"lem:centralMoment_odd_semicircleReal","kind":"lemma","summary":"E[(X - \\mu)^2n + 1] = 0","labels":["lem:centralMoment_odd_semicircleReal"],"detail_key":"p4"},{"id":"n350","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n351","layer":"informal","project":"p4","title":"lem:centralMoment_fun_odd_semicircleReal","kind":"lemma","summary":"E[(X - \\mu)^2n + 1] = 0","labels":["lem:centralMoment_fun_odd_semicircleReal"],"detail_key":"p4"},{"id":"n352","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n353","layer":"informal","project":"p4","title":"def:loop_walk","kind":"definition","summary":"Given a simple graph G, a LoopWalk is a walk on G that is allowed to have consecutive visits to…","labels":["def:loop_walk"],"detail_key":"p4"},{"id":"n354","layer":"informal","project":"p4","title":"def:graph_walk_multi_index","kind":"definition","summary":"Let i \\in[n]^k be a k-index, i=\\left(i_1, i_2, \\ldots, i_k\\right). The path w_i is the closed L…","labels":["def:graph_walk_multi_index"],"detail_key":"p4"},{"id":"n355","layer":"informal","project":"p4","title":"def:graph_walk_edges","kind":"definition","summary":"Given a LoopWalk w_i=((i_1, i_2),(i_2, i_3), \\ldots,(i_k-1, i_k),(i_k, i_1)) on K_n, let E_i= \\…","labels":["def:graph_walk_edges"],"detail_key":"p4"},{"id":"n356","layer":"informal","project":"p4","title":"def:graph_walk_vertices","kind":"definition","summary":"Given a LoopWalk w_i=((i_1, i_2),(i_2, i_3), \\ldots,(i_k-1, i_k),(i_k, i_1)) on K_n, let V_i= \\…","labels":["def:graph_walk_vertices"],"detail_key":"p4"},{"id":"n357","layer":"informal","project":"p4","title":"prop:vertex_edge_inequality","kind":"proposition","summary":"Let w_i be a LoopWalk. Then, |V_i|\\le |E_i|+1.","labels":["prop:vertex_edge_inequality"],"detail_key":"p4"},{"id":"n358","layer":"informal","project":"p4","title":"|V|\\le |E|+1: proof by induction on |V|. Base case |V| = 1 is obvious. For each additiona…","kind":"proof","summary":"|V|\\le |E|+1: proof by induction on |V|. Base case |V| = 1 is obvious. For each additional vert…","labels":[],"detail_key":"p4"},{"id":"n359","layer":"informal","project":"p4","title":"Graph Edge Count","kind":"definition","summary":"[Graph Edge Count] Let i \\in[n]^k be a k-index, i=\\left(i_1, i_2, \\ldots, i_k\\right). For any e…","labels":["def:graph_edge_count"],"detail_key":"p4"},{"id":"n360","layer":"informal","project":"p4","title":"Self Edges","kind":"definition","summary":"[Self Edges] Let i \\in[n]^k be a k-index with walk w_i on K_n. Define the self-edges E_i^s as:…","labels":["def:graph_self_edges"],"detail_key":"p4"},{"id":"n361","layer":"informal","project":"p4","title":"Connecting Edges","kind":"definition","summary":"[Connecting Edges] Let i \\in[n]^k be a k-index with walk w_i on K_n. Define the connecting-edge…","labels":["def:graph_connecting_edges"],"detail_key":"p4"},{"id":"n362","layer":"informal","project":"p4","title":"Length |w_i| : R-1-1 : def:length\\_of\\_w\\_i","kind":"definition","summary":"[Length |w_i| : R-1-1 : def:length\\_of\\_w\\_i] Given a LoopWalk w_i generated by some k-index i,…","labels":["def:length_of_w_i"],"detail_key":"p4"},{"id":"n363","layer":"informal","project":"p4","title":"|w_i| = k : R-1-2 : lem:abs\\_w\\_i\\_eq\\_k","kind":"lemma","summary":"[|w_i| = k : R-1-2 : lem:abs\\_w\\_i\\_eq\\_k] For any k-index i, |V_i| \\leq k and \\[ |w_i| \\equiv…","labels":["lem:abs_w_i_eq_k"],"detail_key":"p4"},{"id":"n364","layer":"informal","project":"p4","title":"Foremost, since the number of vertices are the number of distinct elements of the k-index…","kind":"proof","summary":"Foremost, since the number of vertices are the number of distinct elements of the k-index i, it…","labels":[],"detail_key":"p4"},{"id":"n365","layer":"informal","project":"p4","title":"def:graph_walk_equiv","kind":"definition","summary":"Given two LoopWalks w_j = ((j_1, j_2), \\dots (j_k-1, j_k), (j_k, j_1)) and w_i=((i_1, i_2), \\do…","labels":["def:graph_walk_equiv"],"detail_key":"p4"},{"id":"n366","layer":"informal","project":"p4","title":"lem:graph_walk_equiv","kind":"lemma","summary":"The relation in \\refdef:graph_walk_equiv is indeed an equivalence relation.","labels":["lem:graph_walk_equiv"],"detail_key":"p4"},{"id":"n367","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n368","layer":"informal","project":"p4","title":"lem:walk_vertex_card_equiv","kind":"lemma","summary":"If w_i \\sim w_j, then |V_i| = |V_j|.","labels":["lem:walk_vertex_card_equiv"],"detail_key":"p4"},{"id":"n369","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n370","layer":"informal","project":"p4","title":"lem:walk_edge_card_equiv","kind":"lemma","summary":"If w_i \\sim w_j, then |E_i| = |E_j|.","labels":["lem:walk_edge_card_equiv"],"detail_key":"p4"},{"id":"n371","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n372","layer":"informal","project":"p4","title":"lem:walk_edge_count_equiv","kind":"lemma","summary":"If w_i \\sim w_j, then the set of edge counts for w_i is the same as w_j. In other words, \\w_i(i…","labels":["lem:walk_edge_count_equiv"],"detail_key":"p4"},{"id":"n373","layer":"informal","project":"p4","title":"G_k : R-1-4 : def:g\\_k","kind":"definition","summary":"[G_k : R-1-4 : def:g\\_k] Let G_k denote the set of all equivalence classes under \\refdef:graph_…","labels":["def:g_k"],"detail_key":"p4"},{"id":"n374","layer":"informal","project":"p4","title":"lem:graph_set_finite","kind":"lemma","summary":"|G_k| is finite.","labels":["lem:graph_set_finite"],"detail_key":"p4"},{"id":"n375","layer":"informal","project":"p4","title":"Requires proof.","kind":"proof","summary":"Requires proof.","labels":[],"detail_key":"p4"},{"id":"n376","layer":"informal","project":"p4","title":"|V_w| : ef:abs.V\\_w","kind":"definition","summary":"[|V_w| : ef:abs.V\\_w] Given a w \\in G_k, we define |V_w| to be |V_i|, where w_i is a LoopWalk i…","labels":["def:abs.V_w"],"detail_key":"p4"},{"id":"n377","layer":"informal","project":"p4","title":"|E_w| : def:abs.E\\_w","kind":"definition","summary":"[|E_w| : def:abs.E\\_w] Given a w \\in G_k, we define |E_w| to be |E_i|, where w_i is a LoopWalk…","labels":["def:abs.E_w"],"detail_key":"p4"},{"id":"n378","layer":"informal","project":"p4","title":"def:edge_count_w","kind":"definition","summary":"Given w \\in G_k, we define EC_w to be the multiset of edge counts of w.","labels":["def:edge_count_w"],"detail_key":"p4"},{"id":"n379","layer":"informal","project":"p4","title":"Lemma 4.3 in \\citeKemp2013RMTNotes : R-1-9 : lem:lem\\_4.3","kind":"lemma","summary":"[Lemma 4.3 in \\citeKemp2013RMTNotes : R-1-9 : lem:lem\\_4.3] Given w \\in G_k, we have \\[ |\\ i \\i…","labels":["lem:lem_4.3"],"detail_key":"p4"},{"id":"n380","layer":"informal","project":"p4","title":"By the way the equivalence relation is defined in Definition \\refdef:graph_walk_equiv, th…","kind":"proof","summary":"By the way the equivalence relation is defined in Definition \\refdef:graph_walk_equiv, the fact…","labels":[],"detail_key":"p4"},{"id":"n381","layer":"informal","project":"p4","title":"G_k, w \\geq 2 : R-1-14 : def:g\\_k\\_ge\\_2","kind":"definition","summary":"[G_k, w \\geq 2 : R-1-14 : def:g\\_k\\_ge\\_2] Let G_k,w \\geq 2 be a subset of G_k in which the wal…","labels":["def:g_k_ge_2"],"detail_key":"p4"},{"id":"n382","layer":"informal","project":"p4","title":"\\#E \\leq k/2 : R-1-17 : lem:edge\\_set\\_order\\_leq\\_k\\_over\\_two","kind":"lemma","summary":"[\\#E \\leq k/2 : R-1-17 : lem:edge\\_set\\_order\\_leq\\_k\\_over\\_two] Given a w \\in G_k,w \\geq 2, w…","labels":["lem:edge_set_order_leq_k_over_two"],"detail_key":"p4"},{"id":"n383","layer":"informal","project":"p4","title":"Since |w| = k, if each edge in G is traversed at least twice, then by construction of w t…","kind":"proof","summary":"Since |w| = k, if each edge in G is traversed at least twice, then by construction of w the num…","labels":[],"detail_key":"p4"},{"id":"n384","layer":"informal","project":"p4","title":"prop:vertex_edge_tree_equality","kind":"proposition","summary":"\\notready Let G=(V,E) be a connected finite graph. Then, |G|=\\#V=\\#E+1 if and only if G is a pl…","labels":["prop:vertex_edge_tree_equality"],"detail_key":"p4"},{"id":"n385","layer":"informal","project":"p4","title":"\\notready |G|=\\#V=\\#E+1 if G is a plane tree is already in Lean: SimpleGraph.IsTree.card\\…","kind":"proof","summary":"\\notready |G|=\\#V=\\#E+1 if G is a plane tree is already in Lean: SimpleGraph.IsTree.card\\_edgeF…","labels":[],"detail_key":"p4"},{"id":"n386","layer":"informal","project":"p4","title":"lem:vertex_bound","kind":"lemma","summary":"Let w \\in G_k,w \\geq 2. Then |V_w| \\le k/2 + 1.","labels":["lem:vertex_bound"],"detail_key":"p4"},{"id":"n387","layer":"informal","project":"p4","title":"Follows directly from earlier lemmas (replacing \\#E with k/2).","kind":"proof","summary":"Follows directly from earlier lemmas (replacing \\#E with k/2).","labels":[],"detail_key":"p4"},{"id":"n388","layer":"informal","project":"p4","title":"lem:odd_vertex_bound","kind":"lemma","summary":"Let w \\in G_k,w \\geq 2. Suppose k odd. Then, |V_w| \\le \\frack2 + \\frac12.","labels":["lem:odd_vertex_bound"],"detail_key":"p4"},{"id":"n389","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n390","layer":"informal","project":"p4","title":"lem:edge_bound_large_w","kind":"lemma","summary":"If w \\in G_k,w \\geq 2, k is even, and there exists e such that w(e) \\ge 3, then |E_w| \\le \\frac…","labels":["lem:edge_bound_large_w"],"detail_key":"p4"},{"id":"n391","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n392","layer":"informal","project":"p4","title":"prop:g_bound_self_edge","kind":"proposition","summary":"Let w\\inG_k,w \\geq 2, and suppose k is even. If there exists a loop in w, then |V_w|\\le k/2.","labels":["prop:g_bound_self_edge"],"detail_key":"p4"},{"id":"n393","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n394","layer":"informal","project":"p4","title":"prop:g_bound_large_w","kind":"proposition","summary":"Let w \\inG_k,w \\geq 2, and suppose k is even. If there exists e\\in EC_w with e \\ge 3, then |V_w…","labels":["prop:g_bound_large_w"],"detail_key":"p4"},{"id":"n395","layer":"informal","project":"p4","title":"The sum of w over all edges E in G is k. Hence, the sum of w over E\\setminus\\e\\ is \\le k-…","kind":"proof","summary":"The sum of w over all edges E in G is k. Hence, the sum of w over E\\setminus\\e\\ is \\le k-3. Sin…","labels":[],"detail_key":"p4"},{"id":"n396","layer":"informal","project":"p4","title":"def:special_set_g","kind":"definition","summary":"Let G^k/2+1_k to be the set of w\\inG_k where |V_w|=k/2+1, contains no self-edges, and the walk…","labels":["def:special_set_g"],"detail_key":"p4"},{"id":"n397","layer":"informal","project":"p4","title":"lem:special_g_edge_count","kind":"lemma","summary":"Elements of G_k^k/2+1 have |E_w| = k/2.","labels":["lem:special_g_edge_count"],"detail_key":"p4"},{"id":"n398","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n399","layer":"informal","project":"p4","title":"lem:special_g_vertex_count","kind":"lemma","summary":"Elements of G_k^k/2+1 have |V_w| = k/2 + 1.","labels":["lem:special_g_vertex_count"],"detail_key":"p4"},{"id":"n400","layer":"informal","project":"p4","title":"By definition.","kind":"proof","summary":"By definition.","labels":[],"detail_key":"p4"},{"id":"n401","layer":"informal","project":"p4","title":"lem:special_g_tree","kind":"lemma","summary":"Elements of G_k^k/2+1 are trees.","labels":["lem:special_g_tree"],"detail_key":"p4"},{"id":"n402","layer":"informal","project":"p4","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p4"},{"id":"n403","layer":"informal","project":"p4","title":"def:Dyck_paths","kind":"definition","summary":"\\notready A Dyck path of length k is a sequence (d_1,...,d_k) \\in \\\\pm 1\\^k such that their par…","labels":["def:Dyck_paths"],"detail_key":"p4"},{"id":"n404","layer":"informal","project":"p4","title":"def:graph_to_Dyck_map","kind":"definition","summary":"\\notready Define a map \\phi whose input is w \\in G^k/2 + 1_k. Then for its output, define a seq…","labels":["def:graph_to_Dyck_map"],"detail_key":"p4"},{"id":"n405","layer":"informal","project":"p4","title":"lem:graph_Dyck_correspondence","kind":"lemma","summary":"\\notready \\phi(w) = d(w) \\in D_k, where D_k denotes the set of Dyck path of order k.","labels":["lem:graph_Dyck_correspondence"],"detail_key":"p4"},{"id":"n406","layer":"informal","project":"p4","title":"\\notready set P_0 = (0,0) and P_j = (j,d_1+\\cdots+d_j) for 1\\le j\\le k; then the piecewis…","kind":"proof","summary":"\\notready set P_0 = (0,0) and P_j = (j,d_1+\\cdots+d_j) for 1\\le j\\le k; then the piecewise line…","labels":[],"detail_key":"p4"},{"id":"n407","layer":"informal","project":"p4","title":"def:Dyck_to_graph_map","kind":"definition","summary":"\\notready Define a map \\psi whose input is a Dyck path of order k: d_k \\in \\\\pm1\\^k. Then the o…","labels":["def:Dyck_to_graph_map"],"detail_key":"p4"},{"id":"n408","layer":"informal","project":"p4","title":"lem:Dyck_graph_correspondence","kind":"lemma","summary":"\\notready \\psi(d_k) \\in G^k/2 + 1_k","labels":["lem:Dyck_graph_correspondence"],"detail_key":"p4"},{"id":"n409","layer":"informal","project":"p4","title":"\\notready Use induction on the order of Dyck path k which is an even number. Assume \\phi(…","kind":"proof","summary":"\\notready Use induction on the order of Dyck path k which is an even number. Assume \\phi(d_k-2)…","labels":[],"detail_key":"p4"},{"id":"n410","layer":"informal","project":"p4","title":"lem:composition1","kind":"lemma","summary":"\\notready \\phi \\circ \\psi = id_D_k","labels":["lem:composition1"],"detail_key":"p4"},{"id":"n411","layer":"informal","project":"p4","title":"\\notready Apply \\psi to a given Dyck path d by definition, then apply \\phi to get a new s…","kind":"proof","summary":"\\notready Apply \\psi to a given Dyck path d by definition, then apply \\phi to get a new sequenc…","labels":[],"detail_key":"p4"},{"id":"n412","layer":"informal","project":"p4","title":"lem:composition2","kind":"lemma","summary":"\\notready \\psi \\circ \\phi = id_G^k/2 + 1_k","labels":["lem:composition2"],"detail_key":"p4"},{"id":"n413","layer":"informal","project":"p4","title":"\\notready The map \\psi recovers the graph walk structure of the input from its Dyck path…","kind":"proof","summary":"\\notready The map \\psi recovers the graph walk structure of the input from its Dyck path by the…","labels":[],"detail_key":"p4"},{"id":"n414","layer":"informal","project":"p4","title":"lem:walk_to_Dyck_paths_bijection","kind":"lemma","summary":"\\notready Let k be even and let D_k denote the set of Dyck paths of length k \\[ D_k = \\(d_1,\\ld…","labels":["lem:walk_to_Dyck_paths_bijection"],"detail_key":"p4"},{"id":"n415","layer":"informal","project":"p4","title":"\\notready obvious from the previous lemmas.","kind":"proof","summary":"\\notready obvious from the previous lemmas.","labels":[],"detail_key":"p4"},{"id":"n416","layer":"informal","project":"p4","title":"def:Catalan_number","kind":"definition","summary":"\\mathlibok \\[C_0 = 1, \\quad \\textand for n \\geq 1, \\quad C_n = \\sum_k=0^n-1 C_k C_n-1-k.\\]","labels":["def:Catalan_number"],"detail_key":"p4"},{"id":"n417","layer":"informal","project":"p4","title":"lem:binary_tree_Catalan_number","kind":"lemma","summary":"\\notready \\#\\binary trees with \\frack2 vertices\\ is given by Catalan number C_k / 2","labels":["lem:binary_tree_Catalan_number"],"detail_key":"p4"},{"id":"n418","layer":"informal","project":"p4","title":"\\notready","kind":"proof","summary":"\\notready","labels":[],"detail_key":"p4"},{"id":"n419","layer":"informal","project":"p4","title":"prop:Catalan_Dyck_samecardinality","kind":"proposition","summary":"\\notready \\[|D_k| = C_k/2 \\] where |D_k| denotes the number of Dyck paths of length k while C_k…","labels":["prop:Catalan_Dyck_samecardinality"],"detail_key":"p4"},{"id":"n420","layer":"informal","project":"p4","title":"\\notready Given a binary tree with k nodes, perform preorder traversal: for each internal…","kind":"proof","summary":"\\notready Given a binary tree with k nodes, perform preorder traversal: for each internal node…","labels":[],"detail_key":"p4"},{"id":"n421","layer":"informal","project":"p4","title":"prop:graph_Catalan_number","kind":"proposition","summary":"\\notready \\[|G^k/2 + 1_k| = C_k/2\\]","labels":["prop:graph_Catalan_number"],"detail_key":"p4"},{"id":"n422","layer":"informal","project":"p4","title":"Graph Union","kind":"definition","summary":"[Graph Union] \\notready Given k-indices i = (i_1, i_2, \\cdots , i_k), j = (j_1, j_2, \\cdots , j…","labels":["def:graph_union"],"detail_key":"p4"},{"id":"n423","layer":"informal","project":"p4","title":"Ordered Triple","kind":"definition","summary":"[Ordered Triple] \\notready Given k-indices i = (i_1, i_2, \\cdots , i_k), j = (j_1, j_2, \\cdots…","labels":["def:ordered_triple"],"detail_key":"p4"},{"id":"n424","layer":"informal","project":"p4","title":"R-2-2 : def:graph\\_walk\\_triple\\_set","kind":"definition","summary":"[R-2-2 : def:graph\\_walk\\_triple\\_set] \\notready We define G_k,k to be the set of connected gra…","labels":["def:graph_walk_triple_set"],"detail_key":"p4"},{"id":"n425","layer":"informal","project":"p4","title":"R-2-3-1 : def:index\\_pair","kind":"definition","summary":"[R-2-3-1 : def:index\\_pair] \\notready Given (G,w,w') \\in G_k,k, we say (i,j) is an ordered pair…","labels":["def:index_pair"],"detail_key":"p4"},{"id":"n426","layer":"informal","project":"p4","title":"R-2-3-3 : def:graph\\_walk\\_triple\\_rel","kind":"definition","summary":"[R-2-3-3 : def:graph\\_walk\\_triple\\_rel] \\notready Let (G_i \\# j,w_i,w_j) be an ordered triple…","labels":["def:graph_walk_triple_rel"],"detail_key":"p4"},{"id":"n427","layer":"informal","project":"p4","title":"R-2-3-5 : lem:common\\_val\\_prod\\_of\\_eq\\_of\\_graph\\_walk\\_triple\\_rel","kind":"lemma","summary":"[R-2-3-5 : lem:common\\_val\\_prod\\_of\\_eq\\_of\\_graph\\_walk\\_triple\\_rel] \\notready Let (G_i_1 \\#…","labels":["lem:common_val_prod_eq_of_graph_walk_triple_rel"],"detail_key":"p4"},{"id":"n428","layer":"informal","project":"p4","title":"This follows from Lemma \\reflem:eq_equiv_eq_expect.","kind":"proof","summary":"This follows from Lemma \\reflem:eq_equiv_eq_expect.","labels":[],"detail_key":"p4"},{"id":"n429","layer":"informal","project":"p4","title":"R-2-6-1 : def:def:graph\\_walk\\_triple\\_single\\_edges","kind":"definition","summary":"[R-2-6-1 : def:def:graph\\_walk\\_triple\\_single\\_edges] \\notready Given a graph G_i \\# j, let E^…","labels":["def:graph_walk_triple_single_edges"],"detail_key":"p4"},{"id":"n430","layer":"informal","project":"p4","title":"R-2-6-2 : def:graph\\_walk\\_triple\\_connected\\_edges","kind":"definition","summary":"[R-2-6-2 : def:graph\\_walk\\_triple\\_connected\\_edges] \\notready Given a graph G_i \\# j, let E^c…","labels":["def:graph_walk_triple_connected_edges"],"detail_key":"p4"},{"id":"n431","layer":"informal","project":"p4","title":"R-2-7 : def:edgeCountPair","kind":"definition","summary":"[R-2-7 : def:edgeCountPair] \\notready Given a graph G_i \\# j, let w_i \\# j(e) denote the number…","labels":["def:edgeCountPair"],"detail_key":"p4"},{"id":"n432","layer":"informal","project":"p4","title":"R-2-11 : lem:sum\\_count\\_edge\\_pair\\_eq\\_length\\_add\\_length","kind":"lemma","summary":"[R-2-11 : lem:sum\\_count\\_edge\\_pair\\_eq\\_length\\_add\\_length] \\notready \\[ \\sum_e \\in E_i \\# j…","labels":["lem:sum_count_edge_pair_eq_length_add_length"],"detail_key":"p4"},{"id":"n433","layer":"informal","project":"p4","title":"By construction of the paths w_i and w_j, \\[ \\sum_e \\in E_i w_i(e) = k = \\sum_e \\in E_j w…","kind":"proof","summary":"By construction of the paths w_i and w_j, \\[ \\sum_e \\in E_i w_i(e) = k = \\sum_e \\in E_j w_j(e).…","labels":[],"detail_key":"p4"},{"id":"n434","layer":"informal","project":"p4","title":"R-2-2 : def:graph\\_walk\\_triple\\_set","kind":"definition","summary":"[R-2-2 : def:graph\\_walk\\_triple\\_set] \\notready (G,w,w')\\in G_k,k,w+w'\\ge 2 if (G,w,w') \\in G_…","labels":["def:graph_walk_triple_set_w_ge_two"],"detail_key":"p4"},{"id":"n435","layer":"informal","project":"p4","title":"lem:g_w_w_count","kind":"proposition","summary":"\\notready For (G, w, w') \\in G_k,k,w+w'\\ge 2, \\#\\left\\(i,j)\\in [n]^2k\\colon (G_i\\#j,w_i,w_j) =…","labels":["lem:g_w_w_count"],"detail_key":"p4"},{"id":"n436","layer":"informal","project":"p4","title":"\\notready The enumeration of the number of 2k-tuples yielding a certain graph with two wa…","kind":"proof","summary":"\\notready The enumeration of the number of 2k-tuples yielding a certain graph with two walks is…","labels":[],"detail_key":"p4"},{"id":"n437","layer":"informal","project":"p4","title":"lem:g_k_k_edge_count_maximum","kind":"lemma","summary":"\\notready In the set G_k,k,w+w'\\ge 2, edge count is at most k.","labels":["lem:g_k_k_edge_count_maximum"],"detail_key":"p4"},{"id":"n438","layer":"informal","project":"p4","title":"\\notready Now, we have the condition w+w'\\ge 2, meaning every edge is traversed at least…","kind":"proof","summary":"\\notready Now, we have the condition w+w'\\ge 2, meaning every edge is traversed at least twice.…","labels":[],"detail_key":"p4"},{"id":"n439","layer":"informal","project":"p4","title":"lem:exactly_k_edges","kind":"lemma","summary":"\\notready G \\in G_k, k. If |G| = k + 1, then G has exactly k edges.","labels":["lem:exactly_k_edges"],"detail_key":"p4"},{"id":"n440","layer":"informal","project":"p4","title":"lem:G_is_tree","kind":"lemma","summary":"\\notready G \\in G_k, k. If |G| = k + 1, then G \\in G_k, k is a tree.","labels":["lem:G_is_tree"],"detail_key":"p4"},{"id":"n441","layer":"informal","project":"p4","title":"lem:subgraph_of_tree","kind":"lemma","summary":"\\notready G_i \\# j is a tree, then its subgraph G_i and G_j are also trees.","labels":["lem:subgraph_of_tree"],"detail_key":"p4"},{"id":"n442","layer":"informal","project":"p4","title":"lem:traverse_exactly_twice","kind":"lemma","summary":"\\notready (G, w, w') \\in G_k, k. If |G| = k + 1, then for a common edge e between the two walks…","labels":["lem:traverse_exactly_twice"],"detail_key":"p4"},{"id":"n443","layer":"informal","project":"p4","title":"Assume for contradiction there is one edge l such that w(l) + w'(l) > 2, then \\sum_e \\in…","kind":"proof","summary":"Assume for contradiction there is one edge l such that w(l) + w'(l) > 2, then \\sum_e \\in G w(e)…","labels":[],"detail_key":"p4"},{"id":"n444","layer":"informal","project":"p4","title":"lem:i_j_traverse_once","kind":"lemma","summary":"\\notready (G, w, w') \\in G_k, k and |G| = k + 1, then for a common edge it is impossible that w…","labels":["lem:i_j_traverse_once"],"detail_key":"p4"},{"id":"n445","layer":"informal","project":"p4","title":"Each walk is a closed walk, which implies their edge needs to be traversed in even number.","kind":"proof","summary":"Each walk is a closed walk, which implies their edge needs to be traversed in even number.","labels":[],"detail_key":"p4"},{"id":"n446","layer":"informal","project":"p4","title":"lem:one_walk_traverse_twice","kind":"lemma","summary":"\\notready (G, w, w') \\in G_k, k and |G| = k + 1, then it can only be the case that w(e) = 2, w'…","labels":["lem:one_walk_traverse_twice"],"detail_key":"p4"},{"id":"n447","layer":"informal","project":"p4","title":"As a common edge, it is impossible that w' does not traverse it.","kind":"proof","summary":"As a common edge, it is impossible that w' does not traverse it.","labels":[],"detail_key":"p4"},{"id":"n448","layer":"informal","project":"p4","title":"lem:no_shared_edges","kind":"lemma","summary":"(G, w, w') \\in G_k, k and |G| = k + 1. Then two walks have no shared edges e which they have tr…","labels":["lem:no_shared_edges"],"detail_key":"p4"},{"id":"n449","layer":"informal","project":"p4","title":"lem:disjoint_edge_set","kind":"lemma","summary":"\\notready The only graph walks (G,w,w')\\in G_k,k with |G|=k+1 must have the edge sets covered b…","labels":["lem:disjoint_edge_set"],"detail_key":"p4"},{"id":"n450","layer":"informal","project":"p4","title":"lem:G_leq_k","kind":"lemma","summary":"\\notready For any (G_i\\#j,w_i,w_j) \\in G_k,k, with k + 1 vertices, \\pi(G_i\\#j,w_i,w_j) = 0","labels":["lem:G_leq_k"],"detail_key":"p4"},{"id":"n451","layer":"informal","project":"p4","title":"\\pi(G_i\\#j,w_i,w_j) = E(X_iX_j) - E(X_i)E(X_j) = 0","kind":"proof","summary":"\\pi(G_i\\#j,w_i,w_j) = E(X_iX_j) - E(X_i)E(X_j) = 0","labels":[],"detail_key":"p4"},{"id":"n452","layer":"informal","project":"p4","title":"Matrix Powers Entries","kind":"lemma","summary":"[Matrix Powers Entries] Let Y be an n\\times n matrix and k \\in N. Then, for each (i, j)-th entr…","labels":["lem:matrix_powers_entries"],"detail_key":"p4"},{"id":"n453","layer":"informal","project":"p4","title":"We proceed by induction on k. Our base case is k=1, then: Y_n^1 = Y_n \\quad \\Rightarrow \\…","kind":"proof","summary":"We proceed by induction on k. Our base case is k=1, then: Y_n^1 = Y_n \\quad \\Rightarrow \\quad […","labels":[],"detail_key":"p4"},{"id":"n454","layer":"informal","project":"p4","title":"Matrix Multi Index","kind":"definition","summary":"[Matrix Multi Index] Let i \\in[n]^k be a k-index, i=\\left(i_1, i_2, \\ldots, i_k\\right). Let Y b…","labels":["def:matrix_multi_index"],"detail_key":"p4"},{"id":"n455","layer":"informal","project":"p4","title":"Matrix Powers Trace","kind":"lemma","summary":"[Matrix Powers Trace] Let Y be an n\\times n matrix and k \\in N. Then, the trace of Y^k is given…","labels":["lem:matrix_powers_trace"],"detail_key":"p4"},{"id":"n456","layer":"informal","project":"p4","title":"We can use the result from Lemma \\reflem:matrix_powers_entries to compute the trace of Y^…","kind":"proof","summary":"We can use the result from Lemma \\reflem:matrix_powers_entries to compute the trace of Y^k: &\\T…","labels":[],"detail_key":"p4"},{"id":"n457","layer":"informal","project":"p4","title":"Trace of Expectation of Matrix","kind":"lemma","summary":"[Trace of Expectation of Matrix] \\bE(\\Tr(Y_n^k)) = \\sum_i \\in [n]^k \\bE(Y_i)","labels":["lem:trace_expectation_of_matrix"],"detail_key":"p4"},{"id":"n458","layer":"informal","project":"p4","title":"By linearity of expectation.","kind":"proof","summary":"By linearity of expectation.","labels":[],"detail_key":"p4"},{"id":"n459","layer":"informal","project":"p4","title":"Matrix Multi Index and Graph Equivalence","kind":"lemma","summary":"[Matrix Multi Index and Graph Equivalence] Let i \\in [n]^k be a k-index, i=\\left(i_1, i_2, \\ldo…","labels":["lem:multi_index_graph_equivalence"],"detail_key":"p4"},{"id":"n460","layer":"informal","project":"p4","title":"We see from definition \\refdef:graph_walk_multi_index that the path is defined as: w_i =…","kind":"proof","summary":"We see from definition \\refdef:graph_walk_multi_index that the path is defined as: w_i = ((i_1,…","labels":[],"detail_key":"p4"},{"id":"n461","layer":"informal","project":"p4","title":"Graph Walk and Graph Count Equivalence","kind":"lemma","summary":"[Graph Walk and Graph Count Equivalence] Let i \\in[n]^k be a k-index, i=\\left(i_1, i_2, \\ldots,…","labels":["lem:graph_walk_count_equivalence"],"detail_key":"p4"},{"id":"n462","layer":"informal","project":"p4","title":"Using lemma \\reflem:multi_index_graph_equivalence, we already have that Y_i = \\prod_w \\in…","kind":"proof","summary":"Using lemma \\reflem:multi_index_graph_equivalence, we already have that Y_i = \\prod_w \\in w_i Y…","labels":[],"detail_key":"p4"},{"id":"n463","layer":"informal","project":"p4","title":"Expectation of Matrix Multi Index","kind":"lemma","summary":"[Expectation of Matrix Multi Index] Let i \\in[n]^k be a k-index, i=\\left(i_1, i_2, \\ldots, i_k\\…","labels":["lem:expectation_matrix_multi_index"],"detail_key":"p4"},{"id":"n464","layer":"informal","project":"p4","title":"Because each Y_ij is independent, we can write: \\bE(Y_i) = \\bE\\left(\\prod_1 \\leq i \\leq j…","kind":"proof","summary":"Because each Y_ij is independent, we can write: \\bE(Y_i) = \\bE\\left(\\prod_1 \\leq i \\leq j \\leq…","labels":[],"detail_key":"p4"},{"id":"n465","layer":"informal","project":"p4","title":"Product of Expectation of Matrix Multi Index","kind":"definition","summary":"[Product of Expectation of Matrix Multi Index] Let i \\in [n]^k be a k-index, i = (i_1, i_2, \\ld…","labels":["def:prod_expectation_matrix_multi_index"],"detail_key":"p4"},{"id":"n466","layer":"informal","project":"p4","title":"i \\sim j \\Rightarrow E(Y_i) = E(Y_j) : R-1-7 : lem:eq\\_equiv\\_eq\\_expect","kind":"lemma","summary":"[i \\sim j \\Rightarrow E(Y_i) = E(Y_j) : R-1-7 : lem:eq\\_equiv\\_eq\\_expect] Given two k-indexes…","labels":["lem:eq_equiv_eq_expect"],"detail_key":"p4"},{"id":"n467","layer":"informal","project":"p4","title":"Let \\varphi be the permutation that maps w_i to w_j in the equivalence relation. Given Y_…","kind":"proof","summary":"Let \\varphi be the permutation that maps w_i to w_j in the equivalence relation. Given Y_i = Y_…","labels":[],"detail_key":"p4"},{"id":"n468","layer":"informal","project":"p4","title":"Partitioning into double summation : R-1-10 : lem:equation\\_4.5\\_1","kind":"lemma","summary":"[Partitioning into double summation : R-1-10 : lem:equation\\_4.5\\_1] \\[ \\bE \\Tr (Y_n^k) = \\sum_…","labels":["lem:equation_4.5_1"],"detail_key":"p4"},{"id":"n469","layer":"informal","project":"p4","title":"This follows from `partitioning' the summation appearing in Lemma \\reflem:trace_expectati…","kind":"proof","summary":"This follows from `partitioning' the summation appearing in Lemma \\reflem:trace_expectation_of_…","labels":[],"detail_key":"p4"},{"id":"n470","layer":"informal","project":"p4","title":"\\Pi (G,w): R-1-11 : def:Pi.G.w","kind":"definition","summary":"[\\Pi (G,w): R-1-11 : def:Pi.G.w] Given w \\in G_k, let w_i be an element of its equivalence clas…","labels":["def:Pi.G.w"],"detail_key":"p4"},{"id":"n471","layer":"informal","project":"p4","title":"Re-indexing the sum with counting argument : R-1-12 : lem:equation\\_4.5\\_2","kind":"lemma","summary":"[Re-indexing the sum with counting argument : R-1-12 : lem:equation\\_4.5\\_2] \\[ \\bE \\Tr (Y_n^k)…","labels":["lem:equation_4.5_2"],"detail_key":"p4"},{"id":"n472","layer":"informal","project":"p4","title":"This follows from re-indexing the sum of Lemma \\reflem:equation_4.5_1 by using Lemma \\ref…","kind":"proof","summary":"This follows from re-indexing the sum of Lemma \\reflem:equation_4.5_1 by using Lemma \\reflem:eq…","labels":[],"detail_key":"p4"},{"id":"n473","layer":"informal","project":"p4","title":"Re-introducing the renormalization factor : R-1-13 : lem:equation\\_4.5\\_3","kind":"lemma","summary":"[Re-introducing the renormalization factor : R-1-13 : lem:equation\\_4.5\\_3] \\[ \\frac1n \\bE \\Tr…","labels":["lem:equation_4.5_3"],"detail_key":"p4"},{"id":"n474","layer":"informal","project":"p4","title":"Combining with the renormalization factor n^-1 of Proposition \\ref??? gives \\[ \\frac1n \\b…","kind":"proof","summary":"Combining with the renormalization factor n^-1 of Proposition \\ref??? gives \\[ \\frac1n \\bE \\Tr…","labels":[],"detail_key":"p4"},{"id":"n475","layer":"informal","project":"p4","title":"\\Pi (G,w) = 0 : R-1-15 : lem:Pi.prod\\_eq\\_zero\\_if\\_w\\_le\\_two","kind":"lemma","summary":"[\\Pi (G,w) = 0 : R-1-15 : lem:Pi.prod\\_eq\\_zero\\_if\\_w\\_le\\_two] Given w \\in G_k, suppose there…","labels":["lem:Pi.prod_eq_zero_if_w_le_two"],"detail_key":"p4"},{"id":"n476","layer":"informal","project":"p4","title":"Let (G,w) \\in G_k and let j be the k-index generated by (G,w). Suppose there exists an ed…","kind":"proof","summary":"Let (G,w) \\in G_k and let j be the k-index generated by (G,w). Suppose there exists an edge e \\…","labels":[],"detail_key":"p4"},{"id":"n477","layer":"informal","project":"p4","title":"Simplifying the summation with the fact \\Pi (G,w) = 0 in certain cases: R-1-16 : lem:equa…","kind":"lemma","summary":"[Simplifying the summation with the fact \\Pi (G,w) = 0 in certain cases: R-1-16 : lem:equation\\…","labels":["lem:equation_4.8"],"detail_key":"p4"},{"id":"n478","layer":"informal","project":"p4","title":"This follows from applying the result of Lemma \\reflem:Pi.prod_eq_zero_if_w_le_two to Lem…","kind":"proof","summary":"This follows from applying the result of Lemma \\reflem:Pi.prod_eq_zero_if_w_le_two to Lemma \\re…","labels":[],"detail_key":"p4"},{"id":"n479","layer":"informal","project":"p4","title":"lem:asc_factorial_product","kind":"lemma","summary":"n(n-1)\\cdots (n-|V|+1) \\le n^|V|.","labels":["lem:asc_factorial_product"],"detail_key":"p4"},{"id":"n480","layer":"informal","project":"p4","title":"Use Nat.ascFactorial\\_eq\\_div. Or prove directly.","kind":"proof","summary":"Use Nat.ascFactorial\\_eq\\_div. Or prove directly.","labels":[],"detail_key":"p4"},{"id":"n481","layer":"informal","project":"p4","title":"lem:bounded_map","kind":"lemma","summary":"The sequence n\\mapsto \\frac1n\\bE\\Tr(X_n^k) is bounded.","labels":["lem:bounded_map"],"detail_key":"p4"},{"id":"n482","layer":"informal","project":"p4","title":"\\notready The only part that depends on n is the big fraction. Since we only care about w…","kind":"proof","summary":"\\notready The only part that depends on n is the big fraction. Since we only care about w \\ge 2…","labels":[],"detail_key":"p4"},{"id":"n483","layer":"informal","project":"p4","title":"lem:odd_ratio_bound","kind":"lemma","summary":"\\notready Suppose k odd. Then, \\fracn(n-1)\\cdots(n-|V_w|+1)n^k/2+1 \\le \\frac1\\sqrtn.","labels":["lem:odd_ratio_bound"],"detail_key":"p4"},{"id":"n484","layer":"informal","project":"p4","title":"\\notready","kind":"proof","summary":"\\notready","labels":[],"detail_key":"p4"},{"id":"n485","layer":"informal","project":"p4","title":"prop:odd_case","kind":"proposition","summary":"\\notready Suppose k odd. Then, \\lim_n\\to\\infty \\frac1n\\bE\\Tr(X_n^k) = 0.","labels":["prop:odd_case"],"detail_key":"p4"},{"id":"n486","layer":"informal","project":"p4","title":"\\notready Since |G_w|\\le \\#E+1 \\le k/2+1 and |G_w| is an integer, it follows that |G_w|\\l…","kind":"proof","summary":"\\notready Since |G_w|\\le \\#E+1 \\le k/2+1 and |G_w| is an integer, it follows that |G_w|\\le (k-1…","labels":[],"detail_key":"p4"},{"id":"n487","layer":"informal","project":"p4","title":"prop:g_difference_bound","kind":"proposition","summary":"\\notready |\\sum_G_k, w \\ge 2 \\Pi (w) \\cdot \\fracn (n-1) \\cdots (n - |G_w| + 1)n^k/2+1 - \\sum_G_…","labels":["prop:g_difference_bound"],"detail_key":"p4"},{"id":"n488","layer":"informal","project":"p4","title":"\\notready","kind":"proof","summary":"\\notready","labels":[],"detail_key":"p4"},{"id":"n489","layer":"informal","project":"p4","title":"prop:trace_ev_special_g","kind":"proposition","summary":"\\notready \\frac1n\\bE\\Tr(X_n^k) = \\sum_w\\inG^k/2+1_k \\Pi(w) \\cdot \\fracn(n-1)\\cdots(n-|G_w|+1)n^…","labels":["prop:trace_ev_special_g"],"detail_key":"p4"},{"id":"n490","layer":"informal","project":"p4","title":"\\notready If |G_w| < k/2 + 1, then there is at least one more n in the denominator than t…","kind":"proof","summary":"\\notready If |G_w| < k/2 + 1, then there is at least one more n in the denominator than the num…","labels":[],"detail_key":"p4"},{"id":"n491","layer":"informal","project":"p4","title":"lem:fraction_limit_one","kind":"proposition","summary":"\\notready \\lim_n\\to\\infty\\fracn^k/2n(n-1)\\cdots(n-k/2+1) = 1.","labels":["lem:fraction_limit_one"],"detail_key":"p4"},{"id":"n492","layer":"informal","project":"p4","title":"\\notready some lower bound stuff + other stuff?","kind":"proof","summary":"\\notready some lower bound stuff + other stuff?","labels":[],"detail_key":"p4"},{"id":"n493","layer":"informal","project":"p4","title":"prop:trace_ev_limit_equals_sum","kind":"proposition","summary":"\\notready \\lim_n\\to\\infty\\bE\\Tr(X_n^k) = \\sum_w\\inG^k/2+1_k \\Pi(w)","labels":["prop:trace_ev_limit_equals_sum"],"detail_key":"p4"},{"id":"n494","layer":"informal","project":"p4","title":"\\notready Proof: use the fact that |G_w|=k/2+1 and n(n-1)\\cdots(n-k/2+1) \\sim n^k/2+1. Li…","kind":"proof","summary":"\\notready Proof: use the fact that |G_w|=k/2+1 and n(n-1)\\cdots(n-k/2+1) \\sim n^k/2+1. Limit as…","labels":[],"detail_key":"p4"},{"id":"n495","layer":"informal","project":"p4","title":"lem:prod_expansion","kind":"lemma","summary":"\\notready For w\\inG^k/2+1_k, \\Pi(w) = \\prod_e_c\\in E^c \\bE(Y_12^w(e_c))","labels":["lem:prod_expansion"],"detail_key":"p4"},{"id":"n496","layer":"informal","project":"p4","title":"\\notready Follows directly.","kind":"proof","summary":"\\notready Follows directly.","labels":[],"detail_key":"p4"},{"id":"n497","layer":"informal","project":"p4","title":"lem:w_2_case","kind":"lemma","summary":"\\notready \\prod_e_c\\in E^c \\bE(Y_12^w(e_c)) = \\prod_e_c\\in E^c \\bE(Y_12^2)","labels":["lem:w_2_case"],"detail_key":"p4"},{"id":"n498","layer":"informal","project":"p4","title":"\\notready Follows directly.","kind":"proof","summary":"\\notready Follows directly.","labels":[],"detail_key":"p4"},{"id":"n499","layer":"informal","project":"p4","title":"lem:def_t_rewrite","kind":"lemma","summary":"\\notready \\bE(Y_12^2) = t","labels":["lem:def_t_rewrite"],"detail_key":"p4"},{"id":"n500","layer":"informal","project":"p4","title":"\\notready Follows directly.","kind":"proof","summary":"\\notready Follows directly.","labels":[],"detail_key":"p4"},{"id":"n501","layer":"informal","project":"p4","title":"lem:e_equals_k_over_two","kind":"lemma","summary":"\\notready t^|E_w| = t^k/2","labels":["lem:e_equals_k_over_two"],"detail_key":"p4"},{"id":"n502","layer":"informal","project":"p4","title":"\\notready Follows directly.","kind":"proof","summary":"\\notready Follows directly.","labels":[],"detail_key":"p4"},{"id":"n503","layer":"informal","project":"p4","title":"prop:product_g_w_to_exponential","kind":"proposition","summary":"\\notready \\Pi(w) = t^k/2.","labels":["prop:product_g_w_to_exponential"],"detail_key":"p4"},{"id":"n504","layer":"informal","project":"p4","title":"\\notready Proof slightly outdated. Let (G,w)\\inG^k/2+1_k. Since w traverses each edge exa…","kind":"proof","summary":"\\notready Proof slightly outdated. Let (G,w)\\inG^k/2+1_k. Since w traverses each edge exactly t…","labels":[],"detail_key":"p4"},{"id":"n505","layer":"informal","project":"p4","title":"prop:trace_ev_limit_equals_t_special_g","kind":"proposition","summary":"\\notready \\lim_n\\to\\infty \\bE\\Tr(X_n^k) = t^k/2\\cdot|G_k^k/2+1|","labels":["prop:trace_ev_limit_equals_t_special_g"],"detail_key":"p4"},{"id":"n506","layer":"informal","project":"p4","title":"\\notready Follows directly from 4.7.5 and 4.8.","kind":"proof","summary":"\\notready Follows directly from 4.7.5 and 4.8.","labels":[],"detail_key":"p4"},{"id":"n507","layer":"informal","project":"p4","title":"Proposition 4.1 in \\citeKemp2013RMTNotes","kind":"proposition","summary":"[Proposition 4.1 in \\citeKemp2013RMTNotes] \\notready Let \\Y_ij\\_1\\le i\\le j be independent rand…","labels":["prop:matrix_moments_convergence"],"detail_key":"p4"},{"id":"n508","layer":"informal","project":"p4","title":"\\notready","kind":"proof","summary":"\\notready","labels":[],"detail_key":"p4"},{"id":"n509","layer":"informal","project":"p4","title":"Variance Trace Expansion","kind":"lemma","summary":"[Variance Trace Expansion] \\notready Let Y_n be an n \\times n symmetric matrix with independent…","labels":["lem:variance_expansion"],"detail_key":"p4"},{"id":"n510","layer":"informal","project":"p4","title":"\\notready Expanding the variance by definition, we get: Var\\left(\\frac1n\\Tr(X_n^k)\\right)…","kind":"proof","summary":"\\notready Expanding the variance by definition, we get: Var\\left(\\frac1n\\Tr(X_n^k)\\right) = \\bE…","labels":[],"detail_key":"p4"},{"id":"n511","layer":"informal","project":"p4","title":"R-2-1 : def:common\\_val\\_prod\\_of","kind":"definition","summary":"[R-2-1 : def:common\\_val\\_prod\\_of] \\notready Given an ordered triple (G_i\\#j,w_i,w_j) generate…","labels":["def:common_val_prod_of"],"detail_key":"p4"},{"id":"n512","layer":"informal","project":"p4","title":"R-2-3-4 : lem:common\\_val\\_eq\\_of\\_index\\_pair\\_rel","kind":"lemma","summary":"[R-2-3-4 : lem:common\\_val\\_eq\\_of\\_index\\_pair\\_rel] \\notready Let i_\\lambda and j_\\lambda be…","labels":["lem:common_val_eq_of_index_pair_rel"],"detail_key":"p4"},{"id":"n513","layer":"informal","project":"p4","title":"This follows a similar reasoning as in Lemma \\reflem:eq_equiv_eq_expect.","kind":"proof","summary":"This follows a similar reasoning as in Lemma \\reflem:eq_equiv_eq_expect.","labels":[],"detail_key":"p4"},{"id":"n514","layer":"informal","project":"p4","title":"This follows from `partitioning' the summation appearing in Lemma \\reflem:variance_expans…","kind":"proof","summary":"This follows from `partitioning' the summation appearing in Lemma \\reflem:variance_expansion us…","labels":[],"detail_key":"p4"},{"id":"n515","layer":"informal","project":"p4","title":"R-2-5-1 : def:common\\_val\\_prod","kind":"definition","summary":"[R-2-5-1 : def:common\\_val\\_prod] \\notready Given (G,w,w'), let (i,j) be an ordered pair of k-i…","labels":["def:common_val_prod"],"detail_key":"p4"},{"id":"n516","layer":"informal","project":"p4","title":"This follows from re-indexing the sum of Lemma \\reflem:sum_eq_sum_over_classes using Lemm…","kind":"proof","summary":"This follows from re-indexing the sum of Lemma \\reflem:sum_eq_sum_over_classes using Lemma \\ref…","labels":[],"detail_key":"p4"},{"id":"n517","layer":"informal","project":"p4","title":"R-2-8 : lem:expect\\_mul\\_eq\\_prod\\_expect\\_edgewise\\_of\\_indep","kind":"lemma","summary":"[R-2-8 : lem:expect\\_mul\\_eq\\_prod\\_expect\\_edgewise\\_of\\_indep] \\notready \\[ E (Y_iY_j) = \\pro…","labels":["lem:expect_mul_eq_prod_expect_edgewise_of_indep"],"detail_key":"p4"},{"id":"n518","layer":"informal","project":"p4","title":"This follows from Lemma \\reflem:expectation_matrix_multi_index and the independency of ra…","kind":"proof","summary":"This follows from Lemma \\reflem:expectation_matrix_multi_index and the independency of random v…","labels":[],"detail_key":"p4"},{"id":"n519","layer":"informal","project":"p4","title":"R-2-12.1 : lem:expect\\_pow\\_edge\\_count\\_pair\\_le","kind":"lemma","summary":"[R-2-12.1 : lem:expect\\_pow\\_edge\\_count\\_pair\\_le] \\notready For any k-indexes i and j, there…","labels":["lem:expect_pow_edge_count_pair_le"],"detail_key":"p4"},{"id":"n520","layer":"informal","project":"p4","title":"Let (G_i\\#j,w_i,w_j) be the ordered triple generated by the two k-indexes i and j. By con…","kind":"proof","summary":"Let (G_i\\#j,w_i,w_j) be the ordered triple generated by the two k-indexes i and j. By construct…","labels":[],"detail_key":"p4"},{"id":"n521","layer":"informal","project":"p4","title":"R-2-12.2 : lem:expect\\_pow\\_edge\\_count\\_pair\\_le\\_off","kind":"lemma","summary":"[R-2-12.2 : lem:expect\\_pow\\_edge\\_count\\_pair\\_le\\_off] \\notready For any k-indexes i and j, t…","labels":["lem:expect_pow_edge_count_pair_le_off"],"detail_key":"p4"},{"id":"n522","layer":"informal","project":"p4","title":"This follows an identical reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le.","kind":"proof","summary":"This follows an identical reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le.","labels":[],"detail_key":"p4"},{"id":"n523","layer":"informal","project":"p4","title":"R-2-13 : lem:expect\\_mul\\_le\\_const","kind":"lemma","summary":"[R-2-13 : lem:expect\\_mul\\_le\\_const] \\notready For any k-indexes i and j, there exists M_1 \\in…","labels":["lem:expect_mul_le_const"],"detail_key":"p4"},{"id":"n524","layer":"informal","project":"p4","title":"By Lemma \\reflem:expect_mul_eq_prod_expect_edgewise_of_indep, we have \\[ E (Y_iY_j) = \\pr…","kind":"proof","summary":"By Lemma \\reflem:expect_mul_eq_prod_expect_edgewise_of_indep, we have \\[ E (Y_iY_j) = \\prod_e_s…","labels":[],"detail_key":"p4"},{"id":"n525","layer":"informal","project":"p4","title":"R-2-14 : lem:expect\\_pow\\_edge\\_count\\_le","kind":"lemma","summary":"[R-2-14 : lem:expect\\_pow\\_edge\\_count\\_le] \\notready For any k-index i, there exists \\lambda \\…","labels":["lem:expect_pow_edge_count_le"],"detail_key":"p4"},{"id":"n526","layer":"informal","project":"p4","title":"This follows a similar reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le with t…","kind":"proof","summary":"This follows a similar reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le with the cou…","labels":[],"detail_key":"p4"},{"id":"n527","layer":"informal","project":"p4","title":"R-2-15 : lem:expect\\_pow\\_edge\\_count\\_le\\_off","kind":"lemma","summary":"[R-2-15 : lem:expect\\_pow\\_edge\\_count\\_le\\_off] \\notready For any k-index i, there exists \\lam…","labels":["lem:expect_pow_edge_count_le_off"],"detail_key":"p4"},{"id":"n528","layer":"informal","project":"p4","title":"This follows a similar reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le with t…","kind":"proof","summary":"This follows a similar reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le with the cou…","labels":[],"detail_key":"p4"},{"id":"n529","layer":"informal","project":"p4","title":"R-2-16 : lem:expect\\_le\\_const","kind":"lemma","summary":"[R-2-16 : lem:expect\\_le\\_const] \\notready For any k-indexes i, there exists M_2 \\in R such tha…","labels":["lem:expect_le_const"],"detail_key":"p4"},{"id":"n530","layer":"informal","project":"p4","title":"This follows an identical reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le.","kind":"proof","summary":"This follows an identical reasoning as in Lemma \\reflem:expect_pow_edge_count_pair_le.","labels":[],"detail_key":"p4"},{"id":"n531","layer":"informal","project":"p4","title":"R-2-17 : lem:abs\\_expect\\_mul\\_sub\\_mul\\_expect\\_le","kind":"lemma","summary":"[R-2-17 : lem:abs\\_expect\\_mul\\_sub\\_mul\\_expect\\_le] \\notready For any k-indexes i and j, ther…","labels":["lem:abs_expect_mul_sub_mul_expect_le"],"detail_key":"p4"},{"id":"n532","layer":"informal","project":"p4","title":"Combining the result of Lemma \\reflem:expect_mul_le_const and Lemma \\reflem:expect_le_con…","kind":"proof","summary":"Combining the result of Lemma \\reflem:expect_mul_le_const and Lemma \\reflem:expect_le_const giv…","labels":[],"detail_key":"p4"},{"id":"n533","layer":"informal","project":"p4","title":"prop:var_trace_as_sum","kind":"proposition","summary":"\\notready \\[ \\Var\\left( \\frac1n\\Tr(X_n^k)\\right) = \\sum_(G,w,w')\\in G_k,k\\atop w+w'\\ge 2 \\pi(G,…","labels":["prop:var_trace_as_sum"],"detail_key":"p4"},{"id":"n534","layer":"informal","project":"p4","title":"\\notready By constr","kind":"proof","summary":"\\notready By constr","labels":[],"detail_key":"p4"},{"id":"n535","layer":"informal","project":"p4","title":"lem:simplified_var_trace_as_sum","kind":"lemma","summary":"\\notready \\[ \\Var\\left( \\frac1n\\Tr(X_n^k)\\right) = \\sum_(G,w,w')\\in G_k,k\\atop w+w'\\ge 2 \\pi(G,…","labels":["lem:simplified_var_trace_as_sum"],"detail_key":"p4"},{"id":"n536","layer":"informal","project":"p4","title":"\\notready Follows directly.","kind":"proof","summary":"\\notready Follows directly.","labels":[],"detail_key":"p4"},{"id":"n537","layer":"informal","project":"p4","title":"prop:var_upper_bound","kind":"proposition","summary":"\\notready \\[ \\Var\\left( \\frac1n\\Tr(X_n^k)\\right) \\le \\sum_(G,w,w')\\inG_k,k\\atop w+w'\\ge 2 \\pi(G…","labels":["prop:var_upper_bound"],"detail_key":"p4"},{"id":"n538","layer":"informal","project":"p4","title":"\\notready Appealing again to \\refprop:vertex_edge_inequality, it follows that |G|\\le k+1.…","kind":"proof","summary":"\\notready Appealing again to \\refprop:vertex_edge_inequality, it follows that |G|\\le k+1. Hence…","labels":[],"detail_key":"p4"},{"id":"n539","layer":"informal","project":"p4","title":"thm:wigner_law_matrix_moments","kind":"theorem","summary":"\\notready Theorem 2.4. (Unsure if it goes in this doc.)","labels":["thm:wigner_law_matrix_moments"],"detail_key":"p4"},{"id":"n540","layer":"informal","project":"p4","title":"\\notready The (potentially enormous) number B_k=2M_2k\\cdot\\#G_k,k is independent of n, an…","kind":"proof","summary":"\\notready The (potentially enormous) number B_k=2M_2k\\cdot\\#G_k,k is independent of n, and so w…","labels":[],"detail_key":"p4"},{"id":"n541","layer":"informal","project":"p4","title":"lem:elimination_kp1","kind":"lemma","summary":"\\notready \\[ \\Var\\left( \\frac1n\\Tr(X_n^k)\\right) \\le \\sum_(G,w,w')\\inG_k,k\\atop w+w'\\ge 2, |G|…","labels":["lem:elimination_kp1"],"detail_key":"p4"},{"id":"n542","layer":"informal","project":"p4","title":"Proposition 4.5 in \\citeKemp2013RMTNotes","kind":"proposition","summary":"[Proposition 4.5 in \\citeKemp2013RMTNotes] \\notready Let \\Y_ij\\_1 \\leq i \\leq j be independent…","labels":["prop:matrix_moments_convergence_probability"],"detail_key":"p4"},{"id":"n543","layer":"informal","project":"p4","title":"This follows from Markov's inequality.","kind":"proof","summary":"This follows from Markov's inequality.","labels":[],"detail_key":"p4"},{"id":"n544","layer":"informal","project":"p4","title":"R-3-3 : def:R-3-3","kind":"definition","summary":"[R-3-3 : def:R-3-3] \\notready We define \\nu to be the random measure \\nu(dx) = |x|^k \\mu_X_n(dx…","labels":["def:R-3-3"],"detail_key":"p4"},{"id":"n545","layer":"informal","project":"p4","title":"R-3-4 : lem:R-3-4","kind":"lemma","summary":"[R-3-4 : lem:R-3-4] \\notready \\[ \\int_|x| > b |x|^k \\mu_X_n(dx) = \\nu\\x : |x|^k > b^k\\. \\]","labels":["lem:R-3-4"],"detail_key":"p4"},{"id":"n546","layer":"informal","project":"p4","title":"\\[ \\int_|x| > b |x|^k \\mu_X_n(dx) = \\nu\\x : |x| > b\\ = \\nu\\x : |x|^k > b^k\\. \\]","kind":"proof","summary":"\\[ \\int_|x| > b |x|^k \\mu_X_n(dx) = \\nu\\x : |x| > b\\ = \\nu\\x : |x|^k > b^k\\. \\]","labels":[],"detail_key":"p4"},{"id":"n547","layer":"informal","project":"p4","title":"R-3-5 : lem:R-3-5","kind":"lemma","summary":"[R-3-5 : lem:R-3-5] \\notready \\[ \\int_|x| > b |x|^k \\mu_X_n(dx) \\leq \\frac1b^k \\int |x|^2k \\mu_…","labels":["lem:R-3-5"],"detail_key":"p4"},{"id":"n548","layer":"informal","project":"p4","title":"First, applying Markov's inequality on Lemma \\reflem:R-3-4 gives \\[ \\int_|x| > b |x|^k \\m…","kind":"proof","summary":"First, applying Markov's inequality on Lemma \\reflem:R-3-4 gives \\[ \\int_|x| > b |x|^k \\mu_X_n(…","labels":[],"detail_key":"p4"},{"id":"n549","layer":"informal","project":"p4","title":"Catalan Number bound","kind":"lemma","summary":"[Catalan Number bound] \\notready Let C_k be the Catalan number. Then, for all k \\in N, we have:…","labels":["lem:Catalan_bound"],"detail_key":"p4"},{"id":"n550","layer":"informal","project":"p4","title":"Bound for \\lim \\sup_n\\to\\infty\\frac1nE Tr (X_n^k)","kind":"lemma","summary":"[Bound for \\lim \\sup_n\\to\\infty\\frac1nE Tr (X_n^k)] \\notready Let \\Y_ij\\_1 \\leq i \\leq j be ind…","labels":["lem:bound_for_expectation_trace"],"detail_key":"p4"},{"id":"n551","layer":"informal","project":"p4","title":"From proposition \\refprop:matrix_moments_convergence, we know that \\lim_n\\to\\infty \\frac1…","kind":"proof","summary":"From proposition \\refprop:matrix_moments_convergence, we know that \\lim_n\\to\\infty \\frac1nETr(X…","labels":[],"detail_key":"p4"},{"id":"n552","layer":"informal","project":"p4","title":"New bound for \\lim \\sup_n\\to\\infty P(\\int_|x| > b|x|^k \\mu x_n(dx))","kind":"lemma","summary":"[New bound for \\lim \\sup_n\\to\\infty P(\\int_|x| > b|x|^k \\mu x_n(dx))] \\notready let k \\in N and…","labels":["lem:new_bound_for_lim_sup_p"],"detail_key":"p4"},{"id":"n553","layer":"informal","project":"p4","title":"Increasing Sequence |x|^k","kind":"lemma","summary":"[Increasing Sequence |x|^k] \\notready for x > b > 4 > 1 \\in R, we have k \\mapsto |x|^k is incre…","labels":["lem:increasing_sequence_x_to_k"],"detail_key":"p4"},{"id":"n554","layer":"informal","project":"p4","title":"Increasing Sequence P(\\int_|x| > b|x|^k\\mu x_n(dx))","kind":"lemma","summary":"[Increasing Sequence P(\\int_|x| > b|x|^k\\mu x_n(dx))] \\notready for x > b > 4 > 1 \\in R, we hav…","labels":["lem:increasing_sequence_p_of_int_x_to_k"],"detail_key":"p4"},{"id":"n555","layer":"informal","project":"p4","title":"Increasing Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))","kind":"lemma","summary":"[Increasing Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))] \\notready for x > b >…","labels":["lem:increasing_sequence_lim_sup_p"],"detail_key":"p4"},{"id":"n556","layer":"informal","project":"p4","title":"Nonnegative Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))","kind":"lemma","summary":"[Nonnegative Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))] \\notready for x > b…","labels":["lem:nonnegative_sequence_lim_sup_p"],"detail_key":"p4"},{"id":"n557","layer":"informal","project":"p4","title":"Decreasing Sequence \\frac1\\epsilon(\\frac4b)^k","kind":"lemma","summary":"[Decreasing Sequence \\frac1\\epsilon(\\frac4b)^k] \\notready for x > b > 4 > 1 \\in R, we have k \\m…","labels":["lem:decreasing_sequence_1_over_epsilon_4_over_b_to_k"],"detail_key":"p4"},{"id":"n558","layer":"informal","project":"p4","title":"Limit of Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))","kind":"lemma","summary":"[Limit of Sequence \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx))] \\notready for x > b > 4…","labels":["lem:limit_of_sequence_lim_sup_p"],"detail_key":"p4"},{"id":"n559","layer":"informal","project":"p4","title":"We know that for all k \\in N, \\lim\\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx)) \\geq 0 (…","kind":"proof","summary":"We know that for all k \\in N, \\lim\\sup_n\\to\\inftyP(\\int_|x| > b|x|^k\\mu x_n(dx)) \\geq 0 (lemma…","labels":[],"detail_key":"p4"},{"id":"n560","layer":"informal","project":"p4","title":"All terms of Sequence are Zero conditions","kind":"lemma","summary":"[All terms of Sequence are Zero conditions] \\notready for any sequence (a_k)_k=1^\\infty \\in R,…","labels":["lem:all_terms_zero_conditions"],"detail_key":"p4"},{"id":"n561","layer":"informal","project":"p4","title":"Lemma 4.7 from \\citeKemp2013RMTNotes","kind":"lemma","summary":"[Lemma 4.7 from \\citeKemp2013RMTNotes] \\notready let k \\in N and \\epsilon > 0. Then for any b >…","labels":["lem:convergence_to_zero_of_lim_sup_p"],"detail_key":"p4"},{"id":"n562","layer":"informal","project":"p4","title":"Consider sequence (a_k)_k =1^\\infty defined as a_k = \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x…","kind":"proof","summary":"Consider sequence (a_k)_k =1^\\infty defined as a_k = \\lim \\sup_n\\to\\inftyP(\\int_|x| > b|x|^k \\m…","labels":[],"detail_key":"p4"},{"id":"n563","layer":"informal","project":"p4","title":"Function from Weierstrass Approx Theorem","kind":"lemma","summary":"[Function from Weierstrass Approx Theorem] \\notready Fix a bounded, continuous function f \\in C…","labels":["lem:function_weierstrass_approx"],"detail_key":"p4"},{"id":"n564","layer":"informal","project":"p4","title":"Estimates from Triangle Inequality","kind":"lemma","summary":"[Estimates from Triangle Inequality] \\notready Let f \\in C_b(R), fix \\epsilon > 0, and b > 4, a…","labels":["lem:estimates_from_triangle_inequality"],"detail_key":"p4"},{"id":"n565","layer":"informal","project":"p4","title":"lem:f_pigeonhole","kind":"lemma","summary":"\\notready If \\left|\\int f\\,d\\mu_X_n - \\int f\\,d\\sigma_1\\right|>\\epsilon, then \\left|\\int f\\,d\\m…","labels":["lem:f_pigeonhole"],"detail_key":"p4"},{"id":"n566","layer":"informal","project":"p4","title":"\\notready Pigeonhole.","kind":"proof","summary":"\\notready Pigeonhole.","labels":[],"detail_key":"p4"},{"id":"n567","layer":"informal","project":"p4","title":"lem:f_probability_inequality","kind":"lemma","summary":"\\notready \\bP\\left( \\left|\\int f\\,d\\mu_X_n - \\int f\\,d\\sigma_1\\right|>\\epsilon\\right) \\le \\bP\\l…","labels":["lem:f_probability_inequality"],"detail_key":"p4"},{"id":"n568","layer":"informal","project":"p4","title":"\\notready Triangle equality: use MeasureTheory.lintegral\\_edist\\_triangle. (Not sure if t…","kind":"proof","summary":"\\notready Triangle equality: use MeasureTheory.lintegral\\_edist\\_triangle. (Not sure if this is…","labels":[],"detail_key":"p4"},{"id":"n569","layer":"informal","project":"p4","title":"lem:p_epsilon_minus_f_d_sigma_eq_zero","kind":"lemma","summary":"\\notready \\bP\\left(\\left|\\int P_\\epsilon\\,d\\sigma_1 - \\int f\\,d\\sigma_1\\right|>\\epsilon/3\\right…","labels":["lem:p_epsilon_minus_f_d_sigma_eq_zero"],"detail_key":"p4"},{"id":"n570","layer":"informal","project":"p4","title":"\\notready By construction, |P_\\epsilon-f|<\\epsilon/6 on [-b,b], which includes the suppor…","kind":"proof","summary":"\\notready By construction, |P_\\epsilon-f|<\\epsilon/6 on [-b,b], which includes the support [-2,…","labels":[],"detail_key":"p4"},{"id":"n571","layer":"informal","project":"p4","title":"lem:f_p_epsilon_inequality","kind":"lemma","summary":"\\notready \\[ \\left|\\int (f-P_\\epsilon)\\,d\\mu_X_n\\right| \\le \\int_|x|\\le b |f(x)-P_\\epsilon(x)|\\…","labels":["lem:f_p_epsilon_inequality"],"detail_key":"p4"},{"id":"n572","layer":"informal","project":"p4","title":"\\notready Break up the integral.","kind":"proof","summary":"\\notready Break up the integral.","labels":[],"detail_key":"p4"},{"id":"n573","layer":"informal","project":"p4","title":"lem:f_p_epsilon_pigeonhole","kind":"lemma","summary":"\\notready If \\left|\\int f\\,d\\mu_X_n - \\int P_\\epsilon\\,d\\mu_X_n\\right|>\\epsilon/3, then \\int |f…","labels":["lem:f_p_epsilon_pigeonhole"],"detail_key":"p4"},{"id":"n574","layer":"informal","project":"p4","title":"\\notready Pigeonhole.","kind":"proof","summary":"\\notready Pigeonhole.","labels":[],"detail_key":"p4"},{"id":"n575","layer":"informal","project":"p4","title":"lem:f_p_epsilon_probability_inequality","kind":"lemma","summary":"\\notready \\[ \\bP\\left(\\left|\\int f\\,d\\mu_X_n - \\int P_\\epsilon\\,d\\mu_X_n\\right|>\\epsilon/3\\righ…","labels":["lem:f_p_epsilon_probability_inequality"],"detail_key":"p4"},{"id":"n576","layer":"informal","project":"p4","title":"\\notready Use previous two lemmas.","kind":"proof","summary":"\\notready Use previous two lemmas.","labels":[],"detail_key":"p4"},{"id":"n577","layer":"informal","project":"p4","title":"lem:first_term_of_first_term_zero","kind":"lemma","summary":"\\notready \\limsup_n\\to\\infty \\left(P\\left(\\int |f-P_\\epsilon|1_|x|\\le b\\,d\\mu_X_n>\\epsilon/6\\ri…","labels":["lem:first_term_of_first_term_zero"],"detail_key":"p4"},{"id":"n578","layer":"informal","project":"p4","title":"lem:reestimate","kind":"lemma","summary":"\\notready P\\left( \\left|\\int f\\,d\\mu_X_n - \\int f\\,d\\sigma_1\\right|>\\epsilon\\right) &\\le P\\left…","labels":["lem:reestimate"],"detail_key":"p4"},{"id":"n579","layer":"informal","project":"p4","title":"lem:second_term_estimate","kind":"lemma","summary":"\\notready \\limsup_n \\to \\infty \\left(P\\left(\\left|\\int P_\\epsilon\\,d\\mu_X_n - \\int P_\\epsilon\\,…","labels":["lem:second_term_estimate"],"detail_key":"p4"},{"id":"n580","layer":"informal","project":"p4","title":"lem:final_estimate","kind":"lemma","summary":"\\notready P\\left( \\left|\\int f\\,d\\mu_X_n - \\int f\\,d\\sigma_1\\right|>\\epsilon\\right) \\leq P\\left…","labels":["lem:final_estimate"],"detail_key":"p4"},{"id":"n581","layer":"informal","project":"p4","title":"lem:polynomial_ineq","kind":"lemma","summary":"\\notready P\\left(\\int |f-P_\\epsilon| 1_|x|\\ge b\\,d\\mu_X_n > \\epsilon/6\\right) \\le P\\left(\\int c…","labels":["lem:polynomial_ineq"],"detail_key":"p4"},{"id":"n582","layer":"informal","project":"p4","title":"Let k =degP_\\epsilon, and since f is bounded, |f(x) - P_\\epsilon(x)| \\leq \\|f\\|_\\infty +…","kind":"proof","summary":"Let k =degP_\\epsilon, and since f is bounded, |f(x) - P_\\epsilon(x)| \\leq \\|f\\|_\\infty + |P_\\ep…","labels":[],"detail_key":"p4"},{"id":"n583","layer":"informal","project":"p4","title":"lem:fP_bound","kind":"lemma","summary":"\\notready \\limsup_n \\to \\inftyP\\left(\\int c|x|^k1_|x|\\ge b\\,\\mu_X_n(dx) > \\epsilon/6\\right) = 0","labels":["lem:fP_bound"],"detail_key":"p4"},{"id":"n584","layer":"informal","project":"p4","title":"lem:fP_zero","kind":"lemma","summary":"\\notready \\limsup_n \\to \\infty P\\left(\\int |f-P_\\epsilon| 1_|x|\\ge b\\,d\\mu_X_n > \\epsilon/6\\rig…","labels":["lem:fP_zero"],"detail_key":"p4"},{"id":"n585","layer":"informal","project":"p4","title":"prop:Wigner_Semicircle_Law","kind":"proposition","summary":"\\notready Let X_n=n^-1/2Y_n be a sequence of Wigner matrices, with entries satisfying E(Y_ij)=0…","labels":["prop:Wigner_Semicircle_Law"],"detail_key":"p4"},{"id":"n586","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk","kind":"inductive","summary":"V : Type u → SimpleGraph V → V → V → Type u","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n587","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.LoopWalkEquiv","kind":"def","summary":"(n : Nat) → SimpleGraph.LoopWalk.ClosedLoopWalk (SimpleGraph.LoopWalk.K n) → SimpleGraph.LoopWa…","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.LoopWalkEquiv","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n588","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.LoopWalkSetoid","kind":"def","summary":"(n : Nat) → Setoid (SimpleGraph.LoopWalk.ClosedLoopWalk (SimpleGraph.LoopWalk.K n))","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.LoopWalkSetoid","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n589","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.abs_w_i_eq_k","kind":"theorem","summary":"∀ V : Type u [inst : DecidableEq V] G : SimpleGraph V u v : V (p : G.LoopWalk u v), Eq (Finset.…","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.abs_w_i_eq_k","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n590","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.connectingEdgeSet","kind":"def","summary":"V : Type u → G : SimpleGraph V → u v : V → [DecidableEq V] → G.LoopWalk u v → Finset (Sym2 V)","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.connectingEdgeSet","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n591","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.edgeCount","kind":"def","summary":"V : Type u → [DecidableEq V] → G : SimpleGraph V → u v : V → G.LoopWalk u v → Sym2 V → Nat","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.edgeCount","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n592","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.edgeSet","kind":"def","summary":"V : Type u → G : SimpleGraph V → u v : V → [DecidableEq V] → G.LoopWalk u v → Finset (Sym2 V)","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.edgeSet","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n593","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.graphWalkMultiIndex","kind":"def","summary":"n k : Nat → (hk : GT.gt k 0) → (I : Fin k → Fin n) → (SimpleGraph.LoopWalk.K n).LoopWalk (I ⟨0,…","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.graphWalkMultiIndex","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n594","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.length","kind":"def","summary":"V : Type u → G : SimpleGraph V → u v : V → G.LoopWalk u v → Nat","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.length","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n595","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.selfEdgeSet","kind":"def","summary":"V : Type u → G : SimpleGraph V → u v : V → [DecidableEq V] → G.LoopWalk u v → Finset (Sym2 V)","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.selfEdgeSet","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n596","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.supportSet","kind":"def","summary":"V : Type u → G : SimpleGraph V → u v : V → [DecidableEq V] → G.LoopWalk u v → Finset V","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.supportSet","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n597","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.vertex_edge_inequality","kind":"theorem","summary":"∀ V : Type u [inst : DecidableEq V] G : SimpleGraph V u v : V (p : G.LoopWalk u v), LE.le p.sup…","labels":[],"detail_key":"p4","name":"SimpleGraph.LoopWalk.vertex_edge_inequality","module":"SemicircleLaw.Moments.LoopWalk"},{"id":"n598","layer":"formal","project":"p4","title":"SimpleGraph.LoopWalk.walk_edge_card_equiv","kind":"theorem","summary":"∀ n : Nat (p q : SimpleGraph.LoopWalk.ClosedLoopWalk (SimpleGraph.LoopWalk.K n)), 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We omit the argument for now, although la…","kind":"proof","summary":"This is the main content of Wiles' magnum opus. We omit the argument for now, although later on…","labels":[],"detail_key":"p5"},{"id":"n659","layer":"informal","project":"p5","title":"FreyPackage.false","kind":"corollary","summary":"There is no Frey package.","labels":["FreyPackage.false"],"detail_key":"p5"},{"id":"n660","layer":"informal","project":"p5","title":"Follows immediately from the previous two theorems~\\refMazur_Frey and~\\refWiles_Frey.","kind":"proof","summary":"Follows immediately from the previous two theorems~\\refMazur_Frey and~\\refWiles_Frey.","labels":[],"detail_key":"p5"},{"id":"n661","layer":"informal","project":"p5","title":"FLT","kind":"corollary","summary":"Fermat's Last Theorem is true. In other words, there are no positive integers a,b,c and natural…","labels":["FLT"],"detail_key":"p5"},{"id":"n662","layer":"informal","project":"p5","title":"Assume there is a there is a counterexample a^n+b^n=c^n. By Corollary \\refFermatLastTheor…","kind":"proof","summary":"Assume there is a there is a counterexample a^n+b^n=c^n. By Corollary \\refFermatLastTheorem.of_…","labels":[],"detail_key":"p5"},{"id":"n663","layer":"informal","project":"p5","title":"We make some remarks to orient the reader. \\item Any complete local Noetherian ring with…","kind":"remark","summary":"We make some remarks to orient the reader. \\item Any complete local Noetherian ring with finite…","labels":[],"detail_key":"p5"},{"id":"n664","layer":"informal","project":"p5","title":"hardly_ramified","kind":"definition","summary":"Let R be a coefficient ring with finite residue field of characteristic \\ell\\geq3. Let V be a f…","labels":["hardly_ramified"],"detail_key":"p5"},{"id":"n665","layer":"informal","project":"p5","title":"Frey_curve_hardly_ramified","kind":"theorem","summary":"The \\ell-torsion \\rho:\\Gal(\\overlineQ/Q)\\to\\GL_2(Z/\\ellZ) in the Frey curve associated to a Fre…","labels":["Frey_curve_hardly_ramified"],"detail_key":"p5"},{"id":"n666","layer":"informal","project":"p5","title":"This was well-known in the 1980s. A proof sketch is as follows. First note that \\ell\\geq5…","kind":"proof","summary":"This was well-known in the 1980s. A proof sketch is as follows. First note that \\ell\\geq5>3 by…","labels":[],"detail_key":"p5"},{"id":"n667","layer":"informal","project":"p5","title":"hardly_ramified_reducible","kind":"theorem","summary":"If \\ell\\geq 3 is a prime and \\rho:\\Gal(\\overlineQ/Q)\\to\\GL_2(Z/\\ellZ) is hardly ramified, then…","labels":["hardly_ramified_reducible"],"detail_key":"p5"},{"id":"n668","layer":"informal","project":"p5","title":"Wiles_Frey_again","kind":"theorem","summary":"If \\overline\\rho is the mod p Galois representation associated to a Frey package (a,b,c,p) then…","labels":["Wiles_Frey_again"],"detail_key":"p5"},{"id":"n669","layer":"informal","project":"p5","title":"Indeed, \\rho is hardly ramified by theorem~\\refFrey_curve_hardly_ramified and thus reduci…","kind":"proof","summary":"Indeed, \\rho is hardly ramified by theorem~\\refFrey_curve_hardly_ramified and thus reducible by…","labels":[],"detail_key":"p5"},{"id":"n670","layer":"informal","project":"p5","title":"hardly_ramified_lifts","kind":"theorem","summary":"If \\ell\\geq3 is prime and \\overline\\rho:\\Gal(\\overlineQ/Q)\\to\\GL_2(Z/\\ellZ) is hardly ramified…","labels":["hardly_ramified_lifts"],"detail_key":"p5"},{"id":"n671","layer":"informal","project":"p5","title":"Omitted for now \\bf TODO","kind":"proof","summary":"Omitted for now \\bf TODO","labels":[],"detail_key":"p5"},{"id":"n672","layer":"informal","project":"p5","title":"hardly_ramified_spreads_out","kind":"theorem","summary":"If \\ell\\geq3 is prime, K is a finite extension of Q_\\ell with integers O and if \\rho:\\Gal(\\over…","labels":["hardly_ramified_spreads_out"],"detail_key":"p5"},{"id":"n673","layer":"informal","project":"p5","title":"Omitted for now \\bf TODO","kind":"proof","summary":"Omitted for now \\bf TODO","labels":[],"detail_key":"p5"},{"id":"n674","layer":"informal","project":"p5","title":"hardly_ramified_mod3_reducible","kind":"theorem","summary":"Suppose k is a finite field of characteristic 3, and suppose \\overlinerho:\\Gal(\\overlineQ/Q)\\to…","labels":["hardly_ramified_mod3_reducible"],"detail_key":"p5"},{"id":"n675","layer":"informal","project":"p5","title":"Omitted for now. \\bf TODO","kind":"proof","summary":"Omitted for now. \\bf TODO","labels":[],"detail_key":"p5"},{"id":"n676","layer":"informal","project":"p5","title":"hardly_ramified_3adic_reducible","kind":"theorem","summary":"Suppose L/Q_3 is a finite extension, with integer ring O_L, and suppose \\rho_3:\\Gal(\\overlineQ/…","labels":["hardly_ramified_3adic_reducible"],"detail_key":"p5"},{"id":"n677","layer":"informal","project":"p5","title":"Omitted for now \\bf TODO","kind":"proof","summary":"Omitted for now \\bf TODO","labels":[],"detail_key":"p5"},{"id":"n678","layer":"informal","project":"p5","title":"Assume for a contradiction that \\overline\\rho is irreducible. By theorem~\\refhardly_ramif…","kind":"proof","summary":"Assume for a contradiction that \\overline\\rho is irreducible. By theorem~\\refhardly_ramified_li…","labels":[],"detail_key":"p5"},{"id":"n679","layer":"informal","project":"p5","title":"modularity_lifting_theorem","kind":"theorem","summary":"\\notready If \\overline\\rho is modular of level \\Gamma_1(S) and \\rho:G_F\\to\\GL_2(O) is an S-good…","labels":["modularity_lifting_theorem"],"detail_key":"p5"},{"id":"n680","layer":"informal","project":"p5","title":"(Sketch) The proof is a two-stage procedure and has a nontrivial analytic input. First on…","kind":"proof","summary":"(Sketch) The proof is a two-stage procedure and has a nontrivial analytic input. First one uses…","labels":[],"detail_key":"p5"},{"id":"n681","layer":"informal","project":"p5","title":"ZHat","kind":"definition","summary":"The profinite completion \\widehatZ of Z is the set of all compatible collections c=(c_N)_N of e…","labels":["ZHat"],"detail_key":"p5"},{"id":"n682","layer":"informal","project":"p5","title":"ZHat.commRing","kind":"lemma","summary":"\\widehatZ is a subring of \\prod_N\\geq1(Z/NZ) and in particular is a ring.","labels":["ZHat.commRing"],"detail_key":"p5"},{"id":"n683","layer":"informal","project":"p5","title":"Follow your nose.","kind":"proof","summary":"Follow your nose.","labels":[],"detail_key":"p5"},{"id":"n684","layer":"informal","project":"p5","title":"ZHat.nontrivial","kind":"lemma","summary":"0\\not=1 in \\widehatZ.","labels":["ZHat.nontrivial"],"detail_key":"p5"},{"id":"n685","layer":"informal","project":"p5","title":"Recall that you can evaluate an element of \\widehatZ at a positive integer. Evaluating 0…","kind":"proof","summary":"Recall that you can evaluate an element of \\widehatZ at a positive integer. Evaluating 0 at 2 g…","labels":[],"detail_key":"p5"},{"id":"n686","layer":"informal","project":"p5","title":"ZHat.charZero","kind":"lemma","summary":"The map from the naturals into \\widehatZ sending n to n is injective.","labels":["ZHat.charZero"],"detail_key":"p5"},{"id":"n687","layer":"informal","project":"p5","title":"Generalise the above idea. Feel free to write up a LaTeX proof and PR it.","kind":"proof","summary":"Generalise the above idea. Feel free to write up a LaTeX proof and PR it.","labels":[],"detail_key":"p5"},{"id":"n688","layer":"informal","project":"p5","title":"ZHat.e","kind":"definition","summary":"The infinite sum 0!+1!+2!+3!+4!+5!+\\cdots looks like it makes no sense at all; it is the sum of…","labels":["ZHat.e"],"detail_key":"p5"},{"id":"n689","layer":"informal","project":"p5","title":"ZHat.e_def","kind":"lemma","summary":"The collection (e_N)_N is an element of \\widehatZ.","labels":["ZHat.e_def"],"detail_key":"p5"},{"id":"n690","layer":"informal","project":"p5","title":"This boils down to checking that D!+(D+1)!+\\cdots+(N-1)! is a multiple of~D.","kind":"proof","summary":"This boils down to checking that D!+(D+1)!+\\cdots+(N-1)! is a multiple of~D.","labels":[],"detail_key":"p5"},{"id":"n691","layer":"informal","project":"p5","title":"ZHat.e_not_in_Int","kind":"lemma","summary":"The element (e_N)_N of \\widehatZ is not in Z.","labels":["ZHat.e_not_in_Int"],"detail_key":"p5"},{"id":"n692","layer":"informal","project":"p5","title":"First imagine that e=n with n\\inZ and 0\\leq n. In this case, choose j such that 0!+1!+2!+…","kind":"proof","summary":"First imagine that e=n with n\\inZ and 0\\leq n. In this case, choose j such that 0!+1!+2!+\\cdots…","labels":[],"detail_key":"p5"},{"id":"n693","layer":"informal","project":"p5","title":"ZHat.torsionfree","kind":"lemma","summary":"If 0<N is an integer then multiplication by N is injective on \\widehatZ.","labels":["ZHat.torsionfree"],"detail_key":"p5"},{"id":"n694","layer":"informal","project":"p5","title":"Suppose that (z_i)_i\\in\\widehatZ and Nz=0. This means that Nz_i=0\\inZ/iZ for all i. Let u…","kind":"proof","summary":"Suppose that (z_i)_i\\in\\widehatZ and Nz=0. This means that Nz_i=0\\inZ/iZ for all i. Let us fix…","labels":[],"detail_key":"p5"},{"id":"n695","layer":"informal","project":"p5","title":"ZHat.multiples","kind":"lemma","summary":"The multiples of~N in \\widehatZ are precisely the compatible collections (z_i)_i\\in\\widehatZ wi…","labels":["ZHat.multiples"],"detail_key":"p5"},{"id":"n696","layer":"informal","project":"p5","title":"Clearly z_N=0 is a necessary condition to be a multiple of~N. To see it is sufficient, ta…","kind":"proof","summary":"Clearly z_N=0 is a necessary condition to be a multiple of~N. To see it is sufficient, take a g…","labels":[],"detail_key":"p5"},{"id":"n697","layer":"informal","project":"p5","title":"QHat","kind":"definition","summary":"The profinite completion \\widehatQ of Q is the tensor product Q\\otimes_Z\\widehatZ, or \\widehatQ…","labels":["QHat"],"detail_key":"p5"},{"id":"n698","layer":"informal","project":"p5","title":"Recall that the sum of all the factorials is an element e\\in\\widehatZ, and 22/7 is certai…","kind":"example","summary":"Recall that the sum of all the factorials is an element e\\in\\widehatZ, and 22/7 is certainly a…","labels":[],"detail_key":"p5"},{"id":"n699","layer":"informal","project":"p5","title":"A summary of the situation: if A and B are abelian groups, then every element of A\\otimes…","kind":"remark","summary":"A summary of the situation: if A and B are abelian groups, then every element of A\\otimes B can…","labels":[],"detail_key":"p5"},{"id":"n700","layer":"informal","project":"p5","title":"QHat.canonicalForm","kind":"lemma","summary":"Every element of \\widehatQ:=Q\\otimes\\widehatZ can be written as q\\otimes_t z with q\\inQ and z\\i…","labels":["QHat.canonicalForm"],"detail_key":"p5"},{"id":"n701","layer":"informal","project":"p5","title":"A proof I would write on the board would look like the following. Take a general element…","kind":"proof","summary":"A proof I would write on the board would look like the following. Take a general element of \\wi…","labels":[],"detail_key":"p5"},{"id":"n702","layer":"informal","project":"p5","title":"QHat.IsCoprime","kind":"definition","summary":"If N\\inN^+ and z\\in\\widehatZ then we say that N and z are \\emphcoprime if z_N\\in(Z/NZ)^\\times.…","labels":["QHat.IsCoprime"],"detail_key":"p5"},{"id":"n703","layer":"informal","project":"p5","title":"QHat.lowestTerms","kind":"lemma","summary":"Every element of \\widehatQ can be uniquely written as z/N with z\\in\\widehatZ, N\\inN^+, and with…","labels":["QHat.lowestTerms"],"detail_key":"p5"},{"id":"n704","layer":"informal","project":"p5","title":"Existence: by the previous lemma, an arbitrary element can be written as z/N; let D be th…","kind":"proof","summary":"Existence: by the previous lemma, an arbitrary element can be written as z/N; let D be the grea…","labels":[],"detail_key":"p5"},{"id":"n705","layer":"informal","project":"p5","title":"QHat.injective_rat","kind":"lemma","summary":"The ring homomorphism Q\\to\\widehatQ sending q to q\\otimes_t 1 is injective.","labels":["QHat.injective_rat"],"detail_key":"p5"},{"id":"n706","layer":"informal","project":"p5","title":"We have seen that the map from Z to \\widehatZ is injective. Now Q is a flat Z-module, bec…","kind":"proof","summary":"We have seen that the map from Z to \\widehatZ is injective. Now Q is a flat Z-module, because i…","labels":[],"detail_key":"p5"},{"id":"n707","layer":"informal","project":"p5","title":"QHat.injective_zHat","kind":"lemma","summary":"The ring homomorphism \\widehatZ\\to\\widehatQ sending z to 1\\otimes_t z is injective.","labels":["QHat.injective_zHat"],"detail_key":"p5"},{"id":"n708","layer":"informal","project":"p5","title":"The map from Z to Q is injective, and we have seen that \\widehatZ is a torsion-free and t…","kind":"proof","summary":"The map from Z to Q is injective, and we have seen that \\widehatZ is a torsion-free and thus fl…","labels":[],"detail_key":"p5"},{"id":"n709","layer":"informal","project":"p5","title":"QHat.rat_meet_zHat","kind":"lemma","summary":"The intersection of Q and \\widehatZ in \\widehatQ is Z.","labels":["QHat.rat_meet_zHat"],"detail_key":"p5"},{"id":"n710","layer":"informal","project":"p5","title":"Clearly Z\\subseteqQ\\cap\\widehatZ. Now suppose that x\\inQ\\cap\\widehatZ. Because x is ratio…","kind":"proof","summary":"Clearly Z\\subseteqQ\\cap\\widehatZ. Now suppose that x\\inQ\\cap\\widehatZ. Because x is rational we…","labels":[],"detail_key":"p5"},{"id":"n711","layer":"informal","project":"p5","title":"QHat.rat_join_zHat","kind":"lemma","summary":"The sum of Q and \\widehatZ in \\widehatQ is \\widehatQ. More precisely, every element of \\widehat…","labels":["QHat.rat_join_zHat"],"detail_key":"p5"},{"id":"n712","layer":"informal","project":"p5","title":"Write x\\in\\widehatQ as x=z/N in lowest terms. Lift z_N to an integer t and observe that (…","kind":"proof","summary":"Write x\\in\\widehatQ as x=z/N in lowest terms. Lift z_N to an integer t and observe that (z-t)_N…","labels":[],"detail_key":"p5"},{"id":"n713","layer":"informal","project":"p5","title":"Qhat.unitsrat_meet_unitszHat","kind":"lemma","summary":"The intersection of Q^\\times and \\widehatZ^\\times in \\widehatQ^\\times is Z^\\times.","labels":["Qhat.unitsrat_meet_unitszHat"],"detail_key":"p5"},{"id":"n714","layer":"informal","project":"p5","title":"Clearly the intersection is contained within Q\\cap\\widehatZ=Z. If n\\inZ is in \\widehatZ^\\…","kind":"proof","summary":"Clearly the intersection is contained within Q\\cap\\widehatZ=Z. If n\\inZ is in \\widehatZ^\\times…","labels":[],"detail_key":"p5"},{"id":"n715","layer":"informal","project":"p5","title":"QHat.unitsrat_join_unitszHat","kind":"lemma","summary":"The product of Q^\\times and \\widehatZ^\\times in \\widehatQ^\\times is all of \\widehatQ^\\times. Mo…","labels":["QHat.unitsrat_join_unitszHat"],"detail_key":"p5"},{"id":"n716","layer":"informal","project":"p5","title":"We already know that a general element of \\widehatQ^\\times can be written as x/N with N p…","kind":"proof","summary":"We already know that a general element of \\widehatQ^\\times can be written as x/N with N positiv…","labels":[],"detail_key":"p5"},{"id":"n717","layer":"informal","project":"p5","title":"Hurwitz","kind":"definition","summary":"The Hurwitz quaternions are the set O:= Z\\oplusZ\\omega\\oplusZi\\oplus Zi\\omega (as an abstract a…","labels":["Hurwitz"],"detail_key":"p5"},{"id":"n718","layer":"informal","project":"p5","title":"Hurwitz.ring","kind":"lemma","summary":"The Hurwitz quaternions form a ring.","labels":["Hurwitz.ring"],"detail_key":"p5"},{"id":"n719","layer":"informal","project":"p5","title":"Follow your nose.","kind":"proof","summary":"Follow your nose.","labels":[],"detail_key":"p5"},{"id":"n720","layer":"informal","project":"p5","title":"Hurwitz.starRing","kind":"definition","summary":"There's a conjugation map (which we'll call \"star\") from the Hurwitz quaternions to themselves,…","labels":["Hurwitz.starRing"],"detail_key":"p5"},{"id":"n721","layer":"informal","project":"p5","title":"Hurwitz.norm","kind":"definition","summary":"The Hurwitz quaternions come equipped with an integer-valued norm, which is a^2+b^2+c^2+d^2 on…","labels":["Hurwitz.norm"],"detail_key":"p5"},{"id":"n722","layer":"informal","project":"p5","title":"Hurwitz.norm_eq_mul_conj","kind":"lemma","summary":"We have N(x)=x\\overlinex.","labels":["Hurwitz.norm_eq_mul_conj"],"detail_key":"p5"},{"id":"n723","layer":"informal","project":"p5","title":"Easy calculation.","kind":"proof","summary":"Easy calculation.","labels":[],"detail_key":"p5"},{"id":"n724","layer":"informal","project":"p5","title":"Hurwitz.norm_zero","kind":"lemma","summary":"The norm of 0 is 0.","labels":["Hurwitz.norm_zero"],"detail_key":"p5"},{"id":"n725","layer":"informal","project":"p5","title":"A calculation.","kind":"proof","summary":"A calculation.","labels":[],"detail_key":"p5"},{"id":"n726","layer":"informal","project":"p5","title":"Hurwitz.norm_one","kind":"lemma","summary":"The norm of 1 is 1.","labels":["Hurwitz.norm_one"],"detail_key":"p5"},{"id":"n727","layer":"informal","project":"p5","title":"A calculation.","kind":"proof","summary":"A calculation.","labels":[],"detail_key":"p5"},{"id":"n728","layer":"informal","project":"p5","title":"Hurwitz.norm_mul","kind":"lemma","summary":"The norm of a product is the product of the norms.","labels":["Hurwitz.norm_mul"],"detail_key":"p5"},{"id":"n729","layer":"informal","project":"p5","title":"A calculation.","kind":"proof","summary":"A calculation.","labels":[],"detail_key":"p5"},{"id":"n730","layer":"informal","project":"p5","title":"Hurwitz.norm_nonneg","kind":"lemma","summary":"The norm of an element is nonnegative.","labels":["Hurwitz.norm_nonneg"],"detail_key":"p5"},{"id":"n731","layer":"informal","project":"p5","title":"It's a sum of rational squares.","kind":"proof","summary":"It's a sum of rational squares.","labels":[],"detail_key":"p5"},{"id":"n732","layer":"informal","project":"p5","title":"Hurwitz.norm_eq_zero","kind":"lemma","summary":"The norm of an element is zero if and only if the element is zero.","labels":["Hurwitz.norm_eq_zero"],"detail_key":"p5"},{"id":"n733","layer":"informal","project":"p5","title":"It's a sum of rational squares.","kind":"proof","summary":"It's a sum of rational squares.","labels":[],"detail_key":"p5"},{"id":"n734","layer":"informal","project":"p5","title":"Hurwitz.exists_near","kind":"lemma","summary":"Given a ``usual'' quaternion a=x+yi+zj+wk with x,y,z,w\\inR, there exists a Hurwitz quaternion q…","labels":["Hurwitz.exists_near"],"detail_key":"p5"},{"id":"n735","layer":"informal","project":"p5","title":"If [r] denotes the nearest integer to the real number r, then |r-[r]|\\leq \\frac12. Hence…","kind":"proof","summary":"If [r] denotes the nearest integer to the real number r, then |r-[r]|\\leq \\frac12. Hence if q=[…","labels":[],"detail_key":"p5"},{"id":"n736","layer":"informal","project":"p5","title":"Hurwitz.quot_rem","kind":"lemma","summary":"Given two Hurwitz quaternions a and b with b nonzero, there exists q and r such that a=qb+r and…","labels":["Hurwitz.quot_rem"],"detail_key":"p5"},{"id":"n737","layer":"informal","project":"p5","title":"Let q be the Hurwitz quaternion obtained by applying Lemma~\\refHurwitz.exists_near to a/b…","kind":"proof","summary":"Let q be the Hurwitz quaternion obtained by applying Lemma~\\refHurwitz.exists_near to a/b := ab…","labels":[],"detail_key":"p5"},{"id":"n738","layer":"informal","project":"p5","title":"Hurwitz.left_ideal_princ","kind":"corollary","summary":"All left ideals of O are principal.","labels":["Hurwitz.left_ideal_princ"],"detail_key":"p5"},{"id":"n739","layer":"informal","project":"p5","title":"If the ideal is 0, use 0. Otherwise, choose a nonzero element of smallest norm.","kind":"proof","summary":"If the ideal is 0, use 0. Otherwise, choose a nonzero element of smallest norm.","labels":[],"detail_key":"p5"},{"id":"n740","layer":"informal","project":"p5","title":"All right ideals are principal too, because there's another version of Euclid saying a=bq…","kind":"remark","summary":"All right ideals are principal too, because there's another version of Euclid saying a=bq+r.","labels":[],"detail_key":"p5"},{"id":"n741","layer":"informal","project":"p5","title":"Hurwitz.surjective_pnat_quotient","kind":"theorem","summary":"If N is a positive natural then the obvious map O\\to\\widehatO/N\\widehatO is surjective.","labels":["Hurwitz.surjective_pnat_quotient"],"detail_key":"p5"},{"id":"n742","layer":"informal","project":"p5","title":"This is just four copies of the surjection Z\\to\\widehatZ/N\\widehatZ. Note that this latte…","kind":"proof","summary":"This is just four copies of the surjection Z\\to\\widehatZ/N\\widehatZ. Note that this latter map…","labels":[],"detail_key":"p5"},{"id":"n743","layer":"informal","project":"p5","title":"HurwitzRatHat.canonicalForm","kind":"lemma","summary":"Every element of \\widehatD can be written as z/N with z\\in\\widehatO and N\\inN^+.","labels":["HurwitzRatHat.canonicalForm"],"detail_key":"p5"},{"id":"n744","layer":"informal","project":"p5","title":"Same as the proof for \\widehatQ.","kind":"proof","summary":"Same as the proof for \\widehatQ.","labels":[],"detail_key":"p5"},{"id":"n745","layer":"informal","project":"p5","title":"HurwitzRatHat.completed_units","kind":"theorem","summary":"The group of units of \\widehatD is D^\\times\\widehatO^\\times. More precisely, every element of \\…","labels":["HurwitzRatHat.completed_units"],"detail_key":"p5"},{"id":"n746","layer":"informal","project":"p5","title":"Given an element x of \\widehatD^\\times, we can use lemma~\\refHurwitzRatHat.canonicalForm…","kind":"proof","summary":"Given an element x of \\widehatD^\\times, we can use lemma~\\refHurwitzRatHat.canonicalForm to wri…","labels":[],"detail_key":"p5"},{"id":"n747","layer":"informal","project":"p5","title":"MatrixRing.isCentralSimple","kind":"lemma","summary":"\\discussion47 If n\\geq1 then the n\\times n matrices M_n(K) are a central simple algebra over~K.","labels":["MatrixRing.isCentralSimple"],"detail_key":"p5"},{"id":"n748","layer":"informal","project":"p5","title":"We prove more generally that matrices with coefficients in~K and indexed by an arbitrary…","kind":"proof","summary":"We prove more generally that matrices with coefficients in~K and indexed by an arbitrary nonemp…","labels":[],"detail_key":"p5"},{"id":"n749","layer":"informal","project":"p5","title":"IsCentralSimple.baseChange","kind":"lemma","summary":"If D is a central simple algebra over~K and L/K is a field extension, then L\\otimes_KD is a cen…","labels":["IsCentralSimple.baseChange"],"detail_key":"p5"},{"id":"n750","layer":"informal","project":"p5","title":"This is not too hard: it's lemma b of section 12.4 in Peirce's \"Associative algebras\".","kind":"proof","summary":"This is not too hard: it's lemma b of section 12.4 in Peirce's \"Associative algebras\".","labels":[],"detail_key":"p5"},{"id":"n751","layer":"informal","project":"p5","title":"AutomorphicForm.GLn.IsSmooth","kind":"definition","summary":"A function f:\\GL_n(A_Q^f)\\times\\GL_n(R)\\toC is \\emphsmooth if it has the following three proper…","labels":["AutomorphicForm.GLn.IsSmooth"],"detail_key":"p5"},{"id":"n752","layer":"informal","project":"p5","title":"AutomorphicForm.GLn.IsSlowlyIncreasing","kind":"definition","summary":"We say that a function f:\\GL_n(R)\\toC is \\emphslowly-increasing if there's some real constant C…","labels":["AutomorphicForm.GLn.IsSlowlyIncreasing"],"detail_key":"p5"},{"id":"n753","layer":"informal","project":"p5","title":"AutomorphicForm.GLn.Weight","kind":"definition","summary":"The \\emphweight of an automorphic form for \\GL_n/Q can be thought of as a finite-dimensional co…","labels":["AutomorphicForm.GLn.Weight"],"detail_key":"p5"},{"id":"n754","layer":"informal","project":"p5","title":"instLieAlgebraAction","kind":"definition","summary":"There is a natural action of the real Lie algebra of \\GL_n(R) on the complex vector space of sm…","labels":["instLieAlgebraAction"],"detail_key":"p5"},{"id":"n755","layer":"informal","project":"p5","title":"instComplexLieAlgebraAction","kind":"definition","summary":"This extends to is a natural complex Lie algebra action of the complexification of the real Lie…","labels":["instComplexLieAlgebraAction"],"detail_key":"p5"},{"id":"n756","layer":"informal","project":"p5","title":"instUniversalEnvelopingAlgebraAction","kind":"definition","summary":"By functoriality, we get an action of the universal enveloping algebra of this complexified Lie…","labels":["instUniversalEnvelopingAlgebraAction"],"detail_key":"p5"},{"id":"n757","layer":"informal","project":"p5","title":"instCentreAction","kind":"definition","summary":"Thus the \\emphcentre Z_n of this universal enveloping algebra also acts on the smooth complex f…","labels":["instCentreAction"],"detail_key":"p5"},{"id":"n758","layer":"informal","project":"p5","title":"The centre we just defined is a commutative ring which contains a copy of C. Note that Ha…","kind":"remark","summary":"The centre we just defined is a commutative ring which contains a copy of C. Note that Harish-C…","labels":[],"detail_key":"p5"},{"id":"n759","layer":"informal","project":"p5","title":"AutomorphicForm.GLn.AutomorphicFormForGLnOverQ","kind":"definition","summary":"A smooth function f:\\GL_n(A_Q^f)\\times\\GL_n(R)\\toC is an O_n(R)-\\emphautomorphic form on \\GL_n(…","labels":["AutomorphicForm.GLn.AutomorphicFormForGLnOverQ"],"detail_key":"p5"},{"id":"n760","layer":"informal","project":"p5","title":"The group \\GL_n(A_Q^f) acts (on the left) on the space of automorphic forms for \\GL_n(A_Q…","kind":"lemma","summary":"The group \\GL_n(A_Q^f) acts (on the left) on the space of automorphic forms for \\GL_n(A_Q) by t…","labels":[],"detail_key":"p5"},{"id":"n761","layer":"informal","project":"p5","title":"This is obvious. Note that the conjugate of a compact open subgroup is still compact and…","kind":"proof","summary":"This is obvious. Note that the conjugate of a compact open subgroup is still compact and open.","labels":[],"detail_key":"p5"},{"id":"n762","layer":"informal","project":"p5","title":"This function is well-defined, i.e., it sends a U-invariant form to a U-invariant form wh…","kind":"lemma","summary":"This function is well-defined, i.e., it sends a U-invariant form to a U-invariant form which is…","labels":[],"detail_key":"p5"},{"id":"n763","layer":"informal","project":"p5","title":"Easy group theory.","kind":"proof","summary":"Easy group theory.","labels":[],"detail_key":"p5"},{"id":"n764","layer":"informal","project":"p5","title":"IsFractionRing.stabilizerHom","kind":"definition","summary":"Choose g\\in D_Q. Then the action of g on B gives us an induced A/P-algebra automorphism of B/Q…","labels":["IsFractionRing.stabilizerHom"],"detail_key":"p5"},{"id":"n765","layer":"informal","project":"p5","title":"IsFractionRing.stabilizerHom_surjective","kind":"theorem","summary":"\\mathlibok The map g\\mapsto \\phi_g from D_Q to \\Aut(L/K) defined above is surjective.","labels":["IsFractionRing.stabilizerHom_surjective"],"detail_key":"p5"},{"id":"n766","layer":"informal","project":"p5","title":"MulSemiringAction.charpoly","kind":"definition","summary":"\\mathlibok If b\\in B then define the \\emphcharacteristic polynomial F_b(X) \\in B[X] of b to be…","labels":["MulSemiringAction.charpoly"],"detail_key":"p5"},{"id":"n767","layer":"informal","project":"p5","title":"MulSemiringAction.monic_charpoly","kind":"lemma","summary":"\\mathlibok F_b is monic.","labels":["MulSemiringAction.monic_charpoly"],"detail_key":"p5"},{"id":"n768","layer":"informal","project":"p5","title":"\\mathlibok Obvious.","kind":"proof","summary":"\\mathlibok Obvious.","labels":[],"detail_key":"p5"},{"id":"n769","layer":"informal","project":"p5","title":"Algebra.IsInvariant.charpoly_mem_lifts","kind":"lemma","summary":"\\mathlibok F_b is the lift of some monic polynomial M_b in A[X].","labels":["Algebra.IsInvariant.charpoly_mem_lifts"],"detail_key":"p5"},{"id":"n770","layer":"informal","project":"p5","title":"\\mathlibok The coefficients of F_b are G-invariant, and thus lie in the image of A.","kind":"proof","summary":"\\mathlibok The coefficients of F_b are G-invariant, and thus lie in the image of A.","labels":[],"detail_key":"p5"},{"id":"n771","layer":"informal","project":"p5","title":"Algebra.IsInvariant.isIntegral","kind":"theorem","summary":"\\mathlibok B/A is integral.","labels":["Algebra.IsInvariant.isIntegral"],"detail_key":"p5"},{"id":"n772","layer":"informal","project":"p5","title":"\\mathlibok Use M_b.","kind":"proof","summary":"\\mathlibok Use M_b.","labels":[],"detail_key":"p5"},{"id":"n773","layer":"informal","project":"p5","title":"fixed_of_fixed1_aux1","kind":"lemma","summary":"\\mathlibok There exist elements a,b \\in B, with a \\notin Q and a in the image of A such that fo…","labels":["fixed_of_fixed1_aux1"],"detail_key":"p5"},{"id":"n774","layer":"informal","project":"p5","title":"The ideals g \\cdot Q \\neq Q are not contained in Q. Since Q is a prime ideal, this implie…","kind":"proof","summary":"The ideals g \\cdot Q \\neq Q are not contained in Q. Since Q is a prime ideal, this implies that…","labels":[],"detail_key":"p5"},{"id":"n775","layer":"informal","project":"p5","title":"fixed_of_fixed1_aux2","kind":"lemma","summary":"\\mathlibok Let b_0 \\in B. Suppose that the image of b_0 in the quotient B/Q is fixed by the sta…","labels":["fixed_of_fixed1_aux2"],"detail_key":"p5"},{"id":"n776","layer":"informal","project":"p5","title":"Multiply the b from~\\reffixed_of_fixed1_aux1 by b_0. \\mathlibok","kind":"proof","summary":"Multiply the b from~\\reffixed_of_fixed1_aux1 by b_0. \\mathlibok","labels":[],"detail_key":"p5"},{"id":"n777","layer":"informal","project":"p5","title":"FixedPoints.toAlgAut_surjective","kind":"theorem","summary":"Let H be a finite group acting on a field F by field automorphisms. Then the map H \\to \\Aut(F/F…","labels":["FixedPoints.toAlgAut_surjective"],"detail_key":"p5"},{"id":"n778","layer":"informal","project":"p5","title":"This is a general fact of Galois theory and was already in mathlib. \\mathlibok","kind":"proof","summary":"This is a general fact of Galois theory and was already in mathlib. \\mathlibok","labels":[],"detail_key":"p5"},{"id":"n779","layer":"informal","project":"p5","title":"fixed_of_fixed1","kind":"proposition","summary":"\\mathlibok Let b_0 \\in B/Q. Suppose that b_0 is fixed by the stabilizer subgroup D_Q. Then b_0…","labels":["fixed_of_fixed1"],"detail_key":"p5"},{"id":"n780","layer":"informal","project":"p5","title":"Let a,b\\in B be elements from~\\reffixed_of_fixed1_aux2. Let M_b \\in A[X] be the character…","kind":"proof","summary":"Let a,b\\in B be elements from~\\reffixed_of_fixed1_aux2. Let M_b \\in A[X] be the characteristic…","labels":[],"detail_key":"p5"},{"id":"n781","layer":"informal","project":"p5","title":"IsAlgebraic.exists_smul_eq_mul","kind":"lemma","summary":"\\mathlibok If R/S is an algebraic extension of integral domains, then any fraction a/b with a,b…","labels":["IsAlgebraic.exists_smul_eq_mul"],"detail_key":"p5"},{"id":"n782","layer":"informal","project":"p5","title":"If f\\in S[X] satisfies f(b)=0, then f(0)\\in S is a multiple of b. If f(0)=bx\\in S, then a…","kind":"proof","summary":"If f\\in S[X] satisfies f(b)=0, then f(0)\\in S is a multiple of b. If f(0)=bx\\in S, then a/b=(ax…","labels":[],"detail_key":"p5"},{"id":"n783","layer":"informal","project":"p5","title":"fixed_of_fixed2","kind":"proposition","summary":"\\mathlibok Let x \\in L. Suppose that x is fixed by the stabilizer subgroup D_Q. Then x is fixed…","labels":["fixed_of_fixed2"],"detail_key":"p5"},{"id":"n784","layer":"informal","project":"p5","title":"Since (B/Q)/(A/Q) is algebraic by~\\refAlgebra.IsInvariant.isIntegral, ~\\refIsAlgebraic.ex…","kind":"proof","summary":"Since (B/Q)/(A/Q) is algebraic by~\\refAlgebra.IsInvariant.isIntegral, ~\\refIsAlgebraic.exists_s…","labels":[],"detail_key":"p5"},{"id":"n785","layer":"informal","project":"p5","title":"Proof of main theorem","kind":"proof","summary":"[Proof of main theorem] The map D_Q \\to \\Aut(L/L^D_Q) is surjective by~\\refFixedPoints.toAlgAut…","labels":[],"detail_key":"p5"},{"id":"n786","layer":"informal","project":"p5","title":"NumberField.instCompactSpaceAdicCompletionIntegers","kind":"theorem","summary":"\\discussion451 If K is a number field and v is a nonzero prime ideal of the integers of K, then…","labels":["NumberField.instCompactSpaceAdicCompletionIntegers"],"detail_key":"p5"},{"id":"n787","layer":"informal","project":"p5","title":"Openness should follow from the fact that the integers are \\x : v(x)<v(1/\\pi)\\ where \\pi…","kind":"proof","summary":"Openness should follow from the fact that the integers are \\x : v(x)<v(1/\\pi)\\ where \\pi is a u…","labels":[],"detail_key":"p5"},{"id":"n788","layer":"informal","project":"p5","title":"NumberField.AdeleRing.locallyCompactSpace","kind":"theorem","summary":"\\discussion253 The adeles of a number field are locally compact.","labels":["NumberField.AdeleRing.locallyCompactSpace"],"detail_key":"p5"},{"id":"n789","layer":"informal","project":"p5","title":"The adeles of a number field are a product of the finite adeles and the infinite adeles s…","kind":"proof","summary":"The adeles of a number field are a product of the finite adeles and the infinite adeles so it s…","labels":[],"detail_key":"p5"},{"id":"n790","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.valuation_comap","kind":"lemma","summary":"If i:K\\to L denotes the inclusion then for k\\in K we have e\\times w(i(k))=v(k), where e is the…","labels":["IsDedekindDomain.HeightOneSpectrum.valuation_comap"],"detail_key":"p5"},{"id":"n791","layer":"informal","project":"p5","title":"Standard (and formalized).","kind":"proof","summary":"Standard (and formalized).","labels":[],"detail_key":"p5"},{"id":"n792","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom","kind":"definition","summary":"There's a natural ring map K_v\\to L_w extending the map K\\to L. It is defined by completing the…","labels":["IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom"],"detail_key":"p5"},{"id":"n793","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.Extension.valued_adicCompletionSemialgHom","kind":"lemma","summary":"If i_v:K_v\\to L_w denotes the map of the previous definition then for x\\in K_v we have e\\times…","labels":["IsDedekindDomain.HeightOneSpectrum.Extension.valued_adicCompletionSemialgHom"],"detail_key":"p5"},{"id":"n794","layer":"informal","project":"p5","title":"Follows by continuity from lemma~\\refIsDedekindDomain.HeightOneSpectrum.valuation_comap.","kind":"proof","summary":"Follows by continuity from lemma~\\refIsDedekindDomain.HeightOneSpectrum.valuation_comap.","labels":[],"detail_key":"p5"},{"id":"n795","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom_image_adicCompletio…","kind":"lemma","summary":"The map i_v:K_v\\to L_w sends the integer ring A_v into B_w.","labels":["IsDedekindDomain.HeightOneSpectrum.Extension.adicCompletionSemialgHom_image_adicCompletionIntegers"],"detail_key":"p5"},{"id":"n796","layer":"informal","project":"p5","title":"The integer ring is defined by v\\geq0 (or v\\leq 1 in mathlib, which uses multiplicative v…","kind":"proof","summary":"The integer ring is defined by v\\geq0 (or v\\leq 1 in mathlib, which uses multiplicative valuati…","labels":[],"detail_key":"p5"},{"id":"n797","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopology","kind":"theorem","summary":"\\discussion326 Giving L_w the K_v-algebra structure coming from the natural map K_v\\to L_w, the…","labels":["IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopology"],"detail_key":"p5"},{"id":"n798","layer":"informal","project":"p5","title":"Any basis for L as a K-vector space spans L_w as a K_v-module, so L_w is finite-dimension…","kind":"proof","summary":"Any basis for L as a K-vector space spans L_w as a K_v-module, so L_w is finite-dimensional ove…","labels":[],"detail_key":"p5"},{"id":"n799","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.Extension.finite","kind":"lemma","summary":"There are only finitely many primes w of B lying above v.","labels":["IsDedekindDomain.HeightOneSpectrum.Extension.finite"],"detail_key":"p5"},{"id":"n800","layer":"informal","project":"p5","title":"This is a standard fact about Dedekind domains. The key input is mathlib's theorem \\tt pr…","kind":"proof","summary":"This is a standard fact about Dedekind domains. The key input is mathlib's theorem \\tt primesOv…","labels":[],"detail_key":"p5"},{"id":"n801","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.semialgHomPi","kind":"definition","summary":"The product of the maps K_v\\to L_w for w|v is a natural ring map K_v\\to\\prod_w|vL_w lying over…","labels":["IsDedekindDomain.HeightOneSpectrum.adicCompletion.semialgHomPi"],"detail_key":"p5"},{"id":"n802","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeAlgEquiv","kind":"theorem","summary":"The induced L-algebra homomorphism L\\otimes_KK_v\\to\\prod_w|vL_w is an isomorphism of rings.","labels":["IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeAlgEquiv"],"detail_key":"p5"},{"id":"n803","layer":"informal","project":"p5","title":"My current proposal to formalize this is as follows. The map is surjective because the im…","kind":"proof","summary":"My current proposal to formalize this is as follows. The map is surjective because the image is…","labels":[],"detail_key":"p5"},{"id":"n804","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopologyPi","kind":"theorem","summary":"For v fixed, the product topology on \\prod_w|vL_w is the K_v-module topology.","labels":["IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsModuleTopologyPi"],"detail_key":"p5"},{"id":"n805","layer":"informal","project":"p5","title":"This is a finite product of K_v-modules each of which has the K_v-module topology by~\\ref…","kind":"proof","summary":"This is a finite product of K_v-modules each of which has the K_v-module topology by~\\refIsDede…","labels":[],"detail_key":"p5"},{"id":"n806","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeContinuousAlgEquiv","kind":"theorem","summary":"If we give L\\otimes_KK_v the K_v-module topology then the L-algebra isomorphism L\\otimes_K K_v\\…","labels":["IsDedekindDomain.HeightOneSpectrum.adicCompletion.baseChangeContinuousAlgEquiv"],"detail_key":"p5"},{"id":"n807","layer":"informal","project":"p5","title":"Indeed, is a K_v-algebra isomorphism between two modules each of which have the module to…","kind":"proof","summary":"Indeed, is a K_v-algebra isomorphism between two modules each of which have the module topology…","labels":[],"detail_key":"p5"},{"id":"n808","layer":"informal","project":"p5","title":"IsDedekindDomain.HeightOneSpectrum.range_baseChange_comp_tensorAdicCompletionTo_eq_pi","kind":"theorem","summary":"The isomorphism L\\otimes_KK_v\\to\\prod_w|vL_w induces an isomorphism B\\otimes_AA_v\\to \\prod_w|vB…","labels":["IsDedekindDomain.HeightOneSpectrum.range_baseChange_comp_tensorAdicCompletionTo_eq_pi"],"detail_key":"p5"},{"id":"n809","layer":"informal","project":"p5","title":"Certainly the image of the integral elements are integral. The argument in the other dire…","kind":"proof","summary":"Certainly the image of the integral elements are integral. The argument in the other direction…","labels":[],"detail_key":"p5"},{"id":"n810","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.mapSemialgHom","kind":"definition","summary":"There's a natural ring homomorphism A_A,K^\\infty\\toA_B,L^\\infty lying over K\\to L.","labels":["IsDedekindDomain.FiniteAdeleRing.mapSemialgHom"],"detail_key":"p5"},{"id":"n811","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv","kind":"theorem","summary":"\\discussion243 This natural map L\\otimes_KA_A,K^\\infty\\toA_B,L^\\infty is an isomorphism.","labels":["IsDedekindDomain.FiniteAdeleRing.baseChangeAlgEquiv"],"detail_key":"p5"},{"id":"n812","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv","kind":"theorem","summary":"The induced L-algebra morphism L\\otimes_KA_A,K^\\infty\\toA_B,L^\\infty is a topological isomorphi…","labels":["IsDedekindDomain.FiniteAdeleRing.baseChangeContinuousAlgEquiv"],"detail_key":"p5"},{"id":"n813","layer":"informal","project":"p5","title":"IsDedekindDomain.dvd_norm","kind":"lemma","summary":"If 0\\not=b\\in B then there exists 0\\not=a\\in A such that b divides the image of a in B.","labels":["IsDedekindDomain.dvd_norm"],"detail_key":"p5"},{"id":"n814","layer":"informal","project":"p5","title":"Is this already in mathlib?","kind":"remark","summary":"Is this already in mathlib?","labels":[],"detail_key":"p5"},{"id":"n815","layer":"informal","project":"p5","title":"Let a=N_L/K(b), the norm. This is known to take nonzero elements of L to nonzero elements…","kind":"proof","summary":"Let a=N_L/K(b), the norm. This is known to take nonzero elements of L to nonzero elements of K…","labels":[],"detail_key":"p5"},{"id":"n816","layer":"informal","project":"p5","title":"IsDedekindDomain.AKLB.surjective_tensorProduct_map","kind":"corollary","summary":"The A-bilinear map B\\times K\\to L sending (b,k) to bk is surjective.","labels":["IsDedekindDomain.AKLB.surjective_tensorProduct_map"],"detail_key":"p5"},{"id":"n817","layer":"informal","project":"p5","title":"Given \\lambda\\in L write it as n/d with 0\\not=d\\in B. Choose 0\\not=a\\in A and b\\in B with…","kind":"proof","summary":"Given \\lambda\\in L write it as n/d with 0\\not=d\\in B. Choose 0\\not=a\\in A and b\\in B with db=a…","labels":[],"detail_key":"p5"},{"id":"n818","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","kind":"corollary","summary":"The natural map B\\otimes_AK\\to L is a B-algebra isomorphism.","labels":["IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv"],"detail_key":"p5"},{"id":"n819","layer":"informal","project":"p5","title":"We write down an inverse. Regard B\\otimes_AK as a B-algebra via the action on the left. N…","kind":"proof","summary":"We write down an inverse. Regard B\\otimes_AK as a B-algebra via the action on the left. Note th…","labels":[],"detail_key":"p5"},{"id":"n820","layer":"informal","project":"p5","title":"IsDedekindDomain.AKLB.tensorProduct_module_algEquiv","kind":"corollary","summary":"If M is any K-module then the canonical map B\\otimes_A M\\to L\\otimes_K M is an isomorphism.","labels":["IsDedekindDomain.AKLB.tensorProduct_module_algEquiv"],"detail_key":"p5"},{"id":"n821","layer":"informal","project":"p5","title":"We can factor this map as B\\otimes_AM\\cong B\\otimes_A(K\\otimes_KM)\\cong (B\\otimes_A K)\\co…","kind":"proof","summary":"We can factor this map as B\\otimes_AM\\cong B\\otimes_A(K\\otimes_KM)\\cong (B\\otimes_A K)\\cong_KM\\…","labels":[],"detail_key":"p5"},{"id":"n822","layer":"informal","project":"p5","title":"IsDedekindDomain.AKLB.finitePresentation","kind":"theorem","summary":"B is a finitely-presented A-module.","labels":["IsDedekindDomain.AKLB.finitePresentation"],"detail_key":"p5"},{"id":"n823","layer":"informal","project":"p5","title":"A is Noetherian as it is a Dedekind domain, so it suffices to prove that B is finitely-ge…","kind":"proof","summary":"A is Noetherian as it is a Dedekind domain, so it suffices to prove that B is finitely-generate…","labels":[],"detail_key":"p5"},{"id":"n824","layer":"informal","project":"p5","title":"pi_tensorProduct_of_finitePresentation","kind":"theorem","summary":"If R is a commutative ring, if M is a finitely presented R-module and if N_i are a collection o…","labels":["pi_tensorProduct_of_finitePresentation"],"detail_key":"p5"},{"id":"n825","layer":"informal","project":"p5","title":"If M is finite and free then Maddy Crim has already formalized this in FLT. For the gener…","kind":"proof","summary":"If M is finite and free then Maddy Crim has already formalized this in FLT. For the general cas…","labels":[],"detail_key":"p5"},{"id":"n826","layer":"informal","project":"p5","title":"IsDedekindDomain.pi_tensorProduct","kind":"corollary","summary":"If S is a finite set of nonzero primes of A then the natural map B\\otimes((\\prod_v\\in SK_v)\\tim…","labels":["IsDedekindDomain.pi_tensorProduct"],"detail_key":"p5"},{"id":"n827","layer":"informal","project":"p5","title":"Follows from the previous two theorems.","kind":"proof","summary":"Follows from the previous two theorems.","labels":[],"detail_key":"p5"},{"id":"n828","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","kind":"corollary","summary":"The natural map B\\otimes_AA_K^\\infty\\to R is a B-algebra isomorphism.","labels":["IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1"],"detail_key":"p5"},{"id":"n829","layer":"informal","project":"p5","title":"This follows from the previous corollary and the fact that tensor products commute with f…","kind":"proof","summary":"This follows from the previous corollary and the fact that tensor products commute with filtere…","labels":[],"detail_key":"p5"},{"id":"n830","layer":"informal","project":"p5","title":"RestrictedProduct.relabelIso","kind":"definition","summary":"Let V and W be index sets, and let f:W\\to V be a map with finite fibres. Let X_v be sets, with…","labels":["RestrictedProduct.relabelIso"],"detail_key":"p5"},{"id":"n831","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2","kind":"corollary","summary":"The ring R introduced above (the restricted product of the B\\otimes_A K_v with respect to the B…","labels":["IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2"],"detail_key":"p5"},{"id":"n832","layer":"informal","project":"p5","title":"Let V be the finite places of K and W the finite places of L, let X_v be B\\otimes_A K_v,…","kind":"proof","summary":"Let V be the finite places of K and W the finite places of L, let X_v be B\\otimes_A K_v, let C_…","labels":[],"detail_key":"p5"},{"id":"n833","layer":"informal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","kind":"theorem","summary":"The natural map B\\otimes_AA_K^\\infty\\toA_L^\\infty is a B-algebra isomorphism.","labels":["IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv"],"detail_key":"p5"},{"id":"n834","layer":"informal","project":"p5","title":"This map factors through the auxiliary ring~R so the result follows from the previous two…","kind":"proof","summary":"This map factors through the auxiliary ring~R so the result follows from the previous two const…","labels":[],"detail_key":"p5"},{"id":"n835","layer":"informal","project":"p5","title":"Follows immediately from theorem~\\refIsDedekindDomain.FiniteAdeleRing.baseChangeIntegralA…","kind":"proof","summary":"Follows immediately from theorem~\\refIsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEqui…","labels":[],"detail_key":"p5"},{"id":"n836","layer":"informal","project":"p5","title":"If X_v and Y_v are families of topological spaces indexed by v\\in V, if f_v:X_v\\to Y_v is…","kind":"definition","summary":"If X_v and Y_v are families of topological spaces indexed by v\\in V, if f_v:X_v\\to Y_v is a con…","labels":[],"detail_key":"p5"},{"id":"n837","layer":"informal","project":"p5","title":"If all the f_v are homeomorphisms identifying C_v with D_v then the induced map on restri…","kind":"definition","summary":"If all the f_v are homeomorphisms identifying C_v with D_v then the induced map on restricted p…","labels":[],"detail_key":"p5"},{"id":"n838","layer":"informal","project":"p5","title":"In the same setup as definition~\\refRestrictedProduct.relabelIso (V,W index sets, f:W\\to…","kind":"theorem","summary":"In the same setup as definition~\\refRestrictedProduct.relabelIso (V,W index sets, f:W\\to V, C_v…","labels":[],"detail_key":"p5"},{"id":"n839","layer":"informal","project":"p5","title":"I have only thought about the cofinite filter case, where this should follow easily from…","kind":"proof","summary":"I have only thought about the cofinite filter case, where this should follow easily from the de…","labels":[],"detail_key":"p5"},{"id":"n840","layer":"informal","project":"p5","title":"A_L^\\infty is homeomorphic to \\prod_v(B\\otimes_AK_v,B\\otimes_AA_v).","kind":"corollary","summary":"A_L^\\infty is homeomorphic to \\prod_v(B\\otimes_AK_v,B\\otimes_AA_v).","labels":[],"detail_key":"p5"},{"id":"n841","layer":"informal","project":"p5","title":"Follows from the previous theorem with X_v=B\\otimes_AK_v D_w=L_w etc.","kind":"proof","summary":"Follows from the previous theorem with X_v=B\\otimes_AK_v D_w=L_w etc.","labels":[],"detail_key":"p5"},{"id":"n842","layer":"informal","project":"p5","title":"If X_v and Y_v are topological spaces with open subspaces C_v and D_v, then the obvious b…","kind":"lemma","summary":"If X_v and Y_v are topological spaces with open subspaces C_v and D_v, then the obvious bijecti…","labels":[],"detail_key":"p5"},{"id":"n843","layer":"informal","project":"p5","title":"This should hopefully be straightforward using \\tt RestrictedProduct.continuous\\_dom\\_prod","kind":"proof","summary":"This should hopefully be straightforward using \\tt RestrictedProduct.continuous\\_dom\\_prod","labels":[],"detail_key":"p5"},{"id":"n844","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","kind":"theorem","summary":"Let S be a set of infinite places of K. The image of K under the embedding K\\hookrightarrow (K_…","labels":["NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi"],"detail_key":"p5"},{"id":"n845","layer":"informal","project":"p5","title":"Let (K, v) denote K equipped with the topology induced by the infinite place v. It suffic…","kind":"proof","summary":"Let (K, v) denote K equipped with the topology induced by the infinite place v. It suffices to…","labels":[],"detail_key":"p5"},{"id":"n846","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","kind":"theorem","summary":"For a fixed infinite place v of K, we have \\textdim_K_v \\prod_w\\mid v L_w = \\textdim_K_v L\\otim…","labels":["NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct"],"detail_key":"p5"},{"id":"n847","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.piExtension","kind":"definition","summary":"Let v be an infinite place of K. There is a continuous K-algebra homomorphism K_v \\to \\prod_w\\m…","labels":["NumberField.InfinitePlace.Completion.piExtension"],"detail_key":"p5"},{"id":"n848","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange","kind":"definition","summary":"Let v be an infinite place of K. There is a natural L-algebra homomorphism L\\otimes_K K_v \\to \\…","labels":["NumberField.InfinitePlace.Completion.baseChange"],"detail_key":"p5"},{"id":"n849","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange_surjective","kind":"theorem","summary":"For a fixed infinite place v of K, the map L\\otimes_K K_v \\to\\prod_w\\mid vL_w is surjective.","labels":["NumberField.InfinitePlace.Completion.baseChange_surjective"],"detail_key":"p5"},{"id":"n850","layer":"informal","project":"p5","title":"Let (x_i)_i be a K_v-basis of \\prod_w\\mid vL_w. By the density of L in \\prod_w\\mid vL_w (…","kind":"proof","summary":"Let (x_i)_i be a K_v-basis of \\prod_w\\mid vL_w. By the density of L in \\prod_w\\mid vL_w (Theore…","labels":[],"detail_key":"p5"},{"id":"n851","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange_injective","kind":"theorem","summary":"For a fixed infinite place v of K, the map L\\otimes_K K_v \\to\\prod_w\\mid vL_w is injective.","labels":["NumberField.InfinitePlace.Completion.baseChange_injective"],"detail_key":"p5"},{"id":"n852","layer":"informal","project":"p5","title":"The L-algebra map L\\otimes_K K_v \\to\\prod_w\\mid vL_w can equivalently be thought of as K_…","kind":"proof","summary":"The L-algebra map L\\otimes_K K_v \\to\\prod_w\\mid vL_w can equivalently be thought of as K_v-line…","labels":[],"detail_key":"p5"},{"id":"n853","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","kind":"theorem","summary":"If w \\mid v is an infinite place of L lying above the infinite place v of K, then L_w has the K…","labels":["NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT"],"detail_key":"p5"},{"id":"n854","layer":"informal","project":"p5","title":"Because L_w is a finite-dimensional normed K_v vector space, there exists a K_v-linear li…","kind":"proof","summary":"Because L_w is a finite-dimensional normed K_v vector space, there exists a K_v-linear linear h…","labels":[],"detail_key":"p5"},{"id":"n855","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChangeEquiv","kind":"theorem","summary":"Let v be an infinite place of K. There is a natural L-algebra homeomorphism L\\otimes_K K_v \\con…","labels":["NumberField.InfinitePlace.Completion.baseChangeEquiv"],"detail_key":"p5"},{"id":"n856","layer":"informal","project":"p5","title":"The map in~\\refNumberField.InfinitePlace.Completion.baseChange is an L-algebra isomorphis…","kind":"proof","summary":"The map in~\\refNumberField.InfinitePlace.Completion.baseChange is an L-algebra isomorphism by T…","labels":[],"detail_key":"p5"},{"id":"n857","layer":"informal","project":"p5","title":"NumberField.InfinitePlace.Completion.piEquiv","kind":"theorem","summary":"Let v be an infinite place of K. There is a natural K_v-linear homeomorphism K_v^[L:K] \\cong_K_…","labels":["NumberField.InfinitePlace.Completion.piEquiv"],"detail_key":"p5"},{"id":"n858","layer":"informal","project":"p5","title":"Compose the K_v-linear isomorphism K_v^[L:K] \\cong \\prod_w\\mid vL_w with the K_v-linear v…","kind":"proof","summary":"Compose the K_v-linear isomorphism K_v^[L:K] \\cong \\prod_w\\mid vL_w with the K_v-linear version…","labels":[],"detail_key":"p5"},{"id":"n859","layer":"informal","project":"p5","title":"NumberField.InfiniteAdeleRing.piEquiv","kind":"theorem","summary":"There is a natural K_\\infty-linear homeomorphism K_\\infty^[L:K] \\cong_K_\\infty L_\\infty.","labels":["NumberField.InfiniteAdeleRing.piEquiv"],"detail_key":"p5"},{"id":"n860","layer":"informal","project":"p5","title":"Using the isomorphisms K_v^[L:K] \\cong_K_v \\prod_w\\mid vL_w from Theorem~\\refNumberField.…","kind":"proof","summary":"Using the isomorphisms K_v^[L:K] \\cong_K_v \\prod_w\\mid vL_w from Theorem~\\refNumberField.Infini…","labels":[],"detail_key":"p5"},{"id":"n861","layer":"informal","project":"p5","title":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","kind":"theorem","summary":"L_\\infty has the K_\\infty-module topology.","labels":["NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT"],"detail_key":"p5"},{"id":"n862","layer":"informal","project":"p5","title":"Since L_\\infty is homeomorphic to a finite product of K_\\infty as a K_\\infty-vector space…","kind":"proof","summary":"Since L_\\infty is homeomorphic to a finite product of K_\\infty as a K_\\infty-vector space, it h…","labels":[],"detail_key":"p5"},{"id":"n863","layer":"informal","project":"p5","title":"NumberField.InfiniteAdeleRing.baseChangeAlgEquiv","kind":"theorem","summary":"There is a natural L-algebra isomorphism L \\otimes_K K_\\infty \\cong_L L_\\infty.","labels":["NumberField.InfiniteAdeleRing.baseChangeAlgEquiv"],"detail_key":"p5"},{"id":"n864","layer":"informal","project":"p5","title":"This follows from the following chain of isomorphisms: \\[ L \\otimes_K K_\\infty \\cong_L \\p…","kind":"proof","summary":"This follows from the following chain of isomorphisms: \\[ L \\otimes_K K_\\infty \\cong_L \\prod_v…","labels":[],"detail_key":"p5"},{"id":"n865","layer":"informal","project":"p5","title":"NumberField.InfiniteAdeleRing.baseChangeEquiv","kind":"theorem","summary":"If K\\to L is a ring homomorphism between two number fields then there is a natural isomorphism…","labels":["NumberField.InfiniteAdeleRing.baseChangeEquiv"],"detail_key":"p5"},{"id":"n866","layer":"informal","project":"p5","title":"Since both sides of the L-algebra isomorphism in~\\refNumberField.InfiniteAdeleRing.baseCh…","kind":"proof","summary":"Since both sides of the L-algebra isomorphism in~\\refNumberField.InfiniteAdeleRing.baseChangeAl…","labels":[],"detail_key":"p5"},{"id":"n867","layer":"informal","project":"p5","title":"NumberField.AdeleRing.baseChangeEquiv","kind":"theorem","summary":"If K\\to L is a ring homomorphism between two number fields then there is a natural isomorphism…","labels":["NumberField.AdeleRing.baseChangeEquiv"],"detail_key":"p5"},{"id":"n868","layer":"informal","project":"p5","title":"Follows from the previous results.","kind":"proof","summary":"Follows from the previous results.","labels":[],"detail_key":"p5"},{"id":"n869","layer":"informal","project":"p5","title":"NumberField.AdeleRing.baseChange_moduleTopology","kind":"theorem","summary":"If K\\to L is a ring homomorphism between two number fields then the topology on A_L is the A_K-…","labels":["NumberField.AdeleRing.baseChange_moduleTopology"],"detail_key":"p5"},{"id":"n870","layer":"informal","project":"p5","title":"Indeed A_L\\cong L\\otimes_KA_K is a homeomorphism, and the right hand side has the A_K-mod…","kind":"proof","summary":"Indeed A_L\\cong L\\otimes_KA_K is a homeomorphism, and the right hand side has the A_K-module to…","labels":[],"detail_key":"p5"},{"id":"n871","layer":"informal","project":"p5","title":"Rat.AdeleRing.zero_discrete","kind":"theorem","summary":"There's an open subset of A_Q whose intersection with Q is \\0\\.","labels":["Rat.AdeleRing.zero_discrete"],"detail_key":"p5"},{"id":"n872","layer":"informal","project":"p5","title":"Use \\prod_pZ_p\\times(-1,1). Any rational q in this set is a p-adic integer for all primes…","kind":"proof","summary":"Use \\prod_pZ_p\\times(-1,1). Any rational q in this set is a p-adic integer for all primes p and…","labels":[],"detail_key":"p5"},{"id":"n873","layer":"informal","project":"p5","title":"NumberField.AdeleRing.zero_discrete","kind":"theorem","summary":"There's an open subset of A_K whose intersection with K is \\0\\.","labels":["NumberField.AdeleRing.zero_discrete"],"detail_key":"p5"},{"id":"n874","layer":"informal","project":"p5","title":"By a previous result, we have A_K=K\\otimes_QA_Q. Choose a basis of K/Q; then K can be ide…","kind":"proof","summary":"By a previous result, we have A_K=K\\otimes_QA_Q. Choose a basis of K/Q; then K can be identifie…","labels":[],"detail_key":"p5"},{"id":"n875","layer":"informal","project":"p5","title":"NumberField.AdeleRing.discrete","kind":"theorem","summary":"The additive subgroup K of A_K is discrete.","labels":["NumberField.AdeleRing.discrete"],"detail_key":"p5"},{"id":"n876","layer":"informal","project":"p5","title":"If x\\in K and U is the open subset in the previous lemma, then it's easily checked that K…","kind":"proof","summary":"If x\\in K and U is the open subset in the previous lemma, then it's easily checked that K\\cap U…","labels":[],"detail_key":"p5"},{"id":"n877","layer":"informal","project":"p5","title":"Rat.AdeleRing.cocompact","kind":"theorem","summary":"The quotient A_Q/Q is compact.","labels":["Rat.AdeleRing.cocompact"],"detail_key":"p5"},{"id":"n878","layer":"informal","project":"p5","title":"The space \\prod_pZ_p\\times[0,1]\\subseteqA_Q is a product of compact spaces and is hence c…","kind":"proof","summary":"The space \\prod_pZ_p\\times[0,1]\\subseteqA_Q is a product of compact spaces and is hence compact…","labels":[],"detail_key":"p5"},{"id":"n879","layer":"informal","project":"p5","title":"NumberField.AdeleRing.cocompact","kind":"theorem","summary":"The quotient A_K/K is compact.","labels":["NumberField.AdeleRing.cocompact"],"detail_key":"p5"},{"id":"n880","layer":"informal","project":"p5","title":"We proceed as in the discreteness proof above, by reducing to Q. As before, choosing a Q-…","kind":"proof","summary":"We proceed as in the discreteness proof above, by reducing to Q. As before, choosing a Q-basis…","labels":[],"detail_key":"p5"},{"id":"n881","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar","kind":"definition","summary":"If A is a locally compact topological additive abelian group, if \\mu is a regular Haar measure…","labels":["MeasureTheory.addEquivAddHaarChar"],"detail_key":"p5"},{"id":"n882","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_eq","kind":"lemma","summary":"\\discussion508 d_A(\\phi) is independent of choice of regular Haar measure.","labels":["MeasureTheory.addEquivAddHaarChar_eq"],"detail_key":"p5"},{"id":"n883","layer":"informal","project":"p5","title":"If \\mu' is a second choice then \\mu'=\\lambda\\mu for some positive real \\lambda, and the \\…","kind":"proof","summary":"If \\mu' is a second choice then \\mu'=\\lambda\\mu for some positive real \\lambda, and the \\lambda…","labels":[],"detail_key":"p5"},{"id":"n884","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_smul_map","kind":"lemma","summary":"If \\mu is any regular Haar measure on A then d_A(\\phi)(\\phi_*\\mu) = \\mu.","labels":["MeasureTheory.addEquivAddHaarChar_smul_map"],"detail_key":"p5"},{"id":"n885","layer":"informal","project":"p5","title":"This is a restatement of the previous result.","kind":"proof","summary":"This is a restatement of the previous result.","labels":[],"detail_key":"p5"},{"id":"n886","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_comap","kind":"corollary","summary":"If \\mu is any regular Haar measure on A then d_A(\\phi)\\mu = \\phi^*\\mu.","labels":["MeasureTheory.addEquivAddHaarChar_comap"],"detail_key":"p5"},{"id":"n887","layer":"informal","project":"p5","title":"This follows from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map applied to the reg…","kind":"proof","summary":"This follows from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map applied to the regular H…","labels":[],"detail_key":"p5"},{"id":"n888","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_smul_preimage","kind":"lemma","summary":"\\discussion509 If X is a Borel set then \\mu(X)=d_A(\\phi)\\mu(\\phi^-1X).","labels":["MeasureTheory.addEquivAddHaarChar_smul_preimage"],"detail_key":"p5"},{"id":"n889","layer":"informal","project":"p5","title":"This follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map and th…","kind":"proof","summary":"This follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map and the defi…","labels":[],"detail_key":"p5"},{"id":"n890","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_smul_integral_map","kind":"lemma","summary":"\\discussion510 If f:A\\toR is a Borel measurable function then d_A(\\phi)\\int f(x)d\\phi_*\\mu(x)=\\…","labels":["MeasureTheory.addEquivAddHaarChar_smul_integral_map"],"detail_key":"p5"},{"id":"n891","layer":"informal","project":"p5","title":"This also follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map.","kind":"proof","summary":"This also follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_map.","labels":[],"detail_key":"p5"},{"id":"n892","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_smul_integral_comap","kind":"lemma","summary":"If f:A\\toR is a Borel measurable function then d_A(\\phi)\\int f(x)d\\mu(x)=\\int f(x)d\\phi^*\\mu(x).","labels":["MeasureTheory.addEquivAddHaarChar_smul_integral_comap"],"detail_key":"p5"},{"id":"n893","layer":"informal","project":"p5","title":"This is immediate from corollary~\\refMeasureTheory.addEquivAddHaarChar_comap.","kind":"proof","summary":"This is immediate from corollary~\\refMeasureTheory.addEquivAddHaarChar_comap.","labels":[],"detail_key":"p5"},{"id":"n894","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_refl","kind":"lemma","summary":"d_A(id)=1.","labels":["MeasureTheory.addEquivAddHaarChar_refl"],"detail_key":"p5"},{"id":"n895","layer":"informal","project":"p5","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p5"},{"id":"n896","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_trans","kind":"lemma","summary":"\\discussion511 d_A(\\phi\\circ\\psi)=d_A(\\phi)d_A(\\psi).","labels":["MeasureTheory.addEquivAddHaarChar_trans"],"detail_key":"p5"},{"id":"n897","layer":"informal","project":"p5","title":"Here's one way: it suffices to prove that d_A(\\phi\\circ\\psi)(\\phi\\circ\\psi)_*\\mu=d_A(\\phi…","kind":"proof","summary":"Here's one way: it suffices to prove that d_A(\\phi\\circ\\psi)(\\phi\\circ\\psi)_*\\mu=d_A(\\phi)d_A(\\…","labels":[],"detail_key":"p5"},{"id":"n898","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar","kind":"definition","summary":"We define \\delta_R(u) (or just \\delta(u) when the ring R is clear) to be d_R(\\ell_u).","labels":["MeasureTheory.ringHaarChar"],"detail_key":"p5"},{"id":"n899","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_mul_integral","kind":"lemma","summary":"\\discussion514 If f:R\\toR is a Borel measurable function and u\\in R^\\times then \\delta_R(u)\\int…","labels":["MeasureTheory.ringHaarChar_mul_integral"],"detail_key":"p5"},{"id":"n900","layer":"informal","project":"p5","title":"A short calculation using lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_integral_map.","kind":"proof","summary":"A short calculation using lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_integral_map.","labels":[],"detail_key":"p5"},{"id":"n901","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_mul_volume","kind":"lemma","summary":"\\discussion515 If X is a Borel subset of R and r\\in R^\\times then \\mu(rX)=\\delta_R(r)\\mu(X).","labels":["MeasureTheory.ringHaarChar_mul_volume"],"detail_key":"p5"},{"id":"n902","layer":"informal","project":"p5","title":"Immediate from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_preimage.","kind":"proof","summary":"Immediate from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_preimage.","labels":[],"detail_key":"p5"},{"id":"n903","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_continuous","kind":"corollary","summary":"\\discussion516 The function \\delta_R:R^\\times\\toR_>0 is continuous.","labels":["MeasureTheory.ringHaarChar_continuous"],"detail_key":"p5"},{"id":"n904","layer":"informal","project":"p5","title":"Fix a Haar measure \\mu on R and a continuous real-valued function f on R with compact sup…","kind":"proof","summary":"Fix a Haar measure \\mu on R and a continuous real-valued function f on R with compact support a…","labels":[],"detail_key":"p5"},{"id":"n905","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_real","kind":"lemma","summary":"If R=R then \\delta_R(u)=|u|.","labels":["MeasureTheory.ringHaarChar_real"],"detail_key":"p5"},{"id":"n906","layer":"informal","project":"p5","title":"Take \\mu to be Lebesgue measure and X=[0,1]. We have \\delta(u)=\\mu(uX). If u>0 then u[0,1…","kind":"proof","summary":"Take \\mu to be Lebesgue measure and X=[0,1]. We have \\delta(u)=\\mu(uX). If u>0 then u[0,1]=[0,u…","labels":[],"detail_key":"p5"},{"id":"n907","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_complex","kind":"lemma","summary":"If R=C then \\delta_R(u)=|u|^2.","labels":["MeasureTheory.ringHaarChar_complex"],"detail_key":"p5"},{"id":"n908","layer":"informal","project":"p5","title":"Multiplication by a positive real r sends a unit square to a square of area r^2=|r|^2. Mu…","kind":"proof","summary":"Multiplication by a positive real r sends a unit square to a square of area r^2=|r|^2. Multipli…","labels":[],"detail_key":"p5"},{"id":"n909","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_padic","kind":"lemma","summary":"If R=Q_p then \\delta_R(u)=|u|_p, the usual p-adic norm.","labels":["MeasureTheory.ringHaarChar_padic"],"detail_key":"p5"},{"id":"n910","layer":"informal","project":"p5","title":"Normalise Haar measure so that \\mu(Z_p)=1. If u is a p-adic unit then uZ_p=Z_p so multipl…","kind":"proof","summary":"Normalise Haar measure so that \\mu(Z_p)=1. If u is a p-adic unit then uZ_p=Z_p so multiplicatio…","labels":[],"detail_key":"p5"},{"id":"n911","layer":"informal","project":"p5","title":"If R is a finite extension of Q_p then \\delta_R(u) is the norm on R normalised in the fol…","kind":"remark","summary":"If R is a finite extension of Q_p then \\delta_R(u) is the norm on R normalised in the following…","labels":[],"detail_key":"p5"},{"id":"n912","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det","kind":"lemma","summary":"\\discussion517 Assume that there's an F-basis for V such that \\phi is a product of elementary a…","labels":["{MeasureTheory.addEquivAddHaarChar_eq_ringHaarChar_det"],"detail_key":"p5"},{"id":"n913","layer":"informal","project":"p5","title":"The proof is a generalization of \\hrefhttps://leanprover-community.github.io/mathlib4\\_do…","kind":"proof","summary":"The proof is a generalization of \\hrefhttps://leanprover-community.github.io/mathlib4\\_docs/Mat…","labels":[],"detail_key":"p5"},{"id":"n914","layer":"informal","project":"p5","title":"MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det","kind":"corollary","summary":"If u\\in R^\\times then \\delta_R(u)=\\delta_F(\\det(\\ell_u)).","labels":["MeasureTheory.algebra_ringHaarChar_eq_ringHaarChar_det"],"detail_key":"p5"},{"id":"n915","layer":"informal","project":"p5","title":"Follows immediately from the preceding lemma.","kind":"proof","summary":"Follows immediately from the preceding lemma.","labels":[],"detail_key":"p5"},{"id":"n916","layer":"informal","project":"p5","title":"IsSimpleRing.mulLeft_det_eq_mulRight_det","kind":"lemma","summary":"\\discussion518 Say B is a finite-dimensional central simple algebra over a field~k, and u\\in B^…","labels":["IsSimpleRing.mulLeft_det_eq_mulRight_det"],"detail_key":"p5"},{"id":"n917","layer":"informal","project":"p5","title":"Determinants are unchanged by base extension, so WLOG k is algebraically closed. Then it'…","kind":"proof","summary":"Determinants are unchanged by base extension, so WLOG k is algebraically closed. Then it's know…","labels":[],"detail_key":"p5"},{"id":"n918","layer":"informal","project":"p5","title":"IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight","kind":"corollary","summary":"If B is a central simple algebra over a locally compact field F, and if u\\in B^\\times, then d_B…","labels":["IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight"],"detail_key":"p5"},{"id":"n919","layer":"informal","project":"p5","title":"If \\ell_u and r_u denote left and right multiplication by u on B, then we have seen in le…","kind":"proof","summary":"If \\ell_u and r_u denote left and right multiplication by u on B, then we have seen in lemma~\\r…","labels":[],"detail_key":"p5"},{"id":"n920","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_prodCongr","kind":"lemma","summary":"\\discussion520 If (A,+) and (B,+) are locally compact topological abelian groups, and if \\phi:A…","labels":["MeasureTheory.addEquivAddHaarChar_prodCongr"],"detail_key":"p5"},{"id":"n921","layer":"informal","project":"p5","title":"We only need this result in the case where both A and B are second-countable, in which ca…","kind":"proof","summary":"We only need this result in the case where both A and B are second-countable, in which case \\tt…","labels":[],"detail_key":"p5"},{"id":"n922","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_piCongrRight","kind":"lemma","summary":"\\discussion521 If A_i are a finite collection of locally compact topological abelian groups, wi…","labels":["MeasureTheory.addEquivAddHaarChar_piCongrRight"],"detail_key":"p5"},{"id":"n923","layer":"informal","project":"p5","title":"Induction on the size of the finite set, using the previous lemma.","kind":"proof","summary":"Induction on the size of the finite set, using the previous lemma.","labels":[],"detail_key":"p5"},{"id":"n924","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_prod","kind":"lemma","summary":"If R and S are locally compact topological rings, then \\delta_R\\times S(r,s)=\\delta_R(r)\\times\\…","labels":["MeasureTheory.ringHaarChar_prod"],"detail_key":"p5"},{"id":"n925","layer":"informal","project":"p5","title":"Follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_prodCongr.","kind":"proof","summary":"Follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_prodCongr.","labels":[],"detail_key":"p5"},{"id":"n926","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_pi","kind":"lemma","summary":"If R_i are a finite collection of locally compact topological rings, and u_i\\in R_i^\\times then…","labels":["MeasureTheory.ringHaarChar_pi"],"detail_key":"p5"},{"id":"n927","layer":"informal","project":"p5","title":"Follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_piCongrRight.","kind":"proof","summary":"Follows immediately from lemma~\\refMeasureTheory.addEquivAddHaarChar_piCongrRight.","labels":[],"detail_key":"p5"},{"id":"n928","layer":"informal","project":"p5","title":"Topology.IsOpenEmbedding.isHaarMeasure_comap","kind":"lemma","summary":"\\discussion507 Let A and B be locally compact topological groups and let f:A\\to B be both a gro…","labels":["Topology.IsOpenEmbedding.isHaarMeasure_comap"],"detail_key":"p5"},{"id":"n929","layer":"informal","project":"p5","title":"Translation-invariance is easy, compact sets are finite because continuous image of compa…","kind":"proof","summary":"Translation-invariance is easy, compact sets are finite because continuous image of compact is…","labels":[],"detail_key":"p5"},{"id":"n930","layer":"informal","project":"p5","title":"Topology.IsOpenEmbedding.regular_comap","kind":"lemma","summary":"\\discussion513 The pullback of a regular Borel measure along an open embedding is a regular Bor…","labels":["Topology.IsOpenEmbedding.regular_comap"],"detail_key":"p5"},{"id":"n931","layer":"informal","project":"p5","title":"Again this is because the image of compact is compact and the image of open is open, so a…","kind":"proof","summary":"Again this is because the image of compact is compact and the image of open is open, so all the…","labels":[],"detail_key":"p5"},{"id":"n932","layer":"informal","project":"p5","title":"MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace","kind":"lemma","summary":"\\discussion532 Say A is a compact topological additive group and \\phi:A\\to A is an additive iso…","labels":["MeasureTheory.mulEquivHaarChar_eq_one_of_compactSpace"],"detail_key":"p5"},{"id":"n933","layer":"informal","project":"p5","title":"We have d_A(\\phi)\\mu(A)=\\mu(A) from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_prei…","kind":"proof","summary":"We have d_A(\\phi)\\mu(A)=\\mu(A) from lemma~\\refMeasureTheory.addEquivAddHaarChar_smul_preimage a…","labels":[],"detail_key":"p5"},{"id":"n934","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding","kind":"lemma","summary":"\\discussion551 If f:A\\to B is a group homomorphism and open embedding between locally compact t…","labels":["MeasureTheory.addEquivAddHaarChar_eq_addEquivAddHaarChar_of_isOpenEmbedding"],"detail_key":"p5"},{"id":"n935","layer":"informal","project":"p5","title":"Choose a regular Haar measure \\mu_B on B. We just saw in lemmas~\\refTopology.IsOpenEmbedd…","kind":"proof","summary":"Choose a regular Haar measure \\mu_B on B. We just saw in lemmas~\\refTopology.IsOpenEmbedding.is…","labels":[],"detail_key":"p5"},{"id":"n936","layer":"informal","project":"p5","title":"Continuous.restrictedProduct_congrRight","kind":"lemma","summary":"\\discussion531 If the A_i and B_i are topological spaces and the \\phi_i are continuous function…","labels":["Continuous.restrictedProduct_congrRight"],"detail_key":"p5"},{"id":"n937","layer":"informal","project":"p5","title":"We use the universal property \\tt RestrictedProduct.continuous\\_dom of the topology in ma…","kind":"proof","summary":"We use the universal property \\tt RestrictedProduct.continuous\\_dom of the topology in mathlib…","labels":[],"detail_key":"p5"},{"id":"n938","layer":"informal","project":"p5","title":"MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight","kind":"theorem","summary":"\\discussion552 With A, A_i, C_i, \\phi_i, \\phi defined as above, we have \\delta_A(\\phi)=\\prod_i\\…","labels":["MeasureTheory.addEquivAddHaarChar_restrictedProductCongrRight"],"detail_key":"p5"},{"id":"n939","layer":"informal","project":"p5","title":"In the Lean file we make the additional assumption that the index set over which we're ta…","kind":"remark","summary":"In the Lean file we make the additional assumption that the index set over which we're taking t…","labels":[],"detail_key":"p5"},{"id":"n940","layer":"informal","project":"p5","title":"Assume \\phi_i(C_i)=C_i for all i\\not \\in S, a finite set, and work in the open subgroup U…","kind":"proof","summary":"Assume \\phi_i(C_i)=C_i for all i\\not \\in S, a finite set, and work in the open subgroup U:=\\pro…","labels":[],"detail_key":"p5"},{"id":"n941","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_restrictedProduct","kind":"corollary","summary":"\\discussion554 If u=(u_i)_i\\in R^\\times then \\delta_R(u)=\\prod_i\\delta_R_i(u_i).","labels":["MeasureTheory.ringHaarChar_restrictedProduct"],"detail_key":"p5"},{"id":"n942","layer":"informal","project":"p5","title":"By definition of restricted product we have u_i\\in C_i for all but finitely many i. Note…","kind":"proof","summary":"By definition of restricted product we have u_i\\in C_i for all but finitely many i. Note also t…","labels":[],"detail_key":"p5"},{"id":"n943","layer":"informal","project":"p5","title":"IsModuleTopology.continuous_bilinear_of_finite_left","kind":"lemma","summary":"Say R and S are topological rings, and S is an R-algebra, finite as an R-module. Assume that th…","labels":["IsModuleTopology.continuous_bilinear_of_finite_left"],"detail_key":"p5"},{"id":"n944","layer":"informal","project":"p5","title":"Let i:R\\to S denote the structure map. First observe that S has the R-module topology so…","kind":"proof","summary":"Let i:R\\to S denote the structure map. First observe that S has the R-module topology so the R-…","labels":[],"detail_key":"p5"},{"id":"n945","layer":"informal","project":"p5","title":"NumberField.AdeleRing.ModuleBaseChangeContinuousLinearEquiv","kind":"corollary","summary":"If K is a number field and V is an K-module, then the natural isomorphism V\\otimes_KA_K=V\\otime…","labels":["NumberField.AdeleRing.ModuleBaseChangeContinuousLinearEquiv"],"detail_key":"p5"},{"id":"n946","layer":"informal","project":"p5","title":"Lemma~\\refIsModuleTopology.continuous_bilinear_of_finite_left tells us that V\\otimes_KA_K…","kind":"proof","summary":"Lemma~\\refIsModuleTopology.continuous_bilinear_of_finite_left tells us that V\\otimes_KA_K has t…","labels":[],"detail_key":"p5"},{"id":"n947","layer":"informal","project":"p5","title":"NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul","kind":"theorem","summary":"Let B be a finite-dimensional central simple K-algebra. Say u\\in B_A^\\times, and define \\ell_u…","labels":["NumberField.AdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul"],"detail_key":"p5"},{"id":"n948","layer":"informal","project":"p5","title":"We think of B_A as B\\otimes_KA_K. If u=(u_v) as v runs through the places of K then d_B_A…","kind":"proof","summary":"We think of B_A as B\\otimes_KA_K. If u=(u_v) as v runs through the places of K then d_B_A(\\ell_…","labels":[],"detail_key":"p5"},{"id":"n949","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_adeles_rat","kind":"lemma","summary":"If x\\inA_Q^\\times then \\delta_A_Q(x)=\\prod_v|x_v|_v.","labels":["MeasureTheory.ringHaarChar_adeles_rat"],"detail_key":"p5"},{"id":"n950","layer":"informal","project":"p5","title":"By theorem~\\refMeasureTheory.addEquivAddHaarChar_prodCongr we have \\delta_A_Q(x)=\\delta_A…","kind":"proof","summary":"By theorem~\\refMeasureTheory.addEquivAddHaarChar_prodCongr we have \\delta_A_Q(x)=\\delta_A_Q^\\in…","labels":[],"detail_key":"p5"},{"id":"n951","layer":"informal","project":"p5","title":"MeasureTheory.ringHaarChar_adeles_units_rat_eq_one","kind":"lemma","summary":"If x\\inQ^\\times\\subseteqA_Q^\\times then \\delta_A_Q(x)=1.","labels":["MeasureTheory.ringHaarChar_adeles_units_rat_eq_one"],"detail_key":"p5"},{"id":"n952","layer":"informal","project":"p5","title":"By lemma~\\refMeasureTheory.ringHaarChar_adeles_rat we have \\delta_A_Q(x)=\\prod_v|x|_v. Bu…","kind":"proof","summary":"By lemma~\\refMeasureTheory.ringHaarChar_adeles_rat we have \\delta_A_Q(x)=\\prod_v|x|_v. But the…","labels":[],"detail_key":"p5"},{"id":"n953","layer":"informal","project":"p5","title":"MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one","kind":"theorem","summary":"In the above situation (V a finite-dimensional Q-vector space, \\phi:V\\cong V is Q-linear, \\phi_…","labels":["MeasureTheory.addHaarScalarFactor_tensor_adeles_eq_one"],"detail_key":"p5"},{"id":"n954","layer":"informal","project":"p5","title":"The original blueprint proof of this was that \\phi_A : V_A\\to V_A could be written as a r…","kind":"proof","summary":"The original blueprint proof of this was that \\phi_A : V_A\\to V_A could be written as a restric…","labels":[],"detail_key":"p5"},{"id":"n955","layer":"informal","project":"p5","title":"NumberField.AdeleRing.units_mem_ringHaarCharacter_ker","kind":"corollary","summary":"If B is a finite-dimensional Q-algebra (for example a number field, or a quaternion algebra ove…","labels":["NumberField.AdeleRing.units_mem_ringHaarCharacter_ker"],"detail_key":"p5"},{"id":"n956","layer":"informal","project":"p5","title":"Follows immediately from the previous theorem.","kind":"proof","summary":"Follows immediately from the previous theorem.","labels":[],"detail_key":"p5"},{"id":"n957","layer":"informal","project":"p5","title":"NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one","kind":"corollary","summary":"If B is a finite-dimensional Q-algebra and if b\\in B^\\times then right multiplication by b does…","labels":["NumberField.AdeleRing.addEquivAddHaarChar_mulRight_unit_eq_one"],"detail_key":"p5"},{"id":"n958","layer":"informal","project":"p5","title":"Follows immediately from the previous theorem.","kind":"proof","summary":"Follows immediately from the previous theorem.","labels":[],"detail_key":"p5"},{"id":"n959","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.compact_quotient","kind":"theorem","summary":"If D is a division algebra then the quotient D^\\times\\backslash D_A^(1) with its quotient topol…","labels":["NumberField.AdeleRing.DivisionAlgebra.compact_quotient"],"detail_key":"p5"},{"id":"n960","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.existsE","kind":"lemma","summary":"There's a compact subset E of D_A with the property that for all x\\in D_A^(1), the obvious map…","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.existsE"],"detail_key":"p5"},{"id":"n961","layer":"informal","project":"p5","title":"We know that if we pick a Q-basis for D of size d then this identifies D with Q^d, D_A wi…","kind":"proof","summary":"We know that if we pick a Q-basis for D of size d then this identifies D with Q^d, D_A with A_Q…","labels":[],"detail_key":"p5"},{"id":"n962","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.E","kind":"definition","summary":"We let E denote any compact set satisfying the hypothesis of the previous lemma.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.E"],"detail_key":"p5"},{"id":"n963","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X","kind":"definition","summary":"Define X:=E-E:=\\e-f:e,f\\in E\\\\subseteq D_A.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.X"],"detail_key":"p5"},{"id":"n964","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y","kind":"definition","summary":"Define Y:=X.X:=\\xy:x,y\\in X\\\\subseteq D_A.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.Y"],"detail_key":"p5"},{"id":"n965","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact","kind":"lemma","summary":"X is a compact subset of D_A.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact"],"detail_key":"p5"},{"id":"n966","layer":"informal","project":"p5","title":"It's the continuous image of the compact set~E\\times E.","kind":"proof","summary":"It's the continuous image of the compact set~E\\times E.","labels":[],"detail_key":"p5"},{"id":"n967","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact","kind":"lemma","summary":"Y is a compact subset of D_A.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.Y_compact"],"detail_key":"p5"},{"id":"n968","layer":"informal","project":"p5","title":"It's the continuous image of the compact set~X\\times X.","kind":"proof","summary":"It's the continuous image of the compact set~X\\times X.","labels":[],"detail_key":"p5"},{"id":"n969","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel","kind":"lemma","summary":"If \\beta\\in D_A^(1) then \\beta X\\cap D^\\times\\not=\\emptyset.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel"],"detail_key":"p5"},{"id":"n970","layer":"informal","project":"p5","title":"Indeed by lemma~\\refNumberField.AdeleRing.DivisionAlgebra.Aux.existsE, the map \\beta E\\to…","kind":"proof","summary":"Indeed by lemma~\\refNumberField.AdeleRing.DivisionAlgebra.Aux.existsE, the map \\beta E\\to D\\bac…","labels":[],"detail_key":"p5"},{"id":"n971","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'","kind":"lemma","summary":"Similarly, if \\beta\\in D_A^(1) then X\\beta^-1\\cap D^\\times\\not=\\emptyset.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'"],"detail_key":"p5"},{"id":"n972","layer":"informal","project":"p5","title":"Indeed, \\beta^-1\\in D_A^(1), and so left multiplication by \\beta^-1 doesn't change Haar m…","kind":"proof","summary":"Indeed, \\beta^-1\\in D_A^(1), and so left multiplication by \\beta^-1 doesn't change Haar measure…","labels":[],"detail_key":"p5"},{"id":"n973","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.T","kind":"definition","summary":"Let T:=Y\\cap D^\\times.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.T"],"detail_key":"p5"},{"id":"n974","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","kind":"lemma","summary":"T is finite.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite"],"detail_key":"p5"},{"id":"n975","layer":"informal","project":"p5","title":"It suffices to prove that Y\\cap D is finite. But D\\subseteq D_A is a discrete additive su…","kind":"proof","summary":"It suffices to prove that Y\\cap D is finite. But D\\subseteq D_A is a discrete additive subgroup…","labels":[],"detail_key":"p5"},{"id":"n976","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.C","kind":"definition","summary":"Define C:= (T^-1.X) \\times X\\subset D_A\\times D_A.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.C"],"detail_key":"p5"},{"id":"n977","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","kind":"lemma","summary":"C is compact.","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact"],"detail_key":"p5"},{"id":"n978","layer":"informal","project":"p5","title":"X is compact and T is finite.","kind":"proof","summary":"X is compact and T is finite.","labels":[],"detail_key":"p5"},{"id":"n979","layer":"informal","project":"p5","title":"NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","kind":"lemma","summary":"For every \\beta\\in D_A^(1), there exists b\\in D^\\times and \\nu\\in D_A^(1) such that \\beta=b\\nu…","labels":["NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C"],"detail_key":"p5"},{"id":"n980","layer":"informal","project":"p5","title":"By lemma~\\refNumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel, \\beta X\\cap D^\\tim…","kind":"proof","summary":"By lemma~\\refNumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel, \\beta X\\cap D^\\times\\not…","labels":[],"detail_key":"p5"},{"id":"n981","layer":"informal","project":"p5","title":"Indeed, if M is the preimage of C under the inclusion D_A^(1) \\to D_A\\times D_A sending \\…","kind":"proof","summary":"Indeed, if M is the preimage of C under the inclusion D_A^(1) \\to D_A\\times D_A sending \\nu to…","labels":[],"detail_key":"p5"},{"id":"n982","layer":"informal","project":"p5","title":"NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact","kind":"theorem","summary":"D^\\times\\backslash(D\\otimes_KA_K^\\infty)^\\times is compact.","labels":["NumberField.FiniteAdeleRing.DivisionAlgebra.units_cocompact"],"detail_key":"p5"},{"id":"n983","layer":"informal","project":"p5","title":"There's a natural map \\alpha from D^\\times\\backslash D_A^(1) to D^\\times\\backslash (D\\oti…","kind":"proof","summary":"There's a natural map \\alpha from D^\\times\\backslash D_A^(1) to D^\\times\\backslash (D\\otimes_K…","labels":[],"detail_key":"p5"},{"id":"n984","layer":"informal","project":"p5","title":"In this generality the quotient might not be Hausdorff.","kind":"remark","summary":"In this generality the quotient might not be Hausdorff.","labels":[],"detail_key":"p5"},{"id":"n985","layer":"informal","project":"p5","title":"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset","kind":"theorem","summary":"If U is an open subgroup of (D\\otimes_K A_K^\\infty)^\\times then the double coset space D^\\times…","labels":["NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset"],"detail_key":"p5"},{"id":"n986","layer":"informal","project":"p5","title":"The double cosets give a disjoint open cover of (D\\otimes_K A_K^\\infty) which descends to…","kind":"proof","summary":"The double cosets give a disjoint open cover of (D\\otimes_K A_K^\\infty) which descends to a dis…","labels":[],"detail_key":"p5"},{"id":"n987","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","kind":"definition","summary":"The space of R-valued \\emphautomorphic forms for D^\\times is the set of functions f:D_A^\\infty^…","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm"],"detail_key":"p5"},{"id":"n988","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup","kind":"definition","summary":"Pointwise addition (f_1+f_2)(g):=f_1(g)+f_2(g) makes S^D(R) into an additive abelian group.","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup"],"detail_key":"p5"},{"id":"n989","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module","kind":"definition","summary":"If R is a commutative ring then pointwise scalar multiplication (r\\cdot f)(g):= r\\cdot(f(g)) ma…","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module"],"detail_key":"p5"},{"id":"n990","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction","kind":"definition","summary":"The group D_A^f^\\times acts on the additive abelian group S^D(R) by (g\\cdot f)(x)=f(xg).","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction"],"detail_key":"p5"},{"id":"n991","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel","kind":"definition","summary":"The quaternionic modular forms of level U, with notation S^D(U;R), are the U-invariants for the…","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel"],"detail_key":"p5"},{"id":"n992","layer":"informal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional","kind":"theorem","summary":"Let k be a field. Then the space S^D(U;k) is a finite-dimensional k-vector space.","labels":["TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional"],"detail_key":"p5"},{"id":"n993","layer":"informal","project":"p5","title":"The finite-dimensionality theorem is in fact an easy consequence of Fujisaki's lemma, pro…","kind":"proof","summary":"The finite-dimensionality theorem is in fact an easy consequence of Fujisaki's lemma, proved in…","labels":[],"detail_key":"p5"},{"id":"n994","layer":"informal","project":"p5","title":"AbstractHeckeOperator.HeckeOperatorToFun","kind":"definition","summary":"Assuming UgV is a finite union of cosets g_iV, we define [UgV]:A^V\\to A^U to be the map sending…","labels":["AbstractHeckeOperator.HeckeOperatorToFun"],"detail_key":"p5"},{"id":"n995","layer":"informal","project":"p5","title":"AbstractHeckeOperator.HeckeOperator","kind":"lemma","summary":"This function is well-defined (that is, independent of the choice of g_i), has image in A^U and…","labels":["AbstractHeckeOperator.HeckeOperator"],"detail_key":"p5"},{"id":"n996","layer":"informal","project":"p5","title":"Well-definedness is because if we change g_i to g'_i:=g_iv for some v\\in V then g_ia=g_i'…","kind":"proof","summary":"Well-definedness is because if we change g_i to g'_i:=g_iv for some v\\in V then g_ia=g_i'a beca…","labels":[],"detail_key":"p5"},{"id":"n997","layer":"informal","project":"p5","title":"AbstractHeckeOperator.comm","kind":"lemma","summary":"Say g,h\\in G and we have UgU=\\coprod_i g_iU and UhU=\\coprod_j h_j and we have g_ih_j=h_jg_i for…","labels":["AbstractHeckeOperator.comm"],"detail_key":"p5"},{"id":"n998","layer":"informal","project":"p5","title":"We have [UgU][UhU]a=\\sum_ig_i(\\sum_jh_ja)=\\sum_i,jg_ih_ja and [UhU][UgU]a=\\sum_jh_j\\sum_i…","kind":"proof","summary":"We have [UgU][UhU]a=\\sum_ig_i(\\sum_jh_ja)=\\sum_i,jg_ih_ja and [UhU][UgU]a=\\sum_jh_j\\sum_ig_ia=\\…","labels":[],"detail_key":"p5"},{"id":"n999","layer":"informal","project":"p5","title":"QuotientGroup.mk_image_finite_of_compact_of_open","kind":"lemma","summary":"\\discussion563 If U and V are compact subgroups of a topological group~G, if V is also open, an…","labels":["QuotientGroup.mk_image_finite_of_compact_of_open"],"detail_key":"p5"},{"id":"n1000","layer":"informal","project":"p5","title":"The subset UgV of G is a continuous image of the compact set U\\times V and is hence compa…","kind":"proof","summary":"The subset UgV of G is a continuous image of the compact set U\\times V and is hence compact, an…","labels":[],"detail_key":"p5"},{"id":"n1001","layer":"informal","project":"p5","title":"Homeomorph.restrictedProductProd","kind":"lemma","summary":"\\discussion568 If A_i is a family of topological spaces equipped with open subsets B_i, and if…","labels":["Homeomorph.restrictedProductProd"],"detail_key":"p5"},{"id":"n1002","layer":"informal","project":"p5","title":"This may well not be true if B_i and D_i are not open, because filtered colimits and bina…","kind":"remark","summary":"This may well not be true if B_i and D_i are not open, because filtered colimits and binary pro…","labels":[],"detail_key":"p5"},{"id":"n1003","layer":"informal","project":"p5","title":"We need to check continuity in both directions. The easy way is continuity of the map fro…","kind":"proof","summary":"We need to check continuity in both directions. The easy way is continuity of the map from the…","labels":[],"detail_key":"p5"},{"id":"n1004","layer":"informal","project":"p5","title":"Homeomorph.restrictedProductPi","kind":"corollary","summary":"\\discussion570 Restricted products (with respect to open subspaces) commute with finite product…","labels":["Homeomorph.restrictedProductPi"],"detail_key":"p5"},{"id":"n1005","layer":"informal","project":"p5","title":"Induction on the size of the finite set, using lemma~\\refHomeomorph.restrictedProductProd…","kind":"proof","summary":"Induction on the size of the finite set, using lemma~\\refHomeomorph.restrictedProductProd to ge…","labels":[],"detail_key":"p5"},{"id":"n1006","layer":"informal","project":"p5","title":"Homeomorph.restrictedProductMatrix","kind":"corollary","summary":"\\discussion571 If X_i are topological spaces and the Y_i are open subspaces, then the obvious m…","labels":["Homeomorph.restrictedProductMatrix"],"detail_key":"p5"},{"id":"n1007","layer":"informal","project":"p5","title":"Immediate from the previous corollary~\\refHomeomorph.restrictedProductPi.","kind":"proof","summary":"Immediate from the previous corollary~\\refHomeomorph.restrictedProductPi.","labels":[],"detail_key":"p5"},{"id":"n1008","layer":"informal","project":"p5","title":"Submonoid.units_isOpen","kind":"lemma","summary":"\\discussion587 If M is a topological monoid and U is an open submonoid, then the units U^\\times…","labels":["Submonoid.units_isOpen"],"detail_key":"p5"},{"id":"n1009","layer":"informal","project":"p5","title":"Note that M^\\times doesn't get the subspace topology from~M, it is embedded into M\\times…","kind":"remark","summary":"Note that M^\\times doesn't get the subspace topology from~M, it is embedded into M\\times M via…","labels":[],"detail_key":"p5"},{"id":"n1010","layer":"informal","project":"p5","title":"We have U\\times U is an open subset of M\\times M, and if we imagine M^\\times embedded in…","kind":"proof","summary":"We have U\\times U is an open subset of M\\times M, and if we imagine M^\\times embedded in M\\time…","labels":[],"detail_key":"p5"},{"id":"n1011","layer":"informal","project":"p5","title":"Submonoid.units_isCompact","kind":"lemma","summary":"\\discussion588 If M is a Hausdorff topological monoid and U is a compact submonoid, then the un…","labels":["Submonoid.units_isCompact"],"detail_key":"p5"},{"id":"n1012","layer":"informal","project":"p5","title":"Is Hausdorffness necessary?","kind":"remark","summary":"Is Hausdorffness necessary?","labels":[],"detail_key":"p5"},{"id":"n1013","layer":"informal","project":"p5","title":"First I claim that M^\\times embedded in M\\times M via g\\mapsto (g,g^-1) is a closed subse…","kind":"proof","summary":"First I claim that M^\\times embedded in M\\times M via g\\mapsto (g,g^-1) is a closed subset of M…","labels":[],"detail_key":"p5"},{"id":"n1014","layer":"informal","project":"p5","title":"ContinuousMulEquiv.piUnits","kind":"lemma","summary":"\\discussion581 If U_i are topological monoids then the canonical group isomorphism (\\prod_i U_i…","labels":["ContinuousMulEquiv.piUnits"],"detail_key":"p5"},{"id":"n1015","layer":"informal","project":"p5","title":"We prove that the maps in both directions are continuous. Let's start with the map from l…","kind":"proof","summary":"We prove that the maps in both directions are continuous. Let's start with the map from left to…","labels":[],"detail_key":"p5"},{"id":"n1016","layer":"informal","project":"p5","title":"ContinuousMulEquiv.restrictedProductUnits","kind":"theorem","summary":"\\discussion582 If M_i are a family of topological monoids equipped with open submonoids U_i, th…","labels":["ContinuousMulEquiv.restrictedProductUnits"],"detail_key":"p5"},{"id":"n1017","layer":"informal","project":"p5","title":"I don't know a clean way of showing that the map from left to right is continuous, so her…","kind":"proof","summary":"I don't know a clean way of showing that the map from left to right is continuous, so here is a…","labels":[],"detail_key":"p5"},{"id":"n1018","layer":"informal","project":"p5","title":"NumberField.isOpenAdicCompletionIntegers","kind":"lemma","summary":"O_v is an open subring of K_v.","labels":["NumberField.isOpenAdicCompletionIntegers"],"detail_key":"p5"},{"id":"n1019","layer":"informal","project":"p5","title":"Openness is already in mathlib.","kind":"proof","summary":"Openness is already in mathlib.","labels":[],"detail_key":"p5"},{"id":"n1020","layer":"informal","project":"p5","title":"M2.localFullLevel.isOpen","kind":"lemma","summary":"M_2(O_v) is an open subring of M_2(K_v).","labels":["M2.localFullLevel.isOpen"],"detail_key":"p5"},{"id":"n1021","layer":"informal","project":"p5","title":"Topologically M_2(O_v)\\cong O_v^4 as a subset of K_v^4 so this follows because a product…","kind":"proof","summary":"Topologically M_2(O_v)\\cong O_v^4 as a subset of K_v^4 so this follows because a product of com…","labels":[],"detail_key":"p5"},{"id":"n1022","layer":"informal","project":"p5","title":"M2.localFullLevel.isCompact","kind":"lemma","summary":"M_2(O_v) is a compact subring of M_2(K_v).","labels":["M2.localFullLevel.isCompact"],"detail_key":"p5"},{"id":"n1023","layer":"informal","project":"p5","title":"Topologically M_2(O_v)\\cong O_v^4 as a subset of K_v^4 so this follows because a product…","kind":"proof","summary":"Topologically M_2(O_v)\\cong O_v^4 as a subset of K_v^4 so this follows because a product of com…","labels":[],"detail_key":"p5"},{"id":"n1024","layer":"informal","project":"p5","title":"nolean-compactopen-GL2","kind":"lemma","summary":"GL_2(O_v) is a compact open subgroup of GL_2(K_v).","labels":["nolean-compactopen-GL2"],"detail_key":"p5"},{"id":"n1025","layer":"informal","project":"p5","title":"K_v is known to be Hausdorff, so M_2(K_v) is Hausdorff and the results follow from lemmas…","kind":"proof","summary":"K_v is known to be Hausdorff, so M_2(K_v) is Hausdorff and the results follow from lemmas~\\refS…","labels":[],"detail_key":"p5"},{"id":"n1026","layer":"informal","project":"p5","title":"nolean-compactopen-U1p","kind":"lemma","summary":"U_v is a compact open subgroup of GL_2(K_v).","labels":["nolean-compactopen-U1p"],"detail_key":"p5"},{"id":"n1027","layer":"informal","project":"p5","title":"\\Gamma_v is a group and hence its preimage U_v is a subgroup of the monoid M_2(K_v). It i…","kind":"proof","summary":"\\Gamma_v is a group and hence its preimage U_v is a subgroup of the monoid M_2(K_v). It is comp…","labels":[],"detail_key":"p5"},{"id":"n1028","layer":"informal","project":"p5","title":"bijOn_unipotent_mul_diagU1_U1diagU1","kind":"lemma","summary":"The double coset space UgU is the disjoint union of g_tU as t ranges through O_v/\\alphaO_v and…","labels":["bijOn_unipotent_mul_diagU1_U1diagU1"],"detail_key":"p5"},{"id":"n1029","layer":"informal","project":"p5","title":"We first manipulate the statement into a statement about finite groups. We have UgU=\\copr…","kind":"proof","summary":"We first manipulate the statement into a statement about finite groups. We have UgU=\\coprod_t g…","labels":[],"detail_key":"p5"},{"id":"n1030","layer":"informal","project":"p5","title":"GL2.restrictedProduct","kind":"theorem","summary":"G is isomorphic and homeomorphic to the restricted product of GL_2(K_v) with respect to the com…","labels":["GL2.restrictedProduct"],"detail_key":"p5"},{"id":"n1031","layer":"informal","project":"p5","title":"This follows from lemma~\\refContinuousMulEquiv.restrictedProductUnits and lemma~\\refHomeo…","kind":"proof","summary":"This follows from lemma~\\refContinuousMulEquiv.restrictedProductUnits and lemma~\\refHomeomorph.…","labels":[],"detail_key":"p5"},{"id":"n1032","layer":"informal","project":"p5","title":"nolean-hecke-algebra-commutative-noetherian","kind":"theorem","summary":"Say~R is a Noetherian ring. Then the subalgebra of the R-linear endomorphisms of A^U generated…","labels":["nolean-hecke-algebra-commutative-noetherian"],"detail_key":"p5"},{"id":"n1033","layer":"informal","project":"p5","title":"nolean-product-of-U-alpha","kind":"lemma","summary":"If v\\in S and 1&*\\\\0&1 \\subseteq\\Gamma_v\\subseteq *&*\\\\0&* then U_\\alpha,vU_\\beta,v=U_\\alpha\\be…","labels":["nolean-product-of-U-alpha"],"detail_key":"p5"},{"id":"n1034","layer":"informal","project":"p5","title":"Follows easily from the explicit double coset decomposition proved above.","kind":"proof","summary":"Follows easily from the explicit double coset decomposition proved above.","labels":[],"detail_key":"p5"},{"id":"n1035","layer":"informal","project":"p5","title":"maximal_unramified_extension_of_p-adic_field","kind":"theorem","summary":"\\notready The maximal unramified extension K^un in a given algebraic closure of K is Galois","labels":["maximal_unramified_extension_of_p-adic_field"],"detail_key":"p5"},{"id":"n1036","layer":"informal","project":"p5","title":"local_Weil_group","kind":"definition","summary":"\\notready The topological group described above is called the \\emphWeil group of K.","labels":["local_Weil_group"],"detail_key":"p5"},{"id":"n1037","layer":"informal","project":"p5","title":"local_class_field_theory","kind":"theorem","summary":"\\notready If K is a finite extension of Q_p then there are two ``canonical'' isomorphisms of to…","labels":["local_class_field_theory"],"detail_key":"p5"},{"id":"n1038","layer":"informal","project":"p5","title":"This is the main theorem of local class field theory; see for example the relevant articl…","kind":"proof","summary":"This is the main theorem of local class field theory; see for example the relevant articles in~…","labels":[],"detail_key":"p5"},{"id":"n1039","layer":"informal","project":"p5","title":"local_galois_coh_finite","kind":"theorem","summary":"\\notready If M is finite then the cohomology groups H^i(G_K,M) all finite.","labels":["local_galois_coh_finite"],"detail_key":"p5"},{"id":"n1040","layer":"informal","project":"p5","title":"This is Proposition~14 in section~5.2 of~\\citeserre-galcoh.","kind":"proof","summary":"This is Proposition~14 in section~5.2 of~\\citeserre-galcoh.","labels":[],"detail_key":"p5"},{"id":"n1041","layer":"informal","project":"p5","title":"\"the dimension is 2\"","kind":"theorem","summary":"[\"the dimension is 2\"] \\notready If M is torsion then H^i(G_K,M)=0 if i>2.","labels":["local_galois_coh_dim_two"],"detail_key":"p5"},{"id":"n1042","layer":"informal","project":"p5","title":"This follows from Proposition~15 in~section 5.3 of~\\citeserre-galcoh.","kind":"proof","summary":"This follows from Proposition~15 in~section 5.3 of~\\citeserre-galcoh.","labels":[],"detail_key":"p5"},{"id":"n1043","layer":"informal","project":"p5","title":"\"top degree\"","kind":"theorem","summary":"[\"top degree\"] \\notready H^2(G_K,\\mu_n) is ``canonically'' isomorphic to Z/nZ.","labels":["local_galois_coh_top_degree"],"detail_key":"p5"},{"id":"n1044","layer":"informal","project":"p5","title":"This is also included in Lemma 2 of section 5.2 of \\citeserre-galcoh (Serre just writes t…","kind":"proof","summary":"This is also included in Lemma 2 of section 5.2 of \\citeserre-galcoh (Serre just writes that th…","labels":[],"detail_key":"p5"},{"id":"n1045","layer":"informal","project":"p5","title":"\\notready There is a ``canonical'' isomorphism H^2(K,\\mu_\\infty)=Q/Z.","kind":"theorem","summary":"\\notready There is a ``canonical'' isomorphism H^2(K,\\mu_\\infty)=Q/Z.","labels":[],"detail_key":"p5"},{"id":"n1046","layer":"informal","project":"p5","title":"\\notready This is in Theorem II.5.2 in~\\citeserre-galcoh.","kind":"proof","summary":"\\notready This is in Theorem II.5.2 in~\\citeserre-galcoh.","labels":[],"detail_key":"p5"},{"id":"n1047","layer":"informal","project":"p5","title":"\"Poincar\\'e duality\"","kind":"theorem","summary":"[\"Poincar\\'e duality\"] \\notready If \\mu=\\bigcup_n\\geq1\\mu_n and M':=\\Hom(M,\\mu) is the dual of…","labels":["local_galois_coh_poincare"],"detail_key":"p5"},{"id":"n1048","layer":"informal","project":"p5","title":"\\notready This is Theorem 2 in section 5.2 in \\citeserre-galcoh. Note again the dubious (…","kind":"proof","summary":"\\notready This is Theorem 2 in section 5.2 in \\citeserre-galcoh. Note again the dubious (as far…","labels":[],"detail_key":"p5"},{"id":"n1049","layer":"informal","project":"p5","title":"\"Euler-Poincar\\'e characteristic\"","kind":"theorem","summary":"[\"Euler-Poincar\\'e characteristic\"] \\notready If h^i(M) denotes the order of H^i(G_K,M) then h^…","labels":["local_galois_coh_euler_poincare"],"detail_key":"p5"},{"id":"n1050","layer":"informal","project":"p5","title":"global_class_field_theory","kind":"theorem","summary":"","labels":["global_class_field_theory"],"detail_key":"p5"},{"id":"n1051","layer":"informal","project":"p5","title":"\\notready This is the main theorem of global class field theory; see for example Tate's a…","kind":"proof","summary":"\\notready This is the main theorem of global class field theory; see for example Tate's article…","labels":[],"detail_key":"p5"},{"id":"n1052","layer":"informal","project":"p5","title":"Skinner_Wiles_CFT_trick","kind":"theorem","summary":"\\notready Let S be a finite set of places of a number field K . For each v \\in S let L_v/K_v be…","labels":["Skinner_Wiles_CFT_trick"],"detail_key":"p5"},{"id":"n1053","layer":"informal","project":"p5","title":"topology_on_affine_variety_points","kind":"definition","summary":"If X is an affine scheme of finite type over K, and if R is a K-algebra which is also a topolog…","labels":["topology_on_affine_variety_points"],"detail_key":"p5"},{"id":"n1054","layer":"informal","project":"p5","title":"topology_on_affine_variety_computation","kind":"theorem","summary":"If X is as above and X\\toA^n_K is a closed immersion, then the induced map from X(R) with its t…","labels":["topology_on_affine_variety_computation"],"detail_key":"p5"},{"id":"n1055","layer":"informal","project":"p5","title":"See \\hrefhttps://math.stanford.edu/~conrad/papers/adelictop.pdfConrad's notes.","kind":"proof","summary":"See \\hrefhttps://math.stanford.edu/~conrad/papers/adelictop.pdfConrad's notes.","labels":[],"detail_key":"p5"},{"id":"n1056","layer":"informal","project":"p5","title":"manifold_on_algebraic_variety_points","kind":"definition","summary":"\\notready Let K be a field equipped with an isomorphism to the reals, complexes, or a finite ex…","labels":["manifold_on_algebraic_variety_points"],"detail_key":"p5"},{"id":"n1057","layer":"informal","project":"p5","title":"Probably this is fine for a broader class of fields K.","kind":"remark","summary":"Probably this is fine for a broader class of fields K.","labels":[],"detail_key":"p5"},{"id":"n1058","layer":"informal","project":"p5","title":"manifold_on_algebraic_variety_computation","kind":"theorem","summary":"\\notready If X is as in the previous definition and X\\toA^n_K is a closed immersion, then the i…","labels":["manifold_on_algebraic_variety_computation"],"detail_key":"p5"},{"id":"n1059","layer":"informal","project":"p5","title":"I'm assuming this is standard, if true.","kind":"proof","summary":"I'm assuming this is standard, if true.","labels":[],"detail_key":"p5"},{"id":"n1060","layer":"informal","project":"p5","title":"lie_group_from_algebraic_group","kind":"corollary","summary":"\\notready If G is an affine algebraic group of finite type over K=R or C then G(K) is naturally…","labels":["lie_group_from_algebraic_group"],"detail_key":"p5"},{"id":"n1061","layer":"informal","project":"p5","title":"The coro","kind":"remark","summary":"The coro","labels":[],"detail_key":"p5"},{"id":"n1062","layer":"informal","project":"p5","title":"connected_reductive_group","kind":"definition","summary":"\\notready An affine algebraic group~G of finite type over a field~k is said to be \\emphconnecte…","labels":["connected_reductive_group"],"detail_key":"p5"},{"id":"n1063","layer":"informal","project":"p5","title":"slowly_increasing","kind":"definition","summary":"\\notready A function f : G(N_\\infty)\\toC is \\emphslowly-increasing if there exists some C>0 and…","labels":["slowly_increasing"],"detail_key":"p5"},{"id":"n1064","layer":"informal","project":"p5","title":"slowly_increasing_well_defined","kind":"theorem","summary":"\\notready This is independent of the choice of \\rho as above.","labels":["slowly_increasing_well_defined"],"detail_key":"p5"},{"id":"n1065","layer":"informal","project":"p5","title":"Follows from the above.","kind":"proof","summary":"Follows from the above.","labels":[],"detail_key":"p5"},{"id":"n1066","layer":"informal","project":"p5","title":"automorphic_form","kind":"definition","summary":"\\notready An \\emphautomorphic form is a function \\phi:G(A_N)\\toC satisfying the following condi…","labels":["automorphic_form"],"detail_key":"p5"},{"id":"n1067","layer":"informal","project":"p5","title":"cuspidal_automorphic_form","kind":"definition","summary":"","labels":["cuspidal_automorphic_form"],"detail_key":"p5"},{"id":"n1068","layer":"informal","project":"p5","title":"automorphic_form_actions","kind":"definition","summary":"\\notready The group G(A_N) acts on itself on the right, and this induces a left action of its s…","labels":["automorphic_form_actions"],"detail_key":"p5"},{"id":"n1069","layer":"informal","project":"p5","title":"cuspidal_automorphic_form_decomposition","kind":"theorem","summary":"The cusp forms decompose as a (typically infinite) direct sum of irreducible (G(A_N^f)\\times U_…","labels":["cuspidal_automorphic_form_decomposition"],"detail_key":"p5"},{"id":"n1070","layer":"informal","project":"p5","title":"cuspidal_automorphic_representation","kind":"definition","summary":"\\notready A cuspidal automorphic representation is an irreducible (G(A_N^f)\\times U_\\infty,\\mat…","labels":["cuspidal_automorphic_representation"],"detail_key":"p5"},{"id":"n1071","layer":"informal","project":"p5","title":"automorphic_representation","kind":"definition","summary":"\\notready An automorphic representation is an irreducible (G(A_N^f)\\times U_\\infty,\\mathfrakg)-…","labels":["automorphic_representation"],"detail_key":"p5"},{"id":"n1072","layer":"informal","project":"p5","title":"automorphic_representation_local_decomposition","kind":"theorem","summary":"","labels":["automorphic_representation_local_decomposition"],"detail_key":"p5"},{"id":"n1073","layer":"informal","project":"p5","title":"See Flath's article in~\\citecorvallis2.","kind":"proof","summary":"See Flath's article in~\\citecorvallis2.","labels":[],"detail_key":"p5"},{"id":"n1074","layer":"informal","project":"p5","title":"compatible_family","kind":"definition","summary":"\\discussion23 Let N be a number field. A \\emphcompatible family of d-dimensional Galois represe…","labels":["compatible_family"],"detail_key":"p5"},{"id":"n1075","layer":"informal","project":"p5","title":"Galois_representation_from_automorphic_representation_on_GL_2_form","kind":"theorem","summary":"","labels":["Galois_representation_from_automorphic_representation_on_GL_2_form"],"detail_key":"p5"},{"id":"n1076","layer":"informal","project":"p5","title":"Shimura_varieties","kind":"definition","summary":"\\notready We need the definition of (the canonical model over F of) the Shimura curve attached…","labels":["Shimura_varieties"],"detail_key":"p5"},{"id":"n1077","layer":"informal","project":"p5","title":"moret-bailly","kind":"theorem","summary":"\\notready Let K^\\avoid/K be a Galois extension of number fields. Suppose also that S is a finit…","labels":["moret-bailly"],"detail_key":"p5"},{"id":"n1078","layer":"formal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","kind":"inductive","summary":"(F : Type u_1) → [inst : Field F] → [inst_1 : NumberField F] → (D : Type u_2) → [inst_2 : Ring…","labels":[],"detail_key":"p5","name":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm","module":"FLT.AutomorphicForm.QuaternionAlgebra.Basic"},{"id":"n1079","layer":"formal","project":"p5","title":"TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.LevelStruct.form","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst_1 : NumberField F] → (D : Type u_2) → [inst_2 : Ring 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(IsOpen…","labels":[],"detail_key":"p5","name":"NumberField.AdeleRing.discrete","module":"FLT.NumberField.AdeleRing"},{"id":"n1205","layer":"formal","project":"p5","title":"NumberField.AdeleRing.instIsModuleTopology","kind":"theorem","summary":"∀ K : Type u_3 L : Type u_4 [inst : Field K] [inst_1 : Field L] [inst_2 : NumberField K] [inst_…","labels":[],"detail_key":"p5","name":"NumberField.AdeleRing.instIsModuleTopology","module":"FLT.NumberField.AdeleRing"},{"id":"n1206","layer":"formal","project":"p5","title":"NumberField.AdeleRing.zero_discrete","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K], Exists fun U => And (IsOpen U) (Eq…","labels":[],"detail_key":"p5","name":"NumberField.AdeleRing.zero_discrete","module":"FLT.NumberField.AdeleRing"},{"id":"n1207","layer":"formal","project":"p5","title":"Rat.AdeleRing.cocompact","kind":"theorem","summary":"CompactSpace (HasQuotient.Quotient (NumberField.AdeleRing (NumberField.RingOfIntegers Rat) Rat)…","labels":[],"detail_key":"p5","name":"Rat.AdeleRing.cocompact","module":"FLT.NumberField.AdeleRing"},{"id":"n1208","layer":"formal","project":"p5","title":"Rat.AdeleRing.zero_discrete","kind":"theorem","summary":"Exists fun U => And (IsOpen U) (Eq (Set.preimage (⇑(algebraMap Rat (NumberField.AdeleRing (Numb…","labels":[],"detail_key":"p5","name":"Rat.AdeleRing.zero_discrete","module":"FLT.NumberField.AdeleRing"},{"id":"n1209","layer":"formal","project":"p5","title":"NumberField.instCompactSpaceAdicCompletionIntegers","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] (v : IsDedekindDomain.HeightOneSpect…","labels":[],"detail_key":"p5","name":"NumberField.instCompactSpaceAdicCompletionIntegers","module":"FLT.NumberField.Completion.Finite"},{"id":"n1210","layer":"formal","project":"p5","title":"NumberField.isOpenAdicCompletionIntegers","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] (v : IsDedekindDomain.HeightOneSpect…","labels":[],"detail_key":"p5","name":"NumberField.isOpenAdicCompletionIntegers","module":"FLT.NumberField.Completion.Finite"},{"id":"n1211","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange","kind":"def","summary":"K : Type u_1 → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L]…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.baseChange","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1212","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChangeEquiv","kind":"def","summary":"K : Type u_1 → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L]…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.baseChangeEquiv","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1213","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange_injective","kind":"theorem","summary":"∀ K : Type u_1 (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (v : N…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.baseChange_injective","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1214","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.baseChange_surjective","kind":"theorem","summary":"∀ K : Type u_1 (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (v : N…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.baseChange_surjective","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1215","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] (p : NumberField.InfinitePlace K → Prop) [NumberField K], Den…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.denseRange_algebraMap_subtype_pi","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1216","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","kind":"theorem","summary":"∀ K : Type u_1 (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (v : N…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.finrank_pi_eq_finrank_tensorProduct","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1217","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","kind":"theorem","summary":"∀ K : Type u_1 (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (v : N…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.instIsModuleTopologyValEqComapAlgebraMap_fLT","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1218","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.piEquiv","kind":"def","summary":"K : Type u_1 → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L]…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.piEquiv","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1219","layer":"formal","project":"p5","title":"NumberField.InfinitePlace.Completion.piExtension","kind":"def","summary":"K : Type u_1 → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L]…","labels":[],"detail_key":"p5","name":"NumberField.InfinitePlace.Completion.piExtension","module":"FLT.NumberField.Completion.Infinite"},{"id":"n1220","layer":"formal","project":"p5","title":"NumberField.InfiniteAdeleRing.baseChangeAlgEquiv","kind":"def","summary":"(K : Type u_1) → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L…","labels":[],"detail_key":"p5","name":"NumberField.InfiniteAdeleRing.baseChangeAlgEquiv","module":"FLT.NumberField.InfiniteAdeleRing"},{"id":"n1221","layer":"formal","project":"p5","title":"NumberField.InfiniteAdeleRing.baseChangeEquiv","kind":"def","summary":"(K : Type u_1) → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L…","labels":[],"detail_key":"p5","name":"NumberField.InfiniteAdeleRing.baseChangeEquiv","module":"FLT.NumberField.InfiniteAdeleRing"},{"id":"n1222","layer":"formal","project":"p5","title":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","kind":"theorem","summary":"∀ (K : Type u_1) (L : Type u_2) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] [Num…","labels":[],"detail_key":"p5","name":"NumberField.InfiniteAdeleRing.instIsModuleTopology_fLT","module":"FLT.NumberField.InfiniteAdeleRing"},{"id":"n1223","layer":"formal","project":"p5","title":"NumberField.InfiniteAdeleRing.piEquiv","kind":"def","summary":"(K : Type u_1) → (L : Type u_2) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algebra K L…","labels":[],"detail_key":"p5","name":"NumberField.InfiniteAdeleRing.piEquiv","module":"FLT.NumberField.InfiniteAdeleRing"},{"id":"n1224","layer":"formal","project":"p5","title":"FLT.Bosses.B3_proof","kind":"theorem","summary":"FLT.Bosses.B3","labels":[],"detail_key":"p5","name":"FLT.Bosses.B3_proof","module":"FLT.Proof"},{"id":"n1225","layer":"formal","project":"p5","title":"FLT.Bosses.B4_proof","kind":"theorem","summary":"FLT.Bosses.B4","labels":[],"detail_key":"p5","name":"FLT.Bosses.B4_proof","module":"FLT.Proof"},{"id":"n1226","layer":"formal","project":"p5","title":"flt","kind":"theorem","summary":"FermatLastTheorem","labels":[],"detail_key":"p5","name":"flt","module":"FLT.Proof"},{"id":"n1227","layer":"formal","project":"p5","title":"IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst_1 : NumberField F] → ContinuousMulEquiv (Matrix.General…","labels":[],"detail_key":"p5","name":"IsDedekindDomain.FiniteAdeleRing.GL2.restrictedProduct","module":"FLT.QuaternionAlgebra.NumberField"},{"id":"n1228","layer":"formal","project":"p5","title":"IsDedekindDomain.M2.localFullLevel.isCompact","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : NumberField F] (v : IsDedekindDomain.HeightOneSpectru…","labels":[],"detail_key":"p5","name":"IsDedekindDomain.M2.localFullLevel.isCompact","module":"FLT.QuaternionAlgebra.NumberField"},{"id":"n1229","layer":"formal","project":"p5","title":"IsDedekindDomain.M2.localFullLevel.isOpen","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : NumberField F] (v : IsDedekindDomain.HeightOneSpectru…","labels":[],"detail_key":"p5","name":"IsDedekindDomain.M2.localFullLevel.isOpen","module":"FLT.QuaternionAlgebra.NumberField"},{"id":"n1230","layer":"informal","project":"p6","title":"Set aRb","kind":"definition","summary":"[Set aRb] For a, b \\in R, denote by aRb the set \\arb| r \\in R\\.","labels":["def:aRb"],"detail_key":"p6"},{"id":"n1231","layer":"informal","project":"p6","title":"Left/Right Ideal","kind":"definition","summary":"[Left/Right Ideal] A left/right ideal I of a ring R is an additive subgroup of R such that rI \\…","labels":["def:left_right_ideal"],"detail_key":"p6"},{"id":"n1232","layer":"informal","project":"p6","title":"Two-sided Ideal","kind":"definition","summary":"[Two-sided Ideal] A two-sided ideal is a subset of R that is both left and right ideal of R.","labels":["def:two_sided_ideal"],"detail_key":"p6"},{"id":"n1233","layer":"informal","project":"p6","title":"Product of Ideals","kind":"definition","summary":"[Product of Ideals] A product of (left/right/two-sided) ideals I and J is the ideal IJ generate…","labels":["def:product_of_ideals"],"detail_key":"p6"},{"id":"n1234","layer":"informal","project":"p6","title":"Prime Ring","kind":"definition","summary":"[Prime Ring] A ring is prime if we have I = 0 or J = 0 whenever IJ = 0 for some left ideals I a…","labels":["def:is_prime_ring"],"detail_key":"p6"},{"id":"n1235","layer":"informal","project":"p6","title":"thm:prime_ring_equiv","kind":"theorem","summary":"A ring is prime if and only if for all a, b \\in R, aRb = 0 implies a = 0 or b = 0.","labels":["thm:prime_ring_equiv"],"detail_key":"p6"},{"id":"n1236","layer":"informal","project":"p6","title":"(\\Rightarrow) Suppose aRb = 0. Then (Ra)(Rb) = 0, thus by primality Ra = 0 or Rb = 0. In…","kind":"proof","summary":"(\\Rightarrow) Suppose aRb = 0. Then (Ra)(Rb) = 0, thus by primality Ra = 0 or Rb = 0. In the fo…","labels":[],"detail_key":"p6"},{"id":"n1237","layer":"informal","project":"p6","title":"thm:prime_ring_equiv'","kind":"theorem","summary":"A ring is prime if and only if for all two-sided ideals I and J, IJ = 0 implies I = 0 or J = 0.","labels":["thm:prime_ring_equiv'"],"detail_key":"p6"},{"id":"n1238","layer":"informal","project":"p6","title":"(\\Rightarrow) Two-sided ideals are left ideals, so the result follows directly from defin…","kind":"proof","summary":"(\\Rightarrow) Two-sided ideals are left ideals, so the result follows directly from definiton.…","labels":[],"detail_key":"p6"},{"id":"n1239","layer":"informal","project":"p6","title":"Simple Ring","kind":"definition","summary":"[Simple Ring] A ring is simple if it has no nontrivial two-sided ideals.","labels":["def:IsSimpleRing"],"detail_key":"p6"},{"id":"n1240","layer":"informal","project":"p6","title":"thm:simple_ring_is_prime","kind":"theorem","summary":"A simple ring is prime.","labels":["thm:simple_ring_is_prime"],"detail_key":"p6"},{"id":"n1241","layer":"informal","project":"p6","title":"Suppose IJ = 0. If both I and J are nonzero, they must be equal to R by simplicity. But R…","kind":"proof","summary":"Suppose IJ = 0. If both I and J are nonzero, they must be equal to R by simplicity. But RR = R…","labels":[],"detail_key":"p6"},{"id":"n1242","layer":"informal","project":"p6","title":"Orthogonal Elements","kind":"definition","summary":"[Orthogonal Elements] Two elements a, b \\in R are orthogonal if ab = ba = 0.","labels":["def:IsOrthogonal"],"detail_key":"p6"},{"id":"n1243","layer":"informal","project":"p6","title":"thm:one_sub_e_larger_span_on_sub_e_sub_f","kind":"theorem","summary":"If e, f \\in R are orthogonal idempotents and f \\neq 0, then the left ideal generated by 1 - e -…","labels":["thm:one_sub_e_larger_span_on_sub_e_sub_f"],"detail_key":"p6"},{"id":"n1244","layer":"informal","project":"p6","title":"Note that (1 - e - f)(1 - e) = 1 - e - f, and hence x(1 - e - f) = x(1 - e - f)(1 - e) \\i…","kind":"proof","summary":"Note that (1 - e - f)(1 - e) = 1 - e - f, and hence x(1 - e - f) = x(1 - e - f)(1 - e) \\in R(1…","labels":[],"detail_key":"p6"},{"id":"n1245","layer":"informal","project":"p6","title":"Corner Ring Set","kind":"definition","summary":"[Corner Ring Set] The set of the corner ring is eRe.","labels":["def:corner_ring"],"detail_key":"p6"},{"id":"n1246","layer":"informal","project":"p6","title":"thm:characterization_of_corner_elements","kind":"theorem","summary":"An element x is in the set e R f if and only if x = e x f.","labels":["thm:characterization_of_corner_elements"],"detail_key":"p6"},{"id":"n1247","layer":"informal","project":"p6","title":"(\\Rightarrow) Suppose x \\in e R f. Then x = e y f for some y \\in R. But then e x f = e e…","kind":"proof","summary":"(\\Rightarrow) Suppose x \\in e R f. Then x = e y f for some y \\in R. But then e x f = e e y f f…","labels":[],"detail_key":"p6"},{"id":"n1248","layer":"informal","project":"p6","title":"thm:characterization_of_corner_ring_elements","kind":"theorem","summary":"An element x of R is in the corner ring if and only if x = e x e.","labels":["thm:characterization_of_corner_ring_elements"],"detail_key":"p6"},{"id":"n1249","layer":"informal","project":"p6","title":"Application of theorem \\refthm:characterization_of_corner_elements.","kind":"proof","summary":"Application of theorem \\refthm:characterization_of_corner_elements.","labels":[],"detail_key":"p6"},{"id":"n1250","layer":"informal","project":"p6","title":"thm:characterization_of_corner_ring_elements'","kind":"theorem","summary":"An element x of the corner ring is of the form e y e for some y \\in R.","labels":["thm:characterization_of_corner_ring_elements'"],"detail_key":"p6"},{"id":"n1251","layer":"informal","project":"p6","title":"Clear from the theorem \\refthm:characterization_of_corner_ring_elements","kind":"proof","summary":"Clear from the theorem \\refthm:characterization_of_corner_ring_elements","labels":[],"detail_key":"p6"},{"id":"n1252","layer":"informal","project":"p6","title":"thm:corner_ring_is_ring","kind":"theorem","summary":"The corner ring is a (non-unital) subring of R. It has its own unit e.","labels":["thm:corner_ring_is_ring"],"detail_key":"p6"},{"id":"n1253","layer":"informal","project":"p6","title":"If a, b \\in eRe, then a + b = e a e + e b e = e (a + b) e so eRe is closed under addition…","kind":"proof","summary":"If a, b \\in eRe, then a + b = e a e + e b e = e (a + b) e so eRe is closed under addition. If a…","labels":[],"detail_key":"p6"},{"id":"n1254","layer":"informal","project":"p6","title":"thm:corner_ring_artinian","kind":"theorem","summary":"If R is left artinian, then the corner ring is left artinian.","labels":["thm:corner_ring_artinian"],"detail_key":"p6"},{"id":"n1255","layer":"informal","project":"p6","title":"Let L_1 \\supseteq L_2 \\supseteq \\ldots be a descending chain of left ideals in eRe. Then…","kind":"proof","summary":"Let L_1 \\supseteq L_2 \\supseteq \\ldots be a descending chain of left ideals in eRe. Then RL_1 \\…","labels":[],"detail_key":"p6"},{"id":"n1256","layer":"informal","project":"p6","title":"thm:corner_ring_prime","kind":"theorem","summary":"If R is a prime ring, then the corner ring is prime.","labels":["thm:corner_ring_prime"],"detail_key":"p6"},{"id":"n1257","layer":"informal","project":"p6","title":"Suppose aeReb = 0 for a, b \\in eRe. Then ae = a = 0 or eb = b = 0 by \\refthm:prime_ring_e…","kind":"proof","summary":"Suppose aeReb = 0 for a, b \\in eRe. Then ae = a = 0 or eb = b = 0 by \\refthm:prime_ring_equiv,…","labels":[],"detail_key":"p6"},{"id":"n1258","layer":"informal","project":"p6","title":"thm:all_left_inv_div_ring","kind":"theorem","summary":"If all elements in a ring are left invertible, then the ring is a division ring.","labels":["thm:all_left_inv_div_ring"],"detail_key":"p6"},{"id":"n1259","layer":"informal","project":"p6","title":"Let x \\in R be arbitrary. Then yx = 1 for some y \\in R. Since y is left invertible, there…","kind":"proof","summary":"Let x \\in R be arbitrary. Then yx = 1 for some y \\in R. Since y is left invertible, there exist…","labels":[],"detail_key":"p6"},{"id":"n1260","layer":"informal","project":"p6","title":"Brauer's lemma","kind":"theorem","summary":"[Brauer's lemma] Suppose L is a minimal (left) ideal of R and L^2 \\neq 0. Then there exists an…","labels":["thm:brauer_lemma"],"detail_key":"p6"},{"id":"n1261","layer":"informal","project":"p6","title":"By assumption, there exists y \\in L such that Ly \\neq 0. By minimality L = Ly. Thus, ther…","kind":"proof","summary":"By assumption, there exists y \\in L such that Ly \\neq 0. By minimality L = Ly. Thus, there exis…","labels":[],"detail_key":"p6"},{"id":"n1262","layer":"informal","project":"p6","title":"J is a left ideal of R contained in L.","kind":"claim","summary":"J is a left ideal of R contained in L.","labels":[],"detail_key":"p6"},{"id":"n1263","layer":"informal","project":"p6","title":"Let a, b \\in J. Then (a + b) y = a y + b y = 0, so (a + b) \\in J. For any x \\in R, x a y…","kind":"proof","summary":"Let a, b \\in J. Then (a + b) y = a y + b y = 0, so (a + b) \\in J. For any x \\in R, x a y = 0 so…","labels":[],"detail_key":"p6"},{"id":"n1264","layer":"informal","project":"p6","title":"thm:artinian_has_minimal_left_ideal","kind":"theorem","summary":"(Already proven in Mathlib) A nonzero left artinian ring has a minimal left ideal.","labels":["thm:artinian_has_minimal_left_ideal"],"detail_key":"p6"},{"id":"n1265","layer":"informal","project":"p6","title":"If minimal left ideal does not exist, then starting with any nonzero left ideal, we can a…","kind":"proof","summary":"If minimal left ideal does not exist, then starting with any nonzero left ideal, we can allways…","labels":[],"detail_key":"p6"},{"id":"n1266","layer":"informal","project":"p6","title":"Set of Matrix Units","kind":"definition","summary":"[Set of Matrix Units] A set e_ij for i, j \\in [1, n] is a set of matrix units of R if e_ije_kl…","labels":["def:matrixunits"],"detail_key":"p6"},{"id":"n1267","layer":"informal","project":"p6","title":"thm:ring_with_matrix_units","kind":"theorem","summary":"If R has a set of matrix units e_ij, then R is isomorphic to the ring of n \\times n matrices ov…","labels":["thm:ring_with_matrix_units"],"detail_key":"p6"},{"id":"n1268","layer":"informal","project":"p6","title":"For a \\in R, denote a_ij = e_1iae_j1. Then e_11a_ije_11 = e_11e_1iae_j1e_11 = e_1iae_j1 b…","kind":"proof","summary":"For a \\in R, denote a_ij = e_1iae_j1. Then e_11a_ije_11 = e_11e_1iae_j1e_11 = e_1iae_j1 by the…","labels":[],"detail_key":"p6"},{"id":"n1269","layer":"informal","project":"p6","title":"\\phi is additive.","kind":"claim","summary":"\\phi is additive.","labels":[],"detail_key":"p6"},{"id":"n1270","layer":"informal","project":"p6","title":"For a, b \\in R, we have: ((a + b)_ij)_i,j=1^n = (e_1i(a + b)e_j1)_i,j=1^n = (e_1iae_j1 +…","kind":"proof","summary":"For a, b \\in R, we have: ((a + b)_ij)_i,j=1^n = (e_1i(a + b)e_j1)_i,j=1^n = (e_1iae_j1 + e_1ibe…","labels":[],"detail_key":"p6"},{"id":"n1271","layer":"informal","project":"p6","title":"The map is multiplicative.","kind":"claim","summary":"The map is multiplicative.","labels":[],"detail_key":"p6"},{"id":"n1272","layer":"informal","project":"p6","title":"The (i,j) entry of \\phi(a)\\phi(b) is equal to \\sum_k=1^n e_1iae_k1e_1kbe_j1 = e_1ia \\sum_…","kind":"proof","summary":"The (i,j) entry of \\phi(a)\\phi(b) is equal to \\sum_k=1^n e_1iae_k1e_1kbe_j1 = e_1ia \\sum_k=1^n…","labels":[],"detail_key":"p6"},{"id":"n1273","layer":"informal","project":"p6","title":"The map is injective.","kind":"claim","summary":"The map is injective.","labels":[],"detail_key":"p6"},{"id":"n1274","layer":"informal","project":"p6","title":"Suppose a_ij = 0 for all i, j. Then e_iiae_jj = e_1ia_ije_j1 = 0. Therefore, a = a(\\sum_i…","kind":"proof","summary":"Suppose a_ij = 0 for all i, j. Then e_iiae_jj = e_1ia_ije_j1 = 0. Therefore, a = a(\\sum_i=1^n e…","labels":[],"detail_key":"p6"},{"id":"n1275","layer":"informal","project":"p6","title":"The map is surjective.","kind":"claim","summary":"The map is surjective.","labels":[],"detail_key":"p6"},{"id":"n1276","layer":"informal","project":"p6","title":"Note the \\phi(e_k1ae_1l)_kl = e_1ke_k1a e_1le_l1 = e_11ae_11 and \\phi(e_k1ae_1l)_ab = e_1…","kind":"proof","summary":"Note the \\phi(e_k1ae_1l)_kl = e_1ke_k1a e_1le_l1 = e_11ae_11 and \\phi(e_k1ae_1l)_ab = e_1ae_k1a…","labels":[],"detail_key":"p6"},{"id":"n1277","layer":"informal","project":"p6","title":"thm:criterion_for_matrix_units","kind":"theorem","summary":"If a ring R has a set of pairwise orthogonal idempotents e_ii and \\item e_1i \\in e_11Re_ii for…","labels":["thm:criterion_for_matrix_units"],"detail_key":"p6"},{"id":"n1278","layer":"informal","project":"p6","title":"Define f_ij = e_i1e_1j. For i = 1, we have f_1j = e_1j. f_1j = e_11e_1j. Since e_1j \\in e…","kind":"proof","summary":"Define f_ij = e_i1e_1j. For i = 1, we have f_1j = e_1j. f_1j = e_11e_1j. Since e_1j \\in e_11Re_…","labels":[],"detail_key":"p6"},{"id":"n1279","layer":"informal","project":"p6","title":"For i = 1, we have f_1j = e_1j.","kind":"claim","summary":"For i = 1, we have f_1j = e_1j.","labels":[],"detail_key":"p6"},{"id":"n1280","layer":"informal","project":"p6","title":"f_1j = e_11e_1j. Since e_1j \\in e_11Re_jj, we have e_11e_1j = e_1j by theorem \\refthm:cha…","kind":"proof","summary":"f_1j = e_11e_1j. Since e_1j \\in e_11Re_jj, we have e_11e_1j = e_1j by theorem \\refthm:character…","labels":[],"detail_key":"p6"},{"id":"n1281","layer":"informal","project":"p6","title":"For j = 1, we have f_i1 = e_i1 for all i.","kind":"claim","summary":"For j = 1, we have f_i1 = e_i1 for all i.","labels":[],"detail_key":"p6"},{"id":"n1282","layer":"informal","project":"p6","title":"f_i1 = e_i1e_11. Since e_i1 \\in e_iiRe_11, we have e_i1e_11 = e_i1.","kind":"proof","summary":"f_i1 = e_i1e_11. Since e_i1 \\in e_iiRe_11, we have e_i1e_11 = e_i1.","labels":[],"detail_key":"p6"},{"id":"n1283","layer":"informal","project":"p6","title":"f_1j f_k1 = \\delta_jk f_11 for all j, k","kind":"claim","summary":"f_1j f_k1 = \\delta_jk f_11 for all j, k","labels":[],"detail_key":"p6"},{"id":"n1284","layer":"informal","project":"p6","title":"f_1j f_k1 = e_11e_1je_k1e_11 = e_1je_k1 = e_11 r e_jj e_kk r' e_11 = \\delta_jk e_11 for s…","kind":"proof","summary":"f_1j f_k1 = e_11e_1je_k1e_11 = e_1je_k1 = e_11 r e_jj e_kk r' e_11 = \\delta_jk e_11 for some r,…","labels":[],"detail_key":"p6"},{"id":"n1285","layer":"informal","project":"p6","title":"f_ij f_kl = \\delta_jk f_il.","kind":"claim","summary":"f_ij f_kl = \\delta_jk f_il.","labels":[],"detail_key":"p6"},{"id":"n1286","layer":"informal","project":"p6","title":"By definition, f_ij f_kl = e_i1e_1j e_k1e_1l = f_i1f_1j f_k1f_1l = \\delta_jk f_i1 f_1l =…","kind":"proof","summary":"By definition, f_ij f_kl = e_i1e_1j e_k1e_1l = f_i1f_1j f_k1f_1l = \\delta_jk f_i1 f_1l = \\delta…","labels":[],"detail_key":"p6"},{"id":"n1287","layer":"informal","project":"p6","title":"thm:orthogonal_idempotents_division_ring","kind":"theorem","summary":"Let e, f \\in R be nonzero orthogonal idempotents and R a prime ring. Also let eRe and fRf be di…","labels":["thm:orthogonal_idempotents_division_ring"],"detail_key":"p6"},{"id":"n1288","layer":"informal","project":"p6","title":"There exists a, b \\in R such that eafbe \\neq 0. Suppose eRf = 0. By theorem \\refthm:prime…","kind":"proof","summary":"There exists a, b \\in R such that eafbe \\neq 0. Suppose eRf = 0. By theorem \\refthm:prime_ring_…","labels":[],"detail_key":"p6"},{"id":"n1289","layer":"informal","project":"p6","title":"There exists a, b \\in R such that eafbe \\neq 0.","kind":"claim","summary":"There exists a, b \\in R such that eafbe \\neq 0.","labels":[],"detail_key":"p6"},{"id":"n1290","layer":"informal","project":"p6","title":"Suppose eRf = 0. By theorem \\refthm:prime_ring_equiv', eRf = 0 implies e = 0 or f = 0, a…","kind":"proof","summary":"Suppose eRf = 0. By theorem \\refthm:prime_ring_equiv', eRf = 0 implies e = 0 or f = 0, a contra…","labels":[],"detail_key":"p6"},{"id":"n1291","layer":"informal","project":"p6","title":"vu = f.","kind":"claim","summary":"vu = f.","labels":[],"detail_key":"p6"},{"id":"n1292","layer":"informal","project":"p6","title":"Suppose not. Then vu - f \\neq 0, but vu - f is left invertible since fRf is a division ri…","kind":"proof","summary":"Suppose not. Then vu - f \\neq 0, but vu - f is left invertible since fRf is a division ring. Mu…","labels":[],"detail_key":"p6"},{"id":"n1293","layer":"informal","project":"p6","title":"thm:orthogonal_idempotents_division_ring_matrix","kind":"theorem","summary":"If a prime ring R contains pairwise orthogonal idempotents e_ii with sum 1 and e_iiRe_ii is a d…","labels":["thm:orthogonal_idempotents_division_ring_matrix"],"detail_key":"p6"},{"id":"n1294","layer":"informal","project":"p6","title":"Applying the theorem \\refthm:orthogonal_idempotents_division_ring for e_11 and each e_ii,…","kind":"proof","summary":"Applying the theorem \\refthm:orthogonal_idempotents_division_ring for e_11 and each e_ii, we de…","labels":[],"detail_key":"p6"},{"id":"n1295","layer":"informal","project":"p6","title":"The defined elements satisfy the conditions of theorem \\refthm:criterion_for_matrix_units.","kind":"claim","summary":"The defined elements satisfy the conditions of theorem \\refthm:criterion_for_matrix_units.","labels":[],"detail_key":"p6"},{"id":"n1296","layer":"informal","project":"p6","title":"By the conclusion of theorem \\refthm:orthogonal_idempotents_division_ring, e_1ie_i1 = e_i…","kind":"proof","summary":"By the conclusion of theorem \\refthm:orthogonal_idempotents_division_ring, e_1ie_i1 = e_ii and…","labels":[],"detail_key":"p6"},{"id":"n1297","layer":"informal","project":"p6","title":"thm:idempotents_orthogonal","kind":"theorem","summary":"If e, f \\in R are idempotents and f \\in (1-e)R(1-e) they are orthogonal. Further fRf = f(1 - e)…","labels":["thm:idempotents_orthogonal"],"detail_key":"p6"},{"id":"n1298","layer":"informal","project":"p6","title":"f = f(1 - e) + fe = f + fe. Thus fe = 0. Similarly, ef = 0. Therefore, f and e are orthog…","kind":"proof","summary":"f = f(1 - e) + fe = f + fe. Thus fe = 0. Similarly, ef = 0. Therefore, f and e are orthogonal.…","labels":[],"detail_key":"p6"},{"id":"n1299","layer":"informal","project":"p6","title":"Artin Wedderburn for prime rings","kind":"theorem","summary":"[Artin Wedderburn for prime rings] If R is a prime ring and artinian, then R is isomorphic to M…","labels":["thm:artin_wedderburn_for_prime"],"detail_key":"p6"},{"id":"n1300","layer":"informal","project":"p6","title":"Since R is artinian, it contains a minimal nonzero left ideal L. If L^2 = 0, this would i…","kind":"proof","summary":"Since R is artinian, it contains a minimal nonzero left ideal L. If L^2 = 0, this would imply b…","labels":[],"detail_key":"p6"},{"id":"n1301","layer":"informal","project":"p6","title":"Artin Wedderburn for simple rings","kind":"theorem","summary":"[Artin Wedderburn for simple rings] If R is a simple ring, then R is isomorphic to M_n(D) for s…","labels":["thm:artin_wedderburn_for_simple"],"detail_key":"p6"},{"id":"n1302","layer":"informal","project":"p6","title":"Since R is simple, it is prime. By theorem \\refthm:artin_wedderburn_for_prime, R is isomo…","kind":"proof","summary":"Since R is simple, it is prime. By theorem \\refthm:artin_wedderburn_for_prime, R is isomorphic…","labels":[],"detail_key":"p6"},{"id":"n1303","layer":"informal","project":"p6","title":"thm:semisimple_direct_product_simple_artinian","kind":"theorem","summary":"Let R be a semisimple ring. Then, R is isomorphic to a direct product of simple, artinian rings.","labels":["thm:semisimple_direct_product_simple_artinian"],"detail_key":"p6"},{"id":"n1304","layer":"informal","project":"p6","title":"WLOG, suppose, R is not simple. We know that R is (left) artinian, which is a stronger co…","kind":"proof","summary":"WLOG, suppose, R is not simple. We know that R is (left) artinian, which is a stronger conditio…","labels":[],"detail_key":"p6"},{"id":"n1305","layer":"informal","project":"p6","title":"I J = 0.","kind":"claim","summary":"I J = 0.","labels":[],"detail_key":"p6"},{"id":"n1306","layer":"informal","project":"p6","title":"Suppose x \\in I J. Then x \\in I since I is a twosided ideal. Also x \\in J since J is a le…","kind":"proof","summary":"Suppose x \\in I J. Then x \\in I since I is a twosided ideal. Also x \\in J since J is a left ide…","labels":[],"detail_key":"p6"},{"id":"n1307","layer":"informal","project":"p6","title":"i is an idempotent.","kind":"claim","summary":"i is an idempotent.","labels":[],"detail_key":"p6"},{"id":"n1308","layer":"informal","project":"p6","title":"i = i 1 = i(i + j) = i i + i j = i i by the previous claim.","kind":"proof","summary":"i = i 1 = i(i + j) = i i + i j = i i by the previous claim.","labels":[],"detail_key":"p6"},{"id":"n1309","layer":"informal","project":"p6","title":"I I = I.","kind":"claim","summary":"I I = I.","labels":[],"detail_key":"p6"},{"id":"n1310","layer":"informal","project":"p6","title":"By simplicity of I, I I = 0 or I. Since i i = i, the first case is impossible.","kind":"proof","summary":"By simplicity of I, I I = 0 or I. Since i i = i, the first case is impossible.","labels":[],"detail_key":"p6"},{"id":"n1311","layer":"informal","project":"p6","title":"J I = 0.","kind":"claim","summary":"J I = 0.","labels":[],"detail_key":"p6"},{"id":"n1312","layer":"informal","project":"p6","title":"Note that JI is spanned by the set of all pairwise products of elements of J and I. Since…","kind":"proof","summary":"Note that JI is spanned by the set of all pairwise products of elements of J and I. Since J is…","labels":[],"detail_key":"p6"},{"id":"n1313","layer":"informal","project":"p6","title":"J is a two-sided ideal.","kind":"claim","summary":"J is a two-sided ideal.","labels":[],"detail_key":"p6"},{"id":"n1314","layer":"informal","project":"p6","title":"We know that it is a left ideal. For arbitrary x \\in R, write x = x i + x j. Let y \\in J…","kind":"proof","summary":"We know that it is a left ideal. For arbitrary x \\in R, write x = x i + x j. Let y \\in J be arb…","labels":[],"detail_key":"p6"},{"id":"n1315","layer":"informal","project":"p6","title":"R = I \\times J as rings.","kind":"claim","summary":"R = I \\times J as rings.","labels":[],"detail_key":"p6"},{"id":"n1316","layer":"informal","project":"p6","title":"Let x = x_i + x_j where x_i = x i \\in I and x j \\in J. Similarly, let y = y_i + y_j. Then…","kind":"proof","summary":"Let x = x_i + x_j where x_i = x i \\in I and x j \\in J. Similarly, let y = y_i + y_j. Then x y =…","labels":[],"detail_key":"p6"},{"id":"n1317","layer":"informal","project":"p6","title":"Let K \\subseteq I be a left I-submodule of I, where I is treated as a unital ring. Then K…","kind":"claim","summary":"Let K \\subseteq I be a left I-submodule of I, where I is treated as a unital ring. Then K is a…","labels":[],"detail_key":"p6"},{"id":"n1318","layer":"informal","project":"p6","title":"Let r \\in R and k \\in K. Then r k = r 1 k = r (i + j) k = r i k + r j k = r i k \\in K sin…","kind":"proof","summary":"Let r \\in R and k \\in K. Then r k = r 1 k = r (i + j) k = r i k + r j k = r i k \\in K since k \\…","labels":[],"detail_key":"p6"},{"id":"n1319","layer":"informal","project":"p6","title":"Both I and J are artinian (as rings).","kind":"claim","summary":"Both I and J are artinian (as rings).","labels":[],"detail_key":"p6"},{"id":"n1320","layer":"informal","project":"p6","title":"They are both submodules of R which is assumed to be artinian. Submodules of artinian mod…","kind":"proof","summary":"They are both submodules of R which is assumed to be artinian. Submodules of artinian modules a…","labels":[],"detail_key":"p6"},{"id":"n1321","layer":"informal","project":"p6","title":"J is (left) semisimple.","kind":"claim","summary":"J is (left) semisimple.","labels":[],"detail_key":"p6"},{"id":"n1322","layer":"informal","project":"p6","title":"A submodu","kind":"proof","summary":"A submodu","labels":[],"detail_key":"p6"},{"id":"n1323","layer":"formal","project":"p6","title":"ArtinWedderburnForPrime","kind":"theorem","summary":"∀ R : Type u [inst : Ring R] [h_nontriv : Nontrivial R], IsPrimeRing R → IsArtinian R R → Exist…","labels":[],"detail_key":"p6","name":"ArtinWedderburnForPrime","module":"ArtinWedderburn.ArtinWedderburnTheorem"},{"id":"n1324","layer":"formal","project":"p6","title":"ArtinWedderburnForSimple","kind":"theorem","summary":"∀ R : Type u [inst : Ring R] [inst_1 : IsSimpleRing R] [h_art : IsArtinian R R], Exists fun n =…","labels":[],"detail_key":"p6","name":"ArtinWedderburnForSimple","module":"ArtinWedderburn.ArtinWedderburnTheorem"},{"id":"n1325","layer":"formal","project":"p6","title":"CornerRingSet","kind":"def","summary":"R : Type u_1 → [inst : Ring R] → R → Set R","labels":[],"detail_key":"p6","name":"CornerRingSet","module":"ArtinWedderburn.CornerRing"},{"id":"n1326","layer":"formal","project":"p6","title":"corner_ring_artinian","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] e : R (idem_e : IsIdempotentElem e) [h_ar : IsArtinian R R], IsA…","labels":[],"detail_key":"p6","name":"corner_ring_artinian","module":"ArtinWedderburn.CornerRing"},{"id":"n1327","layer":"formal","project":"p6","title":"corner_ring_prime","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] e : R (idem_e : IsIdempotentElem e), IsPrimeRing R → IsPrimeRing…","labels":[],"detail_key":"p6","name":"corner_ring_prime","module":"ArtinWedderburn.CornerRing"},{"id":"n1328","layer":"formal","project":"p6","title":"corner_ring_set_mem","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] e x : R, IsIdempotentElem e → Iff (Membership.mem (CornerRingSet…","labels":[],"detail_key":"p6","name":"corner_ring_set_mem","module":"ArtinWedderburn.CornerRing"},{"id":"n1329","layer":"formal","project":"p6","title":"x_in_corner_x_eq_e_y_e","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] e x : R, Membership.mem (CornerRingSet e) x → Exists fun y => Eq…","labels":[],"detail_key":"p6","name":"x_in_corner_x_eq_e_y_e","module":"ArtinWedderburn.CornerRing"},{"id":"n1330","layer":"formal","project":"p6","title":"IdealProd.ring_subset_prod_ideal","kind":"def","summary":"R : Type u_1 → [inst : Ring R] → Set R → Set R → Ideal R","labels":[],"detail_key":"p6","name":"IdealProd.ring_subset_prod_ideal","module":"ArtinWedderburn.IdealProd"},{"id":"n1331","layer":"formal","project":"p6","title":"both_mul","kind":"def","summary":"R : Type u_1 → [inst : Ring R] → R → R → Set R","labels":[],"detail_key":"p6","name":"both_mul","module":"ArtinWedderburn.IdealProd"},{"id":"n1332","layer":"formal","project":"p6","title":"IsOrthogonal","kind":"def","summary":"R : Type u_1 → [inst : Ring R] → R → R → Prop","labels":[],"detail_key":"p6","name":"IsOrthogonal","module":"ArtinWedderburn.Idempotents"},{"id":"n1333","layer":"formal","project":"p6","title":"OrtIdem_imply_MatUnits","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] n : Nat (hn : LT.lt 0 n) (diag_es : Fin n → R), (∀ (i : Fin n),…","labels":[],"detail_key":"p6","name":"OrtIdem_imply_MatUnits","module":"ArtinWedderburn.Idempotents"},{"id":"n1334","layer":"formal","project":"p6","title":"artinian_ring_has_minimal_left_ideal_of_element","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] [inst_1 : IsArtinian R R] [inst_2 : Nontrivial R], Exists fun I…","labels":[],"detail_key":"p6","name":"artinian_ring_has_minimal_left_ideal_of_element","module":"ArtinWedderburn.Idempotents"},{"id":"n1335","layer":"formal","project":"p6","title":"one_sub_e_larger_span_on_sub_e_sub_f","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] (e f : R), AreOrthogonalIdempotents e f → Ne f 0 → LT.lt (Ideal.…","labels":[],"detail_key":"p6","name":"one_sub_e_larger_span_on_sub_e_sub_f","module":"ArtinWedderburn.Idempotents"},{"id":"n1336","layer":"formal","project":"p6","title":"hasMatrixUnits","kind":"inductive","summary":"(R : Type u_1) → [inst : Ring R] → Nat → Type u_1","labels":[],"detail_key":"p6","name":"hasMatrixUnits","module":"ArtinWedderburn.MatrixUnits"},{"id":"n1337","layer":"formal","project":"p6","title":"ring_with_matrix_units_isomorphic_to_matrix_ring","kind":"def","summary":"(R : Type u_1) → [inst : Ring R] → (n : Nat) → (hn : LT.lt 0 n) → (mu : hasMatrixUnits R n) → R…","labels":[],"detail_key":"p6","name":"ring_with_matrix_units_isomorphic_to_matrix_ring","module":"ArtinWedderburn.MatrixUnits"},{"id":"n1338","layer":"formal","project":"p6","title":"minimal_ideal_I_sq_nonzero_exists_idem","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] (I : Ideal R), IsAtom I → Ne (HMul.hMul I I) Bot.bot → Exists fu…","labels":[],"detail_key":"p6","name":"minimal_ideal_I_sq_nonzero_exists_idem","module":"ArtinWedderburn.MinIdeals"},{"id":"n1339","layer":"formal","project":"p6","title":"IsPrimeRing","kind":"def","summary":"(R : Type u_2) → [inst : Ring R] → Prop","labels":[],"detail_key":"p6","name":"IsPrimeRing","module":"ArtinWedderburn.PrimeRing"},{"id":"n1340","layer":"formal","project":"p6","title":"prime_ring_equiv","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R], Iff (IsPrimeRing R) (∀ (a b : R), Eq (both_mul a b) (singleton…","labels":[],"detail_key":"p6","name":"prime_ring_equiv","module":"ArtinWedderburn.PrimeRing"},{"id":"n1341","layer":"formal","project":"p6","title":"prime_ring_equiv'","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R], Iff (IsPrimeRing R) (∀ (I J : TwoSidedIdeal R), Eq (HMul.hMul I…","labels":[],"detail_key":"p6","name":"prime_ring_equiv'","module":"ArtinWedderburn.PrimeRing"},{"id":"n1342","layer":"formal","project":"p6","title":"simple_ring_is_prime","kind":"theorem","summary":"∀ R : Type u_1 [inst : Ring R] [inst_1 : IsSimpleRing R], IsPrimeRing R","labels":[],"detail_key":"p6","name":"simple_ring_is_prime","module":"ArtinWedderburn.PrimeRing"},{"id":"n1343","layer":"formal","project":"p7","title":"Combine.combine","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] L : Finset F dsta…","labels":[],"detail_key":"p7","name":"Combine.combine","module":"STIR.Combine"},{"id":"n1344","layer":"formal","project":"p7","title":"Combine.combineFinal","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst : DecidableEq F] → L : Finset F → m : Nat → (dstar : Na…","labels":[],"detail_key":"p7","name":"Combine.combineFinal","module":"STIR.Combine"},{"id":"n1345","layer":"formal","project":"p7","title":"Combine.combineInterm","kind":"def","summary":"F : Type u_1 → [inst : Field F] → (L : Finset F) → (m dstar : Nat) → F → (Fin m → (Subtype fun…","labels":[],"detail_key":"p7","name":"Combine.combineInterm","module":"STIR.Combine"},{"id":"n1346","layer":"formal","project":"p7","title":"Combine.degreeCorrFinal","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst : DecidableEq F] → (L : Finset F) → (dstar : Nat) → F →…","labels":[],"detail_key":"p7","name":"Combine.degreeCorrFinal","module":"STIR.Combine"},{"id":"n1347","layer":"formal","project":"p7","title":"Combine.degreeCorrInterm","kind":"def","summary":"F : Type u_1 → [inst : Field F] → (L : Finset F) → (dstar : Nat) → F → ((Subtype fun x => Membe…","labels":[],"detail_key":"p7","name":"Combine.degreeCorrInterm","module":"STIR.Combine"},{"id":"n1348","layer":"formal","project":"p7","title":"Combine.geometric_sum_units","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : DecidableEq F] (r : Units F) (a : Nat), Eq (Finset.un…","labels":[],"detail_key":"p7","name":"Combine.geometric_sum_units","module":"STIR.Combine"},{"id":"n1349","layer":"formal","project":"p7","title":"Folding.degree_bound_bivariate","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : Fintype F] (q : Polynomial F), GT.gt q.natDegree 0 →…","labels":[],"detail_key":"p7","name":"Folding.degree_bound_bivariate","module":"STIR.Folding"},{"id":"n1350","layer":"formal","project":"p7","title":"Folding.exists_unique_bivariate","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : Fintype F] (q : Polynomial F), GT.gt q.natDegree 0 →…","labels":[],"detail_key":"p7","name":"Folding.exists_unique_bivariate","module":"STIR.Folding"},{"id":"n1351","layer":"formal","project":"p7","title":"Folding.fold","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst_1 : Fintype F] → [inst_2 : DecidableEq F] → L : Finset…","labels":[],"detail_key":"p7","name":"Folding.fold","module":"STIR.Folding"},{"id":"n1352","layer":"formal","project":"p7","title":"Folding.folding","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] L : Finset F d :…","labels":[],"detail_key":"p7","name":"Folding.folding","module":"STIR.Folding"},{"id":"n1353","layer":"formal","project":"p7","title":"Folding.polyFold","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst_1 : Fintype F] → [inst_2 : DecidableEq F] → Polynomial…","labels":[],"detail_key":"p7","name":"Folding.polyFold","module":"STIR.Folding"},{"id":"n1354","layer":"formal","project":"p7","title":"johnson_bound","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] L : Finset F d 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Com…","labels":[],"detail_key":"p8","name":"MPSTensor.isIrreducibleTensor_smul_conj","module":"TNLean.MPS.CanonicalForm.Reduction"},{"id":"n9318","layer":"formal","project":"p8","title":"MPSTensor.HasPrimitiveIrreducibleCyclicSectors","kind":"def","summary":"d D : Nat → MPSTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.HasPrimitiveIrreducibleCyclicSectors","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorConstruction"},{"id":"n9319","layer":"formal","project":"p8","title":"MPSTensor.exists_commonBlockedCyclicSectorFamily_of_commonMultiple","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor d (dim k)) (hcyc : ∀ (k : Fin r…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_commonBlockedCyclicSectorFamily_of_commonMultiple","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorConstruction"},{"id":"n9320","layer":"formal","project":"p8","title":"MPSTensor.exists_commonBlockedCyclicSectorFamily_of_hasPrimitiveIrreducibleCyclicSectors","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), (bl…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_commonBlockedCyclicSectorFamily_of_hasPrimitiveIrreducibleCyclicSectors","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorConstruction"},{"id":"n9321","layer":"formal","project":"p8","title":"MPSTensor.CommonBlockedCyclicSectorFamily.blockTensor_sameMPV₂_commonReindexedBlock","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat blocks : (k : Fin r) → MPSTensor d (dim k) (F : MPSTensor.CommonB…","labels":[],"detail_key":"p8","name":"MPSTensor.CommonBlockedCyclicSectorFamily.blockTensor_sameMPV₂_commonReindexedBlock","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorFamily"},{"id":"n9322","layer":"formal","project":"p8","title":"MPSTensor.CommonBlockedCyclicSectorFamily.derived_properties","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat blocks : (k : Fin r) → MPSTensor d (dim k) (F : MPSTensor.CommonB…","labels":[],"detail_key":"p8","name":"MPSTensor.CommonBlockedCyclicSectorFamily.derived_properties","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorFamily"},{"id":"n9323","layer":"formal","project":"p8","title":"MPSTensor.CommonBlockedCyclicSectorFamily.reindexed_nonzero_part","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat blocks : (k : Fin r) → MPSTensor d (dim k) (F : MPSTensor.CommonB…","labels":[],"detail_key":"p8","name":"MPSTensor.CommonBlockedCyclicSectorFamily.reindexed_nonzero_part","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorFamily"},{"id":"n9324","layer":"formal","project":"p8","title":"MPSTensor.CommonBlockedCyclicSectorFamily.reindexed_sameMPV₂","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat blocks : (k : Fin r) → MPSTensor d (dim k) (F : MPSTensor.CommonB…","labels":[],"detail_key":"p8","name":"MPSTensor.CommonBlockedCyclicSectorFamily.reindexed_sameMPV₂","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonBlockedCyclicSectorFamily"},{"id":"n9325","layer":"formal","project":"p8","title":"MPSTensor.afterBlocking_commonLengthCommonSectorData_of_sameMPV₂Pos","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.SameMPV₂Pos B → Exists fun p => An…","labels":[],"detail_key":"p8","name":"MPSTensor.afterBlocking_commonLengthCommonSectorData_of_sameMPV₂Pos","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonSectorData"},{"id":"n9326","layer":"formal","project":"p8","title":"MPSTensor.afterBlocking_perBlockCyclicData_of_sameMPV₂Pos","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.SameMPV₂Pos B → Exists fun rA => E…","labels":[],"detail_key":"p8","name":"MPSTensor.afterBlocking_perBlockCyclicData_of_sameMPV₂Pos","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonSectorData"},{"id":"n9327","layer":"formal","project":"p8","title":"MPSTensor.unconditional_commonPrimitiveIrreducibleBlocks","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.SameMPV₂Pos B → Exists fun p => An…","labels":[],"detail_key":"p8","name":"MPSTensor.unconditional_commonPrimitiveIrreducibleBlocks","module":"TNLean.MPS.CanonicalForm.SectorComparison.CommonSectorTransport"},{"id":"n9328","layer":"formal","project":"p8","title":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_of_TP_of_isIrreducibleTensor","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_of_TP_of_isIrreducibleTensor","module":"TNLean.MPS.CanonicalForm.SectorComparison.CyclicSectorDecomposition"},{"id":"n9329","layer":"formal","project":"p8","title":"MPSTensor.exists_primitive_irreducible_cyclic_sector_decomp_of_TP_of_isIrreducibleTensor","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_primitive_irreducible_cyclic_sector_decomp_of_TP_of_isIrreducibleTensor","module":"TNLean.MPS.CanonicalForm.SectorComparison.CyclicSectorDecomposition"},{"id":"n9330","layer":"formal","project":"p8","title":"MPSTensor.primitive_and_irreducible_sectorBlocks_of_cyclic_decomp_after_blocking","kind":"theorem","summary":"∀ d D m : Nat [NeZero D] [inst : NeZero m] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HM…","labels":[],"detail_key":"p8","name":"MPSTensor.primitive_and_irreducible_sectorBlocks_of_cyclic_decomp_after_blocking","module":"TNLean.MPS.CanonicalForm.SectorComparison.CyclicSectorDecomposition"},{"id":"n9331","layer":"formal","project":"p8","title":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_with_letter_and_isometry","kind":"theorem","summary":"∀ d D m : Nat [NeZero D] [inst : NeZero m] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HM…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_with_letter_and_isometry","module":"TNLean.MPS.CanonicalForm.SectorComparison.CyclicSectorRelation"},{"id":"n9332","layer":"formal","project":"p8","title":"MPSTensor.IsNormalCanonicalFormBNT.exists_common_blockTensor_isInjective","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat μ : Fin r → Complex blocks : (k : Fin r) → MPSTensor d (dim k) [∀…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNormalCanonicalFormBNT.exists_common_blockTensor_isInjective","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9333","layer":"formal","project":"p8","title":"MPSTensor.exists_common_blockTensor_isInjective_two_of_isNormalCanonicalFormBNT","kind":"theorem","summary":"∀ d rA rB : Nat dimA : Fin rA → Nat dimB : Fin rB → Nat [∀ (j : Fin rA), NeZero (dimA j)] [∀ (k…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_common_blockTensor_isInjective_two_of_isNormalCanonicalFormBNT","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9334","layer":"formal","project":"p8","title":"MPSTensor.exists_pos_blockTensor_isInjective_le_pow_four_of_isNormal_leftCanonical","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pos_blockTensor_isInjective_le_pow_four_of_isNormal_leftCanonical","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9335","layer":"formal","project":"p8","title":"MPSTensor.exists_pos_blockTensor_isInjective_of_tp_primitive_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pos_blockTensor_isInjective_of_tp_primitive_irreducible","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9336","layer":"formal","project":"p8","title":"MPSTensor.isNBlkInjective_pow_four_of_isNormal_leftCanonical","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.isNBlkInjective_pow_four_of_isNormal_leftCanonical","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9337","layer":"formal","project":"p8","title":"MPSTensor.isNormal_blockTensor_of_isNormal","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) P : Nat, LT.lt 0 P → Kraus.IsNormal A → Kraus.IsNormal (MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.isNormal_blockTensor_of_isNormal","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9338","layer":"formal","project":"p8","title":"MPSTensor.isNormal_of_tp_primitive_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i).conjTr…","labels":[],"detail_key":"p8","name":"MPSTensor.isNormal_of_tp_primitive_irreducible","module":"TNLean.MPS.CanonicalForm.SectorComparison.NormalityChain"},{"id":"n9339","layer":"formal","project":"p8","title":"MPSTensor.exists_common_injective_blocking_of_tp_primitive_irr_family","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), 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(Y…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_tensorProduct_kronecker","module":"TNLean.MPS.CanonicalForm.TensorProduct"},{"id":"n9346","layer":"formal","project":"p8","title":"MPSTensor.exists_unitaryConj_of_intertwines_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat B C : MPSTensor d D L₀ : Nat, Eq (Finset.univ.sum fun i => HMul.hMul (B i) (B i).co…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitaryConj_of_intertwines_of_isNBlkInjective","module":"TNLean.MPS.CanonicalForm.TranslationInvariantUniqueness"},{"id":"n9347","layer":"formal","project":"p8","title":"MPSTensor.isUnit_of_intertwines_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat [NeZero D] B C : MPSTensor d D L : Nat, Kraus.IsNBlkInjective B L → ∀ R : Matrix 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(MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSChainTensor.fundamentalTheorem_injective_chain","module":"TNLean.MPS.Chain.FundamentalTheorem"},{"id":"n9382","layer":"formal","project":"p8","title":"MPSChainTensor.fundamentalTheorem_injective_chain_gaugePhase","kind":"theorem","summary":"∀ d D n : Nat (A B : MPSChainTensor d D n), LT.lt 0 n → LT.lt 0 D → A.IsInjective → (MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSChainTensor.fundamentalTheorem_injective_chain_gaugePhase","module":"TNLean.MPS.Chain.FundamentalTheorem"},{"id":"n9383","layer":"formal","project":"p8","title":"MPSTensor.chainCombinedTensor_smul_chain","kind":"theorem","summary":"∀ d D n : Nat (A : Fin n → MPSTensor d D) (ζ : Complex), Eq (MPSTensor.chainCombinedTensor fun…","labels":[],"detail_key":"p8","name":"MPSTensor.chainCombinedTensor_smul_chain","module":"TNLean.MPS.Chain.FundamentalTheorem"},{"id":"n9384","layer":"formal","project":"p8","title":"Kraus.IsInjective.exists_decomposition","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → ∀ (X : Matrix (Fin D) (Fin D) Complex), Ex…","labels":[],"detail_key":"p8","name":"Kraus.IsInjective.exists_decomposition","module":"TNLean.MPS.Chain.OneSidedInverse"},{"id":"n9385","layer":"formal","project":"p8","title":"Kraus.IsInjective.exists_rightInverse","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → Exists fun Ψ => ∀ (X : Matrix (Fin D) (Fin…","labels":[],"detail_key":"p8","name":"Kraus.IsInjective.exists_rightInverse","module":"TNLean.MPS.Chain.OneSidedInverse"},{"id":"n9386","layer":"formal","project":"p8","title":"Kraus.decompositionMap","kind":"def","summary":"d D : Nat → A : MPSTensor d D → Kraus.IsInjective A → LinearMap (RingHom.id 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B₁ B₂ : MPSTensor d D), LT.lt 0 D → Kraus.IsInjective A₁ → Kraus.IsInjective…","labels":[],"detail_key":"p8","name":"MPSTensor.tensor_proportional","module":"TNLean.MPS.Chain.TensorEquality"},{"id":"n9390","layer":"formal","project":"p8","title":"MPSChainTensor.exists_gauge_to_first_of_cyclicShift_gaugeEquiv","kind":"theorem","summary":"∀ d D n : Nat [inst : NeZero n] A : MPSChainTensor d D n, A.GaugeEquiv A.cyclicShift → ∀ (k : F…","labels":[],"detail_key":"p8","name":"MPSChainTensor.exists_gauge_to_first_of_cyclicShift_gaugeEquiv","module":"TNLean.MPS.Chain.TranslationInvariance"},{"id":"n9391","layer":"formal","project":"p8","title":"MPSChainTensor.ti_reduction_corollary","kind":"theorem","summary":"∀ d D n : Nat (A B : MPSTensor d D), LT.lt 0 n → Kraus.IsInjective A → (MPSTensor.chainCombined…","labels":[],"detail_key":"p8","name":"MPSChainTensor.ti_reduction_corollary","module":"TNLean.MPS.Chain.TranslationInvariance"},{"id":"n9392","layer":"formal","project":"p8","title":"MPSChainTensor.ti_tensors_collapse_to_single_gauge","kind":"theorem","summary":"∀ d D n : Nat (A B : MPSTensor d D), LT.lt 0 n → Kraus.IsInjective A → (MPSTensor.chainCombined…","labels":[],"detail_key":"p8","name":"MPSChainTensor.ti_tensors_collapse_to_single_gauge","module":"TNLean.MPS.Chain.TranslationInvariance"},{"id":"n9393","layer":"formal","project":"p8","title":"MPSChainTensor.ti_tensors_single_gauge","kind":"theorem","summary":"∀ d D n : Nat (A B : MPSTensor d D), LT.lt 0 n → Kraus.IsInjective A → (MPSTensor.chainCombined…","labels":[],"detail_key":"p8","name":"MPSChainTensor.ti_tensors_single_gauge","module":"TNLean.MPS.Chain.TranslationInvariance"},{"id":"n9394","layer":"formal","project":"p8","title":"MPSChainTensor.coeff_cyclicShift","kind":"theorem","summary":"∀ d D N : Nat (A : MPSChainTensor d D N) [NeZero N] (σ : Fin N → Fin d), Eq (A.cyclicShift.coef…","labels":[],"detail_key":"p8","name":"MPSChainTensor.coeff_cyclicShift","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9395","layer":"formal","project":"p8","title":"MPSChainTensor.coeff_eq_sum_cyclic","kind":"theorem","summary":"∀ d D N : Nat (A : MPSChainTensor d D N) [inst : NeZero N] (σ : Fin N → Fin d), Eq (A.coeff σ)…","labels":[],"detail_key":"p8","name":"MPSChainTensor.coeff_eq_sum_cyclic","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9396","layer":"formal","project":"p8","title":"OBCChainTensor","kind":"inductive","summary":"Nat → Nat → Nat → Type","labels":[],"detail_key":"p8","name":"OBCChainTensor","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9397","layer":"formal","project":"p8","title":"OBCChainTensor.IsTranslationInvariantState","kind":"def","summary":"d D N : Nat → OBCChainTensor d D N → 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1","labels":[],"detail_key":"p8","name":"OBCChainTensor.coeff_zero","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9401","layer":"formal","project":"p8","title":"OBCChainTensor.coeff_zeroPad","kind":"theorem","summary":"∀ d D N : Nat (A : OBCChainTensor d D N) [NeZero N] (σ : Fin N → Fin d), Eq (A.zeroPad.coeff σ)…","labels":[],"detail_key":"p8","name":"OBCChainTensor.coeff_zeroPad","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9402","layer":"formal","project":"p8","title":"OBCChainTensor.coeff_zeroPad_eq_zero_of_bondDim_eq_zero","kind":"theorem","summary":"∀ d D N : Nat (A : OBCChainTensor d D N) [NeZero N] (σ : Fin N → Fin d) (k : Fin (HAdd.hAdd N 1…","labels":[],"detail_key":"p8","name":"OBCChainTensor.coeff_zeroPad_eq_zero_of_bondDim_eq_zero","module":"TNLean.MPS.Chain.VaryingBondOBC"},{"id":"n9403","layer":"formal","project":"p8","title":"OBCChainTensor.coeff_zeroPad_zero","kind":"theorem","summary":"∀ d D : Nat (A : OBCChainTensor d D 0) (σ : Fin 0 → Fin d), Eq 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Matrix…","labels":[],"detail_key":"p8","name":"MPSTensor.physRealize","module":"TNLean.MPS.Chain.VirtualInsertion"},{"id":"n9407","layer":"formal","project":"p8","title":"MPSTensor.physRealize_mul","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (hA : Kraus.IsInjective A) (X Y : Matrix (Fin D) (Fin D) Comple…","labels":[],"detail_key":"p8","name":"MPSTensor.physRealize_mul","module":"TNLean.MPS.Chain.VirtualInsertion"},{"id":"n9408","layer":"formal","project":"p8","title":"MPSTensor.physRealize_spec","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (hA : Kraus.IsInjective A) (X : Matrix (Fin D) (Fin D) Complex)…","labels":[],"detail_key":"p8","name":"MPSTensor.physRealize_spec","module":"TNLean.MPS.Chain.VirtualInsertion"},{"id":"n9409","layer":"formal","project":"p8","title":"MPSTensor.SameMPV.blockTensor","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.SameMPV B → ∀ (L : Nat), MPSTensor.SameMPV (MPSTensor.blockT…","labels":[],"detail_key":"p8","name":"MPSTensor.SameMPV.blockTensor","module":"TNLean.MPS.Core.Blocking"},{"id":"n9410","layer":"formal","project":"p8","title":"MPSTensor.blockTensor","kind":"def","summary":"d D : Nat → (Fin d → Matrix (Fin D) (Fin D) Complex) → (L : Nat) → Fin (MPSTensor.blockPhysDim…","labels":[],"detail_key":"p8","name":"MPSTensor.blockTensor","module":"TNLean.MPS.Core.Blocking"},{"id":"n9411","layer":"formal","project":"p8","title":"MPSTensor.blockedConfigEquiv","kind":"def","summary":"(d N L : Nat) → Equiv (Fin N → Fin (MPSTensor.blockPhysDim d L)) (Fin (HMul.hMul N L) → Fin d)","labels":[],"detail_key":"p8","name":"MPSTensor.blockedConfigEquiv","module":"TNLean.MPS.Core.Blocking"},{"id":"n9412","layer":"formal","project":"p8","title":"MPSTensor.evalWord_blockTensor","kind":"theorem","summary":"∀ d D : Nat (A : Fin d → Matrix (Fin D) (Fin D) Complex) (L : Nat) (w : List (Fin (MPSTensor.bl…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_blockTensor","module":"TNLean.MPS.Core.Blocking"},{"id":"n9413","layer":"formal","project":"p8","title":"MPSTensor.isNBlkInjective_iff_blockTensor_isInjective","kind":"theorem","summary":"∀ d D : Nat (A : Fin d → Matrix (Fin D) (Fin D) Complex) (N : Nat), Iff (Kraus.IsNBlkInjective…","labels":[],"detail_key":"p8","name":"MPSTensor.isNBlkInjective_iff_blockTensor_isInjective","module":"TNLean.MPS.Core.Blocking"},{"id":"n9414","layer":"formal","project":"p8","title":"MPSTensor.isNBlkInjective_mul_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) N m : Nat, LT.lt 0 m → Kraus.IsNBlkInjective A N → Kraus.IsNBlk…","labels":[],"detail_key":"p8","name":"MPSTensor.isNBlkInjective_mul_of_isNBlkInjective","module":"TNLean.MPS.Core.Blocking"},{"id":"n9415","layer":"formal","project":"p8","title":"MPSTensor.leftCanonical_blockTensor","kind":"theorem","summary":"∀ d D : Nat (A : Fin d → Matrix (Fin 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(…","labels":[],"detail_key":"p8","name":"MPSTensor.isPrimitive_transferMap_blockTensor_of_dvd","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9426","layer":"formal","project":"p8","title":"MPSTensor.lcmPeriod","kind":"def","summary":"k : Nat → (Fin k → Nat) → Nat","labels":[],"detail_key":"p8","name":"MPSTensor.lcmPeriod","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9427","layer":"formal","project":"p8","title":"MPSTensor.mpv_blockTensor_eq_mpv_blockedFlatConfig","kind":"theorem","summary":"∀ d D' : Nat (T : MPSTensor d D') (p : Nat) N : Nat (σ : Fin N → Fin (MPSTensor.blockPhysDim d…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_blockTensor_eq_mpv_blockedFlatConfig","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9428","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂Pos_blockTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.SameMPV₂Pos B → ∀ (p : Nat), LT.lt…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂Pos_blockTensor","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9429","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂Pos_blockTensor_toTensorFromBlocks","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (A : MPSTensor d D) (μ : Fin r → Complex) (blocks : (k : Fin r)…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂Pos_blockTensor_toTensorFromBlocks","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9430","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂Pos_toTensorFromBlocks_blockPower","kind":"theorem","summary":"∀ d rA rB : Nat dimA : Fin rA → Nat dimB : Fin rB → Nat (μA : Fin rA → Complex) (blocksA : (k :…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂Pos_toTensorFromBlocks_blockPower","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9431","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂_blockTensor_toTensorFromBlocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂_blockTensor_toTensorFromBlocks","module":"TNLean.MPS.Core.BlockingInfrastructure"},{"id":"n9432","layer":"formal","project":"p8","title":"MPSTensor.transferMap_blockTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), Eq (Kraus.transferMap (MPSTensor.blockTensor A L)) (…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_blockTensor","module":"TNLean.MPS.Core.BlockingTransfer"},{"id":"n9433","layer":"formal","project":"p8","title":"MPSTensor.transferMap_blockTensor_fixedPoint","kind":"theorem","summary":"∀ d D : Nat (A : 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(K…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_blockTensor_quasi_idempotent","module":"TNLean.MPS.Core.BlockingTransfer"},{"id":"n9436","layer":"formal","project":"p8","title":"MPSTensor.IsLeftCanonical","kind":"def","summary":"d D : Nat → MPSTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.IsLeftCanonical","module":"TNLean.MPS.Core.CanonicalNormalization"},{"id":"n9437","layer":"formal","project":"p8","title":"MPSTensor.connectedCorrelator_eq_trace_transfer_tracelessPart","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρ X Y : MPSTensor.Mat D) (n : Nat), Eq ((Kraus.transferMap A)…","labels":[],"detail_key":"p8","name":"MPSTensor.connectedCorrelator_eq_trace_transfer_tracelessPart","module":"TNLean.MPS.Core.CorrelationReduction"},{"id":"n9438","layer":"formal","project":"p8","title":"MPSTensor.trace_tracelessPart","kind":"theorem","summary":"∀ D : Nat (ρ X : MPSTensor.Mat D), Eq (Matrix.trace ρ) 1 → Eq (Matrix.trace (MPSTensor.traceles…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_tracelessPart","module":"TNLean.MPS.Core.CorrelationReduction"},{"id":"n9439","layer":"formal","project":"p8","title":"MPSTensor.tracelessPart","kind":"def","summary":"D : Nat → MPSTensor.Mat D → MPSTensor.Mat D → MPSTensor.Mat D","labels":[],"detail_key":"p8","name":"MPSTensor.tracelessPart","module":"TNLean.MPS.Core.CorrelationReduction"},{"id":"n9440","layer":"formal","project":"p8","title":"MPSTensor.connectedCorrelator","kind":"def","summary":"d D : Nat → MPSTensor d D → MPSTensor.Mat D → MPSTensor.Mat D → MPSTensor.Mat D → Nat → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.connectedCorrelator","module":"TNLean.MPS.Core.Correlations"},{"id":"n9441","layer":"formal","project":"p8","title":"MPSTensor.connectedCorrelator_bound","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρR X Y : MPSTensor.Mat D) (CXY : Real) (lam₂ : Complex), (∀ (n…","labels":[],"detail_key":"p8","name":"MPSTensor.connectedCorrelator_bound","module":"TNLean.MPS.Core.Correlations"},{"id":"n9442","layer":"formal","project":"p8","title":"MPSTensor.connectedCorrelator_eq_sum","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρR X Y : MPSTensor.Mat D) (c lam : Fin (HSub.hSub (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPSTensor.connectedCorrelator_eq_sum","module":"TNLean.MPS.Core.Correlations"},{"id":"n9443","layer":"formal","project":"p8","title":"MPSTensor.correlationLength","kind":"def","summary":"Complex → Real","labels":[],"detail_key":"p8","name":"MPSTensor.correlationLength","module":"TNLean.MPS.Core.Correlations"},{"id":"n9444","layer":"formal","project":"p8","title":"MPSTensor.correlationLength_pos","kind":"theorem","summary":"∀ lam₂ : Complex, LT.lt 0 (norm lam₂) → LT.lt (norm lam₂) 1 → LT.lt 0 (MPSTensor.correlationLen…","labels":[],"detail_key":"p8","name":"MPSTensor.correlationLength_pos","module":"TNLean.MPS.Core.Correlations"},{"id":"n9445","layer":"formal","project":"p8","title":"MPSTensor.onePointExpectation","kind":"def","summary":"D : Nat → MPSTensor.Mat D → MPSTensor.Mat D → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.onePointExpectation","module":"TNLean.MPS.Core.Correlations"},{"id":"n9446","layer":"formal","project":"p8","title":"MPSTensor.twoPointExpectation","kind":"def","summary":"d D : Nat → MPSTensor d D → MPSTensor.Mat D → MPSTensor.Mat D → MPSTensor.Mat D → Nat → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.twoPointExpectation","module":"TNLean.MPS.Core.Correlations"},{"id":"n9447","layer":"formal","project":"p8","title":"MPSTensor.evalWord_ofFn_apply","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat) (x : Fin N → Fin d) (a b : Fin D), Eq (Kraus.evalWord…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_ofFn_apply","module":"TNLean.MPS.Core.CyclicTrace"},{"id":"n9448","layer":"formal","project":"p8","title":"MPSTensor.evalWord_ofFn_eq_prod","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) N : Nat (σ : Fin N → Fin d), Eq (Kraus.evalWord A (List.ofFn σ)…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_ofFn_eq_prod","module":"TNLean.MPS.Core.CyclicTrace"},{"id":"n9449","layer":"formal","project":"p8","title":"MPSTensor.trace_evalWord_eq_sum_cyclic","kind":"theorem","summary":"∀ d D n : Nat [inst : NeZero n] (A : MPSTensor d D) (σ : Fin n → Fin d), Eq (Kraus.evalWord A (…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_evalWord_eq_sum_cyclic","module":"TNLean.MPS.Core.CyclicTrace"},{"id":"n9450","layer":"formal","project":"p8","title":"MPSTensor.CanonicalForm","kind":"inductive","summary":"Nat → 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W.conjTra…","labels":[],"detail_key":"p8","name":"MPSTensor.isInjective_kraus_isometry_iff","module":"TNLean.MPS.Core.PhysicalIndexMixing"},{"id":"n9454","layer":"formal","project":"p8","title":"MPSTensor.isInjective_of_kraus_mixing_isInjective","kind":"theorem","summary":"∀ d D m : Nat (B : MPSTensor d D) (W : Matrix (Fin m) (Fin d) Complex), (Kraus.IsInjective fun…","labels":[],"detail_key":"p8","name":"MPSTensor.isInjective_of_kraus_mixing_isInjective","module":"TNLean.MPS.Core.PhysicalIndexMixing"},{"id":"n9455","layer":"formal","project":"p8","title":"MPSTensor.isLeftCanonical_kraus_isometry","kind":"theorem","summary":"∀ d D m : Nat (B : MPSTensor d D) (W : Matrix (Fin m) (Fin d) Complex), Eq (HMul.hMul W.conjTra…","labels":[],"detail_key":"p8","name":"MPSTensor.isLeftCanonical_kraus_isometry","module":"TNLean.MPS.Core.PhysicalIndexMixing"},{"id":"n9456","layer":"formal","project":"p8","title":"MPSTensor.transferMap_kraus_isometry","kind":"theorem","summary":"∀ d D m : Nat (B : MPSTensor d D) (W : Matrix (Fin m) (Fin d) Complex), Eq (HMul.hMul W.conjTra…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_kraus_isometry","module":"TNLean.MPS.Core.PhysicalIndexMixing"},{"id":"n9457","layer":"formal","project":"p8","title":"MPSTensor.leftCanonical_reindexPhysical_equiv","kind":"theorem","summary":"∀ d₁ d₂ D : Nat (e : Equiv (Fin d₁) (Fin d₂)) (A : MPSTensor d₂ D), Iff (Eq (Finset.univ.sum fu…","labels":[],"detail_key":"p8","name":"MPSTensor.leftCanonical_reindexPhysical_equiv","module":"TNLean.MPS.Core.PhysicalReindexTransport"},{"id":"n9458","layer":"formal","project":"p8","title":"MPSTensor.transferMap_reindexPhysical_equiv","kind":"theorem","summary":"∀ d₁ d₂ D : Nat (e : Equiv (Fin d₁) (Fin d₂)) (A : MPSTensor d₂ D), Eq (Kraus.transferMap (Krau…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_reindexPhysical_equiv","module":"TNLean.MPS.Core.PhysicalReindexTransport"},{"id":"n9459","layer":"formal","project":"p8","title":"MPSTensor.evalWord_rotatePhysical_ofFn","kind":"theorem","summary":"∀ d D : Nat (M : Matrix (Fin d) (Fin d) Complex) (A : MPSTensor d D) (N : Nat) (s : Fin N → Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_rotatePhysical_ofFn","module":"TNLean.MPS.Core.PhysicalRotation"},{"id":"n9460","layer":"formal","project":"p8","title":"MPSTensor.mpv_rotatePhysical","kind":"theorem","summary":"∀ d D : Nat (M : Matrix (Fin d) (Fin d) Complex) (A : MPSTensor d D) N : Nat (s : Fin N → Fin d…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_rotatePhysical","module":"TNLean.MPS.Core.PhysicalRotation"},{"id":"n9461","layer":"formal","project":"p8","title":"MPSTensor.rotatePhysical","kind":"def","summary":"d D : Nat → Matrix (Fin d) (Fin d) Complex → MPSTensor d D → MPSTensor d 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(Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction","module":"TNLean.MPS.Core.Reduction"},{"id":"n9465","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.bondDim_le","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.bondDim_le","module":"TNLean.MPS.Core.Reduction"},{"id":"n9466","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.evalWord","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.evalWord","module":"TNLean.MPS.Core.Reduction"},{"id":"n9467","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.evalWord_nil","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.evalWord_nil","module":"TNLean.MPS.Core.Reduction"},{"id":"n9468","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.evalWord_smul_target","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.evalWord_smul_target","module":"TNLean.MPS.Core.Reduction"},{"id":"n9469","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.iff_forall_evalWord","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.iff_forall_evalWord","module":"TNLean.MPS.Core.Reduction"},{"id":"n9470","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.mul_eq_one","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.mul_eq_one","module":"TNLean.MPS.Core.Reduction"},{"id":"n9471","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.reciprocal_smul","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.reciprocal_smul","module":"TNLean.MPS.Core.Reduction"},{"id":"n9472","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.smul","kind":"theorem","summary":"∀ d D₁ D₂ : Nat B : MPSTensor d D₂ A : MPSTensor d D₁ V : Matrix (Fin D₁) (Fin D₂) Complex W :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.smul","module":"TNLean.MPS.Core.Reduction"},{"id":"n9473","layer":"formal","project":"p8","title":"MPSTensor.IsReduction.blockTensor","kind":"theorem","summary":"∀ d D_A D_B : Nat B : MPSTensor d D_B A : MPSTensor d D_A V : Matrix (Fin D_A) (Fin D_B) 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Com…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReduction.reindexPhysical","module":"TNLean.MPS.Core.ReductionBlocking"},{"id":"n9478","layer":"formal","project":"p8","title":"MPSTensor.IsReductionExteriorBufferLength","kind":"def","summary":"d D_A D_B : Nat → MPSTensor d D_B → MPSTensor d D_A → Matrix (Fin D_A) (Fin D_B) Complex → Matr…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReductionExteriorBufferLength","module":"TNLean.MPS.Core.ReductionBlocking"},{"id":"n9479","layer":"formal","project":"p8","title":"MPSTensor.IsReductionExteriorBufferLength.blockTensor","kind":"theorem","summary":"∀ d D_A D_B : Nat B : MPSTensor d D_B A : MPSTensor d D_A V : Matrix (Fin D_A) (Fin D_B) Comple…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReductionExteriorBufferLength.blockTensor","module":"TNLean.MPS.Core.ReductionBlocking"},{"id":"n9480","layer":"formal","project":"p8","title":"MPSTensor.IsReductionExteriorBufferLength.mono","kind":"theorem","summary":"∀ d D_A D_B : Nat B : MPSTensor d D_B A : MPSTensor d D_A V : Matrix (Fin D_A) (Fin D_B) Comple…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReductionExteriorBufferLength.mono","module":"TNLean.MPS.Core.ReductionBlocking"},{"id":"n9481","layer":"formal","project":"p8","title":"MPSTensor.IsReductionExteriorBufferLength.reciprocal_smul","kind":"theorem","summary":"∀ d D_A D_B : Nat B : MPSTensor d D_B A : MPSTensor d D_A V : Matrix (Fin D_A) (Fin D_B) Comple…","labels":[],"detail_key":"p8","name":"MPSTensor.IsReductionExteriorBufferLength.reciprocal_smul","module":"TNLean.MPS.Core.ReductionBlocking"},{"id":"n9482","layer":"formal","project":"p8","title":"MPSTensor.IsReductionExteriorBufferLength.reciprocal_smul_iff","kind":"theorem","summary":"∀ d D_A D_B : Nat B : MPSTensor d D_B A : MPSTensor d D_A V : Matrix (Fin D_A) (Fin D_B) 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D_A)…","labels":[],"detail_key":"p8","name":"MPSTensor.reductionCrossMatrix_apply","module":"TNLean.MPS.Core.ReductionCrossMatrix"},{"id":"n9488","layer":"formal","project":"p8","title":"MPSTensor.reductionCrossMatrix_coefficientDualInverse","kind":"theorem","summary":"∀ d D_A : Nat (A : MPSTensor d D_A) (hA : Kraus.IsInjective A), Eq (A.reductionCrossMatrix (A.c…","labels":[],"detail_key":"p8","name":"MPSTensor.reductionCrossMatrix_coefficientDualInverse","module":"TNLean.MPS.Core.ReductionCrossMatrix"},{"id":"n9489","layer":"formal","project":"p8","title":"MPSTensor.reductionCrossMatrix_pow_apply","kind":"theorem","summary":"∀ d D_A D_B : Nat (B : MPSTensor d D_B) (C : MPSTensor d D_A) (n : Nat) (b b' : Fin D_B) (a a'…","labels":[],"detail_key":"p8","name":"MPSTensor.reductionCrossMatrix_pow_apply","module":"TNLean.MPS.Core.ReductionCrossMatrix"},{"id":"n9490","layer":"formal","project":"p8","title":"MPSTensor.reductionCrossMatrix_pow_eq_sum","kind":"theorem","summary":"∀ d D_A D_B : Nat (B : MPSTensor d D_B) (C : MPSTensor d D_A) (n : Nat), Eq (HPow.hPow (B.reduc…","labels":[],"detail_key":"p8","name":"MPSTensor.reductionCrossMatrix_pow_eq_sum","module":"TNLean.MPS.Core.ReductionCrossMatrix"},{"id":"n9491","layer":"formal","project":"p8","title":"MPSTensor.reductionOpenBoundaryContraction","kind":"def","summary":"d D_A D_B : Nat → MPSTensor d D_B → MPSTensor d D_A → Nat → List (Fin d) → Fin D_A → Fin D_A →…","labels":[],"detail_key":"p8","name":"MPSTensor.reductionOpenBoundaryContraction","module":"TNLean.MPS.Core.ReductionCrossMatrix"},{"id":"n9492","layer":"formal","project":"p8","title":"MPSTensor.reductionOpenBoundaryContraction_of_sameMPV₂Pos","kind":"theorem","summary":"∀ d D_A D_B : Nat (A : MPSTensor d D_A) (B : MPSTensor d D_B) (hA : Kraus.IsInjective A), 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MPSTensor.SameMPV…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV_of_sameMPVFrom_of_injective","module":"TNLean.MPS.FundamentalTheorem.FiniteLength"},{"id":"n9693","layer":"formal","project":"p8","title":"MPSTensor.CanonicalForm.toTensor_eq_toTensorFromBlocks","kind":"theorem","summary":"∀ d : Nat (C : MPSTensor.CanonicalForm d), Eq C.toTensor (MPSTensor.toTensorFromBlocks C.μ C.bl…","labels":[],"detail_key":"p8","name":"MPSTensor.CanonicalForm.toTensor_eq_toTensorFromBlocks","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9694","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_canonicalForm_sameStructure","kind":"theorem","summary":"∀ d : Nat (C : MPSTensor.CanonicalForm d) (B : (k : Fin C.numBlocks) → MPSTensor d (C.blockDim…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_canonicalForm_sameStructure","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9695","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_multiBlock_blocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A B : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), Kraus.…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_multiBlock_blocks","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9696","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_multiBlock_global","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)),…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_multiBlock_global","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9697","layer":"formal","project":"p8","title":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockConj","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)) (…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockConj","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9698","layer":"formal","project":"p8","title":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockGauge","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)),…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockGauge","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9699","layer":"formal","project":"p8","title":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockGaugePhase_weight","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μA μB : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugeEquiv_toTensorFromBlocks_of_blockGaugePhase_weight","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9700","layer":"formal","project":"p8","title":"MPSTensor.sameMPV_toTensorFromBlocks_of_blockSameMPV","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)),…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV_toTensorFromBlocks_of_blockSameMPV","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9701","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂_toTensorFromBlocks_cast","kind":"theorem","summary":"∀ d r : Nat dimA dimB : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dimA…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂_toTensorFromBlocks_cast","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9702","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂_toTensorFromBlocks_perm","kind":"theorem","summary":"∀ d rA rB : Nat dim : Fin rB → Nat (μ : Fin rB → Complex) (A : (k : Fin rB) → MPSTensor d (dim…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂_toTensorFromBlocks_perm","module":"TNLean.MPS.FundamentalTheorem.Multi"},{"id":"n9703","layer":"formal","project":"p8","title":"MPSTensor.exists_unitary_conj_of_positive_perBlockLinearExtension","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (A B : (k : Fin r) → MPSTensor d…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitary_conj_of_positive_perBlockLinearExtension","module":"TNLean.MPS.FundamentalTheorem.PositiveLinearExtension"},{"id":"n9704","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_multiBlock_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [inst : ∀ (k : Fin r), NeZero (dim k)] (A B : (k : Fin r) → MPSTe…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_multiBlock_decomposition","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9705","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_multiBlock_explicit","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A B : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), Kraus.…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_multiBlock_explicit","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9706","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_multiBlock_full","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)),…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_multiBlock_full","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9707","layer":"formal","project":"p8","title":"MPSTensor.perBlockLinearExtension","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → (A B : (k : Fin r) → MPSTensor d (dim k)) → (∀ (k : Fin r), Kra…","labels":[],"detail_key":"p8","name":"MPSTensor.perBlockLinearExtension","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9708","layer":"formal","project":"p8","title":"MPSTensor.perBlockLinearExtension_bijective","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (A B : (k : Fin r) → MPSTensor d…","labels":[],"detail_key":"p8","name":"MPSTensor.perBlockLinearExtension_bijective","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9709","layer":"formal","project":"p8","title":"MPSTensor.perBlockLinearExtension_mul","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A B : (k : Fin r) → MPSTensor d (dim k)) (hA : ∀ (k : Fin r), Kr…","labels":[],"detail_key":"p8","name":"MPSTensor.perBlockLinearExtension_mul","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9710","layer":"formal","project":"p8","title":"MPSTensor.perBlockLinearExtension_spec","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A B : (k : Fin r) → MPSTensor d (dim k)) (hA : ∀ (k : Fin r), Kr…","labels":[],"detail_key":"p8","name":"MPSTensor.perBlockLinearExtension_spec","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9711","layer":"formal","project":"p8","title":"MPSTensor.perBlock_sameMPV_iff_gaugeEquiv","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A B : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), Kraus.…","labels":[],"detail_key":"p8","name":"MPSTensor.perBlock_sameMPV_iff_gaugeEquiv","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9712","layer":"formal","project":"p8","title":"MPSTensor.piAlgEquiv","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → [∀ (k : Fin r), NeZero (dim k)] → (A B : (k : Fin r) → MPSTenso…","labels":[],"detail_key":"p8","name":"MPSTensor.piAlgEquiv","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9713","layer":"formal","project":"p8","title":"MPSTensor.piAlgEquiv_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [inst : ∀ (k : Fin r), NeZero (dim k)] (A B : (k : Fin r) → MPSTe…","labels":[],"detail_key":"p8","name":"MPSTensor.piAlgEquiv_decomposition","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9714","layer":"formal","project":"p8","title":"MPSTensor.piTraceMulRightPi_ker_eq_bot","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), Kraus.Is…","labels":[],"detail_key":"p8","name":"MPSTensor.piTraceMulRightPi_ker_eq_bot","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9715","layer":"formal","project":"p8","title":"MPSTensor.piTrace_mul_right_eq_zero","kind":"theorem","summary":"∀ r : Nat dim : Fin r → Nat (M : (k : Fin r) → Matrix (Fin (dim k)) (Fin (dim k)) Complex), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.piTrace_mul_right_eq_zero","module":"TNLean.MPS.FundamentalTheorem.ProductAlgebra"},{"id":"n9716","layer":"formal","project":"p8","title":"MPSTensor.exists_ge_not_forall_mpv_eq_mul_of_not_gaugePhaseEquiv_of_irreducible_TP","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsIrreducibleFamily A → Kraus.IsIrreducible…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_ge_not_forall_mpv_eq_mul_of_not_gaugePhaseEquiv_of_irreducible_TP","module":"TNLean.MPS.FundamentalTheorem.Proportional"},{"id":"n9717","layer":"formal","project":"p8","title":"MPSTensor.mpvOverlap_norm_tendsto_one_of_eventually_proportionalMPV₂","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Filter.Tendsto (fun N => A.mpvOverla…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvOverlap_norm_tendsto_one_of_eventually_proportionalMPV₂","module":"TNLean.MPS.FundamentalTheorem.Proportional"},{"id":"n9718","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.combined_family_eventually_li","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.combined_family_eventually_li","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Api"},{"id":"n9719","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.cross_overlap_basis_tendsto_zero","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ j k : Fin P.b…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.cross_overlap_basis_tendsto_zero","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Api"},{"id":"n9720","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_eq_sum_weight_pow","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (N : Nat) (j : Fin P.basisCount), Eq (P.coeff N…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_eq_sum_weight_pow","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Api"},{"id":"n9721","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm","kind":"inductive","summary":"d : Nat → MPSTensor.SectorDecomposition d → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Basic"},{"id":"n9722","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.weight_unit_exists","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → Exists fun j =>…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.weight_unit_exists","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Basic"},{"id":"n9723","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.IsBNTCanonicalForm.blockTensor","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ p : Nat, LT.l…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.IsBNTCanonicalForm.blockTensor","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9724","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.IsBNTCanonicalForm.reindexPhysical","kind":"theorem","summary":"∀ d d' : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (e : Equiv…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.IsBNTCanonicalForm.reindexPhysical","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9725","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor","kind":"def","summary":"d : Nat → MPSTensor.SectorDecomposition d → (p : Nat) → MPSTensor.SectorDecomposition (MPSTenso…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9726","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_basis","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat) (j : Fin P.basisCount), Eq ((P.blockT…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_basis","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9727","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_copies","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), Eq (P.blockTensor p).copies P.copies","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_copies","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9728","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_flatBasis","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat) (s : Fin P.totalCopies), Eq ((P.block…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_flatBasis","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9729","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_flatDim","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), Eq (P.blockTensor p).flatDim P.flatD…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_flatDim","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9730","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_flatWeight","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat) (s : Fin P.totalCopies), Eq ((P.block…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_flatWeight","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9731","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_totalCopies","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), Eq (P.blockTensor p).totalCopies P.t…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_totalCopies","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9732","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_totalDim","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), Eq (P.blockTensor p).totalDim P.tota…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_totalDim","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9733","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.blockTensor_weight","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat) (j : Fin P.basisCount) (q : Fin (P.co…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.blockTensor_weight","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9734","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_blockTensor","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p N : Nat) (j : Fin P.basisCount), Eq ((P.bloc…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_blockTensor","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9735","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.reindexPhysical","kind":"def","summary":"d d' : Nat → MPSTensor.SectorDecomposition d → Equiv (Fin d') (Fin d) → MPSTensor.SectorDecompo…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.reindexPhysical","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9736","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.sameMPV₂_blockTensor_toTensor","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), MPSTensor.SameMPV₂ (MPSTensor.blockT…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.sameMPV₂_blockTensor_toTensor","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Blocking"},{"id":"n9737","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.exists_isBNTCanonicalForm","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), data.IsWeightNormalized → Ne A…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.exists_isBNTCanonicalForm","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CanonicalFormBridge"},{"id":"n9738","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.exists_isBNTCanonicalForm_exact","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), data.IsWeightNormalized → Ne A…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.exists_isBNTCanonicalForm_exact","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CanonicalFormBridge"},{"id":"n9739","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.norm_phase_of_matched_mpv","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.norm_phase_of_matched_mpv","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CoeffIdentity"},{"id":"n9740","layer":"formal","project":"p8","title":"MPSTensor.coeff_identity_via_global_gauge","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.coeff_identity_via_global_gauge","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CoeffIdentity"},{"id":"n9741","layer":"formal","project":"p8","title":"MPSTensor.coeff_identity_via_matched_mpv_phase","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → P.toTensor.Sa…","labels":[],"detail_key":"p8","name":"MPSTensor.coeff_identity_via_matched_mpv_phase","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CoeffIdentity"},{"id":"n9742","layer":"formal","project":"p8","title":"MPSTensor.coeff_identity_via_matched_mpv_phase_proportional","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → P.toTensor.Ev…","labels":[],"detail_key":"p8","name":"MPSTensor.coeff_identity_via_matched_mpv_phase_proportional","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CoeffIdentity"},{"id":"n9743","layer":"formal","project":"p8","title":"MPSTensor.SectorBNTCopyWeightMatching","kind":"inductive","summary":"d : Nat → P Q : MPSTensor.SectorDecomposition d → Equiv (Fin Q.basisCount) (Fin P.basisCount) →…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorBNTCopyWeightMatching","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CopyWeightMatching"},{"id":"n9744","layer":"formal","project":"p8","title":"MPSTensor.SectorBNTCopyWeightMatching.of_coeff_identity","kind":"def","summary":"d : Nat → P Q : MPSTensor.SectorDecomposition d → (β : Equiv (Fin Q.basisCount) (Fin P.basisCou…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorBNTCopyWeightMatching.of_coeff_identity","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.CopyWeightMatching"},{"id":"n9745","layer":"formal","project":"p8","title":"MPSTensor.exists_block_match_exact","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → 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MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_global_gauge","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9749","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_equal_matched_copy_weight_witnesses","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_matched_copy_weight_witnesses","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9750","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_equal_sector_data","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_sector_data","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9751","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_proportional_global_gauge_of_coeff_identity","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_proportional_global_gauge_of_coeff_identity","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9752","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_proportional_global_gauge_of_copy_weight_matching","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_proportional_global_gauge_of_copy_weight_matching","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9753","layer":"formal","project":"p8","title":"MPSTensor.matched_p_basis","kind":"def","summary":"d : Nat → P Q : MPSTensor.SectorDecomposition d → (β : Equiv (Fin Q.basisCount) (Fin P.basisCou…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_p_basis","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9754","layer":"formal","project":"p8","title":"MPSTensor.matched_p_weight","kind":"def","summary":"d : Nat → P Q : MPSTensor.SectorDecomposition d → (β : Equiv (Fin Q.basisCount) (Fin P.basisCou…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_p_weight","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9755","layer":"formal","project":"p8","title":"MPSTensor.sector_bnt_global_gauge_of_copy_weight_matching","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d (β : Equiv (Fin Q.basisCount) (Fin P.basisCount…","labels":[],"detail_key":"p8","name":"MPSTensor.sector_bnt_global_gauge_of_copy_weight_matching","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9756","layer":"formal","project":"p8","title":"MPSTensor.sector_bnt_global_gauge_of_matched_weights","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d (β : Equiv (Fin Q.basisCount) (Fin P.basisCount…","labels":[],"detail_key":"p8","name":"MPSTensor.sector_bnt_global_gauge_of_matched_weights","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Fundamental"},{"id":"n9757","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_equal_mps_gaugeEquiv_literal","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_mps_gaugeEquiv_literal","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9758","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_equal_mps_gaugeEquiv_witnesses","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_mps_gaugeEquiv_witnesses","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9759","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_equal_canonicalForm","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_equal_canonicalForm","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9760","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_equal_canonicalForm_unitary","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_equal_canonicalForm_unitary","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9761","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_proportional_canonicalForm","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_proportional_canonicalForm","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9762","layer":"formal","project":"p8","title":"MPSTensor.permGL","kind":"def","summary":"n : Type u_1 → [inst : DecidableEq n] → [inst_1 : Fintype n] → Equiv.Perm n → Matrix.GeneralLin…","labels":[],"detail_key":"p8","name":"MPSTensor.permGL","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9763","layer":"formal","project":"p8","title":"MPSTensor.permGL_inv_val","kind":"theorem","summary":"∀ n : Type u_1 [inst : DecidableEq n] [inst_1 : Fintype n] (σ : Equiv.Perm n), Eq (↑(Inv.inv (M…","labels":[],"detail_key":"p8","name":"MPSTensor.permGL_inv_val","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9764","layer":"formal","project":"p8","title":"MPSTensor.permGL_val","kind":"theorem","summary":"∀ n : Type u_1 [inst : DecidableEq n] [inst_1 : Fintype n] (σ : Equiv.Perm n), Eq (↑(MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.permGL_val","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9765","layer":"formal","project":"p8","title":"MPSTensor.permMatrix_conj_eq_submatrix","kind":"theorem","summary":"∀ n : Type u_1 [inst : DecidableEq n] [inst_1 : Fintype n] (σ : Equiv.Perm n) (M : Matrix n n C…","labels":[],"detail_key":"p8","name":"MPSTensor.permMatrix_conj_eq_submatrix","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.FundamentalCoord"},{"id":"n9766","layer":"formal","project":"p8","title":"MPSTensor.bijection_from_matches","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.bijection_from_matches","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.MatchAux"},{"id":"n9767","layer":"formal","project":"p8","title":"MPSTensor.exists_isBNTCanonicalForm_exact_of_tp_primitive_irr_blocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_isBNTCanonicalForm_exact_of_tp_primitive_irr_blocks","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.PreparedReconstruction"},{"id":"n9768","layer":"formal","project":"p8","title":"MPSTensor.exists_block_match_exact_of_eventuallyProportional","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_block_match_exact_of_eventuallyProportional","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.ProportionalMatch.Core"},{"id":"n9769","layer":"formal","project":"p8","title":"MPSTensor.forall_k_exists_j_nondecaying_overlap_of_eventuallyProportional","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.forall_k_exists_j_nondecaying_overlap_of_eventuallyProportional","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.ProportionalMatch.Core"},{"id":"n9770","layer":"formal","project":"p8","title":"MPSTensor.bijective_match_of_eventuallyProportional","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.bijective_match_of_eventuallyProportional","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.ProportionalMatch"},{"id":"n9771","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_proportional_sector_match_witnesses","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_proportional_sector_match_witnesses","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.ProportionalMatch"},{"id":"n9772","layer":"formal","project":"p8","title":"MPSTensor.IsNormalTensor.exists_gaugeEquiv_singleSectorDecomposition","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNormalTensor → Exists fun B => And (A.GaugeEquiv B) (And (MP…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNormalTensor.exists_gaugeEquiv_singleSectorDecomposition","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.SingleSector"},{"id":"n9773","layer":"formal","project":"p8","title":"MPSTensor.isBNTCanonicalForm_singleSectorDecomposition","kind":"theorem","summary":"∀ d D : Nat [NeZero D] A : MPSTensor d D, Kraus.IsIrreducibleFamily A → A.IsLeftCanonical → Fil…","labels":[],"detail_key":"p8","name":"MPSTensor.isBNTCanonicalForm_singleSectorDecomposition","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.SingleSector"},{"id":"n9774","layer":"formal","project":"p8","title":"MPSTensor.singleSectorDecomposition","kind":"def","summary":"d D : Nat → MPSTensor d D → MPSTensor.SectorDecomposition d","labels":[],"detail_key":"p8","name":"MPSTensor.singleSectorDecomposition","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.SingleSector"},{"id":"n9775","layer":"formal","project":"p8","title":"MPSTensor.bijective_match_of_sameMPV","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.bijective_match_of_sameMPV","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.StrongMatch"},{"id":"n9776","layer":"formal","project":"p8","title":"MPSTensor.forall_k_exists_j_nondecaying_overlap_of_sameMPV","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.forall_k_exists_j_nondecaying_overlap_of_sameMPV","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.StrongMatch"},{"id":"n9777","layer":"formal","project":"p8","title":"MPSTensor.collapsedBntSectorDecomp_totalDim_eq_sum_dim_of_tp_primitive_irr","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.collapsedBntSectorDecomp_totalDim_eq_sum_dim_of_tp_primitive_irr","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9778","layer":"formal","project":"p8","title":"MPSTensor.dim_eq_of_MPVBlockPhaseEquiv_of_tp_primitive_irr","kind":"theorem","summary":"∀ d DX DY : Nat [NeZero DX] [NeZero DY] X : MPSTensor d DX Y : MPSTensor d DY, X.IsLeftCanonica…","labels":[],"detail_key":"p8","name":"MPSTensor.dim_eq_of_MPVBlockPhaseEquiv_of_tp_primitive_irr","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9779","layer":"formal","project":"p8","title":"MPSTensor.exists_gauge_choose_MPVBlockPhaseEquiv_of_tp_primitive_irr","kind":"theorem","summary":"∀ d DX DY : Nat [NeZero DX] [NeZero DY] X : MPSTensor d DX Y : MPSTensor d DY, X.IsLeftCanonica…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_gauge_choose_MPVBlockPhaseEquiv_of_tp_primitive_irr","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9780","layer":"formal","project":"p8","title":"MPSTensor.exists_isBNTCanonicalForm_of_tp_primitive_irr_blocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_isBNTCanonicalForm_of_tp_primitive_irr_blocks","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9781","layer":"formal","project":"p8","title":"MPSTensor.exists_isBNTCanonicalForm_of_tp_primitive_irr_blocks_and_totalDim","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_isBNTCanonicalForm_of_tp_primitive_irr_blocks_and_totalDim","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9782","layer":"formal","project":"p8","title":"MPSTensor.exists_prepared_BNT_blocks_afterBlocking_pos","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Exists fun p => And (LT.lt 0 p) (Exists fun r => Exists fun di…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_prepared_BNT_blocks_afterBlocking_pos","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9783","layer":"formal","project":"p8","title":"MPSTensor.gaugePhaseEquiv_of_MPVBlockPhaseEquiv_of_tp_primitive_irr","kind":"theorem","summary":"∀ d DX DY : Nat [NeZero DX] [NeZero DY] X : MPSTensor d DX Y : MPSTensor d DY, X.IsLeftCanonica…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhaseEquiv_of_MPVBlockPhaseEquiv_of_tp_primitive_irr","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9784","layer":"formal","project":"p8","title":"MPSTensor.isBNTCanonicalForm_collapsedBntSectorDecomp_of_tp_primitive_irr_blocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k : Fin r) → MPSTensor d (dim k)…","labels":[],"detail_key":"p8","name":"MPSTensor.isBNTCanonicalForm_collapsedBntSectorDecomp_of_tp_primitive_irr_blocks","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9785","layer":"formal","project":"p8","title":"MPSTensor.mpvPhaseClassData_dim_eq_of_tp_primitive_irr","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), LT.…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvPhaseClassData_dim_eq_of_tp_primitive_irr","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Supplier"},{"id":"n9786","layer":"formal","project":"p8","title":"MPSTensor.exists_isBNTCanonicalForm_afterBlocking_pos_normalized","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), (Exists fun N => And (LT.lt 0 N) (Exists fun σ => Ne (A.mpv σ)…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_isBNTCanonicalForm_afterBlocking_pos_normalized","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.SupplierNormalized"},{"id":"n9787","layer":"formal","project":"p8","title":"MPSTensor.exists_weight_normalization","kind":"theorem","summary":"∀ r : Nat, LT.lt 0 r → ∀ (μ : Fin r → Complex), (∀ (k : Fin r), Ne (μ k) 0) → Exists fun m => A…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_weight_normalization","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.SupplierNormalized"},{"id":"n9788","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_equal_unitary_global_gauge_witnesses","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_equal_unitary_global_gauge_witnesses","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Unitary"},{"id":"n9789","layer":"formal","project":"p8","title":"MPSTensor.ft_sector_bnt_proportional_unitary_sector_match_witnesses","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → MPSTensor.IsB…","labels":[],"detail_key":"p8","name":"MPSTensor.ft_sector_bnt_proportional_unitary_sector_match_witnesses","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.Unitary"},{"id":"n9790","layer":"formal","project":"p8","title":"MPSTensor.matched_sector_weight_equiv","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d (j₀ : Fin P.basisCount) (k₀' : Fin Q.basisCount…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_sector_weight_equiv","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.WeightEquiv"},{"id":"n9791","layer":"formal","project":"p8","title":"MPSTensor.matched_sector_weight_multiset_eq","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d (j₀ : Fin P.basisCount) (k₀' : Fin Q.basisCount…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_sector_weight_multiset_eq","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.WeightEquiv"},{"id":"n9792","layer":"formal","project":"p8","title":"MPSTensor.matched_sector_weight_pow_equiv_of_period_multiple","kind":"theorem","summary":"∀ d : Nat P Q : MPSTensor.SectorDecomposition d (j₀ : Fin P.basisCount) (k₀' : Fin Q.basisCount…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_sector_weight_pow_equiv_of_period_multiple","module":"TNLean.MPS.FundamentalTheorem.SectorBNT.WeightEquiv"},{"id":"n9793","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_mul_not_eventually_zero","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (p : Nat), LT.lt 0 p → ∀ (j : Fin P.basisCount)…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_mul_not_eventually_zero","module":"TNLean.MPS.FundamentalTheorem.SectorWeightComparison"},{"id":"n9794","layer":"formal","project":"p8","title":"MPSTensor.SectorWeightData.power_sums_eq_of_eventually_eq_hetero","kind":"theorem","summary":"∀ m n : Nat (a : Fin m → Complex) (b : Fin n → Complex), (∀ (i : Fin m), Ne (a i) 0) → (∀ (i :…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorWeightData.power_sums_eq_of_eventually_eq_hetero","module":"TNLean.MPS.FundamentalTheorem.SectorWeightComparison"},{"id":"n9795","layer":"formal","project":"p8","title":"MPSTensor.exists_unitaryConj_gaugePhase_of_leftCanonical_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] A B : MPSTensor d D, A.GaugePhaseEquiv B → A.IsLeftCanonical → B.IsLeftC…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitaryConj_gaugePhase_of_leftCanonical_irreducible","module":"TNLean.MPS.FundamentalTheorem.UnitaryGauge"},{"id":"n9796","layer":"formal","project":"p8","title":"MPSTensor.exists_unitaryConj_of_gaugePhase_data_of_leftCanonical_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] A B : MPSTensor d D (X : Matrix.GeneralLinearGroup (Fin D) Complex) (ζ :…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitaryConj_of_gaugePhase_data_of_leftCanonical_irreducible","module":"TNLean.MPS.FundamentalTheorem.UnitaryGauge"},{"id":"n9797","layer":"formal","project":"p8","title":"MPSTensor.gaugePhase_scalar_norm_eq_one_of_leftCanonical_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] A B : MPSTensor d D, A.IsLeftCanonical → B.IsLeftCanonical → Kraus.IsIrr…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhase_scalar_norm_eq_one_of_leftCanonical_irreducible","module":"TNLean.MPS.FundamentalTheorem.UnitaryGauge"},{"id":"n9798","layer":"formal","project":"p8","title":"MPSTensor.exists_unitary_diag_posDef_adjointFixedPoint_of_unital_of_isIrreducibleTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul.hMul (A i) (A i).conjTranspo…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitary_diag_posDef_adjointFixedPoint_of_unital_of_isIrreducibleTensor","module":"TNLean.MPS.Irreducible.Adjoint"},{"id":"n9799","layer":"formal","project":"p8","title":"MPSTensor.exists_singular_posSemidef_fixedPoint_of_unital_nonScalar_fixedPoint","kind":"theorem","summary":"∀ d D : Nat [Nonempty (Fin D)] (A : MPSTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((K…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_singular_posSemidef_fixedPoint_of_unital_nonScalar_fixedPoint","module":"TNLean.MPS.Irreducible.FixedPointProjection"},{"id":"n9800","layer":"formal","project":"p8","title":"MPSTensor.exists_twoBlock_decomp_of_posSemidef_fixedPoint_strict","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex), ρ.PosSemidef → Eq ((Kraus…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_twoBlock_decomp_of_posSemidef_fixedPoint_strict","module":"TNLean.MPS.Irreducible.FixedPointProjection"},{"id":"n9801","layer":"formal","project":"p8","title":"MPSTensor.exists_twoBlock_decomp_of_unital_nonScalar_fixedPoint","kind":"theorem","summary":"∀ d D : Nat [Nonempty (Fin D)] (A : MPSTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((K…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_twoBlock_decomp_of_unital_nonScalar_fixedPoint","module":"TNLean.MPS.Irreducible.FixedPointProjection"},{"id":"n9802","layer":"formal","project":"p8","title":"MPSTensor.exists_unitary_diag_posDef_fixedPoint_of_TP_of_isIrreducibleTensor","kind":"theorem","summary":"∀ d D : Nat [inst : DecidableEq (Fin D)] (A : MPSTensor d D), Eq (Finset.univ.sum fun i => HMul…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unitary_diag_posDef_fixedPoint_of_TP_of_isIrreducibleTensor","module":"TNLean.MPS.Irreducible.FormII"},{"id":"n9803","layer":"formal","project":"p8","title":"MPSTensor.exists_posDef_adjoint_eigenvector","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → (Exists fun i => Ne (…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_posDef_adjoint_eigenvector","module":"TNLean.MPS.Irreducible.PerronGauge"},{"id":"n9804","layer":"formal","project":"p8","title":"MPSTensor.exists_tp_data_of_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → (Exists fun i => Ne (…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_tp_data_of_irreducible","module":"TNLean.MPS.Irreducible.PerronGauge"},{"id":"n9805","layer":"formal","project":"p8","title":"MPSTensor.exists_unital_data_of_irreducible","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → (Exists fun i => Ne (…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_unital_data_of_irreducible","module":"TNLean.MPS.Irreducible.PerronGauge"},{"id":"n9806","layer":"formal","project":"p8","title":"MPSTensor.fixed_eq_scalar_of_isIrreducibleTensor_unital","kind":"theorem","summary":"∀ d D : Nat [Nonempty (Fin D)] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → Eq ((Kraus.tr…","labels":[],"detail_key":"p8","name":"MPSTensor.fixed_eq_scalar_of_isIrreducibleTensor_unital","module":"TNLean.MPS.Irreducible.ScalarFixedPoint"},{"id":"n9807","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.crossSectorMatrix","kind":"def","summary":"Fin 4 → Fin 4 → Matrix (Fin 2) (Fin 2) 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MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.normalizedFourSiteTail_tensor","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9812","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.sectorMatrix_eq_crossSectorMatrix","kind":"theorem","summary":"∀ (i : Fin 4), Eq (MPOTensor.ActiveSectorSpanningCounterexample.sectorMatrix i) (MPOTensor.Acti…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.sectorMatrix_eq_crossSectorMatrix","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9813","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.sectorTensorL_hayashiData","kind":"theorem","summary":"∀ (k : Fin 4) (beta : Fin 2), Eq (MPOTensor.ActiveSectorSpanningCounterexample.tensor.sectorTen…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.sectorTensorL_hayashiData","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9814","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.sectorTensorR_hayashiData","kind":"theorem","summary":"∀ (k : Fin 4) (alpha : Fin 2), Eq (MPOTensor.ActiveSectorSpanningCounterexample.tensor.sectorTe…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.sectorTensorR_hayashiData","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9815","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.threeSiteState","kind":"def","summary":"Matrix (Prod (Fin 4) (Prod (Fin 4) (Fin 4))) (Prod (Fin 4) (Prod (Fin 4) (Fin 4))) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.threeSiteState","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9816","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.trace_crossSectorMatrix_mul_transfer","kind":"theorem","summary":"∀ (i k : Fin 4), Eq (HMul.hMul (MPOTensor.ActiveSectorSpanningCounterexample.crossSectorMatrix…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.trace_crossSectorMatrix_mul_transfer","module":"TNLean.MPS.MPDO.ActiveSectorInverseMapProvenance"},{"id":"n9817","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.tensor_isSAL","kind":"theorem","summary":"MPOTensor.ActiveSectorSpanningCounterexample.tensor.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.tensor_isSAL","module":"TNLean.MPS.MPDO.ActiveSectorSpanningAreaLaw"},{"id":"n9818","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.factorization","kind":"def","summary":"MPOTensor.ActiveSectorSpanningCounterexample.tensor.PhysicalSectorFactorization","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.factorization","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9819","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorEquiv","kind":"def","summary":"Equiv (Fin 4) (Sigma fun _k => Prod (Fin 2) (Fin 2))","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorEquiv","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9820","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorFactorization","kind":"def","summary":"MPOTensor.ActiveSectorSpanningCounterexample.tensor.PhysicalSectorFactorization","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorFactorization","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9821","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorLeftTensor","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorLeftTensor","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9822","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorLeftTensor_apply","kind":"theorem","summary":"∀ (beta i j : Fin 2), Eq (MPOTensor.ActiveSectorSpanningCounterexample.oneSectorLeftTensor beta…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorLeftTensor_apply","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9823","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorRightTensor","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorRightTensor","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9824","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorRightTensor_apply","kind":"theorem","summary":"∀ (alpha i j : Fin 2), Eq (MPOTensor.ActiveSectorSpanningCounterexample.oneSectorRightTensor al…","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorRightTensor_apply","module":"TNLean.MPS.MPDO.ActiveSectorSpanningCounterexample"},{"id":"n9825","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.oneSectorSign","kind":"def","summary":"Fin 2 → 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2","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.normalizedTensor","module":"TNLean.MPS.MPDO.ActiveSectorSpanningRFP"},{"id":"n9828","layer":"formal","project":"p8","title":"MPOTensor.ActiveSectorSpanningCounterexample.tensor_isRFPViaTS","kind":"theorem","summary":"MPOTensor.ActiveSectorSpanningCounterexample.tensor.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.ActiveSectorSpanningCounterexample.tensor_isRFPViaTS","module":"TNLean.MPS.MPDO.ActiveSectorSpanningRFP"},{"id":"n9829","layer":"formal","project":"p8","title":"MPOTensor.exists_hasBlockedAdjointFixedPointAlgebraTower_not_isZCL","kind":"theorem","summary":"Exists fun M => Exists fun ρ => And (Kraus.IsTP M.toMPSTensor) (And ρ.PosDef (And (Eq (M.transf…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_hasBlockedAdjointFixedPointAlgebraTower_not_isZCL","module":"TNLean.MPS.MPDO.AlgebraFusionCounterexample"},{"id":"n9830","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9831","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.BlockedStructureChiFamily","kind":"inductive","summary":"d D : Nat → MPOTensor.AlgebraStructureData d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.BlockedStructureChiFamily","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9832","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.BlockedStructureChiFamily.trace_matrix_pow","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D (χ : data.BlockedStructureChiFamily) (n :…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.BlockedStructureChiFamily.trace_matrix_pow","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9833","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.HasBlockedStructureChiTracePowerForm","kind":"def","summary":"d D : Nat → (data : MPOTensor.AlgebraStructureData d D) → data.BlockedStructureChiFamily → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.HasBlockedStructureChiTracePowerForm","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9834","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.HasBlockedStructureChiTracePowerForm.eq_trace_matrix_pow","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D χ : data.BlockedStructureChiFamily, data.…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.HasBlockedStructureChiTracePowerForm.eq_trace_matrix_pow","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9835","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.PositiveBlockedStructureChiTracePowerForm","kind":"inductive","summary":"d D : Nat → MPOTensor.AlgebraStructureData d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.PositiveBlockedStructureChiTracePowerForm","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9836","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.PositiveBlockedStructureChiTracePowerForm.eq_trace_pow","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D (h : data.PositiveBlockedStructureChiTrac…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.PositiveBlockedStructureChiTracePowerForm.eq_trace_pow","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9837","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.blockedInclusionCoefficients","kind":"def","summary":"d D : Nat → (data : MPOTensor.AlgebraStructureData d D) → (n : Nat) → ↑(data.BlockedIndex n) →…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.blockedInclusionCoefficients","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9838","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.blockedStructureCoefficients","kind":"def","summary":"d D : Nat → (data : MPOTensor.AlgebraStructureData d D) → (n : Nat) → ↑(data.BlockedIndex n) →…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.blockedStructureCoefficients","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9839","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.coe_mul_eq_sum_blockedStructureCoefficients","kind":"theorem","summary":"∀ d D : Nat (data : MPOTensor.AlgebraStructureData d D) (n : Nat) (i j : ↑(data.BlockedIndex n)…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.coe_mul_eq_sum_blockedStructureCoefficients","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9840","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.reconstructFromBlockedCoefficients_apply","kind":"theorem","summary":"∀ d D : Nat (data : MPOTensor.AlgebraStructureData d D) (n : Nat) (a : data.BlockedCoefficients…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.reconstructFromBlockedCoefficients_apply","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9841","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.reconstructFromBlockedCoefficients_of_fixedPoint","kind":"theorem","summary":"∀ d D : Nat (data : MPOTensor.AlgebraStructureData d D) M : MPOTensor d D (hCompat : data.Compa…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.reconstructFromBlockedCoefficients_of_fixedPoint","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9842","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.reconstructFromBlockedInclusionCoefficients","kind":"theorem","summary":"∀ d D : Nat (data : MPOTensor.AlgebraStructureData d D) (n : Nat) (i : ↑(data.BlockedIndex n)),…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.reconstructFromBlockedInclusionCoefficients","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9843","layer":"formal","project":"p8","title":"MPOTensor.AlgebraStructureData.toBlockedCoefficients","kind":"def","summary":"d D : Nat → (data : MPOTensor.AlgebraStructureData d D) → (n : Nat) → LinearEquiv (RingHom.id C…","labels":[],"detail_key":"p8","name":"MPOTensor.AlgebraStructureData.toBlockedCoefficients","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9844","layer":"formal","project":"p8","title":"MPOTensor.DiagonalChiFamily","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.DiagonalChiFamily","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9845","layer":"formal","project":"p8","title":"MPOTensor.DiagonalChiFamily.trace_matrix_pow","kind":"theorem","summary":"∀ I : Type u_1 (χ : MPOTensor.DiagonalChiFamily I) (α β γ : I) (L : Nat), Eq (HPow.hPow (χ.matr…","labels":[],"detail_key":"p8","name":"MPOTensor.DiagonalChiFamily.trace_matrix_pow","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9846","layer":"formal","project":"p8","title":"MPOTensor.HasBlockedAdjointFixedPointAlgebraTower","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasBlockedAdjointFixedPointAlgebraTower","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9847","layer":"formal","project":"p8","title":"MPOTensor.HasChiTracePowerForm","kind":"def","summary":"I : Type u_1 → (Nat → I → I → I → Complex) → MPOTensor.DiagonalChiFamily I → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasChiTracePowerForm","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9848","layer":"formal","project":"p8","title":"MPOTensor.adjoint_blockedTransferMap_apply_iff_of_hasBlockedAdjointFixedPointAlgebraTower","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBlockedAdjointFixedPointAlgebraTower → ∀ n : Nat, LT.lt 0 n…","labels":[],"detail_key":"p8","name":"MPOTensor.adjoint_blockedTransferMap_apply_iff_of_hasBlockedAdjointFixedPointAlgebraTower","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9849","layer":"formal","project":"p8","title":"MPOTensor.adjoint_blockedTransferMap_apply_of_adjoint_transferMap_apply","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (n : Nat) X : Matrix (Fin D) (Fin D) Complex, Eq ((LinearMap.adjo…","labels":[],"detail_key":"p8","name":"MPOTensor.adjoint_blockedTransferMap_apply_of_adjoint_transferMap_apply","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9850","layer":"formal","project":"p8","title":"MPOTensor.adjoint_blockedTransferMap_apply_of_adjoint_transferMap_eigenvector","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (n : Nat) lam : Complex X : Matrix (Fin D) (Fin D) Complex, Eq ((…","labels":[],"detail_key":"p8","name":"MPOTensor.adjoint_blockedTransferMap_apply_of_adjoint_transferMap_eigenvector","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9851","layer":"formal","project":"p8","title":"MPOTensor.adjoint_transferMap_apply_of_hasBlockedAdjointFixedPointAlgebraTower","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBlockedAdjointFixedPointAlgebraTower → ∀ n : Nat, LT.lt 0 n…","labels":[],"detail_key":"p8","name":"MPOTensor.adjoint_transferMap_apply_of_hasBlockedAdjointFixedPointAlgebraTower","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9852","layer":"formal","project":"p8","title":"MPOTensor.adjoint_transferMap_eigenvalue_eq_one_of_hasBlockedAdjointFixedPointAlgebraTower","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBlockedAdjointFixedPointAlgebraTower → ∀ n : Nat, LT.lt 0 n…","labels":[],"detail_key":"p8","name":"MPOTensor.adjoint_transferMap_eigenvalue_eq_one_of_hasBlockedAdjointFixedPointAlgebraTower","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9853","layer":"formal","project":"p8","title":"MPOTensor.hasBlockedAdjointFixedPointAlgebraTower_of_adjointFixedPoints_eq_of_isTP_of_pos…","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, Kraus.IsTP M.toMPSTensor → ∀ ρ : Matrix (Fin D) (Fin D) Complex,…","labels":[],"detail_key":"p8","name":"MPOTensor.hasBlockedAdjointFixedPointAlgebraTower_of_adjointFixedPoints_eq_of_isTP_of_posDef_fixed","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9854","layer":"formal","project":"p8","title":"MPOTensor.hasBlockedAdjointFixedPointAlgebraTower_of_isZCL_of_isTP_of_posDef_fixed","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsZCL → Kraus.IsTP M.toMPSTensor → ∀ ρ : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.hasBlockedAdjointFixedPointAlgebraTower_of_isZCL_of_isTP_of_posDef_fixed","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9855","layer":"formal","project":"p8","title":"MPOTensor.stationaryOfFaithfulFixedPoint_compatible_of_hasBlockedAdjointFixedPointAlgebra…","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBlockedAdjointFixedPointAlgebraTower → ∀ (h_tp : Kraus.IsTP…","labels":[],"detail_key":"p8","name":"MPOTensor.stationaryOfFaithfulFixedPoint_compatible_of_hasBlockedAdjointFixedPointAlgebraTower","module":"TNLean.MPS.MPDO.AlgebraStructure"},{"id":"n9856","layer":"formal","project":"p8","title":"MPOTensor.IsSAL","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsSAL","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9857","layer":"formal","project":"p8","title":"MPOTensor.blockEntropy","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → (N L : Nat) → LE.le L N → (M.mpo N).PosSemidef → Real","labels":[],"detail_key":"p8","name":"MPOTensor.blockEntropy","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9858","layer":"formal","project":"p8","title":"MPOTensor.blockEntropy_of_charpoly_roots_eq","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N L : Nat (hL : LE.le L N) (hM : (M.mpo N).PosSemidef) (s : Mul…","labels":[],"detail_key":"p8","name":"MPOTensor.blockEntropy_of_charpoly_roots_eq","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9859","layer":"formal","project":"p8","title":"MPOTensor.mpo_submatrix_rotateConfig","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat), Eq ((M.mpo N).submatrix ⇑(MPOTensor.rotateConfig N d…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_submatrix_rotateConfig","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9860","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → (N L : Nat) → LE.le L N → (M.mpo N).PosSemidef → Real","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9861","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain_eq","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N L : Nat) (hL : LE.le L N) (hM : (M.mpo N).PosSemidef), Eq (M…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_eq","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9862","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain_eq_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hSAL : M.IsSAL) N L L' : Nat, LE.le 1 L → ∀ (hLN : LE.le L (HD…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_eq_of_isSAL","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9863","layer":"formal","project":"p8","title":"MPOTensor.rotateConfig","kind":"def","summary":"(N d : Nat) → Equiv (Fin N → Fin d) (Fin N → Fin 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hL).…","labels":[],"detail_key":"p8","name":"MPSTensor.reducedPureBlockState_posSemidef","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9867","layer":"formal","project":"p8","title":"MPSTensor.reducedPureBlockState_trace","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Ne (A.pureState N).trace 0 → Eq (…","labels":[],"detail_key":"p8","name":"MPSTensor.reducedPureBlockState_trace","module":"TNLean.MPS.MPDO.AreaLaw"},{"id":"n9868","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClause"},{"id":"n9869","layer":"formal","project":"p8","title":"MPOTensor.HasBNTAlgebraTensorClause","kind":"def","summary":"d D : Nat → MPOTensor d D → 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MPOTensor.BNTLabelTrace…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalBNTTraceScalarFamily","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClause"},{"id":"n9873","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.BNTAlgebraTensorClause → Matrix (Fin d) (Fin d) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseConditionalPhysicalMaps"},{"id":"n9874","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection_isHermitian","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause), H.oneSiteRetainedProjection.IsHer…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection_isHermitian","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseConditionalPhysicalMaps"},{"id":"n9875","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection_mul_self","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause), Eq (HMul.hMul H.oneSiteRetainedPr…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteRetainedProjection_mul_self","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseConditionalPhysicalMaps"},{"id":"n9876","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTAlgebraTensorClause) → LinearMap (RingHom.id Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9877","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetractionExtension","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTAlgebraTensorClause) → LinearMap (RingHom.id Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetractionExtension","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9878","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetractionExtension_isKrausCP","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause), IsKrausCP H.oneSiteAmbientSectorR…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetractionExtension_isKrausCP","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9879","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction_apply","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause) (X : Matrix (Fin d) (Fin d) Comple…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction_apply","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9880","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction_verticalTensor","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause) (v : Fin (HMul.hMul D D)), Eq (H.o…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteAmbientSectorRetraction_verticalTensor","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9881","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestoration","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTAlgebraTensorClause) → LinearMap (RingHom.id Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestoration","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9882","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestorationExtension","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTAlgebraTensorClause) → LinearMap (RingHom.id Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestorationExtension","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9883","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestoration_apply","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause) (X : MPOTensor.VerticalSectorAlgeb…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.oneSiteNormalizedAmbientSectorRestoration_apply","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9884","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.trace_oneSiteNormalizedAmbientSectorRestoration","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause) (X : 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Comple…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.verticalSectorTrace_oneSiteAmbientSectorRetraction_le","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseOneSiteAmbientSectorMaps"},{"id":"n9887","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTAlgebraTensorClause), M.toMPSTensor.IsCPSVCanonicalForm…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.isRFPViaTS","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseReflectedTarget"},{"id":"n9888","layer":"formal","project":"p8","title":"MPOTensor.HasBNTAlgebraTensorClause.isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBNTAlgebraTensorClause → M.toMPSTensor.IsCPSVCanonicalForm…","labels":[],"detail_key":"p8","name":"MPOTensor.HasBNTAlgebraTensorClause.isRFPViaTS","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseReflectedTarget"},{"id":"n9889","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.TwoSiteExactSectorGauge","kind":"inductive","summary":"d D : Nat → M : MPOTensor d D → M.BNTAlgebraTensorClause → Type","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.TwoSiteExactSectorGauge","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseSpectrum"},{"id":"n9890","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraTensorClause.TwoSiteMultiplicitySpectrum","kind":"inductive","summary":"d D : Nat → M : MPOTensor d D → M.BNTAlgebraTensorClause → Type","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraTensorClause.TwoSiteMultiplicitySpectrum","module":"TNLean.MPS.MPDO.BNTAlgebraTensorClauseSpectrum"},{"id":"n9891","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.EventuallyLinearIndependent","kind":"def","summary":"Λ : Type u_1 → O : Nat → Type u_2 → MPOTensor.BNTLabelOperatorFamily Λ O → [inst : (L : Nat) →…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.EventuallyLinearIndependent","module":"TNLean.MPS.MPDO.BNTAssociativity"},{"id":"n9892","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.LinearIndependentAt","kind":"def","summary":"Λ : Type u_1 → O : Nat → Type u_2 → MPOTensor.BNTLabelOperatorFamily Λ O → (L : Nat) → [inst :…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.LinearIndependentAt","module":"TNLean.MPS.MPDO.BNTAssociativity"},{"id":"n9893","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_eq_sum_commonWeightAbsorbedBasis_of_gauge_toTensor","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_eq_sum_commonWeightAbsorbedBasis_of_gauge_toTensor","module":"TNLean.MPS.MPDO.BNTBoundaryDecomposition"},{"id":"n9894","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_eq_sum_flatBasis_of_gauge_toTensor","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (M : MPOTensor d S.totalDim) (G :…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_eq_sum_flatBasis_of_gauge_toTensor","module":"TNLean.MPS.MPDO.BNTBoundaryDecomposition"},{"id":"n9895","layer":"formal","project":"p8","title":"MPOTensor.blockTwo_isRFPViaTS_of_chainCoordinateRFP","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : LinearMap (RingHom.id Complex) (Matrix (Fin 4 → Fin d) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo_isRFPViaTS_of_chainCoordinateRFP","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9896","layer":"formal","project":"p8","title":"MPOTensor.blockTwo_isRFPViaTS_of_commonWeightAbsorbedBNTChannels","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo_isRFPViaTS_of_commonWeightAbsorbedBNTChannels","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9897","layer":"formal","project":"p8","title":"MPOTensor.blockTwo_isRFPViaTS_of_commonWeightAbsorbedBNTChannels_of_orthogonalSupports","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo_isRFPViaTS_of_commonWeightAbsorbedBNTChannels_of_orthogonalSupports","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9898","layer":"formal","project":"p8","title":"MPOTensor.blockedFirstSiteProjection","kind":"def","summary":"d : Nat → Matrix (Fin d) (Fin d) Complex → Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)) C…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedFirstSiteProjection","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9899","layer":"formal","project":"p8","title":"MPOTensor.blockedFirstSiteProjection_eq_reindex_kronecker_one","kind":"theorem","summary":"∀ d : Nat (P : Matrix (Fin d) (Fin d) Complex), Eq (MPOTensor.blockedFirstSiteProjection P) ((M…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedFirstSiteProjection_eq_reindex_kronecker_one","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9900","layer":"formal","project":"p8","title":"MPOTensor.blockedOneChainEquiv","kind":"def","summary":"(d : Nat) → Equiv (Fin 2 → Fin d) (Fin (HMul.hMul d d))","labels":[],"detail_key":"p8","name":"MPOTensor.blockedOneChainEquiv","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9901","layer":"formal","project":"p8","title":"MPOTensor.blockedTwoChainEquiv","kind":"def","summary":"(d : Nat) → Equiv (Fin 4 → Fin d) (Prod (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)))","labels":[],"detail_key":"p8","name":"MPOTensor.blockedTwoChainEquiv","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9902","layer":"formal","project":"p8","title":"MPOTensor.coarseningMapInChainCoordinates","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Prod (Fin (HMul.hMul d d)) (Fin (HMul.hMul d…","labels":[],"detail_key":"p8","name":"MPOTensor.coarseningMapInChainCoordinates","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9903","layer":"formal","project":"p8","title":"MPOTensor.coarseningMapInChainCoordinates_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin (HMul.hMul d d)) (Fin (HMul.hMu…","labels":[],"detail_key":"p8","name":"MPOTensor.coarseningMapInChainCoordinates_isKrausCPTP","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9904","layer":"formal","project":"p8","title":"MPOTensor.coarseningMapInChainCoordinates_physCloseN","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin (HMul.hM…","labels":[],"detail_key":"p8","name":"MPOTensor.coarseningMapInChainCoordinates_physCloseN","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9905","layer":"formal","project":"p8","title":"MPOTensor.equivReindexMap_physCloseN_four_eq_physClose2_blockTwo","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((Matrix.equivReindexM…","labels":[],"detail_key":"p8","name":"MPOTensor.equivReindexMap_physCloseN_four_eq_physClose2_blockTwo","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9906","layer":"formal","project":"p8","title":"MPOTensor.equivReindexMap_physCloseN_two_eq_physClose1_blockTwo","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((Matrix.equivReindexM…","labels":[],"detail_key":"p8","name":"MPOTensor.equivReindexMap_physCloseN_two_eq_physClose1_blockTwo","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9907","layer":"formal","project":"p8","title":"MPOTensor.exists_chainCoordinateRFP_of_orthogonalSectorDecomposition","kind":"theorem","summary":"∀ d D : Nat ι : Type u_1 [inst : Fintype ι] bondDim : ι → Nat (M : MPOTensor d D) (K : (s : ι)…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_chainCoordinateRFP_of_orthogonalSectorDecomposition","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9908","layer":"formal","project":"p8","title":"MPOTensor.exists_chainCoordinateRFP_of_projectiveSectorDecomposition","kind":"theorem","summary":"∀ d D : Nat ι : Type u_1 [inst : Fintype ι] bondDim : ι → Nat (M : MPOTensor d D) (K : (s : ι)…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_chainCoordinateRFP_of_projectiveSectorDecomposition","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9909","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_isStarProjection","kind":"theorem","summary":"∀ d : Nat P : Matrix (Fin d) (Fin d) Complex, IsOrthogonalProjection P → ∀ (N : Nat), IsStarPro…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_isStarProjection","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9910","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_mul_physCloseN_eq_zero_of_mul_physicalSlice_eq_zero","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), (∀ (β α : Fin D), Eq (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_mul_physCloseN_eq_zero_of_mul_physicalSlice_eq_zero","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9911","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_mul_physCloseN_of_mul_physicalSlice","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), (∀ (β α : Fin D), Eq (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_mul_physCloseN_of_mul_physicalSlice","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9912","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_mul_firstSiteMatrix_of_physicalSlice_mul","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), (∀ (β α : Fin D), Eq (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_mul_firstSiteMatrix_of_physicalSlice_mul","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9913","layer":"formal","project":"p8","title":"MPOTensor.refinementMapInChainCoordinates","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)) Co…","labels":[],"detail_key":"p8","name":"MPOTensor.refinementMapInChainCoordinates","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9914","layer":"formal","project":"p8","title":"MPOTensor.refinementMapInChainCoordinates_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat T : LinearMap (RingHom.id Complex) (Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)…","labels":[],"detail_key":"p8","name":"MPOTensor.refinementMapInChainCoordinates_isKrausCPTP","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9915","layer":"formal","project":"p8","title":"MPOTensor.refinementMapInChainCoordinates_physCloseN","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (T : LinearMap (RingHom.id Complex) (Matrix (Fin (HMul.hMul d d…","labels":[],"detail_key":"p8","name":"MPOTensor.refinementMapInChainCoordinates_physCloseN","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9916","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_firstSiteMatrix_physCloseN_eq_ite","kind":"theorem","summary":"∀ d : Nat ι : Type u_1 [inst : DecidableEq ι] bondDim : ι → Nat (K : (s : ι) → MPOTensor d (bon…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_firstSiteMatrix_physCloseN_eq_ite","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9917","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_firstSiteMatrix_physCloseN_of_twoSidedSupport","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), P.IsHermitian → (∀ (β α :…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_firstSiteMatrix_physCloseN_of_twoSidedSupport","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9918","layer":"formal","project":"p8","title":"MPOTensor.sum_firstSiteMatrix","kind":"theorem","summary":"∀ d : Nat ι : Type u_1 [inst : Fintype ι] (P : ι → Matrix (Fin d) (Fin d) Complex) (N : Nat), E…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_firstSiteMatrix","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9919","layer":"formal","project":"p8","title":"MPOTensor.sum_firstSiteMatrix_eq_one","kind":"theorem","summary":"∀ d : Nat ι : Type u_1 [inst : Fintype ι] (P : ι → Matrix (Fin d) (Fin d) Complex), Eq (Finset.…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_firstSiteMatrix_eq_one","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9920","layer":"formal","project":"p8","title":"MPOTensor.sum_projection_twoSidedSupport_physicalSlice","kind":"theorem","summary":"∀ d : Nat ι : Type u_1 [inst : Fintype ι] bondDim : ι → Nat (K : (s : ι) → MPOTensor d (bondDim…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_projection_twoSidedSupport_physicalSlice","module":"TNLean.MPS.MPDO.BNTChannelComposition"},{"id":"n9921","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison","kind":"inductive","summary":"d D : Nat → MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → MPOTensor.BNTLabelCoefficientFa…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9922","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.blockedLabel","kind":"def","summary":"d D : Nat → data : MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → c : MPOTensor.BNTLabelCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.blockedLabel","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9923","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.blocked_coeff_eq","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.blocked_coeff_eq","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9924","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.blocked_coeff_eq_trace_pow","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.blocked_coeff_eq_trace_pow","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9925","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChiFamily","kind":"def","summary":"d D : Nat → data : MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → c : MPOTensor.BNTLabelCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChiFamily","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9926","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChiFamily_toDiagonal_of_pos","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChiFamily_toDiagonal_of_pos","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9927","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_dim_of_pos","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_dim_of_pos","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9928","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_tracePowerCoeff_of_pos","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_tracePowerCoeff_of_pos","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9929","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_trace_matrix_pow_of_pos","kind":"theorem","summary":"∀ d D : Nat data : MPOTensor.AlgebraStructureData d D Λ : Type u_1 c : MPOTensor.BNTLabelCoeffi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.pulledBlockedChi_trace_matrix_pow_of_pos","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9930","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.sourceLabel","kind":"def","summary":"d D : Nat → data : MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → c : MPOTensor.BNTLabelCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.sourceLabel","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9931","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisCoefficientComparison.targetLabel","kind":"def","summary":"d D : Nat → data : MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → c : MPOTensor.BNTLabelCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisCoefficientComparison.targetLabel","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9932","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisLabelAssignment","kind":"inductive","summary":"d D : Nat → MPOTensor.AlgebraStructureData d D → Type u_1 → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisLabelAssignment","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9933","layer":"formal","project":"p8","title":"MPOTensor.BNTBlockedBasisLabelAssignment.blockedLabel","kind":"def","summary":"d D : Nat → data : MPOTensor.AlgebraStructureData d D → Λ : Type u_1 → MPOTensor.BNTBlockedBasi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTBlockedBasisLabelAssignment.blockedLabel","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9934","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9935","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.HasPositiveLengthChiTracePowerForm","kind":"def","summary":"Λ : Type u_1 → MPOTensor.BNTLabelCoefficientFamily Λ → MPOTensor.DiagonalChiFamily Λ → 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Λ","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.ofChi","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9938","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.ofChi_coeff","kind":"theorem","summary":"∀ Λ : Type u_1 (χ : MPOTensor.DiagonalChiFamily Λ) (L : Nat) (α β γ : Λ), Eq ((MPOTensor.BNTLab…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.ofChi_coeff","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9939","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.ofChi_coeff_eq_trace_matrix_pow","kind":"theorem","summary":"∀ Λ : Type u_1 (χ : MPOTensor.DiagonalChiFamily Λ) (L : Nat) (α β γ : Λ), Eq ((MPOTensor.BNTLab…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.ofChi_coeff_eq_trace_matrix_pow","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9940","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.ofChi_hasPositiveLengthChiTracePowerForm","kind":"theorem","summary":"∀ Λ : Type u_1 (χ : MPOTensor.DiagonalChiFamily Λ), (MPOTensor.BNTLabelCoefficientFamily.ofChi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.ofChi_hasPositiveLengthChiTracePowerForm","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9941","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily","kind":"inductive","summary":"Type u_1 → (Nat → Type u_2) → Type (max u_1 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Λ…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.HasSameLengthProductForm.eq_sum","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9944","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.HasSameLengthProductForm.eq_sum_chi_trace_pow","kind":"theorem","summary":"∀ Λ : Type u_1 O : Nat → Type u_2 c : MPOTensor.BNTLabelCoefficientFamily Λ op : MPOTensor.BNTL…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.HasSameLengthProductForm.eq_sum_chi_trace_pow","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9945","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.HasSameLengthProductForm.eq_sum_ofChi_trace_pow","kind":"theorem","summary":"∀ Λ : Type u_1 O : Nat → Type u_2 op : MPOTensor.BNTLabelOperatorFamily Λ O [inst : Fintype Λ]…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.HasSameLengthProductForm.eq_sum_ofChi_trace_pow","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9946","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTraceScalarFamily","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelTraceScalarFamily","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9947","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm","kind":"def","summary":"Λ : Type u_1 → MPOTensor.BNTLabelTraceScalarFamily Λ → [Fintype Λ] → MPOTensor.BNTLabelCoeffici…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9948","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum","kind":"theorem","summary":"∀ Λ : Type u_1 (m : MPOTensor.BNTLabelTraceScalarFamily Λ) [inst : Fintype Λ] c : MPOTensor.BNT…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9949","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum_chi_trace","kind":"theorem","summary":"∀ Λ : Type u_1 c : MPOTensor.BNTLabelCoefficientFamily Λ m : MPOTensor.BNTLabelTraceScalarFamil…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum_chi_trace","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9950","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum_ofChi_trace","kind":"theorem","summary":"∀ Λ : Type u_1 m : MPOTensor.BNTLabelTraceScalarFamily Λ [inst : Fintype Λ] χ : MPOTensor.Diago…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelTraceScalarFamily.HasIdempotentCoefficientForm.eq_sum_ofChi_trace","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9951","layer":"formal","project":"p8","title":"MPOTensor.PositiveBNTLabelChiTracePowerForm","kind":"inductive","summary":"Λ : Type u_1 → MPOTensor.BNTLabelCoefficientFamily Λ → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.PositiveBNTLabelChiTracePowerForm","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9952","layer":"formal","project":"p8","title":"MPOTensor.PositiveBNTLabelChiTracePowerForm.eq_trace_pow","kind":"theorem","summary":"∀ Λ : Type u_1 c : MPOTensor.BNTLabelCoefficientFamily Λ (h : MPOTensor.PositiveBNTLabelChiTrac…","labels":[],"detail_key":"p8","name":"MPOTensor.PositiveBNTLabelChiTracePowerForm.eq_trace_pow","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9953","layer":"formal","project":"p8","title":"MPOTensor.PositiveBNTLabelChiTracePowerForm.ofChi","kind":"def","summary":"Λ : Type u_1 → (χ : MPOTensor.DiagonalChiFamily Λ) → χ.PosEntries → MPOTensor.PositiveBNTLabelC…","labels":[],"detail_key":"p8","name":"MPOTensor.PositiveBNTLabelChiTracePowerForm.ofChi","module":"TNLean.MPS.MPDO.BNTCoefficients"},{"id":"n9954","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofOneSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Eq F.sectorCount 1 → (∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofOneSector","module":"TNLean.MPS.MPDO.BNTFactorizationChannels"},{"id":"n9955","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.LeftFinalMultiplicity","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.BNTFusionIsome…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.LeftFinalMultiplicity","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9956","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.RightFinalMultiplicity","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.BNTFusionIsome…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.RightFinalMultiplicity","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9957","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftFinalFusionMap","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFinalFusionMap","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9958","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftFinalFusion_apply","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIsom…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFinalFusion_apply","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9959","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftFinalIndexEquiv","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFinalIndexEquiv","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9960","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFinalFusionMap","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFinalFusionMap","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9961","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFinalFusion_apply","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIsom…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFinalFusion_apply","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9962","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFinalIndexEquiv","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFinalIndexEquiv","module":"TNLean.MPS.MPDO.BNTFinalSectorFusion"},{"id":"n9963","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftTripleFinalEquiv","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftTripleFinalEquiv","module":"TNLean.MPS.MPDO.BNTFixedFinalUnitarity"},{"id":"n9964","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightTripleFinalEquiv","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightTripleFinalEquiv","module":"TNLean.MPS.MPDO.BNTFixedFinalUnitarity"},{"id":"n9965","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily","kind":"inductive","summary":"(Λ : Type u_1) → [Fintype Λ] → [DecidableEq Λ] → Nat → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily","module":"TNLean.MPS.MPDO.BNTFusionCoisometries"},{"id":"n9966","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.mpo_mul_mpo_eq_sum","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.mpo_mul_mpo_eq_sum","module":"TNLean.MPS.MPDO.BNTFusionCoisometries"},{"id":"n9967","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily","kind":"inductive","summary":"(Λ : Type u_1) → [Fintype Λ] → [DecidableEq Λ] → Nat → Type u_1","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9968","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.fusionIsometry_mul_mulTensor","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.fusionIsometry_mul_mulTensor","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9969","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.mpo_mul_mpo_eq_sum","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.mpo_mul_mpo_eq_sum","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9970","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.toBNTAlgebraClause","kind":"def","summary":"Λ : Type u_1 → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFu…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.toBNTAlgebraClause","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9971","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily","kind":"def","summary":"Λ : Type u_1 → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.BNTFusionIso…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9972","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily_hasSameLengthProductForm","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily_hasSameLengthProductForm","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9973","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily_operator","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.toOperatorFamily_operator","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9974","layer":"formal","project":"p8","title":"Matrix.blockDiagonal'_conj","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] [inst_1 : DecidableEq ι] m : ι → Type u_2 n : ι → Type u_3 [i…","labels":[],"detail_key":"p8","name":"Matrix.blockDiagonal'_conj","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9975","layer":"formal","project":"p8","title":"Matrix.submatrix_left_conj_equiv","kind":"theorem","summary":"∀ l : Type u_1 m : Type u_2 u : Type u_3 [inst : Fintype l] (A : Matrix m l Complex) (r : Equiv…","labels":[],"detail_key":"p8","name":"Matrix.submatrix_left_conj_equiv","module":"TNLean.MPS.MPDO.BNTFusionIsometries"},{"id":"n9976","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause","module":"TNLean.MPS.MPDO.BNTFusionTensorClause"},{"id":"n9977","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.toBNTAlgebraTensorClause","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.BNTFusionTensorClause → M.BNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.toBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.BNTFusionTensorClause"},{"id":"n9978","layer":"formal","project":"p8","title":"MPOTensor.HasBNTFusionTensorClause","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasBNTFusionTensorClause","module":"TNLean.MPS.MPDO.BNTFusionTensorClause"},{"id":"n9979","layer":"formal","project":"p8","title":"MPOTensor.HasBNTFusionTensorClause.to_hasBNTAlgebraTensorClause","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasBNTFusionTensorClause → M.HasBNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.HasBNTFusionTensorClause.to_hasBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.BNTFusionTensorClause"},{"id":"n9980","layer":"formal","project":"p8","title":"MPOTensor.HasBNTFusionTensorClause.of_isRFPViaTS_of_horizontalCF","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → M.HasBNTFusionTen…","labels":[],"detail_key":"p8","name":"MPOTensor.HasBNTFusionTensorClause.of_isRFPViaTS_of_horizontalCF","module":"TNLean.MPS.MPDO.BNTFusionTensorClauseFromRFP"},{"id":"n9981","layer":"formal","project":"p8","title":"MPOTensor.hasIdempotentCoefficientForm_of_blockedRepresentations","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (D₁ : M.CPSVVerticalDecomposition) (D₂ : M.blockTwo.CPSVVerticalD…","labels":[],"detail_key":"p8","name":"MPOTensor.hasIdempotentCoefficientForm_of_blockedRepresentations","module":"TNLean.MPS.MPDO.BNTFusionTensorClauseFromRFP"},{"id":"n9982","layer":"formal","project":"p8","title":"MPOTensor.BNTLayerOrthogonalityCounterexample.family_isBNTLayerOrthogonal","kind":"theorem","summary":"MPOTensor.IsBNTLayerOrthogonal MPOTensor.BNTLayerOrthogonalityCounterexample.family","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLayerOrthogonalityCounterexample.family_isBNTLayerOrthogonal","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9983","layer":"formal","project":"p8","title":"MPOTensor.BNTLayerOrthogonalityCounterexample.no_pairwise_orthogonal_twoSided_support","kind":"theorem","summary":"Not (Exists fun P => And (∀ (x : Fin 2), IsOrthogonalProjection (P x)) (And (∀ x y : Fin 2, Ne…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLayerOrthogonalityCounterexample.no_pairwise_orthogonal_twoSided_support","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9984","layer":"formal","project":"p8","title":"MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix_mul_self","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix_mul_self","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9985","layer":"formal","project":"p8","title":"MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix_ne_zero","kind":"theorem","summary":"Ne MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix 0","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLayerOrthogonalityCounterexample.squareZeroPhysicalMatrix_ne_zero","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9986","layer":"formal","project":"p8","title":"MPOTensor.IsBNTLayerOrthogonal","kind":"def","summary":"d g : Nat → bondDim : Fin g → Nat → ((x : Fin g) → MPOTensor d (bondDim x)) → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsBNTLayerOrthogonal","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9987","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.sameMPV_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D, K.IsMPDO → K.toMPSTensor.SameMPV K.physicalAdjointTensor.toMPSTe…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.sameMPV_physicalAdjointTensor","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9988","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.sameMPV₂Pos_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D, K.IsMPDO → K.toMPSTensor.SameMPV₂Pos K.physicalAdjointTensor.toM…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.sameMPV₂Pos_physicalAdjointTensor","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9989","layer":"formal","project":"p8","title":"MPOTensor.exists_pairwise_orthogonal_twoSided_physicalSupport","kind":"theorem","summary":"∀ d g : Nat bondDim : Fin g → Nat (K : (x : Fin g) → MPOTensor d (bondDim x)), (∀ (x : Fin g),…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_pairwise_orthogonal_twoSided_physicalSupport","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9990","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSlice_conjTranspose_eq_sum_of_isInjective_isMPDO","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), Kraus.IsInjective K.toMPSTensor → K.IsMPDO → Exists fun X => ∀…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSlice_conjTranspose_eq_sum_of_isInjective_isMPDO","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9991","layer":"formal","project":"p8","title":"MPOTensor.isBNTLayerOrthogonal_iff_physicalSlice_mul_apply_eq_zero","kind":"theorem","summary":"∀ d g : Nat bondDim : Fin g → Nat (K : (x : Fin g) → MPOTensor d (bondDim x)), Iff (MPOTensor.I…","labels":[],"detail_key":"p8","name":"MPOTensor.isBNTLayerOrthogonal_iff_physicalSlice_mul_apply_eq_zero","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9992","layer":"formal","project":"p8","title":"MPOTensor.isBNTLayerOrthogonal_iff_physicalSlice_mul_eq_zero","kind":"theorem","summary":"∀ d g : Nat bondDim : Fin g → Nat (K : (x : Fin g) → MPOTensor d (bondDim x)), Iff (MPOTensor.I…","labels":[],"detail_key":"p8","name":"MPOTensor.isBNTLayerOrthogonal_iff_physicalSlice_mul_eq_zero","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9993","layer":"formal","project":"p8","title":"MPOTensor.isBNTLayerOrthogonal_of_pairwise_orthogonal_twoSided_support","kind":"theorem","summary":"∀ d g : Nat bondDim : Fin g → Nat (K : (x : Fin g) → MPOTensor d (bondDim x)) (P : Fin g → Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.isBNTLayerOrthogonal_of_pairwise_orthogonal_twoSided_support","module":"TNLean.MPS.MPDO.BNTLayerOrthogonality"},{"id":"n9994","layer":"formal","project":"p8","title":"MPOTensor.physicalSliceColumns","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin (HMul.hMul (HMul.hMul D D) d)) 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DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFu…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFusionIsometry","module":"TNLean.MPS.MPDO.BNTLeftTripleFusion"},{"id":"n10000","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftFusionIsometry_isometry","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFusionIsometry_isometry","module":"TNLean.MPS.MPDO.BNTLeftTripleFusion"},{"id":"n10001","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.leftFusion_apply","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.leftFusion_apply","module":"TNLean.MPS.MPDO.BNTLeftTripleFusion"},{"id":"n10002","layer":"formal","project":"p8","title":"MPOTensor.bntMarkovLeftFactor","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.bntMarkovLeftFactor","module":"TNLean.MPS.MPDO.BNTMarkovKeyFormula"},{"id":"n10003","layer":"formal","project":"p8","title":"MPOTensor.bntMarkovRightFactor","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.bntMarkovRightFactor","module":"TNLean.MPS.MPDO.BNTMarkovKeyFormula"},{"id":"n10004","layer":"formal","project":"p8","title":"MPOTensor.isMPOBlockLeftInverse_bnt_markov_key_formula","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.isMPOBlockLeftInverse_bnt_markov_key_formula","module":"TNLean.MPS.MPDO.BNTMarkovKeyFormula"},{"id":"n10005","layer":"formal","project":"p8","title":"MPOTensor.isMPOBlockLeftInverse_bnt_markov_offdiagonal","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.isMPOBlockLeftInverse_bnt_markov_offdiagonal","module":"TNLean.MPS.MPDO.BNTMarkovKeyFormula"},{"id":"n10006","layer":"formal","project":"p8","title":"MPOTensor.outerInverseContraction_markov_block","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (P…","labels":[],"detail_key":"p8","name":"MPOTensor.outerInverseContraction_markov_block","module":"TNLean.MPS.MPDO.BNTMarkovKeyFormula"},{"id":"n10007","layer":"formal","project":"p8","title":"MPOTensor.BNTMarkovBlockNonzero","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTMarkovBlockNonzero","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10008","layer":"formal","project":"p8","title":"MPOTensor.activeMarkovProjection","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.activeMarkovProjection","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10009","layer":"formal","project":"p8","title":"MPOTensor.bntMarkovBlock","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.bntMarkovBlock","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10010","layer":"formal","project":"p8","title":"MPOTensor.bntMarkovBlockNonzero_eq_of_probability_ne_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.bntMarkovBlockNonzero_eq_of_probability_ne_zero","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10011","layer":"formal","project":"p8","title":"MPOTensor.bntSectorLabel","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → R : (s : Fin g) → Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorLabel","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10012","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProjection","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → R : (s : Fin g) → Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProjection","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10013","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProjection_isOrthogonal","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProjection_isOrthogonal","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10014","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProjection_mul_eq_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProjection_mul_eq_zero","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10015","layer":"formal","project":"p8","title":"MPOTensor.existsUnique_bntMarkovBlockNonzero_of_probability_ne_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.existsUnique_bntMarkovBlockNonzero_of_probability_ne_zero","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10016","layer":"formal","project":"p8","title":"MPOTensor.exists_bntMarkovBlockNonzero_of_probability_ne_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_bntMarkovBlockNonzero_of_probability_ne_zero","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10017","layer":"formal","project":"p8","title":"MPOTensor.sum_bntSectorProjection","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_bntSectorProjection","module":"TNLean.MPS.MPDO.BNTMarkovSectorProjectors"},{"id":"n10018","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraClause.twoMultiplicityTrace_eq_traceScalar","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] O : Nat → Type u_2 [inst_1 : (L : Nat) → AddCommMonoid (O L)]…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraClause.twoMultiplicityTrace_eq_traceScalar","module":"TNLean.MPS.MPDO.BNTMultiplicityNormalization"},{"id":"n10019","layer":"formal","project":"p8","title":"MPOTensor.BNTMultiplicitySpectrumComparison","kind":"inductive","summary":"Λ : Type u_1 → [Fintype Λ] → MPOTensor.DiagonalChiFamily Λ → MPOTensor.BNTLabelTraceScalarFamil…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTMultiplicitySpectrumComparison","module":"TNLean.MPS.MPDO.BNTMultiplicityNormalization"},{"id":"n10020","layer":"formal","project":"p8","title":"MPOTensor.BNTMultiplicitySpectrumComparison.sum_products_eq","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] χ : MPOTensor.DiagonalChiFamily Λ m : 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MPOTensor.BNTLabelTrace…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTMultiplicitySpectrumComparison.twoTrace_eq_traceScalar","module":"TNLean.MPS.MPDO.BNTMultiplicityNormalization"},{"id":"n10023","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProjection_mul_physicalSlice_mul_eq_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProjection_mul_physicalSlice_mul_eq_zero","module":"TNLean.MPS.MPDO.BNTProjectorSelection"},{"id":"n10024","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProjection_mul_physicalSlice_mul_self","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix 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q'…","labels":[],"detail_key":"p8","name":"MPOTensor.commonWeightAbsorbedBasisMPOTensor","module":"TNLean.MPS.MPDO.BNTProjectorSelection"},{"id":"n10027","layer":"formal","project":"p8","title":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_toMPSTensor","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_toMPSTensor","module":"TNLean.MPS.MPDO.BNTProjectorSelection"},{"id":"n10028","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_sitewise_bntSectorProjection_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_sitewise_bntSectorProjection_mpo","module":"TNLean.MPS.MPDO.BNTProjectorSelection"},{"id":"n10029","layer":"formal","project":"p8","title":"MPOTensor.sitewise_bntSectorProjection_mul_mpo_mul_self","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewise_bntSectorProjection_mul_mpo_mul_self","module":"TNLean.MPS.MPDO.BNTProjectorSelection"},{"id":"n10030","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFusionIsometry","kind":"def","summary":"Λ : Type u_1 → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFu…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFusionIsometry","module":"TNLean.MPS.MPDO.BNTRightTripleFusion"},{"id":"n10031","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFusionIsometry_isometry","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFusionIsometry_isometry","module":"TNLean.MPS.MPDO.BNTRightTripleFusion"},{"id":"n10032","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.rightFusion_apply","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionIs…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.rightFusion_apply","module":"TNLean.MPS.MPDO.BNTRightTripleFusion"},{"id":"n10033","layer":"formal","project":"p8","title":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hTotal : E…","labels":[],"detail_key":"p8","name":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_isSourceZCL","module":"TNLean.MPS.MPDO.BNTSectorAnalyticProperties"},{"id":"n10034","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_commonWeightAbsorbedBasisMPOTensor","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin 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M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.blockEntropy_eq_sum_bntSectorProbability","module":"TNLean.MPS.MPDO.BNTSectorAreaLaw"},{"id":"n10037","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProbability","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) → (hWeigh…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProbability","module":"TNLean.MPS.MPDO.BNTSectorAreaLaw"},{"id":"n10038","layer":"formal","project":"p8","title":"MPOTensor.bntSectorProbability_pos","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight :…","labels":[],"detail_key":"p8","name":"MPOTensor.bntSectorProbability_pos","module":"TNLean.MPS.MPDO.BNTSectorAreaLaw"},{"id":"n10039","layer":"formal","project":"p8","title":"MPOTensor.normalizedMPO_eq_sum_bntSectorProbability_smul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedMPO_eq_sum_bntSectorProbability_smul","module":"TNLean.MPS.MPDO.BNTSectorAreaLaw"},{"id":"n10040","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_eq_sum_bntSectorProbability_smul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_eq_sum_bntSectorProbability_smul","module":"TNLean.MPS.MPDO.BNTSectorAreaLaw"},{"id":"n10041","layer":"formal","project":"p8","title":"MPOTensor.changePhysicalBasis_eq_ite_of_pairwise_orthogonal_twoSided_physicalSupport","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat (K : (s : Fin g) → MPOTensor d (dim s)) (P : Fin g → Matrix (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.changePhysicalBasis_eq_ite_of_pairwise_orthogonal_twoSided_physicalSupport","module":"TNLean.MPS.MPDO.BNTSectorCoefficientPositivity"},{"id":"n10042","layer":"formal","project":"p8","title":"MPOTensor.exists_nonnegative_real_eq_sectorCoefficient_of_orthogonalSupport","kind":"theorem","summary":"∀ d g N : Nat dim : Fin g → Nat (K : (s : Fin g) → MPOTensor d (dim s)) (P : Fin g → Matrix (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_nonnegative_real_eq_sectorCoefficient_of_orthogonalSupport","module":"TNLean.MPS.MPDO.BNTSectorCoefficientPositivity"},{"id":"n10043","layer":"formal","project":"p8","title":"MPOTensor.exists_nonnegative_sectorCoefficient_of_orthogonalSupport","kind":"theorem","summary":"∀ d g N : Nat dim : Fin g → Nat (K : (s : Fin g) → MPOTensor d (dim s)) (P : Fin g → Matrix (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_nonnegative_sectorCoefficient_of_orthogonalSupport","module":"TNLean.MPS.MPDO.BNTSectorCoefficientPositivity"},{"id":"n10044","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_sitewise_eq_coefficient_smul_of_orthogonalSupport","kind":"theorem","summary":"∀ d g N : Nat dim : Fin g → Nat (K : (s : Fin g) → MPOTensor d (dim s)) (P : Fin g → Matrix (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_sitewise_eq_coefficient_smul_of_orthogonalSupport","module":"TNLean.MPS.MPDO.BNTSectorCoefficientPositivity"},{"id":"n10045","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHForm_of_bntSectorSAL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHForm_of_bntSectorSAL","module":"TNLean.MPS.MPDO.BNTSectorCommutingFamily"},{"id":"n10046","layer":"formal","project":"p8","title":"MPOTensor.nonempty_orthogonalCommutingSectorFamily_of_bntSectorSAL","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_orthogonalCommutingSectorFamily_of_bntSectorSAL","module":"TNLean.MPS.MPDO.BNTSectorCommutingFamily"},{"id":"n10047","layer":"formal","project":"p8","title":"MPOTensor.nonempty_orthogonalCommutingSectorFamily_of_twoSidedPhysicalSlice","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat (K : (s : Fin g) → MPOTensor d (dim s)) (P : Fin g → Matrix (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_orthogonalCommutingSectorFamily_of_twoSidedPhysicalSlice","module":"TNLean.MPS.MPDO.BNTSectorCommutingFamily"},{"id":"n10048","layer":"formal","project":"p8","title":"MPOTensor.completedBntSectorLabel","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → R : (s : Fin g) → Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.completedBntSectorLabel","module":"TNLean.MPS.MPDO.BNTSeparatingProjectors"},{"id":"n10049","layer":"formal","project":"p8","title":"MPOTensor.completedBntSectorProjection","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → R : (s : Fin g) → Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.completedBntSectorProjection","module":"TNLean.MPS.MPDO.BNTSeparatingProjectors"},{"id":"n10050","layer":"formal","project":"p8","title":"MPOTensor.completedBntSectorProjection_isOrthogonal","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.completedBntSectorProjection_isOrthogonal","module":"TNLean.MPS.MPDO.BNTSeparatingProjectors"},{"id":"n10051","layer":"formal","project":"p8","title":"MPOTensor.completedBntSectorProjection_mul_eq_zero","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.completedBntSectorProjection_mul_eq_zero","module":"TNLean.MPS.MPDO.BNTSeparatingProjectors"},{"id":"n10052","layer":"formal","project":"p8","title":"MPOTensor.sum_completedBntSectorProjection","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) R : (s : Fin g) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_completedBntSectorProjection","module":"TNLean.MPS.MPDO.BNTSeparatingProjectors"},{"id":"n10053","layer":"formal","project":"p8","title":"MPOTensor.exists_bntSectorProjectors_four_of_sameMPV₂Pos_isSAL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hM : M.toM…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_bntSectorProjectors_four_of_sameMPV₂Pos_isSAL","module":"TNLean.MPS.MPDO.BNTSourceSectorProjectors"},{"id":"n10054","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_four_reindex_eq_submatrix","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((Matrix.reindex (finThreeArrowEquiv (Fin d)) (finThreeArro…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_four_reindex_eq_submatrix","module":"TNLean.MPS.MPDO.BNTSourceSectorProjectors"},{"id":"n10055","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraClause","kind":"inductive","summary":"Λ : Type u_1 → [Fintype Λ] → O : Nat → Type u_2 → [inst : (L : Nat) → AddCommMonoid (O L)] → [(…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraClause","module":"TNLean.MPS.MPDO.BNTTheoremData"},{"id":"n10056","layer":"formal","project":"p8","title":"MPOTensor.BNTAlgebraClause.idempotent_coefficient_form_ofChi","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] O : Nat → Type u_2 [inst_1 : (L : Nat) → AddCommMonoid (O L)]…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTAlgebraClause.idempotent_coefficient_form_ofChi","module":"TNLean.MPS.MPDO.BNTTheoremData"},{"id":"n10057","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelTheoremData","kind":"inductive","summary":"d D : Nat → 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(S : MPSTensor.SectorDecomposition (HMul.hMul d d)) → Nat → (j : Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.normalizedThreeSiteClosingMatrix","module":"TNLean.MPS.MPDO.BNTThreeSiteReducedClosure"},{"id":"n10141","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.tripleFusionComparison","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.tripleFusionComparison","module":"TNLean.MPS.MPDO.BNTTripleFusionComparison"},{"id":"n10142","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.HasFinalLabelSelectorWords","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.BNTFusionIsome…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.HasFinalLabelSelectorWords","module":"TNLean.MPS.MPDO.BNTTripleFusionSeparation"},{"id":"n10143","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.exists_basis_wordTupleSpanTop_le_three_totalDim_pow_five","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → Exists fun N =>…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.exists_basis_wordTupleSpanTop_le_three_totalDim_pow_five","module":"TNLean.MPS.MPDO.BiCFDerivation.BNTDirectSum"},{"id":"n10144","layer":"formal","project":"p8","title":"MPSTensor.three_mul_pred_mul_pow_four_add_one_le_three_mul_pow_five","kind":"theorem","summary":"∀ D : Nat, LT.lt 0 D → LE.le (HMul.hMul 3 (HMul.hMul (HSub.hSub D 1) (HAdd.hAdd (HPow.hPow D 4)…","labels":[],"detail_key":"p8","name":"MPSTensor.three_mul_pred_mul_pow_four_add_one_le_three_mul_pow_five","module":"TNLean.MPS.MPDO.BiCFDerivation.BNTDirectSum"},{"id":"n10145","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_blockTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) (p N : Nat), A.PairTraceSeparatingAt…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_blockTensor","module":"TNLean.MPS.MPDO.BiCFDerivation.Blocking"},{"id":"n10146","layer":"formal","project":"p8","title":"MPSTensor.BlockEntryIndex","kind":"def","summary":"r : Nat → (Fin r → Nat) → Type","labels":[],"detail_key":"p8","name":"MPSTensor.BlockEntryIndex","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10147","layer":"formal","project":"p8","title":"MPSTensor.HasBlockSelectorOn","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((k : Fin r) → MPSTensor d (dim k)) → Fin r → Nat → Finset (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.HasBlockSelectorOn","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10148","layer":"formal","project":"p8","title":"MPSTensor.HasBlockSelectorWords","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((k : Fin r) → MPSTensor d (dim k)) → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.HasBlockSelectorWords","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10149","layer":"formal","project":"p8","title":"MPSTensor.HasFiniteWordTraceSeparation","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((k : Fin r) → MPSTensor d (dim k)) → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.HasFiniteWordTraceSeparation","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10150","layer":"formal","project":"p8","title":"MPSTensor.HasPairBlockSeparatingWords","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((k : Fin r) → MPSTensor d (dim k)) → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.HasPairBlockSeparatingWords","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10151","layer":"formal","project":"p8","title":"MPSTensor.IsMPOBlockLeftInverse","kind":"def","summary":"r : Nat → dim : Fin r → Nat → p : Nat → ((k : Fin r) → MPOTensor p (dim k)) → Matrix (MPSTensor…","labels":[],"detail_key":"p8","name":"MPSTensor.IsMPOBlockLeftInverse","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10152","layer":"formal","project":"p8","title":"MPSTensor.PairAllWordsSpanTop","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairAllWordsSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10153","layer":"formal","project":"p8","title":"MPSTensor.PairCumulativeWordTupleSpanTop","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairCumulativeWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10154","layer":"formal","project":"p8","title":"MPSTensor.PairTraceSeparatingAll","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10155","layer":"formal","project":"p8","title":"MPSTensor.PairTraceSeparatingAt","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10156","layer":"formal","project":"p8","title":"MPSTensor.PairTraceSeparatingUpTo","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairTraceSeparatingUpTo","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10157","layer":"formal","project":"p8","title":"MPSTensor.PairTraceSeparatingUpTo.mono","kind":"theorem","summary":"∀ d D₁ D₂ : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ S T : Nat, A.PairTraceSeparatingUpTo B S…","labels":[],"detail_key":"p8","name":"MPSTensor.PairTraceSeparatingUpTo.mono","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10158","layer":"formal","project":"p8","title":"MPSTensor.PairWordTupleSpanTop","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.PairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10159","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.exists_mpo_block_left_inverse","kind":"theorem","summary":"∀ r : Nat dim : Fin r → Nat p : Nat M : (k : Fin r) → MPOTensor p (dim k), MPSTensor.WordTupleS…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.exists_mpo_block_left_inverse","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10160","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.isInjective_one","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k), MPSTensor.WordTupleSpanTop…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.isInjective_one","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10161","layer":"formal","project":"p8","title":"MPSTensor.exists_forall_pairTraceSeparatingUpTo_of_forall_pairTraceSeparatingAll","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), (∀ (k j : Fin r), Ne j k…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_forall_pairTraceSeparatingUpTo_of_forall_pairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10162","layer":"formal","project":"p8","title":"MPSTensor.exists_pairCumulativeWordTupleSpanTop_of_pairAllWordsSpanTop","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.PairAllWordsSpanTop B → Exists fun…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairCumulativeWordTupleSpanTop_of_pairAllWordsSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10163","layer":"formal","project":"p8","title":"MPSTensor.exists_pairTraceSeparatingUpTo_iff_pairTraceSeparatingAll","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Iff (Exists fun S => A.PairTraceSepa…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairTraceSeparatingUpTo_iff_pairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10164","layer":"formal","project":"p8","title":"MPSTensor.exists_pairTraceSeparatingUpTo_of_pairTraceSeparatingAll","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.PairTraceSeparatingAll B → Exists…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairTraceSeparatingUpTo_of_pairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10165","layer":"formal","project":"p8","title":"MPSTensor.exists_wordTupleSpanTop_of_hasFiniteWordTraceSeparation","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), MPSTensor.HasFiniteWordT…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_wordTupleSpanTop_of_hasFiniteWordTraceSeparation","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10166","layer":"formal","project":"p8","title":"MPSTensor.hasFiniteWordTraceSeparation_iff_exists_wordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), Iff (MPSTensor.HasFinite…","labels":[],"detail_key":"p8","name":"MPSTensor.hasFiniteWordTraceSeparation_iff_exists_wordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10167","layer":"formal","project":"p8","title":"MPSTensor.hasFiniteWordTraceSeparation_of_wordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) L : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.hasFiniteWordTraceSeparation_of_wordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10168","layer":"formal","project":"p8","title":"MPSTensor.pairAllWordsSpan","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Submodule Complex (Prod (Matrix (Fin D₁) (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAllWordsSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10169","layer":"formal","project":"p8","title":"MPSTensor.pairAllWordsSpanTop_iff_pairTraceSeparatingAll","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Iff (A.PairAllWordsSpanTop B) (A.Pai…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAllWordsSpanTop_iff_pairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10170","layer":"formal","project":"p8","title":"MPSTensor.pairAllWordsSpanTop_of_pairTraceSeparatingAll","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.PairTraceSeparatingAll B → A.PairA…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAllWordsSpanTop_of_pairTraceSeparatingAll","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10171","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeSpan","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Submodule Complex (Prod (Matrix (Fin D₁…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10172","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeSpan_mono","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S T : Nat, LE.le S T → LE.le (A.pairC…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeSpan_mono","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10173","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeWordTupleSpanTop_iff_pairTraceSeparatingUpTo","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, Iff (A.PairCumulativeWordTup…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeWordTupleSpanTop_iff_pairTraceSeparatingUpTo","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10174","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairTraceSeparatingAt","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairTraceSeparatingAt B S…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10175","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairTraceSeparatingUpTo","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairTraceSeparatingUpTo B…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairTraceSeparatingUpTo","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10176","layer":"formal","project":"p8","title":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairWordTupleSpanTop","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairWordTupleSpanTop B S →…","labels":[],"detail_key":"p8","name":"MPSTensor.pairCumulativeWordTupleSpanTop_of_pairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10177","layer":"formal","project":"p8","title":"MPSTensor.pairEvalWordTuple","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → List (Fin d) → Prod (Matrix (Fin D₁) (Fin D₁)…","labels":[],"detail_key":"p8","name":"MPSTensor.pairEvalWordTuple","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10178","layer":"formal","project":"p8","title":"MPSTensor.pairEvalWordTuple_mem_pairCumulativeSpan","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) w : List (Fin d) S : Nat, LE.le w.len…","labels":[],"detail_key":"p8","name":"MPSTensor.pairEvalWordTuple_mem_pairCumulativeSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10179","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_of_pairAllWordsSpanTop","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), A.PairAllWordsSpanTop B → A.PairTrac…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAll_of_pairAllWordsSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10180","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_of_pairWordTupleSpanTop","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairWordTupleSpanTop B S →…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_of_pairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10181","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingUpTo_of_pairCumulativeWordTupleSpanTop","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairCumulativeWordTupleSpa…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingUpTo_of_pairCumulativeWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10182","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingUpTo_of_pairTraceSeparatingAt","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairTraceSeparatingAt B S…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingUpTo_of_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10183","layer":"formal","project":"p8","title":"MPSTensor.pairWordTuple","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → (S : Nat) → (Fin S → Fin d) → Prod (Matrix (F…","labels":[],"detail_key":"p8","name":"MPSTensor.pairWordTuple","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10184","layer":"formal","project":"p8","title":"MPSTensor.pairWordTupleSpanTop_iff_pairTraceSeparatingAt","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, Iff (A.PairWordTupleSpanTop…","labels":[],"detail_key":"p8","name":"MPSTensor.pairWordTupleSpanTop_iff_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10185","layer":"formal","project":"p8","title":"MPSTensor.pairWordTupleSpanTop_of_pairTraceSeparatingAt","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairTraceSeparatingAt B S…","labels":[],"detail_key":"p8","name":"MPSTensor.pairWordTupleSpanTop_of_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10186","layer":"formal","project":"p8","title":"MPSTensor.wordEntryFamily","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((k : Fin r) → MPSTensor d (dim k)) → (L : Nat) → MPSTensor.Blo…","labels":[],"detail_key":"p8","name":"MPSTensor.wordEntryFamily","module":"TNLean.MPS.MPDO.BiCFDerivation.Core"},{"id":"n10187","layer":"formal","project":"p8","title":"MPSTensor.horizontalCFData_diagBlock_not_isNormal","kind":"theorem","summary":"Exists fun weight => Exists fun block => And (MPOTensor.HorizontalCFData weight block) (Not (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.horizontalCFData_diagBlock_not_isNormal","module":"TNLean.MPS.MPDO.BiCFDerivation.DiagonalRestrictionCounterexample"},{"id":"n10188","layer":"formal","project":"p8","title":"MPSTensor.bondDim_eq_and_groundSpace_eq_of_groundSpace_le_of_isNBlkInjective_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.bondDim_eq_and_groundSpace_eq_of_groundSpace_le_of_isNBlkInjective_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10189","layer":"formal","project":"p8","title":"MPSTensor.bondDim_eq_and_groundSpace_eq_of_three_block_trace_relation_left_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.bondDim_eq_and_groundSpace_eq_of_three_block_trace_relation_left_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10190","layer":"formal","project":"p8","title":"MPSTensor.bondDim_eq_of_groundSpace_eq_of_isNBlkInjective","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.bondDim_eq_of_groundSpace_eq_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10191","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMap_eq_leftTraceWordMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), Eq (A.groundSpaceMap L) (A.leftTraceWordMap L)","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMap_eq_leftTraceWordMap","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10192","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_eq_leftTraceWordMap_range","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), Eq (A.groundSpace L) (A.leftTraceWordMap L).range","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_eq_leftTraceWordMap_range","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10193","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_eq_of_three_block_trace_relation","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_eq_of_three_block_trace_relation","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10194","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_finrank_eq_of_isNBlkInjective","kind":"theorem","summary":"∀ d L D : Nat A : MPSTensor d D, Kraus.IsNBlkInjective A L → Eq (Module.finrank Complex (Subtyp…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_finrank_eq_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10195","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_inf_eq_bot_of_not_bondDim_eq_and_groundSpace_eq_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_inf_eq_bot_of_not_bondDim_eq_and_groundSpace_eq_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10196","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_le_of_three_block_trace_relation_left","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_le_of_three_block_trace_relation_left","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10197","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_injective_groundSpaceMap","kind":"theorem","summary":"∀ d D₁ D₂ : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ S : Nat, Eq (min (A.groundSpace S) (B.gro…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_injective_groundSpaceMap","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10198","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_isNBlkInjective","kind":"theorem","summary":"∀ d D₁ D₂ : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ S : Nat, Eq (min (A.groundSpace S) (B.gro…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_of_groundSpace_inf_eq_bot_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumGroundSpace"},{"id":"n10199","layer":"formal","project":"p8","title":"MPSTensor.exists_left_trace_test_of_three_block_trace_relation_right","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_left_trace_test_of_three_block_trace_relation_right","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10200","layer":"formal","project":"p8","title":"MPSTensor.exists_right_trace_test_of_three_block_trace_relation_left","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_right_trace_test_of_three_block_trace_relation_left","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10201","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap","kind":"def","summary":"d D : Nat → MPSTensor d D → (L : Nat) → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D)…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10202","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_injective_of_isNBlkInjective","kind":"theorem","summary":"∀ d L D : Nat A : MPSTensor d D, Kraus.IsNBlkInjective A L → Function.Injective ⇑(A.leftTraceWo…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_injective_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10203","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_range_eq_of_range_le_of_isNBlkInjective_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_range_eq_of_range_le_of_isNBlkInjective_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10204","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_range_eq_of_three_block_trace_relation","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_range_eq_of_three_block_trace_relation","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10205","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_range_eq_of_three_block_trace_relation_left_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_range_eq_of_three_block_trace_relation_left_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10206","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_range_finrank_eq_of_isNBlkInjective","kind":"theorem","summary":"∀ d L D : Nat A : MPSTensor d D, Kraus.IsNBlkInjective A L → Eq (Module.finrank Complex (Subtyp…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_range_finrank_eq_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10207","layer":"formal","project":"p8","title":"MPSTensor.leftTraceWordMap_range_le_of_three_block_trace_relation_left","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.leftTraceWordMap_range_le_of_three_block_trace_relation_left","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10208","layer":"formal","project":"p8","title":"MPSTensor.not_three_block_trace_relation_left_of_isNBlkInjective_of_dim_gt","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ ΔA : Matrix (Fin D₁) (Fin D₁) Complex Δ…","labels":[],"detail_key":"p8","name":"MPSTensor.not_three_block_trace_relation_left_of_isNBlkInjective_of_dim_gt","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10209","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_gt","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_gt","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10210","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_ne","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Kraus.IsNBlkInjective A L → Kraus.IsNB…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_isNBlkInjective_of_dim_ne","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumInput"},{"id":"n10211","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_inf_eq_bot_of_exists_not_forall_mpv_eq_mul_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ [NeZero D₁] [NeZero D₂], Kraus.IsNBlkIn…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_inf_eq_bot_of_exists_not_forall_mpv_eq_mul_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumUniqueness"},{"id":"n10212","layer":"formal","project":"p8","title":"MPSTensor.mpvSubmodule_ne_of_not_exists_mpv_eq_smul","kind":"theorem","summary":"∀ d D₁ D₂ N : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Not (Exists fun c => Eq A.mpv (HSMul.h…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvSubmodule_ne_of_not_exists_mpv_eq_smul","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumUniqueness"},{"id":"n10213","layer":"formal","project":"p8","title":"MPSTensor.not_bondDim_eq_and_groundSpace_eq_of_mpvSubmodule_ne","kind":"theorem","summary":"∀ d D₁ D₂ L N : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ [NeZero D₁] [NeZero D₂], Kraus.IsInje…","labels":[],"detail_key":"p8","name":"MPSTensor.not_bondDim_eq_and_groundSpace_eq_of_mpvSubmodule_ne","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumUniqueness"},{"id":"n10214","layer":"formal","project":"p8","title":"MPSTensor.not_exists_mpv_eq_smul_of_not_exists_forall_mpv_eq_mul","kind":"theorem","summary":"∀ d D₁ D₂ N : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, Not (Exists fun c => ∀ (σ : Fin N → Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.not_exists_mpv_eq_smul_of_not_exists_forall_mpv_eq_mul","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumUniqueness"},{"id":"n10215","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_exists_not_forall_mpv_eq_mul_of_dim_ge","kind":"theorem","summary":"∀ d D₁ D₂ L : Nat A : MPSTensor d D₁ B : MPSTensor d D₂ [NeZero D₁] [NeZero D₂], Kraus.IsNBlkIn…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_threeBlock_of_exists_not_forall_mpv_eq_mul_of_dim_ge","module":"TNLean.MPS.MPDO.BiCFDerivation.DirectSumUniqueness"},{"id":"n10216","layer":"formal","project":"p8","title":"MPSTensor.gaugePhaseEquiv_of_pairAlgSpan_eq_graph_algEquiv","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D) (φ : AlgEquiv Complex (MPSTensor.MatD D) (MPSTenso…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhaseEquiv_of_pairAlgSpan_eq_graph_algEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10217","layer":"formal","project":"p8","title":"MPSTensor.matrixPairSubalgebra_eq_graph_algEquiv_of_axisIdeals_eq_bot","kind":"theorem","summary":"∀ D : Nat (S : Subalgebra Complex (Prod (MPSTensor.MatD D) (MPSTensor.MatD D))) (hfst : Eq (Sub…","labels":[],"detail_key":"p8","name":"MPSTensor.matrixPairSubalgebra_eq_graph_algEquiv_of_axisIdeals_eq_bot","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10218","layer":"formal","project":"p8","title":"MPSTensor.matrixPairSubalgebra_eq_top_of_axes","kind":"theorem","summary":"∀ D₁ D₂ : Nat (S : Subalgebra Complex (Prod (MPSTensor.MatD D₁) (MPSTensor.MatD D₂))), (∀ (x :…","labels":[],"detail_key":"p8","name":"MPSTensor.matrixPairSubalgebra_eq_top_of_axes","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10219","layer":"formal","project":"p8","title":"MPSTensor.matrixPairSubalgebra_eq_top_of_leftAxisIdeal_eq_top","kind":"theorem","summary":"∀ D₁ D₂ : Nat (S : Subalgebra Complex (Prod (MPSTensor.MatD D₁) (MPSTensor.MatD D₂))) (hfst : E…","labels":[],"detail_key":"p8","name":"MPSTensor.matrixPairSubalgebra_eq_top_of_leftAxisIdeal_eq_top","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10220","layer":"formal","project":"p8","title":"MPSTensor.matrixPairSubalgebra_eq_top_of_rightAxisIdeal_eq_top","kind":"theorem","summary":"∀ D₁ D₂ : Nat (S : Subalgebra Complex (Prod (MPSTensor.MatD D₁) (MPSTensor.MatD D₂))), Eq (Suba…","labels":[],"detail_key":"p8","name":"MPSTensor.matrixPairSubalgebra_eq_top_of_rightAxisIdeal_eq_top","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10221","layer":"formal","project":"p8","title":"MPSTensor.not_pairAlgSpan_eq_graph_algEquiv_of_not_gaugePhaseEquiv","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Not (A.GaugePhaseEquiv B) → ∀ (φ : AlgEquiv Compl…","labels":[],"detail_key":"p8","name":"MPSTensor.not_pairAlgSpan_eq_graph_algEquiv_of_not_gaugePhaseEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10222","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Subalgebra Complex (Prod (Matrix (Fin D₁) (Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10223","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan_map_fst_eq_top_of_isInjective","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsInjective A → Eq (Subalgebra…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan_map_fst_eq_top_of_isInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10224","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan_map_snd_eq_top_of_isInjective","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsInjective B → Eq (Subalgebra…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan_map_snd_eq_top_of_isInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10225","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan_toSubmodule_le_pairAllWordsSpan","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), LE.le (Subalgebra.toSubmodule (A.pai…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan_toSubmodule_le_pairAllWordsSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10226","layer":"formal","project":"p8","title":"MPSTensor.pairAllWordsSpanTop_of_pairAlgSpan_eq_top","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Eq (A.pairAlgSpan B) Top.top → A.Pai…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAllWordsSpanTop_of_pairAlgSpan_eq_top","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10227","layer":"formal","project":"p8","title":"MPSTensor.pairEvalWordTuple_mem_pairAlgSpan","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) (w : List (Fin d)), Membership.mem (A…","labels":[],"detail_key":"p8","name":"MPSTensor.pairEvalWordTuple_mem_pairAlgSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10228","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_of_pairAlgSpan_eq_top","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Eq (A.pairAlgSpan B) Top.top → A.Pai…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAll_of_pairAlgSpan_eq_top","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10229","layer":"formal","project":"p8","title":"MPSTensor.pair_mul_mem_pairAllWordsSpan","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) M N : Prod (Matrix (Fin D₁) (Fin D₁)…","labels":[],"detail_key":"p8","name":"MPSTensor.pair_mul_mem_pairAllWordsSpan","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10230","layer":"formal","project":"p8","title":"MPSTensor.subdirect_matrix_pair_eq_top_of_dim_ne","kind":"theorem","summary":"∀ D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (S : Subalgebra Complex (Prod (MPSTensor.MatD D₁) (MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.subdirect_matrix_pair_eq_top_of_dim_ne","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10231","layer":"formal","project":"p8","title":"MPSTensor.subdirect_matrix_pair_eq_top_or_eq_graph_algEquiv","kind":"theorem","summary":"∀ D : Nat [NeZero D] (S : Subalgebra Complex (Prod (MPSTensor.MatD D) (MPSTensor.MatD D))), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.subdirect_matrix_pair_eq_top_or_eq_graph_algEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Algebra"},{"id":"n10232","layer":"formal","project":"p8","title":"MPSTensor.exists_forall_pairTraceSeparatingAt_of_pairSpanTop_period_windows","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), (∀ (k j : Fin r), Ne j k…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_forall_pairTraceSeparatingAt_of_pairSpanTop_period_windows","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10233","layer":"formal","project":"p8","title":"MPSTensor.exists_forall_pairTraceSeparatingAt_of_pairTraceSeparatingAll_of_identity_perio…","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)), (∀ (k j : Fin r), Ne j k…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_forall_pairTraceSeparatingAt_of_pairTraceSeparatingAll_of_identity_period_windows","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10234","layer":"formal","project":"p8","title":"MPSTensor.exists_pairTraceSeparatingAt_of_injective_dim_ne_of_pairWordTupleSpanTop_period…","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsInje…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairTraceSeparatingAt_of_injective_dim_ne_of_pairWordTupleSpanTop_period_window","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10235","layer":"formal","project":"p8","title":"MPSTensor.exists_pairTraceSeparatingAt_of_not_gaugePhaseEquiv_of_pairWordTupleSpanTop_per…","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsInjective A → Kraus.IsInjective B → Eq (F…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairTraceSeparatingAt_of_not_gaugePhaseEquiv_of_pairWordTupleSpanTop_period_window","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10236","layer":"formal","project":"p8","title":"MPSTensor.exists_pairTraceSeparatingAt_of_pairTraceSeparatingUpTo_of_identity_period_wind…","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S : Nat, A.PairTraceSeparatingUpTo B…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_pairTraceSeparatingAt_of_pairTraceSeparatingUpTo_of_identity_period_window","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10237","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan_eq_top_of_injective_dim_ne","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsInje…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan_eq_top_of_injective_dim_ne","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10238","layer":"formal","project":"p8","title":"MPSTensor.pairAlgSpan_eq_top_of_injective_not_gaugePhaseEquiv","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsInjective A → Kraus.IsInjective B → Not (…","labels":[],"detail_key":"p8","name":"MPSTensor.pairAlgSpan_eq_top_of_injective_not_gaugePhaseEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10239","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_and_exists_pairTraceSeparatingUpTo_of_injective_not_gaug…","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsInjective A → Kraus.IsInjective B → Eq (F…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAll_and_exists_pairTraceSeparatingUpTo_of_injective_not_gaugePhaseEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10240","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_of_injective_dim_ne","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsInje…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAll_of_injective_dim_ne","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10241","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_of_injective_not_gaugePhaseEquiv","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsInjective A → Kraus.IsInjective B → Eq (F…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAll_of_injective_not_gaugePhaseEquiv","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.BurnsideJacobson"},{"id":"n10242","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAll_of_injective_not_gaugePhaseEquiv_cast_left","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (h : Eq D₁ D₂) (A : MPSTensor d D₁) (B : MPSTensor d 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(S…","labels":[],"detail_key":"p8","name":"MPSTensor.pairEvalWordTuple_mem_span_pairWordTuple_length","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10247","layer":"formal","project":"p8","title":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_add","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) m n : Nat, Membership.mem (Submodule.…","labels":[],"detail_key":"p8","name":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_add","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10248","layer":"formal","project":"p8","title":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_add_mul","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) base period : Nat, Membership.mem 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(Submodu…","labels":[],"detail_key":"p8","name":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_mul","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10253","layer":"formal","project":"p8","title":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_zero","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂), Membership.mem (Submodule.span Compl…","labels":[],"detail_key":"p8","name":"MPSTensor.pairIdentity_mem_pairWordTupleSpan_zero","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10254","layer":"formal","project":"p8","title":"MPSTensor.pairTraceSeparatingAt_of_pairTraceSeparatingUpTo_of_identity_padding","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S T : Nat, LE.le S T → A.PairTraceSep…","labels":[],"detail_key":"p8","name":"MPSTensor.pairTraceSeparatingAt_of_pairTraceSeparatingUpTo_of_identity_padding","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10255","layer":"formal","project":"p8","title":"MPSTensor.pairWordTupleSpanTop_of_pairCumulativeWordTupleSpanTop_of_identity_padding","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) S T : Nat, LE.le S T → A.PairCumulati…","labels":[],"detail_key":"p8","name":"MPSTensor.pairWordTupleSpanTop_of_pairCumulativeWordTupleSpanTop_of_identity_padding","module":"TNLean.MPS.MPDO.BiCFDerivation.PairHomogenization.Span"},{"id":"n10256","layer":"formal","project":"p8","title":"MPSTensor.pair_mul_mem_span_pairWordTuple_add","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) L S : Nat M N : Prod (Matrix (Fin 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(…","labels":[],"detail_key":"p8","name":"MPOTensor.horizontalCFData_of_wordEntryFamily_linearIndependent","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10259","layer":"formal","project":"p8","title":"MPOTensor.horizontalCFData_of_wordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat μ : Fin r → Complex (A : (k : Fin r) → MPSTensor d (dim k)), (∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.horizontalCFData_of_wordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10260","layer":"formal","project":"p8","title":"MPSTensor.HasBlockSelectorOn.mul","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k) k : Fin r L S : Nat targets…","labels":[],"detail_key":"p8","name":"MPSTensor.HasBlockSelectorOn.mul","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10261","layer":"formal","project":"p8","title":"MPSTensor.PropBlockInjective","kind":"def","summary":"d r : Nat → dim : 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k…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_mem_span_wordTuple_eq_and_eq_zero_of_pairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10264","layer":"formal","project":"p8","title":"MPSTensor.hasBlockSelectorOn_finset_of_pairBlockSeparatingWords","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) S : Nat, MPSTensor.HasPai…","labels":[],"detail_key":"p8","name":"MPSTensor.hasBlockSelectorOn_finset_of_pairBlockSeparatingWords","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10265","layer":"formal","project":"p8","title":"MPSTensor.hasBlockSelectorOn_of_pairTraceSeparatingAt","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) S : Nat k j : Fin r, (A k…","labels":[],"detail_key":"p8","name":"MPSTensor.hasBlockSelectorOn_of_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10266","layer":"formal","project":"p8","title":"MPSTensor.hasBlockSelectorOn_of_pairWordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) S : Nat k j : Fin r, (A k…","labels":[],"detail_key":"p8","name":"MPSTensor.hasBlockSelectorOn_of_pairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10267","layer":"formal","project":"p8","title":"MPSTensor.hasBlockSelectorWords_of_forall_hasBlockSelectorOn_univ_erase","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) S : Nat, (∀ (k : Fin 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(…","labels":[],"detail_key":"p8","name":"MPSTensor.wordTupleSpanTop_mul_pred_of_forall_pairTraceSeparatingAt","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10278","layer":"formal","project":"p8","title":"MPSTensor.wordTupleSpanTop_mul_pred_of_forall_pairWordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) S : Nat, LE.le 2 r → (∀ (…","labels":[],"detail_key":"p8","name":"MPSTensor.wordTupleSpanTop_mul_pred_of_forall_pairWordTupleSpanTop","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10279","layer":"formal","project":"p8","title":"MPSTensor.wordTupleSpanTop_of_card_eq_one_of_isNBlkInjective","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) L : Nat, Eq r 1 → (∀ (k :…","labels":[],"detail_key":"p8","name":"MPSTensor.wordTupleSpanTop_of_card_eq_one_of_isNBlkInjective","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10280","layer":"formal","project":"p8","title":"MPSTensor.wordTupleSpanTop_of_common_blockInjective_of_blockSelectorWords","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) L S : Nat, (∀ (k : Fin r)…","labels":[],"detail_key":"p8","name":"MPSTensor.wordTupleSpanTop_of_common_blockInjective_of_blockSelectorWords","module":"TNLean.MPS.MPDO.BiCFDerivation.Selectors"},{"id":"n10281","layer":"formal","project":"p8","title":"MPSTensor.wordTupleSpanTop_of_common_blockInjective_of_pairBlockSeparatingWords","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) L S : Nat, (∀ (k : Fin 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(Sigma…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.blockedFusionAnalysis","module":"TNLean.MPS.MPDO.BlockedCompleteZipper"},{"id":"n10289","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.blockedFusionSynthesis","kind":"def","summary":"p g : Nat → (Fam : MPOTensor.BNTFusionIsometryFamily (Fin g) p) → (α β : Fin g) → Matrix (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.blockedFusionSynthesis","module":"TNLean.MPS.MPDO.BlockedCompleteZipper"},{"id":"n10290","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.factorization","kind":"def","summary":"d : Nat → [NeZero d] → (K : MPOTensor d 1) → K.PhysicalSectorFactorization","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.factorization","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10291","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_eq_submatrix","kind":"theorem","summary":"∀ d : Nat [inst : NeZero d] (K : MPOTensor d 1) (k h : Fin (MPOTensor.BondOnePhysicalSectorFact…","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_eq_submatrix","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10292","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_pos","kind":"theorem","summary":"∀ d : Nat [inst : NeZero d] (K : MPOTensor d 1), K.IsSAL → ∀ (k h : Fin (MPOTensor.BondOnePhysi…","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_pos","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10293","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_trace","kind":"theorem","summary":"∀ d : Nat [inst : NeZero d] (K : MPOTensor d 1), K.IsSAL → K.IsPhysicalTraceIdempotent → ∀ (k h…","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringOperator_trace","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10294","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringTraceFactorization","kind":"def","summary":"d : Nat → [inst : NeZero d] → (K : MPOTensor d 1) → K.IsSAL → K.IsPhysicalTraceIdempotent → (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.neighboringTraceFactorization","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10295","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.physTraceTransfer_eq_one","kind":"theorem","summary":"∀ d : Nat (K : MPOTensor d 1), K.IsSAL → K.IsPhysicalTraceIdempotent → Eq K.physTraceTransfer 1","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.physTraceTransfer_eq_one","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10296","layer":"formal","project":"p8","title":"MPOTensor.BondOnePhysicalSectorFactorization.sectorEquiv","kind":"def","summary":"d : Nat → Equiv (Fin d) (Sigma fun _k => Prod (Fin d) (Fin 1))","labels":[],"detail_key":"p8","name":"MPOTensor.BondOnePhysicalSectorFactorization.sectorEquiv","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10297","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSectorFactorization_of_bondDim_one","kind":"theorem","summary":"∀ d : Nat [NeZero d] (K : MPOTensor d 1), K.IsSAL → K.IsPhysicalTraceIdempotent → Exists fun F…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSectorFactorization_of_bondDim_one","module":"TNLean.MPS.MPDO.BondOnePhysicalSectorFactorization"},{"id":"n10298","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseMPO","kind":"def","summary":"MPOTensor 2 4","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseMPO","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10299","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseMPOBNTAlgebraTensorClause","kind":"def","summary":"MPOTensor.BondTwoSingletonBaseModel.baseMPO.BNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseMPOBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10300","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseMPO_hasBNTAlgebraTensorClause","kind":"theorem","summary":"MPOTensor.BondTwoSingletonBaseModel.baseMPO.HasBNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseMPO_hasBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10301","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseMPO_isMPDO","kind":"theorem","summary":"MPOTensor.BondTwoSingletonBaseModel.baseMPO.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseMPO_isMPDO","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10302","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolBlocks","kind":"def","summary":"Fin 4 → MPSTensor 4 1","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolBlocks","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10303","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolTensor","kind":"def","summary":"Fin 4 → MPSTensor 4 1","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10304","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolTensor_isNormalTensor","kind":"theorem","summary":"∀ (a : Fin 4), (MPOTensor.BondTwoSingletonBaseModel.baseSymbolTensor a).IsNormalTensor","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.baseSymbolTensor_isNormalTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10305","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.constantPairWeight","kind":"def","summary":"MPOTensor.BondTwoSingletonBaseModel.I → MPOTensor.BondTwoSingletonBaseModel.I → MPOTensor.BondT…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.constantPairWeight","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10306","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.edgeSource","kind":"def","summary":"Fin 4 → MPOTensor.BondTwoSingletonBaseModel.I","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.edgeSource","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10307","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.edgeTarget","kind":"def","summary":"Fin 4 → MPOTensor.BondTwoSingletonBaseModel.I","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.edgeTarget","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10308","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.ghzAmplitude","kind":"def","summary":"(N : Nat) → (Fin N → MPOTensor.BondTwoSingletonBaseModel.I) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.ghzAmplitude","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10309","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.mpo_baseMPO_zero","kind":"theorem","summary":"Eq (MPOTensor.BondTwoSingletonBaseModel.baseMPO.mpo 0) (HSMul.hSMul 4 1)","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.mpo_baseMPO_zero","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10310","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonChi","kind":"def","summary":"MPOTensor.DiagonalChiFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonChi","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10311","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonCoeffs","kind":"def","summary":"MPOTensor.BNTLabelCoefficientFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonCoeffs","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10312","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonTensor","kind":"def","summary":"MPSTensor (HMul.hMul 4 4) 2","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.normalizedSingletonTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10313","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.retainedCyclicIndicator","kind":"def","summary":"(N : Nat) → (Fin N → Fin 4) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.retainedCyclicIndicator","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10314","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.singletonRetainedCoordinateEquiv","kind":"def","summary":"Equiv MPOTensor.BondTwoSingletonBaseModel.SingletonRetainedCoordinate (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.singletonRetainedCoordinateEquiv","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10315","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.singletonScale","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.singletonScale","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10316","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.singletonSectorCoordinateEquiv","kind":"def","summary":"Equiv (Sigma fun alpha => Prod (Fin (MPOTensor.BondTwoSingletonBaseModel.singletonMultiplicity…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.singletonSectorCoordinateEquiv","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10317","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.singletonVerticalCoisometry","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonBaseModel.SingletonRetainedCoordinate (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.singletonVerticalCoisometry","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10318","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.unnormalizedGHZRankOne","kind":"def","summary":"(N : Nat) → Matrix (Fin N → MPOTensor.BondTwoSingletonBaseModel.I) (Fin N → MPOTensor.BondTwoSi…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.unnormalizedGHZRankOne","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10319","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.unnormalizedGHZRankOne_zero","kind":"theorem","summary":"Eq (MPOTensor.BondTwoSingletonBaseModel.unnormalizedGHZRankOne 0) 1","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.unnormalizedGHZRankOne_zero","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10320","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.verticalTensor_baseMPO","kind":"theorem","summary":"Eq MPOTensor.BondTwoSingletonBaseModel.baseMPO.verticalTensor MPOTensor.BondTwoSingletonGramBou…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.verticalTensor_baseMPO","module":"TNLean.MPS.MPDO.BondTwoSingletonBaseModel"},{"id":"n10321","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.bellProjector","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonGramBoundary.I₂ MPOTensor.BondTwoSingletonGramBoundary.I₂ Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.bellProjector","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10322","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.bellProjector_posSemidef","kind":"theorem","summary":"MPOTensor.BondTwoSingletonGramBoundary.bellProjector.PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.bellProjector_posSemidef","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10323","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.bellVector","kind":"def","summary":"MPOTensor.BondTwoSingletonGramBoundary.I₂ → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.bellVector","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10324","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.deformedCompanionBell","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonGramBoundary.I₂ MPOTensor.BondTwoSingletonGramBoundary.I₂ Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.deformedCompanionBell","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10325","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.deformedCompanionBell_not_isHermitian","kind":"theorem","summary":"Not MPOTensor.BondTwoSingletonGramBoundary.deformedCompanionBell.IsHermitian","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.deformedCompanionBell_not_isHermitian","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10326","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonGramBoundary.I MPOTensor.BondTwoSingletonGramBoundary.I Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10327","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal_eq_terminalJ","kind":"theorem","summary":"Eq MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal MPOTensor.BondTwoSingletonGramBounda…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal_eq_terminalJ","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10328","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal_posSemidef","kind":"theorem","summary":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal.PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.deformedTerminal_posSemidef","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10329","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.gauge","kind":"def","summary":"Matrix.GeneralLinearGroup MPOTensor.BondTwoSingletonGramBoundary.I Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.gauge","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10330","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.gaugeMatrix","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonGramBoundary.I MPOTensor.BondTwoSingletonGramBoundary.I Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.gaugeMatrix","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10331","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.gauge_commutes_terminalJ","kind":"theorem","summary":"Eq (HMul.hMul (↑MPOTensor.BondTwoSingletonGramBoundary.gauge) MPOTensor.BondTwoSingletonGramBou…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.gauge_commutes_terminalJ","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10332","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.gauge_gram_ne_one","kind":"theorem","summary":"Ne (HMul.hMul (↑MPOTensor.BondTwoSingletonGramBoundary.gauge).conjTranspose ↑MPOTensor.BondTwoS…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.gauge_gram_ne_one","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10333","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.gaugedSingletonTensor","kind":"def","summary":"MPSTensor (HMul.hMul 4 4) 2","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.gaugedSingletonTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10334","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.physTraceTransfer_singletonTensor","kind":"theorem","summary":"Eq (MPOTensor.verticalBNTMPO MPOTensor.BondTwoSingletonGramBoundary.singletonTensor).physTraceT…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.physTraceTransfer_singletonTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10335","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor","kind":"def","summary":"MPSTensor (HMul.hMul 4 4) 2","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10336","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor_isInjective","kind":"theorem","summary":"Kraus.IsInjective MPOTensor.BondTwoSingletonGramBoundary.singletonTensor","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor_isInjective","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10337","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor_isNormal","kind":"theorem","summary":"Kraus.IsNormal MPOTensor.BondTwoSingletonGramBoundary.singletonTensor","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.singletonTensor_isNormal","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10338","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.terminalJ","kind":"def","summary":"Matrix MPOTensor.BondTwoSingletonGramBoundary.I MPOTensor.BondTwoSingletonGramBoundary.I Complex","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.terminalJ","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10339","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonGramBoundary.terminalJ_posSemidef","kind":"theorem","summary":"MPOTensor.BondTwoSingletonGramBoundary.terminalJ.PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonGramBoundary.terminalJ_posSemidef","module":"TNLean.MPS.MPDO.BondTwoSingletonGramBoundary"},{"id":"n10340","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.gaugeDeformedBaseMPO","kind":"def","summary":"Matrix.GeneralLinearGroup MPOTensor.BondTwoSingletonBaseModel.I Complex → MPOTensor 2 4","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.gaugeDeformedBaseMPO","module":"TNLean.MPS.MPDO.BondTwoSingletonPhysicalGauge"},{"id":"n10341","layer":"formal","project":"p8","title":"MPOTensor.BondTwoSingletonBaseModel.gauge_gram_ne_smul_one","kind":"theorem","summary":"∀ (ω : Real), Ne (HMul.hMul (↑MPOTensor.BondTwoSingletonGramBoundary.gauge).conjTranspose ↑MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.BondTwoSingletonBaseModel.gauge_gram_ne_smul_one","module":"TNLean.MPS.MPDO.BondTwoSingletonPhysicalGauge"},{"id":"n10342","layer":"formal","project":"p8","title":"MPOTensor.HasBNTFusionTensorClause.of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → M.IsRFPViaTS →…","labels":[],"detail_key":"p8","name":"MPOTensor.HasBNTFusionTensorClause.of_isRFPViaTS","module":"TNLean.MPS.MPDO.CPSVBNTFusionTensorClauseFromRFP"},{"id":"n10343","layer":"formal","project":"p8","title":"MPOTensor.isRFPViaTS_iff_hasBNTAlgebraTensorClause","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → Iff M.IsRFPViaT…","labels":[],"detail_key":"p8","name":"MPOTensor.isRFPViaTS_iff_hasBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.CPSVBNTTheoremEquivalence"},{"id":"n10344","layer":"formal","project":"p8","title":"MPOTensor.isRFPViaTS_iff_hasBNTFusionTensorClause","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → Iff M.IsRFPViaT…","labels":[],"detail_key":"p8","name":"MPOTensor.isRFPViaTS_iff_hasBNTFusionTensorClause","module":"TNLean.MPS.MPDO.CPSVBNTTheoremEquivalence"},{"id":"n10345","layer":"formal","project":"p8","title":"MPOTensor.IsCPSVCanonicalForm_toMPSTensor_blockTensor","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.toMPSTensor.IsCPSVCanonicalForm → ∀ (p : Nat), LT.lt 0 p → (M.…","labels":[],"detail_key":"p8","name":"MPOTensor.IsCPSVCanonicalForm_toMPSTensor_blockTensor","module":"TNLean.MPS.MPDO.CPSVBlocking"},{"id":"n10346","layer":"formal","project":"p8","title":"MPOTensor.IsCPSVCanonicalForm_toMPSTensor_blockTwo","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.toMPSTensor.IsCPSVCanonicalForm → M.blockTwo.toMPSTensor.IsCPS…","labels":[],"detail_key":"p8","name":"MPOTensor.IsCPSVCanonicalForm_toMPSTensor_blockTwo","module":"TNLean.MPS.MPDO.CPSVBlocking"},{"id":"n10347","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.bondPattern","kind":"def","summary":"Prod Bool (Prod Bool (Prod Bool Bool)) → Prod Bool (Prod Bool (Prod Bool Bool))","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.bondPattern","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10348","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.bondWeight","kind":"def","summary":"Prod Bool (Prod Bool (Prod Bool Bool)) → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.bondWeight","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10349","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.bondWeightValue","kind":"def","summary":"Prod Bool (Prod Bool (Prod Bool Bool)) → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.bondWeightValue","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10350","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.bondWeight_eq_bondWeightValue","kind":"theorem","summary":"Eq CPSVExample410CorrelatedFlip.bondWeight CPSVExample410CorrelatedFlip.bondWeightValue","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.bondWeight_eq_bondWeightValue","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10351","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.bondWeight_nonneg","kind":"theorem","summary":"∀ (t : Prod Bool (Prod Bool (Prod Bool Bool))), LE.le 0 (CPSVExample410CorrelatedFlip.bondWeigh…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.bondWeight_nonneg","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10352","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyFour","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyFour","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10353","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyFour_eq","kind":"theorem","summary":"Eq CPSVExample410CorrelatedFlip.entropyFour (HSub.hSub (HSub.hSub (HSub.hSub (HMul.hMul 7 (Real…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyFour_eq","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10354","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyOne","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyOne","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10355","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyOne_eq","kind":"theorem","summary":"Eq CPSVExample410CorrelatedFlip.entropyOne (HMul.hMul 2 (Real.log 2))","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyOne_eq","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10356","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyThree","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyThree","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10357","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyThree_eq","kind":"theorem","summary":"Eq CPSVExample410CorrelatedFlip.entropyThree (HSub.hSub (HSub.hSub (HMul.hMul 6 (Real.log 2)) (…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyThree_eq","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10358","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyTwo","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyTwo","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10359","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.entropyTwo_eq","kind":"theorem","summary":"Eq CPSVExample410CorrelatedFlip.entropyTwo (HSub.hSub (HSub.hSub (HMul.hMul 5 (Real.log 2)) (HM…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.entropyTwo_eq","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10360","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.flipProb","kind":"def","summary":"Prod Bool (Prod Bool (Prod Bool Bool)) → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.flipProb","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10361","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.flipProb_nonneg","kind":"theorem","summary":"∀ (s : Prod Bool (Prod Bool (Prod Bool Bool))), LE.le 0 (CPSVExample410CorrelatedFlip.flipProb…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.flipProb_nonneg","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10362","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.flipWeight","kind":"def","summary":"Bool → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.flipWeight","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10363","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.mutualInfoOne","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.mutualInfoOne","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10364","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.mutualInfoTwo","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.mutualInfoTwo","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10365","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.mutualInfoTwo_sub_mutualInfoOne_eq","kind":"theorem","summary":"Eq (HSub.hSub CPSVExample410CorrelatedFlip.mutualInfoTwo CPSVExample410CorrelatedFlip.mutualInf…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.mutualInfoTwo_sub_mutualInfoOne_eq","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10366","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.mutualInfoTwo_sub_mutualInfoOne_pos","kind":"theorem","summary":"LT.lt 0 (HSub.hSub CPSVExample410CorrelatedFlip.mutualInfoTwo CPSVExample410CorrelatedFlip.mutu…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.mutualInfoTwo_sub_mutualInfoOne_pos","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10367","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.oneBondWeight","kind":"def","summary":"Bool → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.oneBondWeight","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10368","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.oneBondWeight_eq_marginal","kind":"theorem","summary":"∀ (u : Bool), Eq (CPSVExample410CorrelatedFlip.oneBondWeight u) (Finset.univ.sum fun v => Finse…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.oneBondWeight_eq_marginal","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10369","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.oneBondWeight_false","kind":"theorem","summary":"Eq (CPSVExample410CorrelatedFlip.oneBondWeight false) (5 / 8)","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.oneBondWeight_false","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10370","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.oneBondWeight_true","kind":"theorem","summary":"Eq (CPSVExample410CorrelatedFlip.oneBondWeight true) (3 / 8)","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.oneBondWeight_true","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10371","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.sum_bondWeight","kind":"theorem","summary":"Eq (Finset.univ.sum fun t => CPSVExample410CorrelatedFlip.bondWeight t) 1","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.sum_bondWeight","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10372","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.sum_flipProb","kind":"theorem","summary":"Eq (Finset.univ.sum fun s => CPSVExample410CorrelatedFlip.flipProb s) 1","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.sum_flipProb","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10373","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.sum_windowWeightOne","kind":"theorem","summary":"Eq (Finset.univ.sum fun x => CPSVExample410CorrelatedFlip.windowWeightOne x) 1","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.sum_windowWeightOne","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10374","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.sum_windowWeightThree","kind":"theorem","summary":"Eq (Finset.univ.sum fun x => CPSVExample410CorrelatedFlip.windowWeightThree x) 1","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.sum_windowWeightThree","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10375","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.sum_windowWeightTwo","kind":"theorem","summary":"Eq (Finset.univ.sum fun x => CPSVExample410CorrelatedFlip.windowWeightTwo x) 1","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.sum_windowWeightTwo","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10376","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.twoBondWeight","kind":"def","summary":"Bool → Bool → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.twoBondWeight","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10377","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.twoBondWeight_eq_marginal","kind":"theorem","summary":"∀ (u v : Bool), Eq (CPSVExample410CorrelatedFlip.twoBondWeight u v) (Finset.univ.sum fun w => F…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.twoBondWeight_eq_marginal","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10378","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.twoBondWeight_false_false","kind":"theorem","summary":"Eq (CPSVExample410CorrelatedFlip.twoBondWeight false false) (7 / 16)","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.twoBondWeight_false_false","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10379","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.twoBondWeight_of_pair_ne_false_false","kind":"theorem","summary":"∀ (u v : Bool), Ne (Prod.mk u v) (Prod.mk false false) → Eq (CPSVExample410CorrelatedFlip.twoBo…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.twoBondWeight_of_pair_ne_false_false","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10380","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightOne","kind":"def","summary":"Prod Bool Bool → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightOne","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10381","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightOne_nonneg","kind":"theorem","summary":"∀ (x : Prod Bool Bool), LE.le 0 (CPSVExample410CorrelatedFlip.windowWeightOne x)","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightOne_nonneg","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10382","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightThree","kind":"def","summary":"Prod Bool (Prod Bool (Prod Bool Bool)) → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightThree","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10383","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightThree_nonneg","kind":"theorem","summary":"∀ (x : Prod Bool (Prod Bool (Prod Bool Bool))), LE.le 0 (CPSVExample410CorrelatedFlip.windowWei…","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightThree_nonneg","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10384","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightTwo","kind":"def","summary":"Prod Bool (Prod Bool Bool) → Real","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightTwo","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10385","layer":"formal","project":"p8","title":"CPSVExample410CorrelatedFlip.windowWeightTwo_nonneg","kind":"theorem","summary":"∀ (x : Prod Bool (Prod Bool Bool)), LE.le 0 (CPSVExample410CorrelatedFlip.windowWeightTwo x)","labels":[],"detail_key":"p8","name":"CPSVExample410CorrelatedFlip.windowWeightTwo_nonneg","module":"TNLean.MPS.MPDO.CPSVExample410CorrelatedFlip"},{"id":"n10386","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.M_isSourceZCL_and_not_isSAL","kind":"theorem","summary":"And MPOTensor.CPSVExample410Operator.M.IsSourceZCL (Not MPOTensor.CPSVExample410Operator.M.IsSA…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.M_isSourceZCL_and_not_isSAL","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10387","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.M_not_isSAL","kind":"theorem","summary":"Not MPOTensor.CPSVExample410Operator.M.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.M_not_isSAL","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10388","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.blockEntropy_four","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.blockEntropy 4 4 ⋯ ⋯) CPSVExample410CorrelatedFlip.entro…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.blockEntropy_four","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10389","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.blockEntropy_one","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.blockEntropy 4 1 ⋯ ⋯) CPSVExample410CorrelatedFlip.entro…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.blockEntropy_one","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10390","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.blockEntropy_three","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.blockEntropy 4 3 ⋯ ⋯) CPSVExample410CorrelatedFlip.entro…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.blockEntropy_three","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10391","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.blockEntropy_two","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.blockEntropy 4 2 ⋯ ⋯) CPSVExample410CorrelatedFlip.entro…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.blockEntropy_two","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10392","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.mutualInfo_two_sub_one","kind":"theorem","summary":"Eq (HSub.hSub (MPOTensor.CPSVExample410Operator.M.mutualInfoChain 4 2 ⋯ ⋯) (MPOTensor.CPSVExamp…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.mutualInfo_two_sub_one","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10393","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Entropy.mutualInfo_two_sub_one_pos","kind":"theorem","summary":"LT.lt 0 (HSub.hSub (MPOTensor.CPSVExample410Operator.M.mutualInfoChain 4 2 ⋯ ⋯) (MPOTensor.CPSV…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Entropy.mutualInfo_two_sub_one_pos","module":"TNLean.MPS.MPDO.CPSVExample410Entropy"},{"id":"n10394","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.M","kind":"def","summary":"MPOTensor 4 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.M","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10395","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.M_isLPDO","kind":"theorem","summary":"MPOTensor.CPSVExample410Operator.M.IsLPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.M_isLPDO","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10396","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.M_isMPDO","kind":"theorem","summary":"MPOTensor.CPSVExample410Operator.M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.M_isMPDO","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10397","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.M_isSourceZCL","kind":"theorem","summary":"MPOTensor.CPSVExample410Operator.M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.M_isSourceZCL","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10398","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.channelCoeff","kind":"def","summary":"Fin 2 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.channelCoeff","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10399","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.correlatedFlip","kind":"def","summary":"Fin 4 → Fin 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.correlatedFlip","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10400","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.pairEquiv","kind":"def","summary":"Equiv (Fin 4) (Prod (Fin 2) (Fin 2))","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.pairEquiv","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10401","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M","kind":"theorem","summary":"Eq MPOTensor.CPSVExample410Operator.M.physTraceTransfer MPOTensor.CPSVExample410Operator.traceP…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10402","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M_ne_zero","kind":"theorem","summary":"Ne MPOTensor.CPSVExample410Operator.M.physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M_ne_zero","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10403","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M_sq","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CPSVExample410Operator.M.physTraceTransfer MPOTensor.CPSVExample410Oper…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.physTraceTransfer_M_sq","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10404","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.purifier","kind":"def","summary":"Fin 4 → Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.purifier","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10405","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.purifierTerm","kind":"def","summary":"Fin 4 → Fin 4 → Fin 2 → Matrix (Prod (Fin 2) (Fin 2)) (Prod (Fin 2) (Fin 2)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.purifierTerm","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10406","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.traceProjector","kind":"def","summary":"Matrix (Fin 4) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.traceProjector","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10407","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Operator.traceProjector_sq","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CPSVExample410Operator.traceProjector MPOTensor.CPSVExample410Operator.…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Operator.traceProjector_sq","module":"TNLean.MPS.MPDO.CPSVExample410Operator"},{"id":"n10408","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_four","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.reducedBlockState 4 4 ⋯).charpoly.roots (HAdd.hAdd (HAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_four","module":"TNLean.MPS.MPDO.CPSVExample410Spectrum"},{"id":"n10409","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_one","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.reducedBlockState 4 1 ⋯).charpoly.roots (HSMul.hSMul 4 (…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_one","module":"TNLean.MPS.MPDO.CPSVExample410Spectrum"},{"id":"n10410","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_three","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.reducedBlockState 4 3 ⋯).charpoly.roots (HAdd.hAdd (HAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_three","module":"TNLean.MPS.MPDO.CPSVExample410Spectrum"},{"id":"n10411","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_two","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample410Operator.M.reducedBlockState 4 2 ⋯).charpoly.roots (HAdd.hAdd (HAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample410Spectrum.charpoly_roots_two","module":"TNLean.MPS.MPDO.CPSVExample410Spectrum"},{"id":"n10412","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.ambientM","kind":"def","summary":"MPOTensor 4 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.ambientM","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10413","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.ambientM_left_unsupported","kind":"theorem","summary":"∀ (i : Fin 4), (∀ (k : Fin 2), Ne i (MPOTensor.CPSVExample411Ambient.siteEmbedding k)) → ∀ (j :…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.ambientM_left_unsupported","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10414","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.ambientM_right_unsupported","kind":"theorem","summary":"∀ (j : Fin 4), (∀ (k : Fin 2), Ne j (MPOTensor.CPSVExample411Ambient.siteEmbedding k)) → ∀ (i :…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.ambientM_right_unsupported","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10415","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.ambientM_supported","kind":"theorem","summary":"∀ (k h : Fin 2), Eq (MPOTensor.CPSVExample411Ambient.ambientM (MPOTensor.CPSVExample411Ambient.…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.ambientM_supported","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10416","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.embedConfig","kind":"def","summary":"N : Nat → (Fin N → Fin 2) → Fin N → Fin 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.embedConfig","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10417","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.inclusion","kind":"def","summary":"Matrix (Fin 4) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.inclusion","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10418","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.inclusion_isometry","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CPSVExample411Ambient.inclusion.conjTranspose MPOTensor.CPSVExample411A…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.inclusion_isometry","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10419","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.mpo_ambientM_eq_singleKrausMap","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.CPSVExample411Ambient.ambientM.mpo N) ((singleKrausMap (MPOTensor.si…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.mpo_ambientM_eq_singleKrausMap","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10420","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.mpo_ambientM_supported_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (σ τ : Fin N → Fin 2), Eq (MPOTensor.CPSVExample411Ambient.ambientM…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.mpo_ambientM_supported_apply","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10421","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.siteEmbedding","kind":"def","summary":"Fin 2 → Fin 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.siteEmbedding","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10422","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.siteEmbedding_injective","kind":"theorem","summary":"Function.Injective MPOTensor.CPSVExample411Ambient.siteEmbedding","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.siteEmbedding_injective","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10423","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.CPSVExample411Ambient.ambientM.mpo N).trace (MPOTensor.CPSVExample41…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10424","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM_closed_form","kind":"theorem","summary":"∀ N : Nat [NeZero N], Eq (MPOTensor.CPSVExample411Ambient.ambientM.mpo N).trace (HDiv.hDiv (HAd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM_closed_form","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10425","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM_eq_partitionFunction","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], Eq (MPOTensor.CPSVExample411Ambient.ambientM.mpo N).trace (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.trace_mpo_ambientM_eq_partitionFunction","module":"TNLean.MPS.MPDO.CPSVExample411Ambient"},{"id":"n10426","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.M","kind":"def","summary":"MPOTensor 2 4","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.M","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10427","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.cycleWeight","kind":"def","summary":"N : Nat → [NeZero N] → (Fin N → Fin 2) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.cycleWeight","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10428","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.edgeWeight","kind":"def","summary":"Fin 2 → Fin 2 → Fin 2 → Fin 2 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.edgeWeight","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10429","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.internalEdgeProduct_eq","kind":"theorem","summary":"∀ L : Nat (x : Fin L → Fin 2), Eq (MPOTensor.CPSVExample411BinarySupport.internalEdgeProduct x)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.internalEdgeProduct_eq","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10430","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.internalTransitionCount","kind":"def","summary":"L : Nat → (Fin L → Fin 2) → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.internalTransitionCount","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10431","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.mpo_M_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (σ τ : Fin N → Fin 2), Eq (MPOTensor.CPSVExample411BinarySupport.M.…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.mpo_M_apply","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10432","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.mpo_M_eq_diagonal","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], Eq (MPOTensor.CPSVExample411BinarySupport.M.mpo N) (Matrix.diagona…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.mpo_M_eq_diagonal","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10433","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.partitionFunction","kind":"def","summary":"(N : Nat) → [NeZero N] → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.partitionFunction","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10434","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.partitionFunction_closed_form","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], Eq (MPOTensor.CPSVExample411BinarySupport.partitionFunction N) (HD…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.partitionFunction_closed_form","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10435","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.partitionFunction_eq_trace_pow","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], Eq (MPOTensor.CPSVExample411BinarySupport.partitionFunction N) (HP…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.partitionFunction_eq_trace_pow","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10436","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_apply","kind":"theorem","summary":"∀ N L : Nat [NeZero N], LE.le 1 L → ∀ (hLN : LE.le L N) (x y : Fin L → Fin 2), Eq (MPOTensor.CP…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_apply","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10437","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_apply_eq_internal_mul_pow","kind":"theorem","summary":"∀ N L : Nat [inst : NeZero N] (hL : LE.le 1 L) (hLN : LE.le L N) (x y : Fin L → Fin 2), Eq (MPO…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_apply_eq_internal_mul_pow","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10438","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.sum_complementPathWeight_eq_weightMatrix_pow","kind":"theorem","summary":"∀ (K : Nat) (a b : Fin 2), Eq (Finset.univ.sum fun w => MPOTensor.CPSVExample411BinarySupport.c…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.sum_complementPathWeight_eq_weightMatrix_pow","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10439","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.sum_cycleWeight_append_eq_internalEdgeProduct_mul_p…","kind":"theorem","summary":"∀ L K : Nat (hL : LE.le 1 L) (x : Fin L → Fin 2), Eq (Finset.univ.sum fun w => MPOTensor.CPSVEx…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.sum_cycleWeight_append_eq_internalEdgeProduct_mul_pow","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10440","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.trace_weightMatrix_pow","kind":"theorem","summary":"∀ (N : Nat), Eq (HPow.hPow MPOTensor.CPSVExample411BinarySupport.weightMatrix N).trace (HDiv.hD…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.trace_weightMatrix_pow","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10441","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.weightMatrix","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.weightMatrix","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10442","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.weightMatrix_pow_apply","kind":"theorem","summary":"∀ (N : Nat) (a b : Fin 2), Eq (HPow.hPow MPOTensor.CPSVExample411BinarySupport.weightMatrix N a…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.weightMatrix_pow_apply","module":"TNLean.MPS.MPDO.CPSVExample411BinarySupport"},{"id":"n10443","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.M_isMPDO","kind":"theorem","summary":"MPOTensor.CPSVExample411BinarySupport.M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.M_isMPDO","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10444","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.M_not_isSAL","kind":"theorem","summary":"Not MPOTensor.CPSVExample411BinarySupport.M.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.M_not_isSAL","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10445","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.ambientM_not_isSAL","kind":"theorem","summary":"Not MPOTensor.CPSVExample411Ambient.ambientM.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.ambientM_not_isSAL","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10446","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.blockEntropy_four","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.blockEntropy 4 4 ⋯ ⋯) (HAdd.hAdd (HAdd.hAdd (HMul.h…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.blockEntropy_four","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10447","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.blockEntropy_one","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.blockEntropy 4 1 ⋯ ⋯) (Real.log 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.blockEntropy_one","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10448","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.blockEntropy_three","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.blockEntropy 4 3 ⋯ ⋯) (HAdd.hAdd (HAdd.hAdd (HMul.h…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.blockEntropy_three","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10449","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.blockEntropy_two","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.blockEntropy 4 2 ⋯ ⋯) (HAdd.hAdd (HMul.hMul 2 (14 /…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.blockEntropy_two","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10450","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.mutualInfo_two_sub_one","kind":"theorem","summary":"Eq (HSub.hSub (MPOTensor.CPSVExample411BinarySupport.M.mutualInfoChain 4 2 ⋯ ⋯) (MPOTensor.CPSV…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.mutualInfo_two_sub_one","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10451","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Entropy.mutualInfo_two_sub_one_pos","kind":"theorem","summary":"LT.lt 0 (HSub.hSub (MPOTensor.CPSVExample411BinarySupport.M.mutualInfoChain 4 2 ⋯ ⋯) (MPOTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Entropy.mutualInfo_two_sub_one_pos","module":"TNLean.MPS.MPDO.CPSVExample411Entropy"},{"id":"n10452","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.ambientM_not_isSourceZCL","kind":"theorem","summary":"Not MPOTensor.CPSVExample411Ambient.ambientM.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.ambientM_not_isSourceZCL","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10453","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.physTraceTransfer_ambientM_ne_zero","kind":"theorem","summary":"Ne MPOTensor.CPSVExample411Ambient.ambientM.physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.physTraceTransfer_ambientM_ne_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10454","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.physTraceTransfer_ambientM_not_idempotent","kind":"theorem","summary":"Not (Eq (HMul.hMul MPOTensor.CPSVExample411Ambient.ambientM.physTraceTransfer MPOTensor.CPSVExa…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.physTraceTransfer_ambientM_not_idempotent","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10455","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.reducedBlockState_four_two_zero","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 2 ⋯ (MPOTensor.CPSVExample411A…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.reducedBlockState_four_two_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10456","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.reducedBlockState_supported_apply","kind":"theorem","summary":"∀ (N L : Nat) (hL : LE.le L N) (x y : Fin L → Fin 2), Eq (MPOTensor.CPSVExample411Ambient.ambie…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.reducedBlockState_supported_apply","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10457","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Ambient.reducedBlockState_three_two_zero","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 3 2 ⋯ (MPOTensor.CPSVExample411A…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Ambient.reducedBlockState_three_two_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10458","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.M_not_isSourceZCL","kind":"theorem","summary":"Not MPOTensor.CPSVExample411BinarySupport.M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.M_not_isSourceZCL","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10459","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.physTraceTransfer_M_ne_zero","kind":"theorem","summary":"Ne MPOTensor.CPSVExample411BinarySupport.M.physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.physTraceTransfer_M_ne_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10460","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.physTraceTransfer_M_not_idempotent","kind":"theorem","summary":"Not (Eq (HMul.hMul MPOTensor.CPSVExample411BinarySupport.M.physTraceTransfer MPOTensor.CPSVExam…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.physTraceTransfer_M_not_idempotent","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10461","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_four_two_zero","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 4 2 ⋯ (fun x => 0) fun x => 0) (1…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_four_two_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10462","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_three_two_zero","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 3 2 ⋯ (fun x => 0) fun x => 0) (5…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411BinarySupport.reducedBlockState_three_two_zero","module":"TNLean.MPS.MPDO.CPSVExample411SourceZCL"},{"id":"n10463","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_eq_X_pow_mul_effective","kind":"theorem","summary":"∀ L : Nat (hL4 : LE.le L 4), Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 L…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_eq_X_pow_mul_effective","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10464","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_eq_zero_add_effective","kind":"theorem","summary":"∀ L : Nat (hL4 : LE.le L 4), Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 L…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_eq_zero_add_effective","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10465","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_four","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 4 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_four","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10466","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_one","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 1 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_one","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10467","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_three","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 3 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_three","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10468","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_two","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411Ambient.ambientM.reducedBlockState 4 2 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.ambient_charpoly_roots_two","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10469","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_four","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 4 4 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_four","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10470","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_one","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 4 1 ⋯).charpoly.roots (HSMul.hSMu…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_one","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10471","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_three","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 4 3 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_three","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10472","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_two","kind":"theorem","summary":"Eq (MPOTensor.CPSVExample411BinarySupport.M.reducedBlockState 4 2 ⋯).charpoly.roots (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.effective_charpoly_roots_two","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10473","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.effective_reducedBlockState_four_eq_diagonal","kind":"theorem","summary":"∀ L : Nat, LE.le 1 L → ∀ (hL4 : LE.le L 4), Eq (MPOTensor.CPSVExample411BinarySupport.M.reduced…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.effective_reducedBlockState_four_eq_diagonal","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10474","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.transition_counts_four","kind":"theorem","summary":"Eq (Multiset.map MPOTensor.CPSVExample411BinarySupport.internalTransitionCount Finset.univ.val)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.transition_counts_four","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10475","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.transition_counts_one","kind":"theorem","summary":"Eq (Multiset.map MPOTensor.CPSVExample411BinarySupport.internalTransitionCount Finset.univ.val)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.transition_counts_one","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10476","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.transition_counts_three","kind":"theorem","summary":"Eq (Multiset.map MPOTensor.CPSVExample411BinarySupport.internalTransitionCount Finset.univ.val)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.transition_counts_three","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10477","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample411Spectrum.transition_counts_two","kind":"theorem","summary":"Eq (Multiset.map MPOTensor.CPSVExample411BinarySupport.internalTransitionCount Finset.univ.val)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample411Spectrum.transition_counts_two","module":"TNLean.MPS.MPDO.CPSVExample411Spectrum"},{"id":"n10478","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleMiddleBits","kind":"def","summary":"Fin 4 → Prod (Fin 2) (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleMiddleBits","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10479","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteEquiv","kind":"def","summary":"Equiv (Prod (Fin 2) (Prod (Fin 4) (Fin 2))) (Fin 4 → Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteEquiv","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10480","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState","kind":"def","summary":"Matrix (Prod (Fin 2) (Prod (Fin 4) (Fin 2))) (Prod (Fin 2) (Prod (Fin 4) (Fin 2))) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10481","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_entropy","kind":"theorem","summary":"Eq (vonNeumannEntropy MPOTensor.CPSVExample412Literal.fourCycleTripartiteState MPOTensor.CPSVEx…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_entropy","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10482","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_eq_diagonal","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.fourCycleTripartiteState (Matrix.diagonal fun x => ↑(MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_eq_diagonal","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10483","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_isHermitian","kind":"theorem","summary":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.IsHermitian","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_isHermitian","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10484","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_not_isSSAEquality","kind":"theorem","summary":"Not (IsSSAEquality MPOTensor.CPSVExample412Literal.fourCycleTripartiteState MPOTensor.CPSVExamp…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_not_isSSAEquality","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10485","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleWeight","kind":"def","summary":"Prod (Fin 2) (Prod (Fin 4) (Fin 2)) → Real","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleWeight","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10486","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.siteSignReal","kind":"def","summary":"Fin 2 → Real","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.siteSignReal","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10487","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceAC_fourCycleTripartiteState","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceAC_ABC (HSMul.hSMul (Inv.inv 4…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceAC_fourCycleTripartiteState","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10488","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceAC_fourCycleTripartiteState_entropy","kind":"theorem","summary":"Eq (vonNeumannEntropy MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceAC_ABC ⋯) (…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceAC_fourCycleTripartiteState_entropy","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10489","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceA_fourCycleTripartiteState","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceA_ABC (HSMul.hSMul (Inv.inv 8)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceA_fourCycleTripartiteState","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10490","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceA_fourCycleTripartiteState_entropy","kind":"theorem","summary":"Eq (vonNeumannEntropy MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceA_ABC ⋯) (H…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceA_fourCycleTripartiteState_entropy","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10491","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceC_fourCycleTripartiteState","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceC_ABC (HSMul.hSMul (Inv.inv 8)…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceC_fourCycleTripartiteState","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10492","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.traceC_fourCycleTripartiteState_entropy","kind":"theorem","summary":"Eq (vonNeumannEntropy MPOTensor.CPSVExample412Literal.fourCycleTripartiteState.traceC_ABC ⋯) (H…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.traceC_fourCycleTripartiteState_entropy","module":"TNLean.MPS.MPDO.CPSVExample412FourCycleEntropy"},{"id":"n10493","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M","kind":"def","summary":"MPOTensor 2 2","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10494","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M_isMPDO","kind":"theorem","summary":"MPOTensor.CPSVExample412Literal.M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M_isMPDO","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10495","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M_isSAL","kind":"theorem","summary":"MPOTensor.CPSVExample412Literal.M.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M_isSAL","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10496","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M_isSourceZCL","kind":"theorem","summary":"MPOTensor.CPSVExample412Literal.M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M_isSourceZCL","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10497","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M_not_isRFPViaTS","kind":"theorem","summary":"Not MPOTensor.CPSVExample412Literal.M.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M_not_isRFPViaTS","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10498","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.one_site_trace_loses_sigmaZ","kind":"theorem","summary":"∀ (L : Nat), LE.le 1 L → Eq (MPOTensor.CPSVExample412Literal.M.reducedBlockState (HAdd.hAdd L 1…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.one_site_trace_loses_sigmaZ","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10499","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.M.physTraceTransfer (Matrix.of (Matrix.vecCons (Matrix.vecCo…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10500","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M_not_idempotent","kind":"theorem","summary":"Ne (HMul.hMul MPOTensor.CPSVExample412Literal.M.physTraceTransfer MPOTensor.CPSVExample412Liter…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M_not_idempotent","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10501","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M_sq","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CPSVExample412Literal.M.physTraceTransfer MPOTensor.CPSVExample412Liter…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.physTraceTransfer_M_sq","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10502","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.reducedBlockState_M_eq_scaled_one","kind":"theorem","summary":"∀ N L : Nat, LE.le 1 L → ∀ (hLN : LT.lt L N) (hL : optParam (LE.le L N) ⋯), Eq (MPOTensor.CPSVE…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.reducedBlockState_M_eq_scaled_one","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10503","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.rho_eq_diagonal","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.CPSVExample412Literal.M.mpo N) (Matrix.diagonal fun σ => HAdd.hAdd 1…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.rho_eq_diagonal","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10504","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.rho_eq_finKronecker","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.CPSVExample412Literal.M.mpo N) (HAdd.hAdd (Matrix.finKro…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.rho_eq_finKronecker","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10505","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.sigmaZ","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.sigmaZ","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10506","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.trace_rho","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.CPSVExample412Literal.M.mpo N).trace (HPow.hPow 2 N)","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.trace_rho","module":"TNLean.MPS.MPDO.CPSVExample412Literal"},{"id":"n10507","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_not_isGSNNCH","kind":"theorem","summary":"Not MPOTensor.CPSVExample412NormalizedRFP.Mhat.IsGSNNCH","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_not_isGSNNCH","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedGSNNCH"},{"id":"n10508","layer":"formal","project":"p8","title":"MPOTensor.isGSNNCH_smul_iff","kind":"theorem","summary":"∀ d D : Nat c : Complex, Ne c 0 → ∀ (M : MPOTensor d D), Iff (HSMul.hSMul c M).IsGSNNCH M.IsGSN…","labels":[],"detail_key":"p8","name":"MPOTensor.isGSNNCH_smul_iff","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedGSNNCH"},{"id":"n10509","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.Mhat","kind":"def","summary":"MPOTensor 2 2","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.Mhat","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10510","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_isMPDO","kind":"theorem","summary":"MPOTensor.CPSVExample412NormalizedRFP.Mhat.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_isMPDO","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10511","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_isRFPViaTS","kind":"theorem","summary":"MPOTensor.CPSVExample412NormalizedRFP.Mhat.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.Mhat_isRFPViaTS","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10512","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.coarseKraus","kind":"def","summary":"Prod (Fin 2) (Fin 2) → Matrix (Fin 2) (Prod (Fin 2) (Fin 2)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.coarseKraus","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10513","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Prod (Fin 2) (Fin 2)) (Prod (Fin 2) (Fin 2)) Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10514","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.CPSVExample412NormalizedRFP.coarseMap","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap_isKrausCPTP","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10515","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap_physClose2","kind":"theorem","summary":"∀ (X : Matrix (Fin 2) (Fin 2) Complex), Eq (MPOTensor.CPSVExample412NormalizedRFP.coarseMap (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.coarseMap_physClose2","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10516","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.mpo_Mhat","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.CPSVExample412NormalizedRFP.Mhat.mpo N) (HSMul.hSMul (HPow.hPow (↑(1…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.mpo_Mhat","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10517","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.mpo_Mhat_eq_normalizedMPO_M","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.CPSVExample412NormalizedRFP.Mhat.mpo N) (MPOTensor.CPSVE…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.mpo_Mhat_eq_normalizedMPO_M","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10518","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.pairParity","kind":"def","summary":"Prod (Fin 2) (Fin 2) → Fin 2","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.pairParity","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10519","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.refineKraus","kind":"def","summary":"Prod (Fin 2) (Fin 2) → Matrix (Prod (Fin 2) (Fin 2)) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.refineKraus","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10520","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.refineMap","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Fin 2) (Fin 2) Complex) (Matrix (Prod (Fin 2) (Fin 2))…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.refineMap","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10521","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.refineMap_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.CPSVExample412NormalizedRFP.refineMap","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.refineMap_isKrausCPTP","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10522","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.refineMap_physClose1","kind":"theorem","summary":"∀ (X : Matrix (Fin 2) (Fin 2) Complex), Eq (MPOTensor.CPSVExample412NormalizedRFP.refineMap (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.refineMap_physClose1","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10523","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.refinementAmplitude","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.refinementAmplitude","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10524","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412NormalizedRFP.trace_mpo_Mhat","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.CPSVExample412NormalizedRFP.Mhat.mpo N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412NormalizedRFP.trace_mpo_Mhat","module":"TNLean.MPS.MPDO.CPSVExample412NormalizedRFP"},{"id":"n10525","layer":"formal","project":"p8","title":"CPSVExamples410411Arithmetic.example410_integer_inequality","kind":"theorem","summary":"GT.gt (HMul.hMul (HPow.hPow 2 32) (HPow.hPow 7 7)) (HMul.hMul (HPow.hPow 3 3) (HPow.hPow 5 20))","labels":[],"detail_key":"p8","name":"CPSVExamples410411Arithmetic.example410_integer_inequality","module":"TNLean.MPS.MPDO.CPSVExamples410411Arithmetic"},{"id":"n10526","layer":"formal","project":"p8","title":"CPSVExamples410411Arithmetic.example410_log_ratio_pos","kind":"theorem","summary":"LT.lt 0 (Real.log (HDiv.hDiv (HMul.hMul (HPow.hPow 2 32) (HPow.hPow 7 7)) (HMul.hMul (HPow.hPow…","labels":[],"detail_key":"p8","name":"CPSVExamples410411Arithmetic.example410_log_ratio_pos","module":"TNLean.MPS.MPDO.CPSVExamples410411Arithmetic"},{"id":"n10527","layer":"formal","project":"p8","title":"CPSVExamples410411Arithmetic.example411_log_ratio_pos","kind":"theorem","summary":"LT.lt 0 (Real.log (HDiv.hDiv (HMul.hMul (HPow.hPow 41 82) (HPow.hPow 5 50)) (HMul.hMul (HMul.hM…","labels":[],"detail_key":"p8","name":"CPSVExamples410411Arithmetic.example411_log_ratio_pos","module":"TNLean.MPS.MPDO.CPSVExamples410411Arithmetic"},{"id":"n10528","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.gramDressing_eq_of_two_grouped_corners","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → ∀ (A : MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.gramDressing_eq_of_two_grouped_corners","module":"TNLean.MPS.MPDO.CPSVFigureEight"},{"id":"n10529","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.gram_eq_pos_smul_gram_of_two_grouped_corners","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → ∀ (A : MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.gram_eq_pos_smul_gram_of_two_grouped_corners","module":"TNLean.MPS.MPDO.CPSVFigureEight"},{"id":"n10530","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_gram_conj_eq","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_gram_conj_eq","module":"TNLean.MPS.MPDO.CPSVGroupedFigureEight"},{"id":"n10531","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_exists_unitary_normalization","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_exists_unitary_normalization","module":"TNLean.MPS.MPDO.CPSVGroupedGramNormalization"},{"id":"n10532","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_gram_eq_pos_smul_one","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.grouped_sector_gram_eq_pos_smul_one","module":"TNLean.MPS.MPDO.CPSVGroupedGramNormalization"},{"id":"n10533","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.exists_normalized_grouped_sector_maps","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.exists_normalized_grouped_sector_maps","module":"TNLean.MPS.MPDO.CPSVNormalizedGroupedSectors"},{"id":"n10534","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.insertedTensor_eq_of_firstSiteActionAgree","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsCPSVCanonicalForm → ∀ Y Z : Matrix (Fin d) (Fin d) Complex…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.insertedTensor_eq_of_firstSiteActionAgree","module":"TNLean.MPS.MPDO.CPSVOriginalSpaceLemmaL"},{"id":"n10535","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.linearMarkedTensor_eq_of_trace_agree","kind":"theorem","summary":"∀ d e D : Nat (A : MPSTensor d D), A.IsCPSVCanonicalForm → ∀ (f g : Fin e → Fin d → Complex), (…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.linearMarkedTensor_eq_of_trace_agree","module":"TNLean.MPS.MPDO.CPSVOriginalSpaceLemmaL"},{"id":"n10536","layer":"formal","project":"p8","title":"MPSTensor.linearMarkedTensor_coisometry_reconstruction","kind":"theorem","summary":"∀ d e D n : Nat (f : Fin e → Fin d → Complex) (A : MPSTensor d D) (B : MPSTensor d n) (V : Matr…","labels":[],"detail_key":"p8","name":"MPSTensor.linearMarkedTensor_coisometry_reconstruction","module":"TNLean.MPS.MPDO.CPSVOriginalSpaceLemmaL"},{"id":"n10537","layer":"formal","project":"p8","title":"MPSTensor.trace_linearMarkedTensor_mul_evalWord_of_coisometry_reconstruction","kind":"theorem","summary":"∀ d e D n : Nat (f : Fin e → Fin d → Complex) (A : MPSTensor d D) (B : MPSTensor d n) (V : Matr…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_linearMarkedTensor_mul_evalWord_of_coisometry_reconstruction","module":"TNLean.MPS.MPDO.CPSVOriginalSpaceLemmaL"},{"id":"n10538","layer":"formal","project":"p8","title":"MPSTensor.trace_marked_mul_evalWord_of_coisometry_reconstruction","kind":"theorem","summary":"∀ d D n : Nat (A : MPSTensor d D) (B : MPSTensor d n) (V : Matrix (Fin n) (Fin D) Complex), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_marked_mul_evalWord_of_coisometry_reconstruction","module":"TNLean.MPS.MPDO.CPSVOriginalSpaceLemmaL"},{"id":"n10539","layer":"formal","project":"p8","title":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_cpsvCanonicalForm","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → M.toMPSTensor.IsCPSVCanonicalForm → M.verticalTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_cpsvCanonicalForm","module":"TNLean.MPS.MPDO.CPSVPeriodicExclusion"},{"id":"n10540","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.exists_not_commute_of_displaced","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → ∀ Q : Matrix (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.exists_not_commute_of_displaced","module":"TNLean.MPS.MPDO.CPSVPeriodicExclusion"},{"id":"n10541","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.sum_representative_dim_le","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), LE.le (Finset.univ.sum fun j =>…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.sum_representative_dim_le","module":"TNLean.MPS.MPDO.CPSVSharpBlocking"},{"id":"n10542","layer":"formal","project":"p8","title":"MPOTensor.cpsvRFPBNTFusionTensorClause","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → M.IsRFPViaTS →…","labels":[],"detail_key":"p8","name":"MPOTensor.cpsvRFPBNTFusionTensorClause","module":"TNLean.MPS.MPDO.CPSVTopologicalPhysicalGibbs"},{"id":"n10543","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.exists_sectorCompression_ne_zero_of_corner","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → ∀ (P : Matrix (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.exists_sectorCompression_ne_zero_of_corner","module":"TNLean.MPS.MPDO.CPSVVerticalBNT"},{"id":"n10544","layer":"formal","project":"p8","title":"MPOTensor.verticalCF_of_cpsvCanonicalForm","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → M.IsVerticalCF","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCF_of_cpsvCanonicalForm","module":"TNLean.MPS.MPDO.CPSVVerticalCanonicalForm"},{"id":"n10545","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.exists_cpsvVerticalDecomposition","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → Nonempty M.CPSV…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.exists_cpsvVerticalDecomposition","module":"TNLean.MPS.MPDO.CPSVVerticalDecomposition"},{"id":"n10546","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.exists_retainedProductSpectralFamily","kind":"theorem","summary":"∀ g d D : Nat M : MPOTensor d D, M.toMPSTensor.IsCPSVCanonicalForm → ∀ (dim mult : Fin g → Nat)…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.exists_retainedProductSpectralFamily","module":"TNLean.MPS.MPDO.CPSVVerticalProductSpectralFamily"},{"id":"n10547","layer":"formal","project":"p8","title":"MPOTensor.CZX.actedSector","kind":"def","summary":"Fin 2 → MPSTensor 4 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actedSector","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10548","layer":"formal","project":"p8","title":"MPOTensor.CZX.actedSector_apply","kind":"theorem","summary":"∀ (x : Fin 2) (i : Fin 4), Eq (MPOTensor.CZX.actedSector x i) (ite (Eq i (MPOTensor.CZX.actionL…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actedSector_apply","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10549","layer":"formal","project":"p8","title":"MPOTensor.CZX.actionLetter","kind":"def","summary":"Fin 2 → Fin 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actionLetter","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10550","layer":"formal","project":"p8","title":"MPOTensor.CZX.actionMatrix","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actionMatrix","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10551","layer":"formal","project":"p8","title":"MPOTensor.CZX.actionMatrix_coordinates","kind":"theorem","summary":"And (Eq (MPOTensor.CZX.actionMatrix 0) (Matrix.of (Matrix.vecCons (Matrix.vecCons 0 (Matrix.vec…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actionMatrix_coordinates","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10552","layer":"formal","project":"p8","title":"MPOTensor.CZX.actionTarget","kind":"def","summary":"Fin 2 → MPSTensor 4 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actionTarget","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10553","layer":"formal","project":"p8","title":"MPOTensor.CZX.actionTarget_apply","kind":"theorem","summary":"∀ (x : Fin 2) (i : Fin 4), Eq (MPOTensor.CZX.actionTarget x i) (ite (Eq i (MPOTensor.CZX.action…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.actionTarget_apply","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10554","layer":"formal","project":"p8","title":"MPOTensor.CZX.blockedSector","kind":"def","summary":"Fin 2 → MPSTensor 4 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.blockedSector","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10555","layer":"formal","project":"p8","title":"MPOTensor.CZX.dressedActionBra","kind":"def","summary":"Fin 2 → Matrix (Fin 1) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.dressedActionBra","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10556","layer":"formal","project":"p8","title":"MPOTensor.CZX.dressedActionBra_absorption","kind":"theorem","summary":"∀ (x : Fin 2) (b : List (Fin 4)), Ne b List.nil → Eq (HMul.hMul (MPOTensor.CZX.dressedActionBra…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.dressedActionBra_absorption","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10557","layer":"formal","project":"p8","title":"MPOTensor.CZX.dressedAction_isReduction","kind":"theorem","summary":"∀ (x : Fin 2), (MPOTensor.CZX.actedSector x).IsReduction (MPOTensor.CZX.actionTarget x) (MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.dressedAction_isReduction","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10558","layer":"formal","project":"p8","title":"MPOTensor.CZX.dressedAction_isReductionExteriorBufferLength","kind":"theorem","summary":"∀ (x : Fin 2), (MPOTensor.CZX.actedSector x).IsReductionExteriorBufferLength (MPOTensor.CZX.act…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.dressedAction_isReductionExteriorBufferLength","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10559","layer":"formal","project":"p8","title":"MPOTensor.CZX.evalWord_actedSector","kind":"theorem","summary":"∀ (x : Fin 2) (w : List (Fin 4)), Ne w List.nil → Eq (Kraus.evalWord (MPOTensor.CZX.actedSector…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.evalWord_actedSector","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10560","layer":"formal","project":"p8","title":"MPOTensor.CZX.evalWord_actionTarget","kind":"theorem","summary":"∀ (x : Fin 2) (w : List (Fin 4)), Ne w List.nil → Eq (Kraus.evalWord (MPOTensor.CZX.actionTarge…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.evalWord_actionTarget","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10561","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedActionBra","kind":"def","summary":"Fin 2 → Matrix (Fin 1) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedActionBra","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10562","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedActionKet","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 1) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedActionKet","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10563","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedAction_empty_ne","kind":"theorem","summary":"∀ (x : Fin 2), Ne (HMul.hMul (HMul.hMul (MPOTensor.CZX.printedActionBra x) (Kraus.evalWord (MPO…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedAction_empty_ne","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10564","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedAction_exterior","kind":"theorem","summary":"∀ (x : Fin 2) (a c b : List (Fin 4)), Ne a List.nil → Ne c List.nil → Ne b List.nil → Eq (HMul.…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedAction_exterior","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10565","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedAction_finite_identities","kind":"theorem","summary":"∀ (x : Fin 2), And (Eq (HMul.hMul (MPOTensor.CZX.actionMatrix x) (MPOTensor.CZX.actionMatrix x)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedAction_finite_identities","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10566","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedAction_interior","kind":"theorem","summary":"∀ (x : Fin 2) (w : List (Fin 4)), Ne w List.nil → Eq (HMul.hMul (HMul.hMul (MPOTensor.CZX.print…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedAction_interior","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10567","layer":"formal","project":"p8","title":"MPOTensor.CZX.printedAction_not_isReduction","kind":"theorem","summary":"∀ (x : Fin 2), Not ((MPOTensor.CZX.actedSector x).IsReduction (MPOTensor.CZX.actionTarget x) (M…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.printedAction_not_isReduction","module":"TNLean.MPS.MPDO.CZXActionTensors"},{"id":"n10568","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedComparison_coordinates","kind":"theorem","summary":"And (Eq MPOTensor.CZX.displayedRawComparison (HSMul.hSMul (-2) 1)) (And (Eq MPOTensor.CZX.displ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedComparison_coordinates","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10569","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedRawComparison","kind":"def","summary":"Matrix (Fin 4) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedRawComparison","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10570","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedRawComparison_gram","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CZX.displayedRawComparison.conjTranspose MPOTensor.CZX.displayedRawComp…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedRawComparison_gram","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10571","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedRawComparison_ne_rightOverlap","kind":"theorem","summary":"Ne MPOTensor.CZX.displayedRawComparison (HMul.hMul MPOTensor.CZX.displayedY₁ MPOTensor.CZX.disp…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedRawComparison_ne_rightOverlap","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10572","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedRawComparison_not_isUnitaryBetween","kind":"theorem","summary":"Not MPOTensor.CZX.displayedRawComparison.IsUnitaryBetween","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedRawComparison_not_isUnitaryBetween","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10573","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedReflectedX₁","kind":"def","summary":"Matrix (Prod (Fin 4) (Fin 2)) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedReflectedX₁","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10574","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedReflectedY₁","kind":"def","summary":"Matrix (Fin 4) (Prod (Fin 2) (Fin 4)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedReflectedY₁","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10575","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedReflected_eq_neg","kind":"theorem","summary":"And (Eq MPOTensor.CZX.displayedReflectedX₁ (Neg.neg MPOTensor.CZX.displayedX₁)) (Eq MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedReflected_eq_neg","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10576","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedWeightedComparison","kind":"def","summary":"Matrix (Fin 4) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedWeightedComparison","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10577","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedWeightedComparison_spec","kind":"theorem","summary":"And MPOTensor.CZX.displayedWeightedComparison.IsUnitaryBetween (And (Eq MPOTensor.CZX.displayed…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedWeightedComparison_spec","module":"TNLean.MPS.MPDO.CZXComparisonNormalization"},{"id":"n10578","layer":"formal","project":"p8","title":"MPOTensor.CZX.bddAbove_range_finrank_gaugeInvariantSubspace","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N), BddAbove (Set.range fun R => iSup fun x => Module.finrank Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bddAbove_range_finrank_gaugeInvariantSubspace","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10579","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_gen_gen_mulVec_defectVector_one","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTupleStar MPOTensor.CZX.gen MPOTensor.CZX.gen)).mulVec (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_gen_gen_mulVec_defectVector_one","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10580","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_gen_gen_mulVec_defectVector_zero","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTupleStar MPOTensor.CZX.gen MPOTensor.CZX.gen)).mulVec (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_gen_gen_mulVec_defectVector_zero","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10581","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_mem_completionClass","kind":"theorem","summary":"Membership.mem MPOTensor.CZX.defectMaps.completionClass MPOTensor.CZX.circuitTupleStar","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_mem_completionClass","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10582","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_mulVec_defectVector","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (x : ZMod 2), Eq ((↑(MPOTensor.CZX.circuitTupleStar a b)).mul…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_mulVec_defectVector","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10583","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_gen_mulVec_defectVector_one","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen MPOTensor.CZX.gen)).mulVec (MPOTensor.CZX.d…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_gen_mulVec_defectVector_one","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10584","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_gen_mulVec_defectVector_zero","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen MPOTensor.CZX.gen)).mulVec (MPOTensor.CZX.d…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_gen_mulVec_defectVector_zero","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10585","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_one_mulVec_defectVector_one","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen 1)).mulVec (MPOTensor.CZX.defectVector MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_one_mulVec_defectVector_one","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10586","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_one_mulVec_defectVector_zero","kind":"theorem","summary":"Eq ((↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen 1)).mulVec (MPOTensor.CZX.defectVector MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_one_mulVec_defectVector_zero","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10587","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_mem_completionClass","kind":"theorem","summary":"Membership.mem MPOTensor.CZX.defectMaps.completionClass MPOTensor.CZX.circuitTuple","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_mem_completionClass","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10588","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_mulVec_defectVector","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (x : ZMod 2), Eq ((↑(MPOTensor.CZX.circuitTuple a b)).mulVec…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_mulVec_defectVector","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10589","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_one_mulVec_defectVector","kind":"theorem","summary":"∀ (b : Multiplicative (ZMod 2)) (x : ZMod 2), Eq ((↑(MPOTensor.CZX.circuitTuple 1 b)).mulVec (M…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_one_mulVec_defectVector","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10590","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTupleStar_four","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem (MPOTensor.gaugeInvariantSubspace 4…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTupleStar_four","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10591","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTupleStar_three","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem (MPOTensor.gaugeInvariantSubspace 4…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTupleStar_three","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10592","layer":"formal","project":"p8","title":"MPOTensor.CZX.fourteen_le_iSup_finrank_gaugeInvariantSubspace_completionClass","kind":"theorem","summary":"LE.le 14 (iSup fun R => iSup fun h => Module.finrank Complex (Subtype fun x => Membership.mem (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fourteen_le_iSup_finrank_gaugeInvariantSubspace_completionClass","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10593","layer":"formal","project":"p8","title":"MPOTensor.CZX.isCompletion_circuitTuple","kind":"theorem","summary":"MPOTensor.CZX.defectMaps.IsCompletion MPOTensor.CZX.circuitTuple","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.isCompletion_circuitTuple","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10594","layer":"formal","project":"p8","title":"MPOTensor.CZX.isCompletion_circuitTupleStar","kind":"theorem","summary":"MPOTensor.CZX.defectMaps.IsCompletion MPOTensor.CZX.circuitTupleStar","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.isCompletion_circuitTupleStar","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10595","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_monomial_mulVec_matterKet","kind":"theorem","summary":"∀ (σ : Equiv.Perm (Fin 4 → ZMod 2)) (φ : (Fin 4 → ZMod 2) → Complex) (x : Fin 4 → ZMod 2), Eq (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_monomial_mulVec_matterKet","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10596","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_tildeLambdaStar_mulVec_matterKet_one","kind":"theorem","summary":"Eq ((MPOTensor.CZX.matterMatrix MPOTensor.CZX.tildeLambdaStar).mulVec (MPOTensor.CZX.matterKet…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_tildeLambdaStar_mulVec_matterKet_one","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10597","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_tildeLambdaStar_mulVec_matterKet_zero","kind":"theorem","summary":"Eq ((MPOTensor.CZX.matterMatrix MPOTensor.CZX.tildeLambdaStar).mulVec (MPOTensor.CZX.matterKet…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_tildeLambdaStar_mulVec_matterKet_zero","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10598","layer":"formal","project":"p8","title":"MPOTensor.CZX.thirtySix_le_iSup_finrank_gaugeInvariantSubspace_completionClass","kind":"theorem","summary":"LE.le 36 (iSup fun R => iSup fun h => Module.finrank Complex (Subtype fun x => Membership.mem (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.thirtySix_le_iSup_finrank_gaugeInvariantSubspace_completionClass","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10599","layer":"formal","project":"p8","title":"MPOTensor.CZX.two_pow_le_iSup_finrank_gaugeInvariantSubspace_completionClass","kind":"theorem","summary":"∀ N : Nat (hN : LE.le 3 N), LE.le (HPow.hPow 2 N) (iSup fun R => iSup fun h => Module.finrank C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.two_pow_le_iSup_finrank_gaugeInvariantSubspace_completionClass","module":"TNLean.MPS.MPDO.CZXCompletion"},{"id":"n10600","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10601","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGaugeUnitary","kind":"def","summary":"Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin 2) Complex) x","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGaugeUnitary","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10602","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge_conjTranspose","kind":"theorem","summary":"Eq MPOTensor.CZX.daggerGauge.conjTranspose MPOTensor.CZX.daggerGauge","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge_conjTranspose","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10603","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge_map_star","kind":"theorem","summary":"Eq (MPOTensor.CZX.daggerGauge.map ⇑(starRingEnd Complex)) (Neg.neg MPOTensor.CZX.daggerGauge)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge_map_star","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10604","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge_mem_unitaryGroup","kind":"theorem","summary":"Membership.mem (Matrix.unitaryGroup (Fin 2) Complex) MPOTensor.CZX.daggerGauge","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge_mem_unitaryGroup","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10605","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge_mul_map_star","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CZX.daggerGauge (MPOTensor.CZX.daggerGauge.map ⇑(starRingEnd Complex)))…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge_mul_map_star","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10606","layer":"formal","project":"p8","title":"MPOTensor.CZX.daggerGauge_mul_self","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CZX.daggerGauge MPOTensor.CZX.daggerGauge) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.daggerGauge_mul_self","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10607","layer":"formal","project":"p8","title":"MPOTensor.CZX.physicalAdjointTensor_tensor","kind":"theorem","summary":"∀ (u v : Fin 4), Eq (MPOTensor.CZX.tensor.physicalAdjointTensor u v) (HMul.hMul (HMul.hMul MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.physicalAdjointTensor_tensor","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10608","layer":"formal","project":"p8","title":"MPOTensor.CZX.physicalAdjointTensor_tensor_eq_unitaryGauge","kind":"theorem","summary":"∀ (u v : Fin 4), Eq (MPOTensor.CZX.tensor.physicalAdjointTensor u v) (HMul.hMul (HMul.hMul (↑MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.physicalAdjointTensor_tensor_eq_unitaryGauge","module":"TNLean.MPS.MPDO.CZXDaggerGauge"},{"id":"n10609","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain","kind":"def","summary":"Multiplicative (ZMod 2) → Multiplicative (ZMod 2) → Submodule Complex ((Fin 2 → Fin 4) → Comple…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10610","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain_eq","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.defectDomain a b) (ite (Eq (Multiplicative…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain_eq","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10611","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain_gen_gen","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectDomain MPOTensor.CZX.gen MPOTensor.CZX.gen) MPOTensor.CZX.oddTarget","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain_gen_gen","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10612","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain_gen_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectDomain MPOTensor.CZX.gen 1) MPOTensor.CZX.evenTarget","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain_gen_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10613","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain_one_gen","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectDomain 1 MPOTensor.CZX.gen) MPOTensor.CZX.oddTarget","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain_one_gen","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10614","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectDomain_one_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectDomain 1 1) MPOTensor.CZX.evenTarget","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectDomain_one_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10615","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectMaps","kind":"def","summary":"TNLean.Algebra.DefectMaps (Multiplicative (ZMod 2)) (Fin 2 → Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectMaps","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10616","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectMaps_domain","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.defectMaps.domain a b) (MPOTensor.CZX.defe…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectMaps_domain","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10617","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectMaps_prescribed","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.defectMaps.prescribed a b) (MPOTensor.CZX.…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectMaps_prescribed","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10618","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector","kind":"def","summary":"Multiplicative (ZMod 2) → Multiplicative (ZMod 2) → ZMod 2 → (Fin 2 → Fin 4) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10619","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_gen","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen MPOTensor.CZX.gen x) (MPOTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_gen","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10620","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_gen_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen MPOTensor.CZX.gen 1) (HSMul.hSMul (Neg.neg Com…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_gen_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10621","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_gen_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen MPOTensor.CZX.gen 0) (HSMul.hSMul (Neg.neg Com…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_gen_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10622","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_one","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen 1 x) (MPOTensor.CZX.twoSiteTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10623","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_one_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen 1 1) (HSMul.hSMul (Neg.neg Complex.I) (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_one_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10624","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_gen_one_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector MPOTensor.CZX.gen 1 0) (MPOTensor.CZX.matterKet (Matrix.vecCons…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_gen_one_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10625","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_gen","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.defectVector 1 MPOTensor.CZX.gen x) (MPOTensor.CZX.twoSiteTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_gen","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10626","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_gen_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector 1 MPOTensor.CZX.gen 1) (HSMul.hSMul (Neg.neg Complex.I) (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_gen_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10627","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_gen_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector 1 MPOTensor.CZX.gen 0) (MPOTensor.CZX.matterKet (Matrix.vecCons…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_gen_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10628","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_one","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.defectVector 1 1 x) (MPOTensor.CZX.twoSiteTensor (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10629","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_one_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector 1 1 1) (MPOTensor.CZX.matterKet (Matrix.vecCons 1 (Matrix.vecCon…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_one_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10630","layer":"formal","project":"p8","title":"MPOTensor.CZX.defectVector_one_one_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.defectVector 1 1 0) (MPOTensor.CZX.matterKet (Matrix.vecCons 0 (Matrix.vecCon…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.defectVector_one_one_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10631","layer":"formal","project":"p8","title":"MPOTensor.CZX.evenTarget","kind":"def","summary":"Submodule Complex ((Fin 2 → Fin 4) → Complex)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.evenTarget","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10632","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_defectDomain","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)), Eq (Module.finrank Complex (Subtype fun x => Membership.mem…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_defectDomain","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10633","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_evenTarget","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem MPOTensor.CZX.evenTarget x)) 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_evenTarget","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10634","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_oddTarget","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem MPOTensor.CZX.oddTarget x)) 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_oddTarget","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10635","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipDefect","kind":"def","summary":"ZMod 2 → Fin 4 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipDefect","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10636","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipDefect_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.flipDefect 1) (HSMul.hSMul (Neg.neg Complex.I) (MPOTensor.CZX.siteKet 1 1))","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipDefect_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10637","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipDefect_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.flipDefect 0) (MPOTensor.CZX.siteKet 0 0)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipDefect_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10638","layer":"formal","project":"p8","title":"MPOTensor.CZX.identityDefect","kind":"def","summary":"ZMod 2 → Fin 4 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.identityDefect","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10639","layer":"formal","project":"p8","title":"MPOTensor.CZX.identityDefect_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.identityDefect 1) (MPOTensor.CZX.siteKet 1 1)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.identityDefect_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10640","layer":"formal","project":"p8","title":"MPOTensor.CZX.identityDefect_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.identityDefect 0) (MPOTensor.CZX.siteKet 0 0)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.identityDefect_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10641","layer":"formal","project":"p8","title":"MPOTensor.CZX.isometry_on_defectDomain","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (ξ : (Fin 2 → Fin 4) → Complex), Membership.mem (MPOTensor.CZ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.isometry_on_defectDomain","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10642","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterKet","kind":"def","summary":"(Fin 4 → ZMod 2) → (Fin 2 → Fin 4) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterKet","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10643","layer":"formal","project":"p8","title":"MPOTensor.CZX.oddTarget","kind":"def","summary":"Submodule Complex ((Fin 2 → Fin 4) → Complex)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.oddTarget","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10644","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap","kind":"def","summary":"Multiplicative (ZMod 2) → Multiplicative (ZMod 2) → Matrix (Fin 2 → Fin 4) (Fin 2 → Fin 4) Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10645","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_gen_gen_mulVec_one","kind":"theorem","summary":"Eq ((MPOTensor.CZX.prescribedMap MPOTensor.CZX.gen MPOTensor.CZX.gen).mulVec (MPOTensor.CZX.mat…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_gen_gen_mulVec_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10646","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_gen_gen_mulVec_zero","kind":"theorem","summary":"Eq ((MPOTensor.CZX.prescribedMap MPOTensor.CZX.gen MPOTensor.CZX.gen).mulVec (MPOTensor.CZX.mat…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_gen_gen_mulVec_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10647","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_gen_one_mulVec_one","kind":"theorem","summary":"Eq ((MPOTensor.CZX.prescribedMap MPOTensor.CZX.gen 1).mulVec (MPOTensor.CZX.matterKet (Matrix.v…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_gen_one_mulVec_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10648","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_gen_one_mulVec_zero","kind":"theorem","summary":"Eq ((MPOTensor.CZX.prescribedMap MPOTensor.CZX.gen 1).mulVec (MPOTensor.CZX.matterKet (Matrix.v…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_gen_one_mulVec_zero","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10649","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_mem_defectDomain","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (ξ : (Fin 2 → Fin 4) → Complex), Membership.mem (MPOTensor.CZ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_mem_defectDomain","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10650","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_mulVec_defectVector","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (x : ZMod 2), Eq ((MPOTensor.CZX.prescribedMap a b).mulVec (M…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_mulVec_defectVector","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10651","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_one_gen_mulVec","kind":"theorem","summary":"∀ (ξ : (Fin 2 → Fin 4) → Complex), Membership.mem MPOTensor.CZX.oddTarget ξ → Eq ((MPOTensor.CZ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_one_gen_mulVec","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10652","layer":"formal","project":"p8","title":"MPOTensor.CZX.prescribedMap_one_one_mulVec","kind":"theorem","summary":"∀ (ξ : (Fin 2 → Fin 4) → Complex), Membership.mem MPOTensor.CZX.evenTarget ξ → Eq ((MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.prescribedMap_one_one_mulVec","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10653","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteDefect","kind":"def","summary":"Multiplicative (ZMod 2) → ZMod 2 → Fin 4 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteDefect","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10654","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteDefect_gen","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.siteDefect MPOTensor.CZX.gen x) (MPOTensor.CZX.flipDefect x)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteDefect_gen","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10655","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteDefect_one","kind":"theorem","summary":"∀ (x : ZMod 2), Eq (MPOTensor.CZX.siteDefect 1 x) (MPOTensor.CZX.identityDefect x)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteDefect_one","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10656","layer":"formal","project":"p8","title":"MPOTensor.CZX.star_defectVector_dotProduct_defectVector","kind":"theorem","summary":"∀ (a b : Multiplicative (ZMod 2)) (x y : ZMod 2), Eq (dotProduct (star (MPOTensor.CZX.defectVec…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.star_defectVector_dotProduct_defectVector","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10657","layer":"formal","project":"p8","title":"MPOTensor.CZX.star_dotProduct_eq_zero_of_mem_evenTarget_of_mem_oddTarget","kind":"theorem","summary":"∀ ξ η : (Fin 2 → Fin 4) → Complex, Membership.mem MPOTensor.CZX.evenTarget ξ → Membership.mem M…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.star_dotProduct_eq_zero_of_mem_evenTarget_of_mem_oddTarget","module":"TNLean.MPS.MPDO.CZXDefectMaps"},{"id":"n10658","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionActionCoefficient","kind":"def","summary":"Fin 2 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionActionCoefficient","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10659","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionActionCoefficient_coordinates","kind":"theorem","summary":"And (Eq (MPOTensor.CZX.fusionActionCoefficient 0) 1) (Eq (MPOTensor.CZX.fusionActionCoefficient…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionActionCoefficient_coordinates","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10660","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionActionLetter","kind":"def","summary":"Fin 2 → Matrix (Fin 4) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionActionLetter","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10661","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionAction_finite_identities","kind":"theorem","summary":"∀ (x : Fin 2), And (Eq (HMul.hMul (HMul.hMul (MPOTensor.CZX.sequentialActionBra x) (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionAction_finite_identities","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10662","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionAction_positive_power","kind":"theorem","summary":"∀ (x : Fin 2) (n : Nat), And (Eq (HPow.hPow (MPOTensor.CZX.fusionActionLetter x) (HAdd.hAdd n 1…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionAction_positive_power","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10663","layer":"formal","project":"p8","title":"MPOTensor.CZX.sequentialActionBra","kind":"def","summary":"Fin 2 → Matrix (Fin 1) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.sequentialActionBra","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10664","layer":"formal","project":"p8","title":"MPOTensor.CZX.sequentialActionKet","kind":"def","summary":"Fin 2 → Matrix (Fin 4) (Fin 1) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.sequentialActionKet","module":"TNLean.MPS.MPDO.CZXFusionActionCoordinates"},{"id":"n10665","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionV","kind":"def","summary":"Matrix (Fin 1) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionV","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10666","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusionW","kind":"def","summary":"Matrix (Fin 4) (Fin 1) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusionW","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10667","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusion_caps_apply","kind":"theorem","summary":"∀ (a b : Fin 2), And (Eq (MPOTensor.CZX.fusionV 0 (finProdFinEquiv (Prod.mk a b))) (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusion_caps_apply","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10668","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusion_isReduction","kind":"theorem","summary":"(MPOTensor.CZX.tensor.mulTensor MPOTensor.CZX.tensor).toMPSTensor.IsReduction (MPOTensor.identi…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusion_isReduction","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10669","layer":"formal","project":"p8","title":"MPOTensor.CZX.fusion_isReductionExteriorBufferLength_one","kind":"theorem","summary":"(MPOTensor.CZX.tensor.mulTensor MPOTensor.CZX.tensor).toMPSTensor.IsReductionExteriorBufferLeng…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fusion_isReductionExteriorBufferLength_one","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10670","layer":"formal","project":"p8","title":"MPOTensor.CZX.mulTensor_diagonal_letter_coordinates","kind":"theorem","summary":"∀ (x : Fin 2), Eq (MPOTensor.CZX.tensor.mulTensor MPOTensor.CZX.tensor (finProdFinEquiv (Prod.m…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mulTensor_diagonal_letter_coordinates","module":"TNLean.MPS.MPDO.CZXFusionTensors"},{"id":"n10671","layer":"formal","project":"p8","title":"MPOTensor.CZX.blockTensor_ghzSectorTensor_apply","kind":"theorem","summary":"∀ (x : Fin 2) (i : Fin 4), Eq (MPSTensor.blockTensor (MPSTensor.ghzSectorTensor x) 2 ((MPOTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.blockTensor_ghzSectorTensor_apply","module":"TNLean.MPS.MPDO.CZXGHZAction"},{"id":"n10672","layer":"formal","project":"p8","title":"MPOTensor.CZX.blockedGHZTensor","kind":"def","summary":"MPSTensor 4 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.blockedGHZTensor","module":"TNLean.MPS.MPDO.CZXGHZAction"},{"id":"n10673","layer":"formal","project":"p8","title":"MPOTensor.CZX.mpo_tensor_mulVec_blockedGHZ","kind":"theorem","summary":"∀ N : Nat [NeZero N], Eq ((MPOTensor.CZX.tensor.mpo N).mulVec fun s => MPSTensor.mpv (MPSTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mpo_tensor_mulVec_blockedGHZ","module":"TNLean.MPS.MPDO.CZXGHZAction"},{"id":"n10674","layer":"formal","project":"p8","title":"MPOTensor.CZX.mpv_blockedGHZTensor","kind":"theorem","summary":"∀ N : Nat, Eq MPOTensor.CZX.blockedGHZTensor.mpv (HAdd.hAdd (Pi.single (fun x => 0) 1) (Pi.sing…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mpv_blockedGHZTensor","module":"TNLean.MPS.MPDO.CZXGHZAction"},{"id":"n10675","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip","kind":"def","summary":"Equiv.Perm (Fin 4 → ZMod 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10676","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_apply","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.barFlip x) (HAdd.hAdd x (HAdd.hAdd (Pi.single 0 1) (P…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_apply","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10677","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_mul_barFlip","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CZX.barFlip MPOTensor.CZX.barFlip) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_mul_barFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10678","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_symm","kind":"theorem","summary":"Eq (Equiv.symm MPOTensor.CZX.barFlip) MPOTensor.CZX.barFlip","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_symm","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10679","layer":"formal","project":"p8","title":"MPOTensor.CZX.bitFlip","kind":"def","summary":"Fin 4 → Equiv.Perm (Fin 4 → ZMod 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bitFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10680","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple","kind":"def","summary":"Multiplicative (ZMod 2) → Multiplicative (ZMod 2) → Subtype fun x => Membership.mem (Matrix.uni…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10681","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_gen","kind":"theorem","summary":"Eq (↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen MPOTensor.CZX.gen)) (MPOTensor.CZX.matterMat…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_gen","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10682","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_gen_one","kind":"theorem","summary":"Eq (↑(MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen 1)) (MPOTensor.CZX.matterMatrix MPOTensor.CZ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_gen_one","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10683","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTuple_one","kind":"theorem","summary":"∀ (b : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.circuitTuple 1 b) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTuple_one","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10684","layer":"formal","project":"p8","title":"MPOTensor.CZX.controlledZ","kind":"def","summary":"Fin 4 → Fin 4 → Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.controlledZ","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10685","layer":"formal","project":"p8","title":"MPOTensor.CZX.eExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.eExponent","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10686","layer":"formal","project":"p8","title":"MPOTensor.CZX.eExponent_barFlip","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.eExponent (MPOTensor.CZX.barFlip x)) (HAdd.hAdd (MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.eExponent_barFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10687","layer":"formal","project":"p8","title":"MPOTensor.CZX.fExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fExponent","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10688","layer":"formal","project":"p8","title":"MPOTensor.CZX.fExponent_barFlip","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.fExponent (MPOTensor.CZX.barFlip x)) (HAdd.hAdd (MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fExponent_barFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10689","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussOperator_circuitTuple_gen","kind":"theorem","summary":"Eq (TNLean.Algebra.gaussOperator MPOTensor.CZX.circuitTuple MPOTensor.CZX.gen) (Matrix.monomial…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussOperator_circuitTuple_gen","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10690","layer":"formal","project":"p8","title":"MPOTensor.CZX.gen","kind":"def","summary":"Multiplicative (ZMod 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gen","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10691","layer":"formal","project":"p8","title":"MPOTensor.CZX.gen_inv","kind":"theorem","summary":"Eq (Inv.inv MPOTensor.CZX.gen) MPOTensor.CZX.gen","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gen_inv","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10692","layer":"formal","project":"p8","title":"MPOTensor.CZX.gen_mul_gen","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.CZX.gen MPOTensor.CZX.gen) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gen_mul_gen","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10693","layer":"formal","project":"p8","title":"MPOTensor.CZX.hExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.hExponent","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10694","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow","kind":"def","summary":"ZMod 4 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10695","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_add","kind":"theorem","summary":"∀ (a b : ZMod 4), Eq (MPOTensor.CZX.iPow (HAdd.hAdd a b)) (HMul.hMul (MPOTensor.CZX.iPow a) (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_add","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10696","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_natCast","kind":"theorem","summary":"∀ (n : Nat), Eq (MPOTensor.CZX.iPow ↑n) (HPow.hPow Complex.I n)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_natCast","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10697","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_three","kind":"theorem","summary":"Eq (MPOTensor.CZX.iPow 3) (Neg.neg Complex.I)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_three","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10698","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_two","kind":"theorem","summary":"Eq (MPOTensor.CZX.iPow 2) (-1)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_two","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10699","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_two_mul_cast","kind":"theorem","summary":"∀ (m : ZMod 2), Eq (MPOTensor.CZX.iPow (HMul.hMul 2 m.cast)) (HPow.hPow (-1) m.val)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_two_mul_cast","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10700","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_two_mul_val","kind":"theorem","summary":"∀ (m : ZMod 2), Eq (MPOTensor.CZX.iPow (HMul.hMul 2 ↑m.val)) (HPow.hPow (-1) m.val)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_two_mul_val","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10701","layer":"formal","project":"p8","title":"MPOTensor.CZX.iPow_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.iPow 0) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.iPow_zero","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10702","layer":"formal","project":"p8","title":"MPOTensor.CZX.lambda","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.lambda","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10703","layer":"formal","project":"p8","title":"MPOTensor.CZX.lambda_eq","kind":"theorem","summary":"Eq MPOTensor.CZX.lambda (Matrix.monomial MPOTensor.CZX.barFlip fun x => HMul.hMul (Neg.neg Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.lambda_eq","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10704","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits","kind":"def","summary":"Equiv (Fin 2 → Fin 4) (Fin 4 → ZMod 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10705","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits_apply_one","kind":"theorem","summary":"∀ (i : Fin 2 → Fin 4), Eq (MPOTensor.CZX.localBits i 1) (MPOTensor.CZX.siteBits (i 0)).2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits_apply_one","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10706","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits_apply_three","kind":"theorem","summary":"∀ (i : Fin 2 → Fin 4), Eq (MPOTensor.CZX.localBits i 3) (MPOTensor.CZX.siteBits (i 1)).2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits_apply_three","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10707","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits_apply_two","kind":"theorem","summary":"∀ (i : Fin 2 → Fin 4), Eq (MPOTensor.CZX.localBits i 2) (MPOTensor.CZX.siteBits (i 1)).1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits_apply_two","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10708","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits_apply_zero","kind":"theorem","summary":"∀ (i : Fin 2 → Fin 4), Eq (MPOTensor.CZX.localBits i 0) (MPOTensor.CZX.siteBits (i 0)).1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits_apply_zero","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10709","layer":"formal","project":"p8","title":"MPOTensor.CZX.localBits_matterBarFlip","kind":"theorem","summary":"∀ (i : Fin 2 → Fin 4), Eq (MPOTensor.CZX.localBits (MPOTensor.CZX.matterBarFlip i)) (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localBits_matterBarFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10710","layer":"formal","project":"p8","title":"MPOTensor.CZX.localPerm","kind":"def","summary":"Equiv.Perm (Prod (Fin 2 → Fin 4) (Prod (Multiplicative (ZMod 2)) (Multiplicative (ZMod 2))))","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localPerm","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10711","layer":"formal","project":"p8","title":"MPOTensor.CZX.localPhase","kind":"def","summary":"Prod (Fin 2 → Fin 4) (Prod (Multiplicative (ZMod 2)) (Multiplicative (ZMod 2))) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localPhase","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10712","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterBarFlip","kind":"def","summary":"Equiv.Perm (Fin 2 → Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterBarFlip","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10713","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterBarFlip_symm","kind":"theorem","summary":"Eq (Equiv.symm MPOTensor.CZX.matterBarFlip) MPOTensor.CZX.matterBarFlip","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterBarFlip_symm","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10714","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex → Matrix (Fin 2 → Fin 4) (Fin 2 → Fin 4) Compl…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10715","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_monomial","kind":"theorem","summary":"∀ (σ : Equiv.Perm (Fin 4 → ZMod 2)) (φ : (Fin 4 → ZMod 2) → Complex), Eq (MPOTensor.CZX.matterM…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_monomial","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10716","layer":"formal","project":"p8","title":"MPOTensor.CZX.neg_one_pow_val_add","kind":"theorem","summary":"∀ (a b : ZMod 2), Eq (HPow.hPow (-1) (HAdd.hAdd a b).val) (HMul.hMul (HPow.hPow (-1) a.val) (HP…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.neg_one_pow_val_add","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10717","layer":"formal","project":"p8","title":"MPOTensor.CZX.pauliX","kind":"def","summary":"Fin 4 → Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.pauliX","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10718","layer":"formal","project":"p8","title":"MPOTensor.CZX.pauliZ","kind":"def","summary":"Fin 4 → Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.pauliZ","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10719","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2 → ZMod 2 → ZMod 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponent","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10720","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponent_one_one","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponent x 1 1) (HAdd.hAdd 3 (HMul.hMul 2 ↑(MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponent_one_one","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10721","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponent_one_zero","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponent x 1 0) (HMul.hMul 2 ↑(MPOTensor.CZX.eEx…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponent_one_zero","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10722","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponent_zero_one","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponent x 0 1) (HAdd.hAdd 2 (HMul.hMul 2 ↑(MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponent_zero_one","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10723","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponent_zero_zero","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponent x 0 0) (HAdd.hAdd 3 (HMul.hMul 2 ↑(MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponent_zero_zero","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10724","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteBits","kind":"def","summary":"Equiv (Fin 4) (Prod (ZMod 2) (ZMod 2))","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteBits","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10725","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambda","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambda","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10726","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambda_eq","kind":"theorem","summary":"Eq MPOTensor.CZX.tildeLambda (Matrix.monomial MPOTensor.CZX.barFlip fun x => HMul.hMul (Neg.neg…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambda_eq","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10727","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambda_mem_unitaryGroup","kind":"theorem","summary":"Membership.mem (Matrix.unitaryGroup (Fin 2 → Fin 4) Complex) (MPOTensor.CZX.matterMatrix MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambda_mem_unitaryGroup","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10728","layer":"formal","project":"p8","title":"MPOTensor.CZX.toAdd_gen_ne_zero","kind":"theorem","summary":"Ne (Multiplicative.toAdd MPOTensor.CZX.gen) 0","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.toAdd_gen_ne_zero","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10729","layer":"formal","project":"p8","title":"MPOTensor.CZX.uExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.uExponent","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10730","layer":"formal","project":"p8","title":"MPOTensor.CZX.w","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.w","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10731","layer":"formal","project":"p8","title":"MPOTensor.CZX.w_eq","kind":"theorem","summary":"Eq MPOTensor.CZX.w (Matrix.monomial MPOTensor.CZX.barFlip fun x => HPow.hPow (-1) (MPOTensor.CZ…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.w_eq","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10732","layer":"formal","project":"p8","title":"MPOTensor.CZX.w_mem_unitaryGroup","kind":"theorem","summary":"Membership.mem (Matrix.unitaryGroup (Fin 2 → Fin 4) Complex) (MPOTensor.CZX.matterMatrix MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.w_mem_unitaryGroup","module":"TNLean.MPS.MPDO.CZXGaussCircuitTuple"},{"id":"n10733","layer":"formal","project":"p8","title":"MPOTensor.CZX.Site","kind":"def","summary":"Type","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.Site","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10734","layer":"formal","project":"p8","title":"MPOTensor.CZX.bLabel","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → (Fin N → MPOTensor.CZX.Site) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bLabel","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10735","layer":"formal","project":"p8","title":"MPOTensor.CZX.bLabel_fiberEquiv","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (x : Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bLabel_fiberEquiv","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10736","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_apply_one","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.barFlip x 1) (HAdd.hAdd (x 1) 1)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_apply_one","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10737","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_apply_three","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.barFlip x 3) (x 3)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_apply_three","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10738","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_apply_two","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.barFlip x 2) (x 2)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_apply_two","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10739","layer":"formal","project":"p8","title":"MPOTensor.CZX.barFlip_apply_zero","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.barFlip x 0) (HAdd.hAdd (x 0) 1)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.barFlip_apply_zero","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10740","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponent","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → (Fin N → MPOTensor.CZX.Site) → ZMod 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponent","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10741","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponent_add_bondExponent_chainFlip","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (s : Fin N → MPOTensor.CZX.Site), Eq (HA…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponent_add_bondExponent_chainFlip","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10742","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponent_chainFlip_of_disjoint","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j k : Fin N), Ne k j → Ne k (HAdd.hAdd j 1) → Ne (HAdd.hAdd k 1) j…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponent_chainFlip_of_disjoint","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10743","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponent_holonomy","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (j : Fin N) (s : Fin N → MPOTensor.CZX.Site), Eq (HA…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponent_holonomy","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10744","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainDecode","kind":"def","summary":"(N : Nat) → Equiv (Fin N → Fin (Fintype.card (Prod (Fin 4) (Multiplicative (ZMod 2))))) (Fin N…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainDecode","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10745","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainDecode_apply","kind":"theorem","summary":"∀ (N : Nat) (t : Fin N → Fin (Fintype.card (Prod (Fin 4) (Multiplicative (ZMod 2))))) (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainDecode_apply","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10746","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainDecode_window","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (t : Fin N → Fin (Fintype.card (Prod (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainDecode_window","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10747","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainDecode_windowPerm","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (hN : LE.le 2 N) (j : Fin N) (t : Fin N → Fin (Fintype.card (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainDecode_windowPerm","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10748","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainFlip","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → Equiv.Perm (Fin N → MPOTensor.CZX.Site)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainFlip","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10749","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainFlip_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j : Fin N) (s : Fin N → MPOTensor.CZX.Site), Eq ((MPOTensor.CZX.ch…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainFlip_apply","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10750","layer":"formal","project":"p8","title":"MPOTensor.CZX.chainFlip_fiberEquiv","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMo…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.chainFlip_fiberEquiv","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10751","layer":"formal","project":"p8","title":"MPOTensor.CZX.commonFixedSubmodule_placedGaussProjector_circuitTuple","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N), Eq (LinearMap.commonFixedSubmodule fun j => Matrix.toLin' (MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.commonFixedSubmodule_placedGaussProjector_circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10752","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberCoordinates","kind":"def","summary":"(N : Nat) → [NeZero N] → Equiv (Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2))…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberCoordinates","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10753","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberEquiv","kind":"def","summary":"(N : Nat) → [NeZero N] → Equiv (Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2))…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberEquiv","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10754","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberEquiv_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (x : Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberEquiv_apply","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10755","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhase","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2) →…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhase","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10756","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhase_flip_of_disjoint","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j k : Fin N), Ne k j → Ne k (HAdd.hAdd j 1) → Ne (H…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhase_flip_of_disjoint","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10757","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhase_holonomy","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMo…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhase_holonomy","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10758","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhase_mul_fiberPhase_flip","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMo…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhase_mul_fiberPhase_flip","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10759","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhase_ne_zero","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j : Fin N) (x : Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhase_ne_zero","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10760","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTuple","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 3 N), Eq (Module.finrank Complex (Subtype fun x => Membership.mem (Line…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10761","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTuple","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 3 N), Eq (Module.finrank Complex (Subtype fun x => Membership.mem (MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_gaugeInvariantSubspace_circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10762","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipPattern","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → Fin N → MPOTensor.CZX.Site","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipPattern","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10763","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipPattern_apply_of_ne","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j k : Fin N), Ne k j → Ne k (HAdd.hAdd j 1) → Eq (MPOTensor.CZX.fl…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipPattern_apply_of_ne","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10764","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipPattern_apply_self","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N), Eq (MPOTensor.CZX.flipPattern j j) (Pro…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipPattern_apply_self","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10765","layer":"formal","project":"p8","title":"MPOTensor.CZX.flipPattern_apply_succ","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N), Eq (MPOTensor.CZX.flipPattern j (HAdd.h…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.flipPattern_apply_succ","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10766","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussLegAction_fst","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] (g : G) (l : Prod G G), Eq ((TNLean.Algebra.gaussLegAction g) l…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussLegAction_fst","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10767","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussLegAction_snd","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] (g : G) (l : Prod G G), Eq ((TNLean.Algebra.gaussLegAction g) l…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussLegAction_snd","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10768","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussProjector_circuitTuple","kind":"theorem","summary":"Eq (TNLean.Algebra.gaussProjector MPOTensor.CZX.circuitTuple) (HSMul.hSMul (Inv.inv 2) (HAdd.hA…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussProjector_circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10769","layer":"formal","project":"p8","title":"MPOTensor.CZX.inv_two_smul_add_mulVec_eq_iff","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] [inst_1 : DecidableEq ι] (A : Matrix ι ι Complex) (v : ι → Co…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.inv_two_smul_add_mulVec_eq_iff","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10770","layer":"formal","project":"p8","title":"MPOTensor.CZX.isTrivialHolonomy_fiberPhase_iff","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (pb : Prod (Fin N → ZMod 2) (Fin N → ZMod 2)), Iff (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.isTrivialHolonomy_fiberPhase_iff","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10771","layer":"formal","project":"p8","title":"MPOTensor.CZX.localPhase_window","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (t : Fin N → Fin (Fintype.card (Prod (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localPhase_window","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10772","layer":"formal","project":"p8","title":"MPOTensor.CZX.pLabel","kind":"def","summary":"N : Nat → Fin N → (Fin N → MPOTensor.CZX.Site) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.pLabel","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10773","layer":"formal","project":"p8","title":"MPOTensor.CZX.pLabel_fiberEquiv","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (x : Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.pLabel_fiberEquiv","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10774","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperator","kind":"def","summary":"(N : Nat) → LE.le 2 N → Fin N → MPOTensor.ChainOperator (Fintype.card (Prod (Fin 4) (Multiplica…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperator","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10775","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperator_eq_monomial","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.CZX.placedGaussOperator N hN j) (Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperator_eq_monomial","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10776","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperator_eq_reindex","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.CZX.placedGaussOperat…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperator_eq_reindex","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10777","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussProjector_circuitTuple","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.placedGaussProjector 4 (Multiplicative…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussProjector_circuitTuple","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10778","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteBits_eq_localBits_one","kind":"theorem","summary":"∀ (m : Fin 2 → Fin 4), Eq (MPOTensor.CZX.siteBits (m 1)) (Prod.mk (MPOTensor.CZX.localBits m 2)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteBits_eq_localBits_one","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10779","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteBits_eq_localBits_zero","kind":"theorem","summary":"∀ (m : Fin 2 → Fin 4), Eq (MPOTensor.CZX.siteBits (m 0)) (Prod.mk (MPOTensor.CZX.localBits m 0)…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteBits_eq_localBits_zero","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10780","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteDecode","kind":"def","summary":"Equiv (Fin (Fintype.card (Prod (Fin 4) (Multiplicative (ZMod 2))))) MPOTensor.CZX.Site","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteDecode","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10781","layer":"formal","project":"p8","title":"MPOTensor.CZX.siteDecode_equivFin","kind":"theorem","summary":"∀ (i : Fin 4) (a : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.siteDecode ((Fintype.equivFin (P…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.siteDecode_equivFin","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10782","layer":"formal","project":"p8","title":"MPOTensor.CZX.succ_ne_self","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N), Ne (HAdd.hAdd j 1) j","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.succ_ne_self","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10783","layer":"formal","project":"p8","title":"MPOTensor.CZX.succ_succ_ne_self","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (j : Fin N), Ne (HAdd.hAdd (HAdd.hAdd j 1) 1) j","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.succ_succ_ne_self","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10784","layer":"formal","project":"p8","title":"MPOTensor.CZX.sum_multiplicative_zmod_two","kind":"theorem","summary":"∀ M : Type u_1 [inst : AddCommMonoid M] (f : Multiplicative (ZMod 2) → M), Eq (Finset.univ.sum…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.sum_multiplicative_zmod_two","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10785","layer":"formal","project":"p8","title":"MPOTensor.CZX.toAdd_gen_mul","kind":"theorem","summary":"∀ (a : Multiplicative (ZMod 2)), Eq (Multiplicative.toAdd (HMul.hMul MPOTensor.CZX.gen a)) (HAd…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.toAdd_gen_mul","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10786","layer":"formal","project":"p8","title":"MPOTensor.CZX.toAdd_mul_gen_inv","kind":"theorem","summary":"∀ (a : Multiplicative (ZMod 2)), Eq (Multiplicative.toAdd (HMul.hMul a (Inv.inv MPOTensor.CZX.g…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.toAdd_mul_gen_inv","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10787","layer":"formal","project":"p8","title":"MPOTensor.CZX.windowCoordinates","kind":"def","summary":"Equiv (Prod (Fin 2 → Fin 4) (Prod (Multiplicative (ZMod 2)) (Multiplicative (ZMod 2)))) (Fin 2…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.windowCoordinates","module":"TNLean.MPS.MPDO.CZXGaussInvariantSubspace"},{"id":"n10788","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedSourceFactors_sourceWL_coordinates","kind":"theorem","summary":"∀ (l i j k : Fin 4), Eq (MPOTensor.SourceFactors.sourceWL MPOTensor.CZX.tensor MPOTensor.CZX.di…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedSourceFactors_sourceWL_coordinates","module":"TNLean.MPS.MPDO.CZXMovementCoordinates"},{"id":"n10789","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedSourceFactors_sourceWR_coordinates","kind":"theorem","summary":"∀ (i r a j : Fin 4), Eq (MPOTensor.SourceFactors.sourceWR MPOTensor.CZX.tensor MPOTensor.CZX.di…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedSourceFactors_sourceWR_coordinates","module":"TNLean.MPS.MPDO.CZXMovementCoordinates"},{"id":"n10790","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponentStar","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → (Fin N → MPOTensor.CZX.Site) → ZMod 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponentStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10791","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponentStar_add_bondExponentStar_chainFlip","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (s : Fin N → MPOTensor.CZX.Site), Eq (HA…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponentStar_add_bondExponentStar_chainFlip","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10792","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponentStar_chainFlip_of_disjoint","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j k : Fin N), Ne k j → Ne k (HAdd.hAdd j 1) → Ne (HAdd.hAdd k 1) j…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponentStar_chainFlip_of_disjoint","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10793","layer":"formal","project":"p8","title":"MPOTensor.CZX.bondExponentStar_holonomy","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (j : Fin N) (s : Fin N → MPOTensor.CZX.Site), Eq (HA…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.bondExponentStar_holonomy","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10794","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar","kind":"def","summary":"Multiplicative (ZMod 2) → Multiplicative (ZMod 2) → Subtype fun x => Membership.mem (Matrix.uni…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10795","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_gen_gen","kind":"theorem","summary":"Eq (↑(MPOTensor.CZX.circuitTupleStar MPOTensor.CZX.gen MPOTensor.CZX.gen)) (MPOTensor.CZX.matte…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_gen_gen","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10796","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_gen_one","kind":"theorem","summary":"Eq (↑(MPOTensor.CZX.circuitTupleStar MPOTensor.CZX.gen 1)) (MPOTensor.CZX.matterMatrix MPOTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_gen_one","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10797","layer":"formal","project":"p8","title":"MPOTensor.CZX.circuitTupleStar_one","kind":"theorem","summary":"∀ (b : Multiplicative (ZMod 2)), Eq (MPOTensor.CZX.circuitTupleStar 1 b) 1","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.circuitTupleStar_one","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10798","layer":"formal","project":"p8","title":"MPOTensor.CZX.commonFixedSubmodule_placedGaussProjector_circuitTupleStar","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N), Eq (LinearMap.commonFixedSubmodule fun j => Matrix.toLin' (MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.commonFixedSubmodule_placedGaussProjector_circuitTupleStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10799","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhaseStar","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin N → ZMod 2) →…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhaseStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10800","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhaseStar_flip_of_disjoint","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j k : Fin N), Ne k j → Ne k (HAdd.hAdd j 1) → Ne (H…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhaseStar_flip_of_disjoint","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10801","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhaseStar_holonomy","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMo…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhaseStar_holonomy","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10802","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhaseStar_mul_fiberPhaseStar_flip","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMo…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhaseStar_mul_fiberPhaseStar_flip","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10803","layer":"formal","project":"p8","title":"MPOTensor.CZX.fiberPhaseStar_ne_zero","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j : Fin N) (x : Prod (Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.fiberPhaseStar_ne_zero","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10804","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (hN : LE.le 3 N), Eq (Module.finrank Complex (Subtype fun x => Me…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10805","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar_four","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem (LinearMap.commonFixedSubmodule fun…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar_four","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10806","layer":"formal","project":"p8","title":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar_three","kind":"theorem","summary":"Eq (Module.finrank Complex (Subtype fun x => Membership.mem (LinearMap.commonFixedSubmodule fun…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.finrank_commonFixedSubmodule_placedGaussProjector_circuitTupleStar_three","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10807","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussOperator_circuitTupleStar_gen","kind":"theorem","summary":"Eq (TNLean.Algebra.gaussOperator MPOTensor.CZX.circuitTupleStar MPOTensor.CZX.gen) (Matrix.mono…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussOperator_circuitTupleStar_gen","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10808","layer":"formal","project":"p8","title":"MPOTensor.CZX.gaussProjector_circuitTupleStar","kind":"theorem","summary":"Eq (TNLean.Algebra.gaussProjector MPOTensor.CZX.circuitTupleStar) (HSMul.hSMul (Inv.inv 2) (HAd…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.gaussProjector_circuitTupleStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10809","layer":"formal","project":"p8","title":"MPOTensor.CZX.isTrivialHolonomy_fiberPhaseStar_iff","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 3 N → ∀ (pb : Prod (Fin N → ZMod 2) (Fin N → ZMod 2)), Iff (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.isTrivialHolonomy_fiberPhaseStar_iff","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10810","layer":"formal","project":"p8","title":"MPOTensor.CZX.localPhaseStar","kind":"def","summary":"Prod (Fin 2 → Fin 4) (Prod (Multiplicative (ZMod 2)) (Multiplicative (ZMod 2))) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localPhaseStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10811","layer":"formal","project":"p8","title":"MPOTensor.CZX.localPhaseStar_window","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], LE.le 2 N → ∀ (j : Fin N) (t : Fin N → Fin (Fintype.card (Prod (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.localPhaseStar_window","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10812","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponentStar","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2 → ZMod 2 → ZMod 4","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponentStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10813","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponentStar_one_one","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponentStar x 1 1) (HAdd.hAdd (HAdd.hAdd 3 (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponentStar_one_one","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10814","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponentStar_one_zero","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponentStar x 1 0) (HMul.hMul 2 ↑(MPOTensor.CZX…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponentStar_one_zero","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10815","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponentStar_zero_one","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponentStar x 0 1) (HAdd.hAdd 2 (HMul.hMul 2 ↑(…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponentStar_zero_one","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10816","layer":"formal","project":"p8","title":"MPOTensor.CZX.phaseExponentStar_zero_zero","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.phaseExponentStar x 0 0) (HAdd.hAdd (HAdd.hAdd 3 (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.phaseExponentStar_zero_zero","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10817","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperatorStar","kind":"def","summary":"(N : Nat) → LE.le 2 N → Fin N → MPOTensor.ChainOperator (Fintype.card (Prod (Fin 4) (Multiplica…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperatorStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10818","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperatorStar_eq_monomial","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.CZX.placedGaussOperatorStar N hN j) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperatorStar_eq_monomial","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10819","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussOperatorStar_eq_reindex","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.CZX.placedGaussOperat…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussOperatorStar_eq_reindex","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10820","layer":"formal","project":"p8","title":"MPOTensor.CZX.placedGaussProjector_circuitTupleStar","kind":"theorem","summary":"∀ (N : Nat) (hN : LE.le 2 N) (j : Fin N), Eq (MPOTensor.placedGaussProjector 4 (Multiplicative…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.placedGaussProjector_circuitTupleStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10821","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStar","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10822","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStarExponent","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStarExponent","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10823","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStarExponent_barFlip","kind":"theorem","summary":"∀ (x : Fin 4 → ZMod 2), Eq (MPOTensor.CZX.rStarExponent (MPOTensor.CZX.barFlip x)) (HAdd.hAdd (…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStarExponent_barFlip","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10824","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStarExponent_one_one_zero_zero","kind":"theorem","summary":"Eq (MPOTensor.CZX.rStarExponent (Matrix.vecCons 1 (Matrix.vecCons 1 (Matrix.vecCons 0 (Matrix.v…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStarExponent_one_one_zero_zero","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10825","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStarExponent_zero_zero_one_one","kind":"theorem","summary":"Eq (MPOTensor.CZX.rStarExponent (Matrix.vecCons 0 (Matrix.vecCons 0 (Matrix.vecCons 1 (Matrix.v…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStarExponent_zero_zero_one_one","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10826","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStar_eq","kind":"theorem","summary":"Eq MPOTensor.CZX.rStar (Matrix.monomial 1 fun x => HPow.hPow (-1) (MPOTensor.CZX.rStarExponent…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStar_eq","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10827","layer":"formal","project":"p8","title":"MPOTensor.CZX.rStar_mulVec_eq_self","kind":"theorem","summary":"∀ v : (Fin 4 → ZMod 2) → Complex, (∀ (x : Fin (Nat.succ 0).succ.succ.succ → ZMod 2), Ne x (Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.rStar_mulVec_eq_self","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10828","layer":"formal","project":"p8","title":"MPOTensor.CZX.starDefect","kind":"def","summary":"(Fin 4 → ZMod 2) → ZMod 2 → ZMod 2 → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.starDefect","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10829","layer":"formal","project":"p8","title":"MPOTensor.CZX.starFiberLabel","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → Prod (Fin N → ZMod 2) (Fin N → ZMod 2) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.starFiberLabel","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10830","layer":"formal","project":"p8","title":"MPOTensor.CZX.starHolonomyLabel","kind":"def","summary":"N : Nat → [NeZero N] → Fin N → (Fin N → MPOTensor.CZX.Site) → ZMod 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.starHolonomyLabel","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10831","layer":"formal","project":"p8","title":"MPOTensor.CZX.starHolonomyLabel_fiberEquiv","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (j : Fin N) (pb : Prod (Fin N → ZMod 2) (Fin N → ZMod 2)) (γ : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.starHolonomyLabel_fiberEquiv","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10832","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambdaStar","kind":"def","summary":"Matrix (Fin 4 → ZMod 2) (Fin 4 → ZMod 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambdaStar","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10833","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambdaStar_eq","kind":"theorem","summary":"Eq MPOTensor.CZX.tildeLambdaStar (Matrix.monomial MPOTensor.CZX.barFlip fun x => HMul.hMul (Neg…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambdaStar_eq","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10834","layer":"formal","project":"p8","title":"MPOTensor.CZX.tildeLambdaStar_mem_unitaryGroup","kind":"theorem","summary":"Membership.mem (Matrix.unitaryGroup (Fin 2 → Fin 4) Complex) (MPOTensor.CZX.matterMatrix MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tildeLambdaStar_mem_unitaryGroup","module":"TNLean.MPS.MPDO.CZXSecondTupleCertificate"},{"id":"n10835","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedLeftEquiv","kind":"def","summary":"Equiv (Fin MPOTensor.CZX.tensor.leftRank) (Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedLeftEquiv","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10836","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedRightEquiv","kind":"def","summary":"Equiv (Fin MPOTensor.CZX.tensor.rightRank) (Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedRightEquiv","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10837","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedSourceFactors","kind":"def","summary":"MPOTensor.CZX.tensor.SourceFactors (HSMul.hSMul (1 / 2) 1)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedSourceFactors","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10838","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedSourceFactors_coordinates","kind":"theorem","summary":"∀ (k i : Fin 4) (α : Fin 2), And (Eq (MPOTensor.CZX.displayedSourceFactors.X₁ (Prod.mk i α) (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedSourceFactors_coordinates","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10839","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedSourceFactors_inverse_coordinates","kind":"theorem","summary":"∀ (k i : Fin 4) (β : Fin 2), And (Eq (MPOTensor.CZX.displayedSourceFactors.X₁ (Prod.mk i β) (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedSourceFactors_inverse_coordinates","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10840","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedX₁","kind":"def","summary":"Matrix (Prod (Fin 4) (Fin 2)) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedX₁","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10841","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedX₂","kind":"def","summary":"Matrix (Prod (Fin 2) (Fin 4)) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedX₂","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10842","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedY₁","kind":"def","summary":"Matrix (Fin 4) (Prod (Fin 2) (Fin 4)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedY₁","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10843","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedY₂","kind":"def","summary":"Matrix (Fin 4) (Prod (Fin 4) (Fin 2)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedY₂","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10844","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedZ₁","kind":"def","summary":"Matrix (Prod (Fin 2) (Fin 4)) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedZ₁","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10845","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayedZ₂","kind":"def","summary":"Matrix (Prod (Fin 4) (Fin 2)) (Fin 4) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayedZ₂","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10846","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayed_cuts","kind":"theorem","summary":"And (Eq (HMul.hMul MPOTensor.CZX.displayedX₁ MPOTensor.CZX.displayedY₁) MPOTensor.CZX.tensor.so…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayed_cuts","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10847","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayed_grams","kind":"theorem","summary":"And (Eq (HMul.hMul MPOTensor.CZX.displayedX₁.conjTranspose MPOTensor.CZX.displayedX₁) (HSMul.hS…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayed_grams","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10848","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayed_normalizations","kind":"theorem","summary":"And (Eq (HMul.hMul MPOTensor.CZX.displayedY₁ MPOTensor.CZX.displayedZ₁) 1) (And (Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayed_normalizations","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10849","layer":"formal","project":"p8","title":"MPOTensor.CZX.displayed_ranks","kind":"theorem","summary":"And (Eq MPOTensor.CZX.tensor.rightRank 4) (Eq MPOTensor.CZX.tensor.leftRank 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.displayed_ranks","module":"TNLean.MPS.MPDO.CZXSourceFactors"},{"id":"n10850","layer":"formal","project":"p8","title":"MPOTensor.CZX.complement","kind":"def","summary":"(N : Nat) → Equiv.Perm (Fin N → Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.complement","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10851","layer":"formal","project":"p8","title":"MPOTensor.CZX.complementSite","kind":"def","summary":"Equiv.Perm (Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.complementSite","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10852","layer":"formal","project":"p8","title":"MPOTensor.CZX.cyclicExponent","kind":"def","summary":"N : Nat → [NeZero N] → (Fin N → Fin 4) → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.cyclicExponent","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10853","layer":"formal","project":"p8","title":"MPOTensor.CZX.decoratedSiteTensor","kind":"def","summary":"MPOTensor 2 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.decoratedSiteTensor","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10854","layer":"formal","project":"p8","title":"MPOTensor.CZX.edgeExponent","kind":"def","summary":"Fin 4 → Fin 2 → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.edgeExponent","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10855","layer":"formal","project":"p8","title":"MPOTensor.CZX.mpo_tensor","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N], Eq (MPOTensor.CZX.tensor.mpo N) (Matrix.monomial (MPOTensor.CZX.co…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mpo_tensor","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10856","layer":"formal","project":"p8","title":"MPOTensor.CZX.mpo_tensor_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] (s t : Fin N → Fin 4), Eq (MPOTensor.CZX.tensor.mpo N s t) (ite (Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mpo_tensor_apply","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10857","layer":"formal","project":"p8","title":"MPOTensor.CZX.mpo_tensor_mem_unitaryGroup","kind":"theorem","summary":"∀ N : Nat [NeZero N], Membership.mem (Matrix.unitaryGroup (Fin N → Fin 4) Complex) (MPOTensor.C…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.mpo_tensor_mem_unitaryGroup","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10858","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor","kind":"def","summary":"MPOTensor 4 2","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10859","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor_apply","kind":"theorem","summary":"∀ (i j : Fin 4) (l r : Fin 2), Eq (MPOTensor.CZX.tensor i j l r) (ite (And (Eq i (MPOTensor.CZX…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor_apply","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10860","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor_apply_bits","kind":"theorem","summary":"∀ (a b c d l r : ZMod 2), Eq (MPOTensor.CZX.tensor (MPOTensor.CZX.siteBits.symm (Prod.mk a b))…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor_apply_bits","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10861","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor_eq_blockTensor","kind":"theorem","summary":"Eq MPOTensor.CZX.tensor fun i j => MPOTensor.CZX.decoratedSiteTensor.blockTensor 2 ((MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor_eq_blockTensor","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10862","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor_isMPU","kind":"theorem","summary":"MPOTensor.CZX.tensor.IsMPU","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor_isMPU","module":"TNLean.MPS.MPDO.CZXTensor"},{"id":"n10863","layer":"formal","project":"p8","title":"MPOTensor.CZX.tensor_isInjective","kind":"theorem","summary":"Kraus.IsInjective MPOTensor.CZX.tensor.toMPSTensor","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.tensor_isInjective","module":"TNLean.MPS.MPDO.CZXTensorInjectivity"},{"id":"n10864","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_lambda_mulVec_defectVector_one","kind":"theorem","summary":"Eq ((MPOTensor.CZX.matterMatrix MPOTensor.CZX.lambda).mulVec (MPOTensor.CZX.defectVector MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_lambda_mulVec_defectVector_one","module":"TNLean.MPS.MPDO.CZXUnmodifiedFusion"},{"id":"n10865","layer":"formal","project":"p8","title":"MPOTensor.CZX.matterMatrix_lambda_mulVec_defectVector_zero","kind":"theorem","summary":"Eq ((MPOTensor.CZX.matterMatrix MPOTensor.CZX.lambda).mulVec (MPOTensor.CZX.defectVector MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.CZX.matterMatrix_lambda_mulVec_defectVector_zero","module":"TNLean.MPS.MPDO.CZXUnmodifiedFusion"},{"id":"n10866","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_eq_weighted_basis_blocks","kind":"theorem","summary":"∀ (i j : Fin 3), Eq (MPOTensor.CaseIIAbsorptionCounterexample.ambient i j) (Matrix.diagonal fun…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_eq_weighted_basis_blocks","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10867","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isHorizontalCF","kind":"theorem","summary":"MPOTensor.CaseIIAbsorptionCounterexample.ambient.IsHorizontalCF","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isHorizontalCF","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10868","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isMPDO","kind":"theorem","summary":"MPOTensor.CaseIIAbsorptionCounterexample.ambient.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isMPDO","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10869","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSAL","kind":"theorem","summary":"MPOTensor.CaseIIAbsorptionCounterexample.ambient.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSAL","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10870","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSAL_isSimpleCanonicalForm_and_firstAbs…","kind":"theorem","summary":"And MPOTensor.CaseIIAbsorptionCounterexample.ambient.IsMPDO (And (∀ (N : Nat), LT.lt 0 N → LT.l…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSAL_isSimpleCanonicalForm_and_firstAbsorbed_not_isNormalTensor","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10871","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSimpleCanonicalForm","kind":"theorem","summary":"MPOTensor.CaseIIAbsorptionCounterexample.ambient.IsSimpleCanonicalForm","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_isSimpleCanonicalForm","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10872","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_toMPSTensor_eq_sectors_toTensor","kind":"theorem","summary":"Eq MPOTensor.CaseIIAbsorptionCounterexample.ambient.toMPSTensor MPOTensor.CaseIIAbsorptionCount…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.ambient_toMPSTensor_eq_sectors_toTensor","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10873","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.basis_isNormalTensor","kind":"theorem","summary":"∀ (s : Fin 2), (MPOTensor.CaseIIAbsorptionCounterexample.basis s).IsNormalTensor","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.basis_isNormalTensor","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10874","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.printed_absorbed_normality_step_is_false","kind":"theorem","summary":"And (Eq (MPOTensor.CaseIIAbsorptionCounterexample.weight 0) MPOTensor.CaseIIAbsorptionCounterex…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.printed_absorbed_normality_step_is_false","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10875","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_basis_doubledPhysTraceTransfer_not_isNil…","kind":"theorem","summary":"∀ (j : Fin MPOTensor.CaseIIAbsorptionCounterexample.sectors.basisCount), Not (IsNilpotent (MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_basis_doubledPhysTraceTransfer_not_isNilpotent","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10876","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_isBNTCanonicalForm","kind":"theorem","summary":"MPSTensor.IsBNTCanonicalForm MPOTensor.CaseIIAbsorptionCounterexample.sectors","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_isBNTCanonicalForm","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10877","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_wordTupleSpanTop_one","kind":"theorem","summary":"MPSTensor.WordTupleSpanTop MPOTensor.CaseIIAbsorptionCounterexample.sectors.basis 1","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.sectors_wordTupleSpanTop_one","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10878","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.trace_mpo_ambient","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.CaseIIAbsorptionCounterexample.ambient.mpo N).trace 2","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.trace_mpo_ambient","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10879","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.trace_mpo_ambient_pos","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 (MPOTensor.CaseIIAbsorptionCounterexample.ambient.mpo N).trace","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.trace_mpo_ambient_pos","module":"TNLean.MPS.MPDO.CaseIIAbsorptionCounterexample"},{"id":"n10880","layer":"formal","project":"p8","title":"MPOTensor.exists_pairwise_orthogonal_twoSided_physicalSupport_commonWeightAbsorbedBasis","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_pairwise_orthogonal_twoSided_physicalSupport_commonWeightAbsorbedBasis","module":"TNLean.MPS.MPDO.CommonWeightAbsorbedBNTSupport"},{"id":"n10881","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_eq_copies_mul_commonWeight_pow","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d) (hWeight : ∀ (j : Fin S.basisCount) (q q' : Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_eq_copies_mul_commonWeight_pow","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10882","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_ne_zero_of_weight_copy_independent","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d), (∀ (j : Fin S.basisCount) (q q' : Fin (S.copie…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_ne_zero_of_weight_copy_independent","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10883","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.commonWeight","kind":"def","summary":"d : Nat → (S : MPSTensor.SectorDecomposition d) → (∀ (j : Fin S.basisCount) (q q' : Fin (S.copi…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.commonWeight","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10884","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.commonWeightAbsorbedBasis","kind":"def","summary":"d : Nat → (S : MPSTensor.SectorDecomposition d) → (∀ (j : Fin S.basisCount) (q q' : Fin (S.copi…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.commonWeightAbsorbedBasis","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10885","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.commonWeight_ne_zero","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d) (hWeight : ∀ (j : Fin S.basisCount) (q q' : Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.commonWeight_ne_zero","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10886","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.mpv_toTensor_eq_sum_copies_mul_commonWeightAbsorbedBasis","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d) (hWeight : ∀ (j : Fin S.basisCount) (q q' : Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.mpv_toTensor_eq_sum_copies_mul_commonWeightAbsorbedBasis","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10887","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.weight_eq_commonWeight","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d) (hWeight : ∀ (j : Fin S.basisCount) (q q' : Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.weight_eq_commonWeight","module":"TNLean.MPS.MPDO.CommonWeightAbsorption"},{"id":"n10888","layer":"formal","project":"p8","title":"Matrix.EtaRankOneTraceFactorization","kind":"inductive","summary":"K : Nat → (dl dr : Fin K → Nat) → ((q h : Fin K) → Matrix (Matrix.EtaEdgeIndex dl dr q h) (Matr…","labels":[],"detail_key":"p8","name":"Matrix.EtaRankOneTraceFactorization","module":"TNLean.MPS.MPDO.CommutingBondEtaBoundaryTrace"},{"id":"n10889","layer":"formal","project":"p8","title":"Matrix.etaTwoBoundaryContraction_eq_separated","kind":"theorem","summary":"∀ K : Nat (dl dr : Fin K → Nat) (η : (q h : Fin K) → Matrix (Matrix.EtaEdgeIndex dl dr q h) (Ma…","labels":[],"detail_key":"p8","name":"Matrix.etaTwoBoundaryContraction_eq_separated","module":"TNLean.MPS.MPDO.CommutingBondEtaBoundaryTrace"},{"id":"n10890","layer":"formal","project":"p8","title":"Matrix.eta_fourth_region_trace_formula","kind":"theorem","summary":"∀ K L : Nat (dl dr : Fin K → Nat) (η : (q h : Fin K) → Matrix (Matrix.EtaEdgeIndex dl dr q h) (…","labels":[],"detail_key":"p8","name":"Matrix.eta_fourth_region_trace_formula","module":"TNLean.MPS.MPDO.CommutingBondEtaBoundaryTrace"},{"id":"n10891","layer":"formal","project":"p8","title":"Matrix.etaPairSpatialBlockEquiv","kind":"def","summary":"d K : Nat → dl dr : Fin K → Nat → Equiv (Sigma fun q => Prod (Fin (dr q)) (Fin (dl q))) (Fin d)…","labels":[],"detail_key":"p8","name":"Matrix.etaPairSpatialBlockEquiv","module":"TNLean.MPS.MPDO.CommutingBondEtaDecomposition"},{"id":"n10892","layer":"formal","project":"p8","title":"Matrix.exists_positive_etaPair_decomposition_of_overlappingLifts_commute","kind":"theorem","summary":"∀ d : Nat (B : Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin d)) Complex), B.PosSemidef → Eq…","labels":[],"detail_key":"p8","name":"Matrix.exists_positive_etaPair_decomposition_of_overlappingLifts_commute","module":"TNLean.MPS.MPDO.CommutingBondEtaDecomposition"},{"id":"n10893","layer":"formal","project":"p8","title":"MPOTensor.CommutingFormData","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.CommutingFormData","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10894","layer":"formal","project":"p8","title":"MPOTensor.CommutingFormData.hasGSNNCHFormAt_of_realizes","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.CommutingFormData d N) ρ : MPOTensor.ChainOperator d N, data.Real…","labels":[],"detail_key":"p8","name":"MPOTensor.CommutingFormData.hasGSNNCHFormAt_of_realizes","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10895","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10896","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.ofCommutingFormData","kind":"def","summary":"d N : Nat → MPOTensor.CommutingFormData d N → MPOTensor.GSNNCHData d N","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.ofCommutingFormData","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10897","layer":"formal","project":"p8","title":"MPOTensor.HasCommutingForm","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasCommutingForm","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10898","layer":"formal","project":"p8","title":"MPOTensor.HasGSNNCHForm","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasGSNNCHForm","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10899","layer":"formal","project":"p8","title":"MPOTensor.HasGSNNCHFormAt","kind":"def","summary":"d N : Nat → MPOTensor.ChainOperator d N → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasGSNNCHFormAt","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10900","layer":"formal","project":"p8","title":"MPOTensor.IsGSNNCH","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsGSNNCH","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10901","layer":"formal","project":"p8","title":"MPOTensor.IsGSNNCHAt","kind":"def","summary":"d N : Nat → MPOTensor.ChainOperator d N → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsGSNNCHAt","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10902","layer":"formal","project":"p8","title":"MPOTensor.agreesOutsideWindow_iff","kind":"theorem","summary":"∀ d : Nat (L : Nat) N : Nat (hLN : LE.le L N) (i : Fin N) (σ τ : Fin N → Fin d), Iff (MPOTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.agreesOutsideWindow_iff","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10903","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_commute_of_not_cyclicWindowsOverlap","kind":"theorem","summary":"∀ d : Nat (L N : Nat) (hLN : LE.le L N) i j : Fin N, Not (MPSTensor.cyclicWindowsOverlap N L i…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_commute_of_not_cyclicWindowsOverlap","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10904","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_mul","kind":"theorem","summary":"∀ d : Nat (L N : Nat) (hLN : LE.le L N) (i : Fin N) (B C : Matrix (Fin L → Fin d) (Fin L → Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_mul","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10905","layer":"formal","project":"p8","title":"MPOTensor.reindex_embedLocalOperator_windowComplement","kind":"theorem","summary":"∀ d : Nat (L N : Nat) (hLN : LE.le L N) (i : Fin N) (B : Matrix (Fin L → Fin d) (Fin L → Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_embedLocalOperator_windowComplement","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10906","layer":"formal","project":"p8","title":"MPOTensor.windowComplementEquiv","kind":"def","summary":"d : Nat → (L N : Nat) → LE.le L N → Fin N → Equiv (Fin N → Fin d) (Prod (Fin L → Fin d) (Fin (H…","labels":[],"detail_key":"p8","name":"MPOTensor.windowComplementEquiv","module":"TNLean.MPS.MPDO.CommutingForm"},{"id":"n10907","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10908","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.exists_positive_scalar_mpo_eq_product","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData) (N : Nat) (hN : LE.le 2 N), Exis…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.exists_positive_scalar_mpo_eq_product","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10909","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.formAt_bondAt_comm","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData) (N : Nat) (hN : LE.le 2 N) (i j…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.formAt_bondAt_comm","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10910","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.formAt_realizes","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData) (N : Nat) (hN : LE.le 2 N), (dat…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.formAt_realizes","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10911","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.hasCommutingForm","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData), M.HasCommutingForm","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.hasCommutingForm","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10912","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.hasGSNNCHForm","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData), M.HasGSNNCHForm","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.hasGSNNCHForm","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10913","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.positive_commuting_product_form","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData) (N : Nat) (hN : LE.le 2 N), And…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.positive_commuting_product_form","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10914","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.translationInvariantBondData","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (∀ (k h : Fin F.sectorCou…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.translationInvariantBondData","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10915","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10916","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.toCommutingFormData","kind":"def","summary":"d : Nat → MPOTensor.TranslationInvariantBondData d → N : Nat → LE.le 2 N → MPOTensor.CommutingF…","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.toCommutingFormData","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10917","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_bond","kind":"theorem","summary":"∀ d : Nat (data : MPOTensor.TranslationInvariantBondData d) N : Nat (hN : LE.le 2 N), Eq (data.…","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_bond","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10918","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_bondAt","kind":"theorem","summary":"∀ d : Nat (data : MPOTensor.TranslationInvariantBondData d) N : Nat (hN : LE.le 2 N) (i : Fin N…","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_bondAt","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10919","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_hN","kind":"theorem","summary":"∀ d : Nat (data : MPOTensor.TranslationInvariantBondData d) N : Nat (hN : LE.le 2 N), Eq ⋯ hN","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.toCommutingFormData_hN","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10920","layer":"formal","project":"p8","title":"MPOTensor.hasCommutingForm_of_etaLocalStructure","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (hEta : M.EtaLocalStructureData), M.HasCommutingForm","labels":[],"detail_key":"p8","name":"MPOTensor.hasCommutingForm_of_etaLocalStructure","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10921","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHForm_of_etaLocalStructure","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (hEta : M.EtaLocalStructureData), M.HasGSNNCHForm","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHForm_of_etaLocalStructure","module":"TNLean.MPS.MPDO.CommutingFormBridge"},{"id":"n10922","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.pairBond","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.EtaLocalStructureData → Matrix (Prod (Fin d) (Fin d)) (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.pairBond","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10923","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.pairBond","kind":"def","summary":"d : Nat → MPOTensor.TranslationInvariantBondData d → Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.pairBond","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10924","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.pairBond_isHermitian","kind":"theorem","summary":"∀ d : Nat (data : MPOTensor.TranslationInvariantBondData d), data.pairBond.IsHermitian","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.pairBond_isHermitian","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10925","layer":"formal","project":"p8","title":"MPOTensor.pairBondMatrix","kind":"def","summary":"d : Nat → Matrix (Fin 2 → Fin d) (Fin 2 → Fin d) Complex → Matrix (Prod (Fin d) (Fin d)) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.pairBondMatrix","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10926","layer":"formal","project":"p8","title":"MPOTensor.reindex_embedLocalOperator_one_eq_rightOverlappingLift","kind":"theorem","summary":"∀ d : Nat (B : Matrix (Fin 2 → Fin d) (Fin 2 → Fin d) Complex), Eq ((Matrix.reindex (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_embedLocalOperator_one_eq_rightOverlappingLift","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10927","layer":"formal","project":"p8","title":"MPOTensor.reindex_embedLocalOperator_zero_eq_leftOverlappingLift","kind":"theorem","summary":"∀ d : Nat (B : Matrix (Fin 2 → Fin d) (Fin 2 → Fin d) Complex), Eq ((Matrix.reindex (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_embedLocalOperator_zero_eq_leftOverlappingLift","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10928","layer":"formal","project":"p8","title":"MPOTensor.threeSiteOverlappingEquiv","kind":"def","summary":"(n : Type u_1) → Equiv (Fin 3 → n) (Prod (Prod n n) n)","labels":[],"detail_key":"p8","name":"MPOTensor.threeSiteOverlappingEquiv","module":"TNLean.MPS.MPDO.CommutingOverlappingCoordinates"},{"id":"n10929","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily","kind":"inductive","summary":"(Λ : Type u) → [Fintype Λ] → [DecidableEq Λ] → Nat → Type u","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily","module":"TNLean.MPS.MPDO.CompleteZipperFusionDefs"},{"id":"n10930","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.FourfoldMiddleMultiplicity","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.CompleteZipper…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.FourfoldMiddleMultiplicity","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10931","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.FourfoldPairMultiplicity","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.CompleteZipper…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.FourfoldPairMultiplicity","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10932","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.FourfoldRightAssocMultiplicity","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.CompleteZipper…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.FourfoldRightAssocMultiplicity","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10933","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.leftAssocFourfoldSynthesis","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.leftAssocFourfoldSynthesis","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10934","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.leftInnerFourfoldSynthesis","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.leftInnerFourfoldSynthesis","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10935","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.middleFourfoldSynthesis","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.middleFourfoldSynthesis","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10936","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.pairFourfoldSynthesis","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.pairFourfoldSynthesis","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10937","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.rightAssocFourfoldSynthesis","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.rightAssocFourfoldSynthesis","module":"TNLean.MPS.MPDO.CompleteZipperFusionFourfold"},{"id":"n10938","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.inversePrintedFMatrix_pentagon","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fus : MPOTensor.CompleteZippe…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.inversePrintedFMatrix_pentagon","module":"TNLean.MPS.MPDO.CompleteZipperFusionPentagon"},{"id":"n10939","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.leftAssocToPairPrintedFMatrix","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.leftAssocToPairPrintedFMatrix","module":"TNLean.MPS.MPDO.CompleteZipperFusionPentagon"},{"id":"n10940","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.pairToRightAssocPrintedFMatrix","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.pairToRightAssocPrintedFMatrix","module":"TNLean.MPS.MPDO.CompleteZipperFusionPentagon"},{"id":"n10941","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.threeEdgePrintedFMatrix","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.threeEdgePrintedFMatrix","module":"TNLean.MPS.MPDO.CompleteZipperFusionPentagon"},{"id":"n10942","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.twoEdgePrintedFMatrix","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fus : MPOTensor.Complet…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.twoEdgePrintedFMatrix","module":"TNLean.MPS.MPDO.CompleteZipperFusionPentagon"},{"id":"n10943","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.CompleteZipper…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector","module":"TNLean.MPS.MPDO.CompleteZipperFusionSupport"},{"id":"n10944","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector_apply","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fus : MPOTensor.CompleteZippe…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector_apply","module":"TNLean.MPS.MPDO.CompleteZipperFusionSupport"},{"id":"n10945","layer":"formal","project":"p8","title":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector_sum","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fus : MPOTensor.CompleteZippe…","labels":[],"detail_key":"p8","name":"MPOTensor.CompleteZipperFusionFamily.finalBlockSelector_sum","module":"TNLean.MPS.MPDO.CompleteZipperFusionSupport"},{"id":"n10946","layer":"formal","project":"p8","title":"MPOTensor.periodicTwoPointCorrelation","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → Matrix (Fin d) (Fin d) Complex → N…","labels":[],"detail_key":"p8","name":"MPOTensor.periodicTwoPointCorrelation","module":"TNLean.MPS.MPDO.Correlations"},{"id":"n10947","layer":"formal","project":"p8","title":"MPOTensor.periodicTwoPointCorrelation_positiveGaps_independent","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsPhysicalTraceIdempotent → ∀ (O₁ O₂ : Matrix (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.periodicTwoPointCorrelation_positiveGaps_independent","module":"TNLean.MPS.MPDO.Correlations"},{"id":"n10948","layer":"formal","project":"p8","title":"MPOTensor.periodicTwoPointCorrelation_positiveWrapGaps_independent","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsPhysicalTraceIdempotent → ∀ (O₁ O₂ : Matrix (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.periodicTwoPointCorrelation_positiveWrapGaps_independent","module":"TNLean.MPS.MPDO.Correlations"},{"id":"n10949","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.isSAL_of_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D, K.IsMPDO → K.IsInjective → ∀ (data : K.EtaLocalStructureData), K…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.isSAL_of_isSourceZCL","module":"TNLean.MPS.MPDO.CyclicActiveAreaLaw"},{"id":"n10950","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isSAL_of_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [NeZero D], K.IsInjective → (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isSAL_of_isSourceZCL","module":"TNLean.MPS.MPDO.CyclicActiveAreaLaw"},{"id":"n10951","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.suffixSectorContraction","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (R : Nat) → [NeZero R] →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.suffixSectorContraction","module":"TNLean.MPS.MPDO.CyclicActiveFourthRegionContraction"},{"id":"n10952","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.retainedBulkProduct","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → n : Nat → (k : Fin (HAdd.…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.retainedBulkProduct","module":"TNLean.MPS.MPDO.CyclicActiveRetainedCoordinates"},{"id":"n10953","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.CyclicActiveSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.CyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10954","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.IsCyclicActiveSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin F.sectorCount → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.IsCyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10955","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → K.IsInjective → K.PhysicalSecto…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10956","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_leftTensor","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (hK : K.IsInjective) (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_leftTensor","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10957","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (hK : K.IsInjective) (k h : F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_neighboringOperator","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10958","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_rightTensor","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (hK : K.IsInjective) (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveLeftRestriction_rightTensor","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10959","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorOneSiteMatrixFamily_span_eq_top","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → Eq (Submodul…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorOneSiteMatrixFamily_span_eq_top","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10960","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Matrix F.CyclicActiveSect…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10961","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_isIrreducible","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → (∀ (k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_isIrreducible","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10962","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_isPrimitive","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → (∀ (k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_isPrimitive","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10963","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_normalized_relations_…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → (∀ (k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_normalized_relations_of_isSourceZCL","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10964","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_pow_three_diag_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → (∀ (k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_pow_three_diag_pos","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10965","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_pow_two_diag_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → (∀ (k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_pow_two_diag_pos","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10966","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_rank_pow_two_eq_one","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [NeZero D], K.IsInjective → (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSectorTraceMatrix_rank_pow_two_eq_one","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10967","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSector_exists_sectorVirtualMatrix_ne_ze…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : F.CyclicActiveSector), E…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveSector_exists_sectorVirtualMatrix_ne_zero","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10968","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveWeight","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin F.sectorCount → Real","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveWeight","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10969","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicActiveWeight_ne_zero_iff","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Iff…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicActiveWeight_ne_zero_iff","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10970","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct_eq_zero_of_not_forall_isCy…","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct_eq_zero_of_not_forall_isCyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10971","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct_eq_zero_of_not_isCyclicAct…","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct_eq_zero_of_not_isCyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10972","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_cyclicActive_two_edge_return","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → ∀ k h : F.Cy…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_cyclicActive_two_edge_return","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10973","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_normalized_cyclicActiveSectorTraceMatrix_ran…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [NeZero D], K.IsInjective → (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_normalized_cyclicActiveSectorTraceMatrix_rank_pow_two_eq_one_of_isSourceZCL","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10974","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_sectorVirtualMatrix_ne_zero_of_isCyclicActiv…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) k : Fin F.sectorCount, F.IsCy…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_sectorVirtualMatrix_ne_zero_of_isCyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10975","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_cyclicNeighboringProduct_ne…","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_cyclicNeighboringProduct_ne_zero","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10976","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_cyclic_edges","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_cyclic_edges","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10977","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_sectorVirtualMatrix_ne_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq (Submodule.span Complex (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isCyclicActiveSector_of_sectorVirtualMatrix_ne_zero","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10978","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_ne_zero_of_cyclicNeighboringPro…","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_ne_zero_of_cyclicNeighboringProduct_ne_zero","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10979","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_trace_re_pos_iff","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_trace_re_pos_iff","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10980","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.nonempty_cyclicActiveSector","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [NeZero D], K.IsInjective → N…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.nonempty_cyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10981","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.normalized_cyclicActiveSectorTraceMatrix_rank_pow_t…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [NeZero D], K.IsInjective → (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.normalized_cyclicActiveSectorTraceMatrix_rank_pow_two_eq_one","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10982","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.reindex_cyclicActiveLeftRestriction_activeSectorTra…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (hK : K.IsInjective), Eq ((Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.reindex_cyclicActiveLeftRestriction_activeSectorTraceMatrix","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10983","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix_eq_zero_of_not_isCyclicActiveSe…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → ∀ (k : Fin F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix_eq_zero_of_not_isCyclicActiveSector","module":"TNLean.MPS.MPDO.CyclicActiveSectorRestriction"},{"id":"n10984","layer":"formal","project":"p8","title":"MPOTensor.cyclicEdgeWeightTensor","kind":"def","summary":"d : Nat → (Fin d → Fin d → Fin d → Fin d → Complex) → MPOTensor d (HMul.hMul d d)","labels":[],"detail_key":"p8","name":"MPOTensor.cyclicEdgeWeightTensor","module":"TNLean.MPS.MPDO.CyclicEdgeWeightTensor"},{"id":"n10985","layer":"formal","project":"p8","title":"MPOTensor.cyclicEdgeWeightTensor_apply_eq","kind":"theorem","summary":"∀ d : Nat (w : Fin d → Fin d → Fin d → Fin d → Complex) (i j i' j' : Fin d), Eq (MPOTensor.cycl…","labels":[],"detail_key":"p8","name":"MPOTensor.cyclicEdgeWeightTensor_apply_eq","module":"TNLean.MPS.MPDO.CyclicEdgeWeightTensor"},{"id":"n10986","layer":"formal","project":"p8","title":"MPOTensor.mpo_cyclicEdgeWeightTensor","kind":"theorem","summary":"∀ d : Nat (w : Fin d → Fin d → Fin d → Fin d → Complex) N : Nat [inst : NeZero N] (sigma tau :…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_cyclicEdgeWeightTensor","module":"TNLean.MPS.MPDO.CyclicEdgeWeightTensor"},{"id":"n10987","layer":"formal","project":"p8","title":"MPOTensor.exists_displaced_invariant_projector_of_periodic_vector","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) n : Nat (V : Matrix (Fin d) (Fin n) Complex) (B : MPSTensor (HM…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_displaced_invariant_projector_of_periodic_vector","module":"TNLean.MPS.MPDO.CyclicProjector"},{"id":"n10988","layer":"formal","project":"p8","title":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_horizontalCF","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → M.IsHorizontalCF → M.verticalTensor.HasNoPeriodicVe…","labels":[],"detail_key":"p8","name":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_horizontalCF","module":"TNLean.MPS.MPDO.CyclicProjector"},{"id":"n10989","layer":"formal","project":"p8","title":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_isInjective","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → Kraus.IsInjective M.toMPSTensor → M.verticalTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.hasNoPeriodicVectors_verticalTensor_of_isInjective","module":"TNLean.MPS.MPDO.CyclicProjector"},{"id":"n10990","layer":"formal","project":"p8","title":"MPSTensor.hasEigenvalue_transferMap_spectralUnitalGauge","kind":"theorem","summary":"∀ d D : Nat (B : MPSTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (rad : Real), ρ.PosDef → L…","labels":[],"detail_key":"p8","name":"MPSTensor.hasEigenvalue_transferMap_spectralUnitalGauge","module":"TNLean.MPS.MPDO.CyclicProjector"},{"id":"n10991","layer":"formal","project":"p8","title":"MPSTensor.spectralUnitalGauge_schwarz_setup","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (B : MPSTensor d D), Kraus.IsIrreducibleFamily B → ∀ (ρ : Matrix (Fin D)…","labels":[],"detail_key":"p8","name":"MPSTensor.spectralUnitalGauge_schwarz_setup","module":"TNLean.MPS.MPDO.CyclicProjector"},{"id":"n10992","layer":"formal","project":"p8","title":"MPOTensor","kind":"def","summary":"Nat → Nat → 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LT.lt 0 N → ∀ (σ τ : Fin N → Fin d), Eq (M…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.mpo_adjointTensor_eq","module":"TNLean.MPS.MPDO.Defs"},{"id":"n10997","layer":"formal","project":"p8","title":"MPOTensor.adjointTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.adjointTensor","module":"TNLean.MPS.MPDO.Defs"},{"id":"n10998","layer":"formal","project":"p8","title":"MPOTensor.adjointTensor_eq_iff_isHermitian","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Iff (Eq M.adjointTensor M) M.IsHermitian","labels":[],"detail_key":"p8","name":"MPOTensor.adjointTensor_eq_iff_isHermitian","module":"TNLean.MPS.MPDO.Defs"},{"id":"n10999","layer":"formal","project":"p8","title":"MPOTensor.evalWord","kind":"def","summary":"d D : Nat → MPOTensor d D → List (Fin d) → List (Fin d) → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11000","layer":"formal","project":"p8","title":"MPOTensor.evalWord_adjointTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (σs τs : List (Fin d)), Eq σs.length τs.length → Eq (M.adjointT…","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord_adjointTensor","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11001","layer":"formal","project":"p8","title":"MPOTensor.evalWord_append","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (l₁ k₁ l₂ k₂ : List (Fin d)), Eq l₁.length k₁.length → Eq (M.ev…","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord_append","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11002","layer":"formal","project":"p8","title":"MPOTensor.evalWord_reindexPhysical","kind":"theorem","summary":"∀ d D d' : Nat (e : Equiv (Fin d') (Fin d)) (U : MPOTensor d D) (is js : List (Fin d')), Eq ((M…","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord_reindexPhysical","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11003","layer":"formal","project":"p8","title":"MPOTensor.mpo","kind":"def","summary":"d D : Nat → MPOTensor d D → (N : Nat) → Matrix (Fin N → Fin d) (Fin N → Fin d) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.mpo","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11004","layer":"formal","project":"p8","title":"MPOTensor.mpo_adjointTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N : Nat (σ τ : Fin N → Fin d), Eq (M.adjointTensor.mpo N σ τ) (…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_adjointTensor","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11005","layer":"formal","project":"p8","title":"MPOTensor.mpo_reindexPhysical","kind":"theorem","summary":"∀ d D d' : Nat (e : Equiv (Fin d') (Fin d)) (U : MPOTensor d D) (N : Nat), Eq ((MPOTensor.reind…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_reindexPhysical","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11006","layer":"formal","project":"p8","title":"MPOTensor.physicalSlice","kind":"def","summary":"d D : Nat → MPOTensor d D → Fin D → Fin D → Matrix (Fin d) (Fin d) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSlice","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11007","layer":"formal","project":"p8","title":"MPOTensor.reindexPhysical","kind":"def","summary":"d D d' : Nat → Equiv (Fin d') (Fin d) → MPOTensor d D → MPOTensor d' D","labels":[],"detail_key":"p8","name":"MPOTensor.reindexPhysical","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11008","layer":"formal","project":"p8","title":"MPOTensor.reindexPhysicalConfigEquiv","kind":"def","summary":"d d' : Nat → (N : Nat) → Equiv (Fin d') (Fin d) → Equiv (Fin N → Fin d') (Fin N → Fin d)","labels":[],"detail_key":"p8","name":"MPOTensor.reindexPhysicalConfigEquiv","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11009","layer":"formal","project":"p8","title":"MPOTensor.toMPSTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → MPSTensor (HMul.hMul d d) D","labels":[],"detail_key":"p8","name":"MPOTensor.toMPSTensor","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11010","layer":"formal","project":"p8","title":"MPOTensor.trace_evalWord_cons_eq_append","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (a b : Fin d) (l k : List (Fin d)), Eq l.length k.length → Eq (…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_evalWord_cons_eq_append","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11011","layer":"formal","project":"p8","title":"MPOTensor.transferMap","kind":"def","summary":"d D : Nat → MPOTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.transferMap","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11012","layer":"formal","project":"p8","title":"MPOTensor.transferMap_eq_toMPSTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq M.transferMap (Kraus.transferMap M.toMPSTensor)","labels":[],"detail_key":"p8","name":"MPOTensor.transferMap_eq_toMPSTensor","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11013","layer":"formal","project":"p8","title":"MPOTensor.transferMap_pos","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) X : Matrix (Fin D) (Fin D) Complex, X.PosSemidef → (M.transferM…","labels":[],"detail_key":"p8","name":"MPOTensor.transferMap_pos","module":"TNLean.MPS.MPDO.Defs"},{"id":"n11014","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutMatrix","kind":"def","summary":"d D : Nat → MPOTensor d D → (L R : Nat) → Matrix (Fin L → Fin d) (Fin R → Fin d) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutMatrix","module":"TNLean.MPS.MPDO.DiagonalCutRank"},{"id":"n11015","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutMatrix_rank_le","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat), LE.le (M.diagonalCutMatrix L R).rank (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutMatrix_rank_le","module":"TNLean.MPS.MPDO.DiagonalCutRank"},{"id":"n11016","layer":"formal","project":"p8","title":"MPOTensor.diagonalCut_classicalMutualInformation_le_two_log","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (M : MPOTensor d D) (L R : Nat) (W P : Matrix (Fin L → Fin d) (Fin R → F…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCut_classicalMutualInformation_le_two_log","module":"TNLean.MPS.MPDO.DiagonalCutRank"},{"id":"n11017","layer":"formal","project":"p8","title":"MPOTensor.IsDiagonal","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsDiagonal","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11018","layer":"formal","project":"p8","title":"MPOTensor.IsDiagonalAt","kind":"def","summary":"d D : Nat → MPOTensor d D → Nat → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsDiagonalAt","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11019","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutMass","kind":"def","summary":"d D : Nat → MPOTensor d D → Nat → Nat → Real","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutMass","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11020","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutMass_eq_trace_re","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat), Eq (M.diagonalCutMass L R) (M.mpo (HAdd.hAdd L R))…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutMass_eq_trace_re","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11021","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutMatrix_apply_eq_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat) (x : Fin L → Fin d) (y : Fin R → Fin d), Eq (M.diag…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutMatrix_apply_eq_mpo","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11022","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutRealMatrix","kind":"def","summary":"d D : Nat → MPOTensor d D → (L R : Nat) → Matrix (Fin L → Fin d) (Fin R → Fin d) Real","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutRealMatrix","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11023","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutRealMatrix_map_ofReal_of_posSemidef","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat), (M.mpo (HAdd.hAdd L R)).PosSemidef → Eq ((M.diagon…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutRealMatrix_map_ofReal_of_posSemidef","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11024","layer":"formal","project":"p8","title":"MPOTensor.diagonalCutRealMatrix_nonneg_of_posSemidef","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat), (M.mpo (HAdd.hAdd L R)).PosSemidef → ∀ (x : Fin L…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalCutRealMatrix_nonneg_of_posSemidef","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11025","layer":"formal","project":"p8","title":"MPOTensor.diagonalFiniteChain_classicalMutualInformation_le_two_log","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (M : MPOTensor d D) (L R : Nat), M.IsDiagonalAt (HAdd.hAdd L R) → (M.mpo…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalFiniteChain_classicalMutualInformation_le_two_log","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11026","layer":"formal","project":"p8","title":"MPOTensor.diagonalMPDO_classicalMutualInformation_le_two_log","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (M : MPOTensor d D), M.IsDiagonal → M.IsMPDO → ∀ (L R : Nat), LT.lt 0 (H…","labels":[],"detail_key":"p8","name":"MPOTensor.diagonalMPDO_classicalMutualInformation_le_two_log","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11027","layer":"formal","project":"p8","title":"MPOTensor.normalizedDiagonalCutDistribution","kind":"def","summary":"d D : Nat → MPOTensor d D → (L R : Nat) → Matrix (Fin L → Fin d) (Fin R → Fin d) Real","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedDiagonalCutDistribution","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11028","layer":"formal","project":"p8","title":"MPOTensor.normalizedDiagonalCutDistribution_isJointDistribution","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L R : Nat), (M.mpo (HAdd.hAdd L R)).PosSemidef → LT.lt 0 (M.di…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedDiagonalCutDistribution_isJointDistribution","module":"TNLean.MPS.MPDO.DiagonalFiniteChain"},{"id":"n11029","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_monomial","kind":"theorem","summary":"∀ d : Nat (L N : Nat) (hLN : LE.le L N) (i : Fin N) (σ : Equiv.Perm (Fin L → Fin d)) (φ : (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_monomial","module":"TNLean.MPS.MPDO.EmbedLocalOperatorMonomial"},{"id":"n11030","layer":"formal","project":"p8","title":"MPOTensor.extractWindow_two","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] α : Type u_1, LE.le 2 N → ∀ (j : Fin N) (t : Fin N → α), Eq (MPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.extractWindow_two","module":"TNLean.MPS.MPDO.EmbedLocalOperatorMonomial"},{"id":"n11031","layer":"formal","project":"p8","title":"MPOTensor.replaceWindow_two_apply","kind":"theorem","summary":"∀ N : Nat [inst : NeZero N] α : Type u_1 (hN : LE.le 2 N) (j : Fin N) (s : Fin N → α) (w : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.replaceWindow_two_apply","module":"TNLean.MPS.MPDO.EmbedLocalOperatorMonomial"},{"id":"n11032","layer":"formal","project":"p8","title":"MPOTensor.windowPerm","kind":"def","summary":"d : Nat → (L : Nat) → N : Nat → LE.le L N → Fin N → Equiv.Perm (Fin L → Fin d) → Equiv.Perm (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.windowPerm","module":"TNLean.MPS.MPDO.EmbedLocalOperatorMonomial"},{"id":"n11033","layer":"formal","project":"p8","title":"MPOTensor.windowPerm_apply","kind":"theorem","summary":"∀ d : Nat (L : Nat) N : Nat (hLN : LE.le L N) (i : Fin N) (σ : Equiv.Perm (Fin L → Fin d)) (s :…","labels":[],"detail_key":"p8","name":"MPOTensor.windowPerm_apply","module":"TNLean.MPS.MPDO.EmbedLocalOperatorMonomial"},{"id":"n11034","layer":"formal","project":"p8","title":"MPOTensor.reindex_embedLocalOperator_two_one","kind":"theorem","summary":"∀ d : Nat (B : Matrix (Fin 2 → Fin d) (Fin 2 → Fin d) Complex), Eq ((Matrix.reindex (finTwoArro…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_embedLocalOperator_two_one","module":"TNLean.MPS.MPDO.EmbedLocalOperatorTwoSite"},{"id":"n11035","layer":"formal","project":"p8","title":"MPOTensor.reindex_embedLocalOperator_two_zero","kind":"theorem","summary":"∀ d : Nat (B : Matrix (Fin 2 → Fin d) (Fin 2 → Fin d) Complex), Eq ((Matrix.reindex (finTwoArro…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_embedLocalOperator_two_zero","module":"TNLean.MPS.MPDO.EmbedLocalOperatorTwoSite"},{"id":"n11036","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.OmegaIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.OmegaIndex","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11037","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.RankOneTraceFactorization","kind":"inductive","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.RankOneTraceFactorization","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11038","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEta","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEta","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11039","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11040","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11041","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap_sameBlock_apply","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEtaControlledMap_sameBlock_apply","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11042","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEtaPreparationMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEtaPreparationMap","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11043","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEtaPreparationMap_eq_normalized","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEtaPreparationMap_eq_normalized","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11044","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEta_eq_normalizedEta","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEta_eq_normalizedEta","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11045","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEta_pos","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEta_pos","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11046","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedEta_preparationMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedEta_preparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11047","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmega","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11048","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11049","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11050","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap_sameBlock_apply","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmegaControlledMap_sameBlock_apply","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11051","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmegaPreparationMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmegaPreparationMap","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11052","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmegaPreparationMap_eq_normalized","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmegaPreparationMap_eq_normalized","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11053","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmega_eq_normalizedOmega","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmega_eq_normalizedOmega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11054","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmega_pos","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmega_pos","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11055","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.completedOmega_preparationMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.completedOmega_preparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11056","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.dL_nonempty","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.dL_nonempty","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11057","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.dR_nonempty","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.dR_nonempty","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11058","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.etaIndex_nonempty","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.etaIndex_nonempty","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11059","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.eta_eq_zero_of_mul_eq_zero","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.eta_eq_zero_of_mul_eq_zero","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11060","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.inactiveEtaDensity","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.inactiveEtaDensity","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11061","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.inactiveOmegaDensity","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.inactiveOmegaDensity","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11062","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedEta","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedEta","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11063","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedEta_pos","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedEta_pos","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11064","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedEta_preparationMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedEta_preparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11065","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedOmega","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedOmega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11066","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedOmega_pos","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedOmega_pos","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11067","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.normalizedOmega_preparationMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.normalizedOmega_preparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11068","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.omega","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.omega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11069","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.omegaIndex_nonempty","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.omegaIndex_nonempty","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11070","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.omega_eq_zero_of_mul_eq_zero","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.omega_eq_zero_of_mul_eq_zero","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11071","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.omega_pos","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.omega_pos","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11072","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.sector_nonempty","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.sector_nonempty","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11073","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_completedEta","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_completedEta","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11074","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_completedOmega","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_completedOmega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11075","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_normalizedEta","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_normalizedEta","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11076","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_normalizedOmega","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_normalizedOmega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11077","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_omega","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_omega","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11078","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.trace_omega_eq_mul","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.trace_omega_eq_mul","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11079","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.weight_nonneg","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.weight_nonneg","module":"TNLean.MPS.MPDO.EtaPreparation"},{"id":"n11080","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.B","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Nat","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.B","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11081","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.ExactlyTwoZero","kind":"def","summary":"Fin 2 → Fin 2 → Fin 2 → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.ExactlyTwoZero","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11082","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.FusionWeights","kind":"inductive","summary":"Type","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.FusionWeights","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11083","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.canonicalFusionWeights","kind":"def","summary":"MPSTensor.FibonacciBoundary.FusionWeights","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.canonicalFusionWeights","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11084","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.canonicalTensor","kind":"def","summary":"MPSTensor 8 2","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.canonicalTensor","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11085","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.openBoundaryOperator","kind":"def","summary":"MPSTensor.FibonacciBoundary.FusionWeights → (n : Nat) → Fin 2 → Fin 2 → Matrix (Fin n → Fin 8)…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.openBoundaryOperator","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11086","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.openBoundaryOperator_apply","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights) (n : Nat) (α β : Fin 2) (σ τ : Fin n → Fin 8)…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.openBoundaryOperator_apply","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11087","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11088","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount_add_two_add","kind":"theorem","summary":"∀ (N : Nat), Eq (HAdd.hAdd (MPSTensor.FibonacciBoundary.periodicTransitionCount (HAdd.hAdd N 2)…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount_add_two_add","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11089","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount_eq_goldenRatio","kind":"theorem","summary":"∀ (N : Nat), Eq (↑(MPSTensor.FibonacciBoundary.periodicTransitionCount N)) (HAdd.hAdd (HPow.hPo…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount_eq_goldenRatio","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11090","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount_not_geometric","kind":"theorem","summary":"Not (Exists fun r => Exists fun s => ∀ (N : Nat), LT.lt 0 N → Eq (MPSTensor.FibonacciBoundary.p…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount_not_geometric","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11091","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount_one","kind":"theorem","summary":"Eq (MPSTensor.FibonacciBoundary.periodicTransitionCount 1) 3","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount_one","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11092","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.periodicTransitionCount_two","kind":"theorem","summary":"Eq (MPSTensor.FibonacciBoundary.periodicTransitionCount 2) 7","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.periodicTransitionCount_two","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11093","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.physicalTriple","kind":"def","summary":"Fin 8 → Prod (Fin 2) (Prod (Fin 2) (Fin 2))","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.physicalTriple","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11094","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.rank_openBoundaryOperator_eq_x","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights) (n : Nat) (α β : Fin 2), Eq (MPSTensor.Fibona…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.rank_openBoundaryOperator_eq_x","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11095","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.tensor","kind":"def","summary":"MPSTensor.FibonacciBoundary.FusionWeights → MPSTensor 8 2","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.tensor","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11096","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.x","kind":"def","summary":"Nat → Fin 2 → Fin 2 → Nat","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.x","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11097","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.x_one","kind":"theorem","summary":"Eq (Prod.mk (MPSTensor.FibonacciBoundary.x 1 0 0) (Prod.mk (MPSTensor.FibonacciBoundary.x 1 0 1…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.x_one","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11098","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.x_succ","kind":"theorem","summary":"∀ (n : Nat) (α : Fin 2), And (Eq (MPSTensor.FibonacciBoundary.x (HAdd.hAdd n 1) α 0) (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.x_succ","module":"TNLean.MPS.MPDO.FibonacciBoundaryRank"},{"id":"n11099","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_eq_goldenRatio","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights) (N : Nat), LT.lt 0 N → Eq (↑((MPSTensor.Fibon…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_eq_goldenRatio","module":"TNLean.MPS.MPDO.FibonacciPeriodicRank"},{"id":"n11100","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_eq_periodicTransitionCount","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights) (N : Nat), LT.lt 0 N → Eq ((MPSTensor.Fibonac…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_eq_periodicTransitionCount","module":"TNLean.MPS.MPDO.FibonacciPeriodicRank"},{"id":"n11101","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_not_geometric","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights), Not (Exists fun r => Exists fun s => ∀ (N :…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.rank_mpo_toMPOTensor_not_geometric","module":"TNLean.MPS.MPDO.FibonacciPeriodicRank"},{"id":"n11102","layer":"formal","project":"p8","title":"MPSTensor.FibonacciBoundary.toMPOTensor_not_isStrongRFP","kind":"theorem","summary":"∀ (W : MPSTensor.FibonacciBoundary.FusionWeights), Not (MPSTensor.FibonacciBoundary.tensor W).t…","labels":[],"detail_key":"p8","name":"MPSTensor.FibonacciBoundary.toMPOTensor_not_isStrongRFP","module":"TNLean.MPS.MPDO.FibonacciPeriodicRank"},{"id":"n11103","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.gramDressing_eq_of_two_grouped_corners","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → ∀ (A : MPSTensor (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.gramDressing_eq_of_two_grouped_corners","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11104","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.markedChainCoefficient_gramDressing_eq_of_two_corners","kind":"theorem","summary":"∀ d D n : Nat M : MPOTensor d D, M.IsMPDO → ∀ (A : MPSTensor (HMul.hMul D D) n) (VX VY : Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.markedChainCoefficient_gramDressing_eq_of_two_corners","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11105","layer":"formal","project":"p8","title":"MPOTensor.cornerGramCoefficients","kind":"def","summary":"d n : Nat → Matrix (Fin d) (Fin n) Complex → Matrix.GeneralLinearGroup (Fin n) Complex → Comple…","labels":[],"detail_key":"p8","name":"MPOTensor.cornerGramCoefficients","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11106","layer":"formal","project":"p8","title":"MPOTensor.gramDressing","kind":"def","summary":"D n : Nat → Matrix.GeneralLinearGroup (Fin n) Complex → MPSTensor (HMul.hMul D D) n → MPSTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.gramDressing","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11107","layer":"formal","project":"p8","title":"MPOTensor.linearMarkedTensor_twoSidedCompressionCoefficients","kind":"theorem","summary":"∀ d D n : Nat M : MPOTensor d D (A : MPSTensor (HMul.hMul D D) n) (L R : Matrix (Fin d) (Fin n)…","labels":[],"detail_key":"p8","name":"MPOTensor.linearMarkedTensor_twoSidedCompressionCoefficients","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11108","layer":"formal","project":"p8","title":"MPOTensor.twoSidedCompressionCoefficients","kind":"def","summary":"d n : Nat → Matrix (Fin d) (Fin n) Complex → Matrix (Fin d) (Fin n) Complex → Complex → Fin (HM…","labels":[],"detail_key":"p8","name":"MPOTensor.twoSidedCompressionCoefficients","module":"TNLean.MPS.MPDO.FigureEightPairwise"},{"id":"n11109","layer":"formal","project":"p8","title":"MPOTensor.braRightMul","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.braRightMul","module":"TNLean.MPS.MPDO.FirstSite"},{"id":"n11110","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix","kind":"def","summary":"d : Nat → Matrix (Fin d) (Fin d) Complex → (N : Nat) → Matrix (Fin (HAdd.hAdd N 1) → Fin d) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix","module":"TNLean.MPS.MPDO.FirstSite"},{"id":"n11111","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_mul_firstSiteMatrix","kind":"theorem","summary":"∀ d : Nat (P Q : Matrix (Fin d) (Fin d) Complex) (N : Nat), Eq (HMul.hMul (MPOTensor.firstSiteM…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_mul_firstSiteMatrix","module":"TNLean.MPS.MPDO.FirstSite"},{"id":"n11112","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_one","kind":"theorem","summary":"∀ d : Nat (N : Nat), Eq (MPOTensor.firstSiteMatrix 1 N) 1","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_one","module":"TNLean.MPS.MPDO.FirstSite"},{"id":"n11113","layer":"formal","project":"p8","title":"MPOTensor.ketLeftMul","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.ketLeftMul","module":"TNLean.MPS.MPDO.FirstSite"},{"id":"n11114","layer":"formal","project":"p8","title":"MPSTensor.FirstSiteActionAgree.blockTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D Y Z : Matrix (Fin d) (Fin d) Complex, A.FirstSiteActionAgree Y Z…","labels":[],"detail_key":"p8","name":"MPSTensor.FirstSiteActionAgree.blockTensor","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11115","layer":"formal","project":"p8","title":"MPSTensor.FirstSiteActionAgree.trace_evalWord","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D Y Z : Matrix (Fin d) (Fin d) Complex, A.FirstSiteActionAgree Y Z…","labels":[],"detail_key":"p8","name":"MPSTensor.FirstSiteActionAgree.trace_evalWord","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11116","layer":"formal","project":"p8","title":"MPSTensor.firstSiteActionAgree_iff_trace","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (Y Z : Matrix (Fin d) (Fin d) Complex), Iff (A.FirstSiteActionA…","labels":[],"detail_key":"p8","name":"MPSTensor.firstSiteActionAgree_iff_trace","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11117","layer":"formal","project":"p8","title":"MPSTensor.firstSiteActionOnBlock","kind":"def","summary":"d : Nat → (L : Nat) → Matrix (Fin d) (Fin d) Complex → Matrix (Fin (MPSTensor.blockPhysDim d (H…","labels":[],"detail_key":"p8","name":"MPSTensor.firstSiteActionOnBlock","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11118","layer":"formal","project":"p8","title":"MPSTensor.insertedTensor_eq_of_firstSiteActionOnBlock_blockTensor_eq","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), Kraus.IsNBlkInjective A L → ∀ Y Z : Matrix (Fin d) (…","labels":[],"detail_key":"p8","name":"MPSTensor.insertedTensor_eq_of_firstSiteActionOnBlock_blockTensor_eq","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11119","layer":"formal","project":"p8","title":"MPSTensor.insertedTensor_firstSiteActionOnBlock_blockTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat) (Y : Matrix (Fin d) (Fin d) Complex) (I : Fin (MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.insertedTensor_firstSiteActionOnBlock_blockTensor","module":"TNLean.MPS.MPDO.FirstSiteBlocking"},{"id":"n11120","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.fixedProductTensorData","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (data : M.EtaLocalStructureData) → data.bondData.FixedProductTe…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.fixedProductTensorData","module":"TNLean.MPS.MPDO.FixedBondProductEtaTensor"},{"id":"n11121","layer":"formal","project":"p8","title":"MPOTensor.EtaLocalStructureData.nonempty_fixedProductTensorData","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (data : M.EtaLocalStructureData), Nonempty data.bondData.FixedPro…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaLocalStructureData.nonempty_fixedProductTensorData","module":"TNLean.MPS.MPDO.FixedBondProductEtaTensor"},{"id":"n11122","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.fixedProductTensorData","kind":"def","summary":"d : Nat → (data : MPOTensor.TranslationInvariantBondData d) → data.FixedProductTensorData","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.fixedProductTensorData","module":"TNLean.MPS.MPDO.FixedBondProductEtaTensor"},{"id":"n11123","layer":"formal","project":"p8","title":"MPOTensor.etaCyclicEdgeWeight","kind":"def","summary":"d K : Nat → (dl dr : Fin K → Nat) → Equiv (Matrix.EtaSiteIndex K dl dr) (Fin d) → ((q h : Fin K…","labels":[],"detail_key":"p8","name":"MPOTensor.etaCyclicEdgeWeight","module":"TNLean.MPS.MPDO.FixedBondProductEtaTensor"},{"id":"n11124","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.CyclicEdgeWeightForm","kind":"inductive","summary":"d : Nat → MPOTensor.TranslationInvariantBondData d → Type","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.CyclicEdgeWeightForm","module":"TNLean.MPS.MPDO.FixedBondProductTensor"},{"id":"n11125","layer":"formal","project":"p8","title":"MPOTensor.TranslationInvariantBondData.FixedProductTensorData","kind":"inductive","summary":"d : Nat → MPOTensor.TranslationInvariantBondData d → Type","labels":[],"detail_key":"p8","name":"MPOTensor.TranslationInvariantBondData.FixedProductTensorData","module":"TNLean.MPS.MPDO.FixedBondProductTensor"},{"id":"n11126","layer":"formal","project":"p8","title":"MPOTensor.TransferRetractData","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.TransferRetractData","module":"TNLean.MPS.MPDO.FusionIsometries"},{"id":"n11127","layer":"formal","project":"p8","title":"MPOTensor.TransferRetractData.isZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (F : M.TransferRetractData 1), M.IsZCL","labels":[],"detail_key":"p8","name":"MPOTensor.TransferRetractData.isZCL","module":"TNLean.MPS.MPDO.FusionIsometries"},{"id":"n11128","layer":"formal","project":"p8","title":"MPOTensor.transferRetractData_of_isZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsZCL → ∀ n : Nat, LT.lt 0 n → Nonempty (M.TransferRetractData…","labels":[],"detail_key":"p8","name":"MPOTensor.transferRetractData_of_isZCL","module":"TNLean.MPS.MPDO.FusionIsometries"},{"id":"n11129","layer":"formal","project":"p8","title":"MPOTensor.transferRetractData_one_iff_isZCL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Iff (Nonempty (M.TransferRetractData 1)) M.IsZCL","labels":[],"detail_key":"p8","name":"MPOTensor.transferRetractData_one_iff_isZCL","module":"TNLean.MPS.MPDO.FusionIsometries"},{"id":"n11130","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.M_not_isGSNNCH","kind":"theorem","summary":"Not MPOTensor.CPSVExample412Literal.M.IsGSNNCH","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.M_not_isGSNNCH","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.ExampleFourCycleObstruction"},{"id":"n11131","layer":"formal","project":"p8","title":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_eq_generic","kind":"theorem","summary":"Eq MPOTensor.CPSVExample412Literal.fourCycleTripartiteState (MPOTensor.fourCycleTripartiteState…","labels":[],"detail_key":"p8","name":"MPOTensor.CPSVExample412Literal.fourCycleTripartiteState_eq_generic","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.ExampleFourCycleObstruction"},{"id":"n11132","layer":"formal","project":"p8","title":"MPOTensor.fourCycleTripartiteEquiv","kind":"def","summary":"(d : Nat) → Equiv (Prod (Fin d) (Prod (Fin (HMul.hMul d d)) (Fin d))) (Fin 4 → Fin d)","labels":[],"detail_key":"p8","name":"MPOTensor.fourCycleTripartiteEquiv","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.FourCycle"},{"id":"n11133","layer":"formal","project":"p8","title":"MPOTensor.fourCycleTripartiteState","kind":"def","summary":"d : Nat → MPOTensor.ChainOperator d 4 → Matrix (Prod (Fin d) (Prod (Fin (HMul.hMul d d)) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.fourCycleTripartiteState","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.FourCycle"},{"id":"n11134","layer":"formal","project":"p8","title":"MPOTensor.fourCycleTripartiteState_posSemidef","kind":"theorem","summary":"∀ d : Nat rho : MPOTensor.ChainOperator d 4, Matrix.PosSemidef rho → (MPOTensor.fourCycleTripar…","labels":[],"detail_key":"p8","name":"MPOTensor.fourCycleTripartiteState_posSemidef","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.FourCycle"},{"id":"n11135","layer":"formal","project":"p8","title":"MPOTensor.isSSAEquality_fourCycle_of_isGSNNCHAt","kind":"theorem","summary":"∀ d : Nat rho : MPOTensor.ChainOperator d 4 (hrho : MPOTensor.IsGSNNCHAt rho), IsSSAEquality (M…","labels":[],"detail_key":"p8","name":"MPOTensor.isSSAEquality_fourCycle_of_isGSNNCHAt","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.FourCycle"},{"id":"n11136","layer":"formal","project":"p8","title":"MPOTensor.nonempty_quantumMarkovDecomposition_fourCycle_of_isGSNNCHAt","kind":"theorem","summary":"∀ d : Nat rho : MPOTensor.ChainOperator d 4, MPOTensor.IsGSNNCHAt rho → Nonempty (Entropy.Quant…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_quantumMarkovDecomposition_fourCycle_of_isGSNNCHAt","module":"TNLean.MPS.MPDO.GSNNCHFourCycleMarkov.FourCycle"},{"id":"n11137","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily","kind":"inductive","summary":"d g : Nat → dim : Fin g → Nat → ((s : Fin g) → MPOTensor d (dim s)) → Type","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11138","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → MPOTensor.OrthogonalCom…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11139","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData_sectorProduct","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) (F : MPOTensor.OrthogonalCo…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData_sectorProduct","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11140","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData_unnormalizedState","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) (F : MPOTensor.OrthogonalCo…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toGSNNCHData_unnormalizedState","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11141","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.toProportional","kind":"def","summary":"d g : Nat → dim : Fin g → Nat → K : (s : Fin g) → MPOTensor d (dim s) → MPOTensor.OrthogonalCom…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toProportional","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11142","layer":"formal","project":"p8","title":"MPOTensor.ProportionalOrthogonalCommutingSectorFamily","kind":"inductive","summary":"d g : Nat → dim : Fin g → Nat → ((s : Fin g) → MPOTensor d (dim s)) → Type","labels":[],"detail_key":"p8","name":"MPOTensor.ProportionalOrthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11143","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHForm_of_commonWeightAbsorbedBasisMPOTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHForm_of_commonWeightAbsorbedBasisMPOTensor","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11144","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHForm_of_orthogonalCommutingSectorFamily","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat D : Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s…","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHForm_of_orthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.GSNNCHOrthogonalSectors"},{"id":"n11145","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.coefficientRoot","kind":"def","summary":"Nat → Nat → Real → 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MPOTensor.OrthogonalCo…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toCoefficientRescaledGSNNCHData_sectorProduct","module":"TNLean.MPS.MPDO.GSNNCHSectorRescaling"},{"id":"n11148","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.toCoefficientRescaledGSNNCHData_unnormalizedSta…","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) (F : MPOTensor.OrthogonalCo…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.toCoefficientRescaledGSNNCHData_unnormalizedState","module":"TNLean.MPS.MPDO.GSNNCHSectorRescaling"},{"id":"n11149","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHForm_of_nonnegative_orthogonalCommutingSectorFamily","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat D : Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s…","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHForm_of_nonnegative_orthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.GSNNCHSectorRescaling"},{"id":"n11150","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.bondAt_comm","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N) (x : Fin data.sectorCount) (i j : Fin N), Eq (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.bondAt_comm","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11151","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.bondAt_posSemidef","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N) (x : Fin data.sectorCount) (i : Fin N), Matrix.Po…","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.bondAt_posSemidef","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11152","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.sectorProduct_posSemidef","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N) (x : Fin data.sectorCount), Matrix.PosSemidef (da…","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.sectorProduct_posSemidef","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11153","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.sectorProduct_submatrix_rotateConfig","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N) (x : Fin data.sectorCount), Eq (Matrix.submatrix…","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.sectorProduct_submatrix_rotateConfig","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11154","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.unnormalizedState_posSemidef","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N), Matrix.PosSemidef data.unnormalizedState","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.unnormalizedState_posSemidef","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11155","layer":"formal","project":"p8","title":"MPOTensor.GSNNCHData.unnormalizedState_submatrix_rotateConfig","kind":"theorem","summary":"∀ d N : Nat (data : MPOTensor.GSNNCHData d N), Eq (Matrix.submatrix data.unnormalizedState ⇑(MP…","labels":[],"detail_key":"p8","name":"MPOTensor.GSNNCHData.unnormalizedState_submatrix_rotateConfig","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11156","layer":"formal","project":"p8","title":"MPOTensor.HasGSNNCHFormAt.posSemidef","kind":"theorem","summary":"∀ d N : Nat ρ : MPOTensor.ChainOperator d N, MPOTensor.HasGSNNCHFormAt ρ → Matrix.PosSemidef ρ","labels":[],"detail_key":"p8","name":"MPOTensor.HasGSNNCHFormAt.posSemidef","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11157","layer":"formal","project":"p8","title":"MPOTensor.HasGSNNCHFormAt.submatrix_rotateConfig","kind":"theorem","summary":"∀ d N : Nat ρ : MPOTensor.ChainOperator d N, MPOTensor.HasGSNNCHFormAt ρ → Eq (Matrix.submatrix…","labels":[],"detail_key":"p8","name":"MPOTensor.HasGSNNCHFormAt.submatrix_rotateConfig","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11158","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_posSemidef","kind":"theorem","summary":"∀ d : Nat (L : Nat) N : Nat (hLN : LE.le L N) (i : Fin N) B : Matrix (Fin L → Fin d) (Fin L → F…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_posSemidef","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11159","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_submatrix_rotateConfig","kind":"theorem","summary":"∀ d : Nat (L : Nat) N : Nat (hLN : LE.le L N) (i : Fin N) (B : Matrix (Fin L → Fin d) (Fin L →…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_submatrix_rotateConfig","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11160","layer":"formal","project":"p8","title":"MPOTensor.hasGSNNCHFormAt_normalizedMPO","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D N : Nat, MPOTensor.HasGSNNCHFormAt (M.mpo N) → Ne (M.mpo N).trace…","labels":[],"detail_key":"p8","name":"MPOTensor.hasGSNNCHFormAt_normalizedMPO","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11161","layer":"formal","project":"p8","title":"MPOTensor.isGSNNCH_of_hasGSNNCHForm","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.HasGSNNCHForm → (∀ (N : Nat), LE.le 2 N → Ne (M.mpo N).trace 0…","labels":[],"detail_key":"p8","name":"MPOTensor.isGSNNCH_of_hasGSNNCHForm","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11162","layer":"formal","project":"p8","title":"MPOTensor.isGSNNCH_of_isInjective_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → K.IsGSNNCH","labels":[],"detail_key":"p8","name":"MPOTensor.isGSNNCH_of_isInjective_of_isSAL","module":"TNLean.MPS.MPDO.GSNNCHSectorSum"},{"id":"n11163","layer":"formal","project":"p8","title":"MPOTensor.HasDefectSupport","kind":"def","summary":"(d : Nat) → (G : Type u_1) → N : Nat → LE.le 2 N → TNLean.Algebra.DefectMaps G (Fin 2 → Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.HasDefectSupport","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11164","layer":"formal","project":"p8","title":"MPOTensor.HasPrescribedDefectCovariance","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → N : Nat → LE.le 2 N → TNLean.Algebra.DefectMaps G (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.HasPrescribedDefectCovariance","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11165","layer":"formal","project":"p8","title":"MPOTensor.chainLabelAction","kind":"def","summary":"G : Type u_1 → [Group G] → N : Nat → LE.le 2 N → Fin N → G → (Fin N → G) → Fin N → G","labels":[],"detail_key":"p8","name":"MPOTensor.chainLabelAction","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11166","layer":"formal","project":"p8","title":"MPOTensor.chainLabelActionEquiv","kind":"def","summary":"G : Type u_1 → [Group G] → N : Nat → LE.le 2 N → Fin N → G → Equiv (Fin N → G) (Fin N → G)","labels":[],"detail_key":"p8","name":"MPOTensor.chainLabelActionEquiv","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11167","layer":"formal","project":"p8","title":"MPOTensor.chainLabelActionEquiv_apply","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] N : Nat (hN : LE.le 2 N) (j : Fin N) (g : G) (α : Fin N → G), E…","labels":[],"detail_key":"p8","name":"MPOTensor.chainLabelActionEquiv_apply","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11168","layer":"formal","project":"p8","title":"MPOTensor.chainLabelAction_chainLabelAction","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] N : Nat (hN : LE.le 2 N) (j : Fin N) (g h : G) (α : Fin N → G),…","labels":[],"detail_key":"p8","name":"MPOTensor.chainLabelAction_chainLabelAction","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11169","layer":"formal","project":"p8","title":"MPOTensor.chainLabelAction_one","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] N : Nat (hN : LE.le 2 N) (j : Fin N) (α : Fin N → G), Eq (MPOTe…","labels":[],"detail_key":"p8","name":"MPOTensor.chainLabelAction_one","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11170","layer":"formal","project":"p8","title":"MPOTensor.embedLocalOperator_mulVec_replaceWindow","kind":"theorem","summary":"∀ D' : Nat (L M : Nat) (hLM : LE.le L M) (i : Fin M) (B : Matrix (Fin L → Fin D') (Fin L → Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.embedLocalOperator_mulVec_replaceWindow","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11171","layer":"formal","project":"p8","title":"MPOTensor.extractWindow_chainLabelAction_one","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] N : Nat (hN : LE.le 2 N) (j : Fin N) (g : G) (α : Fin N → G), E…","labels":[],"detail_key":"p8","name":"MPOTensor.extractWindow_chainLabelAction_one","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11172","layer":"formal","project":"p8","title":"MPOTensor.extractWindow_chainLabelAction_zero","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] N : Nat (hN : LE.le 2 N) (j : Fin N) (g : G) (α : Fin N → G), E…","labels":[],"detail_key":"p8","name":"MPOTensor.extractWindow_chainLabelAction_zero","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11173","layer":"formal","project":"p8","title":"MPOTensor.gaugeInvariantSubspace","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → [inst : Fintype G] → [DecidableEq G] → (N : Nat) → LE.…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugeInvariantSubspace","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11174","layer":"formal","project":"p8","title":"MPOTensor.gaugedLabel","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [inst : Fintype G] → ι : Type u_2 → (ι → Fin (Fintype.card (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugedLabel","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11175","layer":"formal","project":"p8","title":"MPOTensor.gaugedMatter","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [inst : Fintype G] → ι : Type u_2 → (ι → Fin (Fintype.card (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugedMatter","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11176","layer":"formal","project":"p8","title":"MPOTensor.gaugedVector","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [inst : Fintype G] → N : Nat → ((Fin N → G) → (Fin N → Fin d) → Co…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugedVector","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11177","layer":"formal","project":"p8","title":"MPOTensor.gaugedVector_apply","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Fintype G] N : Nat (Ψ : (Fin N → G) → (Fin N → Fin d) → Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugedVector_apply","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11178","layer":"formal","project":"p8","title":"MPOTensor.gaugedVector_mem_gaugeInvariantSubspace","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] N : Nat (…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugedVector_mem_gaugeInvariantSubspace","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11179","layer":"formal","project":"p8","title":"MPOTensor.gaussWindowOperator","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → [inst : Fintype G] → [DecidableEq G] → (G → G → Subtyp…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussWindowOperator","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11180","layer":"formal","project":"p8","title":"MPOTensor.gaussWindowOperator_apply","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (R :…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussWindowOperator_apply","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11181","layer":"formal","project":"p8","title":"MPOTensor.one_le_finrank_gaugeInvariantSubspace","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] N : Nat (…","labels":[],"detail_key":"p8","name":"MPOTensor.one_le_finrank_gaugeInvariantSubspace","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11182","layer":"formal","project":"p8","title":"MPOTensor.placedGaussOperator","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → [inst : Fintype G] → [DecidableEq G] → (N : Nat) → LE.…","labels":[],"detail_key":"p8","name":"MPOTensor.placedGaussOperator","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11183","layer":"formal","project":"p8","title":"MPOTensor.placedGaussOperator_mulVec_gaugedVector","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] N : Nat (…","labels":[],"detail_key":"p8","name":"MPOTensor.placedGaussOperator_mulVec_gaugedVector","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11184","layer":"formal","project":"p8","title":"MPOTensor.placedGaussProjector_eq_average","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] N : Nat (…","labels":[],"detail_key":"p8","name":"MPOTensor.placedGaussProjector_eq_average","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11185","layer":"formal","project":"p8","title":"MPOTensor.placedGaussProjector_mulVec_gaugedVector","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] N : Nat (…","labels":[],"detail_key":"p8","name":"MPOTensor.placedGaussProjector_mulVec_gaugedVector","module":"TNLean.MPS.MPDO.GaugeInvariantSubspace"},{"id":"n11186","layer":"formal","project":"p8","title":"MPOTensor.gaussLocalCoordinateEquiv","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [inst : Fintype G] → Equiv (Prod (Fin 2 → Fin d) (Prod G G)) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussLocalCoordinateEquiv","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11187","layer":"formal","project":"p8","title":"MPOTensor.gaussLocalCoordinateEquiv_apply_one","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Fintype G] (x : Prod (Fin 2 → Fin d) (Prod G G)), Eq ((MPOTe…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussLocalCoordinateEquiv_apply_one","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11188","layer":"formal","project":"p8","title":"MPOTensor.gaussLocalCoordinateEquiv_apply_zero","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Fintype G] (x : Prod (Fin 2 → Fin d) (Prod G G)), Eq ((MPOTe…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussLocalCoordinateEquiv_apply_zero","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11189","layer":"formal","project":"p8","title":"MPOTensor.gaussWindowProjector","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → [inst : Fintype G] → [DecidableEq G] → (G → G → Subtyp…","labels":[],"detail_key":"p8","name":"MPOTensor.gaussWindowProjector","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11190","layer":"formal","project":"p8","title":"MPOTensor.isStarProjection_gaussWindowProjector","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (R :…","labels":[],"detail_key":"p8","name":"MPOTensor.isStarProjection_gaussWindowProjector","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11191","layer":"formal","project":"p8","title":"MPOTensor.isStarProjection_placedGaussProjector","kind":"theorem","summary":"∀ (d : Nat) (G : Type u_1) [inst : Group G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (N :…","labels":[],"detail_key":"p8","name":"MPOTensor.isStarProjection_placedGaussProjector","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11192","layer":"formal","project":"p8","title":"MPOTensor.placedGaussProjector","kind":"def","summary":"(d : Nat) → (G : Type u_1) → [Group G] → [inst : Fintype G] → [DecidableEq G] → (N : Nat) → LE.…","labels":[],"detail_key":"p8","name":"MPOTensor.placedGaussProjector","module":"TNLean.MPS.MPDO.GaussProjectorPlacement"},{"id":"n11193","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.grouped_sector_gram_conj_eq","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.grouped_sector_gram_conj_eq","module":"TNLean.MPS.MPDO.GroupedFigure8"},{"id":"n11194","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.grouped_sector_exists_unitary_normalization","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.grouped_sector_exists_unitary_normalization","module":"TNLean.MPS.MPDO.GroupedGramNormalization"},{"id":"n11195","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.grouped_sector_gram_eq_pos_smul_one","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.grouped_sector_gram_eq_pos_smul_one","module":"TNLean.MPS.MPDO.GroupedGramNormalization"},{"id":"n11196","layer":"formal","project":"p8","title":"MPOTensor.exists_distinguished_grouped_reference_corner","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_distinguished_grouped_reference_corner","module":"TNLean.MPS.MPDO.GroupedReferenceCorner"},{"id":"n11197","layer":"formal","project":"p8","title":"MPOTensor.HasGroupedCornerGramDressing","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasGroupedCornerGramDressing","module":"TNLean.MPS.MPDO.GroupedSectorGram"},{"id":"n11198","layer":"formal","project":"p8","title":"MPOTensor.grouped_sector_gram_conj_eq_of_dressing","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.grouped_sector_gram_conj_eq_of_dressing","module":"TNLean.MPS.MPDO.GroupedSectorGram"},{"id":"n11199","layer":"formal","project":"p8","title":"MPOTensor.grouped_sector_gram_eq_pos_smul_one_of_dressing","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.grouped_sector_gram_eq_pos_smul_one_of_dressing","module":"TNLean.MPS.MPDO.GroupedSectorGram"},{"id":"n11200","layer":"formal","project":"p8","title":"MPOTensor.IsThreeSiteClosure","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex → Matrix (Prod (Fin d) (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.IsThreeSiteClosure","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11201","layer":"formal","project":"p8","title":"MPOTensor.hayashiInverseLeft","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.hayashiInverseLeft","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11202","layer":"formal","project":"p8","title":"MPOTensor.hayashiInverseRight","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.hayashiInverseRight","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11203","layer":"formal","project":"p8","title":"MPOTensor.inverseMap_conj_physicalSlice_expansion","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (hK : K.IsInjective) (R : Matrix (Fin D) (Fin D) Complex) (ρ :…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMap_conj_physicalSlice_expansion","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11204","layer":"formal","project":"p8","title":"MPOTensor.inverseMap_hayashi_sector_comparison","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (hK : K.IsInjective) (R : Matrix (Fin D) (Fin D) Complex) (ρ :…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMap_hayashi_sector_comparison","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11205","layer":"formal","project":"p8","title":"MPOTensor.inverseMap_threeSite_closure_collapse","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (hK : K.IsInjective) (R : Matrix (Fin D) (Fin D) Complex) (ρ :…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMap_threeSite_closure_collapse","module":"TNLean.MPS.MPDO.HayashiSectorComparison"},{"id":"n11206","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure.coordinateSectorProjection","kind":"def","summary":"d : Nat → rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure.coordinateSectorProjection","module":"TNLean.MPS.MPDO.HayashiSectorProjector"},{"id":"n11207","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure.sectorProjection","kind":"def","summary":"d : Nat → rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure.sectorProjection","module":"TNLean.MPS.MPDO.HayashiSectorProjector"},{"id":"n11208","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure.sectorProjection_isOrthogonal","kind":"theorem","summary":"∀ d : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure.sectorProjection_isOrthogonal","module":"TNLean.MPS.MPDO.HayashiSectorProjector"},{"id":"n11209","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure.sectorProjection_mul_eq_zero","kind":"theorem","summary":"∀ d : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure.sectorProjection_mul_eq_zero","module":"TNLean.MPS.MPDO.HayashiSectorProjector"},{"id":"n11210","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure.sum_sectorProjection","kind":"theorem","summary":"∀ d : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure.sum_sectorProjection","module":"TNLean.MPS.MPDO.HayashiSectorProjector"},{"id":"n11211","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11212","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.braRight_eq_ketLeftBraRight_of_invariant","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → ∀ Q : Matrix (Fin d) (Fin d) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.braRight_eq_ketLeftBraRight_of_invariant","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11213","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_not_commute_of_displaced","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → ∀ Q : Matrix (Fin d) (Fin d) Complex, IsIde…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_not_commute_of_displaced","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11214","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_representative_braRight_eq_ketLeftBraRight","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → ∀ Q : Matrix (Fin d) (Fin d) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_representative_braRight_eq_ketLeftBraRight","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11215","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.hasHorizontalCFMPVRepresentation","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsHorizontalCF → M.HasHorizontalCFMPVRepresentation","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.hasHorizontalCFMPVRepresentation","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11216","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.insertedTensor_eq_of_firstSiteActionAgree","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → ∀ Y Z : Matrix (Fin (HMul.hMul d d)) (Fin (…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.insertedTensor_eq_of_firstSiteActionAgree","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11217","layer":"formal","project":"p8","title":"MPOTensor.insertedTensor_braRightAction_toMPSTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (Q : Matrix (Fin d) (Fin d) Complex), Eq (MPSTensor.insertedTen…","labels":[],"detail_key":"p8","name":"MPOTensor.insertedTensor_braRightAction_toMPSTensor","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11218","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.insertedTensor_toTensor_eq_of_basis","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition d) (Y Z : Matrix (Fin d) (Fin d) Complex), (∀ (j :…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.insertedTensor_toTensor_eq_of_basis","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11219","layer":"formal","project":"p8","title":"MPSTensor.insertedTensor_eq_of_gauge","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D (X : Matrix.GeneralLinearGroup (Fin D) Complex), (∀ (i : Fin d)…","labels":[],"detail_key":"p8","name":"MPSTensor.insertedTensor_eq_of_gauge","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11220","layer":"formal","project":"p8","title":"MPSTensor.insertedTensor_toTensorFromBlocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (Y : Matrix (Fin d) (Fin d) Complex) (μ : Fin r → Complex) (A : (…","labels":[],"detail_key":"p8","name":"MPSTensor.insertedTensor_toTensorFromBlocks","module":"TNLean.MPS.MPDO.HorizontalBNT"},{"id":"n11221","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.blockTwo","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsHorizontalCF → M.blockTwo.IsHorizontalCF","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.blockTwo","module":"TNLean.MPS.MPDO.HorizontalBlocking"},{"id":"n11222","layer":"formal","project":"p8","title":"MPOTensor.HasHorizontalCFMPVRepresentation","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasHorizontalCFMPVRepresentation","module":"TNLean.MPS.MPDO.HorizontalCFMPVRepresentation"},{"id":"n11223","layer":"formal","project":"p8","title":"MPOTensor.basis_opposite_insert_eq_of_rotated_mpo_entries","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), MPSTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.basis_opposite_insert_eq_of_rotated_mpo_entries","module":"TNLean.MPS.MPDO.HorizontalCFMPVRepresentation"},{"id":"n11224","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_two_firstSiteActionAgree","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ Yleft Ycorner…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_two_firstSiteActionAgree","module":"TNLean.MPS.MPDO.HorizontalCFMPVRepresentation"},{"id":"n11225","layer":"formal","project":"p8","title":"MPOTensor.basis_braRight_eq_ketLeftBraRight_of_invariant","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ (S : MPSTensor.SectorDecomposition (HMul.hMul d d…","labels":[],"detail_key":"p8","name":"MPOTensor.basis_braRight_eq_ketLeftBraRight_of_invariant","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11226","layer":"formal","project":"p8","title":"MPOTensor.firstSiteActionAgree_braRight_ketLeftBraRight_of_invariant","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ P : Matrix (Fin d) (Fin d) Complex, P.IsHermitian…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteActionAgree_braRight_ketLeftBraRight_of_invariant","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11227","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_mul_mpo_comm","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ P : Matrix (Fin d) (Fin d) Complex, P.IsHermitian…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_mul_mpo_comm","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11228","layer":"formal","project":"p8","title":"MPOTensor.firstSiteMatrix_mul_mpo_of_ketLeftMul_invariant","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), Eq (M.ketLeftMul P) ((M.k…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSiteMatrix_mul_mpo_of_ketLeftMul_invariant","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11229","layer":"formal","project":"p8","title":"MPOTensor.insertedTensor_ketLeftAction_toMPSTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (Q : Matrix (Fin d) (Fin d) Complex), Eq (MPSTensor.insertedTen…","labels":[],"detail_key":"p8","name":"MPOTensor.insertedTensor_ketLeftAction_toMPSTensor","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11230","layer":"formal","project":"p8","title":"MPOTensor.insertedTensor_ketLeftBraRightAction_toMPSTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (Q : Matrix (Fin d) (Fin d) Complex), Eq (MPSTensor.insertedTen…","labels":[],"detail_key":"p8","name":"MPOTensor.insertedTensor_ketLeftBraRightAction_toMPSTensor","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11231","layer":"formal","project":"p8","title":"MPOTensor.ketLeftMul_eq_braRightMul_of_commute_of_isInjective","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Kraus.IsInjective M.toMPSTensor → ∀ Q : Matrix (Fin d) (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.ketLeftMul_eq_braRightMul_of_commute_of_isInjective","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11232","layer":"formal","project":"p8","title":"MPOTensor.mpo_commute_of_commute_pow","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ (N : Nat), LT.lt 0 N → ∀ p : Nat, Ne p 0 → ∀ Q :…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_commute_of_commute_pow","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11233","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_braRightMul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (v : Fin (HMul.hMul D D)),…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_braRightMul","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11234","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_ketLeftMul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (v : Fin (HMul.hMul D D)),…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_ketLeftMul","module":"TNLean.MPS.MPDO.InvariantProjection"},{"id":"n11235","layer":"formal","project":"p8","title":"MPOTensor.rephase_zeroWeightReparameterized_neighboringOperator_eq_zero_of_incident","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.rephase_zeroWeightReparameterized_neighboringOperator_eq_zero_of_incident","module":"TNLean.MPS.MPDO.InverseMapActiveSectorPrimitivity"},{"id":"n11236","layer":"formal","project":"p8","title":"MPOTensor.zeroWeightReparameterized_sectorVirtualMatrix_eq_zero","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.zeroWeightReparameterized_sectorVirtualMatrix_eq_zero","module":"TNLean.MPS.MPDO.InverseMapActiveSectorPrimitivity"},{"id":"n11237","layer":"formal","project":"p8","title":"MPOTensor.conjugated_middle_threeSiteClosure","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.conjugated_middle_threeSiteClosure","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11238","layer":"formal","project":"p8","title":"MPOTensor.exists_active_sectorVirtualMatrix_ne_zero","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_active_sectorVirtualMatrix_ne_zero","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11239","layer":"formal","project":"p8","title":"MPOTensor.exists_positive_inverseMapPhysicalSectorFactorization","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_positive_inverseMapPhysicalSectorFactorization","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11240","layer":"formal","project":"p8","title":"MPOTensor.exists_positive_physicalSectorFactorization_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Exists fun F => ∀ (k h : Fin F.secto…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_positive_physicalSectorFactorization_of_isSAL","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11241","layer":"formal","project":"p8","title":"MPOTensor.exists_rephase_zeroWeightInverseMap_posSemidef","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_rephase_zeroWeightInverseMap_posSemidef","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11242","layer":"formal","project":"p8","title":"MPOTensor.nonempty_etaLocalStructureData_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Nonempty K.EtaLocalStructureData","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_etaLocalStructureData_of_isSAL","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11243","layer":"formal","project":"p8","title":"MPOTensor.probability_ne_zero_of_reparameterized_neighboringOperator_ne_zero","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.probability_ne_zero_of_reparameterized_neighboringOperator_ne_zero","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11244","layer":"formal","project":"p8","title":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization_isRecurrentSuppo…","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization_isRecurrentSupport","module":"TNLean.MPS.MPDO.InverseMapActiveSectorRecurrence"},{"id":"n11245","layer":"formal","project":"p8","title":"MPOTensor.exists_activeSectorTraceMatrix_rank_one_coefficients_of_isSAL_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Eq (HMul.hMul K.physTraceTransfer K.…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_activeSectorTraceMatrix_rank_one_coefficients_of_isSAL_of_literal_ZCL","module":"TNLean.MPS.MPDO.InverseMapLemmaC5CaseI"},{"id":"n11246","layer":"formal","project":"p8","title":"MPOTensor.exists_neighboringOperator_trace_rank_one_coefficients_of_isSAL_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Eq (HMul.hMul K.physTraceTransfer K.…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_neighboringOperator_trace_rank_one_coefficients_of_isSAL_of_literal_ZCL","module":"TNLean.MPS.MPDO.InverseMapLemmaC5CaseI"},{"id":"n11247","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSectorFactorization_rank_one_coefficients_of_isSAL_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Eq (HMul.hMul K.physTraceTransfer K.…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSectorFactorization_rank_one_coefficients_of_isSAL_of_literal_ZCL","module":"TNLean.MPS.MPDO.InverseMapLemmaC5CaseI"},{"id":"n11248","layer":"formal","project":"p8","title":"MPOTensor.exists_rephased_inverseMap_caseI_rank_one_coefficients_witnesses","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Eq (HMul.hMul K.physTraceTransfer K.…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_rephased_inverseMap_caseI_rank_one_coefficients_witnesses","module":"TNLean.MPS.MPDO.InverseMapLemmaC5CaseI"},{"id":"n11249","layer":"formal","project":"p8","title":"MPOTensor.inverseMapPhysicalSectorFactorization","kind":"def","summary":"d D : Nat → rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMapPhysicalSectorFactorization","module":"TNLean.MPS.MPDO.InverseMapPhysicalSectorFactorization"},{"id":"n11250","layer":"formal","project":"p8","title":"MPOTensor.inverseMapPhysicalSectorFactorization_neighboringOperator_eq_sectorEta","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMapPhysicalSectorFactorization_neighboringOperator_eq_sectorEta","module":"TNLean.MPS.MPDO.InverseMapPhysicalSectorFactorization"},{"id":"n11251","layer":"formal","project":"p8","title":"MPOTensor.nonempty_physicalSectorFactorization_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Nonempty K.PhysicalSectorFactorizati…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_physicalSectorFactorization_of_isSAL","module":"TNLean.MPS.MPDO.InverseMapPhysicalSectorFactorization"},{"id":"n11252","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.mpo_tensor_eq_diagonal","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.KatoDeformedRFPObstruction.tensor.mpo N) (Matrix.diagonal fun σ => H…","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.mpo_tensor_eq_diagonal","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11253","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.obstructionBoundary","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.obstructionBoundary","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11254","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.pauliZ","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.pauliZ","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11255","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.physClose1_tensor","kind":"theorem","summary":"∀ (X : Matrix (Fin 2) (Fin 2) Complex), Eq (MPOTensor.KatoDeformedRFPObstruction.tensor.physClo…","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.physClose1_tensor","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11256","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.physClose2_tensor","kind":"theorem","summary":"∀ (X : Matrix (Fin 2) (Fin 2) Complex), Eq (MPOTensor.KatoDeformedRFPObstruction.tensor.physClo…","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.physClose2_tensor","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11257","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.physTraceTransfer_tensor","kind":"theorem","summary":"Eq MPOTensor.KatoDeformedRFPObstruction.tensor.physTraceTransfer (Matrix.of (Matrix.vecCons (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.physTraceTransfer_tensor","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11258","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.tensor","kind":"def","summary":"MPOTensor 2 2","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.tensor","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11259","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.tensor_isMPDO","kind":"theorem","summary":"MPOTensor.KatoDeformedRFPObstruction.tensor.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.tensor_isMPDO","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11260","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.tensor_isSAL","kind":"theorem","summary":"MPOTensor.KatoDeformedRFPObstruction.tensor.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.tensor_isSAL","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11261","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.tensor_not_isRFPViaTS","kind":"theorem","summary":"Not MPOTensor.KatoDeformedRFPObstruction.tensor.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.tensor_not_isRFPViaTS","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11262","layer":"formal","project":"p8","title":"MPOTensor.KatoDeformedRFPObstruction.trace_mpo_tensor","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.KatoDeformedRFPObstruction.tensor.mpo N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.KatoDeformedRFPObstruction.trace_mpo_tensor","module":"TNLean.MPS.MPDO.KatoDeformedRFPObstruction"},{"id":"n11263","layer":"formal","project":"p8","title":"MPOTensor.IsLPDO","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsLPDO","module":"TNLean.MPS.MPDO.LPDO"},{"id":"n11264","layer":"formal","project":"p8","title":"MPOTensor.IsLPDO.isMPDO","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsLPDO → M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsLPDO.isMPDO","module":"TNLean.MPS.MPDO.LPDO"},{"id":"n11265","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceMatrix_pow_two_eq_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) [NeZero D] (F : K.PhysicalSectorFactorization) (p : Fin F.secto…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceMatrix_pow_two_eq_of_literal_ZCL","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11266","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceMatrix_pow_two_pos","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceMatrix_pow_two_pos","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11267","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceMatrix_rank_one_coefficients_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) [NeZero D] (F : K.PhysicalSectorFactorization) (p : Fin F.secto…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceMatrix_rank_one_coefficients_of_literal_ZCL","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11268","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceSqMatrix","kind":"def","summary":"d D : Nat → (K : MPOTensor d D) → (F : K.PhysicalSectorFactorization) → (p : Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceSqMatrix","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11269","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceSqMatrix_le_activeSectorTraceMatrix_sq","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceSqMatrix_le_activeSectorTraceMatrix_sq","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11270","layer":"formal","project":"p8","title":"MPOTensor.activeSectorTraceSqMatrix_nonneg","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.activeSectorTraceSqMatrix_nonneg","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11271","layer":"formal","project":"p8","title":"MPOTensor.card_activeSector_eq_one_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) [NeZero D] (F : K.PhysicalSectorFactorization) (p : Fin F.secto…","labels":[],"detail_key":"p8","name":"MPOTensor.card_activeSector_eq_one_of_literal_ZCL","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11272","layer":"formal","project":"p8","title":"MPOTensor.caseI_rectangular_remainder_eq_zero_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) [NeZero D] (F : K.PhysicalSectorFactorization) (p : Fin F.secto…","labels":[],"detail_key":"p8","name":"MPOTensor.caseI_rectangular_remainder_eq_zero_of_literal_ZCL","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11273","layer":"formal","project":"p8","title":"MPOTensor.mpvOverlap_toMPSTensor_self_eq_ofReal_trace_activeSectorTraceSqMatrix_pow","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.mpvOverlap_toMPSTensor_self_eq_ofReal_trace_activeSectorTraceSqMatrix_pow","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11274","layer":"formal","project":"p8","title":"MPOTensor.neighboringOperator_eq_zero_of_inactive_right","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.neighboringOperator_eq_zero_of_inactive_right","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11275","layer":"formal","project":"p8","title":"MPOTensor.prod_traceSq_eq_zero_of_not_forall_active","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.prod_traceSq_eq_zero_of_not_forall_active","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11276","layer":"formal","project":"p8","title":"MPOTensor.sitewise_prod_conjTranspose_mul_self","kind":"theorem","summary":"∀ d N : Nat (U : Matrix (Fin d) (Fin d) Complex), Eq (HMul.hMul U.conjTranspose U) 1 → Eq (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewise_prod_conjTranspose_mul_self","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11277","layer":"formal","project":"p8","title":"MPOTensor.sum_prod_traceSq_eq_sum_active","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Re…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_prod_traceSq_eq_sum_active","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11278","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_conjugatePhysical_mul_self","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (U : Matrix (Fin d) (Fin d) Complex), Eq (HMul.hMul U.conjTrans…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_conjugatePhysical_mul_self","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11279","layer":"formal","project":"p8","title":"MPOTensor.trace_pow_similarity_squared_diagonal","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (S : Matrix n n Real) (v : n → Real)…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_pow_similarity_squared_diagonal","module":"TNLean.MPS.MPDO.LemmaC5CaseI"},{"id":"n11280","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.LengthIndependent","kind":"def","summary":"Λ : Type u_1 → MPOTensor.BNTLabelCoefficientFamily Λ → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.LengthIndependent","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11281","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelCoefficientFamily.LengthIndependent.coeff_eq","kind":"theorem","summary":"∀ Λ : Type u_1 c : MPOTensor.BNTLabelCoefficientFamily Λ, c.LengthIndependent → ∀ L L' : Nat, L…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelCoefficientFamily.LengthIndependent.coeff_eq","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11282","layer":"formal","project":"p8","title":"MPOTensor.DiagonalChiFamily.matrix_eq_one_of_forall_entry_eq_one","kind":"theorem","summary":"∀ I : Type u_1 χ : MPOTensor.DiagonalChiFamily I, (∀ (α β γ : I) (k : Fin (χ.dim α β γ)), Eq (χ…","labels":[],"detail_key":"p8","name":"MPOTensor.DiagonalChiFamily.matrix_eq_one_of_forall_entry_eq_one","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11283","layer":"formal","project":"p8","title":"MPOTensor.DiagonalChiFamily.tracePowerCoeff_eq_dim_of_forall_entry_eq_one","kind":"theorem","summary":"∀ I : Type u_1 χ : MPOTensor.DiagonalChiFamily I, (∀ (α β γ : I) (k : Fin (χ.dim α β γ)), Eq (χ…","labels":[],"detail_key":"p8","name":"MPOTensor.DiagonalChiFamily.tracePowerCoeff_eq_dim_of_forall_entry_eq_one","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11284","layer":"formal","project":"p8","title":"MPOTensor.PositiveBNTLabelChiTracePowerForm.coeff_eq_dim_of_lengthIndependent","kind":"theorem","summary":"∀ Λ : Type u_1 c : MPOTensor.BNTLabelCoefficientFamily Λ (h : MPOTensor.PositiveBNTLabelChiTrac…","labels":[],"detail_key":"p8","name":"MPOTensor.PositiveBNTLabelChiTracePowerForm.coeff_eq_dim_of_lengthIndependent","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11285","layer":"formal","project":"p8","title":"MPOTensor.PositiveBNTLabelChiTracePowerForm.entry_eq_one_of_lengthIndependent","kind":"theorem","summary":"∀ Λ : Type u_1 c : MPOTensor.BNTLabelCoefficientFamily Λ (h : MPOTensor.PositiveBNTLabelChiTrac…","labels":[],"detail_key":"p8","name":"MPOTensor.PositiveBNTLabelChiTracePowerForm.entry_eq_one_of_lengthIndependent","module":"TNLean.MPS.MPDO.LengthIndependentCoefficients"},{"id":"n11286","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.linearMarkedTensor_eq_of_trace_agree","kind":"theorem","summary":"∀ d e D : Nat (M : MPOTensor d D), M.IsHorizontalCF → ∀ (f g : Fin e → Fin (HMul.hMul d d) → Co…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.linearMarkedTensor_eq_of_trace_agree","module":"TNLean.MPS.MPDO.LinearMarkedTensor"},{"id":"n11287","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.linearMarkedTensor_toTensor_eq_markedTensor","kind":"theorem","summary":"∀ d e : Nat (S : MPSTensor.SectorDecomposition d) (f : Fin e → Fin d → Complex), Eq (MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.linearMarkedTensor_toTensor_eq_markedTensor","module":"TNLean.MPS.MPDO.LinearMarkedTensor"},{"id":"n11288","layer":"formal","project":"p8","title":"MPSTensor.linearMarkedTensor","kind":"def","summary":"d e D : Nat → (Fin e → Fin d → Complex) → MPSTensor d D → MPSTensor e D","labels":[],"detail_key":"p8","name":"MPSTensor.linearMarkedTensor","module":"TNLean.MPS.MPDO.LinearMarkedTensor"},{"id":"n11289","layer":"formal","project":"p8","title":"MPSTensor.linearMarkedTensor_gauge","kind":"theorem","summary":"∀ d e D : Nat A B : MPSTensor d D (f : Fin e → Fin d → Complex) (X : Matrix.GeneralLinearGroup…","labels":[],"detail_key":"p8","name":"MPSTensor.linearMarkedTensor_gauge","module":"TNLean.MPS.MPDO.LinearMarkedTensor"},{"id":"n11290","layer":"formal","project":"p8","title":"MPSTensor.linearMarkedTensor_toTensorFromBlocks","kind":"theorem","summary":"∀ d e r : Nat dim : Fin r → Nat (f : Fin e → Fin d → 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(mu…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_localOrthogonalSum","module":"TNLean.MPS.MPDO.LocalOrthogonalSumAreaLaw"},{"id":"n11293","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_proportionalLocalOrthogonalSum","kind":"theorem","summary":"∀ d D g : Nat dim : Fin g → Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s)) (mu…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_proportionalLocalOrthogonalSum","module":"TNLean.MPS.MPDO.LocalOrthogonalSumAreaLaw"},{"id":"n11294","layer":"formal","project":"p8","title":"MPOTensor.IsLPDO.exists_mutualInfoChain_le_purifyingBond","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (hLPDO : M.IsLPDO) (N L : Nat) (hN : LT.lt 0 N) (hL : LE.le L N),…","labels":[],"detail_key":"p8","name":"MPOTensor.IsLPDO.exists_mutualInfoChain_le_purifyingBond","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11295","layer":"formal","project":"p8","title":"MPOTensor.bipartitionedNormalizedMPO_eq_tensorMapBoth_blockAncillaryTraceMap","kind":"theorem","summary":"∀ d D dK D' : Nat (M : MPOTensor d D) (A : Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.bipartitionedNormalizedMPO_eq_tensorMapBoth_blockAncillaryTraceMap","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11296","layer":"formal","project":"p8","title":"MPOTensor.blockAncillaryTraceMap","kind":"def","summary":"(d dK L : Nat) → LinearMap (RingHom.id Complex) (Matrix (Fin (HPow.hPow (HMul.hMul d dK) L)) 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(finFunct…","labels":[],"detail_key":"p8","name":"MPOTensor.blockSpinAncillaEquiv_symm_apply","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11302","layer":"formal","project":"p8","title":"MPOTensor.blockSpinAncillaEquiv_symm_decode","kind":"theorem","summary":"∀ (d dK L : Nat) (i : Fin (HPow.hPow d L)) (k : Fin (HPow.hPow dK L)), Eq (finFunctionFinEquiv.…","labels":[],"detail_key":"p8","name":"MPOTensor.blockSpinAncillaEquiv_symm_decode","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11303","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain_le_of_bipartitioned_channel_image","kind":"theorem","summary":"∀ d dP D D' : Nat (M : MPOTensor d D) (A : MPSTensor dP D') (N L : Nat) (hL : LE.le L N) (S : L…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_le_of_bipartitioned_channel_image","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11304","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain_le_of_lpdoWitness","kind":"theorem","summary":"∀ d D dK D' : Nat (M : MPOTensor d D) (A : Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_le_of_lpdoWitness","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11305","layer":"formal","project":"p8","title":"MPOTensor.purifyingBondDimension_pos_of_trace_ne_zero","kind":"theorem","summary":"∀ dP DP N : Nat (A : MPSTensor dP DP), Ne (A.pureState N).trace 0 → LT.lt 0 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(MPOTensor.purifi…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_purificationDensity_eq_pureState","module":"TNLean.MPS.MPDO.LocalPurificationAreaLaw"},{"id":"n11308","layer":"formal","project":"p8","title":"MPOTensor.IsNondegeneratePRFP","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsNondegeneratePRFP","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11309","layer":"formal","project":"p8","title":"MPOTensor.IsNondegeneratePRFP.isSourceZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsNondegeneratePRFP → M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.IsNondegeneratePRFP.isSourceZCL","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11310","layer":"formal","project":"p8","title":"MPOTensor.IsPRFP.hasPurificationRFPWitness","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsPRFP → M.HasPurificationRFPWitness","labels":[],"detail_key":"p8","name":"MPOTensor.IsPRFP.hasPurificationRFPWitness","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11311","layer":"formal","project":"p8","title":"MPOTensor.IsPRFP.isLPDO","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsPRFP → M.IsLPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsPRFP.isLPDO","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11312","layer":"formal","project":"p8","title":"MPOTensor.IsPRFP.isMPDO","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsPRFP → M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsPRFP.isMPDO","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11313","layer":"formal","project":"p8","title":"MPOTensor.IsPRFP.isPhysicalTraceIdempotent","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsPRFP → 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M.physTraceTransfer…","labels":[],"detail_key":"p8","name":"MPOTensor.isPRFP_iff_isLPDO_and_physTraceTransfer_sq","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11317","layer":"formal","project":"p8","title":"MPOTensor.isSourceZCL_of_isPRFP","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsPRFP → Ne M.physTraceTransfer 0 → M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.isSourceZCL_of_isPRFP","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11318","layer":"formal","project":"p8","title":"MPOTensor.isSourceZCL_witnessM","kind":"theorem","summary":"MPOTensor.witnessM.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.isSourceZCL_witnessM","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11319","layer":"formal","project":"p8","title":"MPOTensor.mpo_eq_purificationDensity","kind":"theorem","summary":"∀ d D dK D' : Nat (A : Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) (e : Equiv (Fin D) (P…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_eq_purificationDensity","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11320","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_sq_of_isPRFP","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsPRFP → Eq (HMul.hMul M.physTraceTransfer M.physTraceTransf…","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_sq_of_isPRFP","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11321","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_witnessM","kind":"theorem","summary":"Eq MPOTensor.witnessM.physTraceTransfer 1","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_witnessM","module":"TNLean.MPS.MPDO.LocalPurificationRFP"},{"id":"n11322","layer":"formal","project":"p8","title":"MPOTensor.purificationTensor_isTransferIdempotent_iff_physTraceTransfer_sq","kind":"theorem","summary":"∀ d D dK D' : Nat (A : Fin d → Fin dK → Matrix (Fin D') (Fin D') 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(M…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_le_two_log_bondDim","module":"TNLean.MPS.MPDO.MutualInfoAreaLaw"},{"id":"n11325","layer":"formal","project":"p8","title":"MPOTensor.bipartitionedNormalizedMPO","kind":"def","summary":"d D : Nat → MPOTensor d D → (N L K : Nat) → Eq N (HAdd.hAdd L K) → Matrix (Prod (Fin (HPow.hPow…","labels":[],"detail_key":"p8","name":"MPOTensor.bipartitionedNormalizedMPO","module":"TNLean.MPS.MPDO.MutualInfoBridge"},{"id":"n11326","layer":"formal","project":"p8","title":"MPOTensor.bipartitionedNormalizedMPO_hasOperatorSchmidtDecomposition","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N L K : Nat) (h : Eq N (HAdd.hAdd L K)), (M.bipartitionedNorma…","labels":[],"detail_key":"p8","name":"MPOTensor.bipartitionedNormalizedMPO_hasOperatorSchmidtDecomposition","module":"TNLean.MPS.MPDO.MutualInfoBridge"},{"id":"n11327","layer":"formal","project":"p8","title":"MPOTensor.bipartitionedNormalizedMPO_posSemidef","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N L K : Nat) (h : Eq N (HAdd.hAdd L K)), (M.mpo N).PosSemidef…","labels":[],"detail_key":"p8","name":"MPOTensor.bipartitionedNormalizedMPO_posSemidef","module":"TNLean.MPS.MPDO.MutualInfoBridge"},{"id":"n11328","layer":"formal","project":"p8","title":"MPOTensor.bipartitionedNormalizedMPO_trace","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N L K : Nat) (h : Eq N (HAdd.hAdd L K)), Ne (M.mpo N).trace 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(HAdd.hAdd (HAdd.hAdd a b) c) N) (hM…","labels":[],"detail_key":"p8","name":"vonNeumannEntropy_traceC_eq_blockEntropy","module":"TNLean.MPS.MPDO.MutualInfoMonotone"},{"id":"n11349","layer":"formal","project":"p8","title":"vonNeumannEntropy_tripartiteSplit_eq_blockEntropy","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N a b c : Nat (h3 : LE.le (HAdd.hAdd (HAdd.hAdd a b) c) N) (hM…","labels":[],"detail_key":"p8","name":"vonNeumannEntropy_tripartiteSplit_eq_blockEntropy","module":"TNLean.MPS.MPDO.MutualInfoMonotone"},{"id":"n11350","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization","kind":"inductive","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11351","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringC…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → α : Prod (Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringControlledMap","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11352","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringC…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Prod (Fin F.sectorCount) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringControlledMap_isKrausCPTP","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11353","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringC…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Prod (Fin F.sectorCount) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringControlledMap_sameBlock_apply","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11354","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringD…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringDensity","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11355","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringD…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringDensity_eq_normalized","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11356","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringD…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringDensity_pos","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11357","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringP…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → α : Type u_1 → [Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringPreparationMap","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11358","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringP…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Type u_1 [inst : Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringPreparationMap_eq_normalized","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11359","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringP…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Type u_1 [inst : Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringPreparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11360","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDe…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → (k h : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDensity","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11361","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDe…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (k h : Fin F.sectorCount) [inst…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDensity_posDef","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11362","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDe…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (k h : Fin F.sectorCount) [inst…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveNeighboringDensity_trace","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11363","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.neighborIndex_nonempt…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.neighborIndex_nonempty_of_mul_ne_zero","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11364","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.neighboringOperator_e…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.neighboringOperator_eq_zero_of_mul_eq_zero","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11365","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboring…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboringDensity","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11366","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboring…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboringDensity_pos","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11367","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboring…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Type u_1 [inst : Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedNeighboringPreparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11368","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_completedNeighb…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_completedNeighboringDensity","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11369","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_normalizedNeigh…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_normalizedNeighboringDensity","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11370","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.weight_nonneg","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.weight_nonneg","module":"TNLean.MPS.MPDO.NeighboringPreparation"},{"id":"n11371","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.rightTraceMatrix_mul_…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.rightTraceMatrix_mul_leftTraceMatrix_eq_vecMulVec","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11372","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_mpo_eq_one","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_mpo_eq_one","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11373","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_physTraceTransf…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_physTraceTransfer_eq_one","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11374","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_physTraceTransf…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_physTraceTransfer_pow_eq_one","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11375","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.leftTraceMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Matrix (Fin D) (Fin F.sec…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.leftTraceMatrix","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11376","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physTraceTransfer_eq_leftTraceMatrix_mul_rightTrace…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq K.physTraceTransfer (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physTraceTransfer_eq_leftTraceMatrix_mul_rightTraceMatrix","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11377","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rightTraceMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Matrix (Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rightTraceMatrix","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11378","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rightTraceMatrix_mul_leftTraceMatrix_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rightTraceMatrix_mul_leftTraceMatrix_apply","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11379","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.trace_neighboringOperator_eq_sum_trace_mul_trace","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.trace_neighboringOperator_eq_sum_trace_mul_trace","module":"TNLean.MPS.MPDO.NeighboringTraceObstruction"},{"id":"n11380","layer":"formal","project":"p8","title":"MPOTensor.NeighboringTraceObstructionAmbientBlocks.exists_obstruction_terminal_twoBlock_B…","kind":"theorem","summary":"Exists fun mu => Exists fun B => Exists fun hmu => And (LT.lt (norm mu) 1) (And (Kraus.IsInject…","labels":[],"detail_key":"p8","name":"MPOTensor.NeighboringTraceObstructionAmbientBlocks.exists_obstruction_terminal_twoBlock_BNT_witness","module":"TNLean.MPS.MPDO.NeighboringTraceObstructionAmbientBlocks"},{"id":"n11381","layer":"formal","project":"p8","title":"MPOTensor.NeighboringTraceObstructionAmbientBlocks.terminalBlock_isNormalTensor","kind":"theorem","summary":"MPOTensor.NeighboringTraceObstructionAmbientBlocks.terminalBlock.toMPSTensor.IsNormalTensor","labels":[],"detail_key":"p8","name":"MPOTensor.NeighboringTraceObstructionAmbientBlocks.terminalBlock_isNormalTensor","module":"TNLean.MPS.MPDO.NeighboringTraceObstructionAmbientBlocks"},{"id":"n11382","layer":"formal","project":"p8","title":"MPOTensor.NonCartesianActiveSectorCandidate.exists_normalTensor_scalar_representation","kind":"theorem","summary":"Exists fun A => Exists fun mu => And (Ne mu 0) (And (LT.lt (norm mu) 1) (And (Eq MPOTensor.NonC…","labels":[],"detail_key":"p8","name":"MPOTensor.NonCartesianActiveSectorCandidate.exists_normalTensor_scalar_representation","module":"TNLean.MPS.MPDO.NonCartesianActiveSectorCounterexample"},{"id":"n11383","layer":"formal","project":"p8","title":"MPOTensor.NonCartesianActiveSectorCandidate.full_lowLevel_counterexample","kind":"theorem","summary":"Exists fun K => Exists fun A => Exists fun mu => And (Ne mu 0) (And (LT.lt (norm mu) 1) (And (E…","labels":[],"detail_key":"p8","name":"MPOTensor.NonCartesianActiveSectorCandidate.full_lowLevel_counterexample","module":"TNLean.MPS.MPDO.NonCartesianActiveSectorCounterexample"},{"id":"n11384","layer":"formal","project":"p8","title":"MPOTensor.NonCartesianActiveSectorCandidate.transferMap_posDef_eigenvalue_le_one","kind":"theorem","summary":"∀ rho : Matrix (Fin 2) (Fin 2) Complex r : Real, rho.PosDef → Eq ((Kraus.transferMap MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.NonCartesianActiveSectorCandidate.transferMap_posDef_eigenvalue_le_one","module":"TNLean.MPS.MPDO.NonCartesianActiveSectorCounterexample"},{"id":"n11385","layer":"formal","project":"p8","title":"MPOTensor.NonCartesianActiveSectorCandidate.transferMap_posDef_eigenvalue_lt_one","kind":"theorem","summary":"∀ rho : Matrix (Fin 2) (Fin 2) Complex r : Real, rho.PosDef → Eq ((Kraus.transferMap MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.NonCartesianActiveSectorCandidate.transferMap_posDef_eigenvalue_lt_one","module":"TNLean.MPS.MPDO.NonCartesianActiveSectorCounterexample"},{"id":"n11386","layer":"formal","project":"p8","title":"MPOTensor.exists_normalized_grouped_sector_maps_of_gram","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_normalized_grouped_sector_maps_of_gram","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11387","layer":"formal","project":"p8","title":"MPOTensor.normalizedGroupedSectorMap","kind":"def","summary":"d m n : Nat → Eq m n → Matrix (Fin d) (Fin n) Complex → Matrix (Fin n) (Fin n) Complex → Real →…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedGroupedSectorMap","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11388","layer":"formal","project":"p8","title":"MPOTensor.normalizedGroupedSectorMap_corner","kind":"theorem","summary":"∀ d m n : Nat (h : Eq m n) (A : Matrix (Fin m) (Fin m) Complex) (V : Matrix (Fin d) (Fin n) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedGroupedSectorMap_corner","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11389","layer":"formal","project":"p8","title":"MPOTensor.normalizedGroupedSectorMap_intertwining","kind":"theorem","summary":"∀ d m n : Nat (h : Eq m n) (T : Matrix (Fin d) (Fin d) Complex) (A : Matrix (Fin m) (Fin m) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedGroupedSectorMap_intertwining","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11390","layer":"formal","project":"p8","title":"MPOTensor.normalizedGroupedSectorMap_isometry","kind":"theorem","summary":"∀ d m n : Nat (h : Eq m n) (V : Matrix (Fin d) (Fin n) Complex) (X : Matrix (Fin n) (Fin n) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedGroupedSectorMap_isometry","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11391","layer":"formal","project":"p8","title":"MPOTensor.normalizedGroupedSectorMap_orthogonal","kind":"theorem","summary":"∀ d m₁ n₁ m₂ n₂ : Nat (h₁ : Eq m₁ n₁) (h₂ : Eq m₂ n₂) (V₁ : Matrix (Fin d) (Fin n₁) Complex) (V…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedGroupedSectorMap_orthogonal","module":"TNLean.MPS.MPDO.NormalizedGroupedSectorMaps"},{"id":"n11392","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.exists_normalized_grouped_sector_maps","kind":"theorem","summary":"∀ d D r : Nat dim : Fin r → Nat (blocks : (k : Fin r) → MPSTensor (HMul.hMul D D) (dim k)) M :…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.exists_normalized_grouped_sector_maps","module":"TNLean.MPS.MPDO.NormalizedGroupedSectors"},{"id":"n11393","layer":"formal","project":"p8","title":"MPOTensor.normalizedMPO_eq_of_nonzeroProportionalMPV₂_at","kind":"theorem","summary":"∀ d D₁ D₂ N : Nat (M : MPOTensor d D₁) (K : MPOTensor d D₂), (Exists fun c => And (Ne c 0) (∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedMPO_eq_of_nonzeroProportionalMPV₂_at","module":"TNLean.MPS.MPDO.NormalizedMPOProportionality"},{"id":"n11394","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_eq_of_nonzeroProportionalMPV₂_at","kind":"theorem","summary":"∀ d D₁ D₂ N L : Nat (M : MPOTensor d D₁) (K : MPOTensor d D₂) (hL : LE.le L N), (Exists fun c =…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_eq_of_nonzeroProportionalMPV₂_at","module":"TNLean.MPS.MPDO.NormalizedMPOProportionality"},{"id":"n11395","layer":"formal","project":"p8","title":"MPOTensor.evalWord_mulTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (M : MPOTensor d D₁) (N : MPOTensor d D₂) L : Nat (σ τ : Fin L → Fin d), Eq ((M…","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord_mulTensor","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11396","layer":"formal","project":"p8","title":"MPOTensor.listProd_blockDiagonal'_kronecker","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] [inst_1 : DecidableEq ι] m n : ι → Nat (X : (γ : ι) → Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.listProd_blockDiagonal'_kronecker","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11397","layer":"formal","project":"p8","title":"MPOTensor.listProd_conj_of_conjTranspose_mul_self","kind":"theorem","summary":"∀ S : Type u_1 T : Type u_2 [inst : Fintype S] [inst_1 : Fintype T] [inst_2 : DecidableEq S] [i…","labels":[],"detail_key":"p8","name":"MPOTensor.listProd_conj_of_conjTranspose_mul_self","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11398","layer":"formal","project":"p8","title":"MPOTensor.mpo_mulTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (M : MPOTensor d D₁) (N : MPOTensor d D₂) (L : Nat), Eq ((M.mulTensor N).mpo L)…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_mulTensor","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11399","layer":"formal","project":"p8","title":"MPOTensor.mulTensor","kind":"def","summary":"d D₁ D₂ : Nat → MPOTensor d D₁ → MPOTensor d D₂ → MPOTensor d (HMul.hMul D₁ D₂)","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensor","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11400","layer":"formal","project":"p8","title":"MPOTensor.mulTensorAssocEquiv","kind":"def","summary":"(D₁ D₂ D₃ : Nat) → Equiv (Fin (HMul.hMul (HMul.hMul D₁ D₂) D₃)) (Fin (HMul.hMul D₁ (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensorAssocEquiv","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11401","layer":"formal","project":"p8","title":"MPOTensor.mulTensorAssocMatrix","kind":"def","summary":"(D₁ D₂ D₃ : Nat) → Matrix (Fin (HMul.hMul (HMul.hMul D₁ D₂) D₃)) (Fin (HMul.hMul D₁ (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensorAssocMatrix","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11402","layer":"formal","project":"p8","title":"MPOTensor.mulTensor_assoc","kind":"theorem","summary":"∀ d D₁ D₂ D₃ : Nat (M : MPOTensor d D₁) (N : MPOTensor d D₂) (P : MPOTensor d D₃) (i l : Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensor_assoc","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11403","layer":"formal","project":"p8","title":"MPOTensor.mulTensor_mul_assocMatrix","kind":"theorem","summary":"∀ d D₁ D₂ D₃ : Nat (M : MPOTensor d D₁) (N : MPOTensor d D₂) (P : MPOTensor d D₃) (i l : Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensor_mul_assocMatrix","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11404","layer":"formal","project":"p8","title":"Matrix.conjTranspose_mul_eq_one_of_kronecker_one","kind":"theorem","summary":"∀ m : Type u_1 n : Type u_2 [inst : Fintype m] [inst_1 : DecidableEq n] D : Nat, LT.lt 0 D → ∀…","labels":[],"detail_key":"p8","name":"Matrix.conjTranspose_mul_eq_one_of_kronecker_one","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11405","layer":"formal","project":"p8","title":"Matrix.kronecker_one_injective","kind":"theorem","summary":"∀ m : Type u_1 n : Type u_2 D : Nat, LT.lt 0 D → Function.Injective fun A => Matrix.kroneckerMa…","labels":[],"detail_key":"p8","name":"Matrix.kronecker_one_injective","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11406","layer":"formal","project":"p8","title":"Matrix.mul_conjTranspose_eq_one_of_kronecker_one","kind":"theorem","summary":"∀ m : Type u_1 n : Type u_2 [inst : Fintype n] [inst_1 : DecidableEq m] D : Nat, LT.lt 0 D → ∀…","labels":[],"detail_key":"p8","name":"Matrix.mul_conjTranspose_eq_one_of_kronecker_one","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11407","layer":"formal","project":"p8","title":"Matrix.mul_eq_one_of_kronecker_one","kind":"theorem","summary":"∀ m : Type u_1 n : Type u_2 [inst : DecidableEq m] [inst_1 : Fintype n] D : Nat, LT.lt 0 D → ∀…","labels":[],"detail_key":"p8","name":"Matrix.mul_eq_one_of_kronecker_one","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11408","layer":"formal","project":"p8","title":"Matrix.mul_kronecker_one","kind":"theorem","summary":"∀ l : Type u_1 m : Type u_2 n : Type u_3 [inst : Fintype m] (A : Matrix l m Complex) (B : Matri…","labels":[],"detail_key":"p8","name":"Matrix.mul_kronecker_one","module":"TNLean.MPS.MPDO.OperatorProduct"},{"id":"n11409","layer":"formal","project":"p8","title":"MPOTensor.OrthogonalCommutingSectorFamily.firstSiteMatrix_mul_reducedBlockState","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat K : (s : Fin g) → MPOTensor d (dim s) (F : MPOTensor.OrthogonalCo…","labels":[],"detail_key":"p8","name":"MPOTensor.OrthogonalCommutingSectorFamily.firstSiteMatrix_mul_reducedBlockState","module":"TNLean.MPS.MPDO.OrthogonalSectorAreaLaw"},{"id":"n11410","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_commonWeightAbsorbedBasisMPOTensor_of_orthogonalCommutingSectorFamily","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), M.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_commonWeightAbsorbedBasisMPOTensor_of_orthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.OrthogonalSectorAreaLaw"},{"id":"n11411","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_orthogonalCommutingSectorFamily","kind":"theorem","summary":"∀ d D g : Nat dim : Fin g → Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s)) (mu…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_orthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.OrthogonalSectorAreaLaw"},{"id":"n11412","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_proportionalOrthogonalCommutingSectorFamily","kind":"theorem","summary":"∀ d D g : Nat dim : Fin g → Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s)) (mu…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_proportionalOrthogonalCommutingSectorFamily","module":"TNLean.MPS.MPDO.OrthogonalSectorAreaLaw"},{"id":"n11413","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_proportionalOrthogonalCommutingSectorFamily_of_sectorwise_isSourceZCL","kind":"theorem","summary":"∀ d D g : Nat dim : Fin g → Nat (M : MPOTensor d D) (K : (s : Fin g) → MPOTensor d (dim s)) (mu…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_proportionalOrthogonalCommutingSectorFamily_of_sectorwise_isSourceZCL","module":"TNLean.MPS.MPDO.OrthogonalSectorAreaLaw"},{"id":"n11414","layer":"formal","project":"p8","title":"MPOTensor.HasGlobalPurificationEquation","kind":"def","summary":"d D : Nat → MPOTensor d D → dK D' : Nat → (Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) →…","labels":[],"detail_key":"p8","name":"MPOTensor.HasGlobalPurificationEquation","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11415","layer":"formal","project":"p8","title":"MPOTensor.HasPurificationRFPWitness","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasPurificationRFPWitness","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11416","layer":"formal","project":"p8","title":"MPOTensor.HasTracePreservingSpinReduction","kind":"def","summary":"Nat → Nat → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasTracePreservingSpinReduction","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11417","layer":"formal","project":"p8","title":"MPOTensor.IsPRFP","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsPRFP","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11418","layer":"formal","project":"p8","title":"MPOTensor.ancillaryTraceMap","kind":"def","summary":"(d dK : Nat) → LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin dK)) (Prod (Fin d) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.ancillaryTraceMap","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11419","layer":"formal","project":"p8","title":"MPOTensor.ancillaryTraceMap_isKrausCPTP","kind":"theorem","summary":"∀ (d dK : Nat), IsKrausCPTP (MPOTensor.ancillaryTraceMap d dK)","labels":[],"detail_key":"p8","name":"MPOTensor.ancillaryTraceMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11420","layer":"formal","project":"p8","title":"MPOTensor.hasTracePreservingSpinReduction","kind":"theorem","summary":"∀ (d dK : Nat), MPOTensor.HasTracePreservingSpinReduction d dK","labels":[],"detail_key":"p8","name":"MPOTensor.hasTracePreservingSpinReduction","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11421","layer":"formal","project":"p8","title":"MPOTensor.purificationDensity","kind":"def","summary":"d dK D' : Nat → (Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) → (N : Nat) → Matrix (Fin N…","labels":[],"detail_key":"p8","name":"MPOTensor.purificationDensity","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11422","layer":"formal","project":"p8","title":"MPOTensor.purificationTensor","kind":"def","summary":"d dK D' : Nat → (Fin d → Fin dK → Matrix (Fin D') (Fin D') Complex) → MPSTensor (HMul.hMul d dK…","labels":[],"detail_key":"p8","name":"MPOTensor.purificationTensor","module":"TNLean.MPS.MPDO.PRFP"},{"id":"n11423","layer":"formal","project":"p8","title":"MPOTensor.blockwise_insert_eq_of_mpv_agree","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat μ : Fin r → Complex (A : (k : Fin r) → MPSTensor d (dim k)), MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.blockwise_insert_eq_of_mpv_agree","module":"TNLean.MPS.MPDO.PerCopyHorizontalCF"},{"id":"n11424","layer":"formal","project":"p8","title":"MPOTensor.PeriodicSectorProjector","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PeriodicSectorProjector","module":"TNLean.MPS.MPDO.PeriodicExclusion"},{"id":"n11425","layer":"formal","project":"p8","title":"MPOTensor.PeriodicVectorYieldsProjector","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.PeriodicVectorYieldsProjector","module":"TNLean.MPS.MPDO.PeriodicExclusion"},{"id":"n11426","layer":"formal","project":"p8","title":"MPOTensor.hasNoPeriodicVectors_verticalTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → M.PeriodicVectorYieldsProjector → M.verticalTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.hasNoPeriodicVectors_verticalTensor","module":"TNLean.MPS.MPDO.PeriodicExclusion"},{"id":"n11427","layer":"formal","project":"p8","title":"MPOTensor.isEmpty_periodicSectorProjector","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ p : Nat, IsEmpty (M.PeriodicSectorProjector p)","labels":[],"detail_key":"p8","name":"MPOTensor.isEmpty_periodicSectorProjector","module":"TNLean.MPS.MPDO.PeriodicExclusion"},{"id":"n11428","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwap","kind":"def","summary":"D : Nat → Fin (HMul.hMul D D) → Fin (HMul.hMul D D)","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwap","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11429","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwapEquiv","kind":"def","summary":"(D : Nat) → Equiv (Fin (HMul.hMul D D)) (Fin (HMul.hMul D D))","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwapEquiv","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11430","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwapEquiv_apply","kind":"theorem","summary":"∀ D : Nat (ab : Fin (HMul.hMul D D)), Eq ((MPOTensor.bondPairSwapEquiv D) ab) (MPOTensor.bondPa…","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwapEquiv_apply","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11431","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwapEquiv_symm","kind":"theorem","summary":"∀ D : Nat, Eq (MPOTensor.bondPairSwapEquiv D).symm (MPOTensor.bondPairSwapEquiv D)","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwapEquiv_symm","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11432","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwap_finProdFinEquiv","kind":"theorem","summary":"∀ D : Nat (a b : Fin D), Eq (MPOTensor.bondPairSwap (finProdFinEquiv (Prod.mk a b))) (finProdFi…","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwap_finProdFinEquiv","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11433","layer":"formal","project":"p8","title":"MPOTensor.bondPairSwap_involutive","kind":"theorem","summary":"∀ D : Nat (ab : Fin (HMul.hMul D D)), Eq (MPOTensor.bondPairSwap (MPOTensor.bondPairSwap ab)) ab","labels":[],"detail_key":"p8","name":"MPOTensor.bondPairSwap_involutive","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11434","layer":"formal","project":"p8","title":"MPOTensor.mpo_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (N : Nat) (σ τ : Fin N → Fin d), Eq (K.physicalAdjointTensor.mp…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_physicalAdjointTensor","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11435","layer":"formal","project":"p8","title":"MPOTensor.physicalAdjointTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.physicalAdjointTensor","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11436","layer":"formal","project":"p8","title":"MPOTensor.physicalAdjointTensor_mulTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (U : MPOTensor d D₁) (V : MPOTensor d D₂), Eq (U.mulTensor V).physicalAdjointTe…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalAdjointTensor_mulTensor","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11437","layer":"formal","project":"p8","title":"MPOTensor.physicalSlice_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (β α : Fin D), Eq (K.physicalAdjointTensor.physicalSlice β α) (…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSlice_physicalAdjointTensor","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11438","layer":"formal","project":"p8","title":"MPOTensor.productBondSwapEquiv","kind":"def","summary":"(D₁ D₂ : Nat) → Equiv (Fin (HMul.hMul D₁ D₂)) (Fin (HMul.hMul D₂ D₁))","labels":[],"detail_key":"p8","name":"MPOTensor.productBondSwapEquiv","module":"TNLean.MPS.MPDO.PhysicalAdjoint"},{"id":"n11439","layer":"formal","project":"p8","title":"Continuous.blockTensor","kind":"theorem","summary":"∀ X : Type u_1 [inst : TopologicalSpace X] d D : Nat M : X → MPOTensor d D, Continuous M → ∀ (L…","labels":[],"detail_key":"p8","name":"Continuous.blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11440","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.blockTensor","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsMPDO → ∀ (L : Nat), LT.lt 0 L → (M.blockTensor L).IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11441","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.blockTwo","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsMPDO → M.blockTwo.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.blockTwo","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11442","layer":"formal","project":"p8","title":"MPOTensor.blockTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → (L : Nat) → MPOTensor (MPSTensor.blockPhysDim d L) D","labels":[],"detail_key":"p8","name":"MPOTensor.blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11443","layer":"formal","project":"p8","title":"MPOTensor.blockTensor_apply","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L : Nat) (i j : Fin (MPSTensor.blockPhysDim d L)), Eq (M.block…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTensor_apply","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11444","layer":"formal","project":"p8","title":"MPOTensor.blockTensor_mulTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (M : MPOTensor d D₁) (N : MPOTensor d D₂) L : Nat, Eq ((M.mulTensor N).blockTen…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTensor_mulTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11445","layer":"formal","project":"p8","title":"MPOTensor.blockTwo","kind":"def","summary":"d D : Nat → MPOTensor d D → MPOTensor (HMul.hMul d d) D","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11446","layer":"formal","project":"p8","title":"MPOTensor.blockTwo_eq_blockTensor_reindex","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq M.blockTwo fun i j => M.blockTensor 2 ((MPOTensor.twoSiteBl…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo_eq_blockTensor_reindex","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11447","layer":"formal","project":"p8","title":"MPOTensor.blockedDoubledIndexEquiv","kind":"def","summary":"(d L : Nat) → Equiv (Fin (HMul.hMul (MPSTensor.blockPhysDim d L) (MPSTensor.blockPhysDim d L)))…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedDoubledIndexEquiv","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11448","layer":"formal","project":"p8","title":"MPOTensor.continuous_blockTensor","kind":"theorem","summary":"∀ d D : Nat (L : Nat), Continuous fun M => M.blockTensor L","labels":[],"detail_key":"p8","name":"MPOTensor.continuous_blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11449","layer":"formal","project":"p8","title":"MPOTensor.continuous_evalWord","kind":"theorem","summary":"∀ d D : Nat (is js : List (Fin d)), Continuous fun M => M.evalWord is js","labels":[],"detail_key":"p8","name":"MPOTensor.continuous_evalWord","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11450","layer":"formal","project":"p8","title":"MPOTensor.decodeBlock_blockedDoubledIndexEquiv","kind":"theorem","summary":"∀ (d L : Nat) (ij : Fin (HMul.hMul (MPSTensor.blockPhysDim d L) (MPSTensor.blockPhysDim d L)))…","labels":[],"detail_key":"p8","name":"MPOTensor.decodeBlock_blockedDoubledIndexEquiv","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11451","layer":"formal","project":"p8","title":"MPOTensor.equivReindexMap_symm_apply_self","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 (e : Equiv α β) (X : Matrix α α Complex), Eq ((Matrix.equivReindexM…","labels":[],"detail_key":"p8","name":"MPOTensor.equivReindexMap_symm_apply_self","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11452","layer":"formal","project":"p8","title":"MPOTensor.isInjective_toMPSTensor_blockTensor_iff","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) L : Nat, Iff (Kraus.IsInjective (M.blockTensor L).toMPSTensor)…","labels":[],"detail_key":"p8","name":"MPOTensor.isInjective_toMPSTensor_blockTensor_iff","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11453","layer":"formal","project":"p8","title":"MPOTensor.mpo_blockTensor_eq_reindex","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L N : Nat), Eq ((M.blockTensor L).mpo N) ((Matrix.reindex (MPS…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_blockTensor_eq_reindex","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11454","layer":"formal","project":"p8","title":"MPOTensor.mpo_blockTwo_eq_reindex_blockTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat), Eq (M.blockTwo.mpo N) ((Matrix.reindex ((Equiv.refl…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_blockTwo_eq_reindex_blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11455","layer":"formal","project":"p8","title":"MPOTensor.physClose1_blockTwo_eq_physClose2","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (MPOTensor.bl…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose1_blockTwo_eq_physClose2","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11456","layer":"formal","project":"p8","title":"MPOTensor.physClose1_blockTwo_eq_physClose2_apply","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((Matrix.equivReindexM…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose1_blockTwo_eq_physClose2_apply","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11457","layer":"formal","project":"p8","title":"MPOTensor.physClose2_blockTwo_eq_physClose4","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (MPOTensor.bl…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose2_blockTwo_eq_physClose4","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11458","layer":"formal","project":"p8","title":"MPOTensor.physClose2_eq_physClose1_blockTwo_apply","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((Matrix.equivReindexM…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose2_eq_physClose1_blockTwo_apply","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11459","layer":"formal","project":"p8","title":"MPOTensor.physClose4","kind":"def","summary":"d D : Nat → MPOTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose4","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11460","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_four_eq_physClose4","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (finFourArrow…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_four_eq_physClose4","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11461","layer":"formal","project":"p8","title":"MPOTensor.physicalAdjointTensor_blockTensor","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (L : Nat), Eq (U.blockTensor L).physicalAdjointTensor (U.physic…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalAdjointTensor_blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11462","layer":"formal","project":"p8","title":"MPOTensor.toMPSTensor_blockTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) L : Nat, Eq (M.blockTensor L).toMPSTensor (Kraus.reindexPhysica…","labels":[],"detail_key":"p8","name":"MPOTensor.toMPSTensor_blockTensor","module":"TNLean.MPS.MPDO.PhysicalBlocking"},{"id":"n11463","layer":"formal","project":"p8","title":"MPOTensor.physClose1","kind":"def","summary":"d D : Nat → MPOTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose1","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11464","layer":"formal","project":"p8","title":"MPOTensor.physClose2","kind":"def","summary":"d D : Nat → MPOTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose2","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11465","layer":"formal","project":"p8","title":"MPOTensor.physClose3","kind":"def","summary":"d D : Nat → MPOTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose3","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11466","layer":"formal","project":"p8","title":"MPOTensor.physClose3_apply","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (X : Matrix (Fin D) (Fin D) Complex) (i j : Prod (Fin d) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose3_apply","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11467","layer":"formal","project":"p8","title":"MPOTensor.physCloseN","kind":"def","summary":"d D : Nat → MPOTensor d D → (N : Nat) → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D)…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11468","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_apply","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat) (X : Matrix (Fin D) (Fin D) Complex) (σ τ : Fin N → F…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_apply","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11469","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_identity_eq_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat), Eq ((M.physCloseN N) 1) (M.mpo N)","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_identity_eq_mpo","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11470","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_one_eq_physClose1","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (Equiv.funUni…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_one_eq_physClose1","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11471","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_three_eq_physClose3","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (finThreeArro…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_three_eq_physClose3","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11472","layer":"formal","project":"p8","title":"MPOTensor.physCloseN_two_eq_physClose2","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq ((↑(Matrix.reindexLinearEquiv Complex Complex (finTwoArrowE…","labels":[],"detail_key":"p8","name":"MPOTensor.physCloseN_two_eq_physClose2","module":"TNLean.MPS.MPDO.PhysicalClosure"},{"id":"n11473","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat K : MPOTensor d D, K.IsMPDO → ∀ (V : Matrix (Fin e) (Fin d) Complex), (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11474","layer":"formal","project":"p8","title":"MPOTensor.IsSAL.changePhysicalBasis_of_isometry","kind":"theorem","summary":"∀ d e D : Nat K : MPOTensor d D, K.IsSAL → ∀ (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMu…","labels":[],"detail_key":"p8","name":"MPOTensor.IsSAL.changePhysicalBasis_of_isometry","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11475","layer":"formal","project":"p8","title":"MPOTensor.doubledPhysicalMatrix","kind":"def","summary":"d e : Nat → Matrix (Fin e) (Fin d) Complex → Matrix (Fin (HMul.hMul e e)) (Fin (HMul.hMul d d))…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledPhysicalMatrix","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11476","layer":"formal","project":"p8","title":"MPOTensor.doubledPhysicalMatrix_isometry","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → Eq (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledPhysicalMatrix_isometry","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11477","layer":"formal","project":"p8","title":"MPOTensor.gaugeEquiv_toMPSTensor_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex) K L : MPOTensor d D, K.toMPSTensor.GaugeEqui…","labels":[],"detail_key":"p8","name":"MPOTensor.gaugeEquiv_toMPSTensor_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11478","layer":"formal","project":"p8","title":"MPOTensor.isInjective_toMPSTensor_changePhysicalBasis_iff","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.isInjective_toMPSTensor_changePhysicalBasis_iff","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11479","layer":"formal","project":"p8","title":"MPOTensor.isLeftCanonical_toMPSTensor_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.isLeftCanonical_toMPSTensor_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11480","layer":"formal","project":"p8","title":"MPOTensor.isMPDO_changePhysicalBasis_iff","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.isMPDO_changePhysicalBasis_iff","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11481","layer":"formal","project":"p8","title":"MPOTensor.isNormalTensor_toMPSTensor_changePhysicalBasis_iff","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.isNormalTensor_toMPSTensor_changePhysicalBasis_iff","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11482","layer":"formal","project":"p8","title":"MPOTensor.isSAL_changePhysicalBasis_iff","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_changePhysicalBasis_iff","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11483","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11484","layer":"formal","project":"p8","title":"MPOTensor.physicalSliceColumnCoisometry","kind":"def","summary":"d e D : Nat → Matrix (Fin e) (Fin d) Complex → Matrix (Fin (HMul.hMul (HMul.hMul D D) d)) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSliceColumnCoisometry","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11485","layer":"formal","project":"p8","title":"MPOTensor.physicalSliceColumnCoisometry_mul_conjTranspose","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → Eq (HM…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSliceColumnCoisometry_mul_conjTranspose","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11486","layer":"formal","project":"p8","title":"MPOTensor.physicalSliceColumns_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex) (K : MPOTensor d D), Eq (MPOTensor.PhysicalS…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSliceColumns_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11487","layer":"formal","project":"p8","title":"MPOTensor.physicalSliceColumns_mul_conjTranspose_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSliceColumns_mul_conjTranspose_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11488","layer":"formal","project":"p8","title":"MPOTensor.physicalSupportProj_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSupportProj_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11489","layer":"formal","project":"p8","title":"MPOTensor.toMPSTensor_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex) (K : MPOTensor d D), Eq (MPOTensor.PhysicalS…","labels":[],"detail_key":"p8","name":"MPOTensor.toMPSTensor_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11490","layer":"formal","project":"p8","title":"MPOTensor.transferMap_toMPSTensor_changePhysicalBasis","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (K :…","labels":[],"detail_key":"p8","name":"MPOTensor.transferMap_toMPSTensor_changePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalIsometricEmbedding"},{"id":"n11491","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.IsActiveSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin F.sectorCount → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.IsActiveSector","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11492","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSector_factorSupport_twoSided","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Kraus.IsInjective K.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSector_factorSupport_twoSided","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11493","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_activeSector_factorSupportIsometries","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) k : Fin F.sectorCount, F.IsAc…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_activeSector_factorSupportIsometries","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11494","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isActiveSector_iff_exists_sectorProductFamily_ne_ze…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Iff…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isActiveSector_iff_exists_sectorProductFamily_ne_zero","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11495","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.leftFactorSupportProj","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.leftFactorSupportProj","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11496","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.leftTensor_conjTranspose_mem_span_of_isActiveSector","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Kraus.IsInjective K.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.leftTensor_conjTranspose_mem_span_of_isActiveSector","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11497","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_bondDim_zero","kind":"theorem","summary":"∀ d : Nat K : MPOTensor d 0 (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Not (F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_bondDim_zero","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11498","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_iff","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Iff…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_iff","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11499","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_iff_left_or_right_eq_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Iff…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.not_isActiveSector_iff_left_or_right_eq_zero","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11500","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rightFactorSupportProj","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rightFactorSupportProj","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11501","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rightTensor_conjTranspose_mem_span_of_isActiveSector","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Kraus.IsInjective K.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rightTensor_conjTranspose_mem_span_of_isActiveSector","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11502","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_isInjective","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Kraus.IsInjective K.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_isInjective","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11503","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_isMPDO_of_neighboringOperato…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_isMPDO_of_neighboringOperator_pos","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11504","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11505","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily_conjTranspose_mem_span","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Kraus.IsInjective K.toMPSTen…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily_conjTranspose_mem_span","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11506","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily_supportProj","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount), Eq (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorProductFamily_supportProj","module":"TNLean.MPS.MPDO.PhysicalSectorActiveFactorSupport"},{"id":"n11507","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.FactorActiveSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.FactorActiveSector","module":"TNLean.MPS.MPDO.PhysicalSectorActiveNeighboring"},{"id":"n11508","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalSupportInclusion","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (A : F.ActiveFactorSuppor…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalSupportInclusion","module":"TNLean.MPS.MPDO.PhysicalSectorActiveRestriction"},{"id":"n11509","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalSupportInclusion_isometry","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (A : F.ActiveFactorSupportDat…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalSupportInclusion_isometry","module":"TNLean.MPS.MPDO.PhysicalSectorActiveRestriction"},{"id":"n11510","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ActiveFactorSupportData","kind":"inductive","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ActiveFactorSupportData","module":"TNLean.MPS.MPDO.PhysicalSectorActiveSupportData"},{"id":"n11511","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.blockTwo_sectorCoordi…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.blockTwo_sectorCoordinateTensor_isRFPViaTS","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11512","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11513","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap_f…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap_fourSiteClosure_eq_twoSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11514","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap_i…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSSquaredMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11515","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11516","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap_i…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11517","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap_t…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTSquaredMap_twoSiteClosure_eq_fourSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorBlockedRFP"},{"id":"n11518","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.adjacentSectorBonds_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount),…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.adjacentSectorBonds_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondCommutativity"},{"id":"n11519","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.leftPairMatrix","kind":"def","summary":"n : Type u_1 → [Fintype n] → [DecidableEq n] → Matrix (Prod n n) (Prod n n) Complex → Matrix (P…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.leftPairMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorBondCommutativity"},{"id":"n11520","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_one","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq ((Matrix.reindex (finThre…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_one","module":"TNLean.MPS.MPDO.PhysicalSectorBondCommutativity"},{"id":"n11521","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq ((Matrix.reindex (finThre…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_zero","module":"TNLean.MPS.MPDO.PhysicalSectorBondCommutativity"},{"id":"n11522","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rightPairMatrix","kind":"def","summary":"n : Type u_1 → [Fintype n] → [DecidableEq n] → Matrix (Prod n n) (Prod n n) Complex → Matrix (P…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rightPairMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorBondCommutativity"},{"id":"n11523","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_pairwise_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) N : Nat (hN : LE.le 2 N) (i j…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_pairwise_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondPairwise"},{"id":"n11524","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_adjacent_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) N : Nat (hN : LE.le 3 N) (i :…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_adjacent_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondTransport"},{"id":"n11525","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_disjoint_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) N : Nat (hN : LE.le 2 N) i j…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_disjoint_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondTransport"},{"id":"n11526","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_zero_one_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq (HMul.hMul (MPOTensor.emb…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_zero_one_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondTransport"},{"id":"n11527","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_two_zero_one_comm","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq (HMul.hMul (MPOTensor.emb…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_two_zero_one_comm","module":"TNLean.MPS.MPDO.PhysicalSectorBondTwoSite"},{"id":"n11528","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_two_one","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq ((Matrix.reindex (finTwoA…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_two_one","module":"TNLean.MPS.MPDO.PhysicalSectorBondTwoSite"},{"id":"n11529","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_two_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq ((Matrix.reindex (finTwoA…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.reindex_embedLocalOperator_two_zero","module":"TNLean.MPS.MPDO.PhysicalSectorBondTwoSite"},{"id":"n11530","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.SectorChainFiber","kind":"def","summary":"d D N : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (Fin N → Fin F.sectorCo…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.SectorChainFiber","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11531","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.SectorChainIndex","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.SectorChainIndex","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11532","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct","kind":"def","summary":"d D N : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → [NeZero N] → (k : Fin N…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.cyclicNeighboringProduct","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11533","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.mpo_reindex_sectorChainEquiv_eq_blockDiagonal","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N], Eq ((Mat…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.mpo_reindex_sectorChainEquiv_eq_blockDiagonal","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11534","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.mpo_submatrix_sector_eq_cyclicNeighboringProduct","kind":"theorem","summary":"∀ d D N : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) [inst : NeZero N] (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.mpo_submatrix_sector_eq_cyclicNeighboringProduct","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11535","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorChainEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (N : Nat) → Equiv (Fin N…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorChainEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorChainDecomposition"},{"id":"n11536","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0Regrouped_threeSiteClosure_apply_of_outer_ne","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0Regrouped_threeSiteClosure_apply_of_outer_ne","module":"TNLean.MPS.MPDO.PhysicalSectorClosureBlocks"},{"id":"n11537","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0Regrouped_threeSiteClosure_block_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0Regrouped_threeSiteClosure_block_eq","module":"TNLean.MPS.MPDO.PhysicalSectorClosureBlocks"},{"id":"n11538","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorCoordinateEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod (Fin (Fintype…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorCoordinateEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureCoordinates"},{"id":"n11539","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorCoordinateEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorCoordinateEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureCoordinates"},{"id":"n11540","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorCoordinateEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod (Fin (Fintype…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorCoordinateEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureCoordinates"},{"id":"n11541","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorCoordinateEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (x…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorCoordinateEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureCoordinates"},{"id":"n11542","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalClosure_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalClosure_eq","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11543","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalRegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod (Fin (Fintype…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalRegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11544","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalRegroupEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteGlobalRegroupEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11545","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalClosure_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalClosure_eq","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11546","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalRegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod (Fin (Fintype…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalRegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11547","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalRegroupEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (x…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteGlobalRegroupEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureGlobal"},{"id":"n11548","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteRegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k l h : Fin F.sectorCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteRegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureThree"},{"id":"n11549","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteRegroupEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteRegroupEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureThree"},{"id":"n11550","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorClosure","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k l h : Fin F.sectorCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorClosure","module":"TNLean.MPS.MPDO.PhysicalSectorClosureThree"},{"id":"n11551","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorClosure_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorClosure_eq","module":"TNLean.MPS.MPDO.PhysicalSectorClosureThree"},{"id":"n11552","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorEmbedding","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k l h : Fin F.sectorCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteSectorEmbedding","module":"TNLean.MPS.MPDO.PhysicalSectorClosureThree"},{"id":"n11553","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteRegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteRegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorClosureTwo"},{"id":"n11554","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteRegroupEquiv_symm_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (x…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteRegroupEquiv_symm_apply","module":"TNLean.MPS.MPDO.PhysicalSectorClosureTwo"},{"id":"n11555","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorClosure","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorClosure","module":"TNLean.MPS.MPDO.PhysicalSectorClosureTwo"},{"id":"n11556","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorClosure_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (X…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorClosure_eq","module":"TNLean.MPS.MPDO.PhysicalSectorClosureTwo"},{"id":"n11557","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorEmbedding","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.twoSiteSectorEmbedding","module":"TNLean.MPS.MPDO.PhysicalSectorClosureTwo"},{"id":"n11558","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGraining"},{"id":"n11559","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGraining"},{"id":"n11560","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGraining"},{"id":"n11561","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGraining"},{"id":"n11562","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringP…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedNeighboringPreparationMap_smul","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingAction"},{"id":"n11563","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s0Map_block_eq_smul","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s0Map_block_eq_smul","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingAction"},{"id":"n11564","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_block_eq_kronec…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_block_eq_kronecker","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingAction"},{"id":"n11565","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0Map_sameBlock_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Prod F.SectorSit…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0Map_sameBlock_apply","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingAction"},{"id":"n11566","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s2Map_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix F.S2PreparationIn…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s2Map_apply","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingAction"},{"id":"n11567","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_apply_of_ne","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.s1Map_apply_of_ne","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingIdentity"},{"id":"n11568","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap_threeSiteClosure…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sMap_threeSiteClosure_eq_twoSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorCoarseGrainingIdentity"},{"id":"n11569","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofChangePhysicalBasis","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (V : Matrix (Fin d) (Fin d) Complex) → (hV : Eq (HMul.hMul V.co…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofChangePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11570","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.toChangePhysicalBasis","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (V : Matrix (Fin d) (Fin d) Complex) → (hV : Eq (HMul.hMul V.co…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.toChangePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11571","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_changePhysical…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (V : Matrix (Fin d) (Fin d) Complex), Eq (HMul.hMul V.conjTranspo…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_changePhysicalBasis_iff","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11572","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_changePhysical…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D e : Nat (V : Matrix (Fin e) (Fin d) Complex), V.IsUnitaryBetween…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_changePhysicalBasis_iff_of_isUnitaryBetween","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11573","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofChangePhysicalBasis","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (V : Matrix (Fin d) (Fin d) Complex) → Eq (HMul.hMul V.conjTran…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofChangePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11574","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofChangePhysicalBasis_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (V : Matrix (Fin d) (Fin d) Complex) (hV : Eq (HMul.hMul V.conjTr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofChangePhysicalBasis_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11575","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.toChangePhysicalBasis","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (V : Matrix (Fin d) (Fin d) Complex) → Eq (HMul.hMul V.conjTran…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.toChangePhysicalBasis","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11576","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.toChangePhysicalBasis_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (V : Matrix (Fin d) (Fin d) Complex) (hV : Eq (HMul.hMul V.conjTr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.toChangePhysicalBasis_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorCoordinateTransport"},{"id":"n11577","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorDirectedCut"},{"id":"n11578","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorSpace","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Set (Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorSpace","module":"TNLean.MPS.MPDO.PhysicalSectorDirectedCut"},{"id":"n11579","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorSpace_mul_eq_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (S C : Set (Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.oneSiteSectorSpace_mul_eq_zero","module":"TNLean.MPS.MPDO.PhysicalSectorDirectedCut"},{"id":"n11580","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.etaLocalStructureData","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (∀ (k h : Fin F.sectorCou…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.etaLocalStructureData","module":"TNLean.MPS.MPDO.PhysicalSectorEtaLocalStructure"},{"id":"n11581","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11582","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.boundaryOperator","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.boundaryOperator","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11583","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_add","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (X…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_add","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11584","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (X…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_apply","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11585","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_smul","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (c…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_smul","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11586","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.boundaryOperator_zero","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11587","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighborIndex_nonempty","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), No…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighborIndex_nonempty","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11588","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperator","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11589","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (x…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_apply","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11590","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → MPOTensor (Fintype.card (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11591","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (i j : Fin (Fintype.card (Sig…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11592","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply_ne","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) k h : Fin F.sectorCount, Ne k…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply_ne","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11593","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply_same","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k : Fin F.sectorCount) (x y…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_apply_same","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11594","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorFinEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Fin (Fintype.card…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorFinEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11595","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin D → Fin D → Matrix (S…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11596","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_apply_ne","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (beta alpha : Fin D) k h : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_apply_ne","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11597","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_apply_same","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (beta alpha : Fin D) (k : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_apply_same","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11598","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (beta alpha : Fin D), Eq (F.t…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.transformedPhysicalSlice_eq","module":"TNLean.MPS.MPDO.PhysicalSectorFactorization"},{"id":"n11599","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofGaugeEquiv","kind":"def","summary":"d D : Nat → K L : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactori…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.ofGaugeEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11600","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_iff_of_gaugeEq…","kind":"theorem","summary":"∀ d D : Nat K L : MPOTensor d D, K.toMPSTensor.GaugeEquiv L.toMPSTensor → Iff (Exists fun F =>…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_neighboringTraceFactorization_iff_of_gaugeEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11601","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv","kind":"def","summary":"d D : Nat → K L : MPOTensor d D → K.PhysicalSectorFactorization → K.toMPSTensor.GaugeEquiv L.to…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11602","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_eq_ofVirtualMatrices","kind":"theorem","summary":"∀ d D : Nat K L : MPOTensor d D (F : K.PhysicalSectorFactorization) (hGauge : K.toMPSTensor.Gau…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_eq_ofVirtualMatrices","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11603","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K L : MPOTensor d D (F : K.PhysicalSectorFactorization) (hGauge : K.toMPSTensor.Gau…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11604","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_neighboringOperator_posSemidef","kind":"theorem","summary":"∀ d D : Nat K L : MPOTensor d D (F : K.PhysicalSectorFactorization) (hGauge : K.toMPSTensor.Gau…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofGaugeEquiv_neighboringOperator_posSemidef","module":"TNLean.MPS.MPDO.PhysicalSectorGaugeTransport"},{"id":"n11605","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.fixedProductTensorData","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (hη : ∀ (k h : Fin F.sect…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.fixedProductTensorData","module":"TNLean.MPS.MPDO.PhysicalSectorNormalProductRepresentative"},{"id":"n11606","layer":"formal","project":"p8","title":"MPOTensor.exists_normal_fixedProductTensorData_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → ∀ (hSAL : K.IsSAL), K.toMPSTensor.IsNormalTens…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_normal_fixedProductTensorData_of_isSAL","module":"TNLean.MPS.MPDO.PhysicalSectorNormalProductRepresentative"},{"id":"n11607","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteCon…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → α : Prod (Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteControlledMap","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11608","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteCon…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Prod (Fin F.sectorCount) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteControlledMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11609","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteCon…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Prod (Fin F.sectorCount) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteControlledMap_sameBlock_apply","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11610","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNei…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNeighboringDensity","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11611","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNei…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNeighboringDensity_eq_normalized","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11612","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNei…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSiteNeighboringDensity_pos","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11613","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePre…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → α : Type u_1 → [Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePreparationMap","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11614","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePre…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Type u_1 [inst : Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePreparationMap_eq_normalized","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11615","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePre…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization α : Type u_1 [inst : Fintype α]…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePreparationMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11616","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeig…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeighboringDensity","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11617","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeig…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeighboringDensity_posDef","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11618","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeig…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.inactiveThreeSiteNeighboringDensity_trace","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11619","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedThreeSiteNe…","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedThreeSiteNeighboringDensity","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11620","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedThreeSiteNe…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.normalizedThreeSiteNeighboringDensity_pos","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11621","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sector_nonempty","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sector_nonempty","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11622","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteMiddleIndex_…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteMiddleIndex_nonempty","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11623","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteNeighboringO…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteNeighboringOperator_eq_zero_of_mul_eq_zero","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11624","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteNeighboringO…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.threeSiteNeighboringOperator_pos","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11625","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_completedThreeS…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_completedThreeSiteNeighboringDensity","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11626","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_normalizedThree…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_normalizedThreeSiteNeighboringDensity","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11627","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_threeSiteNeighb…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.trace_threeSiteNeighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11628","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.threeSiteNeighboringOperator","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.threeSiteNeighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11629","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.trace_threeSiteNeighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.trace_threeSiteNeighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorOmegaPreparation"},{"id":"n11630","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.FourSiteMiddleIndex","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin F.sectorCount → Fin F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.FourSiteMiddleIndex","module":"TNLean.MPS.MPDO.PhysicalSectorPairPreservingObstruction"},{"id":"n11631","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.fourSiteNeighboringOperator","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.fourSiteNeighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorPairPreservingObstruction"},{"id":"n11632","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringTraceMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Matrix (Fin F.sectorCount…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringTraceMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorPairPreservingObstruction"},{"id":"n11633","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringTraceMatrix_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringTraceMatrix_apply","module":"TNLean.MPS.MPDO.PhysicalSectorPairPreservingObstruction"},{"id":"n11634","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11635","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap_fourSiteClosure_eq_threeSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11636","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalSMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11637","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11638","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11639","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.localizedPhysicalTMap_threeSiteClosure_eq_fourSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11640","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11641","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap_isKrausC…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11642","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap_threeSit…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalSMap_threeSiteClosure_eq_twoSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11643","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11644","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap_isKrausC…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11645","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap_twoSiteC…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.physicalTMap_twoSiteClosure_eq_threeSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalMaps"},{"id":"n11646","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.blockTwo_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.blockTwo_isRFPViaTS","module":"TNLean.MPS.MPDO.PhysicalSectorPhysicalTransport"},{"id":"n11647","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Matrix (Fin 2 → Fin d) (Fin 2 →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11648","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalBond_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalBond_pos","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11649","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalPairBond","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Matrix (Prod (Fin d) (Fin d)) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalPairBond","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11650","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.physicalPairBond_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.physicalPairBond_pos","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11651","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateBond","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Matrix (Prod (Fin (Fintyp…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateBond","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11652","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateBond_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateBond_pos","module":"TNLean.MPS.MPDO.PhysicalSectorPositiveBond"},{"id":"n11653","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.etaSectorFinEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Matrix.EtaSiteInde…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.etaSectorFinEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorProductRealization"},{"id":"n11654","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.isMPDO_of_neighboringOperator_pos","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), (∀ (q h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.isMPDO_of_neighboringOperator_pos","module":"TNLean.MPS.MPDO.PhysicalSectorProductRealization"},{"id":"n11655","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_positive_scalar_mpo_eq_product_physicalBond","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) N : Nat (hN : LE.le 2 N), Exi…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_positive_scalar_mpo_eq_product_physicalBond","module":"TNLean.MPS.MPDO.PhysicalSectorProductTransport"},{"id":"n11656","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.mpo_eq_product_physicalBond","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) N : Nat (hN : LE.le 2 N), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.mpo_eq_product_physicalBond","module":"TNLean.MPS.MPDO.PhysicalSectorProductTransport"},{"id":"n11657","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.zeroRightTensorOnInactive","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (active : Fin F.sectorCou…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.zeroRightTensorOnInactive","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11658","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.zeroRightTensorOnInactive_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (active : Fin F.sectorCount →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.zeroRightTensorOnInactive_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11659","layer":"formal","project":"p8","title":"MPOTensor.sectorEta_eq_zero_of_target_weight_eq_zero","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorEta_eq_zero_of_target_weight_eq_zero","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11660","layer":"formal","project":"p8","title":"MPOTensor.sectorTensorL_eq_zero_of_weight_eq_zero","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorTensorL_eq_zero_of_weight_eq_zero","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11661","layer":"formal","project":"p8","title":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization","kind":"def","summary":"d D : Nat → rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11662","layer":"formal","project":"p8","title":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization_neighboringOpera…","kind":"theorem","summary":"∀ d D : Nat rho : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.zeroWeightReparameterizedInverseMapPhysicalSectorFactorization_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorPruning"},{"id":"n11663","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map","module":"TNLean.MPS.MPDO.PhysicalSectorRefinement"},{"id":"n11664","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorRefinement"},{"id":"n11665","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap","kind":"def","summary":"d D : Nat → K : MPOTensor d D → F : K.PhysicalSectorFactorization → F.NeighboringTraceFactoriza…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap","module":"TNLean.MPS.MPDO.PhysicalSectorRefinement"},{"id":"n11666","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorRefinement"},{"id":"n11667","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePre…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.completedThreeSitePreparationMap_smul","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11668","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t0Map_block_eq_smul","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t0Map_block_eq_smul","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11669","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_apply_of_ne","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_apply_of_ne","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11670","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_block_eq_kronec…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.t1Map_block_eq_kronecker","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11671","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t0Map_sameBlock_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Prod F.SectorSit…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t0Map_sameBlock_apply","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11672","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t0Regrouped_twoSiteClosure_block_eq","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t0Regrouped_twoSiteClosure_block_eq","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11673","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t2Map_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (X : Matrix F.S0PreparationIn…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t2Map_apply","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementAction"},{"id":"n11674","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap_twoSiteClosure_e…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.tMap_twoSiteClosure_eq_threeSiteClosure","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementIdentity"},{"id":"n11675","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t0Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t0Map","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11676","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t0Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), IsKrausCPTP F.t0Map","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t0Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11677","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t0RegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod F.SectorSiteI…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t0RegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11678","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t2Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t2Map","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11679","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t2Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), IsKrausCPTP F.t2Map","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t2Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11680","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.t2ShiftEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv F.S0PreparationInde…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.t2ShiftEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorRefinementRegroupings"},{"id":"n11681","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rephase","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (Fin F.sectorCount → Circ…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rephase","module":"TNLean.MPS.MPDO.PhysicalSectorRephasing"},{"id":"n11682","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rephase_cyclicNeighboringProduct","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (z : Fin F.sectorCount → Circ…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rephase_cyclicNeighboringProduct","module":"TNLean.MPS.MPDO.PhysicalSectorRephasing"},{"id":"n11683","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rephase_neighboringOperator","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (z : Fin F.sectorCount → Circ…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rephase_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorRephasing"},{"id":"n11684","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.rephase_sectorVirtualMatrix","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (z : Fin F.sectorCount → Circ…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.rephase_sectorVirtualMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorRephasing"},{"id":"n11685","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.S0PreparationIndex","kind":"def","summary":"d D : Nat → K : MPOTensor d D → K.PhysicalSectorFactorization → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.S0PreparationIndex","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11686","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ThreeSiteMiddleIndex","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Fin F.sectorCount → Fin F…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ThreeSiteMiddleIndex","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11687","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0Map","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11688","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), IsKrausCPTP F.s0Map","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11689","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s0RegroupEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv (Prod F.SectorSiteI…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s0RegroupEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11690","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s2Map","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s2Map","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11691","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s2Map_isKrausCPTP","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), IsKrausCPTP F.s2Map","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s2Map_isKrausCPTP","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11692","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.s2ShiftEquiv","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Equiv F.S2PreparationInde…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.s2ShiftEquiv","module":"TNLean.MPS.MPDO.PhysicalSectorSubspinMaps"},{"id":"n11693","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_two_edge_return_of_neighboringOperator_ne_ze…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq (Submodule.span Complex (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_two_edge_return_of_neighboringOperator_ne_zero","module":"TNLean.MPS.MPDO.PhysicalSectorSupportRecurrence"},{"id":"n11694","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_reaches_of_ne_zero","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), Eq (Submodule.span Complex (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperator_reaches_of_ne_zero","module":"TNLean.MPS.MPDO.PhysicalSectorSupportRecurrence"},{"id":"n11695","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k : Fin F.sectorCount) →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorSupportRecurrence"},{"id":"n11696","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrixFamily","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → F.SectorEntryIndex → Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrixFamily","module":"TNLean.MPS.MPDO.PhysicalSectorSupportRecurrence"},{"id":"n11697","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix_mul_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (x…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrix_mul_apply","module":"TNLean.MPS.MPDO.PhysicalSectorSupportRecurrence"},{"id":"n11698","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.partialTraceRight_two…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.partialTraceRight_twoSiteSectorClosure","module":"TNLean.MPS.MPDO.PhysicalSectorTraceActions"},{"id":"n11699","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sum_threeSite_trace_c…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D F : K.PhysicalSectorFactorization (H : F.NeighboringTraceFactoriz…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.NeighboringTraceFactorization.sum_threeSite_trace_coefficients","module":"TNLean.MPS.MPDO.PhysicalSectorTraceActions"},{"id":"n11700","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.partialTraceRight_threeSiteSectorClosure","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k l h : Fin F.sectorCount) (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.partialTraceRight_threeSiteSectorClosure","module":"TNLean.MPS.MPDO.PhysicalSectorTraceActions"},{"id":"n11701","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.partialTraceRight_twoSiteSectorClosure","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (X…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.partialTraceRight_twoSiteSectorClosure","module":"TNLean.MPS.MPDO.PhysicalSectorTraceActions"},{"id":"n11702","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ActiveSector","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ActiveSector","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11703","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ActiveSectorEntryIndex","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ActiveSectorEntryIndex","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11704","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteMatrixFamily","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (p : Fin F.sectorCount →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteMatrixFamily","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11705","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteSpace","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (p : Fin F.sectorCount →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteSpace","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11706","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteSpace_ne_bot","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorOneSiteSpace_ne_bot","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11707","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorSet","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (p : Fin F.sectorCount →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorSet","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11708","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (p : Fin F.sectorCount →…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11709","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_isIrreducible","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_isIrreducible","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11710","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_nonneg","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_nonneg","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11711","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_pos_iff","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.activeSectorTraceMatrix_pos_iff","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11712","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.exists_active_two_cycle","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (p : Fin F.sectorCount → Real…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.exists_active_two_cycle","module":"TNLean.MPS.MPDO.PhysicalSectorTraceMatrix"},{"id":"n11713","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofVirtualCompression","kind":"def","summary":"d D E : Nat → K : MPOTensor d D → L : MPOTensor d E → K.PhysicalSectorFactorization → (V : Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofVirtualCompression","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualCompression"},{"id":"n11714","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofVirtualCompression_neighboringOperator","kind":"theorem","summary":"∀ d D E : Nat K : MPOTensor d D L : MPOTensor d E (F : K.PhysicalSectorFactorization) (V : Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofVirtualCompression_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualCompression"},{"id":"n11715","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.changePhysicalBasis_inverse_apply_eq_sum","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex) (M : MPOTensor d D), Eq (HMul.hMul V.conjTra…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.changePhysicalBasis_inverse_apply_eq_sum","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualSpanning"},{"id":"n11716","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateMatrixFamily","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → Prod (Fin (Fintype.card (…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateMatrixFamily","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualSpanning"},{"id":"n11717","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateMatrixFamily_span_eq_top","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → Eq (Submodul…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateMatrixFamily_span_eq_top","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualSpanning"},{"id":"n11718","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_mem_span_sectorVirtualMatrix…","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (i j : Fin (Fintype.card (Sig…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorCoordinateTensor_mem_span_sectorVirtualMatrixFamily","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualSpanning"},{"id":"n11719","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrixFamily_span_eq_top","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization), K.IsInjective → Eq (Submodul…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.sectorVirtualMatrixFamily_span_eq_top","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualSpanning"},{"id":"n11720","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix","kind":"def","summary":"d D : Nat → K : MPOTensor d D → (F : K.PhysicalSectorFactorization) → (k h : Fin F.sectorCount)…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualTransport"},{"id":"n11721","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix_apply","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount) (P…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix_apply","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualTransport"},{"id":"n11722","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix_one","kind":"theorem","summary":"∀ d D : Nat K : MPOTensor d D (F : K.PhysicalSectorFactorization) (k h : Fin F.sectorCount), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.neighboringOperatorWithMatrix_one","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualTransport"},{"id":"n11723","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofVirtualMatrices","kind":"def","summary":"d D E : Nat → K : MPOTensor d D → L : MPOTensor d E → K.PhysicalSectorFactorization → (X : Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofVirtualMatrices","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualTransport"},{"id":"n11724","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSectorFactorization.ofVirtualMatrices_neighboringOperator","kind":"theorem","summary":"∀ d D E : Nat K : MPOTensor d D L : MPOTensor d E (F : K.PhysicalSectorFactorization) (X : Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSectorFactorization.ofVirtualMatrices_neighboringOperator","module":"TNLean.MPS.MPDO.PhysicalSectorVirtualTransport"},{"id":"n11725","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.exists_etaLocalStructureData_lifted_supported","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F : MPOTensor.PhysicalSupport…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.exists_etaLocalStructureData_lifted_supported","module":"TNLean.MPS.MPDO.PhysicalSupportProductTransport"},{"id":"n11726","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.exists_etaLocalStructureData_lifted_supported_of…","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F : MPOTensor.PhysicalSupport…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.exists_etaLocalStructureData_lifted_supported_of_physicalSectorFactorization","module":"TNLean.MPS.MPDO.PhysicalSupportProductTransport"},{"id":"n11727","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_bondProduct_of_unitary","kind":"theorem","summary":"∀ d : Nat (V : Matrix (Fin d) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → Eq (HMul.h…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_bondProduct_of_unitary","module":"TNLean.MPS.MPDO.PhysicalSupportProductTransport"},{"id":"n11728","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData","kind":"inductive","summary":"d D : Nat → Matrix (Fin d) (Fin d) Complex → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData","module":"TNLean.MPS.MPDO.PhysicalSupportRestriction"},{"id":"n11729","layer":"formal","project":"p8","title":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_isInjective","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.commonWeightAbsorbedBasisMPOTensor_isInjective","module":"TNLean.MPS.MPDO.PhysicalSupportRestriction"},{"id":"n11730","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSupportRestrictionData","kind":"theorem","summary":"∀ d D : Nat (P : Matrix (Fin d) (Fin d) Complex) (K : MPOTensor d D), IsOrthogonalProjection P…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSupportRestrictionData","module":"TNLean.MPS.MPDO.PhysicalSupportRestriction"},{"id":"n11731","layer":"formal","project":"p8","title":"MPOTensor.nonempty_physicalSupportRestrictionData_commonWeightAbsorbedBasis","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hWeight : ∀ (j : Fin S.basisCoun…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_physicalSupportRestrictionData_commonWeightAbsorbedBasis","module":"TNLean.MPS.MPDO.PhysicalSupportRestriction"},{"id":"n11732","layer":"formal","project":"p8","title":"MPOTensor.nonempty_physicalSupportRestrictionData_of_twoSided_physicalSlice","kind":"theorem","summary":"∀ d D : Nat (P : Matrix (Fin d) (Fin d) Complex) (K : MPOTensor d D), IsOrthogonalProjection P…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_physicalSupportRestrictionData_of_twoSided_physicalSlice","module":"TNLean.MPS.MPDO.PhysicalSupportRestriction"},{"id":"n11733","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.changePhysicalBasis_supportCoordinateChange","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F G : MPOTensor.PhysicalSuppo…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.changePhysicalBasis_supportCoordinateChange","module":"TNLean.MPS.MPDO.PhysicalSupportRestrictionComparison"},{"id":"n11734","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.exists_neighboringTraceFactorization_restriction…","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F G : MPOTensor.PhysicalSuppo…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.exists_neighboringTraceFactorization_restriction_iff","module":"TNLean.MPS.MPDO.PhysicalSupportRestrictionComparison"},{"id":"n11735","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.supportCoordinateChange","kind":"def","summary":"d D : Nat → P : Matrix (Fin d) (Fin d) Complex → K : MPOTensor d D → (F G : MPOTensor.PhysicalS…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.supportCoordinateChange","module":"TNLean.MPS.MPDO.PhysicalSupportRestrictionComparison"},{"id":"n11736","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.supportCoordinateChange_isUnitaryBetween","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F G : MPOTensor.PhysicalSuppo…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.supportCoordinateChange_isUnitaryBetween","module":"TNLean.MPS.MPDO.PhysicalSupportRestrictionComparison"},{"id":"n11737","layer":"formal","project":"p8","title":"MPOTensor.PhysicalSupportRestrictionData.restricted_isSAL","kind":"theorem","summary":"∀ d D : Nat P : Matrix (Fin d) (Fin d) Complex K : MPOTensor d D (F : MPOTensor.PhysicalSupport…","labels":[],"detail_key":"p8","name":"MPOTensor.PhysicalSupportRestrictionData.restricted_isSAL","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11738","layer":"formal","project":"p8","title":"MPOTensor.blockReducedState_singleKraus_sitewise","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (L K :…","labels":[],"detail_key":"p8","name":"MPOTensor.blockReducedState_singleKraus_sitewise","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11739","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_changePhysicalBasis_isSAL_of_isometry","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_changePhysicalBasis_isSAL_of_isometry","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11740","layer":"formal","project":"p8","title":"MPOTensor.mutualInfoChain_changePhysicalBasis_of_isometry","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInfoChain_changePhysicalBasis_of_isometry","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11741","layer":"formal","project":"p8","title":"MPOTensor.normalizedMPO_changePhysicalBasis_of_isometry","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedMPO_changePhysicalBasis_of_isometry","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11742","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_changePhysicalBasis_of_isometry","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_changePhysicalBasis_of_isometry","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11743","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_changePhysicalBasis_of_isometry","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_changePhysicalBasis_of_isometry","module":"TNLean.MPS.MPDO.PhysicalSupportSALTransport"},{"id":"n11744","layer":"formal","project":"p8","title":"MPSTensor.PositiveMinimalRealizationCounterexample.gaugedTensorFamily","kind":"def","summary":"Fin 1 → MPSTensor 4 2","labels":[],"detail_key":"p8","name":"MPSTensor.PositiveMinimalRealizationCounterexample.gaugedTensorFamily","module":"TNLean.MPS.MPDO.PositiveLinearExtensionCounterexample"},{"id":"n11745","layer":"formal","project":"p8","title":"MPSTensor.PositiveMinimalRealizationCounterexample.tensorFamily","kind":"def","summary":"Fin 1 → MPSTensor 4 2","labels":[],"detail_key":"p8","name":"MPSTensor.PositiveMinimalRealizationCounterexample.tensorFamily","module":"TNLean.MPS.MPDO.PositiveLinearExtensionCounterexample"},{"id":"n11746","layer":"formal","project":"p8","title":"MPSTensor.PositiveMinimalRealizationCounterexample.tensorFamily_isInjective","kind":"theorem","summary":"∀ (k : Fin 1), Kraus.IsInjective (MPSTensor.PositiveMinimalRealizationCounterexample.tensorFami…","labels":[],"detail_key":"p8","name":"MPSTensor.PositiveMinimalRealizationCounterexample.tensorFamily_isInjective","module":"TNLean.MPS.MPDO.PositiveLinearExtensionCounterexample"},{"id":"n11747","layer":"formal","project":"p8","title":"MPSTensor.PositiveMinimalRealizationCounterexample.gaugedTensor_isInjective","kind":"theorem","summary":"Kraus.IsInjective MPSTensor.PositiveMinimalRealizationCounterexample.gaugedTensor","labels":[],"detail_key":"p8","name":"MPSTensor.PositiveMinimalRealizationCounterexample.gaugedTensor_isInjective","module":"TNLean.MPS.MPDO.PositiveMinimalRealizationCounterexample"},{"id":"n11748","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → P.EventuallyRep…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11749","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan_blockTensor","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (p : Nat), LT…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan_blockTensor","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11750","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan_of_basis_injective","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (∀ (j : Fin P.b…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.eventuallyRepresentativeWordTupleSpan_of_basis_injective","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11751","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ Y Z : Matrix…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11752","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree_of_basis_inj…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (∀ (j : Fin P.b…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree_of_basis_injective","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11753","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree_of_common_bl…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (L : Nat), LT…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_firstSiteActionAgree_of_common_blockInjective","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11754","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_sameMPV₂Pos_firstSiteActionAgree","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d D : Nat (A : MPSTensor d D), MPSTensor.IsBNTCanon…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.insertedTensor_basis_eq_of_sameMPV₂Pos_firstSiteActionAgree","module":"TNLean.MPS.MPDO.PostBlockedRepresentativeSpan"},{"id":"n11755","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_empty","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat), Ne (A.pureState N).trace 0 → Eq (A.pureBlockEntropy…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_empty","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11756","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_eq_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsSAL → ∀ N L L' : Nat, LE.le 1 L → ∀ (hLN : LE.le L (HDiv.h…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_eq_of_isSAL","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11757","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_le","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), LT.lt 0 D → Ne (A.pureState N).tr…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_le","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11758","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_nonneg","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Ne (A.pureState N).trace 0 → LE.l…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_nonneg","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11759","layer":"formal","project":"p8","title":"MPSTensor.rank_reducedPureBlockState_le","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), LE.le (A.reducedPureBlockState N…","labels":[],"detail_key":"p8","name":"MPSTensor.rank_reducedPureBlockState_le","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11760","layer":"formal","project":"p8","title":"blockEntropy_doubledTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Eq ((doubledTensor A).blockEntrop…","labels":[],"detail_key":"p8","name":"blockEntropy_doubledTensor","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11761","layer":"formal","project":"p8","title":"doubledTensor","kind":"def","summary":"d D : Nat → MPSTensor d D → MPOTensor d (HMul.hMul D D)","labels":[],"detail_key":"p8","name":"doubledTensor","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11762","layer":"formal","project":"p8","title":"mpo_doubledTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat), Eq ((doubledTensor A).mpo N) (A.pureState N)","labels":[],"detail_key":"p8","name":"mpo_doubledTensor","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11763","layer":"formal","project":"p8","title":"mutualInfoChain_doubledTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Ne (A.pureState N).trace 0 → Eq (…","labels":[],"detail_key":"p8","name":"mutualInfoChain_doubledTensor","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11764","layer":"formal","project":"p8","title":"mutualInfoChain_doubledTensor_le","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), LT.lt 0 D → Ne (A.pureState N).tr…","labels":[],"detail_key":"p8","name":"mutualInfoChain_doubledTensor_le","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11765","layer":"formal","project":"p8","title":"pureBlockEntropy_complement","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Eq (A.pureBlockEntropy N L hL) (A…","labels":[],"detail_key":"p8","name":"pureBlockEntropy_complement","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11766","layer":"formal","project":"p8","title":"pureBlockEntropy_full","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat), Ne (A.pureState N).trace 0 → Eq (A.pureBlockEntropy…","labels":[],"detail_key":"p8","name":"pureBlockEntropy_full","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11767","layer":"formal","project":"p8","title":"pureBlockEntropy_monotone","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) N L : Nat (hN : LE.le (HAdd.hAdd (HMul.hMul 2 L) 1) N), Ne (A.p…","labels":[],"detail_key":"p8","name":"pureBlockEntropy_monotone","module":"TNLean.MPS.MPDO.PureAreaLaw"},{"id":"n11768","layer":"formal","project":"p8","title":"MPSTensor.doubledTensor_isRFPViaTS_iff_hasPhysicalBlockingIsometry","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff (doubledTensor A).IsRFPViaTS A.HasPhysicalBlockingIsometry","labels":[],"detail_key":"p8","name":"MPSTensor.doubledTensor_isRFPViaTS_iff_hasPhysicalBlockingIsometry","module":"TNLean.MPS.MPDO.PureRFPBridge"},{"id":"n11769","layer":"formal","project":"p8","title":"MPSTensor.doubledTensor_isRFPViaTS_iff_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff (doubledTensor A).IsRFPViaTS A.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.doubledTensor_isRFPViaTS_iff_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRFPBridge"},{"id":"n11770","layer":"formal","project":"p8","title":"MPSTensor.isSAL_of_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsTransferIdempotent → A.IsSAL","labels":[],"detail_key":"p8","name":"MPSTensor.isSAL_of_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11771","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_eq_env_charpoly","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N L : Nat) (hL : LE.le L N), Eq (A.pureBlockEntropy N L hL) (M…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_eq_env_charpoly","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11772","layer":"formal","project":"p8","title":"MPSTensor.pureBlockEntropy_eq_of_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsTransferIdempotent → ∀ N L L' : Nat, LE.le 1 L → ∀ (hLN :…","labels":[],"detail_key":"p8","name":"MPSTensor.pureBlockEntropy_eq_of_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11773","layer":"formal","project":"p8","title":"MPSTensor.schmidtLeft","kind":"def","summary":"d D : Nat → MPSTensor d D → (L : Nat) → Matrix (Fin L → Fin d) (Prod (Fin D) (Fin D)) Complex","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtLeft","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11774","layer":"formal","project":"p8","title":"MPSTensor.schmidtLeft_gram_apply","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat) (a b : Prod (Fin D) (Fin D)), Eq (HMul.hMul (A.schmid…","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtLeft_gram_apply","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11775","layer":"formal","project":"p8","title":"MPSTensor.schmidtLeft_gram_eq_of_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsTransferIdempotent → ∀ L : Nat, LE.le 1 L → Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtLeft_gram_eq_of_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11776","layer":"formal","project":"p8","title":"MPSTensor.schmidtRight","kind":"def","summary":"d D : Nat → MPSTensor d D → (M : Nat) → Matrix (Prod (Fin D) (Fin D)) (Fin M → Fin d) Complex","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtRight","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11777","layer":"formal","project":"p8","title":"MPSTensor.schmidtRight_gram_apply","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (M : Nat) (p q : Prod (Fin D) (Fin D)), Eq (HMul.hMul (A.schmid…","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtRight_gram_apply","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11778","layer":"formal","project":"p8","title":"MPSTensor.schmidtRight_gram_eq_of_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsTransferIdempotent → ∀ M : Nat, LE.le 1 M → Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.schmidtRight_gram_eq_of_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRFPSAL"},{"id":"n11779","layer":"formal","project":"p8","title":"MPSTensor.evalWord_toMPOTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (l k : List (Fin d)), Eq (A.toMPOTensor.evalWord l k) (ite (Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_toMPOTensor","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11780","layer":"formal","project":"p8","title":"MPSTensor.mpo_toMPOTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat), Eq (A.toMPOTensor.mpo N) (Matrix.diagonal A.mpv)","labels":[],"detail_key":"p8","name":"MPSTensor.mpo_toMPOTensor","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11781","layer":"formal","project":"p8","title":"MPSTensor.rank_mpo_toMPOTensor","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (N : Nat), Eq (A.toMPOTensor.mpo N).rank (Fintype.card (Subtype…","labels":[],"detail_key":"p8","name":"MPSTensor.rank_mpo_toMPOTensor","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11782","layer":"formal","project":"p8","title":"MPSTensor.toMPOTensor","kind":"def","summary":"d D : Nat → MPSTensor d D → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPSTensor.toMPOTensor","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11783","layer":"formal","project":"p8","title":"MPSTensor.toMPOTensor_isZCL_iff_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.toMPOTensor.IsZCL A.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.toMPOTensor_isZCL_iff_isTransferIdempotent","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11784","layer":"formal","project":"p8","title":"MPSTensor.toMPOTensor_isZCL_iff_mps_isZCL","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.toMPOTensor.IsZCL A.IsZCL","labels":[],"detail_key":"p8","name":"MPSTensor.toMPOTensor_isZCL_iff_mps_isZCL","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11785","layer":"formal","project":"p8","title":"MPSTensor.toMPOTensor_transferMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Eq A.toMPOTensor.transferMap (Kraus.transferMap A)","labels":[],"detail_key":"p8","name":"MPSTensor.toMPOTensor_transferMap","module":"TNLean.MPS.MPDO.PureRecovery"},{"id":"n11786","layer":"formal","project":"p8","title":"MPOTensor.mpvOverlap_toMPSTensor_self","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (N : Nat), Eq (K.toMPSTensor.mpvOverlap K.toMPSTensor N) (HMul.…","labels":[],"detail_key":"p8","name":"MPOTensor.mpvOverlap_toMPSTensor_self","module":"TNLean.MPS.MPDO.Purity"},{"id":"n11787","layer":"formal","project":"p8","title":"MPOTensor.mpvOverlap_toMPSTensor_self_eq_trace_sq","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsMPDO → ∀ N : Nat, LT.lt 0 N → Eq (K.toMPSTensor.mpvOverlap…","labels":[],"detail_key":"p8","name":"MPOTensor.mpvOverlap_toMPSTensor_self_eq_trace_sq","module":"TNLean.MPS.MPDO.Purity"},{"id":"n11788","layer":"formal","project":"p8","title":"MPOTensor.mpvOverlap_toMPSTensor_self_re_eq_frobenius_norm_sq","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (N : Nat), Eq (K.toMPSTensor.mpvOverlap K.toMPSTensor N).re (HP…","labels":[],"detail_key":"p8","name":"MPOTensor.mpvOverlap_toMPSTensor_self_re_eq_frobenius_norm_sq","module":"TNLean.MPS.MPDO.Purity"},{"id":"n11789","layer":"formal","project":"p8","title":"MPOTensor.pairCfgEquiv","kind":"def","summary":"(d N : Nat) → Equiv (Prod (Fin N → Fin d) (Fin N → Fin d)) (Fin N → Fin (HMul.hMul d d))","labels":[],"detail_key":"p8","name":"MPOTensor.pairCfgEquiv","module":"TNLean.MPS.MPDO.Purity"},{"id":"n11790","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_cpsvVerticalDecomposition","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → Nonempty M.CPSVVerticalDecomposi…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_cpsvVerticalDecomposition","module":"TNLean.MPS.MPDO.RFPPositiveFusionDecomposition"},{"id":"n11791","layer":"formal","project":"p8","title":"MPOTensor.exists_positiveFusionDecomposition_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → Exists fun g => E…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_positiveFusionDecomposition_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPPositiveFusionDecomposition"},{"id":"n11792","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.BoundarySubspinIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.BoundarySubspinIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11793","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.EtaPreparationIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.EtaPreparationIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11794","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.NeighborSubspinIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.NeighborSubspinIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11795","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.OmegaPreparationIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.OmegaPreparationIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11796","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.OuterSubspinIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.OuterSubspinIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11797","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.PhysicalSectorIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.PhysicalSectorIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11798","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.SectorSiteIndex","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.SectorSiteIndex","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11799","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.etaShiftEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.etaShiftEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11800","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.omegaShiftEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.omegaShiftEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11801","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s0Map","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s0Map","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11802","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s0Map_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s0Map_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11803","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s0RegroupEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s0RegroupEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11804","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s0SectorMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s0SectorMap","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11805","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s0SectorMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s0SectorMap_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11806","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s2Map","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s2Map","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11807","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s2Map_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s2Map_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11808","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s2SectorMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s2SectorMap","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11809","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s2SectorMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s2SectorMap_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11810","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.s2ShiftEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.s2ShiftEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11811","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t0Map","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t0Map","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11812","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t0Map_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t0Map_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11813","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t0RegroupEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t0RegroupEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11814","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t0SectorMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t0SectorMap","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11815","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t0SectorMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t0SectorMap_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11816","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t2Map","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t2Map","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11817","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t2Map_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t2Map_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11818","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t2SectorMap","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t2SectorMap","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11819","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t2SectorMap_isKrausCPTP","kind":"theorem","summary":"∀ dA dB dC : Nat rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t2SectorMap_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11820","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.t2ShiftEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.t2ShiftEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11821","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.threeSiteTraceRegroupEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.threeSiteTraceRegroupEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11822","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.twoSiteTraceRegroupEquiv","kind":"def","summary":"dA dB dC : Nat → rhoABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.twoSiteTraceRegroupEquiv","module":"TNLean.MPS.MPDO.RFPSubspinMaps"},{"id":"n11823","layer":"formal","project":"p8","title":"MPOTensor.IsRFPViaTS","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTS"},{"id":"n11824","layer":"formal","project":"p8","title":"MPOTensor.isPhysicalTraceIdempotent_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsRFPViaTS → M.IsPhysicalTraceIdempotent","labels":[],"detail_key":"p8","name":"MPOTensor.isPhysicalTraceIdempotent_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTS"},{"id":"n11825","layer":"formal","project":"p8","title":"MPOTensor.isSourceZCL_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsRFPViaTS → Ne M.physTraceTransfer 0 → M.IsSourceZCL","labels":[],"detail_key":"p8","name":"MPOTensor.isSourceZCL_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTS"},{"id":"n11826","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_sq_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsRFPViaTS → Eq (HMul.hMul M.physTraceTransfer M.physTraceTr…","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_sq_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTS"},{"id":"n11827","layer":"formal","project":"p8","title":"MPOTensor.IsRFPViaTS.blockTwo","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsRFPViaTS → M.blockTwo.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.IsRFPViaTS.blockTwo","module":"TNLean.MPS.MPDO.RFPViaTSBlocking"},{"id":"n11828","layer":"formal","project":"p8","title":"MPOTensor.blockTwo_isRFPViaTS_of_two_four_maps","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S₂₄ : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Pr…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTwo_isRFPViaTS_of_two_four_maps","module":"TNLean.MPS.MPDO.RFPViaTSBlocking"},{"id":"n11829","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstTwoSites","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstTwoSites","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11830","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstTwoSites_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstTwoSites_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11831","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstTwoSites_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstTwoSites_mpo","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11832","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstTwoSites_physCloseN","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstTwoSites_physCloseN","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11833","layer":"formal","project":"p8","title":"MPOTensor.exists_global_renormalization_maps","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsRFPViaTS → Exists fun S => Exists fun T => And (IsKrausCPT…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_global_renormalization_maps","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11834","layer":"formal","project":"p8","title":"MPOTensor.refineFirstSite","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Complex) (Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstSite","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11835","layer":"formal","project":"p8","title":"MPOTensor.refineFirstSite_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat T : LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Complex) (Matrix (Prod (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstSite_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11836","layer":"formal","project":"p8","title":"MPOTensor.refineFirstSite_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (T : LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstSite_mpo","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11837","layer":"formal","project":"p8","title":"MPOTensor.refineFirstSite_physCloseN","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (T : LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Com…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstSite_physCloseN","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11838","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_ne_zero_of_isHorizontalCF_isMPDO_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → ∀ (N : Nat), LT.l…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_ne_zero_of_isHorizontalCF_isMPDO_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTSGlobal"},{"id":"n11839","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstBlock","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstBlock","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11840","layer":"formal","project":"p8","title":"MPOTensor.coarsenFirstBlock_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat S : LinearMap (RingHom.id Complex) (Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.coarsenFirstBlock_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11841","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → M.IsSAL","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11842","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_isRFPViaTS_of_trace_ne_zero","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → (∀ (N : Nat), LT.lt 0 N → Ne (M.mpo N).trace 0) → M…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_isRFPViaTS_of_trace_ne_zero","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11843","layer":"formal","project":"p8","title":"MPOTensor.mutualInformation_bipartition_eq_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsRFPViaTS → ∀ (N a b : Nat) (hIn : Eq N (HAdd.hAdd (HAdd.hA…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInformation_bipartition_eq_of_isRFPViaTS","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11844","layer":"formal","project":"p8","title":"MPOTensor.mutualInformation_bipartition_eq_of_local_transfers","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N a b : Nat) (hIn : Eq N (HAdd.hAdd (HAdd.hAdd a 1) (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.mutualInformation_bipartition_eq_of_local_transfers","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11845","layer":"formal","project":"p8","title":"MPOTensor.refineFirstBlock","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Complex) (Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstBlock","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11846","layer":"formal","project":"p8","title":"MPOTensor.refineFirstBlock_isKrausCPTP","kind":"theorem","summary":"∀ d : Nat T : LinearMap (RingHom.id Complex) (Matrix (Fin d) (Fin d) Complex) (Matrix (Prod (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.refineFirstBlock_isKrausCPTP","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11847","layer":"formal","project":"p8","title":"MPOTensor.tensorMapBoth_bipartitionedNormalizedMPO","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N a b : Nat) (hIn : Eq N (HAdd.hAdd (HAdd.hAdd a 1) (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.tensorMapBoth_bipartitionedNormalizedMPO","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11848","layer":"formal","project":"p8","title":"MPOTensor.tensorMapBoth_bipartitionedNormalizedMPO_reverse","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N a b : Nat) (hIn : Eq N (HAdd.hAdd (HAdd.hAdd a 1) (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.tensorMapBoth_bipartitionedNormalizedMPO_reverse","module":"TNLean.MPS.MPDO.RFPViaTSSAL"},{"id":"n11849","layer":"formal","project":"p8","title":"MPOTensor.exists_rephased_inverseMapPhysicalSectorFactorization","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_rephased_inverseMapPhysicalSectorFactorization","module":"TNLean.MPS.MPDO.RecurrentSectorRephasing"},{"id":"n11850","layer":"formal","project":"p8","title":"MPOTensor.exists_vertexPhase_smul_posSemidef","kind":"theorem","summary":"∀ d : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_vertexPhase_smul_posSemidef","module":"TNLean.MPS.MPDO.RecurrentSectorRephasing"},{"id":"n11851","layer":"formal","project":"p8","title":"MPOTensor.nonempty_etaLocalStructureData_of_recurrentSupport","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.nonempty_etaLocalStructureData_of_recurrentSupport","module":"TNLean.MPS.MPDO.RecurrentSectorRephasing"},{"id":"n11852","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.markedChainCoefficient_eq_reflectedAdjoint","kind":"theorem","summary":"∀ d D n : Nat M : MPOTensor d D, M.IsMPDO → ∀ (A : MPSTensor (HMul.hMul D D) n) (V : Matrix (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.markedChainCoefficient_eq_reflectedAdjoint","module":"TNLean.MPS.MPDO.ReflectedMarkedChain"},{"id":"n11853","layer":"formal","project":"p8","title":"MPOTensor.horizontalSlice","kind":"def","summary":"D n : Nat → MPSTensor (HMul.hMul D D) n → Fin n → Fin n → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.horizontalSlice","module":"TNLean.MPS.MPDO.ReflectedMarkedChain"},{"id":"n11854","layer":"formal","project":"p8","title":"MPOTensor.markedChainCoefficient","kind":"def","summary":"d D n : Nat → MPSTensor (HMul.hMul D D) n → MPOTensor d D → Fin n → Fin n → List (Fin d) → List…","labels":[],"detail_key":"p8","name":"MPOTensor.markedChainCoefficient","module":"TNLean.MPS.MPDO.ReflectedMarkedChain"},{"id":"n11855","layer":"formal","project":"p8","title":"MPOTensor.reflectedAdjoint","kind":"def","summary":"D n : Nat → MPSTensor (HMul.hMul D D) n → MPSTensor (HMul.hMul D D) n","labels":[],"detail_key":"p8","name":"MPOTensor.reflectedAdjoint","module":"TNLean.MPS.MPDO.ReflectedMarkedChain"},{"id":"n11856","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.EventuallyRepresentativeWordTupleSpan","kind":"def","summary":"d : Nat → MPSTensor.SectorDecomposition d → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.EventuallyRepresentativeWordTupleSpan","module":"TNLean.MPS.MPDO.RepresentativeGroupedLemmaL"},{"id":"n11857","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.RepresentativeWordTupleSpanAt","kind":"def","summary":"d : Nat → MPSTensor.SectorDecomposition d → Nat → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.RepresentativeWordTupleSpanAt","module":"TNLean.MPS.MPDO.RepresentativeGroupedLemmaL"},{"id":"n11858","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_firstSiteActionAgree","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) Y Z : Matrix (Fin d) (Fin d) Complex, P.Eventua…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_firstSiteActionAgree","module":"TNLean.MPS.MPDO.RepresentativeGroupedLemmaL"},{"id":"n11859","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_firstSiteActionAgree_of_coeff_ne…","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) Y Z : Matrix (Fin d) (Fin d) Complex, P.toTenso…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_firstSiteActionAgree_of_coeff_ne_zero","module":"TNLean.MPS.MPDO.RepresentativeGroupedLemmaL"},{"id":"n11860","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_sameMPV₂Pos_firstSiteActionAgree","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (P : MPSTensor.SectorDecomposition d) Y Z : Matrix (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.insertedTensor_basis_eq_of_sameMPV₂Pos_firstSiteActionAgree","module":"TNLean.MPS.MPDO.RepresentativeGroupedLemmaL"},{"id":"n11861","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.markedTensor_basis_eq_of_trace_agree","kind":"theorem","summary":"∀ d e : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (C E : (j :…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.markedTensor_basis_eq_of_trace_agree","module":"TNLean.MPS.MPDO.RepresentativeGroupedMarkedLemmaL"},{"id":"n11862","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.markedTensor_basis_eq_of_trace_agree","kind":"theorem","summary":"∀ d e : Nat (P : MPSTensor.SectorDecomposition d) (C E : (j : Fin P.basisCount) → MPSTensor e (…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.markedTensor_basis_eq_of_trace_agree","module":"TNLean.MPS.MPDO.RepresentativeGroupedMarkedLemmaL"},{"id":"n11863","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.markedTensor_basis_eq_of_trace_agree_of_coeff_ne_zero","kind":"theorem","summary":"∀ d e : Nat (P : MPSTensor.SectorDecomposition d) (C E : (j : Fin P.basisCount) → MPSTensor e (…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.markedTensor_basis_eq_of_trace_agree_of_coeff_ne_zero","module":"TNLean.MPS.MPDO.RepresentativeGroupedMarkedLemmaL"},{"id":"n11864","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.B_transpose_mul_B","kind":"theorem","summary":"∀ (N : Nat), Eq (HMul.hMul (MPOTensor.RescalingStableLengthDependentRFP.B N).transpose (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.B_transpose_mul_B","module":"TNLean.MPS.MPDO.RescalingStableChiAttachment"},{"id":"n11865","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChiTracePowerForm","kind":"def","summary":"MPOTensor.PositiveBNTLabelChiTracePowerForm MPOTensor.RescalingStableLengthDependentRFP.oneLabe…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChiTracePowerForm","module":"TNLean.MPS.MPDO.RescalingStableChiUniformity"},{"id":"n11866","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.A","kind":"def","summary":"Fin 4 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.A","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11867","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R","kind":"def","summary":"MPOTensor 4 4","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11868","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R_eq_vecMulVec","kind":"theorem","summary":"∀ (p q : Fin 4), Eq (MPOTensor.RescalingStableLengthDependentRFP.R p q) (HSMul.hSMul (25 / 32)…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R_eq_vecMulVec","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11869","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R_isMPDO","kind":"theorem","summary":"MPOTensor.RescalingStableLengthDependentRFP.R.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R_isMPDO","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11870","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.mpo_R_entry_formula","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → ∀ (p q : Fin N → Fin 4), Eq (MPOTensor.RescalingStableLengthDependentRFP…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.mpo_R_entry_formula","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11871","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChi","kind":"def","summary":"MPOTensor.DiagonalChiFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChi","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11872","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChiScaled","kind":"def","summary":"Real → MPOTensor.DiagonalChiFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelChiScaled","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11873","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelCoeffs","kind":"def","summary":"MPOTensor.BNTLabelCoefficientFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelCoeffs","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11874","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelCoeffs_coeff","kind":"theorem","summary":"∀ (L : Nat), Eq (MPOTensor.RescalingStableLengthDependentRFP.oneLabelCoeffs.coeff L 0 0 0) (HAd…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneLabelCoeffs_coeff","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11875","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.rescaledCoeffs","kind":"def","summary":"Real → MPOTensor.BNTLabelCoefficientFamily (Fin 1)","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.rescaledCoeffs","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11876","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.rescaledCoeffs_coeff","kind":"theorem","summary":"∀ (s : Real) (L : Nat), Eq ((MPOTensor.RescalingStableLengthDependentRFP.rescaledCoeffs s).coef…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.rescaledCoeffs_coeff","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11877","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_diag_alt","kind":"theorem","summary":"Eq ((Kraus.transferMap MPOTensor.RescalingStableLengthDependentRFP.A) (Matrix.of (Matrix.vecCon…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_diag_alt","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11878","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_eigen","kind":"theorem","summary":"And (Eq ((Kraus.transferMap MPOTensor.RescalingStableLengthDependentRFP.A) 1) (HSMul.hSMul 1 1)…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_eigen","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11879","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_one","kind":"theorem","summary":"Eq ((Kraus.transferMap MPOTensor.RescalingStableLengthDependentRFP.A) 1) 1","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_one","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11880","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_single01","kind":"theorem","summary":"Eq ((Kraus.transferMap MPOTensor.RescalingStableLengthDependentRFP.A) (Matrix.single 0 1 1)) 0","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_single01","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11881","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_single10","kind":"theorem","summary":"Eq ((Kraus.transferMap MPOTensor.RescalingStableLengthDependentRFP.A) (Matrix.single 1 0 1)) 0","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.transferMap_A_single10","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFP"},{"id":"n11882","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R_toMPSTensor_isCPSVCanonicalForm","kind":"theorem","summary":"MPOTensor.RescalingStableLengthDependentRFP.R.toMPSTensor.IsCPSVCanonicalForm","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R_toMPSTensor_isCPSVCanonicalForm","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFPCanonicalForm"},{"id":"n11883","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R_toMPSTensor_isInjective","kind":"theorem","summary":"Kraus.IsInjective MPOTensor.RescalingStableLengthDependentRFP.R.toMPSTensor","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R_toMPSTensor_isInjective","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFPCanonicalForm"},{"id":"n11884","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.cFun_fixedDiag","kind":"theorem","summary":"Eq (HMul.hMul (HPow.hPow (25 / 32) 2) (Finset.univ.sum fun b => HMul.hMul (HPow.hPow (MPOTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.cFun_fixedDiag","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFPCanonicalForm"},{"id":"n11885","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.toMPSTensor_R_apply","kind":"theorem","summary":"∀ (p q : Fin 4), Eq (MPOTensor.RescalingStableLengthDependentRFP.R.toMPSTensor (finProdFinEquiv…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.toMPSTensor_R_apply","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFPCanonicalForm"},{"id":"n11886","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.isRFPViaTS_R","kind":"theorem","summary":"MPOTensor.RescalingStableLengthDependentRFP.R.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.isRFPViaTS_R","module":"TNLean.MPS.MPDO.RescalingStableLengthDependentRFPViaTS"},{"id":"n11887","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.R_isSimple","kind":"theorem","summary":"MPOTensor.RescalingStableLengthDependentRFP.R.IsSimple","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.R_isSimple","module":"TNLean.MPS.MPDO.RescalingStableSimple"},{"id":"n11888","layer":"formal","project":"p8","title":"MPOTensor.RescalingStableLengthDependentRFP.oneSiteDoubledEquiv","kind":"def","summary":"Equiv (Fin (HMul.hMul (MPSTensor.blockPhysDim 4 1) (MPSTensor.blockPhysDim 4 1))) (Fin (HMul.hM…","labels":[],"detail_key":"p8","name":"MPOTensor.RescalingStableLengthDependentRFP.oneSiteDoubledEquiv","module":"TNLean.MPS.MPDO.RescalingStableSimple"},{"id":"n11889","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_nonempty_verticalBNTGrouping","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → Exists fun r => Exists fun dim =…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_nonempty_verticalBNTGrouping","module":"TNLean.MPS.MPDO.RetainedClass"},{"id":"n11890","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.verticalTensor_ne_zero","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → Ne M.verticalTensor 0","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.verticalTensor_ne_zero","module":"TNLean.MPS.MPDO.RetainedClass"},{"id":"n11891","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_isSSAEquality_tripartite_m","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hMpdo : M.IsMPDO), (∀ (N : Nat), LT.lt 0 N → Ne (M.mpo N).trac…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_isSSAEquality_tripartite_m","module":"TNLean.MPS.MPDO.SALArbitraryCut"},{"id":"n11892","layer":"formal","project":"p8","title":"MPOTensor.isSAL_of_quantumMarkovDecomposition_tripartite_m","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → (∀ (N : Nat), LT.lt 0 N → Ne (M.mpo N).trace 0) → (…","labels":[],"detail_key":"p8","name":"MPOTensor.isSAL_of_quantumMarkovDecomposition_tripartite_m","module":"TNLean.MPS.MPDO.SALArbitraryCut"},{"id":"n11893","layer":"formal","project":"p8","title":"MPOTensor.isSSAEquality_tripartite_m_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hSAL : M.IsSAL) N m : Nat (hm1 : LE.le 1 m) (hmN : LE.le m (HD…","labels":[],"detail_key":"p8","name":"MPOTensor.isSSAEquality_tripartite_m_of_isSAL","module":"TNLean.MPS.MPDO.SALArbitraryCut"},{"id":"n11894","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSectorFactorization_with_cyclicNeighboringProduct_posSemidef_of_…","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → K.IsSAL → Exists fun F => ∀ N : Nat [inst : Ne…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSectorFactorization_with_cyclicNeighboringProduct_posSemidef_of_isSAL","module":"TNLean.MPS.MPDO.SALPhysicalSectorCyclicPositivity"},{"id":"n11895","layer":"formal","project":"p8","title":"MPOTensor.IsSAL.isSourceZCL_of_physTraceTransfer_sq","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSAL → Eq (HMul.hMul M.physTraceTransfer M.physTraceTransfer)…","labels":[],"detail_key":"p8","name":"MPOTensor.IsSAL.isSourceZCL_of_physTraceTransfer_sq","module":"TNLean.MPS.MPDO.SALTraceTransfer"},{"id":"n11896","layer":"formal","project":"p8","title":"MPOTensor.IsSAL.physTraceTransfer_ne_zero","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSAL → Ne M.physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.IsSAL.physTraceTransfer_ne_zero","module":"TNLean.MPS.MPDO.SALTraceTransfer"},{"id":"n11897","layer":"formal","project":"p8","title":"MPOTensor.etaOfSectorTensors","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.etaOfSectorTensors","module":"TNLean.MPS.MPDO.SectorEtaContraction"},{"id":"n11898","layer":"formal","project":"p8","title":"MPOTensor.etaOfSectorTensors_apply","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.etaOfSectorTensors_apply","module":"TNLean.MPS.MPDO.SectorEtaContraction"},{"id":"n11899","layer":"formal","project":"p8","title":"MPOTensor.sum_cyclic_sectorTensor_eq_prod_etaOfSectorTensors","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_cyclic_sectorTensor_eq_prod_etaOfSectorTensors","module":"TNLean.MPS.MPDO.SectorEtaContraction"},{"id":"n11900","layer":"formal","project":"p8","title":"MPOTensor.trace_sector_word_eq_prod_etaOfSectorTensors","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_sector_word_eq_prod_etaOfSectorTensors","module":"TNLean.MPS.MPDO.SectorEtaContraction"},{"id":"n11901","layer":"formal","project":"p8","title":"MPOTensor.SectorChainIndex","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorChainIndex","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11902","layer":"formal","project":"p8","title":"MPOTensor.SectorFiber","kind":"def","summary":"d N : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorFiber","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11903","layer":"formal","project":"p8","title":"MPOTensor.conjugatePhysical","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.conjugatePhysical","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11904","layer":"formal","project":"p8","title":"MPOTensor.cyclicEtaTensorProduct","kind":"def","summary":"d N : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.cyclicEtaTensorProduct","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11905","layer":"formal","project":"p8","title":"MPOTensor.mpo_reindex_sectorChainEquiv_eq_blockDiagonal_cyclicEta","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_reindex_sectorChainEquiv_eq_blockDiagonal_cyclicEta","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11906","layer":"formal","project":"p8","title":"MPOTensor.mpo_submatrix_sector_eq_cyclicEtaTensorProduct","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_submatrix_sector_eq_cyclicEtaTensorProduct","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11907","layer":"formal","project":"p8","title":"MPOTensor.sectorChainEquiv","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorChainEquiv","module":"TNLean.MPS.MPDO.SectorEtaOperator"},{"id":"n11908","layer":"formal","project":"p8","title":"MPOTensor.cyclicEtaTensorProduct_posSemidef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.cyclicEtaTensorProduct_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11909","layer":"formal","project":"p8","title":"MPOTensor.etaOfSectorTensors_kronecker_posSemidef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.etaOfSectorTensors_kronecker_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11910","layer":"formal","project":"p8","title":"MPOTensor.etaOfSectorTensors_self_posSemidef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.etaOfSectorTensors_self_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11911","layer":"formal","project":"p8","title":"MPOTensor.exists_explicitEtaOperators_sectorEta","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_explicitEtaOperators_sectorEta","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11912","layer":"formal","project":"p8","title":"MPOTensor.exists_smul_posSemidef_sectorEta","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_smul_posSemidef_sectorEta","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11913","layer":"formal","project":"p8","title":"MPOTensor.mpo_conjugatePhysical_eq","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (U : Matrix (Fin d) (Fin d) Complex) N : Nat [NeZero N], Eq ((K…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_conjugatePhysical_eq","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11914","layer":"formal","project":"p8","title":"MPOTensor.mpo_conjugatePhysical_posSemidef","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (U : Matrix (Fin d) (Fin d) Complex) N : Nat [NeZero N], (K.mpo…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_conjugatePhysical_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11915","layer":"formal","project":"p8","title":"MPOTensor.sectorEta_kronecker_posSemidef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorEta_kronecker_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11916","layer":"formal","project":"p8","title":"MPOTensor.sectorEta_self_posSemidef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorEta_self_posSemidef","module":"TNLean.MPS.MPDO.SectorEtaPositivity"},{"id":"n11917","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.ofSectorTensors","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.ofSectorTensors","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11918","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.traceMatrix_ofSectorTensors","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.traceMatrix_ofSectorTensors","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11919","layer":"formal","project":"p8","title":"MPOTensor.closedSectorL","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorL","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11920","layer":"formal","project":"p8","title":"MPOTensor.closedSectorR","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorR","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11921","layer":"formal","project":"p8","title":"MPOTensor.exists_physicalSlice_sector_factorization","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → Ne (K.mpo 4).trace 0 → ∀ ρ : Matrix (Prod (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_physicalSlice_sector_factorization","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11922","layer":"formal","project":"p8","title":"MPOTensor.inverseMap_hayashi_sector_offdiagonal","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsInjective → ∀ (R : Matrix (Fin D) (Fin D) Complex) (ρ : Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseMap_hayashi_sector_offdiagonal","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11923","layer":"formal","project":"p8","title":"MPOTensor.isThreeSiteClosure_reducedBlockState","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D), K.IsThreeSiteClosure K.normalizedFourSiteTail ((K.reducedBlock…","labels":[],"detail_key":"p8","name":"MPOTensor.isThreeSiteClosure_reducedBlockState","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11924","layer":"formal","project":"p8","title":"MPOTensor.physicalSlice_neighboring_contraction","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (hK : K.IsInjective) (R : Matrix (Fin D) (Fin D) Complex) (ρ :…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSlice_neighboring_contraction","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11925","layer":"formal","project":"p8","title":"MPOTensor.physicalSlice_sector_factorization","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (hK : K.IsInjective) (R : Matrix (Fin D) (Fin D) Complex) (ρ :…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalSlice_sector_factorization","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11926","layer":"formal","project":"p8","title":"MPOTensor.sectorEta","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorEta","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11927","layer":"formal","project":"p8","title":"MPOTensor.sectorTensorL","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorTensorL","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11928","layer":"formal","project":"p8","title":"MPOTensor.sectorTensorR","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorTensorR","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11929","layer":"formal","project":"p8","title":"MPOTensor.trace_sectorEta","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_sectorEta","module":"TNLean.MPS.MPDO.SectorFactorization"},{"id":"n11930","layer":"formal","project":"p8","title":"MPOTensor.closedSectorPairingOperator","kind":"def","summary":"D m : Nat → dL dR : Fin m → Nat → ((q : Fin m) → Fin D → Matrix (Fin (dL q)) (Fin (dL q)) Compl…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorPairingOperator","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11931","layer":"formal","project":"p8","title":"MPOTensor.closedSectorTraceMatrix","kind":"def","summary":"d D : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorTraceMatrix","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11932","layer":"formal","project":"p8","title":"MPOTensor.closedSectorTraceMatrix_apply","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorTraceMatrix_apply","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11933","layer":"formal","project":"p8","title":"MPOTensor.closedSectorTraceMatrix_normalized_relations","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSectorTraceMatrix_normalized_relations","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11934","layer":"formal","project":"p8","title":"MPOTensor.closedSector_operator_idempotent_of_physTraceTransfer_sq","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSector_operator_idempotent_of_physTraceTransfer_sq","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11935","layer":"formal","project":"p8","title":"MPOTensor.closedSector_operator_normalized_idempotent","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSector_operator_normalized_idempotent","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11936","layer":"formal","project":"p8","title":"MPOTensor.closedSector_operator_quasi_idempotent","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.closedSector_operator_quasi_idempotent","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11937","layer":"formal","project":"p8","title":"MPOTensor.concrete_physTraceTransfer_eq_sum_closedSector","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.concrete_physTraceTransfer_eq_sum_closedSector","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11938","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_eq_sum_closedSector","kind":"theorem","summary":"∀ d D m : Nat dL dR : Fin m → Nat (K : MPOTensor d D) (decompB : Equiv (Fin d) (Sigma fun q =>…","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_eq_sum_closedSector","module":"TNLean.MPS.MPDO.SectorPairingTransfer"},{"id":"n11939","layer":"formal","project":"p8","title":"MPOTensor.CycleFiber","kind":"def","summary":"d N : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.CycleFiber","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11940","layer":"formal","project":"p8","title":"MPOTensor.IsRecurrentSupport","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.IsRecurrentSupport","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11941","layer":"formal","project":"p8","title":"MPOTensor.IsSectorEdge","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.IsSectorEdge","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11942","layer":"formal","project":"p8","title":"MPOTensor.SectorReaches","kind":"def","summary":"d : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorReaches","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11943","layer":"formal","project":"p8","title":"MPOTensor.cycleFiberToSectorFiber","kind":"def","summary":"d N : Nat → ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.cycleFiberToSectorFiber","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11944","layer":"formal","project":"p8","title":"MPOTensor.cyclicEtaTensorProduct_submatrix_eq_finKronecker","kind":"theorem","summary":"∀ d N : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.cyclicEtaTensorProduct_submatrix_eq_finKronecker","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11945","layer":"formal","project":"p8","title":"MPOTensor.exists_pi_smul_posSemidef_of_cyclicEtaTensorProduct_posSemidef","kind":"theorem","summary":"∀ d N : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_pi_smul_posSemidef_of_cyclicEtaTensorProduct_posSemidef","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11946","layer":"formal","project":"p8","title":"MPOTensor.exists_pi_smul_posSemidef_of_sectorEta_cycle","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_pi_smul_posSemidef_of_sectorEta_cycle","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11947","layer":"formal","project":"p8","title":"MPOTensor.isRecurrentSupport_iff_directedWalkReturns","kind":"theorem","summary":"∀ d : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.isRecurrentSupport_iff_directedWalkReturns","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11948","layer":"formal","project":"p8","title":"MPOTensor.sectorReaches_iff_directedWalkReaches","kind":"theorem","summary":"∀ d : Nat ρ : Matrix (Prod (Fin d) (Prod (Fin d) (Fin d))) (Prod (Fin d) (Prod (Fin d) (Fin d))…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorReaches_iff_directedWalkReaches","module":"TNLean.MPS.MPDO.SectorRecurrence"},{"id":"n11949","layer":"formal","project":"p8","title":"MPOTensor.SectorProjectorData","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → Complex → Matrix (Fin D) (Fin D) C…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorProjectorData","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11950","layer":"formal","project":"p8","title":"MPOTensor.SectorProjectorData.trace_sectorCompression","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D P : Matrix (Fin d) (Fin d) Complex μ : Complex Eα : Matrix (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorProjectorData.trace_sectorCompression","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11951","layer":"formal","project":"p8","title":"MPOTensor.SectorProjectorData.weight_mul_blockChainTrace_pos","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D P : Matrix (Fin d) (Fin d) Complex μ : Complex Eα : Matrix (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.SectorProjectorData.weight_mul_blockChainTrace_pos","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11952","layer":"formal","project":"p8","title":"MPOTensor.blockChainTrace","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex → Nat → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.blockChainTrace","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11953","layer":"formal","project":"p8","title":"MPOTensor.blockChainTrace_pos","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D P₁ P : Matrix (Fin d) (Fin d) Complex μ₁ μ : Complex Eα : Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.blockChainTrace_pos","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11954","layer":"formal","project":"p8","title":"MPOTensor.representativeLoop","kind":"def","summary":"D n : Nat → MPSTensor (HMul.hMul D D) n → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.representativeLoop","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11955","layer":"formal","project":"p8","title":"MPOTensor.representativeLoop_cast","kind":"theorem","summary":"∀ D n m : Nat (hdim : Eq n m) (A : MPSTensor (HMul.hMul D D) n), Eq (MPOTensor.representativeLo…","labels":[],"detail_key":"p8","name":"MPOTensor.representativeLoop_cast","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11956","layer":"formal","project":"p8","title":"MPOTensor.sectorCompression","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → (N : Nat) → Matrix (Fin (HAdd.hAdd…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorCompression","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11957","layer":"formal","project":"p8","title":"MPOTensor.sectorCompression_def","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (N : Nat), Eq (M.sectorCom…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorCompression_def","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11958","layer":"formal","project":"p8","title":"MPOTensor.sectorCompression_posSemidef","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ P : Matrix (Fin d) (Fin d) Complex, P.IsHermitian…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorCompression_posSemidef","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11959","layer":"formal","project":"p8","title":"MPOTensor.sectorCompression_trace_pos","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ P : Matrix (Fin d) (Fin d) Complex, P.IsHermitian…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorCompression_trace_pos","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11960","layer":"formal","project":"p8","title":"MPOTensor.sectorProjectorData_of_gauge_corner","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D) (A : MPSTensor (HMul.hMul D D) n) (V : Matrix (Fin d) (Fin n)…","labels":[],"detail_key":"p8","name":"MPOTensor.sectorProjectorData_of_gauge_corner","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11961","layer":"formal","project":"p8","title":"MPOTensor.sector_weight_pos","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D P₁ P : Matrix (Fin d) (Fin d) Complex μ₁ μ : Complex Eα : Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.sector_weight_pos","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11962","layer":"formal","project":"p8","title":"MPOTensor.sum_evalWord_diag_eq_verticalLoop_pow","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat), Eq (Finset.univ.sum fun σ => M.evalWord (List.ofFn σ…","labels":[],"detail_key":"p8","name":"MPOTensor.sum_evalWord_diag_eq_verticalLoop_pow","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11963","layer":"formal","project":"p8","title":"MPOTensor.trace_firstSiteMatrix_mul_mpo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (N : Nat), Eq (HMul.hMul (…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_firstSiteMatrix_mul_mpo","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11964","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_eq_trace_verticalLoop_pow","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (N : Nat), Eq (M.mpo N).trace (HPow.hPow M.verticalLoop N).trace","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_eq_trace_verticalLoop_pow","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11965","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_ne_zero_of_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsSourceZCL → ∀ N : Nat, LT.lt 0 N → Ne (M.mpo N).trace 0","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_ne_zero_of_isSourceZCL","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11966","layer":"formal","project":"p8","title":"MPOTensor.trace_mpo_pos_of_isMPDO_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → M.IsSourceZCL → ∀ N : Nat, LT.lt 0 N → LT.lt 0 (M.m…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_mpo_pos_of_isMPDO_isSourceZCL","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11967","layer":"formal","project":"p8","title":"MPOTensor.trace_sectorCompression_of_idempotent","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) P : Matrix (Fin d) (Fin d) Complex, Eq (HMul.hMul P P) P → ∀ (N…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_sectorCompression_of_idempotent","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11968","layer":"formal","project":"p8","title":"MPOTensor.verticalLoop","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoop","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11969","layer":"formal","project":"p8","title":"MPOTensor.verticalLoopWith","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin d) (Fin d) Complex → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoopWith","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11970","layer":"formal","project":"p8","title":"MPOTensor.verticalLoopWith_apply_eq_trace_mul_verticalTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (a b : Fin D), Eq (M.verti…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoopWith_apply_eq_trace_mul_verticalTensor","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11971","layer":"formal","project":"p8","title":"MPOTensor.verticalLoopWith_one","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq (M.verticalLoopWith 1) M.verticalLoop","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoopWith_one","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11972","layer":"formal","project":"p8","title":"MPOTensor.verticalLoop_braRightMul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), Eq (M.braRightMul P).vert…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoop_braRightMul","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11973","layer":"formal","project":"p8","title":"MPOTensor.verticalLoop_eq_physTraceTransfer","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq M.verticalLoop M.physTraceTransfer","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoop_eq_physTraceTransfer","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11974","layer":"formal","project":"p8","title":"MPOTensor.verticalLoop_ketLeftMul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex), Eq (M.ketLeftMul P).verti…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalLoop_ketLeftMul","module":"TNLean.MPS.MPDO.SectorTrace"},{"id":"n11975","layer":"formal","project":"p8","title":"MPOTensor.IsSimple","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsSimple","module":"TNLean.MPS.MPDO.Simple"},{"id":"n11976","layer":"formal","project":"p8","title":"MPOTensor.IsSimple.exists_mpo_ne_zero","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSimple → Exists fun N => And (LT.lt 0 N) (Ne (M.mpo N) 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(p :…","labels":[],"detail_key":"p8","name":"MPOTensor.physRealize_spec","module":"TNLean.MPS.MPDO.SimpleLocalInverseMaps"},{"id":"n11992","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_four_three_apply","kind":"theorem","summary":"∀ d D : Nat (K : MPOTensor d D) (u v : Fin 3 → Fin d), Eq (K.reducedBlockState 4 3 ⋯ u v) (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_four_three_apply","module":"TNLean.MPS.MPDO.SimpleLocalInverseMaps"},{"id":"n11993","layer":"formal","project":"p8","title":"MPOTensor.EtaStructure","kind":"def","summary":"dA dB dC : Nat → Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod (Fin dB)…","labels":[],"detail_key":"p8","name":"MPOTensor.EtaStructure","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n11994","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators","kind":"inductive","summary":"dA dB dC : Nat → ρ_ABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n11995","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.ofHayashiMarkov","kind":"def","summary":"dA dB dC : Nat → ρ_ABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.ofHayashiMarkov","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n11996","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.traceMatrix","kind":"def","summary":"dA dB dC : Nat → ρ_ABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod (…","labels":[],"detail_key":"p8","name":"MPOTensor.ExplicitEtaOperators.traceMatrix","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n11997","layer":"formal","project":"p8","title":"MPOTensor.ExplicitEtaOperators.traceMatrixRe","kind":"def","summary":"dA dB dC : Nat → ρ_ABC : 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(M.r…","labels":[],"detail_key":"p8","name":"MPOTensor.isSSAEquality_threeSite_of_isSAL","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12016","layer":"formal","project":"p8","title":"MPOTensor.isSSAEquality_tripartite_of_isSAL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hSAL : M.IsSAL) N : Nat (hN : LE.le 4 N), have hM := ⋯; have h…","labels":[],"detail_key":"p8","name":"MPOTensor.isSSAEquality_tripartite_of_isSAL","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12017","layer":"formal","project":"p8","title":"MPOTensor.sal_implies_eta_structure","kind":"theorem","summary":"∀ dA dB dC : Nat (ρ_ABC : Matrix (Prod (Fin dA) (Prod (Fin dB) (Fin dC))) (Prod (Fin dA) (Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.sal_implies_eta_structure","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12018","layer":"formal","project":"p8","title":"MPOTensor.sal_zcl_implies_rank_one_T_of_pairing_idempotent","kind":"theorem","summary":"∀ n : Nat V : Type u_1 [inst : AddCommGroup V] [inst_1 : Module Real V] (T : Matrix (Fin n) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.sal_zcl_implies_rank_one_T_of_pairing_idempotent","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12019","layer":"formal","project":"p8","title":"MPOTensor.sal_zcl_implies_rank_one_T_of_posSemidef","kind":"theorem","summary":"∀ n : Nat (T : Matrix (Fin n) (Fin n) Real), T.IsPrimitive → T.PosSemidef → Eq T.trace 1 → T.Tr…","labels":[],"detail_key":"p8","name":"MPOTensor.sal_zcl_implies_rank_one_T_of_posSemidef","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12020","layer":"formal","project":"p8","title":"MPOTensor.sal_zcl_implies_rank_one_T_of_sector_supports","kind":"theorem","summary":"∀ n : Nat V : Type u_1 [inst : AddCommGroup V] [inst_1 : Module Real V] (T : Matrix (Fin n) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.sal_zcl_implies_rank_one_T_of_sector_supports","module":"TNLean.MPS.MPDO.SimpleLocalStructure"},{"id":"n12021","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.smul_ofReal","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsMPDO → ∀ r : Real, LE.le 0 r → (HSMul.hSMul (↑r) M).IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.smul_ofReal","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12022","layer":"formal","project":"p8","title":"MPOTensor.IsSimple.smul_ofReal","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSimple → ∀ r : Real, LT.lt 0 r → (HSMul.hSMul (↑r) M).IsSimp…","labels":[],"detail_key":"p8","name":"MPOTensor.IsSimple.smul_ofReal","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12023","layer":"formal","project":"p8","title":"MPOTensor.blockTensor_smul","kind":"theorem","summary":"∀ d D : Nat (c : Complex) (M : MPOTensor d D) (L : Nat), Eq ((HSMul.hSMul c M).blockTensor L) (…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTensor_smul","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12024","layer":"formal","project":"p8","title":"MPOTensor.evalWord_smul","kind":"theorem","summary":"∀ d D : Nat (c : Complex) (M : MPOTensor d D) (is js : List (Fin d)), Eq js.length is.length →…","labels":[],"detail_key":"p8","name":"MPOTensor.evalWord_smul","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12025","layer":"formal","project":"p8","title":"MPOTensor.isMPDO_smul_ofReal_iff","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) r : Real, LT.lt 0 r → Iff (HSMul.hSMul (↑r) M).IsMPDO M.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.isMPDO_smul_ofReal_iff","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12026","layer":"formal","project":"p8","title":"MPOTensor.isSimple_smul_ofReal_iff","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) r : Real, LT.lt 0 r → Iff (HSMul.hSMul (↑r) M).IsSimple M.IsSim…","labels":[],"detail_key":"p8","name":"MPOTensor.isSimple_smul_ofReal_iff","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12027","layer":"formal","project":"p8","title":"MPOTensor.mpo_smul","kind":"theorem","summary":"∀ d D : Nat (c : Complex) (M : MPOTensor d D) (N : Nat), Eq ((HSMul.hSMul c M).mpo N) (HSMul.hS…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_smul","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12028","layer":"formal","project":"p8","title":"MPOTensor.normalizedMPO_smul","kind":"theorem","summary":"∀ d D : Nat c : Complex, Ne c 0 → ∀ (M : MPOTensor d D) (N : Nat), Eq ((HSMul.hSMul c M).normal…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedMPO_smul","module":"TNLean.MPS.MPDO.SimpleScaling"},{"id":"n12029","layer":"formal","project":"p8","title":"MPOTensor.IsSimpleCanonicalForm","kind":"def","summary":"d D : Nat → MPOTensor d D → 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M.phys…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledPhysTraceTransfer_toMPSTensor","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12033","layer":"formal","project":"p8","title":"MPOTensor.doubledPhysTraceTransfer_toTensor","kind":"theorem","summary":"∀ d : Nat (S : MPSTensor.SectorDecomposition (HMul.hMul d d)), Eq (MPOTensor.doubledPhysTraceTr…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledPhysTraceTransfer_toTensor","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12034","layer":"formal","project":"p8","title":"MPOTensor.isNilpotent_doubledPhysTraceTransfer_cast_iff","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (hdim : Eq D₁ D₂) (A : MPSTensor (HMul.hMul d d) D₁), Iff (IsNilpotent (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.isNilpotent_doubledPhysTraceTransfer_cast_iff","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12035","layer":"formal","project":"p8","title":"MPOTensor.weight_copy_independent_of_isPhysicalTraceIdempotent","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hTotal : Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.weight_copy_independent_of_isPhysicalTraceIdempotent","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12036","layer":"formal","project":"p8","title":"MPOTensor.weight_copy_independent_of_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hTotal : Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.weight_copy_independent_of_isSourceZCL","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12037","layer":"formal","project":"p8","title":"MPOTensor.weighted_basis_physTraceTransfer_sq_of_literal_ZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (S : MPSTensor.SectorDecomposition (HMul.hMul d d)) (hTotal : Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.weighted_basis_physTraceTransfer_sq_of_literal_ZCL","module":"TNLean.MPS.MPDO.SimpleTensor"},{"id":"n12038","layer":"formal","project":"p8","title":"MPOTensor.isMPDO_of_changePhysicalBasis_isMPDO_of_isometry","kind":"theorem","summary":"∀ d D e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.isMPDO_of_changePhysicalBasis_isMPDO_of_isometry","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12039","layer":"formal","project":"p8","title":"MPOTensor.mpo_ketLeftMul","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (N : Nat), Eq ((M.ketLeftM…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_ketLeftMul","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12040","layer":"formal","project":"p8","title":"MPOTensor.reindex_sitewisePhysicalMatrix_two","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq ((Matrix.reindex (finTwoArrowEquiv (Fin e)…","labels":[],"detail_key":"p8","name":"MPOTensor.reindex_sitewisePhysicalMatrix_two","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12041","layer":"formal","project":"p8","title":"MPOTensor.singleKrausMap_sitewisePhysicalMatrix_mpo","kind":"theorem","summary":"∀ d D e : Nat (V : Matrix (Fin e) (Fin d) Complex) (M : MPOTensor d D) (N : Nat), Eq ((singleKr…","labels":[],"detail_key":"p8","name":"MPOTensor.singleKrausMap_sitewisePhysicalMatrix_mpo","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12042","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalMatrix","kind":"def","summary":"d e : Nat → Matrix (Fin e) (Fin d) Complex → (N : Nat) → Matrix (Fin N → Fin e) (Fin N → Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalMatrix","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12043","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalMatrix_conjTranspose","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex) (N : Nat), Eq (MPOTensor.sitewisePhysicalMatri…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalMatrix_conjTranspose","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12044","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalMatrix_mul_conjTranspose","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin d) (Fin e) Complex) (N : Nat), Eq (HMul.hMul (MPOTensor.sitewisePh…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalMatrix_mul_conjTranspose","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12045","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalMatrix_one","kind":"theorem","summary":"∀ (d N : Nat), Eq (MPOTensor.sitewisePhysicalMatrix 1 N) 1","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalMatrix_one","module":"TNLean.MPS.MPDO.SitewisePhysicalMatrix"},{"id":"n12046","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalRecovery","kind":"def","summary":"d e : Nat → Matrix (Fin e) (Fin d) Complex → (N : Nat) → [NeZero d] → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalRecovery","module":"TNLean.MPS.MPDO.SitewisePhysicalRecovery"},{"id":"n12047","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalRecovery_apply_singleKrausMap","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (N : N…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalRecovery_apply_singleKrausMap","module":"TNLean.MPS.MPDO.SitewisePhysicalRecovery"},{"id":"n12048","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalRecovery_isKrausCPTP","kind":"theorem","summary":"∀ d e : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (N : N…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalRecovery_isKrausCPTP","module":"TNLean.MPS.MPDO.SitewisePhysicalRecovery"},{"id":"n12049","layer":"formal","project":"p8","title":"MPOTensor.sitewisePhysicalRecovery_physCloseN","kind":"theorem","summary":"∀ d e D : Nat (V : Matrix (Fin e) (Fin d) Complex), Eq (HMul.hMul V.conjTranspose V) 1 → ∀ (M :…","labels":[],"detail_key":"p8","name":"MPOTensor.sitewisePhysicalRecovery_physCloseN","module":"TNLean.MPS.MPDO.SitewisePhysicalRecovery"},{"id":"n12050","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVBasisOfNormalTensors.blockTensor","kind":"theorem","summary":"∀ d D g : Nat dim : Fin g → Nat A : MPSTensor d D B : (j : Fin g) → MPSTensor d (dim j), (A.IsC…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVBasisOfNormalTensors.blockTensor","module":"TNLean.MPS.MPDO.SourceBNTBlocking"},{"id":"n12051","layer":"formal","project":"p8","title":"MPSTensor.wordTupleSpanTop_blockTensor_one","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat (B : (j : Fin g) → MPSTensor d (dim j)) L : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.wordTupleSpanTop_blockTensor_one","module":"TNLean.MPS.MPDO.SourceBNTBlocking"},{"id":"n12052","layer":"formal","project":"p8","title":"MPOTensor.reducedBlockState_add_three_eq_succ_of_isSourceZCL","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsSourceZCL → ∀ (L : Nat), Eq (M.reducedBlockState (HAdd.hAd…","labels":[],"detail_key":"p8","name":"MPOTensor.reducedBlockState_add_three_eq_succ_of_isSourceZCL","module":"TNLean.MPS.MPDO.SourceZCLMarginal"},{"id":"n12053","layer":"formal","project":"p8","title":"MPOTensor.IsMPDO.stackedTensor","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsMPDO → ∀ (p : Nat), (M.stackedTensor p).IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPDO.stackedTensor","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12054","layer":"formal","project":"p8","title":"MPOTensor.PeriodicVectorYieldsCyclicProjector","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.PeriodicVectorYieldsCyclicProjector","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12055","layer":"formal","project":"p8","title":"MPOTensor.exists_word_verticalTensor_stackedTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (p : Nat) (v : Fin (HMul.hMul (HPow.hPow D p) (HPow.hPow D p)))…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_word_verticalTensor_stackedTensor","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12056","layer":"formal","project":"p8","title":"MPOTensor.idTensor","kind":"def","summary":"(d : Nat) → MPOTensor d 1","labels":[],"detail_key":"p8","name":"MPOTensor.idTensor","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12057","layer":"formal","project":"p8","title":"MPOTensor.layerMul","kind":"def","summary":"d D D' : Nat → MPOTensor d D → MPOTensor d D' → MPOTensor d (HMul.hMul D D')","labels":[],"detail_key":"p8","name":"MPOTensor.layerMul","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12058","layer":"formal","project":"p8","title":"MPOTensor.mpo_layerMul","kind":"theorem","summary":"∀ d D D' : Nat (M : MPOTensor d D) (M' : MPOTensor d D') (N : Nat), Eq ((M.layerMul M').mpo N)…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_layerMul","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12059","layer":"formal","project":"p8","title":"MPOTensor.mpo_stackedTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (p N : Nat), Eq ((M.stackedTensor p).mpo N) (HPow.hPow (M.mpo N…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_stackedTensor","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12060","layer":"formal","project":"p8","title":"MPOTensor.periodicSectorProjectorOfCyclicData","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → M.IsMPDO → p : Nat → Ne p 0 → Q : Matrix (Fin d) (Fin d) Comp…","labels":[],"detail_key":"p8","name":"MPOTensor.periodicSectorProjectorOfCyclicData","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12061","layer":"formal","project":"p8","title":"MPOTensor.periodicVectorYieldsProjector_of_cyclic","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → M.PeriodicVectorYieldsCyclicProjector → M.PeriodicV…","labels":[],"detail_key":"p8","name":"MPOTensor.periodicVectorYieldsProjector_of_cyclic","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12062","layer":"formal","project":"p8","title":"MPOTensor.stackedTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → (p : Nat) → MPOTensor d (HPow.hPow D p)","labels":[],"detail_key":"p8","name":"MPOTensor.stackedTensor","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12063","layer":"formal","project":"p8","title":"MPOTensor.stackedTensor_ketLeftMul_invariant","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) p : Nat Q : Matrix (Fin d) (Fin d) Complex, (∀ (w : List (Fin (…","labels":[],"detail_key":"p8","name":"MPOTensor.stackedTensor_ketLeftMul_invariant","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12064","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_layerMul","kind":"theorem","summary":"∀ d D D' : Nat (M : MPOTensor d D) (M' : MPOTensor d D') (v : Fin (HMul.hMul (HMul.hMul D D') (…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_layerMul","module":"TNLean.MPS.MPDO.StackedLayers"},{"id":"n12065","layer":"formal","project":"p8","title":"MPOTensor.IsStrongRFP","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsStrongRFP","module":"TNLean.MPS.MPDO.StrongRFP"},{"id":"n12066","layer":"formal","project":"p8","title":"MPOTensor.isStrongRFP_iff_physClose","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Iff M.IsStrongRFP (Exists fun P => Exists fun U => And P.PosSe…","labels":[],"detail_key":"p8","name":"MPOTensor.isStrongRFP_iff_physClose","module":"TNLean.MPS.MPDO.StrongRFP"},{"id":"n12067","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_tensor_relation_iff_physClose","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_tensor_relation_iff_physClose","module":"TNLean.MPS.MPDO.StrongRFP"},{"id":"n12068","layer":"formal","project":"p8","title":"MPOTensor.IsStrongRFP.exists_periodic_rank_factor","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsStrongRFP → Exists fun P => And P.PosSemidef (And (∀ (N : Na…","labels":[],"detail_key":"p8","name":"MPOTensor.IsStrongRFP.exists_periodic_rank_factor","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12069","layer":"formal","project":"p8","title":"MPOTensor.firstSitePreparationEquiv","kind":"def","summary":"d : Nat → (N : Nat) → Equiv (Prod (Prod (Fin d) (Fin N → Fin d)) (Fin d)) (Prod (Prod (Fin d) (…","labels":[],"detail_key":"p8","name":"MPOTensor.firstSitePreparationEquiv","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12070","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_mpo_eq_localized_preparation","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_mpo_eq_localized_preparation","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12071","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_mpo_eq_reindex_singleKrausMap_kronecker","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_mpo_eq_reindex_singleKrausMap_kronecker","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12072","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_physCloseN_eq_localized_preparation","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_physCloseN_eq_localized_preparation","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12073","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_physCloseN_eq_reindex_singleKrausMap_kronecker","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_physCloseN_eq_reindex_singleKrausMap_kronecker","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12074","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_rank_mpo_add_one","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_rank_mpo_add_one","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12075","layer":"formal","project":"p8","title":"MPOTensor.strongRFP_rank_mpo_add_two","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) (U : Matrix (Prod (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.strongRFP_rank_mpo_add_two","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12076","layer":"formal","project":"p8","title":"MPOTensor.tensorMapId_preparationMap_eq_reindex_kronecker","kind":"theorem","summary":"∀ d : Nat (P : Matrix (Fin d) (Fin d) Complex) (N : Nat) (A : Matrix (Prod (Fin d) (Fin N → Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.tensorMapId_preparationMap_eq_reindex_kronecker","module":"TNLean.MPS.MPDO.StrongRFPRankGrowth"},{"id":"n12077","layer":"formal","project":"p8","title":"MPOTensor.CaseIIAbsorptionCounterexample.printed_theorem49_iv_to_v_is_false","kind":"theorem","summary":"And (Eq (MPOTensor.CaseIIAbsorptionCounterexample.repeatedCopyDecomposition.weight 0 0) 1) (And…","labels":[],"detail_key":"p8","name":"MPOTensor.CaseIIAbsorptionCounterexample.printed_theorem49_iv_to_v_is_false","module":"TNLean.MPS.MPDO.Theorem49RepeatedCopyCounterexample"},{"id":"n12078","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.binEntropy_one_div_three_pos","kind":"theorem","summary":"LT.lt 0 (Real.binEntropy (1 / 3))","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.binEntropy_one_div_three_pos","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12079","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.blockEntropy_parityTensor_of_even","kind":"theorem","summary":"∀ N L : Nat, Even N → ∀ (hNpos : LT.lt 0 N), LT.lt 0 L → ∀ (hLN : LE.le L N), Eq (MPOTensor.The…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.blockEntropy_parityTensor_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12080","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.blockEntropy_parityTensor_of_odd","kind":"theorem","summary":"∀ N L : Nat (hN : Odd N) (hLN : LE.le L N), Eq (MPOTensor.ThermodynamicLimitCounterexample.pari…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.blockEntropy_parityTensor_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12081","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.mpo_parityTensor_eq_diagonal","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.mpo N) (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.mpo_parityTensor_eq_diagonal","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12082","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.mutualInfoChain_parityTensor_of_even","kind":"theorem","summary":"∀ N L : Nat, Even N → ∀ (hLpos : LT.lt 0 L) (hLN : LT.lt L N), Eq (MPOTensor.ThermodynamicLimit…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.mutualInfoChain_parityTensor_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12083","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.mutualInfoChain_parityTensor_of_odd","kind":"theorem","summary":"∀ N L : Nat (hN : Odd N) (hLN : LT.lt L N), Eq (MPOTensor.ThermodynamicLimitCounterexample.pari…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.mutualInfoChain_parityTensor_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12084","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_parityTensor_of_even","kind":"theorem","summary":"∀ N : Nat, Even N → LT.lt 0 N → Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.nor…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_parityTensor_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12085","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_parityTensor_of_odd","kind":"theorem","summary":"∀ N : Nat, Odd N → Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.normalizedMPO N)…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_parityTensor_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12086","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_zeroConfig_of_even","kind":"theorem","summary":"∀ N : Nat, Even N → LT.lt 0 N → Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.nor…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_zeroConfig_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12087","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_zeroConfig_of_odd","kind":"theorem","summary":"∀ N : Nat, Odd N → Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.normalizedMPO N…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.normalizedMPO_zeroConfig_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12088","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.not_exists_tendsto_parityMutualInfoAfterCut","kind":"theorem","summary":"∀ (L : Nat), LT.lt 0 L → Not (Exists fun x => Filter.Tendsto (MPOTensor.ThermodynamicLimitCount…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.not_exists_tendsto_parityMutualInfoAfterCut","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12089","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.not_exists_tendsto_reducedBlockState_one_zero_…","kind":"theorem","summary":"Not (Exists fun x => Filter.Tendsto (fun K => (MPOTensor.ThermodynamicLimitCounterexample.parit…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.not_exists_tendsto_reducedBlockState_one_zero_zero","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12090","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.parityMutualInfoAfterCut","kind":"def","summary":"Nat → Nat → Real","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.parityMutualInfoAfterCut","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12091","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.parityTensor","kind":"def","summary":"MPOTensor 2 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No…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.proposition45_limit_counterexample","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12094","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_one_zero_zero_of_even","kind":"theorem","summary":"∀ N : Nat, Even N → ∀ (hNpos : LT.lt 0 N), Eq (MPOTensor.ThermodynamicLimitCounterexample.parit…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_one_zero_zero_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12095","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_one_zero_zero_of_odd","kind":"theorem","summary":"∀ N : Nat (hN : Odd N), Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.reducedBloc…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_one_zero_zero_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12096","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_parityTensor_of_even","kind":"theorem","summary":"∀ N L : Nat, Even N → LT.lt 0 N → LT.lt 0 L → ∀ (hLN : LE.le L N), Eq (MPOTensor.ThermodynamicL…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_parityTensor_of_even","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12097","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_parityTensor_of_odd","kind":"theorem","summary":"∀ N L : Nat, Odd N → ∀ (hLN : LE.le L N), Eq (MPOTensor.ThermodynamicLimitCounterexample.parity…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.reducedBlockState_parityTensor_of_odd","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12098","layer":"formal","project":"p8","title":"MPOTensor.ThermodynamicLimitCounterexample.trace_mpo_parityTensor","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.ThermodynamicLimitCounterexample.parityTensor.mpo N).trace (HAdd.hAd…","labels":[],"detail_key":"p8","name":"MPOTensor.ThermodynamicLimitCounterexample.trace_mpo_parityTensor","module":"TNLean.MPS.MPDO.ThermodynamicLimitCounterexample"},{"id":"n12099","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.HasTopologicalDensityDecomposition","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.BNTFusionTensorClause → 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N…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.topologicalDensityBlock","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12102","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.topologicalDensityOperatorSucc","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → (N : Nat) → Matrix (Fin (HAdd.h…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.topologicalDensityOperatorSucc","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12103","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.topologicalRecursiveFactorSucc","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → (N : Nat) → Matrix (Fin (HAdd.h…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.topologicalRecursiveFactorSucc","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12104","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.verticalCopyChainFusionCoisometry","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → N : Nat → (p : Fin (HAdd.hAdd N…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.verticalCopyChainFusionCoisometry","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12105","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.verticalCopyChainProjectorQ","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → N : Nat → (p : Fin (HAdd.hAdd N…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.verticalCopyChainProjectorQ","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12106","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.verticalMultiplicityChainWeight","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → N : Nat → (Fin (HAdd.hAdd N 1)…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.verticalMultiplicityChainWeight","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12107","layer":"formal","project":"p8","title":"MPOTensor.exists_topologicalDensityDecomposition_and_factorCommutator_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → Exists fun H => A…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_topologicalDensityDecomposition_and_factorCommutator_of_isRFPViaTS","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12108","layer":"formal","project":"p8","title":"MPOTensor.exists_topologicalDensityDecomposition_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → Exists fun H => H…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_topologicalDensityDecomposition_of_isRFPViaTS","module":"TNLean.MPS.MPDO.TopologicalDensityDecomposition"},{"id":"n12109","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.HasTopologicalGibbsDecomposition","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.BNTFusionTensorClause → M.IsMPDO → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.HasTopologicalGibbsDecomposition","module":"TNLean.MPS.MPDO.TopologicalGibbsHamiltonian"},{"id":"n12110","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.hasTopologicalGibbsDecomposition","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTFusionTensorClause) (hM : M.IsMPDO), (MPOTensor.BNTLabe…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.hasTopologicalGibbsDecomposition","module":"TNLean.MPS.MPDO.TopologicalGibbsHamiltonian"},{"id":"n12111","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalSpectralEmbedding","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → Function.Embedding H.TerminalSp…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalSpectralEmbedding","module":"TNLean.MPS.MPDO.TopologicalGibbsHamiltonian"},{"id":"n12112","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.topologicalGibbsLocalTerm","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → Matrix (Fin 2 → Fin H.verticalR…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.topologicalGibbsLocalTerm","module":"TNLean.MPS.MPDO.TopologicalGibbsHamiltonian"},{"id":"n12113","layer":"formal","project":"p8","title":"MPOTensor.topologicalGibbsDecomposition_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hHorizontal : M.IsHorizontalCF) (hM : M.IsMPDO) (hRFP : M.IsRF…","labels":[],"detail_key":"p8","name":"MPOTensor.topologicalGibbsDecomposition_of_isRFPViaTS","module":"TNLean.MPS.MPDO.TopologicalGibbsHamiltonian"},{"id":"n12114","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.retainedMultiplicityEnergyEntry","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → Fin H.verticalRetainedDim → Real","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.retainedMultiplicityEnergyEntry","module":"TNLean.MPS.MPDO.TopologicalMultiplicityEnergy"},{"id":"n12115","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.retainedMultiplicityWeightEntry","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → Fin H.verticalRetainedDim → Com…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.retainedMultiplicityWeightEntry","module":"TNLean.MPS.MPDO.TopologicalMultiplicityEnergy"},{"id":"n12116","layer":"formal","project":"p8","title":"MPOTensor.physicalTopologicalGibbsDecomposition_of_isRFPViaTS","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (hHorizontal : M.IsHorizontalCF) (hM : M.IsMPDO) (hRFP : M.IsRF…","labels":[],"detail_key":"p8","name":"MPOTensor.physicalTopologicalGibbsDecomposition_of_isRFPViaTS","module":"TNLean.MPS.MPDO.TopologicalPhysicalGibbs"},{"id":"n12117","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.FusionHistoryData","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → MPOTensor.BNTFusionCoiso…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.FusionHistoryData","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12118","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.fusionChainTensor","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.fusionChainTensor","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12119","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.fusionHistoryWeight","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.fusionHistoryWeight","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12120","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.recursiveProjectorQ","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.recursiveProjectorQ","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12121","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.recursiveProjectorQ_isStarProjection","kind":"theorem","summary":"∀ Λ : Type u [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionCois…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.recursiveProjectorQ_isStarProjection","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12122","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.sequentialFusionCoisometry","kind":"def","summary":"Λ : Type u → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFusi…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.sequentialFusionCoisometry","module":"TNLean.MPS.MPDO.TopologicalProjectorRecursion"},{"id":"n12123","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.conjugatedProjectorQBlock","kind":"def","summary":"Λ : Type u_1 → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFu…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.conjugatedProjectorQBlock","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12124","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock","kind":"def","summary":"Λ : Type u_1 → [inst : Fintype Λ] → [inst_1 : DecidableEq Λ] → p : Nat → (Fam : MPOTensor.BNTFu…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12125","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock_eq_unweighted","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock_eq_unweighted","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12126","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock_isStarProjection","kind":"theorem","summary":"∀ Λ : Type u_1 [inst : Fintype Λ] [inst_1 : DecidableEq Λ] p : Nat (Fam : MPOTensor.BNTFusionCo…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionCoisometryFamily.projectorQBlock_isStarProjection","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12127","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.conjugatedProjectorQBlock","kind":"def","summary":"g p : Nat → (Fam : MPOTensor.BNTFusionIsometryFamily (Fin g) p) → ((γ : Fin g) → Matrix (Fin (F…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.conjugatedProjectorQBlock","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12128","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock","kind":"def","summary":"g p : Nat → (Fam : MPOTensor.BNTFusionIsometryFamily (Fin g) p) → ((γ : Fin g) → Matrix (Fin (F…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12129","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock_eq_unweighted","kind":"theorem","summary":"∀ g p : Nat (Fam : MPOTensor.BNTFusionIsometryFamily (Fin g) p) (c : MPOTensor.BNTLabelCoeffici…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock_eq_unweighted","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12130","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock_isStarProjection","kind":"theorem","summary":"∀ g p : Nat (Fam : MPOTensor.BNTFusionIsometryFamily (Fin g) p) (c : MPOTensor.BNTLabelCoeffici…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionIsometryFamily.projectorQBlock_isStarProjection","module":"TNLean.MPS.MPDO.TopologicalProjectors"},{"id":"n12131","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.TerminalSpectralIndex","kind":"def","summary":"d D : Nat → M : MPOTensor d D → M.BNTFusionTensorClause → Type","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.TerminalSpectralIndex","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12132","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalEigenProjection","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → M.IsMPDO → (s : H.TerminalSpect…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalEigenProjection","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12133","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalEigenProjectionFamily","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → M.IsMPDO → H.TerminalSpectralIn…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalEigenProjectionFamily","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12134","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalEigenvalue","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → M.IsMPDO → H.TerminalSpectralIn…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalEigenvalue","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12135","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalEigenvalue_nonneg","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D (H : M.BNTFusionTensorClause) (hM : M.IsMPDO) (s : H.TerminalSpec…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalEigenvalue_nonneg","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12136","layer":"formal","project":"p8","title":"MPOTensor.BNTFusionTensorClause.terminalMatrix","kind":"def","summary":"d D : Nat → M : MPOTensor d D → (H : M.BNTFusionTensorClause) → (γ : Fin H.labelCount) → Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTFusionTensorClause.terminalMatrix","module":"TNLean.MPS.MPDO.TopologicalTerminalSpectral"},{"id":"n12137","layer":"formal","project":"p8","title":"MPOTensor.rfpBNTFusionTensorClause","kind":"def","summary":"d D : Nat → (M : MPOTensor d D) → M.IsHorizontalCF → M.IsMPDO → M.IsRFPViaTS → 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X…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physClose1_T","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12147","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physClose2_T","kind":"theorem","summary":"∀ (X : Matrix (Fin 8) (Fin 8) Complex) (i₁ i₂ j₁ j₂ : Fin 8), Eq (MPOTensor.TwistedDimer.T.phys…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physClose2_T","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12148","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physIdx","kind":"def","summary":"Fin 2 → Fin 2 → Fin 2 → Fin 8","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physIdx","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12149","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physTraceTransfer_block_one","kind":"theorem","summary":"Eq (MPOTensor.TwistedDimer.block 1).physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physTraceTransfer_block_one","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12150","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physTraceTransfer_block_zero","kind":"theorem","summary":"Eq (MPOTensor.TwistedDimer.block 0).physTraceTransfer (Finset.univ.sum fun l => Finset.univ.sum…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physTraceTransfer_block_zero","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12151","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.tau","kind":"def","summary":"Fin 2 → Fin 2 → Real","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.tau","module":"TNLean.MPS.MPDO.TwistedDimer"},{"id":"n12152","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_bntAlgebraTensorClause","kind":"def","summary":"MPOTensor.TwistedDimer.T.BNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_bntAlgebraTensorClause","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12153","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_hasBNTAlgebraTensorClause","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.HasBNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_hasBNTAlgebraTensorClause","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12154","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.conjTranspose_flagMPO_mul_self","kind":"theorem","summary":"∀ (f : Fin 2) (a b : Fin 8), Eq (HMul.hMul (MPOTensor.TwistedDimer.flagMPO f a b).conjTranspose…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.conjTranspose_flagMPO_mul_self","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12155","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagAlgebraClause","kind":"def","summary":"MPOTensor.BNTAlgebraClause MPOTensor.TwistedDimer.twoLabelCoeffs (MPOTensor.verticalBNTOperator…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagAlgebraClause","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12156","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isCPSVBasisOfNormalTensors","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.verticalTensor.IsCPSVBasisOfNormalTensors fun f => ⟨MPOTensor.TwistedD…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isCPSVBasisOfNormalTensors","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12157","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isLeftCanonical","kind":"theorem","summary":"∀ (f : Fin 2), (MPOTensor.TwistedDimer.flagFamily f).IsLeftCanonical","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isLeftCanonical","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12158","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isNormalTensor","kind":"theorem","summary":"∀ (f : Fin 2), (MPOTensor.TwistedDimer.flagFamily f).IsNormalTensor","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isNormalTensor","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12159","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorWeight_hasIdempotentCoefficientForm","kind":"theorem","summary":"(MPOTensor.verticalBNTTraceScalarFamily MPOTensor.TwistedDimer.sectorWeight).HasIdempotentCoeff…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorWeight_hasIdempotentCoefficientForm","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12160","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.star_flagCoef_mul_self","kind":"theorem","summary":"∀ (f p p' k q q' k' : Fin 2), Eq (HMul.hMul (star (MPOTensor.TwistedDimer.flagCoef f (MPOTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.star_flagCoef_mul_self","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12161","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelChiTracePowerForm","kind":"def","summary":"MPOTensor.PositiveBNTLabelChiTracePowerForm MPOTensor.TwistedDimer.twoLabelCoeffs","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelChiTracePowerForm","module":"TNLean.MPS.MPDO.TwistedDimerBNTAlgebraClause"},{"id":"n12162","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bondInsertionEquiv","kind":"def","summary":"Equiv (Prod (Fin 4) MPOTensor.TwistedDimer.Bond) (Prod (Fin 4) (Fin 4))","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bondInsertionEquiv","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12163","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bondSiteEquiv","kind":"def","summary":"Equiv MPOTensor.TwistedDimer.Bond (Fin 4)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bondSiteEquiv","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12164","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.decoratedState_eq_mpo_factors","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.decoratedState N) ((Matrix.kroneckerMap (fun…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.decoratedState_eq_mpo_factors","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12165","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.incomingBondEquiv","kind":"def","summary":"(N : Nat) → Equiv (Fin N → Fin 4) (Fin N → MPOTensor.TwistedDimer.Bond)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.incomingBondEquiv","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12166","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_eq_unitary_mpo_factors","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.T.mpo N) (HMul.hMul (HMul.hMul (MPOTensor.Twi…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_eq_unitary_mpo_factors","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12167","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_sigmaDimer_eq_bondProduct","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.sigmaDimer.mpo N) ((MPOTensor.TwistedDimer.po…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_sigmaDimer_eq_bondProduct","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12168","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.onsiteBondFlagEquiv","kind":"def","summary":"(N : Nat) → Equiv (Fin N → Fin 8) (Prod (Fin N → Fin 4) (Fin N → Fin 2))","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.onsiteBondFlagEquiv","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12169","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.onsiteBondFlagEquiv_incoming","kind":"theorem","summary":"∀ (N : Nat), Eq ((MPOTensor.TwistedDimer.onsiteBondFlagEquiv N).trans ((MPOTensor.TwistedDimer.…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.onsiteBondFlagEquiv_incoming","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12170","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimer","kind":"def","summary":"MPOTensor 4 4","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimer","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12171","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerCoarse","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Prod (Fin 4) (Fin 4)) (Prod (Fin 4) (Fin 4)) Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerCoarse","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12172","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerCoarse_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.TwistedDimer.sigmaDimerCoarse","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerCoarse_isKrausCPTP","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12173","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerCoarse_physClose2","kind":"theorem","summary":"∀ (X : Matrix (Fin 4) (Fin 4) Complex), Eq (MPOTensor.TwistedDimer.sigmaDimerCoarse (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerCoarse_physClose2","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12174","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerCoarse_refine","kind":"theorem","summary":"∀ (Y : Matrix (Fin 4) (Fin 4) Complex), Eq (MPOTensor.TwistedDimer.sigmaDimerCoarse (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerCoarse_refine","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12175","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerRefine","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Fin 4) (Fin 4) Complex) (Matrix (Prod (Fin 4) (Fin 4))…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerRefine","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12176","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerRefine_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.TwistedDimer.sigmaDimerRefine","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerRefine_isKrausCPTP","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12177","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimerRefine_physClose1","kind":"theorem","summary":"∀ (X : Matrix (Fin 4) (Fin 4) Complex), Eq (MPOTensor.TwistedDimer.sigmaDimerRefine (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimerRefine_physClose1","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12178","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimer_isMPDO","kind":"theorem","summary":"MPOTensor.TwistedDimer.sigmaDimer.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimer_isMPDO","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12179","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimer_isRFPViaTS","kind":"theorem","summary":"MPOTensor.TwistedDimer.sigmaDimer.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimer_isRFPViaTS","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12180","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimer_mul","kind":"theorem","summary":"∀ (a b c d : MPOTensor.TwistedDimer.Bond), Eq (HMul.hMul (MPOTensor.TwistedDimer.sigmaDimer (MP…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimer_mul","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12181","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigmaDimer_physClose1","kind":"theorem","summary":"∀ (X : Matrix (Fin 4) (Fin 4) Complex) (a b : MPOTensor.TwistedDimer.Bond), Eq (MPOTensor.Twist…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigmaDimer_physClose1","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12182","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_mpo_sigmaDimer","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.sigmaDimer.mpo N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_mpo_sigmaDimer","module":"TNLean.MPS.MPDO.TwistedDimerBondRFP"},{"id":"n12183","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.IsSameChannel","kind":"def","summary":"Fin 2 → Fin 2 → Fin 2 → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.IsSameChannel","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12184","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.alpha","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.alpha","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12185","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.beta","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.beta","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12186","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.rescaledCoeffs","kind":"def","summary":"(Fin 2 → Real) → MPOTensor.BNTLabelCoefficientFamily (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.rescaledCoeffs","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12187","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.rescaledCoeffs_coeff","kind":"theorem","summary":"∀ (s : Fin 2 → Real) (L : Nat) (f f' g : Fin 2), Eq ((MPOTensor.TwistedDimer.rescaledCoeffs s).…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.rescaledCoeffs_coeff","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12188","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelChi","kind":"def","summary":"MPOTensor.DiagonalChiFamily (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelChi","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12189","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelChiScaled","kind":"def","summary":"(Fin 2 → Real) → MPOTensor.DiagonalChiFamily (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelChiScaled","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12190","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelChi_posEntries","kind":"theorem","summary":"MPOTensor.TwistedDimer.twoLabelChi.PosEntries","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelChi_posEntries","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12191","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelCoeffs","kind":"def","summary":"MPOTensor.BNTLabelCoefficientFamily (Fin 2)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelCoeffs","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12192","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelCoeffs_coeff","kind":"theorem","summary":"∀ (L : Nat) (f f' g : Fin 2), Eq (MPOTensor.TwistedDimer.twoLabelCoeffs.coeff L f f' g) (ite (M…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelCoeffs_coeff","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12193","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.twoLabelCoeffs_not_lengthIndependent","kind":"theorem","summary":"Not MPOTensor.TwistedDimer.twoLabelCoeffs.LengthIndependent","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.twoLabelCoeffs_not_lengthIndependent","module":"TNLean.MPS.MPDO.TwistedDimerCoefficients"},{"id":"n12194","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bellProjector_posSemidef","kind":"theorem","summary":"∀ (ε : Fin 2), (MPOTensor.TwistedDimer.bellProjector ε).PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bellProjector_posSemidef","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12195","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.decoratedState_posSemidef","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → (MPOTensor.TwistedDimer.decoratedState N).PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.decoratedState_posSemidef","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12196","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.evenFlagState_eq_mpo_Mhat","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.TwistedDimer.evenFlagState N) (MPOTensor.CPSVExample412NormalizedRFP…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.evenFlagState_eq_mpo_Mhat","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12197","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.evenFlagState_has_rfpRepresentation","kind":"theorem","summary":"Exists fun A => And A.IsMPDO (And A.IsRFPViaTS (∀ (N : Nat), Eq (MPOTensor.TwistedDimer.evenFla…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.evenFlagState_has_rfpRepresentation","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12198","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.evenFlagState_posSemidef","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → (MPOTensor.TwistedDimer.evenFlagState N).PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.evenFlagState_posSemidef","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12199","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagMatrix_one","kind":"theorem","summary":"Eq (MPOTensor.TwistedDimer.flagMatrix 1) MPOTensor.CPSVExample412Literal.sigmaZ","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagMatrix_one","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12200","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagMatrix_zero","kind":"theorem","summary":"Eq (MPOTensor.TwistedDimer.flagMatrix 0) 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagMatrix_zero","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12201","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigma_posSemidef","kind":"theorem","summary":"MPOTensor.TwistedDimer.sigma.PosSemidef","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigma_posSemidef","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12202","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_decoratedState","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.decoratedState N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_decoratedState","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12203","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_evenFlagState","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.evenFlagState N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_evenFlagState","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12204","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_mpo_T","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.T.mpo N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_mpo_T","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12205","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_powN_sigma","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.TwistedDimer.powN MPOTensor.TwistedDimer.sigma N).trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_powN_sigma","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12206","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_sigma","kind":"theorem","summary":"Eq MPOTensor.TwistedDimer.sigma.trace 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_sigma","module":"TNLean.MPS.MPDO.TwistedDimerFactorStates"},{"id":"n12207","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.block_eq_unitTensor","kind":"theorem","summary":"∀ (k : Fin 2), Eq (MPOTensor.TwistedDimer.block k) (MPOTensor.TwistedDimer.unitTensor (MPOTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.block_eq_unitTensor","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12208","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.evalWord_unitTensor_ofFn","kind":"theorem","summary":"∀ (c : Fin 8 → Fin 8 → Complex) (n : Nat) (σ τ : Fin (HAdd.hAdd n 1) → Fin 8), Eq ((MPOTensor.T…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.evalWord_unitTensor_ofFn","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12209","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagCoef","kind":"def","summary":"Fin 2 → Fin 8 → Fin 8 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagCoef","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12210","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily","kind":"def","summary":"Fin 2 → MPSTensor (HMul.hMul 8 8) 4","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12211","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagMPO","kind":"def","summary":"Fin 2 → MPOTensor 8 4","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagMPO","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12212","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagMPO_apply_eq_T","kind":"theorem","summary":"∀ (f : Fin 2) (a b : Fin 8) (l r l' r' : Fin 2), Eq (MPOTensor.TwistedDimer.flagMPO f a b (finP…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagMPO_apply_eq_T","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12213","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagOperatorFamily","kind":"def","summary":"MPOTensor.BNTLabelOperatorFamily (Fin 2) fun L => Matrix (Fin L → Fin 8) (Fin L → Fin 8) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagOperatorFamily","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12214","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagOperatorFamily_operator","kind":"theorem","summary":"∀ (L : Nat) (f : Fin 2), Eq (MPOTensor.TwistedDimer.flagOperatorFamily.operator L f) ((MPOTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagOperatorFamily_operator","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12215","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight","kind":"def","summary":"Fin 2 → Fin 8 → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12216","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight_physIdx","kind":"theorem","summary":"∀ (f p p' k : Fin 2), Eq (MPOTensor.TwistedDimer.flagWeight f (MPOTensor.TwistedDimer.physIdx p…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight_physIdx","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12217","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_flagMPO_apply","kind":"theorem","summary":"∀ (f : Fin 2) L : Nat, LT.lt 0 L → ∀ (σ τ : Fin L → Fin 8), Eq ((MPOTensor.TwistedDimer.flagMPO…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_flagMPO_apply","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12218","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_unitTensor_apply","kind":"theorem","summary":"∀ (c : Fin 8 → Fin 8 → Complex) N : Nat, LT.lt 0 N → ∀ (σ τ : Fin N → Fin 8), Eq ((MPOTensor.Tw…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_unitTensor_apply","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12219","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mu","kind":"def","summary":"Real","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mu","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12220","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.prod_flagCoef","kind":"theorem","summary":"∀ (f : Fin 2) L : Nat (σ τ : Fin L → Fin 8), Eq (Finset.univ.prod fun n => MPOTensor.TwistedDim…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.prod_flagCoef","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12221","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.unitTensor","kind":"def","summary":"(Fin 8 → Fin 8 → Complex) → MPOTensor 8 4","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.unitTensor","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12222","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.unitTensor_mul_single","kind":"theorem","summary":"∀ (c : Fin 8 → Fin 8 → Complex) (i j : Fin 8) (l l' r r' : Fin 2) (u : Complex), Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.unitTensor_mul_single","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12223","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalBNTMPO_flagFamily","kind":"theorem","summary":"∀ (f : Fin 2), Eq (MPOTensor.verticalBNTMPO (MPOTensor.TwistedDimer.flagFamily f)) (MPOTensor.T…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalBNTMPO_flagFamily","module":"TNLean.MPS.MPDO.TwistedDimerFlagSectors"},{"id":"n12224","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_toMPSTensor_isCPSVCanonicalForm","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.toMPSTensor.IsCPSVCanonicalForm","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_toMPSTensor_isCPSVCanonicalForm","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12225","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.ambientCoisometry","kind":"def","summary":"Matrix (Fin (Finset.univ.sum fun _k => 4)) (Fin 8) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.ambientCoisometry","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12226","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.ambientCoisometry_mul_conjTranspose","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.TwistedDimer.ambientCoisometry MPOTensor.TwistedDimer.ambientCoisometry…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.ambientCoisometry_mul_conjTranspose","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12227","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.blockBondEquiv","kind":"def","summary":"Equiv (Sigma fun _k => Fin 4) (Fin 8)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.blockBondEquiv","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12228","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.block_toMPSTensor_isInjective","kind":"theorem","summary":"∀ (k : Fin 2), Kraus.IsInjective (MPOTensor.TwistedDimer.block k).toMPSTensor","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.block_toMPSTensor_isInjective","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12229","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.canonicalFormData","kind":"def","summary":"MPOTensor.TwistedDimer.T.toMPSTensor.CPSVCanonicalFormData","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.canonicalFormData","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12230","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.coef_flag_zero_ne_zero","kind":"theorem","summary":"∀ (k p q p' q' : Fin 2), Ne (MPOTensor.TwistedDimer.coef k (MPOTensor.TwistedDimer.physIdx p q…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.coef_flag_zero_ne_zero","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12231","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.conjTranspose_mul_mul_ambientCoisometry","kind":"theorem","summary":"∀ (Y : Matrix (Fin (Finset.univ.sum fun _k => 4)) (Fin (Finset.univ.sum fun _k => 4)) Complex),…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.conjTranspose_mul_mul_ambientCoisometry","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12232","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.conjTranspose_mul_self_normalizedBlock","kind":"theorem","summary":"∀ (k : Fin 2) (i j : Fin 8), Eq (HMul.hMul (MPOTensor.TwistedDimer.normalizedBlock k i j).conjT…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.conjTranspose_mul_self_normalizedBlock","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12233","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.conjTranspose_toMatrix_toPEquiv","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 [inst : DecidableEq α] [inst_1 : DecidableEq β] (e : Equiv α β), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.conjTranspose_toMatrix_toPEquiv","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12234","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.finSigmaFinEquiv_symm_retainedBondEquiv_symm","kind":"theorem","summary":"∀ (p p' k : Fin 2), Eq (finSigmaFinEquiv.symm (MPOTensor.TwistedDimer.retainedBondEquiv.symm (M…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.finSigmaFinEquiv_symm_retainedBondEquiv_symm","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12235","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.normalizedBlock","kind":"def","summary":"Fin 2 → MPOTensor 8 4","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.normalizedBlock","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12236","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.normalizedBlock_apply","kind":"theorem","summary":"∀ (k : Fin 2) (i j : Fin 8), Eq (MPOTensor.TwistedDimer.normalizedBlock k i j) (Matrix.single (…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.normalizedBlock_apply","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12237","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.normalizedBlock_isLeftCanonical","kind":"theorem","summary":"∀ (k : Fin 2), (MPOTensor.TwistedDimer.normalizedBlock k).toMPSTensor.IsLeftCanonical","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.normalizedBlock_isLeftCanonical","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12238","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.normalizedBlock_isNormalTensor","kind":"theorem","summary":"∀ (k : Fin 2), (MPOTensor.TwistedDimer.normalizedBlock k).toMPSTensor.IsNormalTensor","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.normalizedBlock_isNormalTensor","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12239","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.normalizedBlock_toMPSTensor_isInjective","kind":"theorem","summary":"∀ (k : Fin 2), Kraus.IsInjective (MPOTensor.TwistedDimer.normalizedBlock k).toMPSTensor","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.normalizedBlock_toMPSTensor_isInjective","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12240","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physTraceTransfer_normalizedBlock_one","kind":"theorem","summary":"Eq (MPOTensor.TwistedDimer.normalizedBlock 1).physTraceTransfer 0","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physTraceTransfer_normalizedBlock_one","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12241","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.retainedBondEquiv","kind":"def","summary":"Equiv (Fin (Finset.univ.sum fun _k => 4)) (Fin 8)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.retainedBondEquiv","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12242","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.rightBitEquiv","kind":"def","summary":"Equiv (Prod (Fin 2) (Prod (Fin 2) (Fin 2))) (Fin 8)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.rightBitEquiv","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12243","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.star_coef","kind":"theorem","summary":"∀ (k : Fin 2) (i j : Fin 8), Eq (star (MPOTensor.TwistedDimer.coef k i j)) (MPOTensor.TwistedDi…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.star_coef","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12244","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sum_normalizedCoef_sq","kind":"theorem","summary":"∀ (k r r' : Fin 2), Eq (Finset.univ.sum fun x => Finset.univ.sum fun y => HPow.hPow (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sum_normalizedCoef_sq","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12245","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sum_single_eq_single_sum","kind":"theorem","summary":"∀ ι : Type u_1 m : Type u_2 n : Type u_3 [inst : DecidableEq m] [inst_1 : DecidableEq n] (s : F…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sum_single_eq_single_sum","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12246","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.toMPSTensor_T_reconstruct","kind":"theorem","summary":"∀ (i : Fin (HMul.hMul 8 8)), Eq (MPOTensor.TwistedDimer.T.toMPSTensor i) (HMul.hMul (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.toMPSTensor_T_reconstruct","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12247","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.toMPSTensor_apply_finProdFinEquiv","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (i j : Fin d), Eq (M.toMPSTensor (finProdFinEquiv (Prod.mk i j)…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.toMPSTensor_apply_finProdFinEquiv","module":"TNLean.MPS.MPDO.TwistedDimerHorizontalCF"},{"id":"n12248","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_isMPDO","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_isMPDO","module":"TNLean.MPS.MPDO.TwistedDimerMPDO"},{"id":"n12249","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_T_entry_formula","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → ∀ (σ τ : Fin N → Fin 8), Eq (MPOTensor.TwistedDimer.T.mpo N σ τ) (HMul.h…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_T_entry_formula","module":"TNLean.MPS.MPDO.TwistedDimerMPDO"},{"id":"n12250","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_T_eq_smul_hadamard","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.T.mpo N) (HSMul.hSMul (HPow.hPow (1 / 2) N) (…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_T_eq_smul_hadamard","module":"TNLean.MPS.MPDO.TwistedDimerMPDO"},{"id":"n12251","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_not_isSimple","kind":"theorem","summary":"Not MPOTensor.TwistedDimer.T.IsSimple","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_not_isSimple","module":"TNLean.MPS.MPDO.TwistedDimerNotSimple"},{"id":"n12252","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.blockedCanonicalFormData","kind":"def","summary":"(L : Nat) → LT.lt 0 L → MPSTensor.CPSVCanonicalFormData (Kraus.reindexPhysical (⇑(MPOTensor.blo…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.blockedCanonicalFormData","module":"TNLean.MPS.MPDO.TwistedDimerNotSimple"},{"id":"n12253","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.blockedCanonicalFormData_blocks","kind":"theorem","summary":"∀ (L : Nat) (hL : LT.lt 0 L) (k : Fin 2), Eq ((MPOTensor.TwistedDimer.blockedCanonicalFormData…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.blockedCanonicalFormData_blocks","module":"TNLean.MPS.MPDO.TwistedDimerNotSimple"},{"id":"n12254","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer_blockTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (L : Nat), Eq (M.blockTensor L).physTraceTransfer (HPow.hPow M.…","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer_blockTensor","module":"TNLean.MPS.MPDO.TwistedDimerNotSimple"},{"id":"n12255","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.Cc","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.Cc","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12256","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.Cmat_eq_signs","kind":"theorem","summary":"∀ (k p q : Fin 2), Eq (MPOTensor.TwistedDimer.Cmat k p q) (HDiv.hDiv (HAdd.hAdd MPOTensor.Twist…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.Cmat_eq_signs","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12257","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.alpha_eq_x_div","kind":"theorem","summary":"Eq (HMul.hMul (HDiv.hDiv 1 (HMul.hMul 2 ↑MPOTensor.TwistedDimer.mu)) ↑MPOTensor.TwistedDimer.x)…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.alpha_eq_x_div","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12258","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.beta_eq_y_div","kind":"theorem","summary":"Eq (HMul.hMul (HDiv.hDiv 1 (HMul.hMul 2 ↑MPOTensor.TwistedDimer.mu)) ↑MPOTensor.TwistedDimer.y)…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.beta_eq_y_div","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12259","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagOperatorFamily_hasSameLengthProductForm","kind":"theorem","summary":"MPOTensor.TwistedDimer.flagOperatorFamily.HasSameLengthProductForm MPOTensor.TwistedDimer.twoLa…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagOperatorFamily_hasSameLengthProductForm","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12260","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight_add","kind":"theorem","summary":"∀ (f f' : Fin 2) (a : Fin 8), Eq (MPOTensor.TwistedDimer.flagWeight (HAdd.hAdd f f') a) (HMul.h…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight_add","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12261","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_flagMPO_mul","kind":"theorem","summary":"∀ L : Nat, LT.lt 0 L → ∀ (f f' : Fin 2), Eq (HMul.hMul ((MPOTensor.TwistedDimer.flagMPO f).mpo…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_flagMPO_mul","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12262","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.ofFn_Cc_prod_apply","kind":"theorem","summary":"∀ (L : Nat) (k : Fin L → Fin 2) (p q : Fin 2), Eq ((List.ofFn fun n => MPOTensor.TwistedDimer.C…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.ofFn_Cc_prod_apply","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12263","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.prod_flagWeight_physIdx","kind":"theorem","summary":"∀ (f : Fin 2) L : Nat (k t : Fin L → Fin 2), Eq (Finset.univ.prod fun n => MPOTensor.TwistedDim…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.prod_flagWeight_physIdx","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12264","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sum_cyclic_Cmat","kind":"theorem","summary":"∀ (L : Nat) (k : Fin (HAdd.hAdd L 1) → Fin 2), Eq (Finset.univ.sum fun t => Finset.univ.prod fu…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sum_cyclic_Cmat","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12265","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sum_sector","kind":"theorem","summary":"∀ (f : Fin 2) (L : Nat) (k : Fin L → Fin 2), Eq (Finset.univ.sum fun ρ => ite (And (MPOTensor.T…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sum_sector","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12266","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.tau_add","kind":"theorem","summary":"∀ (k f f' : Fin 2), Eq (MPOTensor.TwistedDimer.tau k (HAdd.hAdd f f')) (HMul.hMul (MPOTensor.Tw…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.tau_add","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12267","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.tau_one_mul_self","kind":"theorem","summary":"∀ (k : Fin 2), Eq (HMul.hMul (MPOTensor.TwistedDimer.tau k 1) (MPOTensor.TwistedDimer.tau k 1))…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.tau_one_mul_self","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12268","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.trace_ofFn_Cc_prod","kind":"theorem","summary":"∀ (L : Nat) (k : Fin L → Fin 2), Eq (List.ofFn fun n => MPOTensor.TwistedDimer.Cc (k n)).prod.t…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.trace_ofFn_Cc_prod","module":"TNLean.MPS.MPDO.TwistedDimerProductLaw"},{"id":"n12269","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.refineMap","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Fin 8) (Fin 8) Complex) (Matrix (Prod (Fin 8) (Fin 8))…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.refineMap","module":"TNLean.MPS.MPDO.TwistedDimerRefine"},{"id":"n12270","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.refineMap_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.TwistedDimer.refineMap","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.refineMap_isKrausCPTP","module":"TNLean.MPS.MPDO.TwistedDimerRefine"},{"id":"n12271","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.refineMap_physClose1","kind":"theorem","summary":"∀ (X : Matrix (Fin 8) (Fin 8) Complex), Eq (MPOTensor.TwistedDimer.refineMap (MPOTensor.Twisted…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.refineMap_physClose1","module":"TNLean.MPS.MPDO.TwistedDimerRefine"},{"id":"n12272","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bellProjector","kind":"def","summary":"Fin 2 → Matrix MPOTensor.TwistedDimer.Bond MPOTensor.TwistedDimer.Bond Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bellProjector","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12273","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bondFlagEquiv","kind":"def","summary":"(N : Nat) → Equiv (Fin N → MPOTensor.TwistedDimer.Cell) (Prod (Fin N → MPOTensor.TwistedDimer.B…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bondFlagEquiv","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12274","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bondState","kind":"def","summary":"Fin 2 → Matrix MPOTensor.TwistedDimer.Bond MPOTensor.TwistedDimer.Bond Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bondState","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12275","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.bondState_apply","kind":"theorem","summary":"∀ (k : Fin 2) (a b : MPOTensor.TwistedDimer.Bond), Eq (MPOTensor.TwistedDimer.bondState k a b)…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.bondState_apply","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12276","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.chainUnitary","kind":"def","summary":"(N : Nat) → Matrix (Fin N → Fin 8) (Fin N → Fin 8) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.chainUnitary","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12277","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.chainUnitary_apply","kind":"theorem","summary":"∀ (N : Nat) (s t : Fin N → Fin 8), Eq (MPOTensor.TwistedDimer.chainUnitary N s t) (Finset.univ.…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.chainUnitary_apply","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12278","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.chainUnitary_conjTranspose","kind":"theorem","summary":"∀ (N : Nat), Eq (MPOTensor.TwistedDimer.chainUnitary N).conjTranspose (MPOTensor.TwistedDimer.c…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.chainUnitary_conjTranspose","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12279","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.chainUnitary_conjTranspose_mul","kind":"theorem","summary":"∀ (N : Nat), Eq (HMul.hMul (MPOTensor.TwistedDimer.chainUnitary N).conjTranspose (MPOTensor.Twi…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.chainUnitary_conjTranspose_mul","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12280","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.chainUnitary_mul_conjTranspose","kind":"theorem","summary":"∀ (N : Nat), Eq (HMul.hMul (MPOTensor.TwistedDimer.chainUnitary N) (MPOTensor.TwistedDimer.chai…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.chainUnitary_mul_conjTranspose","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12281","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.decoratedState","kind":"def","summary":"(N : Nat) → Matrix (Fin N → Fin 8) (Fin N → Fin 8) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.decoratedState","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12282","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.evenFlagState","kind":"def","summary":"(N : Nat) → Matrix (Fin N → Fin 2) (Fin N → Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.evenFlagState","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12283","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFlip","kind":"def","summary":"Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFlip","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12284","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagMatrix","kind":"def","summary":"Fin 2 → Matrix (Fin 2) (Fin 2) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagMatrix","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12285","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.incomingCellEquiv","kind":"def","summary":"(N : Nat) → Equiv (Fin N → Fin 8) (Fin N → MPOTensor.TwistedDimer.Cell)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.incomingCellEquiv","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12286","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.localV","kind":"def","summary":"Matrix MPOTensor.TwistedDimer.Cell MPOTensor.TwistedDimer.Cell Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.localV","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12287","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.localV_conjTranspose","kind":"theorem","summary":"Eq MPOTensor.TwistedDimer.localV.conjTranspose MPOTensor.TwistedDimer.localV","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.localV_conjTranspose","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12288","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.localV_conjugate","kind":"theorem","summary":"∀ (k : Fin 2), Eq (HMul.hMul (HMul.hMul MPOTensor.TwistedDimer.localV (Matrix.kroneckerMap (fun…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.localV_conjugate","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12289","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.localV_mul_self","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.TwistedDimer.localV MPOTensor.TwistedDimer.localV) 1","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.localV_mul_self","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12290","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpo_eq_unitary_factorization","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → Eq (MPOTensor.TwistedDimer.T.mpo N) (HMul.hMul (HMul.hMul (MPOTensor.Twi…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpo_eq_unitary_factorization","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12291","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.powN","kind":"def","summary":"α : Type u_1 → Matrix α α Complex → (N : Nat) → Matrix (Fin N → α) (Fin N → α) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.powN","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12292","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigma","kind":"def","summary":"Matrix MPOTensor.TwistedDimer.Bond MPOTensor.TwistedDimer.Bond Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigma","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12293","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sigma'","kind":"def","summary":"Matrix MPOTensor.TwistedDimer.Bond MPOTensor.TwistedDimer.Bond Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sigma'","module":"TNLean.MPS.MPDO.TwistedDimerUnitaryFactorization"},{"id":"n12294","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.Cmat_ne_zero","kind":"theorem","summary":"∀ (k p q : Fin 2), Ne (MPOTensor.TwistedDimer.Cmat k p q) 0","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.Cmat_ne_zero","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12295","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.RetainedCoordinate","kind":"def","summary":"Type","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.RetainedCoordinate","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12296","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_apply_eq_zero_of_bitF_ne","kind":"theorem","summary":"∀ i j : Fin 8, Ne (MPOTensor.TwistedDimer.bitF i) (MPOTensor.TwistedDimer.bitF j) → Eq (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_apply_eq_zero_of_bitF_ne","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12297","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.T_isVerticalCF","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.IsVerticalCF","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.T_isVerticalCF","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12298","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.conjTranspose_mul_verticalCoisometry","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.TwistedDimer.verticalCoisometry.conjTranspose MPOTensor.TwistedDimer.ve…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.conjTranspose_mul_verticalCoisometry","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12299","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.exists_flagFamily_eq_smul_single","kind":"theorem","summary":"∀ (f : Fin 2) (p q : Fin 4), Exists fun v => Exists fun c => And (Ne c 0) (Eq (MPOTensor.Twiste…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.exists_flagFamily_eq_smul_single","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12300","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_finProdFinEquiv","kind":"theorem","summary":"∀ (f : Fin 2) (a b : Fin 8), Eq (MPOTensor.TwistedDimer.flagFamily f (finProdFinEquiv (Prod.mk…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_finProdFinEquiv","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12301","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isBNT","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.verticalTensor.IsBNT 2 MPOTensor.TwistedDimer.sectorDim MPOTensor.Twis…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isBNT","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12302","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isInjective","kind":"theorem","summary":"∀ (f : Fin 2), Kraus.IsInjective (MPOTensor.TwistedDimer.flagFamily f)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isInjective","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12303","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagFamily_isNormal","kind":"theorem","summary":"∀ (f : Fin 2), Kraus.IsNormal (MPOTensor.TwistedDimer.flagFamily f)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagFamily_isNormal","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12304","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_ne_zero","kind":"theorem","summary":"∀ (f p p' : Fin 2), Ne (MPOTensor.TwistedDimer.flagWeight f (MPOTensor.TwistedDimer.physIdx p p…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_ne_zero","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12305","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_zero_one","kind":"theorem","summary":"∀ (f : Fin 2), Eq (MPOTensor.TwistedDimer.flagWeight f (MPOTensor.TwistedDimer.physIdx 0 0 1))…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_zero_one","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12306","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_zero_zero","kind":"theorem","summary":"∀ (f : Fin 2), Eq (MPOTensor.TwistedDimer.flagWeight f (MPOTensor.TwistedDimer.physIdx 0 0 0))…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.flagWeight_physIdx_zero_zero_zero","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12307","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.linearIndependent_mpvState_flagFamily","kind":"theorem","summary":"∀ (N : Nat), LT.lt 0 N → LinearIndependent Complex fun f => (MPOTensor.TwistedDimer.flagFamily…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.linearIndependent_mpvState_flagFamily","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12308","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.mpv_verticalTensor_T","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → ∀ (σ : Fin N → Fin (HMul.hMul 8 8)), Eq (MPOTensor.TwistedDimer.T.vertic…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.mpv_verticalTensor_T","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12309","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physCoordEquiv","kind":"def","summary":"Equiv MPOTensor.TwistedDimer.RetainedCoordinate (Fin 8)","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physCoordEquiv","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12310","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.physCoordEquiv_sectorCoord","kind":"theorem","summary":"∀ (f l r : Fin 2), Eq (MPOTensor.TwistedDimer.physCoordEquiv (MPOTensor.TwistedDimer.sectorCoor…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.physCoordEquiv_sectorCoord","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12311","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorCoord","kind":"def","summary":"Equiv (Prod (Fin 2) (Fin 4)) MPOTensor.TwistedDimer.RetainedCoordinate","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorCoord","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12312","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorCoord_apply","kind":"theorem","summary":"∀ (f : Fin 2) (q : Fin 4), Eq (MPOTensor.TwistedDimer.sectorCoord (Prod.mk f q)) ((MPOTensor.ve…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorCoord_apply","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12313","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorDim","kind":"def","summary":"Fin 2 → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorDim","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12314","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorMult","kind":"def","summary":"Fin 2 → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorMult","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12315","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.sectorWeight","kind":"def","summary":"(f : Fin 2) → Fin (MPOTensor.TwistedDimer.sectorMult f) → Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.sectorWeight","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12316","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalCoisometry","kind":"def","summary":"Matrix MPOTensor.TwistedDimer.RetainedCoordinate (Fin 8) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalCoisometry","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12317","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalCoisometry_apply","kind":"theorem","summary":"∀ (x : MPOTensor.TwistedDimer.RetainedCoordinate) (i : Fin 8), Eq (MPOTensor.TwistedDimer.verti…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalCoisometry_apply","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12318","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalCoisometry_conj","kind":"theorem","summary":"∀ (M : Matrix (Fin 8) (Fin 8) Complex) (x y : MPOTensor.TwistedDimer.RetainedCoordinate), Eq (H…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalCoisometry_conj","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12319","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalCoisometry_conj_verticalTensor","kind":"theorem","summary":"∀ (v : Fin (HMul.hMul 8 8)), Eq (HMul.hMul (HMul.hMul MPOTensor.TwistedDimer.verticalCoisometry…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalCoisometry_conj_verticalTensor","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12320","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalCoisometry_mul_conjTranspose","kind":"theorem","summary":"Eq (HMul.hMul MPOTensor.TwistedDimer.verticalCoisometry MPOTensor.TwistedDimer.verticalCoisomet…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalCoisometry_mul_conjTranspose","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12321","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.verticalTensor_T_eq_conjTranspose_mul","kind":"theorem","summary":"∀ (v : Fin (HMul.hMul 8 8)), Eq (MPOTensor.TwistedDimer.T.verticalTensor v) (HMul.hMul (HMul.hM…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.verticalTensor_T_eq_conjTranspose_mul","module":"TNLean.MPS.MPDO.TwistedDimerVerticalCF"},{"id":"n12322","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.coarseMap","kind":"def","summary":"LinearMap (RingHom.id Complex) (Matrix (Prod (Fin 8) (Fin 8)) (Prod (Fin 8) (Fin 8)) Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.coarseMap","module":"TNLean.MPS.MPDO.TwistedDimerViaTS"},{"id":"n12323","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.coarseMap_isKrausCPTP","kind":"theorem","summary":"IsKrausCPTP MPOTensor.TwistedDimer.coarseMap","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.coarseMap_isKrausCPTP","module":"TNLean.MPS.MPDO.TwistedDimerViaTS"},{"id":"n12324","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.coarseMap_physClose2","kind":"theorem","summary":"∀ (X : Matrix (Fin 8) (Fin 8) Complex), Eq (MPOTensor.TwistedDimer.coarseMap (MPOTensor.Twisted…","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.coarseMap_physClose2","module":"TNLean.MPS.MPDO.TwistedDimerViaTS"},{"id":"n12325","layer":"formal","project":"p8","title":"MPOTensor.TwistedDimer.isRFPViaTS_T","kind":"theorem","summary":"MPOTensor.TwistedDimer.T.IsRFPViaTS","labels":[],"detail_key":"p8","name":"MPOTensor.TwistedDimer.isRFPViaTS_T","module":"TNLean.MPS.MPDO.TwistedDimerViaTS"},{"id":"n12326","layer":"formal","project":"p8","title":"MPOTensor.verticalCF_and_blockTwo_of_horizontalCF","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → And M.IsVerticalCF M.blockTwo.Is…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCF_and_blockTwo_of_horizontalCF","module":"TNLean.MPS.MPDO.TwoSiteVerticalCanonicalForm"},{"id":"n12327","layer":"formal","project":"p8","title":"MPOTensor.HasVerticalBNTGroupingWithIsometry","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.HasVerticalBNTGroupingWithIsometry","module":"TNLean.MPS.MPDO.VerticalBNT"},{"id":"n12328","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_sectorCompression_ne_zero_of_corner","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → ∀ (P : Matrix (Fin d) (Fin d) Complex), (Ex…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_sectorCompression_ne_zero_of_corner","module":"TNLean.MPS.MPDO.VerticalBNT"},{"id":"n12329","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_verticalBNTGrouping_with_isometry","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.HasVerticalBNTGroupingWithIsom…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_verticalBNTGrouping_with_isometry","module":"TNLean.MPS.MPDO.VerticalBNT"},{"id":"n12330","layer":"formal","project":"p8","title":"MPOTensor.exists_rangeProjection_corner_ne_zero","kind":"theorem","summary":"∀ d D n : Nat [NeZero n] (M : MPOTensor d D) (A : MPSTensor (HMul.hMul D D) n), A.IsNormalTenso…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_rangeProjection_corner_ne_zero","module":"TNLean.MPS.MPDO.VerticalBNT"},{"id":"n12331","layer":"formal","project":"p8","title":"MPOTensor.isBNT_verticalTensor_of_grouping","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (blocks : (k :…","labels":[],"detail_key":"p8","name":"MPOTensor.isBNT_verticalTensor_of_grouping","module":"TNLean.MPS.MPDO.VerticalBNTConstruction"},{"id":"n12332","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂Pos_toTensorFromBlocks_of_reconstruction","kind":"theorem","summary":"∀ d s r : Nat dim : Fin r → Nat (T : MPSTensor s d) (μ : Fin r → Complex) (blocks : (k : Fin r)…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂Pos_toTensorFromBlocks_of_reconstruction","module":"TNLean.MPS.MPDO.VerticalBNTConstruction"},{"id":"n12333","layer":"formal","project":"p8","title":"MPOTensor.blockedVerticalOperatorRepresentations_of_unitaryBlockEquiv","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedVerticalOperatorRepresentations_of_unitaryBlockEquiv","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12334","layer":"formal","project":"p8","title":"MPOTensor.mpo_verticalBNTMPO_eq_pow_smul_of_unitary_reindex","kind":"theorem","summary":"∀ D n₁ n₂ : Nat (A₁ : MPSTensor (HMul.hMul D D) n₁) (A₂ : MPSTensor (HMul.hMul D D) n₂) (c : Co…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_verticalBNTMPO_eq_pow_smul_of_unitary_reindex","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12335","layer":"formal","project":"p8","title":"MPOTensor.mpo_verticalBNTMPO_eq_sum_of_coisometry_reconstruction","kind":"theorem","summary":"∀ d D g : Nat dim mult : Fin g → Nat (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_verticalBNTMPO_eq_sum_of_coisometry_reconstruction","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12336","layer":"formal","project":"p8","title":"MPOTensor.mpo_verticalBNTMPO_verticalAssembledTensor_eq_sum","kind":"theorem","summary":"∀ D g : Nat dim mult : Fin g → Nat (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_verticalBNTMPO_verticalAssembledTensor_eq_sum","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12337","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_exists_blockedOperatorRepresentations","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_exists_blockedOperatorRepresentations","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12338","layer":"formal","project":"p8","title":"MPOTensor.verticalBNTMPO_verticalTensor_blockTwo","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq (MPOTensor.verticalBNTMPO M.blockTwo.verticalTensor) ((MPOT…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalBNTMPO_verticalTensor_blockTwo","module":"TNLean.MPS.MPDO.VerticalBlockedOperatorRepresentations"},{"id":"n12339","layer":"formal","project":"p8","title":"MPOTensor.contractBondMatrix_verticalAssembledTensor","kind":"theorem","summary":"∀ D g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.contractBondMatrix_verticalAssembledTensor","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12340","layer":"formal","project":"p8","title":"MPOTensor.contractBondMatrix_verticalTensor_eq_physClose1","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Eq M.verticalTensor.contractBondMatrix M.physClose1","labels":[],"detail_key":"p8","name":"MPOTensor.contractBondMatrix_verticalTensor_eq_physClose1","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12341","layer":"formal","project":"p8","title":"MPOTensor.mul_physClose1_mul_conjTranspose_of_verticalAssembledTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (m…","labels":[],"detail_key":"p8","name":"MPOTensor.mul_physClose1_mul_conjTranspose_of_verticalAssembledTensor","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12342","layer":"formal","project":"p8","title":"MPOTensor.mul_physClose1_mul_conjTranspose_of_vertical_forward","kind":"theorem","summary":"∀ d D R : Nat (M : MPOTensor d D) (B : MPSTensor (HMul.hMul D D) R) (U : Matrix (Fin R) (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.mul_physClose1_mul_conjTranspose_of_vertical_forward","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12343","layer":"formal","project":"p8","title":"MPOTensor.physClose1_eq_of_verticalAssembledTensor_reconstruction","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (m…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose1_eq_of_verticalAssembledTensor_reconstruction","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12344","layer":"formal","project":"p8","title":"MPOTensor.physClose1_eq_of_vertical_reconstruction","kind":"theorem","summary":"∀ d D R : Nat (M : MPOTensor d D) (B : MPSTensor (HMul.hMul D D) R) (U : Matrix (Fin R) (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.physClose1_eq_of_vertical_reconstruction","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12345","layer":"formal","project":"p8","title":"MPSTensor.contractBondMatrix","kind":"def","summary":"d D : Nat → MPSTensor (HMul.hMul D D) d → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPSTensor.contractBondMatrix","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12346","layer":"formal","project":"p8","title":"MPSTensor.contractBondMatrix_mul_mul","kind":"theorem","summary":"∀ d D R : Nat (A : MPSTensor (HMul.hMul D D) d) (L : Matrix (Fin R) (Fin d) Complex) (K : Matri…","labels":[],"detail_key":"p8","name":"MPSTensor.contractBondMatrix_mul_mul","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12347","layer":"formal","project":"p8","title":"MPSTensor.contractBondMatrix_toTensorFromBlocks","kind":"theorem","summary":"∀ D r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor (HMul.hMul D D…","labels":[],"detail_key":"p8","name":"MPSTensor.contractBondMatrix_toTensorFromBlocks","module":"TNLean.MPS.MPDO.VerticalBoundaryContraction"},{"id":"n12348","layer":"formal","project":"p8","title":"MPOTensor.IsVerticalCF","kind":"def","summary":"d D : Nat → MPOTensor d D → 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Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_compress_trace_pos","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12352","layer":"formal","project":"p8","title":"MPOTensor.mpo_opposite_corner_eq_zero","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsMPDO → ∀ (N : Nat), LT.lt 0 N → ∀ (P : Matrix (Fin N → Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_opposite_corner_eq_zero","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12353","layer":"formal","project":"p8","title":"MPOTensor.mpv_diagonalTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N : Nat (σ : Fin N → Fin d), Eq (M.diagonalTensor.mpv σ) (M.toM…","labels":[],"detail_key":"p8","name":"MPOTensor.mpv_diagonalTensor","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12354","layer":"formal","project":"p8","title":"MPOTensor.mpv_diagonalTensor_eq_blocks","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r…","labels":[],"detail_key":"p8","name":"MPOTensor.mpv_diagonalTensor_eq_blocks","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12355","layer":"formal","project":"p8","title":"MPOTensor.mpv_diagonalTensor_eq_mpo_diag","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N : Nat (σ : Fin N → Fin d), Eq (M.diagonalTensor.mpv σ) (M.mpo…","labels":[],"detail_key":"p8","name":"MPOTensor.mpv_diagonalTensor_eq_mpo_diag","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12356","layer":"formal","project":"p8","title":"MPOTensor.mpv_diagonalTensor_nonneg","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N : Nat, (M.mpo N).PosSemidef → ∀ (σ : Fin N → Fin d), LE.le 0…","labels":[],"detail_key":"p8","name":"MPOTensor.mpv_diagonalTensor_nonneg","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12357","layer":"formal","project":"p8","title":"MPOTensor.mpv_verticalAssembledTensor_eq_sum","kind":"theorem","summary":"∀ d g : Nat (dim mult : Fin g → Nat) (ω : (α : Fin g) → Fin (mult α) → Complex) (A : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.mpv_verticalAssembledTensor_eq_sum","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12358","layer":"formal","project":"p8","title":"MPOTensor.sameMPV₂Pos_toTensorFromBlocks_verticalAssembledTensor_of_equiv","kind":"theorem","summary":"∀ d r g : Nat dim₀ : Fin r → Nat dim : Fin g → Nat (μ : Fin r → Complex) (B : (k : Fin r) → MPS…","labels":[],"detail_key":"p8","name":"MPOTensor.sameMPV₂Pos_toTensorFromBlocks_verticalAssembledTensor_of_equiv","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12359","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyBlocks","kind":"def","summary":"d g : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → MPSTensor d (dim α)) → (q : Fin (Finset.u…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyBlocks","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12360","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyDim","kind":"def","summary":"g : Nat → (Fin g → Nat) → (mult : Fin g → Nat) → Fin (Finset.univ.sum fun α => mult α) → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyDim","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12361","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor","kind":"def","summary":"d D : Nat → MPOTensor d D → MPSTensor (HMul.hMul D D) d","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12362","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_adjointTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (ab : Fin (HMul.hMul D D)), Eq (M.adjointTensor.verticalTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_adjointTensor","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12363","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_finProdFinEquiv","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (a b : Fin D) (i j : Fin d), Eq (M.verticalTensor (finProdFinEq…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_finProdFinEquiv","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12364","layer":"formal","project":"p8","title":"MPSTensor.FirstSiteActionAgree.of_sameMPVPos","kind":"theorem","summary":"∀ d D D' : Nat A : MPSTensor d D B : MPSTensor d D' Y Z : Matrix (Fin d) (Fin d) Complex, A.Sam…","labels":[],"detail_key":"p8","name":"MPSTensor.FirstSiteActionAgree.of_sameMPVPos","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12365","layer":"formal","project":"p8","title":"MPSTensor.braRightAction","kind":"def","summary":"d : Nat → Matrix (Fin d) (Fin d) Complex → Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)) C…","labels":[],"detail_key":"p8","name":"MPSTensor.braRightAction","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12366","layer":"formal","project":"p8","title":"MPSTensor.braRightAction_mpv","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) N : Nat (ρ : Fin (HAdd.hAd…","labels":[],"detail_key":"p8","name":"MPSTensor.braRightAction_mpv","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12367","layer":"formal","project":"p8","title":"MPSTensor.ketLeftAction","kind":"def","summary":"d : Nat → Matrix (Fin d) (Fin d) Complex → Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)) C…","labels":[],"detail_key":"p8","name":"MPSTensor.ketLeftAction","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12368","layer":"formal","project":"p8","title":"MPSTensor.ketLeftAction_mpv","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) N : Nat (ρ : Fin (HAdd.hAd…","labels":[],"detail_key":"p8","name":"MPSTensor.ketLeftAction_mpv","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12369","layer":"formal","project":"p8","title":"MPSTensor.ketLeftBraRightAction","kind":"def","summary":"d : Nat → Matrix (Fin d) (Fin d) Complex → Matrix (Fin (HMul.hMul d d)) (Fin (HMul.hMul d d)) C…","labels":[],"detail_key":"p8","name":"MPSTensor.ketLeftBraRightAction","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12370","layer":"formal","project":"p8","title":"MPSTensor.ketLeftBraRightAction_mpv","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) (P : Matrix (Fin d) (Fin d) Complex) N : Nat (ρ : Fin (HAdd.hAd…","labels":[],"detail_key":"p8","name":"MPSTensor.ketLeftBraRightAction_mpv","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12371","layer":"formal","project":"p8","title":"MPSTensor.mpv_toMPSTensor_pairConfig","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) N : Nat (σ τ : Fin N → Fin d), Eq (M.toMPSTensor.mpv fun n => f…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_toMPSTensor_pairConfig","module":"TNLean.MPS.MPDO.VerticalCF"},{"id":"n12372","layer":"formal","project":"p8","title":"MPOTensor.verticalCF_of_horizontalCF","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.IsVerticalCF","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCF_of_horizontalCF","module":"TNLean.MPS.MPDO.VerticalCanonicalForm"},{"id":"n12373","layer":"formal","project":"p8","title":"MPOTensor.verticalCF_of_grouping_and_gramDressing","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.HasVerticalBNTGroupingWithIsometry → M.HasGroupedCornerGramD…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCF_of_grouping_and_gramDressing","module":"TNLean.MPS.MPDO.VerticalCanonicalFormConstruction"},{"id":"n12374","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftClause","kind":"def","summary":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO.BNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftClause","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12375","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftClause_coeff","kind":"theorem","summary":"∀ (L : Nat) (α β γ : Fin 1), Eq (MPOTensor.VerticalCoefficientPresentationCounterexample.leftCl…","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftClause_coeff","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12376","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO","kind":"def","summary":"MPOTensor 1 2","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12377","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO_isMPDO","kind":"theorem","summary":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.leftMPO_isMPDO","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12378","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightClause","kind":"def","summary":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO.BNTAlgebraTensorClause","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightClause","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12379","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightClause_coeff","kind":"theorem","summary":"∀ (L : Nat) (α β γ : Fin 1), Eq (MPOTensor.VerticalCoefficientPresentationCounterexample.rightC…","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightClause_coeff","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12380","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO","kind":"def","summary":"MPOTensor 1 2","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12381","layer":"formal","project":"p8","title":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO_isMPDO","kind":"theorem","summary":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO.IsMPDO","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalCoefficientPresentationCounterexample.rightMPO_isMPDO","module":"TNLean.MPS.MPDO.VerticalCoefficientPresentationCounterexample"},{"id":"n12382","layer":"formal","project":"p8","title":"MPOTensor.isVerticalCF_of_grouped_orthogonal_sectors","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D) g : Nat (dim mult : Fin g → Nat), (∀ (α : Fin g), LT.lt 0 (mult…","labels":[],"detail_key":"p8","name":"MPOTensor.isVerticalCF_of_grouped_orthogonal_sectors","module":"TNLean.MPS.MPDO.VerticalCoisometry"},{"id":"n12383","layer":"formal","project":"p8","title":"Matrix.sigmaBlockRow","kind":"def","summary":"r d : Nat → dim : Fin r → Nat → ((k : Fin r) → Matrix (Fin d) (Fin (dim k)) Complex) → Matrix (…","labels":[],"detail_key":"p8","name":"Matrix.sigmaBlockRow","module":"TNLean.MPS.MPDO.VerticalCoisometry"},{"id":"n12384","layer":"formal","project":"p8","title":"Matrix.sigmaBlockRow_conjugation","kind":"theorem","summary":"∀ r d s : Nat dim : Fin r → Nat (T : Fin s → Matrix (Fin d) (Fin d) Complex) (B : (k : Fin r) →…","labels":[],"detail_key":"p8","name":"Matrix.sigmaBlockRow_conjugation","module":"TNLean.MPS.MPDO.VerticalCoisometry"},{"id":"n12385","layer":"formal","project":"p8","title":"Matrix.sigmaBlockRow_isCoisometry","kind":"theorem","summary":"∀ r d : Nat dim : Fin r → Nat (W : (k : Fin r) → Matrix (Fin d) (Fin (dim k)) Complex), (∀ (k :…","labels":[],"detail_key":"p8","name":"Matrix.sigmaBlockRow_isCoisometry","module":"TNLean.MPS.MPDO.VerticalCoisometry"},{"id":"n12386","layer":"formal","project":"p8","title":"Matrix.sigmaBlockRow_reconstruction","kind":"theorem","summary":"∀ r d s : Nat dim : Fin r → Nat (T : Fin s → Matrix (Fin d) (Fin d) Complex) (B : (k : Fin r) →…","labels":[],"detail_key":"p8","name":"Matrix.sigmaBlockRow_reconstruction","module":"TNLean.MPS.MPDO.VerticalCoisometry"},{"id":"n12387","layer":"formal","project":"p8","title":"MPOTensor.blockedReferenceInclusion","kind":"def","summary":"g d : Nat → (dim mult : Fin g → Nat) → (∀ (γ : Fin g), LT.lt 0 (mult γ)) → Matrix (Fin (Finset.…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedReferenceInclusion","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12388","layer":"formal","project":"p8","title":"MPOTensor.blockedReferenceInclusion_isometry","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (hMult : ∀ (γ : Fin g), LT.lt 0 (mult γ)) (U : Matrix (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedReferenceInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12389","layer":"formal","project":"p8","title":"MPOTensor.blockedReference_compression","kind":"theorem","summary":"∀ g d D : Nat (M : MPOTensor d D) (dim mult : Fin g → Nat) (hMult : ∀ (γ : Fin g), LT.lt 0 (mul…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedReference_compression","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12390","layer":"formal","project":"p8","title":"MPOTensor.blockedReference_intertwine","kind":"theorem","summary":"∀ g d D : Nat (M : MPOTensor d D) (dim mult : Fin g → Nat) (hMult : ∀ (γ : Fin g), LT.lt 0 (mul…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedReference_intertwine","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12391","layer":"formal","project":"p8","title":"MPOTensor.blockedReference_intertwine_adjoint","kind":"theorem","summary":"∀ g d D : Nat (M : MPOTensor d D) (dim mult : Fin g → Nat) (hMult : ∀ (γ : Fin g), LT.lt 0 (mul…","labels":[],"detail_key":"p8","name":"MPOTensor.blockedReference_intertwine_adjoint","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12392","layer":"formal","project":"p8","title":"MPOTensor.verticalAssembledTensor_mul_verticalCopyBlockInclusion","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalAssembledTensor_mul_verticalCopyBlockInclusion","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12393","layer":"formal","project":"p8","title":"MPOTensor.verticalAssembledTensor_reindex_copyCoordinates","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalAssembledTensor_reindex_copyCoordinates","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12394","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyBlockInclusion","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → (p : Sigma fun α => Fin (mult α)) → Matrix (Fin (Finset.un…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyBlockInclusion","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12395","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyBlockInclusion_compression","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyBlockInclusion_compression","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12396","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyBlockInclusion_conjTranspose_mul_verticalAssembledTensor","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyBlockInclusion_conjTranspose_mul_verticalAssembledTensor","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12397","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyBlockInclusion_isometry","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (p : Sigma fun α => Fin (mult α)), Eq (HMul.hMul (MPOTensor.…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyBlockInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalCopyBlocks"},{"id":"n12398","layer":"formal","project":"p8","title":"MPOTensor.transportSToVerticalCoordinates","kind":"def","summary":"d R₁ R₂ : Nat → Matrix (Fin R₁) (Fin d) Complex → Matrix (Fin R₂) (Fin (HMul.hMul d d)) Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportSToVerticalCoordinates","module":"TNLean.MPS.MPDO.VerticalMapTransport"},{"id":"n12399","layer":"formal","project":"p8","title":"MPOTensor.transportSToVerticalCoordinates_contractBondMatrix","kind":"theorem","summary":"∀ d D R₁ R₂ : Nat (M : MPOTensor d D) (B₁ : MPSTensor (HMul.hMul D D) R₁) (B₂ : MPSTensor (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.transportSToVerticalCoordinates_contractBondMatrix","module":"TNLean.MPS.MPDO.VerticalMapTransport"},{"id":"n12400","layer":"formal","project":"p8","title":"MPOTensor.transportTToVerticalCoordinates","kind":"def","summary":"d R₁ R₂ : Nat → Matrix (Fin R₁) (Fin d) Complex → Matrix (Fin R₂) (Fin (HMul.hMul d d)) Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportTToVerticalCoordinates","module":"TNLean.MPS.MPDO.VerticalMapTransport"},{"id":"n12401","layer":"formal","project":"p8","title":"MPOTensor.transportTToVerticalCoordinates_contractBondMatrix","kind":"theorem","summary":"∀ d D R₁ R₂ : Nat (M : MPOTensor d D) (B₁ : MPSTensor (HMul.hMul D D) R₁) (B₂ : MPSTensor (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.transportTToVerticalCoordinates_contractBondMatrix","module":"TNLean.MPS.MPDO.VerticalMapTransport"},{"id":"n12402","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison.referenceInclusion","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison.referenceInclusion","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12403","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison.reference_compression","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison.reference_compression","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12404","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.ambientInclusion_isometry","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.ambientInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12405","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.ambient_compression","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.ambient_compression","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12406","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.ambient_intertwine","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.ambient_intertwine","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12407","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.ambient_intertwine_adjoint","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.ambient_intertwine_adjoint","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12408","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.exists_originalCornerFamily","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.exists_originalCornerFamily","module":"TNLean.MPS.MPDO.VerticalProductCornerComparison"},{"id":"n12409","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_exists_positiveFusionDecomposition","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_exists_positiveFusionDecomposition","module":"TNLean.MPS.MPDO.VerticalProductFusionDecomposition"},{"id":"n12410","layer":"formal","project":"p8","title":"MPOTensor.verticalCoisometrySquare","kind":"def","summary":"d n : Nat → Matrix (Fin n) (Fin d) Complex → Matrix (Fin (HMul.hMul n n)) (Fin (HMul.hMul d d))…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCoisometrySquare","module":"TNLean.MPS.MPDO.VerticalProductReconstruction"},{"id":"n12411","layer":"formal","project":"p8","title":"MPOTensor.verticalCoisometrySquare_isCoisometry","kind":"theorem","summary":"∀ d n : Nat (U : Matrix (Fin n) (Fin d) Complex), Eq (HMul.hMul U U.conjTranspose) 1 → Eq (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCoisometrySquare_isCoisometry","module":"TNLean.MPS.MPDO.VerticalProductReconstruction"},{"id":"n12412","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyCoordinateEquiv","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → Equiv (Fin (Finset.univ.sum fun q => MPOTensor.verticalCop…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyCoordinateEquiv","module":"TNLean.MPS.MPDO.VerticalProductReconstruction"},{"id":"n12413","layer":"formal","project":"p8","title":"MPOTensor.verticalCopyCoordinateEquiv_symm_apply","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (p : Sigma fun α => Fin (mult α)) (i : Fin (dim p.fst)), Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalCopyCoordinateEquiv_symm_apply","module":"TNLean.MPS.MPDO.VerticalProductReconstruction"},{"id":"n12414","layer":"formal","project":"p8","title":"MPSTensor.projectorClosure_and_noPeriodicVectors_of_coisometry_reconstruction","kind":"theorem","summary":"∀ d D n : Nat (A : MPSTensor d D) (C : MPSTensor d n) (U : Matrix (Fin n) (Fin D) Complex), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.projectorClosure_and_noPeriodicVectors_of_coisometry_reconstruction","module":"TNLean.MPS.MPDO.VerticalProductReconstruction"},{"id":"n12415","layer":"formal","project":"p8","title":"MPOTensor.exists_weightedVerticalProductBlock_normalDecomposition","kind":"theorem","summary":"∀ g d D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_weightedVerticalProductBlock_normalDecomposition","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12416","layer":"formal","project":"p8","title":"MPOTensor.mulTensor_verticalAssembledTensor_reindex","kind":"theorem","summary":"∀ g D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.mulTensor_verticalAssembledTensor_reindex","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12417","layer":"formal","project":"p8","title":"MPOTensor.productRetainedEquiv","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → Equiv (Fin (HMul.hMul (Finset.univ.sum fun q => 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:…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedProductBlockInclusion_conjTranspose_mul_retainedVerticalProductTensor","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12420","layer":"formal","project":"p8","title":"MPOTensor.retainedProductBlockInclusion_isometry","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (p : MPOTensor.VerticalCopyPair mult), Eq (HMul.hMul (MPOTen…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedProductBlockInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12421","layer":"formal","project":"p8","title":"MPOTensor.retainedProductBlockInclusion_orthogonal_of_ne","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) p q : MPOTensor.VerticalCopyPair mult, Ne p q → Eq (HMul.hMu…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedProductBlockInclusion_orthogonal_of_ne","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12422","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalProductTensor","kind":"def","summary":"g D : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → ((α : Fin g) →…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalProductTensor","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12423","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalProductTensor_eq_sum_pairBlocks","kind":"theorem","summary":"∀ g D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalProductTensor_eq_sum_pairBlocks","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12424","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalProductTensor_mul_retainedProductBlockInclusion","kind":"theorem","summary":"∀ g D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalProductTensor_mul_retainedProductBlockInclusion","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12425","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalProduct_projectorClosure_and_noPeriodicVectors","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D) (A : MPSTensor (HMul.hMul D D) n) (U : Matrix (Fin n) (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalProduct_projectorClosure_and_noPeriodicVectors","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12426","layer":"formal","project":"p8","title":"MPOTensor.verticalTensor_blockTwo_squared_coisometry_reconstruction","kind":"theorem","summary":"∀ d D n : Nat (M : MPOTensor d D) (A : MPSTensor (HMul.hMul D D) n) (U : Matrix (Fin n) (Fin d)…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalTensor_blockTwo_squared_coisometry_reconstruction","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12427","layer":"formal","project":"p8","title":"MPOTensor.weightedVerticalProductBlock","kind":"def","summary":"g D : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → ((α : Fin g) →…","labels":[],"detail_key":"p8","name":"MPOTensor.weightedVerticalProductBlock","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12428","layer":"formal","project":"p8","title":"MPOTensor.weightedVerticalProductBlock_projectorClosure_and_noPeriodicVectors","kind":"theorem","summary":"∀ g d D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α…","labels":[],"detail_key":"p8","name":"MPOTensor.weightedVerticalProductBlock_projectorClosure_and_noPeriodicVectors","module":"TNLean.MPS.MPDO.VerticalProductRetainedBlocks"},{"id":"n12429","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily","kind":"inductive","summary":"g D : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → ((α : Fin g) →…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12430","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.ActiveLabel","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.ActiveLabel","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12431","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison","kind":"inductive","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.FlatBlockedBNTComparison","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12432","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.OriginalCornerFamily","kind":"inductive","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.OriginalCornerFamily","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12433","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.activeLabelEquiv","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : 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F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatBlock","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12436","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatBlock_normal","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatBlock_normal","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12437","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatCoefficient","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : 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g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatCoefficient_pos","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12440","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatDim","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatDim","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12441","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatDim_pos","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatDim_pos","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12442","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatInclusion","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatInclusion","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12443","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatInclusion_isometry","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12444","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flatInclusion_orthogonal_of_ne","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flatInclusion_orthogonal_of_ne","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12445","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flat_compression","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flat_compression","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12446","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flat_intertwine","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flat_intertwine","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12447","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flat_intertwine_adjoint","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flat_intertwine_adjoint","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12448","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flat_reconstruction","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flat_reconstruction","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12449","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.flat_sameMPV₂Pos","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.flat_sameMPV₂Pos","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12450","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion","kind":"def","summary":"g D : Nat → dim mult : Fin g → Nat → weight : (α : Fin g) → Fin (mult α) → Complex → B : (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12451","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion_isometry","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion_isometry","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12452","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion_orthogonal_of_ne","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retainedInclusion_orthogonal_of_ne","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12453","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retained_compression","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retained_compression","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12454","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retained_intertwine","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retained_intertwine","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12455","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retained_intertwine_adjoint","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retained_intertwine_adjoint","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12456","layer":"formal","project":"p8","title":"MPOTensor.RetainedProductSpectralFamily.retained_reconstruction_active","kind":"theorem","summary":"∀ g D : Nat dim mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex B : (α : Fin g…","labels":[],"detail_key":"p8","name":"MPOTensor.RetainedProductSpectralFamily.retained_reconstruction_active","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12457","layer":"formal","project":"p8","title":"MPOTensor.exists_retainedProductSpectralFamily","kind":"theorem","summary":"∀ g d D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (B : (α…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_retainedProductSpectralFamily","module":"TNLean.MPS.MPDO.VerticalProductSpectralFamily"},{"id":"n12458","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_complete_verticalReducingProjectors","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → Exists fun r => Exists fun P =>…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_complete_verticalReducingProjectors","module":"TNLean.MPS.MPDO.VerticalReduction"},{"id":"n12459","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_irreducible_verticalBlockDecomp_with_isometry","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → Exists fun r => Exists fun dim =…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_irreducible_verticalBlockDecomp_with_isometry","module":"TNLean.MPS.MPDO.VerticalReduction"},{"id":"n12460","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.hasInvariantProjectorClosure_verticalTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → M.verticalTensor.HasInvariantPro…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.hasInvariantProjectorClosure_verticalTensor","module":"TNLean.MPS.MPDO.VerticalReduction"},{"id":"n12461","layer":"formal","project":"p8","title":"MPSTensor.IsCPSVCanonicalForm.hasInvariantProjectorClosure_verticalTensor","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.toMPSTensor.IsCPSVCanonicalForm → M.IsMPDO → M.verticalTenso…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCPSVCanonicalForm.hasInvariantProjectorClosure_verticalTensor","module":"TNLean.MPS.MPDO.VerticalReduction"},{"id":"n12462","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestoration","kind":"def","summary":"g d : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → Matrix (Fin (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestoration","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12463","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension","kind":"def","summary":"g d : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → Matrix (Fin (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12464","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_factorization","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (U : Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_factorization","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12465","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_isKrausCP","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_isKrausCP","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12466","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_isKrausCPTP","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestorationExtension_isKrausCPTP","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12467","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestoration_apply","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (U : Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestoration_apply","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12468","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorAmbientRestoration_trace_smul","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorAmbientRestoration_trace_smul","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12469","layer":"formal","project":"p8","title":"MPOTensor.trace_normalizedVerticalSectorAmbientRestoration","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_normalizedVerticalSectorAmbientRestoration","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12470","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetraction","kind":"def","summary":"g d : Nat → (dim mult : Fin g → Nat) → Matrix (Fin (Finset.univ.sum fun q => MPOTensor.vertical…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorAmbientRetraction","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12471","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetractionExtension","kind":"def","summary":"g d : Nat → (dim mult : Fin g → Nat) → Matrix (Fin (Finset.univ.sum fun q => MPOTensor.vertical…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorAmbientRetractionExtension","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12472","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetractionExtension_factorization","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (U : Matrix (Fin (Finset.univ.sum fun q => MPOTensor.verti…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorAmbientRetractionExtension_factorization","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12473","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetractionExtension_isKrausCP","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (U : Matrix (Fin (Finset.univ.sum fun q => MPOTensor.verti…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorAmbientRetractionExtension_isKrausCP","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12474","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetraction_apply","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (U : Matrix (Fin (Finset.univ.sum fun q => MPOTensor.verti…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorAmbientRetraction_apply","module":"TNLean.MPS.MPDO.VerticalSectorAmbientMaps"},{"id":"n12475","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorAmbientRetraction_apply_sector","kind":"theorem","summary":"∀ g d : Nat (dim mult : Fin g → Nat) (U : Matrix (Fin (Finset.univ.sum fun q => 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d…","labels":[],"detail_key":"p8","name":"MPOTensor.transportTToVerticalCoordinates_isKrausCP","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12484","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS_isKrausDirectSumMap","kind":"theorem","summary":"∀ g₁ g₂ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (dim₂ mult₂ : Fin g₂ → Nat) (weight₂ : (β : Fin g₂)…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorS_isKrausDirectSumMap","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12485","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorT_isKrausDirectSumMap","kind":"theorem","summary":"∀ g₁ g₂ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorT_isKrausDirectSumMap","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12486","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_composites_traceAdjointSchwarz","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (h : MPOTensor.VerticalSectorHypotheses), And (Matrix.IsSchwarzDirectSumMap (…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_composites_traceAdjointSchwarz","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12487","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlockDiagonal_eq_equivReindexMap_comp_embedding","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat), Eq (MPOTensor.verticalSectorBlockDiagonal dim mult) ((Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockDiagonal_eq_equivReindexMap_comp_embedding","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12488","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlocks_eq_compression_comp_equivReindexMap","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat), Eq (MPOTensor.verticalSectorBlocks dim mult) (Matrix.direct…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlocks_eq_compression_comp_equivReindexMap","module":"TNLean.MPS.MPDO.VerticalSectorCompletePositivity"},{"id":"n12489","layer":"formal","project":"p8","title":"MPOTensor.contractBondMatrix_verticalAssembledTensor_eq_sectorBlockDiagonal","kind":"theorem","summary":"∀ g D : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (A : (α :…","labels":[],"detail_key":"p8","name":"MPOTensor.contractBondMatrix_verticalAssembledTensor_eq_sectorBlockDiagonal","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12490","layer":"formal","project":"p8","title":"MPOTensor.normalizedRetainedVerticalSectorEmbedding","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → LinearMap (RingHo…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedRetainedVerticalSectorEmbedding","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12491","layer":"formal","project":"p8","title":"MPOTensor.normalizedRetainedVerticalSectorEmbedding_trace_smul","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedRetainedVerticalSectorEmbedding_trace_smul","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12492","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalSectorPartialTrace","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → LinearMap (RingHom.id Complex) (MPOTensor.RetainedVertical…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalSectorPartialTrace","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12493","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalSectorPartialTrace_comp_normalizedEmbedding","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalSectorPartialTrace_comp_normalizedEmbedding","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12494","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalSectorPartialTrace_weightedBlockDiagonal","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (X : MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalSectorPartialTrace_weightedBlockDiagonal","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12495","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS","kind":"def","summary":"g₁ g₂ d : Nat → (dim₁ mult₁ : Fin g₁ → Nat) → (dim₂ mult₂ : Fin g₂ → Nat) → ((β : Fin g₂) → 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(MPOTensor.VerticalWeighted…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockDiagonal","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12502","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlockProjection","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → LinearMap (RingHom.id Complex) (MPOTensor.RetainedVertical…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockProjection","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12503","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlockProjection_apply_ne","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Z : MPOTensor.RetainedVerticalSectorMatrix dim mult) α β :…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockProjection_apply_ne","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12504","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlockProjection_apply_same","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Z : MPOTensor.RetainedVerticalSectorMatrix dim mult) (α : F…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockProjection_apply_same","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12505","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlockProjection_comp_self","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat), Eq ((MPOTensor.verticalSectorBlockProjection dim mult).comp…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlockProjection_comp_self","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12506","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlocks","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → LinearMap (RingHom.id Complex) (MPOTensor.RetainedVertical…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlocks","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12507","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorBlocks_comp_verticalSectorBlockDiagonal","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat), Eq ((MPOTensor.verticalSectorBlocks dim mult).comp (MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorBlocks_comp_verticalSectorBlockDiagonal","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12508","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorFinEquiv","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → Equiv (Sigma fun α => Prod (Fin (mult α)) (Fin (dim α))) (…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorFinEquiv","module":"TNLean.MPS.MPDO.VerticalSectorCoordinates"},{"id":"n12509","layer":"formal","project":"p8","title":"MPOTensor.fixedPointProductSpan_finrank_le_of_densityBlocks","kind":"theorem","summary":"∀ g : Nat dim : Fin g → Nat F : LinearMap (RingHom.id Complex) (MPOTensor.VerticalSectorAlgebra…","labels":[],"detail_key":"p8","name":"MPOTensor.fixedPointProductSpan_finrank_le_of_densityBlocks","module":"TNLean.MPS.MPDO.VerticalSectorDensityBlocks"},{"id":"n12510","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS_comp_T_fixed_contractBondMatrix_trace_smul","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (h : MPOTensor.VerticalSectorFixedGeneratorHypotheses) (X : Matrix (Fin D) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorS_comp_T_fixed_contractBondMatrix_trace_smul","module":"TNLean.MPS.MPDO.VerticalSectorFixedGenerators"},{"id":"n12511","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorT_comp_S_fixed_contractBondMatrix_trace_smul","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (h : MPOTensor.VerticalSectorFixedGeneratorHypotheses) (X : Matrix (Fin D) (F…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorT_comp_S_fixed_contractBondMatrix_trace_smul","module":"TNLean.MPS.MPDO.VerticalSectorFixedGenerators"},{"id":"n12512","layer":"formal","project":"p8","title":"MPOTensor.fixedPointProductSpan","kind":"def","summary":"g : Nat → dim : Fin g → Nat → Nat → LinearMap (RingHom.id Complex) (MPOTensor.VerticalSectorAlg…","labels":[],"detail_key":"p8","name":"MPOTensor.fixedPointProductSpan","module":"TNLean.MPS.MPDO.VerticalSectorFixedPointProductSpan"},{"id":"n12513","layer":"formal","project":"p8","title":"MPOTensor.WeightedVerticalBondContractionProductSpanTop","kind":"def","summary":"g D : Nat → dim : Fin g → Nat → (Fin g → Complex) → ((α : Fin g) → MPSTensor (HMul.hMul D D) (d…","labels":[],"detail_key":"p8","name":"MPOTensor.WeightedVerticalBondContractionProductSpanTop","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12514","layer":"formal","project":"p8","title":"MPOTensor.eq_id_of_weightedVerticalBondContractionProducts_fixed","kind":"theorem","summary":"∀ g D L : Nat dim : Fin g → Nat (m : Fin g → Complex) (A : (α : Fin g) → MPSTensor (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPOTensor.eq_id_of_weightedVerticalBondContractionProducts_fixed","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12515","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorScaleEquiv","kind":"def","summary":"g : Nat → dim : Fin g → Nat → (c : Fin g → Complex) → (∀ (α : Fin g), Ne (c α) 0) → LinearEquiv…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorScaleEquiv","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12516","layer":"formal","project":"p8","title":"MPOTensor.weightedVerticalBondContraction","kind":"def","summary":"g D : Nat → dim : Fin g → Nat → (Fin g → Complex) → ((α : Fin g) → MPSTensor (HMul.hMul D D) (d…","labels":[],"detail_key":"p8","name":"MPOTensor.weightedVerticalBondContraction","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12517","layer":"formal","project":"p8","title":"MPOTensor.weightedVerticalBondContractionProduct","kind":"def","summary":"g D L : Nat → dim : Fin g → Nat → (Fin g → Complex) → ((α : Fin g) → MPSTensor (HMul.hMul D D)…","labels":[],"detail_key":"p8","name":"MPOTensor.weightedVerticalBondContractionProduct","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12518","layer":"formal","project":"p8","title":"MPOTensor.weightedVerticalBondContractionProduct_single","kind":"theorem","summary":"∀ g D L : Nat dim : Fin g → Nat (m : Fin g → Complex) (A : (α : Fin g) → MPSTensor (HMul.hMul D…","labels":[],"detail_key":"p8","name":"MPOTensor.weightedVerticalBondContractionProduct_single","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12519","layer":"formal","project":"p8","title":"MPSTensor.contractBondMatrix_single_mod_div","kind":"theorem","summary":"∀ D n : Nat (A : MPSTensor (HMul.hMul D D) n) (ab : Fin (HMul.hMul D D)), Eq (A.contractBondMat…","labels":[],"detail_key":"p8","name":"MPSTensor.contractBondMatrix_single_mod_div","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12520","layer":"formal","project":"p8","title":"MPSTensor.contractBondMatrix_single_transpose","kind":"theorem","summary":"∀ D n : Nat (A : MPSTensor (HMul.hMul D D) n) (a b : Fin D), Eq (A.contractBondMatrix (Matrix.s…","labels":[],"detail_key":"p8","name":"MPSTensor.contractBondMatrix_single_transpose","module":"TNLean.MPS.MPDO.VerticalSectorGeneration"},{"id":"n12521","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_composites_eq_id","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (h : MPOTensor.VerticalSectorHypotheses), And (Eq h.SbarTbar LinearMap.id) (E…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_composites_eq_id","module":"TNLean.MPS.MPDO.VerticalSectorIdentity"},{"id":"n12522","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorS_image_preservation","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorS_image_preservation","module":"TNLean.MPS.MPDO.VerticalSectorImagePreservation"},{"id":"n12523","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorT_image_preservation","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorT_image_preservation","module":"TNLean.MPS.MPDO.VerticalSectorImagePreservation"},{"id":"n12524","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS_trace_of_source_generated","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorS_trace_of_source_generated","module":"TNLean.MPS.MPDO.VerticalSectorImagePreservation"},{"id":"n12525","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorT_trace_of_source_generated","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorT_trace_of_source_generated","module":"TNLean.MPS.MPDO.VerticalSectorImagePreservation"},{"id":"n12526","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_exists_unitaryBlockEquiv","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_exists_unitaryBlockEquiv","module":"TNLean.MPS.MPDO.VerticalSectorRelabeling"},{"id":"n12527","layer":"formal","project":"p8","title":"MPOTensor.VerticalSectorAlgebra","kind":"def","summary":"g : Nat → (Fin g → Nat) → Type","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalSectorAlgebra","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12528","layer":"formal","project":"p8","title":"MPOTensor.VerticalWeightedSectorSpace","kind":"def","summary":"g : Nat → (Fin g → Nat) → (Fin g → Nat) → Type","labels":[],"detail_key":"p8","name":"MPOTensor.VerticalWeightedSectorSpace","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12529","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorEmbedding","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → ((α : Fin g) → Fin (mult α) → Complex) → LinearMap (RingHo…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorEmbedding","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12530","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorEmbedding_apply","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex) (X : MPOTens…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorEmbedding_apply","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12531","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorEmbedding_isKrausDirectSumMap","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorEmbedding_isKrausDirectSumMap","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12532","layer":"formal","project":"p8","title":"MPOTensor.verticalMultiplicityTrace","kind":"def","summary":"g : Nat → mult : Fin g → Nat → ((α : Fin g) → Fin (mult α) → Complex) → Fin g → 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Fin g → Nat) (Y : MPOTensor.VerticalWeightedSectorSpace dim mult) (α : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorPartialTrace_apply","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12536","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorPartialTrace_comp_normalizedVerticalSectorEmbedding","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorPartialTrace_comp_normalizedVerticalSectorEmbedding","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12537","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorPartialTrace_isKrausDirectSumMap","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat), Matrix.IsKrausDirectSumMap (MPOTensor.verticalSectorPartial…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorPartialTrace_isKrausDirectSumMap","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12538","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorSwapEquiv","kind":"def","summary":"g : Nat → (dim mult : Fin g → Nat) → Equiv (Sigma fun α => Prod (Fin (mult α)) (Fin (dim α))) (…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorSwapEquiv","module":"TNLean.MPS.MPDO.VerticalSectorRetractions"},{"id":"n12539","layer":"formal","project":"p8","title":"MPOTensor.IsVerticalSectorPosSemidef","kind":"def","summary":"g : Nat → dim : Fin g → Nat → MPOTensor.VerticalSectorAlgebra dim → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsVerticalSectorPosSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12540","layer":"formal","project":"p8","title":"MPOTensor.normalizedRetainedVerticalSectorEmbedding_posSemidef","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedRetainedVerticalSectorEmbedding_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12541","layer":"formal","project":"p8","title":"MPOTensor.normalizedVerticalSectorEmbedding_posSemidef","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedVerticalSectorEmbedding_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12542","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorS","kind":"def","summary":"g₂ d : Nat → (dim₂ mult₂ : Fin g₂ → Nat) → ((β : Fin g₂) → Fin (mult₂ β) → Complex) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorS","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12543","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorS_posSemidef","kind":"theorem","summary":"∀ g₂ d : Nat (dim₂ mult₂ : Fin g₂ → Nat) (weight₂ : (β : Fin g₂) → Fin (mult₂ β) → Complex), (∀…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorS_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12544","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorT","kind":"def","summary":"g₁ d : Nat → (dim₁ mult₁ : Fin g₁ → Nat) → ((α : Fin g₁) → Fin (mult₁ α) → Complex) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorT","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12545","layer":"formal","project":"p8","title":"MPOTensor.precompressionVerticalSectorT_posSemidef","kind":"theorem","summary":"∀ g₁ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex), (∀…","labels":[],"detail_key":"p8","name":"MPOTensor.precompressionVerticalSectorT_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12546","layer":"formal","project":"p8","title":"MPOTensor.retainedVerticalSectorPartialTrace_posSemidef","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Z : MPOTensor.RetainedVerticalSectorMatrix dim mult), Matri…","labels":[],"detail_key":"p8","name":"MPOTensor.retainedVerticalSectorPartialTrace_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12547","layer":"formal","project":"p8","title":"MPOTensor.trace_normalizedRetainedVerticalSectorEmbedding","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (weight : (α : Fin g) → Fin (mult α) → Complex), (∀ (α : 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(∀…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_precompressionVerticalSectorT","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12550","layer":"formal","project":"p8","title":"MPOTensor.trace_verticalSectorBlockDiagonal","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Y : MPOTensor.VerticalWeightedSectorSpace dim mult), Eq (Ma…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_verticalSectorBlockDiagonal","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12551","layer":"formal","project":"p8","title":"MPOTensor.trace_verticalSectorBlockProjection","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Z : MPOTensor.RetainedVerticalSectorMatrix dim mult), Eq (M…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_verticalSectorBlockProjection","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12552","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS_posSemidef","kind":"theorem","summary":"∀ g₁ g₂ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (dim₂ mult₂ : Fin g₂ → Nat) (weight₂ : (β : Fin g₂)…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorS_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12553","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSectorS_trace_eq_iff","kind":"theorem","summary":"∀ g₁ g₂ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (dim₂ mult₂ : Fin g₂ → Nat) (weight₂ : (β : Fin 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Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSectorT_trace_le","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12558","layer":"formal","project":"p8","title":"MPOTensor.verticalMultiplicityTrace_pos","kind":"theorem","summary":"∀ g : Nat mult : Fin g → Nat weight : (α : Fin g) → Fin (mult α) → Complex, (∀ (α : Fin g), LT.…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalMultiplicityTrace_pos","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12559","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorPartialTrace_posSemidef","kind":"theorem","summary":"∀ g : Nat (dim mult : Fin g → Nat) (Y : MPOTensor.VerticalWeightedSectorSpace dim mult), (∀ (α…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorPartialTrace_posSemidef","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12560","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorTrace","kind":"def","summary":"g : Nat → dim : 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g₂)…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorTrace_transportedVerticalSectorS","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12563","layer":"formal","project":"p8","title":"MPOTensor.verticalSectorTrace_transportedVerticalSectorT","kind":"theorem","summary":"∀ g₁ g₂ d : Nat (dim₁ mult₁ : Fin g₁ → Nat) (weight₁ : (α : Fin g₁) → Fin (mult₁ α) → Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.verticalSectorTrace_transportedVerticalSectorT","module":"TNLean.MPS.MPDO.VerticalSectorTraceLoss"},{"id":"n12564","layer":"formal","project":"p8","title":"MPOTensor.transportedVerticalSector_composites_tracePreserving","kind":"theorem","summary":"∀ g₁ g₂ d D : Nat (h : MPOTensor.VerticalSectorHypotheses), And (Matrix.IsTracePreservingDirect…","labels":[],"detail_key":"p8","name":"MPOTensor.transportedVerticalSector_composites_tracePreserving","module":"TNLean.MPS.MPDO.VerticalSectorTracePreservation"},{"id":"n12565","layer":"formal","project":"p8","title":"MPOTensor.IsHorizontalCF.exists_normal_verticalBlockDecomp_with_isometry","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), M.IsHorizontalCF → M.IsMPDO → Exists fun r => Exists fun dim =…","labels":[],"detail_key":"p8","name":"MPOTensor.IsHorizontalCF.exists_normal_verticalBlockDecomp_with_isometry","module":"TNLean.MPS.MPDO.VerticalSpectral"},{"id":"n12566","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.ExplicitVerticalTransport","kind":"inductive","summary":"Λ : Type u_1 → Λ' : Type u_2 → O : Nat → Type u_3 → O' : Nat → Type u_4 → [inst : (L : Nat) → S…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.ExplicitVerticalTransport","module":"TNLean.MPS.MPDO.VerticalTransportCoefficientCovariance"},{"id":"n12567","layer":"formal","project":"p8","title":"MPOTensor.BNTLabelOperatorFamily.verticalTransportCoefficients","kind":"def","summary":"Λ : Type u_1 → Λ' : Type u_2 → MPOTensor.BNTLabelCoefficientFamily Λ → Equiv Λ' Λ → (Λ' → Compl…","labels":[],"detail_key":"p8","name":"MPOTensor.BNTLabelOperatorFamily.verticalTransportCoefficients","module":"TNLean.MPS.MPDO.VerticalTransportCoefficientCovariance"},{"id":"n12568","layer":"formal","project":"p8","title":"MPOTensor.IsPhysicalTraceIdempotent","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsPhysicalTraceIdempotent","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12569","layer":"formal","project":"p8","title":"MPOTensor.IsPhysicalTraceIdempotent.isSourceZCL","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsPhysicalTraceIdempotent → Ne M.physTraceTransfer 0 → M.IsSou…","labels":[],"detail_key":"p8","name":"MPOTensor.IsPhysicalTraceIdempotent.isSourceZCL","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12570","layer":"formal","project":"p8","title":"MPOTensor.IsSourceZCL","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsSourceZCL","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12571","layer":"formal","project":"p8","title":"MPOTensor.IsSourceZCL.bondDim_ne_zero","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSourceZCL → Ne D 0","labels":[],"detail_key":"p8","name":"MPOTensor.IsSourceZCL.bondDim_ne_zero","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12572","layer":"formal","project":"p8","title":"MPOTensor.IsSourceZCL.normalized_idempotent","kind":"theorem","summary":"∀ d D : Nat M : MPOTensor d D, M.IsSourceZCL → Exists fun lam => And (LT.lt 0 lam) (IsIdempoten…","labels":[],"detail_key":"p8","name":"MPOTensor.IsSourceZCL.normalized_idempotent","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12573","layer":"formal","project":"p8","title":"MPOTensor.IsZCL","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsZCL","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12574","layer":"formal","project":"p8","title":"MPOTensor.isPhysicalTraceIdempotent_iff","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Iff M.IsPhysicalTraceIdempotent (Eq (HMul.hMul M.physTraceTran…","labels":[],"detail_key":"p8","name":"MPOTensor.isPhysicalTraceIdempotent_iff","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12575","layer":"formal","project":"p8","title":"MPOTensor.isZCL_iff_toMPSTensor_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (M : MPOTensor d D), Iff M.IsZCL M.toMPSTensor.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPOTensor.isZCL_iff_toMPSTensor_isTransferIdempotent","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12576","layer":"formal","project":"p8","title":"MPOTensor.physTraceTransfer","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.physTraceTransfer","module":"TNLean.MPS.MPDO.ZCL"},{"id":"n12577","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.adjointSimpleContraction_cross_mul","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.IsMPUSimple → U.physicalAdjointT…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.adjointSimpleContraction_cross_mul","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12578","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.adjointSimpleContraction_eq_smul","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.IsMPUSimple → U.physicalAdjointT…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.adjointSimpleContraction_eq_smul","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12579","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.doubleLayer_boundary_contractions","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.IsMPUSimple → have Φ := fun x =>…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.doubleLayer_boundary_contractions","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12580","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.physicalAdjointTensor","kind":"def","summary":"d D : Nat → U : MPOTensor d D → U.IsMPUCanonicalFormII → U.physicalAdjointTensor.IsMPUCanonical…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.physicalAdjointTensor","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12581","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.physicalAdjoint_doubleLayer_boundary_contractions","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.physicalAdjointTensor.IsMPUSimpl…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.physicalAdjoint_doubleLayer_boundary_contractions","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12582","layer":"formal","project":"p8","title":"MPOTensor.adjointSimpleContraction","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.adjointSimpleContraction","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12583","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_adjointSimpleContraction_raw","kind":"theorem","summary":"∀ d D : Nat (T : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex), Eq (T.adjointSimpleContra…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_adjointSimpleContraction_raw","module":"TNLean.MPS.MPU.AdjointSimpleContraction"},{"id":"n12584","layer":"formal","project":"p8","title":"MPOTensor.IsMPU","kind":"def","summary":"d D : Nat → MPOTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU","module":"TNLean.MPS.MPU.Basic"},{"id":"n12585","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.blockTensor","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ (L : Nat), LT.lt 0 L → (U.blockTensor L).IsMPU","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.blockTensor","module":"TNLean.MPS.MPU.Basic"},{"id":"n12586","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.conjTranspose_mpo_mul_mpo","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ N : Nat, LT.lt 1 N → Eq (HMul.hMul (U.mpo N).conjTra…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.conjTranspose_mpo_mul_mpo","module":"TNLean.MPS.MPU.Basic"},{"id":"n12587","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.mpo_mem_unitaryGroup","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ N : Nat, LT.lt 1 N → Membership.mem (Matrix.unitaryG…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.mpo_mem_unitaryGroup","module":"TNLean.MPS.MPU.Basic"},{"id":"n12588","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.mpo_mul_conjTranspose_mpo","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ N : Nat, LT.lt 1 N → Eq (HMul.hMul (U.mpo N) (U.mpo…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.mpo_mul_conjTranspose_mpo","module":"TNLean.MPS.MPU.Basic"},{"id":"n12589","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.mulTensor","kind":"theorem","summary":"∀ d D₁ D₂ : Nat U : MPOTensor d D₁ V : MPOTensor d D₂, U.IsMPU → V.IsMPU → (U.mulTensor V).IsMPU","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.mulTensor","module":"TNLean.MPS.MPU.Basic"},{"id":"n12590","layer":"formal","project":"p8","title":"MPOTensor.blockingRanks_eq_pow_mul_of_products","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → LT.lt 0 d → Eq (HMul.hMul (U.blockTens…","labels":[],"detail_key":"p8","name":"MPOTensor.blockingRanks_eq_pow_mul_of_products","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12591","layer":"formal","project":"p8","title":"MPOTensor.leftRank_blockTensor_eq_pow_mul_of_products","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → LT.lt 0 d → Eq (HMul.hMul (U.blockTens…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_blockTensor_eq_pow_mul_of_products","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12592","layer":"formal","project":"p8","title":"MPOTensor.leftRank_blockTensor_le_pow_mul","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → LE.le (U.blockTensor k).leftRank (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_blockTensor_le_pow_mul","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12593","layer":"formal","project":"p8","title":"MPOTensor.leftRank_blockTensor_succ_le","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (k : Nat), LE.le (U.blockTensor (HAdd.hAdd k 1)).leftRank (HMul…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_blockTensor_succ_le","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12594","layer":"formal","project":"p8","title":"MPOTensor.rightRank_blockTensor_eq_pow_mul_of_products","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → LT.lt 0 d → Eq (HMul.hMul (U.blockTens…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_blockTensor_eq_pow_mul_of_products","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12595","layer":"formal","project":"p8","title":"MPOTensor.rightRank_blockTensor_le_pow_mul","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → LE.le (U.blockTensor k).rightRank (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_blockTensor_le_pow_mul","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12596","layer":"formal","project":"p8","title":"MPOTensor.rightRank_blockTensor_succ_le","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (k : Nat), LE.le (U.blockTensor (HAdd.hAdd k 1)).rightRank (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_blockTensor_succ_le","module":"TNLean.MPS.MPU.BlockingRanks"},{"id":"n12597","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.isNormalTensor_normalizedFlattening_of_cpsv","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → ∀ (data : U.normalizedFlattening…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.isNormalTensor_normalizedFlattening_of_cpsv","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12598","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Type","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12599","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.hasFullSupport","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), hU.cfii.HasFullSupport","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.hasFullSupport","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12600","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.isNormalTensor_normalizedFlattening","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.normalized…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.isNormalTensor_normalizedFlattening","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12601","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.neZero_bond","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), NeZero D","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.neZero_bond","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12602","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.neZero_phys","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), NeZero d","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.neZero_phys","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12603","layer":"formal","project":"p8","title":"MPOTensor.blockTensorCFIIData","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → U.normalizedFlattening.CPSVCanonicalFormIIData → (p : Nat) →…","labels":[],"detail_key":"p8","name":"MPOTensor.blockTensorCFIIData","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12604","layer":"formal","project":"p8","title":"MPOTensor.normalizedFlattening_blockTensor","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (p : Nat), Eq (U.blockTensor p).normalizedFlattening (Kraus.rei…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedFlattening_blockTensor","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12605","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.HasFullSupport","kind":"def","summary":"d D : Nat → A : 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data.HasFullSuppo…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.dim_eq_of_r_eq_one_of_fullSupport","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12608","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.hasFullSupport_blockTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), data.HasFullSupport → ∀ (p : Na…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.hasFullSupport_blockTensor","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12609","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.hasFullSupport_reindexPhysical","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D d' : Nat (data : A.CPSVCanonicalFormData), data.HasFullSupport →…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.hasFullSupport_reindexPhysical","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12610","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.isIrreducibleTensor_of_dim_eq","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData) (k : Fin data.r), Eq (data.dim k…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.isIrreducibleTensor_of_dim_eq","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12611","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.isIrreducibleTensor_of_r_eq_one_of_fullSupport","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), Eq data.r 1 → data.HasFullSuppo…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.isIrreducibleTensor_of_r_eq_one_of_fullSupport","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12612","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.isNormalTensor_of_r_eq_one_of_fullSupport","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), Eq data.r 1 → data.HasFullSuppo…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.isNormalTensor_of_r_eq_one_of_fullSupport","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12613","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormIIData.hasFullSupport_cast","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D (data : B.CPSVCanonicalFormIIData), data.HasFullSupport → ∀ (h…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormIIData.hasFullSupport_cast","module":"TNLean.MPS.MPU.CanonicalForm"},{"id":"n12614","layer":"formal","project":"p8","title":"MPOTensor.normalizedFlattening_mulTensor_apply","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero d] (U : MPOTensor d D₁) (V : MPOTensor d D₂) (i k : Fin d), Eq ((U.mulT…","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedFlattening_mulTensor_apply","module":"TNLean.MPS.MPU.CompositionFlattening"},{"id":"n12615","layer":"formal","project":"p8","title":"MPOTensor.leftRank_blockTensor_mulTensor_four_le","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (U : MPOTensor d D₁) (V : MPOTensor d D₂), LE.le ((U.mulTensor V).blockTensor 4…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_blockTensor_mulTensor_four_le","module":"TNLean.MPS.MPU.CompositionRanks"},{"id":"n12616","layer":"formal","project":"p8","title":"MPOTensor.rightRank_blockTensor_mulTensor_four_le","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (U : MPOTensor d D₁) (V : MPOTensor d D₂), LE.le ((U.mulTensor V).blockTensor 4…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_blockTensor_mulTensor_four_le","module":"TNLean.MPS.MPU.CompositionRanks"},{"id":"n12617","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.mpo_physicalAdjointTensor_eq_inv","kind":"theorem","summary":"∀ G : Type u d : Nat [inst : Group G] (F : MPOTensor.GroupFamily G d), F.IsRepresentation → ∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.mpo_physicalAdjointTensor_eq_inv","module":"TNLean.MPS.MPU.DaggerInverse"},{"id":"n12618","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.sameMPV₂Pos_physicalAdjointTensor_inv","kind":"theorem","summary":"∀ G : Type u d : Nat [inst : Group G] (F : MPOTensor.GroupFamily G d), F.IsRepresentation → ∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.sameMPV₂Pos_physicalAdjointTensor_inv","module":"TNLean.MPS.MPU.DaggerInverse"},{"id":"n12619","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.bondDim_inv","kind":"theorem","summary":"∀ G : Type u d : Nat [inst : Group G] (F : MPOTensor.GroupFamily G d), F.IsRepresentation → ∀ (…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.bondDim_inv","module":"TNLean.MPS.MPU.DaggerInverseGauge"},{"id":"n12620","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.daggerInverseGauge","kind":"def","summary":"G : Type u → d : Nat → [inst : Group G] → (F : MPOTensor.GroupFamily G d) → F.IsRepresentation…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.daggerInverseGauge","module":"TNLean.MPS.MPU.DaggerInverseGauge"},{"id":"n12621","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.daggerInverseGauge_mul_mapStar_inverse_eq_smul_one","kind":"theorem","summary":"∀ G : Type u d : Nat [inst : Group G] (F : MPOTensor.GroupFamily G d) (hF : 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(MPOTensor.swapTensorSwapMatri…","labels":[],"detail_key":"p8","name":"MPOTensor.identitySwapIdentityMatrix_mul_swapTensorSwapMatrix_isUnitaryBetween","module":"TNLean.MPS.MPU.Examples.ShiftSwapMatrices"},{"id":"n12749","layer":"formal","project":"p8","title":"MPOTensor.swapTensorSwapMatrix_isUnitaryBetween","kind":"theorem","summary":"∀ (d : Nat), (MPOTensor.swapTensorSwapMatrix d).IsUnitaryBetween","labels":[],"detail_key":"p8","name":"MPOTensor.swapTensorSwapMatrix_isUnitaryBetween","module":"TNLean.MPS.MPU.Examples.ShiftSwapMatrices"},{"id":"n12750","layer":"formal","project":"p8","title":"MPOTensor.swapTensorSwapMatrix_mul_identitySwapIdentityMatrix_isUnitaryBetween","kind":"theorem","summary":"∀ (d : Nat), (HMul.hMul (MPOTensor.swapTensorSwapMatrix d) (MPOTensor.identitySwapIdentityMatri…","labels":[],"detail_key":"p8","name":"MPOTensor.swapTensorSwapMatrix_mul_identitySwapIdentityMatrix_isUnitaryBetween","module":"TNLean.MPS.MPU.Examples.ShiftSwapMatrices"},{"id":"n12751","layer":"formal","project":"p8","title":"MPOTensor.ShiftAncillaConfig","kind":"def","summary":"Nat → Nat → Type","labels":[],"detail_key":"p8","name":"MPOTensor.ShiftAncillaConfig","module":"TNLean.MPS.MPU.Examples.ShiftSymmetryPaths"},{"id":"n12752","layer":"formal","project":"p8","title":"MPOTensor.permMatrix_shiftAncillaThreeSwap_conj_swap₁₂","kind":"theorem","summary":"∀ (N d : Nat), Eq (HMul.hMul (HMul.hMul (Equiv.Perm.permMatrix Complex (MPOTensor.shiftAncillaT…","labels":[],"detail_key":"p8","name":"MPOTensor.permMatrix_shiftAncillaThreeSwap_conj_swap₁₂","module":"TNLean.MPS.MPU.Examples.ShiftSymmetryPaths"},{"id":"n12753","layer":"formal","project":"p8","title":"MPOTensor.permMatrix_shiftAncillaThreeSwap_endpoint","kind":"theorem","summary":"∀ (N d : Nat), Eq (Equiv.Perm.permMatrix Complex ((Equiv.trans (MPOTensor.shiftAncillaThreeSwap…","labels":[],"detail_key":"p8","name":"MPOTensor.permMatrix_shiftAncillaThreeSwap_endpoint","module":"TNLean.MPS.MPU.Examples.ShiftSymmetryPaths"},{"id":"n12754","layer":"formal","project":"p8","title":"MPOTensor.shiftAncillaCounterShift","kind":"def","summary":"(N d : Nat) → Equiv.Perm (MPOTensor.ShiftAncillaConfig N d)","labels":[],"detail_key":"p8","name":"MPOTensor.shiftAncillaCounterShift","module":"TNLean.MPS.MPU.Examples.ShiftSymmetryPaths"},{"id":"n12755","layer":"formal","project":"p8","title":"MPOTensor.shiftAncillaSwap₁aNext","kind":"def","summary":"(N d : Nat) → Equiv.Perm (MPOTensor.ShiftAncillaConfig N d)","labels":[],"detail_key":"p8","name":"MPOTensor.shiftAncillaSwap₁aNext","module":"TNLean.MPS.MPU.Examples.ShiftSymmetryPaths"},{"id":"n12756","layer":"formal","project":"p8","title":"MPOTensor.shiftAncillaSwap₁aNext_apply","kind":"theorem","summary":"∀ (N d : Nat) (σ₁ σ₂ a : Fin N → Fin 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(MPOTensor.si…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_shiftExampleTildeU₃","module":"TNLean.MPS.MPU.Examples.ShiftTilde"},{"id":"n12765","layer":"formal","project":"p8","title":"MPOTensor.mpo_shiftExampleTildeU₃_eq_swap_mul_tildeU₂_mul_swap","kind":"theorem","summary":"∀ (d N : Nat) [NeZero N], Eq ((MPOTensor.shiftExampleTildeU₃ d).mpo N) (HMul.hMul (HMul.hMul (M…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_shiftExampleTildeU₃_eq_swap_mul_tildeU₂_mul_swap","module":"TNLean.MPS.MPU.Examples.ShiftTilde"},{"id":"n12766","layer":"formal","project":"p8","title":"MPOTensor.shiftExampleTildeU₁","kind":"def","summary":"(d : Nat) → MPOTensor (HMul.hMul d d) 1","labels":[],"detail_key":"p8","name":"MPOTensor.shiftExampleTildeU₁","module":"TNLean.MPS.MPU.Examples.ShiftTilde"},{"id":"n12767","layer":"formal","project":"p8","title":"MPOTensor.shiftExampleTildeU₁_isMPU","kind":"theorem","summary":"∀ (d : Nat), (MPOTensor.shiftExampleTildeU₁ 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(W.doubledVirtual…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledVirtualContraction_vecMulVec","module":"TNLean.MPS.MPU.FactorFreeSandwich"},{"id":"n12788","layer":"formal","project":"p8","title":"MPOTensor.doubledVirtualContraction_vecMulVec_vec","kind":"theorem","summary":"∀ d D : Nat (W : MPOTensor d D) (L R : Matrix (Fin D) (Fin D) Complex), Eq (W.doubledVirtualCon…","labels":[],"detail_key":"p8","name":"MPOTensor.doubledVirtualContraction_vecMulVec_vec","module":"TNLean.MPS.MPU.FactorFreeSandwich"},{"id":"n12789","layer":"formal","project":"p8","title":"MPOTensor.factorFreeSandwich","kind":"def","summary":"d D : Nat → MPOTensor d D → MPOTensor d D","labels":[],"detail_key":"p8","name":"MPOTensor.factorFreeSandwich","module":"TNLean.MPS.MPU.FactorFreeSandwich"},{"id":"n12790","layer":"formal","project":"p8","title":"MPOTensor.factorFreeSandwich_apply","kind":"theorem","summary":"∀ d D : Nat (W : MPOTensor d D) (i j : Fin d) (a b : Fin D), Eq (W.factorFreeSandwich i j a 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(Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleX₁_weighted_isometry","module":"TNLean.MPS.MPU.InverseCompatibleComparisonUnitarity"},{"id":"n12820","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleComparison","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleComparison","module":"TNLean.MPS.MPU.InverseCompatibleCutComparison"},{"id":"n12821","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleComparisonJ","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D)…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleComparisonJ","module":"TNLean.MPS.MPU.InverseCompatibleCutComparison"},{"id":"n12822","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleComparisonK","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D)…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleComparisonK","module":"TNLean.MPS.MPU.InverseCompatibleCutComparison"},{"id":"n12823","layer":"formal","project":"p8","title":"MPOTensor.rightRank_eq_leftRank_of_physicalAdjointTensor_eq_unitary_gauge","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_eq_leftRank_of_physicalAdjointTensor_eq_unitary_gauge","module":"TNLean.MPS.MPU.InverseCompatibleCutComparison"},{"id":"n12824","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.inverseCompatibleSourceProperties_of_inv_eq","kind":"theorem","summary":"∀ G : Type u_1 [inst : Group G] d : Nat (F : MPOTensor.GroupFamily G d) (hF : F.IsRepresentatio…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.inverseCompatibleSourceProperties_of_inv_eq","module":"TNLean.MPS.MPU.InverseCompatibleFamilyProperties"},{"id":"n12825","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.inverseCompatibleFirstCut_of_inv_eq","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] (F : MPOTensor.GroupFamily G d) (hF : F.IsRepresentatio…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.inverseCompatibleFirstCut_of_inv_eq","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12826","layer":"formal","project":"p8","title":"MPOTensor.GroupFamily.IsRepresentation.physicalAdjointTensor_eq_daggerInverseGauge_of_inv…","kind":"theorem","summary":"∀ d : Nat G : Type u_1 [inst : Group G] (F : MPOTensor.GroupFamily G d) (hF : F.IsRepresentatio…","labels":[],"detail_key":"p8","name":"MPOTensor.GroupFamily.IsRepresentation.physicalAdjointTensor_eq_daggerInverseGauge_of_inv_eq","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12827","layer":"formal","project":"p8","title":"MPOTensor.eq_unitary_mul_physicalAdjointTensor_mul_conjTranspose","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.eq_unitary_mul_physicalAdjointTensor_mul_conjTranspose","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12828","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleLeftGauge","kind":"def","summary":"d D : Nat → (Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D) Complex) x) → Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleLeftGauge","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12829","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleRightGauge","kind":"def","summary":"d D : Nat → (Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D) Complex) x) → Matrix…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleRightGauge","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12830","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleX₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D)…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleX₁","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12831","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleX₁_leftInverse","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => 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(Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleY₁_rightInverse","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12834","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_eq_inverseCompatibleX₁_mul_inverseCompatibleY₁","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_eq_inverseCompatibleX₁_mul_inverseCompatibleY₁","module":"TNLean.MPS.MPU.InverseCompatibleFirstCut"},{"id":"n12835","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_eq_inverseCompatible_dressed_cut","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup 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(Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin D)…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleV","module":"TNLean.MPS.MPU.InverseCompatibleGates"},{"id":"n12843","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleV_eq_sourceV_mul_comparisonK","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.inverseCompatibleV_eq_sourceV_mul_comparisonK","module":"TNLean.MPS.MPU.InverseCompatibleGates"},{"id":"n12844","layer":"formal","project":"p8","title":"MPOTensor.inverseCompatibleSourceFactors_movement_isUnitaryBetween","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (T : Subtype fun x => Membership.mem (Matrix.unitaryGroup 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(…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.residualAlgebra_list_prod_eq_zero","module":"TNLean.MPS.MPU.ResidualAlgebra"},{"id":"n12932","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.trace_eq_zero_of_mem_residualAlgebra","kind":"theorem","summary":"∀ d D : Nat [NeZero d] U : MPOTensor d D, U.IsMPU → ∀ A : Matrix (Fin (HMul.hMul D D)) (Fin (HM…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.trace_eq_zero_of_mem_residualAlgebra","module":"TNLean.MPS.MPU.ResidualAlgebra"},{"id":"n12933","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.trace_list_prod_eq_zero_of_mem_residualGeneratorSet","kind":"theorem","summary":"∀ d D : Nat [NeZero d] U : MPOTensor d D, U.IsMPU → ∀ (l : List (Matrix (Fin (HMul.hMul D D)) (…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.trace_list_prod_eq_zero_of_mem_residualGeneratorSet","module":"TNLean.MPS.MPU.ResidualAlgebra"},{"id":"n12934","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.trace_pow_eq_zero_of_mem_residualAlgebra","kind":"theorem","summary":"∀ d D : Nat [NeZero d] U : MPOTensor d D, U.IsMPU → ∀ A : Matrix (Fin (HMul.hMul D D)) (Fin (HM…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.trace_pow_eq_zero_of_mem_residualAlgebra","module":"TNLean.MPS.MPU.ResidualAlgebra"},{"id":"n12935","layer":"formal","project":"p8","title":"MPOTensor.residualAlgebra","kind":"def","summary":"d D : Nat → MPOTensor d D → NonUnitalSubalgebra Complex (Matrix (Fin (HMul.hMul D D)) (Fin (HMu…","labels":[],"detail_key":"p8","name":"MPOTensor.residualAlgebra","module":"TNLean.MPS.MPU.ResidualAlgebra"},{"id":"n12936","layer":"formal","project":"p8","title":"MPOTensor.residualGeneratorSet","kind":"def","summary":"d D : Nat → MPOTensor d D → Set (Matrix 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U.IsMPU","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUSimple.isMPU","module":"TNLean.MPS.MPU.Simple"},{"id":"n12940","layer":"formal","project":"p8","title":"MPOTensor.IsMPUSimple.mpo_doubleLayerTensor_eq_one","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPUSimple → ∀ (N : Nat), LT.lt 1 N → Eq (U.doubleLayerTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUSimple.mpo_doubleLayerTensor_eq_one","module":"TNLean.MPS.MPU.Simple"},{"id":"n12941","layer":"formal","project":"p8","title":"MPOTensor.IsMPUSimple.simple1","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPUSimple → Exists fun a => Exists fun b => ∀ (i j : Fin d),…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUSimple.simple1","module":"TNLean.MPS.MPU.Simple"},{"id":"n12942","layer":"formal","project":"p8","title":"MPOTensor.IsMPUSimple.simple2","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPUSimple → Exists fun a => Exists fun b => ∀ (i j k l : 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N).…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_doubleLayerTensor","module":"TNLean.MPS.MPU.Simple"},{"id":"n12946","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.blockTensor_isMPUSimple_of_le","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ k k' : Nat, LT.lt 0 k → LE.le k k' → (U.blockTensor…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.blockTensor_isMPUSimple_of_le","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12947","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.blockTensor_one_isMPUSimple_fin_one","kind":"theorem","summary":"∀ d : Nat [NeZero d] U : MPOTensor d 1, U.IsMPU → (U.blockTensor 1).IsMPUSimple","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.blockTensor_one_isMPUSimple_fin_one","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12948","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.blockTensor_pow_four_isMPUSimple","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d 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k)…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.exists_blockTensor_isMPUSimple","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12953","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.exists_blockTensor_isMPUSimple_of_one_lt","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → LT.lt 1 D → Exists fun k => And…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.exists_blockTensor_isMPUSimple_of_one_lt","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12954","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.exists_normalized_fixed_points_simple1","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → Exists fun Φ => Exists fun ρ =>…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.exists_normalized_fixed_points_simple1","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12955","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.isMPUSimple_of_simple2","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ (a b : Fin (HMul.hMul D D) → Complex), (∀ (i j k l :…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.isMPUSimple_of_simple2","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12956","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.simple1","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → Exists fun a => Exists fun b =>…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.simple1","module":"TNLean.MPS.MPU.SimpleBlocking"},{"id":"n12957","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.simple1_of_simple2_supplied","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → ∀ (a b : Fin (HMul.hMul D D) → 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(MPOTensor.reindexPhy…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_reindexPhysical","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12971","layer":"formal","project":"p8","title":"MPOTensor.rightRank","kind":"def","summary":"d D : Nat → MPOTensor d D → Nat","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12972","layer":"formal","project":"p8","title":"MPOTensor.rightRank_bound","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), LE.le U.rightRank (HMul.hMul D d)","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_bound","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12973","layer":"formal","project":"p8","title":"MPOTensor.rightRank_eq","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.rightRank 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finProdFi…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁Fin_eq_reindex","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12980","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (i : Fin d) (β α : Fin D) (j : Fin d), Eq (U.sourceCutM₁ (Prod.…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_apply","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12981","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.physicalAdjointTensor.sourceCutM₁ U.sourceCutM₂.conjTrans…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_physicalAdjointTensor","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12982","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_rank_eq_sourceCutM₁Fin_rank","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.sourceCutM₁.rank U.sourceCutM₁Fin.rank","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_rank_eq_sourceCutM₁Fin_rank","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12983","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_reindexPhysical","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) d' : Nat (e : Equiv (Fin d') (Fin d)), Eq (MPOTensor.reindexPhy…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_reindexPhysical","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12984","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Prod (Fin D) (Fin d)) (Prod (Fin d) (Fin D)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12985","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂Fin","kind":"def","summary":"d D : Nat → MPOTensor d D → Matrix (Fin (HMul.hMul D d)) (Fin (HMul.hMul d D)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂Fin","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12986","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂Fin_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (r : Fin (HMul.hMul D d)) (c : Fin (HMul.hMul d D)), Eq (U.sour…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂Fin_apply","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12987","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂Fin_eq_reindex","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.sourceCutM₂Fin ((Matrix.reindex finProdFinEquiv finProdFi…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂Fin_eq_reindex","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12988","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (α : Fin D) (i j : Fin d) (β : Fin D), Eq (U.sourceCutM₂ (Prod.…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_apply","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12989","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.physicalAdjointTensor.sourceCutM₂ U.sourceCutM₁.conjTrans…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_physicalAdjointTensor","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12990","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_rank_eq_sourceCutM₂Fin_rank","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.sourceCutM₂.rank U.sourceCutM₂Fin.rank","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_rank_eq_sourceCutM₂Fin_rank","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12991","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_reindexPhysical","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) d' : Nat (e : Equiv (Fin d') (Fin d)), Eq (MPOTensor.reindexPhy…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_reindexPhysical","module":"TNLean.MPS.MPU.SourceCuts"},{"id":"n12992","layer":"formal","project":"p8","title":"MPOTensor.exists_reciprocal_unitary_source_gauges","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D X₁ Xt₁ : Matrix (Prod (Fin d) (Fin D)) (Fin U.rightRank) Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.exists_reciprocal_unitary_source_gauges","module":"TNLean.MPS.MPU.SourceDecompositionUniqueness"},{"id":"n12993","layer":"formal","project":"p8","title":"MPOTensor.sourceU_contraction_transport","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D Y₁ Yt₁ : Matrix (Fin U.rightRank) (Prod (Fin D) (Fin d)) Complex…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceU_contraction_transport","module":"TNLean.MPS.MPU.SourceDecompositionUniqueness"},{"id":"n12994","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D, U.IsMPU → U.physicalAdjointTensor.IsMPU","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.physicalAdjointTensor","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n12995","layer":"formal","project":"p8","title":"MPOTensor.doubleLayerTensor_physicalAdjointTensor_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (i k : Fin d) (α γ β δ : Fin D), Eq (U.physicalAdjointTensor.do…","labels":[],"detail_key":"p8","name":"MPOTensor.doubleLayerTensor_physicalAdjointTensor_apply","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n12996","layer":"formal","project":"p8","title":"MPOTensor.mpo_physicalAdjointTensor_eq_conjTranspose","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (N : Nat), Eq (U.physicalAdjointTensor.mpo N) (U.mpo N).conjTra…","labels":[],"detail_key":"p8","name":"MPOTensor.mpo_physicalAdjointTensor_eq_conjTranspose","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n12997","layer":"formal","project":"p8","title":"MPOTensor.physicalAdjointTensor_physicalAdjointTensor","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.physicalAdjointTensor.physicalAdjointTensor U","labels":[],"detail_key":"p8","name":"MPOTensor.physicalAdjointTensor_physicalAdjointTensor","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n12998","layer":"formal","project":"p8","title":"MPOTensor.sourceY₁_eq_sourceX₁_conjTranspose_mul_weight_mul_sourceCutM₁","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef), Eq (U.sou…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₁_eq_sourceX₁_conjTranspose_mul_weight_mul_sourceCutM₁","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n12999","layer":"formal","project":"p8","title":"MPOTensor.sourceY₂_eq_sourceX₂_conjTranspose_mul_sourceCutM₂","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.sourceY₂ (HMul.hMul U.sourceX₂.conjTranspose U.sourceCutM…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₂_eq_sourceX₂_conjTranspose_mul_sourceCutM₂","module":"TNLean.MPS.MPU.SourceFactorContraction"},{"id":"n13000","layer":"formal","project":"p8","title":"MPOTensor.SourceCutSVD","kind":"inductive","summary":"α : Type u_1 → β : Type u_2 → [Fintype α] → [Fintype β] → Matrix α β Complex → Nat → Type (max…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceCutSVD","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13001","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors","kind":"inductive","summary":"d D : Nat → MPOTensor d D → Matrix (Fin D) (Fin D) Complex → Type","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13002","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_eq_sourceX₁_mul_sourceY₁","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef), Eq U.sour…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_eq_sourceX₁_mul_sourceY₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13003","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_eq_sourceX₂_mul_sourceY₂","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq U.sourceCutM₂ (HMul.hMul U.sourceX₂ U.sourceY₂)","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_eq_sourceX₂_mul_sourceY₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13004","layer":"formal","project":"p8","title":"MPOTensor.sourceFactors","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → U.SourceFac…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceFactors","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13005","layer":"formal","project":"p8","title":"MPOTensor.sourceGram₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → Matrix (Fin D) (Fin D) Complex → Matrix (Fin U.rightRank) (Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceGram₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13006","layer":"formal","project":"p8","title":"MPOTensor.sourceGram₁_posDef","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) ρ : Matrix (Fin D) (Fin D) Complex, ρ.PosDef → (U.sourceGram₁ ρ…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceGram₁_posDef","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13007","layer":"formal","project":"p8","title":"MPOTensor.sourceSVD₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → MPOTensor.SourceCutSVD U.sourceCutM₁ U.rightRank","labels":[],"detail_key":"p8","name":"MPOTensor.sourceSVD₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13008","layer":"formal","project":"p8","title":"MPOTensor.sourceSVD₂","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → MPOTensor.SourceCutSVD U.sourceCutM₂ U.leftRank","labels":[],"detail_key":"p8","name":"MPOTensor.sourceSVD₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13009","layer":"formal","project":"p8","title":"MPOTensor.sourceWeight","kind":"def","summary":"d D : Nat → Matrix (Fin D) (Fin D) Complex → Matrix (Prod (Fin d) (Fin D)) (Prod (Fin d) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceWeight","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13010","layer":"formal","project":"p8","title":"MPOTensor.sourceWeight_posDef","kind":"theorem","summary":"∀ d D : Nat ρ : Matrix (Fin D) (Fin D) Complex, ρ.PosDef → (MPOTensor.sourceWeight ρ).PosDef","labels":[],"detail_key":"p8","name":"MPOTensor.sourceWeight_posDef","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13011","layer":"formal","project":"p8","title":"MPOTensor.sourceX₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → Matrix (Pro…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13012","layer":"formal","project":"p8","title":"MPOTensor.sourceX₁_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (a : Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₁_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13013","layer":"formal","project":"p8","title":"MPOTensor.sourceX₁_mul_sourceY₁_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (i : Fin d…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₁_mul_sourceY₁_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13014","layer":"formal","project":"p8","title":"MPOTensor.sourceX₁_weighted_isometry","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef), Eq (HMul.…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₁_weighted_isometry","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13015","layer":"formal","project":"p8","title":"MPOTensor.sourceX₁_weighted_isometry_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (r r' : Fi…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₁_weighted_isometry_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13016","layer":"formal","project":"p8","title":"MPOTensor.sourceX₂","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → Matrix (Prod (Fin D) (Fin d)) (Fin U.leftRank) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13017","layer":"formal","project":"p8","title":"MPOTensor.sourceX₂_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (a : Prod (Fin D) (Fin d)) (q : Fin U.leftRank), Eq (U.sourceX₂…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₂_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13018","layer":"formal","project":"p8","title":"MPOTensor.sourceX₂_isometry","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), U.sourceX₂.IsIsometry","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₂_isometry","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13019","layer":"formal","project":"p8","title":"MPOTensor.sourceX₂_isometry_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (l l' : Fin U.leftRank), Eq (Finset.univ.sum fun β => Finset.un…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₂_isometry_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13020","layer":"formal","project":"p8","title":"MPOTensor.sourceX₂_mul_sourceY₂_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (α : Fin D) (i j : Fin d) (β : Fin D), Eq (HMul.hMul U.sourceX₂…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceX₂_mul_sourceY₂_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13021","layer":"formal","project":"p8","title":"MPOTensor.sourceY₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → Matrix (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13022","layer":"formal","project":"p8","title":"MPOTensor.sourceY₁_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (q : Fin U…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₁_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13023","layer":"formal","project":"p8","title":"MPOTensor.sourceY₁_mul_sourceZ₁","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef), Eq (HMul.…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₁_mul_sourceZ₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13024","layer":"formal","project":"p8","title":"MPOTensor.sourceY₂","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → Matrix (Fin U.leftRank) (Prod (Fin d) (Fin D)) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13025","layer":"formal","project":"p8","title":"MPOTensor.sourceY₂_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (q : Fin U.leftRank) (b : Prod (Fin d) (Fin D)), Eq (U.sourceY₂…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₂_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13026","layer":"formal","project":"p8","title":"MPOTensor.sourceY₂_mul_sourceZ₂","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D), Eq (HMul.hMul U.sourceY₂ U.sourceZ₂) 1","labels":[],"detail_key":"p8","name":"MPOTensor.sourceY₂_mul_sourceZ₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13027","layer":"formal","project":"p8","title":"MPOTensor.sourceZ₁","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → Matrix (Pro…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceZ₁","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13028","layer":"formal","project":"p8","title":"MPOTensor.sourceZ₁_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (b : Prod…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceZ₁_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13029","layer":"formal","project":"p8","title":"MPOTensor.sourceZ₂","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → Matrix (Prod (Fin d) (Fin D)) (Fin U.leftRank) Complex","labels":[],"detail_key":"p8","name":"MPOTensor.sourceZ₂","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13030","layer":"formal","project":"p8","title":"MPOTensor.sourceZ₂_apply","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (b : Prod (Fin d) (Fin D)) (q : Fin U.leftRank), Eq (U.sourceZ₂…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceZ₂_apply","module":"TNLean.MPS.MPU.SourceFactors"},{"id":"n13031","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.independentTensorProductOfIdentityWeight","kind":"def","summary":"d D e E : Nat → U : MPOTensor d D → V : MPOTensor e E → U.SourceFactors 1 → V.SourceFactors 1 →…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.independentTensorProductOfIdentityWeight","module":"TNLean.MPS.MPU.SourceFactorsTensorProduct"},{"id":"n13032","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → LT.lt 0 U.rightRank → LT.lt 0 U.leftRank → Real","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13033","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue_blockTensor_eq_of_products","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) k₀ k : Nat, LE.le k₀ k → ∀ (hd : LT.lt 0 d) (hprod₀ : Eq (HMul.…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue_blockTensor_eq_of_products","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13034","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue_eq_add_of_common_rank_product","kind":"theorem","summary":"∀ d D e E f F : Nat (U : MPOTensor d D) (V : MPOTensor e E) (W : MPOTensor f F) (c : Nat), LT.l…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue_eq_add_of_common_rank_product","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13035","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue_eq_logb_rightRank_div","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (hr : LT.lt 0 U.rightRank) (hℓ : LT.lt 0 U.leftRank), LT.lt 0 d…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue_eq_logb_rightRank_div","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13036","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue_eq_neg_logb_leftRank_div","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (hr : LT.lt 0 U.rightRank) (hℓ : LT.lt 0 U.leftRank), LT.lt 0 d…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue_eq_neg_logb_leftRank_div","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13037","layer":"formal","project":"p8","title":"MPOTensor.sourceIndexValue_eq_of_common_rank_scale","kind":"theorem","summary":"∀ d D e E : Nat (U : MPOTensor d D) (V : MPOTensor e E) (c : Nat) (hc : LT.lt 0 c) (hrU : LT.lt…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceIndexValue_eq_of_common_rank_scale","module":"TNLean.MPS.MPU.SourceIndexValue"},{"id":"n13038","layer":"formal","project":"p8","title":"MPOTensor.sourceRanks_eq_physDim_of_sourceIndexValue_eq_zero","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (hr : LT.lt 0 U.rightRank) (hℓ : LT.lt 0 U.leftRank), LT.lt 0 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(Prod (Fin d) (Fin D)) (Fin r) Complex → Matrix (Prod (F…","labels":[],"detail_key":"p8","name":"MPOTensor.sourcePeriodicArcB","module":"TNLean.MPS.MPU.SourcePeriodicSewing"},{"id":"n13042","layer":"formal","project":"p8","title":"MPOTensor.trace_two_marked_evalWord_eq_sourcePeriodic_sewing","kind":"theorem","summary":"∀ d D r l : Nat (U : MPOTensor d D) (X₁ : Matrix (Prod (Fin d) (Fin D)) (Fin r) Complex) (Y₁ :…","labels":[],"detail_key":"p8","name":"MPOTensor.trace_two_marked_evalWord_eq_sourcePeriodic_sewing","module":"TNLean.MPS.MPU.SourcePeriodicSewing"},{"id":"n13043","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.mul_self_le_rightRank_mul_leftRank","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), LE.le (HMul.hMul d d) (HMul.hMul 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:…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.mul_self_le_rightRank_mul_leftRank_of_isMPU","module":"TNLean.MPS.MPU.SourceUCompleteNetwork"},{"id":"n13046","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.sourceU_gram_eq_closed_doubleLayer_trace_of_isSymm","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) ρ : Matrix (Fin D) (Fin D) Complex (S : U.SourceFactors ρ), ρ.I…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.sourceU_gram_eq_closed_doubleLayer_trace_of_isSymm","module":"TNLean.MPS.MPU.SourceUCompleteNetwork"},{"id":"n13047","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.sourceU_gram_eq_normalized_input_tail","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) [NeZero d] ρ : Matrix (Fin D) (Fin D) Complex (S : 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Eq…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.normalizedDiagonal_mul_mem_residualAlgebra_mul_normalizedDiagonal_eq_zero","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13153","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.residualSlice_doubleLayerTensor_blockTensor_mem_threeFormSubmodule","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → ∀ (ρ Φ : Fin (HMul.hMul D D) → C…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.residualSlice_doubleLayerTensor_blockTensor_mem_threeFormSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13154","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.residualSlice_doubleLayerTensor_blockTensor_single_mem_threeFormSubmodule","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → ∀ (ρ Φ : Fin (HMul.hMul D D) → C…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.residualSlice_doubleLayerTensor_blockTensor_single_mem_threeFormSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13155","layer":"formal","project":"p8","title":"MPOTensor.ordered_prod_smul_projector_add_sub_mem_threeFormSubmodule","kind":"theorem","summary":"∀ R : Type u_1 [inst : CommRing R] B : Type u_2 [inst_1 : Ring B] [inst_2 : Algebra R B] (E : B…","labels":[],"detail_key":"p8","name":"MPOTensor.ordered_prod_smul_projector_add_sub_mem_threeFormSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13156","layer":"formal","project":"p8","title":"MPOTensor.projectorMulResidualSubmodule","kind":"def","summary":"R : Type u_1 → [inst : CommRing R] → B : Type u_2 → [inst_1 : Ring B] → [inst_2 : Algebra R B]…","labels":[],"detail_key":"p8","name":"MPOTensor.projectorMulResidualSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13157","layer":"formal","project":"p8","title":"MPOTensor.residualMulProjectorMulResidualSubmodule","kind":"def","summary":"R : Type u_1 → [inst : CommRing R] → B : Type u_2 → [inst_1 : Ring B] → [inst_2 : Algebra R B]…","labels":[],"detail_key":"p8","name":"MPOTensor.residualMulProjectorMulResidualSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13158","layer":"formal","project":"p8","title":"MPOTensor.residualMulProjectorSubmodule","kind":"def","summary":"R : Type u_1 → [inst : CommRing R] → B : Type u_2 → [inst_1 : Ring B] → [inst_2 : Algebra R B]…","labels":[],"detail_key":"p8","name":"MPOTensor.residualMulProjectorSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13159","layer":"formal","project":"p8","title":"MPOTensor.threeFormSubmodule","kind":"def","summary":"R : Type u_1 → [inst : CommRing R] → B : Type u_2 → [inst_1 : Ring B] → [inst_2 : Algebra R B]…","labels":[],"detail_key":"p8","name":"MPOTensor.threeFormSubmodule","module":"TNLean.MPS.MPU.ThreeFormSpan"},{"id":"n13160","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.normalizedFlattening_charpoly","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → Eq (transferMatrix (Kraus.transf…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.normalizedFlattening_charpoly","module":"TNLean.MPS.MPU.TransferMatrix"},{"id":"n13161","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.normalizedFlattening_nonzero_spectrum","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → Eq (sdiff (spectrum Complex (tra…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.normalizedFlattening_nonzero_spectrum","module":"TNLean.MPS.MPU.TransferMatrix"},{"id":"n13162","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.trace_transferMatrix_normalizedFlattening_pow_eq_one","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → ∀ N : Nat, LT.lt 1 N → Eq (HPow.…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.trace_transferMatrix_normalizedFlattening_pow_eq_one","module":"TNLean.MPS.MPU.TransferMatrix"},{"id":"n13163","layer":"formal","project":"p8","title":"MPOTensor.mpvOverlap_normalizedFlattening_self","kind":"theorem","summary":"∀ d D : Nat [NeZero d] (U : MPOTensor d D) (N : Nat), Eq (U.normalizedFlattening.mpvOverlap U.n…","labels":[],"detail_key":"p8","name":"MPOTensor.mpvOverlap_normalizedFlattening_self","module":"TNLean.MPS.MPU.TransferMatrix"},{"id":"n13164","layer":"formal","project":"p8","title":"MPOTensor.normalizedFlattening","kind":"def","summary":"d D : Nat → MPOTensor d D → MPSTensor (HMul.hMul d d) D","labels":[],"detail_key":"p8","name":"MPOTensor.normalizedFlattening","module":"TNLean.MPS.MPU.TransferMatrix"},{"id":"n13165","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.ambientBlockInclusion","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (data : A.CPSVCanonicalFormData) → (k : Fin data.r) → Matrix (F…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.ambientBlockInclusion","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13166","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.ambientBlockInclusion_conjTranspose_mul_self","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData) (k : Fin data.r), Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.ambientBlockInclusion_conjTranspose_mul_self","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13167","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.mul_ambientBlockInclusion","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData) (k : Fin data.r) (i : Fin d), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.mul_ambientBlockInclusion","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13168","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.r_eq_one_of_shifted_transfer_trace","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), (∀ (N : Nat), LT.lt 1 N → Eq (H…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.r_eq_one_of_shifted_transfer_trace","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13169","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.transferEigenvalue","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (data : A.CPSVCanonicalFormData) → Fin data.r → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.transferEigenvalue","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13170","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.transferEigenvalue_eq_one","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), (∀ (N : Nat), LT.lt 1 N → Eq (H…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.transferEigenvalue_eq_one","module":"TNLean.MPS.MPU.TransferMultiplicity"},{"id":"n13171","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.exists_normalized_transfer_stabilizes_to_rank_one","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → LT.lt 1 D → Exists fun J => And…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.exists_normalized_transfer_stabilizes_to_rank_one","module":"TNLean.MPS.MPU.TransferStabilization"},{"id":"n13172","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.normalized_transfer_matrix_eq_one_fin_one","kind":"theorem","summary":"∀ d : Nat [NeZero d] U : MPOTensor d 1, U.IsMPU → Eq (transferMatrix (Kraus.transferMap U.norma…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.normalized_transfer_matrix_eq_one_fin_one","module":"TNLean.MPS.MPU.TransferStabilization"},{"id":"n13173","layer":"formal","project":"p8","title":"MPOTensor.IsMPU.normalized_transfer_power_eq_vecMulVec_of_reduced_cfii","kind":"theorem","summary":"∀ d D : Nat [NeZero d] [NeZero D] U : MPOTensor d D, U.IsMPU → LT.lt 1 D → ∀ (cfii : U.normaliz…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPU.normalized_transfer_power_eq_vecMulVec_of_reduced_cfii","module":"TNLean.MPS.MPU.TransferStabilization"},{"id":"n13174","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.normalized_transfer_power_eq_vecMulVec","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), Eq (HPow.hPow (transferMatrix (Kra…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.normalized_transfer_power_eq_vecMulVec","module":"TNLean.MPS.MPU.TransferStabilization"},{"id":"n13175","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.vecMul_one_vec_transferMatrix","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), Eq (Matrix.vecMul (Matrix.vec 1) (…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.vecMul_one_vec_transferMatrix","module":"TNLean.MPS.MPU.TransferStabilization"},{"id":"n13176","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.truncatedSymmetry","kind":"def","summary":"d D : Nat → U : MPOTensor d D → ρ : Matrix (Fin D) (Fin D) Complex → U.SourceFactors ρ → (N : N…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.truncatedSymmetry","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13177","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.truncatedSymmetry_cons","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D ρ : Matrix (Fin D) (Fin D) Complex (S : U.SourceFactors ρ) N : Na…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.truncatedSymmetry_cons","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13178","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.truncatedSymmetry_cons_snoc","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D ρ : Matrix (Fin D) (Fin D) Complex (S : U.SourceFactors ρ) N : Na…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.truncatedSymmetry_cons_snoc","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13179","layer":"formal","project":"p8","title":"MPOTensor.SourceFactors.truncatedSymmetry_zero","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D ρ : Matrix (Fin D) (Fin D) Complex (S : U.SourceFactors ρ) (l : F…","labels":[],"detail_key":"p8","name":"MPOTensor.SourceFactors.truncatedSymmetry_zero","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13180","layer":"formal","project":"p8","title":"MPOTensor.truncatedSymmetryOfEndpoints","kind":"def","summary":"d D : Nat → (U : MPOTensor d D) → Matrix (Fin U.leftRank) (Prod (Fin d) (Fin D)) Complex → Matr…","labels":[],"detail_key":"p8","name":"MPOTensor.truncatedSymmetryOfEndpoints","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13181","layer":"formal","project":"p8","title":"MPOTensor.truncatedSymmetryOfEndpoints_cons","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (Y₁ : Matrix (Fin U.rightRank) (Prod (Fin D) (Fin d)) Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.truncatedSymmetryOfEndpoints_cons","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13182","layer":"formal","project":"p8","title":"MPOTensor.truncatedSymmetryOfEndpoints_cons_snoc","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (X₁ : Matrix (Prod (Fin d) (Fin D)) (Fin U.rightRank) Complex)…","labels":[],"detail_key":"p8","name":"MPOTensor.truncatedSymmetryOfEndpoints_cons_snoc","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13183","layer":"formal","project":"p8","title":"MPOTensor.truncatedSymmetryOfEndpoints_zero","kind":"theorem","summary":"∀ d D : Nat (U : MPOTensor d D) (Y₂ : Matrix (Fin U.leftRank) (Prod (Fin d) (Fin D)) Complex) (…","labels":[],"detail_key":"p8","name":"MPOTensor.truncatedSymmetryOfEndpoints_zero","module":"TNLean.MPS.MPU.TruncatedSymmetryGrowth"},{"id":"n13184","layer":"formal","project":"p8","title":"MPOTensor.IsMPUCanonicalFormII.truncatedSymmetry_isUnitaryBetween","kind":"theorem","summary":"∀ d D : Nat U : MPOTensor d D (hU : U.IsMPUCanonicalFormII), U.IsMPUSimple → ∀ (N : Nat), ((U.s…","labels":[],"detail_key":"p8","name":"MPOTensor.IsMPUCanonicalFormII.truncatedSymmetry_isUnitaryBetween","module":"TNLean.MPS.MPU.TruncatedSymmetryUnitarity"},{"id":"n13185","layer":"formal","project":"p8","title":"MPOTensor.continuous_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat X : Type u_1 [inst : TopologicalSpace X] A : X → Matrix (Fin D) (Fin D) Complex U :…","labels":[],"detail_key":"p8","name":"MPOTensor.continuous_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13186","layer":"formal","project":"p8","title":"MPOTensor.leftRank_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat (A : Matrix (Fin D) (Fin D) Complex) (U : MPOTensor d D) (B : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.leftRank_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13187","layer":"formal","project":"p8","title":"MPOTensor.rightRank_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat (A : Matrix (Fin D) (Fin D) Complex) (U : MPOTensor d D) (B : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.rightRank_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13188","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₁_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat (A : Matrix (Fin D) (Fin D) Complex) (U : MPOTensor d D) (B : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₁_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13189","layer":"formal","project":"p8","title":"MPOTensor.sourceCutM₂_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat (A : Matrix (Fin D) (Fin D) Complex) (U : MPOTensor d D) (B : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPOTensor.sourceCutM₂_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13190","layer":"formal","project":"p8","title":"MPOTensor.transferMap_virtualSandwich","kind":"theorem","summary":"∀ d D : Nat (A : Matrix (Fin D) (Fin D) Complex) (U : MPOTensor d D) (B X : Matrix (Fin D) (Fin…","labels":[],"detail_key":"p8","name":"MPOTensor.transferMap_virtualSandwich","module":"TNLean.MPS.MPU.VirtualSandwich"},{"id":"n13191","layer":"formal","project":"p8","title":"MPOTensor.virtualSandwich","kind":"def","summary":"d D : Nat → Matrix (Fin D) (Fin D) Complex → MPOTensor d D → Matrix (Fin 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d…","labels":[],"detail_key":"p8","name":"MPSTensor.openState","module":"TNLean.MPS.OpenBoundary"},{"id":"n13195","layer":"formal","project":"p8","title":"MPSTensor.mpvInner","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.mpvInner","module":"TNLean.MPS.Overlap.Basic"},{"id":"n13196","layer":"formal","project":"p8","title":"MPSTensor.mpvOverlap","kind":"def","summary":"d D₁ D₂ : Nat → MPSTensor d D₁ → MPSTensor d D₂ → Nat → Complex","labels":[],"detail_key":"p8","name":"MPSTensor.mpvOverlap","module":"TNLean.MPS.Overlap.Basic"},{"id":"n13197","layer":"formal","project":"p8","title":"MPSTensor.mpvOverlap_eq_of_pos_mpv_eq","kind":"theorem","summary":"∀ d D₁ D₁' D₂ D₂' : Nat A : MPSTensor d D₁ A' : MPSTensor d D₁' B : MPSTensor d D₂ B' : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvOverlap_eq_of_pos_mpv_eq","module":"TNLean.MPS.Overlap.Basic"},{"id":"n13198","layer":"formal","project":"p8","title":"MPSTensor.mpvOverlap_eq_star_mpvInner","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) (N : Nat), Eq (A.mpvOverlap B N) (sta…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvOverlap_eq_star_mpvInner","module":"TNLean.MPS.Overlap.Basic"},{"id":"n13199","layer":"formal","project":"p8","title":"MPSTensor.mpvOverlap_smul_self","kind":"theorem","summary":"∀ (c : Complex) d D : Nat (A : MPSTensor d D) (N : Nat), Eq (MPSTensor.mpvOverlap (fun i => HSM…","labels":[],"detail_key":"p8","name":"MPSTensor.mpvOverlap_smul_self","module":"TNLean.MPS.Overlap.Basic"},{"id":"n13200","layer":"formal","project":"p8","title":"MPSTensor.mpvState","kind":"def","summary":"d D : Nat → MPSTensor d D → (N : Nat) → MPSTensor.MPVSpace d 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MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.block_matrices_eq_of_wordTupleSpanTop_trace","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionBoundaryDecomposition"},{"id":"n13310","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_iSupIndep_of_wordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_iSupIndep_of_wordTupleSpanTop","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionBoundaryDecomposition"},{"id":"n13311","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07LeftBoundaryComponent_mem_groundSpace","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (C : Fin d → Matrix (Fin D) (Fin D) Complex) (E : Matrix (Fin 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MPSTensor.Wo…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_blockwise_word_compatibility_of_trace_decomposition","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13316","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_complementary_word_boundary_identities_of_trace_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) m K M : Nat, MPSTensor.Wo…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_complementary_word_boundary_identities_of_trace_decomposition","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13317","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_complementary_word_compatibility_of_trace_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) m K M : Nat, MPSTensor.Wo…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_complementary_word_compatibility_of_trace_decomposition","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13318","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_directSum_restriction_intersection_of_wordTupleSpanTop","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_directSum_restriction_intersection_of_wordTupleSpanTop","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13319","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_fixed_complementary_word_compatibility_of_trace_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) m K M : Nat, MPSTensor.Wo…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_fixed_complementary_word_compatibility_of_trace_decomposition","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13320","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_iSup_groundSpace_eq_restriction_intersection","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_iSup_groundSpace_eq_restriction_intersection","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13321","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_iSup_restriction_intersection_eventually_of_period_window","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) start period : Nat, LT.lt…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_iSup_restriction_intersection_eventually_of_period_window","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13322","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_mem_iSup_groundSpace_iff_iSup_restrictions","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_mem_iSup_groundSpace_iff_iSup_restrictions","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13323","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_mem_iSup_groundSpace_of_iSup_restrictions","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_mem_iSup_groundSpace_of_iSup_restrictions","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13324","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_mem_iSup_groundSpace_of_trace_decomposition","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat, MPSTensor.WordTu…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_mem_iSup_groundSpace_of_trace_decomposition","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13325","layer":"formal","project":"p8","title":"MPSTensor.pgvwc07_trace_decompositions_of_iSup_restrictions","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTensor d (dim j)) n : Nat (ψ : MPSTensor.NS…","labels":[],"detail_key":"p8","name":"MPSTensor.pgvwc07_trace_decompositions_of_iSup_restrictions","module":"TNLean.MPS.ParentHamiltonian.BlockIntersectionProperty"},{"id":"n13326","layer":"formal","project":"p8","title":"MPSTensor.commutes_all_of_commutes_long_words_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ m : Nat, Kraus.IsNBlkInjective A L₀ → LT.lt 0 L₀ → LE.le L₀ m…","labels":[],"detail_key":"p8","name":"MPSTensor.commutes_all_of_commutes_long_words_of_isNBlkInjective","module":"TNLean.MPS.ParentHamiltonian.BlockStrip"},{"id":"n13327","layer":"formal","project":"p8","title":"MPSTensor.commutes_block_words_of_commutes_long_words_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ m : Nat, Kraus.IsNBlkInjective A L₀ → LT.lt 0 L₀ → LE.le L₀ 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(L…","labels":[],"detail_key":"p8","name":"MPSTensor.BlockSumGroundSpace.groundSpaceMap_toTensorFromBlocks_eq_sum_blockDiagonal","module":"TNLean.MPS.ParentHamiltonian.BlockSumGroundSpace"},{"id":"n13332","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_block_le_toTensorFromBlocks","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)) j :…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_block_le_toTensorFromBlocks","module":"TNLean.MPS.ParentHamiltonian.BlockSumGroundSpace"},{"id":"n13333","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_toTensorFromBlocks_eq_iSup","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_toTensorFromBlocks_eq_iSup","module":"TNLean.MPS.ParentHamiltonian.BlockSumGroundSpace"},{"id":"n13334","layer":"formal","project":"p8","title":"MPSTensor.blockedConfigLinearIsometryEquiv","kind":"def","summary":"(d N p : Nat) → LinearIsometryEquiv (RingHom.id Complex) (EuclideanSpace Complex (MPSTensor.Cfg…","labels":[],"detail_key":"p8","name":"MPSTensor.blockedConfigLinearIsometryEquiv","module":"TNLean.MPS.ParentHamiltonian.BlockedGroundSpaceTransport"},{"id":"n13335","layer":"formal","project":"p8","title":"MPSTensor.blockedConfigLinearIsometryEquiv_apply_apply","kind":"theorem","summary":"∀ (d N p : Nat) (v : EuclideanSpace Complex (MPSTensor.Cfg (MPSTensor.blockPhysDim d p) N)) (σ…","labels":[],"detail_key":"p8","name":"MPSTensor.blockedConfigLinearIsometryEquiv_apply_apply","module":"TNLean.MPS.ParentHamiltonian.BlockedGroundSpaceTransport"},{"id":"n13336","layer":"formal","project":"p8","title":"MPSTensor.blockedConfigLinearIsometryEquiv_groundSpaceMapES","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (p N : Nat) (x : EuclideanSpace Complex (Prod (Fin D) (Fin D)))…","labels":[],"detail_key":"p8","name":"MPSTensor.blockedConfigLinearIsometryEquiv_groundSpaceMapES","module":"TNLean.MPS.ParentHamiltonian.BlockedGroundSpaceTransport"},{"id":"n13337","layer":"formal","project":"p8","title":"MPSTensor.blockedConfigLinearIsometryEquiv_symm_apply_apply","kind":"theorem","summary":"∀ (d N p : Nat) (v : EuclideanSpace Complex (MPSTensor.Cfg d (HMul.hMul N p))) (σ : 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(hM…","labels":[],"detail_key":"p8","name":"MPSTensor.boundary_closing_product_eq_of_compatible_backgrounds","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13346","layer":"formal","project":"p8","title":"MPSTensor.boundary_closing_product_eq_of_pointwise_compatible_boundary_assignments","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.boundary_closing_product_eq_of_pointwise_compatible_boundary_assignments","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13347","layer":"formal","project":"p8","title":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_closing_restrictions","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_closing_restrictions","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13348","layer":"formal","project":"p8","title":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_mirror_padded_products","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_mirror_padded_products","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13349","layer":"formal","project":"p8","title":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_right_products","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_auxiliary_boundary_product_eq_of_right_products","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13350","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_condition_long_product_of_window_witnesses","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_boundary_condition_long_product_of_window_witnesses","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13351","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_condition_product_of_window_witnesses","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_boundary_condition_product_of_window_witnesses","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13352","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_first_products_of_restrictions","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.lt 0 L₀) (hM…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_boundary_first_products_of_restrictions","module":"TNLean.MPS.ParentHamiltonian.BoundaryClosing"},{"id":"n13353","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_one_sided_products_of_groundSpaceMap","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D] L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : 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r…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwThreeBlockConfigEquiv","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13616","layer":"formal","project":"p8","title":"MPSTensor.fnwThreeBlockConfigEquiv_apply","kind":"theorem","summary":"∀ (d ℓ m r : Nat) (μℓ : MPSTensor.Cfg d ℓ) (μm : MPSTensor.Cfg d m) (μr : MPSTensor.Cfg d r), E…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwThreeBlockConfigEquiv_apply","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13617","layer":"formal","project":"p8","title":"MPSTensor.fnwThreeBlockConfigEquiv_symm_apply_apply","kind":"theorem","summary":"∀ (d ℓ m r : Nat) (μℓ : MPSTensor.Cfg d ℓ) (μm : MPSTensor.Cfg d m) (μr : MPSTensor.Cfg d r), E…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwThreeBlockConfigEquiv_symm_apply_apply","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13618","layer":"formal","project":"p8","title":"MPSTensor.fnwTransferMap_pow_one","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsLeftCanonical → ∀ (N : Nat), Eq ((HPow.hPow A.fnwTransferM…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwTransferMap_pow_one","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13619","layer":"formal","project":"p8","title":"MPSTensor.inner_conjTranspose_mul_mul_conjTranspose_eq_aggregateSummands","kind":"theorem","summary":"∀ D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (U V X Y : Matrix (Fin D) (Fin D…","labels":[],"detail_key":"p8","name":"MPSTensor.inner_conjTranspose_mul_mul_conjTranspose_eq_aggregateSummands","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13620","layer":"formal","project":"p8","title":"MPSTensor.inner_fnwLeftOverlapMap_fnwRightOverlapMap","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D) (ℓ m r : N…","labels":[],"detail_key":"p8","name":"MPSTensor.inner_fnwLeftOverlapMap_fnwRightOverlapMap","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13621","layer":"formal","project":"p8","title":"MPSTensor.inner_overlap_sub_inner_aggregates","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D) (ℓ m r : N…","labels":[],"detail_key":"p8","name":"MPSTensor.inner_overlap_sub_inner_aggregates","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13622","layer":"formal","project":"p8","title":"MPSTensor.norm_fnwLeftFullGroundFamily","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), Eq ((Krau…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_fnwLeftFullGroundFamily","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13623","layer":"formal","project":"p8","title":"MPSTensor.norm_fnwRightFullGroundFamily","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), A.IsLeftC…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_fnwRightFullGroundFamily","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13624","layer":"formal","project":"p8","title":"MPSTensor.sum_inner_overlapFibers_eq_inner_aggregates","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D) (ℓ r : Nat…","labels":[],"detail_key":"p8","name":"MPSTensor.sum_inner_overlapFibers_eq_inner_aggregates","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13625","layer":"formal","project":"p8","title":"MPSTensor.sum_norm_sq_conjTranspose_evalWord_mul","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), A.IsLeftC…","labels":[],"detail_key":"p8","name":"MPSTensor.sum_norm_sq_conjTranspose_evalWord_mul","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13626","layer":"formal","project":"p8","title":"MPSTensor.sum_norm_sq_fnwLeftMiddleBoundary","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), A.IsLeftC…","labels":[],"detail_key":"p8","name":"MPSTensor.sum_norm_sq_fnwLeftMiddleBoundary","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13627","layer":"formal","project":"p8","title":"MPSTensor.sum_norm_sq_fnwRightMiddleBoundary","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), Eq ((Krau…","labels":[],"detail_key":"p8","name":"MPSTensor.sum_norm_sq_fnwRightMiddleBoundary","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13628","layer":"formal","project":"p8","title":"MPSTensor.sum_norm_sq_mul_conjTranspose_evalWord","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D), Eq ((Krau…","labels":[],"detail_key":"p8","name":"MPSTensor.sum_norm_sq_mul_conjTranspose_evalWord","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13629","layer":"formal","project":"p8","title":"MPSTensor.trace_mul_fnwTransferMap_pow","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρ X : Matrix (Fin D) (Fin D) Complex) (N : Nat), Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_mul_fnwTransferMap_pow","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapCoordinates"},{"id":"n13630","layer":"formal","project":"p8","title":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorm_sq_le_leftOverlap","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorm_sq_le_leftOverlap","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13631","layer":"formal","project":"p8","title":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorm_sq_le_rightOverlap","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorm_sq_le_rightOverlap","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13632","layer":"formal","project":"p8","title":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorms_le_overlapNorms","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwLowerBoundaryConstant_mul_familyNorms_le_overlapNorms","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13633","layer":"formal","project":"p8","title":"MPSTensor.norm_fnwLeftOverlapMap_sq","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D) (ℓ m r : N…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_fnwLeftOverlapMap_sq","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13634","layer":"formal","project":"p8","title":"MPSTensor.norm_fnwRightOverlapMap_sq","kind":"theorem","summary":"∀ d D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTensor d D) (ℓ m r : N…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_fnwRightOverlapMap_sq","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13635","layer":"formal","project":"p8","title":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (htr : Eq ρ.trace 1…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13636","layer":"formal","project":"p8","title":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le_div_lowerBoundary","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (htr : Eq ρ.…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le_div_lowerBoundary","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13637","layer":"formal","project":"p8","title":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le_familyNorms","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (htr : Eq ρ.trace 1…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_inner_overlap_sub_inner_aggregates_le_familyNorms","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13638","layer":"formal","project":"p8","title":"MPSTensor.sqrt_fnwLowerBoundaryConstant_mul_familyNorm_le_leftOverlap","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.sqrt_fnwLowerBoundaryConstant_mul_familyNorm_le_leftOverlap","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13639","layer":"formal","project":"p8","title":"MPSTensor.sqrt_fnwLowerBoundaryConstant_mul_familyNorm_le_rightOverlap","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (A : MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.sqrt_fnwLowerBoundaryConstant_mul_familyNorm_le_rightOverlap","module":"TNLean.MPS.ParentHamiltonian.FNWOverlapEstimate"},{"id":"n13640","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.exists_wholeIncrement_groundProjection_defect_le_fnw","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] ρ : Matrix (Fin D) (Fin D) Complex A : MPSTensor d D, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.exists_wholeIncrement_groundProjection_defect_le_fnw","module":"TNLean.MPS.ParentHamiltonian.FNWProjectorDefect"},{"id":"n13641","layer":"formal","project":"p8","title":"MPSTensor.wholeIncrement_groundProjection_defect_le_fnw","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (htr : Eq ρ.…","labels":[],"detail_key":"p8","name":"MPSTensor.wholeIncrement_groundProjection_defect_le_fnw","module":"TNLean.MPS.ParentHamiltonian.FNWProjectorDefect"},{"id":"n13642","layer":"formal","project":"p8","title":"MPSTensor.wholeIncrement_groundProjection_defect_le_fnw_factored","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (htr : Eq ρ.…","labels":[],"detail_key":"p8","name":"MPSTensor.wholeIncrement_groundProjection_defect_le_fnw_factored","module":"TNLean.MPS.ParentHamiltonian.FNWProjectorDefect"},{"id":"n13643","layer":"formal","project":"p8","title":"MPSTensor.fnwTransferMap","kind":"def","summary":"d D : Nat → MPSTensor d D → LinearMap (RingHom.id Complex) (Matrix (Fin D) (Fin D) Complex) (Ma…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwTransferMap","module":"TNLean.MPS.ParentHamiltonian.FNWTransferConvention"},{"id":"n13644","layer":"formal","project":"p8","title":"MPSTensor.fnwTransferMap_eq_traceAdjointMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Eq A.fnwTransferMap (Matrix.traceAdjointMap (Kraus.transferMap…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwTransferMap_eq_traceAdjointMap","module":"TNLean.MPS.ParentHamiltonian.FNWTransferConvention"},{"id":"n13645","layer":"formal","project":"p8","title":"MPSTensor.fnwTransferMap_one","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.IsLeftCanonical → Eq (A.fnwTransferMap 1) 1","labels":[],"detail_key":"p8","name":"MPSTensor.fnwTransferMap_one","module":"TNLean.MPS.ParentHamiltonian.FNWTransferConvention"},{"id":"n13646","layer":"formal","project":"p8","title":"MPSTensor.trace_mul_fnwTransferMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ρ X : Matrix 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A.IsPr…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.fnwRemainder_eigenvalue_norm_lt_one","module":"TNLean.MPS.ParentHamiltonian.FNWTransferDecay"},{"id":"n13649","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.fnwWeightedRemainder_spectralRadius_lt_one","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex (hP : A.IsPr…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.fnwWeightedRemainder_spectralRadius_lt_one","module":"TNLean.MPS.ParentHamiltonian.FNWTransferDecay"},{"id":"n13650","layer":"formal","project":"p8","title":"MPSTensor.fnwMixingQuantity","kind":"def","summary":"D : Nat → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → d : Nat → MPSTensor d D → Eq 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(MPSTensor.fnwTrace…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwTraceInverseFactor_pos","module":"TNLean.MPS.ParentHamiltonian.FNWTransferDecay"},{"id":"n13654","layer":"formal","project":"p8","title":"MPSTensor.fnwWeightedOperatorNorm","kind":"def","summary":"D : Nat → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → Module.End Complex (Matrix (Fin D)…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwWeightedOperatorNorm","module":"TNLean.MPS.ParentHamiltonian.FNWTransferDecay"},{"id":"n13655","layer":"formal","project":"p8","title":"MPSTensor.fnwWeightedOperatorNorm_nonneg","kind":"theorem","summary":"∀ D : Nat (ρ : Matrix (Fin D) (Fin D) Complex) (hρ : ρ.PosDef) (F : Module.End Complex (Matrix…","labels":[],"detail_key":"p8","name":"MPSTensor.fnwWeightedOperatorNorm_nonneg","module":"TNLean.MPS.ParentHamiltonian.FNWTransferDecay"},{"id":"n13656","layer":"formal","project":"p8","title":"MPSTensor.fnwWeightedRemainderSpectralRadius","kind":"def","summary":"D : Nat → (ρ : Matrix (Fin D) (Fin D) Complex) → ρ.PosDef → d : Nat → MPSTensor d D → Ne 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d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.eventually_groundSpaceGram_isUnit_and_inverse_bound","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13662","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.eventually_groundSpaceMapES_injective_and_inverseGram_bound","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.eventually_groundSpaceMapES_injective_and_inverseGram_bound","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13663","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.groundSpaceGram_geometric_inverse_bounds","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.groundSpaceGram_geometric_inverse_bounds","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13664","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.groundSpaceGram_ringInverse_tendsto","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.groundSpaceGram_ringInverse_tendsto","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13665","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.groundSpaceMapES_geometric_inverseGram_bounds","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.groundSpaceMapES_geometric_inverseGram_bounds","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13666","layer":"formal","project":"p8","title":"MPSTensor.geometric_smallness_at_c3_lengths","kind":"theorem","summary":"∀ c r : Real, LT.lt 0 c → LT.lt 0 r → LT.lt r 1 → ∀ l : Nat, LT.lt (HMul.hMul c (HPow.hPow r l)…","labels":[],"detail_key":"p8","name":"MPSTensor.geometric_smallness_at_c3_lengths","module":"TNLean.MPS.ParentHamiltonian.GramInverseConvergence"},{"id":"n13667","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.groundSpace_eq","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → ∀ (L : Nat), Eq (A.groundSpace L) (B.groundSp…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.groundSpace_eq","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13668","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.groundSpace_le","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → ∀ (L : Nat), LE.le (B.groundSpace L) (A.groun…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.groundSpace_le","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13669","layer":"formal","project":"p8","title":"MPSTensor.NSiteSpace","kind":"def","summary":"Nat → Nat → Type","labels":[],"detail_key":"p8","name":"MPSTensor.NSiteSpace","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13670","layer":"formal","project":"p8","title":"MPSTensor.bntMPSVectorSpan","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((j : Fin r) → MPSTensor d (dim j)) → (N : Nat) → Submodule 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Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMap_apply","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13674","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_finrank_le","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), LE.le (Module.finrank Complex (Subtype fun x => Memb…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_finrank_le","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13675","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_ne_top","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), GT.gt (HPow.hPow d L) (HPow.hPow D 2) → Ne (A.ground…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_ne_top","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13676","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_smul_eq","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (ζ : Complex), Ne ζ 0 → ∀ (L : Nat), Eq ((HSMul.hSMul ζ A).grou…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_smul_eq","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13677","layer":"formal","project":"p8","title":"MPSTensor.nSiteSpace_finrank","kind":"theorem","summary":"∀ (d L : Nat), Eq (Module.finrank Complex (MPSTensor.NSiteSpace d L)) (HPow.hPow d L)","labels":[],"detail_key":"p8","name":"MPSTensor.nSiteSpace_finrank","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13678","layer":"formal","project":"p8","title":"MPSTensor.trace_gauge_boundary","kind":"theorem","summary":"∀ D : Nat (X : Matrix.GeneralLinearGroup (Fin D) Complex) (E Y : Matrix (Fin D) (Fin D) Complex…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_gauge_boundary","module":"TNLean.MPS.ParentHamiltonian.GroundSpace"},{"id":"n13679","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceES_starProjection_eq_groundSpaceMapES_comp_inverseGram_comp_adjoint","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceES_starProjection_eq_groundSpaceMapES_comp_inverseGram_comp_adjoint","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13680","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceGram","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceGram","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13681","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceGram_eq_gramReshuffle","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceGram_eq_gramReshuffle","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13682","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMapES","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMapES","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13683","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMapES_frobeniusEquivEuclidean_apply","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMapES_frobeniusEquivEuclidean_apply","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13684","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMapES_injective_of_isNBlkInjective","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMapES_injective_of_isNBlkInjective","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13685","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMapES_injective_of_isNBlkInjective_of_le","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMapES_injective_of_isNBlkInjective_of_le","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13686","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMapES_single","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMapES_single","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13687","layer":"formal","project":"p8","title":"MPSTensor.injectiveRangeProjector_groundSpaceMapES_eq_starProjection","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.injectiveRangeProjector_groundSpaceMapES_eq_starProjection","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13688","layer":"formal","project":"p8","title":"MPSTensor.inner_single_groundSpaceGram_single_eq_rectangularChoi","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.inner_single_groundSpaceGram_single_eq_rectangularChoi","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13689","layer":"formal","project":"p8","title":"MPSTensor.range_groundSpaceMapES","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.range_groundSpaceMapES","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceGram"},{"id":"n13690","layer":"formal","project":"p8","title":"MPSTensor.hasNNCPHGroundSpaces_toTensorFromBlocks_iff_forall_isNNCPH_and_ker_le_bntMPSVec…","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.hasNNCPHGroundSpaces_toTensorFromBlocks_iff_forall_isNNCPH_and_ker_le_bntMPSVectorSpan","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceSpanning"},{"id":"n13691","layer":"formal","project":"p8","title":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_iff_ker_le_bntMPSVec…","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_iff_ker_le_bntMPSVectorSpan","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceSpanning"},{"id":"n13692","layer":"formal","project":"p8","title":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_of_chain_eq_iSup_cha…","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_of_chain_eq_iSup_chain","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceSpanning"},{"id":"n13693","layer":"formal","project":"p8","title":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_of_chain_eq_iSup_mpv","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)), (∀…","labels":[],"detail_key":"p8","name":"MPSTensor.hasParentHamiltonianGroundSpaceSpanning_toTensorFromBlocks_of_chain_eq_iSup_mpv","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceSpanning"},{"id":"n13694","layer":"formal","project":"p8","title":"MPSTensor.ker_parentHamiltonian_toTensorFromBlocks_le_bntMPSVectorSpan","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)) L N…","labels":[],"detail_key":"p8","name":"MPSTensor.ker_parentHamiltonian_toTensorFromBlocks_le_bntMPSVectorSpan","module":"TNLean.MPS.ParentHamiltonian.GroundSpaceSpanning"},{"id":"n13695","layer":"formal","project":"p8","title":"MPSTensor.charpoly_halfChainReducedMatrix","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), Eq (HMul.hMul (HPow.hPow Polynomial.X (HMul.hMul D D…","labels":[],"detail_key":"p8","name":"MPSTensor.charpoly_halfChainReducedMatrix","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13696","layer":"formal","project":"p8","title":"MPSTensor.exists_geometric_bound_halfChainComparison_sub_diagonal","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (lam : Fin D → Complex) (htr : Ne (Matrix.diagonal lam).trace 0…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_geometric_bound_halfChainComparison_sub_diagonal","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13697","layer":"formal","project":"p8","title":"MPSTensor.exists_geometric_bound_halfChainGram_sub_fixedPointProj","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (Λ : Matrix (Fin D) (Fin D) Complex) (htr : Ne Λ.trace 0), IsTr…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_geometric_bound_halfChainGram_sub_fixedPointProj","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13698","layer":"formal","project":"p8","title":"MPSTensor.fixedPointProj_single_entry_diagonal","kind":"theorem","summary":"∀ D : Nat (lam : Fin D → Complex) (htr : Ne (Matrix.diagonal lam).trace 0) (p q : Prod (Fin D)…","labels":[],"detail_key":"p8","name":"MPSTensor.fixedPointProj_single_entry_diagonal","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13699","layer":"formal","project":"p8","title":"MPSTensor.halfChainGram","kind":"def","summary":"d D : Nat → MPSTensor d D → Nat → Matrix (Prod (Fin D) (Fin D)) (Prod (Fin D) (Fin D)) Complex","labels":[],"detail_key":"p8","name":"MPSTensor.halfChainGram","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13700","layer":"formal","project":"p8","title":"MPSTensor.halfChainGram_apply_eq_transferMap_pow","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat) (p q : Prod (Fin D) (Fin D)), Eq (A.halfChainGram L p…","labels":[],"detail_key":"p8","name":"MPSTensor.halfChainGram_apply_eq_transferMap_pow","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13701","layer":"formal","project":"p8","title":"MPSTensor.halfChainKernel","kind":"def","summary":"d D : Nat → MPSTensor d D → (L : Nat) → 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(A.halfChain…","labels":[],"detail_key":"p8","name":"MPSTensor.halfChainSwapMatrix_mul_conjTranspose","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13712","layer":"formal","project":"p8","title":"MPSTensor.posSemidef_halfChainReducedMatrix","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat), (A.halfChainReducedMatrix L).PosSemidef","labels":[],"detail_key":"p8","name":"MPSTensor.posSemidef_halfChainReducedMatrix","module":"TNLean.MPS.ParentHamiltonian.HalfChainSchmidt"},{"id":"n13713","layer":"formal","project":"p8","title":"MPSTensor.InLeftGround","kind":"def","summary":"d D : Nat → MPSTensor d D → (L : Nat) → MPSTensor.NSiteSpace d (HAdd.hAdd L 1) → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.InLeftGround","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13714","layer":"formal","project":"p8","title":"MPSTensor.InRightGround","kind":"def","summary":"d D : Nat → MPSTensor d D → (L : Nat) → MPSTensor.NSiteSpace d 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L : Nat (σ : Fin L → Fin d) (j : Fin d), Eq (Kraus.evalWord A (…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_ofFn_snoc","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13718","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMap_injective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → ∀ L : Nat, LT.lt 0 L → Function.Injective…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMap_injective","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13719","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMap_injective_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L₀ : Nat, Kraus.IsNBlkInjective A L₀ → Function.Injective ⇑(A.gro…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMap_injective_of_isNBlkInjective","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13720","layer":"formal","project":"p8","title":"MPSTensor.groundSpaceMap_injective_of_wordSpan_eq_top","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D L : Nat, Eq (Kraus.wordSpan A L) Top.top → Function.Injective ⇑(A…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpaceMap_injective_of_wordSpan_eq_top","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13721","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_finrank_eq","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → ∀ L : Nat, LT.lt 0 L → Eq (Module.finrank…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_finrank_eq","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13722","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_iff_left_right","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → ∀ L : Nat, LT.lt 1 L → ∀ ψ : MPSTensor.NSi…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_iff_left_right","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13723","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_inLeftGround","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat) ψ : MPSTensor.NSiteSpace d (HAdd.hAdd L 1), Membershi…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_inLeftGround","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13724","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_inRightGround","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (L : Nat) ψ : MPSTensor.NSiteSpace d (HAdd.hAdd L 1), Membershi…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_inRightGround","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13725","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_intersection","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsInjective A → ∀ L : Nat, LT.lt 1 L → ∀ ψ : MPSTensor.NSi…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_intersection","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13726","layer":"formal","project":"p8","title":"MPSTensor.restrictFirst","kind":"def","summary":"d L : Nat → MPSTensor.NSiteSpace d (HAdd.hAdd L 1) → Fin d → MPSTensor.NSiteSpace d 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(MP…","labels":[],"detail_key":"p8","name":"MPSTensor.restrictLastₗ","module":"TNLean.MPS.ParentHamiltonian.IntersectionProperty"},{"id":"n13732","layer":"formal","project":"p8","title":"MPSTensor.chainGroundSpaceES","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.chainGroundSpaceES","module":"TNLean.MPS.ParentHamiltonian.KernelChainGroundSpace"},{"id":"n13733","layer":"formal","project":"p8","title":"MPSTensor.chainGroundSpace_le_ker_parentHamiltonian","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.chainGroundSpace_le_ker_parentHamiltonian","module":"TNLean.MPS.ParentHamiltonian.KernelChainGroundSpace"},{"id":"n13734","layer":"formal","project":"p8","title":"MPSTensor.ker_parentHamiltonian_eq_chainGroundSpace","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.ker_parentHamiltonian_eq_chainGroundSpace","module":"TNLean.MPS.ParentHamiltonian.KernelChainGroundSpace"},{"id":"n13735","layer":"formal","project":"p8","title":"MPSTensor.ker_parentHamiltonian_le_chainGroundSpace","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.ker_parentHamiltonian_le_chainGroundSpace","module":"TNLean.MPS.ParentHamiltonian.KernelChainGroundSpace"},{"id":"n13736","layer":"formal","project":"p8","title":"MPSTensor.parentHamiltonianGroundSpaceES_eq_chainGroundSpaceES","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"MPSTensor.parentHamiltonianGroundSpaceES_eq_chainGroundSpaceES","module":"TNLean.MPS.ParentHamiltonian.KernelChainGroundSpace"},{"id":"n13737","layer":"formal","project":"p8","title":"MPSTensor.HasAppendixD2ParentCommutingHamiltonian","kind":"inductive","summary":"d : Nat → Submodule Complex (MPSTensor.NSiteSpace d 3) → LinearMap (RingHom.id Complex) (MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.HasAppendixD2ParentCommutingHamiltonian","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13738","layer":"formal","project":"p8","title":"MPSTensor.HasAppendixD2ParentCommutingHamiltonian.to_overlapping","kind":"theorem","summary":"∀ d : Nat KAXB : Submodule Complex (MPSTensor.NSiteSpace d 3) QAX QXB : LinearMap (RingHom.id C…","labels":[],"detail_key":"p8","name":"MPSTensor.HasAppendixD2ParentCommutingHamiltonian.to_overlapping","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13739","layer":"formal","project":"p8","title":"MPSTensor.HasOverlappingTwoSiteCommutation","kind":"inductive","summary":"d : Nat → LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace d 2)…","labels":[],"detail_key":"p8","name":"MPSTensor.HasOverlappingTwoSiteCommutation","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13740","layer":"formal","project":"p8","title":"MPSTensor.HasOverlappingTwoSiteCommutation.commute_complement_lifts","kind":"theorem","summary":"∀ d : Nat QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteS…","labels":[],"detail_key":"p8","name":"MPSTensor.HasOverlappingTwoSiteCommutation.commute_complement_lifts","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13741","layer":"formal","project":"p8","title":"MPSTensor.HasOverlappingTwoSiteCommutation.complement","kind":"theorem","summary":"∀ d : Nat QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteS…","labels":[],"detail_key":"p8","name":"MPSTensor.HasOverlappingTwoSiteCommutation.complement","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13742","layer":"formal","project":"p8","title":"MPSTensor.HasOverlappingTwoSiteCommutation.left_complement_idempotent","kind":"theorem","summary":"∀ d : Nat QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteS…","labels":[],"detail_key":"p8","name":"MPSTensor.HasOverlappingTwoSiteCommutation.left_complement_idempotent","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13743","layer":"formal","project":"p8","title":"MPSTensor.HasOverlappingTwoSiteCommutation.right_complement_idempotent","kind":"theorem","summary":"∀ d : Nat QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteS…","labels":[],"detail_key":"p8","name":"MPSTensor.HasOverlappingTwoSiteCommutation.right_complement_idempotent","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13744","layer":"formal","project":"p8","title":"MPSTensor.adjacent_twoSite_cyclicWindowsOverlap","kind":"theorem","summary":"∀ N : Nat (i : Fin N), MPSTensor.cyclicWindowsOverlap N 2 i (MPSTensor.cyclicForwardSite i 1)","labels":[],"detail_key":"p8","name":"MPSTensor.adjacent_twoSite_cyclicWindowsOverlap","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13745","layer":"formal","project":"p8","title":"MPSTensor.axPairCfg","kind":"def","summary":"d : Nat → MPSTensor.Cfg d 3 → MPSTensor.Cfg d 2","labels":[],"detail_key":"p8","name":"MPSTensor.axPairCfg","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13746","layer":"formal","project":"p8","title":"MPSTensor.leftPairLift","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace d 2)…","labels":[],"detail_key":"p8","name":"MPSTensor.leftPairLift","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13747","layer":"formal","project":"p8","title":"MPSTensor.leftPairLift_one_sub","kind":"theorem","summary":"∀ d : Nat (Q : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace…","labels":[],"detail_key":"p8","name":"MPSTensor.leftPairLift_one_sub","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13748","layer":"formal","project":"p8","title":"MPSTensor.leftPairLift_sub","kind":"theorem","summary":"∀ d : Nat (Q R : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpac…","labels":[],"detail_key":"p8","name":"MPSTensor.leftPairLift_sub","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13749","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_one_eq_rightPairLift_parentInteraction","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Eq (A.localTerm 2 3 1) (MPSTensor.rightPairLift (A.parentInter…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_one_eq_rightPairLift_parentInteraction","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13750","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_eq_leftPairLift_parentInteraction","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Eq (A.localTerm 2 3 0) (MPSTensor.leftPairLift (A.parentIntera…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_eq_leftPairLift_parentInteraction","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13751","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_one_commute_of_appendixD2","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) KAXB : Submodule Complex (MPSTensor.NSiteSpace d 3) QAX QXB : L…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_one_commute_of_appendixD2","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13752","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_one_commute_of_appendixD2_complement","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) KAXB : Submodule Complex (MPSTensor.NSiteSpace d 3) QAX QXB : L…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_one_commute_of_appendixD2_complement","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13753","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_commutation","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), MPSTensor.HasOverlappingTwoSiteCommutation (A.parentInteractio…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_commutation","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13754","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_complement","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_complement","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13755","layer":"formal","project":"p8","title":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_projectors","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) QAX QXB : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace…","labels":[],"detail_key":"p8","name":"MPSTensor.localTerm_two_three_zero_one_commute_of_overlapping_two_site_projectors","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13756","layer":"formal","project":"p8","title":"MPSTensor.replaceAXCfg","kind":"def","summary":"d : Nat → MPSTensor.Cfg d 3 → MPSTensor.Cfg d 2 → MPSTensor.Cfg d 3","labels":[],"detail_key":"p8","name":"MPSTensor.replaceAXCfg","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13757","layer":"formal","project":"p8","title":"MPSTensor.replaceXBCfg","kind":"def","summary":"d : Nat → MPSTensor.Cfg d 3 → MPSTensor.Cfg d 2 → MPSTensor.Cfg d 3","labels":[],"detail_key":"p8","name":"MPSTensor.replaceXBCfg","module":"TNLean.MPS.ParentHamiltonian.LocalSupport"},{"id":"n13758","layer":"formal","project":"p8","title":"MPSTensor.rightPairLift","kind":"def","summary":"d : Nat → LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace d 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(A.tailBound…","labels":[],"detail_key":"p8","name":"MPSTensor.reassocTailBoundaryMapES_one","module":"TNLean.MPS.ParentHamiltonian.SpectatorBoundaryGram"},{"id":"n14058","layer":"formal","project":"p8","title":"MPSTensor.tailBoundaryMapES","kind":"def","summary":"d D : Nat → MPSTensor d D → (K L : Nat) → ContinuousLinearMap (RingHom.id Complex) (MPSTensor.B…","labels":[],"detail_key":"p8","name":"MPSTensor.tailBoundaryMapES","module":"TNLean.MPS.ParentHamiltonian.SpectatorBoundaryGram"},{"id":"n14059","layer":"formal","project":"p8","title":"MPSTensor.tailBoundaryMapES_adjoint_comp_self_eq_fiberwise_groundSpaceGram","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K L : Nat), Eq ((ContinuousLinearMap.adjoint (A.tailBoundaryMa…","labels":[],"detail_key":"p8","name":"MPSTensor.tailBoundaryMapES_adjoint_comp_self_eq_fiberwise_groundSpaceGram","module":"TNLean.MPS.ParentHamiltonian.SpectatorBoundaryGram"},{"id":"n14060","layer":"formal","project":"p8","title":"MPSTensor.tailBoundaryMapES_comp_tailVirtualMapES","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K L : Nat), Eq ((A.tailBoundaryMapES K L).comp (A.tailVirtualM…","labels":[],"detail_key":"p8","name":"MPSTensor.tailBoundaryMapES_comp_tailVirtualMapES","module":"TNLean.MPS.ParentHamiltonian.SpectatorBoundaryGram"},{"id":"n14061","layer":"formal","project":"p8","title":"MPSTensor.tailVirtualMapES_adjoint_apply","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K : Nat) (x : MPSTensor.BoundaryFamilySpace (MPSTensor.Cfg d 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Nat A : MPSTensor d D K L₀ : Nat, Kraus.IsNBlkInjective A L₀ → ∀ ψ : MPSTensor.NSiteSpa…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_left_tail_compatibility","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14065","layer":"formal","project":"p8","title":"MPSTensor.groundSpace_inTailGround","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K L : Nat) ψ : MPSTensor.NSiteSpace d (HAdd.hAdd K L), Members…","labels":[],"detail_key":"p8","name":"MPSTensor.groundSpace_inTailGround","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14066","layer":"formal","project":"p8","title":"MPSTensor.restrictFirst_groundSpaceMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) L : Nat (i : Fin d) (X : Matrix (Fin D) (Fin D) Complex), Eq (M…","labels":[],"detail_key":"p8","name":"MPSTensor.restrictFirst_groundSpaceMap","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14067","layer":"formal","project":"p8","title":"MPSTensor.restrictLast_groundSpaceMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) L : Nat (j : Fin d) (X : Matrix (Fin D) (Fin D) Complex), Eq (M…","labels":[],"detail_key":"p8","name":"MPSTensor.restrictLast_groundSpaceMap","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14068","layer":"formal","project":"p8","title":"MPSTensor.tailRestrictₗ","kind":"def","summary":"d K L : Nat → (Fin K → Fin d) → LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d (HAdd.hA…","labels":[],"detail_key":"p8","name":"MPSTensor.tailRestrictₗ","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14069","layer":"formal","project":"p8","title":"MPSTensor.tailRestrictₗ_groundSpaceMap","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) K L : Nat (u : Fin K → Fin d) (X : Matrix (Fin D) (Fin D) Compl…","labels":[],"detail_key":"p8","name":"MPSTensor.tailRestrictₗ_groundSpaceMap","module":"TNLean.MPS.ParentHamiltonian.SuffixWindow"},{"id":"n14070","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.tailVirtualMapES_norm_four_pow_uniform","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.tailVirtualMapES_norm_four_pow_uniform","module":"TNLean.MPS.ParentHamiltonian.TailVirtualGram"},{"id":"n14071","layer":"formal","project":"p8","title":"MPSTensor.IsPrimitiveMPS.tailVirtualMapES_norm_uniform","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] A : MPSTensor d D ρ : Matrix (Fin D) (Fin D) Complex, A.IsPrimiti…","labels":[],"detail_key":"p8","name":"MPSTensor.IsPrimitiveMPS.tailVirtualMapES_norm_uniform","module":"TNLean.MPS.ParentHamiltonian.TailVirtualGram"},{"id":"n14072","layer":"formal","project":"p8","title":"TripartiteDecorrelation.CommutingParentHamiltonian","kind":"structure","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.CommutingParentHamiltonian","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14073","layer":"formal","project":"p8","title":"TripartiteDecorrelation.CommutingParentHamiltonian.hproduct","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.CommutingParentHamiltonian.hproduct","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14074","layer":"formal","project":"p8","title":"TripartiteDecorrelation.CommutingParentHamiltonian.isDecorrelated","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.CommutingParentHamiltonian.isDecorrelated","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14075","layer":"formal","project":"p8","title":"TripartiteDecorrelation.HasCommutingParentHamiltonian","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.HasCommutingParentHamiltonian","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14076","layer":"formal","project":"p8","title":"TripartiteDecorrelation.HasGroundSpaceIntersection","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.HasGroundSpaceIntersection","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14077","layer":"formal","project":"p8","title":"TripartiteDecorrelation.IsDecorrelated.hasCommutingParentHamiltonian","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.IsDecorrelated.hasCommutingParentHamiltonian","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14078","layer":"formal","project":"p8","title":"TripartiteDecorrelation.hasGroundSpaceIntersection_iff_product_eq","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.hasGroundSpaceIntersection_iff_product_eq","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14079","layer":"formal","project":"p8","title":"TripartiteDecorrelation.parentHamiltonian_iff_decorrelated","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.parentHamiltonian_iff_decorrelated","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14080","layer":"formal","project":"p8","title":"TripartiteDecorrelation.parentHamiltonian_iff_observableDecorrelated","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"TripartiteDecorrelation.parentHamiltonian_iff_observableDecorrelated","module":"TNLean.MPS.ParentHamiltonian.TripartiteDecorrelation"},{"id":"n14081","layer":"formal","project":"p8","title":"MPSTensor.HasUniqueGroundState","kind":"def","summary":"V : Type u_1 → [inst : AddCommGroup V] → [inst_1 : Module Complex V] → Submodule Complex V → Pr…","labels":[],"detail_key":"p8","name":"MPSTensor.HasUniqueGroundState","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14082","layer":"formal","project":"p8","title":"MPSTensor.chainGroundSpace_eq_mpvSubmodule_normal","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D], Kraus.IsNormal A → ∀ L₀ : Nat, Kraus.IsNBlkInjective…","labels":[],"detail_key":"p8","name":"MPSTensor.chainGroundSpace_eq_mpvSubmodule_normal","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14083","layer":"formal","project":"p8","title":"MPSTensor.chainGroundSpace_le_groundSpace_of_isNBlkInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D] L₀ L N : Nat, Kraus.IsNBlkInjective A L₀ → LT.lt 0 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LT…","labels":[],"detail_key":"p8","name":"MPSTensor.chainGroundSpace_le_mpvSubmodule_of_isNBlkInjective_of_wrapped_witness_comparison","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14086","layer":"formal","project":"p8","title":"MPSTensor.chainGroundSpace_le_mpvSubmodule_of_normal_range_reduction","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D], Kraus.IsNormal A → ∀ L₀ : Nat, Kraus.IsNBlkInjective…","labels":[],"detail_key":"p8","name":"MPSTensor.chainGroundSpace_le_mpvSubmodule_of_normal_range_reduction","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14087","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_closing_product_eq_of_chainGroundSpace","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D] L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : 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1)…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_boundary_restriction_eq_of_fixed_boundary_letters","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14090","layer":"formal","project":"p8","title":"MPSTensor.closure_property_boundary_right_products_eq_of_chainGroundSpace","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D] L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : LT.l…","labels":[],"detail_key":"p8","name":"MPSTensor.closure_property_boundary_right_products_eq_of_chainGroundSpace","module":"TNLean.MPS.ParentHamiltonian.UniqueGroundState"},{"id":"n14091","layer":"formal","project":"p8","title":"MPSTensor.closure_property_fixed_boundary_letter_eq_of_chainGroundSpace","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D [NeZero D] L₀ M : Nat, Kraus.IsNBlkInjective A L₀ → ∀ (hL₀ : 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(Fin D) Complex) (hρ : ρ.PosDef) (X : Matrix (Fin D) (Fin D) Comp…","labels":[],"detail_key":"p8","name":"Matrix.rhoWeighted_norm_sq","module":"TNLean.MPS.ParentHamiltonian.WeightedVirtualHilbert"},{"id":"n14111","layer":"formal","project":"p8","title":"MPSTensor.inverseGram_reassocTailBoundaryMapES_eq_fiberwise_inverseGram","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K L Q : Nat) (h : Function.Injective ⇑(A.groundSpaceMapES (HAd…","labels":[],"detail_key":"p8","name":"MPSTensor.inverseGram_reassocTailBoundaryMapES_eq_fiberwise_inverseGram","module":"TNLean.MPS.ParentHamiltonian.WholeIncrementCorrectionBounds"},{"id":"n14112","layer":"formal","project":"p8","title":"MPSTensor.norm_reassocTailBoundaryMapES_comp_inverseGram_le_sqrt","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (K L Q : Nat) (h : Function.Injective ⇑(A.groundSpaceMapES 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d D : Nat (M : Matrix (Fin d) (Fin d) Complex) (A : MPSTensor d D) (hA : A.IsIrreducibleForm)…","labels":[],"detail_key":"p8","name":"MPSTensor.zGaugeEquiv_of_isIrreducibleForm_sameMPV_rotatePhysical","module":"TNLean.MPS.Periodic.Applications"},{"id":"n14149","layer":"formal","project":"p8","title":"MPSTensor.cornerProd_blockMatch_pow","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero m] (P Q : Fin m → MatrixAlg D) (A B : MPSTensor d D) (q : Fin m) (…","labels":[],"detail_key":"p8","name":"MPSTensor.cornerProd_blockMatch_pow","module":"TNLean.MPS.Periodic.CornerContraction"},{"id":"n14150","layer":"formal","project":"p8","title":"MPSTensor.cornerProd_contraction","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero m] (P : Fin m → MatrixAlg D) (A : MPSTensor d D) (L : Nat) (Ω : 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:…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_equalCase_sectorDecomposition","module":"TNLean.MPS.Periodic.EqualCaseGlobal"},{"id":"n14173","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.toHetRepeatedBlocks","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → A.HetRepeatedBlocks B","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.toHetRepeatedBlocks","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14174","layer":"formal","project":"p8","title":"MPSTensor.HetRepeatedBlocks.exists_phase_rescaling_sameMPV₂Pos","kind":"theorem","summary":"∀ d D₁ D₂ : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, A.HetRepeatedBlocks B → Exists fun ξ =>…","labels":[],"detail_key":"p8","name":"MPSTensor.HetRepeatedBlocks.exists_phase_rescaling_sameMPV₂Pos","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14175","layer":"formal","project":"p8","title":"MPSTensor.HetRepeatedBlocks.exists_unit_phase_power_mpv","kind":"theorem","summary":"∀ d D₁ D₂ : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, A.HetRepeatedBlocks B → Exists fun ζ =>…","labels":[],"detail_key":"p8","name":"MPSTensor.HetRepeatedBlocks.exists_unit_phase_power_mpv","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14176","layer":"formal","project":"p8","title":"MPSTensor.IsSpectrallyPeriodic.period_eq_of_hetRepeatedBlocks","kind":"theorem","summary":"∀ d D₁ D₂ m n : Nat A : MPSTensor d D₁ B : MPSTensor d D₂, MPSTensor.IsSpectrallyPeriodic m A →…","labels":[],"detail_key":"p8","name":"MPSTensor.IsSpectrallyPeriodic.period_eq_of_hetRepeatedBlocks","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14177","layer":"formal","project":"p8","title":"MPSTensor.PeriodicOverlapHypothesis.of_gaugeEquiv","kind":"theorem","summary":"∀ d rA rB : Nat dimA : Fin rA → Nat dimB : Fin rB → Nat A A' : (j : Fin rA) → MPSTensor d (dimA…","labels":[],"detail_key":"p8","name":"MPSTensor.PeriodicOverlapHypothesis.of_gaugeEquiv","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14178","layer":"formal","project":"p8","title":"MPSTensor.RepeatedBlocks.peripheralEigenvalues_transferMap_eq","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.RepeatedBlocks B → Eq (peripheralEigenvalues (Kraus.transfer…","labels":[],"detail_key":"p8","name":"MPSTensor.RepeatedBlocks.peripheralEigenvalues_transferMap_eq","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14179","layer":"formal","project":"p8","title":"MPSTensor.equalCase_zgauge_of_power_sums","kind":"theorem","summary":"∀ r : Nat (m : Nat) (μ ν : Fin r → Complex), (∀ (i : Fin r), Ne (ν i) 0) → (∀ (i : Fin r), Eq (…","labels":[],"detail_key":"p8","name":"MPSTensor.equalCase_zgauge_of_power_sums","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14180","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_periodic_equalCase","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) (hA : A.IsIrreducibleForm) (hB : B.Is…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_equalCase","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14181","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_periodic_equalCase_matching","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (A : MPSTensor d D₁) (B : MPSTensor d D₂) (hA : A.IsIrreducibleForm) (hB : B.Is…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_equalCase_matching","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14182","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_periodic_proportional","kind":"theorem","summary":"∀ d rA rB : Nat dimA : Fin rA → Nat dimB : Fin rB → Nat (A : (j : Fin rA) → MPSTensor d (dimA j…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_proportional","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14183","layer":"formal","project":"p8","title":"MPSTensor.peripheralProportionalCase_periodicFT_of_sameMPV₂Pos","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂) m_a m_b : Nat…","labels":[],"detail_key":"p8","name":"MPSTensor.peripheralProportionalCase_periodicFT_of_sameMPV₂Pos","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14184","layer":"formal","project":"p8","title":"MPSTensor.zgauge_construction","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (m : Nat) (μ ν : n → Complex), (∀ (i…","labels":[],"detail_key":"p8","name":"MPSTensor.zgauge_construction","module":"TNLean.MPS.Periodic.FundamentalTheorem"},{"id":"n14185","layer":"formal","project":"p8","title":"MPSTensor.globalGauge","kind":"def","summary":"D m : Nat → (Fin m → Matrix (Fin D) (Fin D) Complex) → (Fin m → Matrix (Fin D) (Fin D) Complex)…","labels":[],"detail_key":"p8","name":"MPSTensor.globalGauge","module":"TNLean.MPS.Periodic.GlobalGauge"},{"id":"n14186","layer":"formal","project":"p8","title":"MPSTensor.globalGauge_eq_sum","kind":"theorem","summary":"∀ D m : Nat [NeZero m] P Q U : Fin m → Matrix (Fin D) (Fin D) Complex q : Fin m, (∀ (v : Fin m)…","labels":[],"detail_key":"p8","name":"MPSTensor.globalGauge_eq_sum","module":"TNLean.MPS.Periodic.GlobalGauge"},{"id":"n14187","layer":"formal","project":"p8","title":"MPSTensor.repeatedBlocks_of_globalGauge","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero m] A B : MPSTensor d D P Q U : Fin m → Matrix (Fin D) (Fin D) Comp…","labels":[],"detail_key":"p8","name":"MPSTensor.repeatedBlocks_of_globalGauge","module":"TNLean.MPS.Periodic.GlobalGauge"},{"id":"n14188","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_periodic_equalCase_derivedPeriods","kind":"theorem","summary":"∀ d : Nat (P Q : MPSTensor.SectorDecomposition d), (∀ (j : Fin P.basisCount), Kraus.IsIrreducib…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_equalCase_derivedPeriods","module":"TNLean.MPS.Periodic.IrreducibleFormPeriods"},{"id":"n14189","layer":"formal","project":"p8","title":"MPSTensor.fundamentalTheorem_periodic_proportional_irreducibleForm","kind":"theorem","summary":"∀ d : Nat (P Q : MPSTensor.SectorDecomposition d), (∀ (j : Fin P.basisCount), Kraus.IsIrreducib…","labels":[],"detail_key":"p8","name":"MPSTensor.fundamentalTheorem_periodic_proportional_irreducibleForm","module":"TNLean.MPS.Periodic.IrreducibleFormPeriods"},{"id":"n14190","layer":"formal","project":"p8","title":"MPSTensor.toIsIrreducibleFormOfPhaseNormalized","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → μ : Fin r → Complex → blocks : (k : Fin r) → MPSTensor d (dim k…","labels":[],"detail_key":"p8","name":"MPSTensor.toIsIrreducibleFormOfPhaseNormalized","module":"TNLean.MPS.Periodic.NormalCanonicalPeriodOne"},{"id":"n14191","layer":"formal","project":"p8","title":"MPSTensor.toIsIrreducibleFormOfPhaseNormalized_period_eq_one","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat μ : Fin r → Complex blocks : (k : Fin r) → MPSTensor d (dim k) (h…","labels":[],"detail_key":"p8","name":"MPSTensor.toIsIrreducibleFormOfPhaseNormalized_period_eq_one","module":"TNLean.MPS.Periodic.NormalCanonicalPeriodOne"},{"id":"n14192","layer":"formal","project":"p8","title":"MPSTensor.IsSpectrallyPeriodic.exists_isPeriodic_tpGauge","kind":"theorem","summary":"∀ d D m : Nat A : MPSTensor d D, MPSTensor.IsSpectrallyPeriodic m A → Exists fun σ => And σ.Pos…","labels":[],"detail_key":"p8","name":"MPSTensor.IsSpectrallyPeriodic.exists_isPeriodic_tpGauge","module":"TNLean.MPS.Periodic.Normalization"},{"id":"n14193","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.basis_isNBlkInjective_totalDim_pow_four","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (j : Fin 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=>…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.exists_common_basis_isNBlkInjective","module":"TNLean.MPS.Periodic.NormalizedSelfOverlap"},{"id":"n14196","layer":"formal","project":"p8","title":"MPSTensor.isNormal_of_irreducible_leftCanonical_selfOverlap_tendsto_one","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → A.IsLeftCanonical → F…","labels":[],"detail_key":"p8","name":"MPSTensor.isNormal_of_irreducible_leftCanonical_selfOverlap_tendsto_one","module":"TNLean.MPS.Periodic.NormalizedSelfOverlap"},{"id":"n14197","layer":"formal","project":"p8","title":"MPSTensor.isPrimitive_and_isNormal_of_irreducible_leftCanonical_selfOverlap_tendsto_one","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D), Kraus.IsIrreducibleFamily A → A.IsLeftCanonical → F…","labels":[],"detail_key":"p8","name":"MPSTensor.isPrimitive_and_isNormal_of_irreducible_leftCanonical_selfOverlap_tendsto_one","module":"TNLean.MPS.Periodic.NormalizedSelfOverlap"},{"id":"n14198","layer":"formal","project":"p8","title":"MPSTensor.periodicBasis_eventuallyLinearlyIndependent","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (A : (k : Fin r) → MPSTensor d (dim k)) (period : Fin r → Nat), (…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicBasis_eventuallyLinearlyIndependent","module":"TNLean.MPS.Periodic.Overlap.Dichotomy"},{"id":"n14199","layer":"formal","project":"p8","title":"MPSTensor.periodicOverlapDichotomy","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂) m_a m_b : Nat…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicOverlapDichotomy","module":"TNLean.MPS.Periodic.Overlap.Dichotomy"},{"id":"n14200","layer":"formal","project":"p8","title":"MPSTensor.periodicOverlap_tendsto_zero_of_ne_period","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂) m_a m_b : Nat…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicOverlap_tendsto_zero_of_ne_period","module":"TNLean.MPS.Periodic.Overlap.DifferentPeriod"},{"id":"n14201","layer":"formal","project":"p8","title":"MPSTensor.repeatedBlocks_of_gaugePhaseData_norm_one","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D (X : Matrix.GeneralLinearGroup (Fin D) Complex) ζ : Complex, Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.repeatedBlocks_of_gaugePhaseData_norm_one","module":"TNLean.MPS.Periodic.Overlap.GaugePhase"},{"id":"n14202","layer":"formal","project":"p8","title":"MPSTensor.periodicOverlap_tendsto_zero_of_no_sector_match","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A B : MPSTensor d D) m : Nat [inst_1 : NeZero m], MPSTensor.IsPe…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicOverlap_tendsto_zero_of_no_sector_match","module":"TNLean.MPS.Periodic.Overlap.NoSectorMatch"},{"id":"n14203","layer":"formal","project":"p8","title":"MPSTensor.sectorBlocked_isNormal_of_isPeriodic","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A : MPSTensor d D) m : Nat [inst_1 : NeZero m], MPSTensor.IsPeri…","labels":[],"detail_key":"p8","name":"MPSTensor.sectorBlocked_isNormal_of_isPeriodic","module":"TNLean.MPS.Periodic.Overlap.NoSectorMatch"},{"id":"n14204","layer":"formal","project":"p8","title":"MPSTensor.periodicBasis_nonzero_subfamily_eventuallyLinearlyIndependent","kind":"theorem","summary":"∀ d g : Nat dim : Fin g → Nat (A : (j : Fin g) → MPSTensor d (dim j)) (period : Fin g → Nat), (…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicBasis_nonzero_subfamily_eventuallyLinearlyIndependent","module":"TNLean.MPS.Periodic.Overlap.NonzeroSubfamily"},{"id":"n14205","layer":"formal","project":"p8","title":"MPSTensor.periodicOverlap_gaugeEquiv_of_sector_match","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A B : MPSTensor d D) m : Nat [inst_1 : NeZero m], MPSTensor.IsPe…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicOverlap_gaugeEquiv_of_sector_match","module":"TNLean.MPS.Periodic.Overlap.SectorMatch.Consequences"},{"id":"n14206","layer":"formal","project":"p8","title":"MPSTensor.periodicOverlap_tendsto_zero_of_ne_dim","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂) m_a m_b : Nat…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicOverlap_tendsto_zero_of_ne_dim","module":"TNLean.MPS.Periodic.Overlap.SectorMatch.Consequences"},{"id":"n14207","layer":"formal","project":"p8","title":"MPSTensor.sectorTensor_proportional_of_blockedMatch","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A B : MPSTensor d D) m : Nat [inst_1 : NeZero m], A.IsLeftCanoni…","labels":[],"detail_key":"p8","name":"MPSTensor.sectorTensor_proportional_of_blockedMatch","module":"TNLean.MPS.Periodic.Overlap.SectorMatch.Contraction"},{"id":"n14208","layer":"formal","project":"p8","title":"MPSTensor.sectorMatch_propagation","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A B : MPSTensor d D) m : Nat [inst_1 : NeZero m], MPSTensor.IsPe…","labels":[],"detail_key":"p8","name":"MPSTensor.sectorMatch_propagation","module":"TNLean.MPS.Periodic.Overlap.SectorMatch.Propagation"},{"id":"n14209","layer":"formal","project":"p8","title":"MPSTensor.sectorOverlap_succ_eq_of_cyclicSectorDecomp","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero D] [inst_1 : NeZero m] (A B : MPSTensor d D), A.IsLeftCanonical →…","labels":[],"detail_key":"p8","name":"MPSTensor.sectorOverlap_succ_eq_of_cyclicSectorDecomp","module":"TNLean.MPS.Periodic.Overlap.SectorOverlapTransport"},{"id":"n14210","layer":"formal","project":"p8","title":"MPSTensor.periodicSelfOverlap_tendsto","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D) m : Nat, MPSTensor.IsPeriodic m A → Filter.Tendsto (…","labels":[],"detail_key":"p8","name":"MPSTensor.periodicSelfOverlap_tendsto","module":"TNLean.MPS.Periodic.Overlap.SelfOverlap"},{"id":"n14211","layer":"formal","project":"p8","title":"MPSTensor.offDiag_eigenvector_eq_zero_of_isPeriodic","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D) m : Nat [inst : NeZero m], MPSTensor.IsPeriodic m A…","labels":[],"detail_key":"p8","name":"MPSTensor.offDiag_eigenvector_eq_zero_of_isPeriodic","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapNonrep"},{"id":"n14212","layer":"formal","project":"p8","title":"MPSTensor.IsCyclicSectorDecomp","kind":"def","summary":"d D m : Nat → [NeZero D] → [NeZero m] → MPSTensor d D → dim : Fin m → Nat → ((k : Fin m) → MPST…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCyclicSectorDecomp","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14213","layer":"formal","project":"p8","title":"MPSTensor.IsCyclicSectorDecomp.eq_sum_offDiag","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero D] [inst_1 : NeZero m] A : MPSTensor d D, MPSTensor.IsPeriodic m A…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCyclicSectorDecomp.eq_sum_offDiag","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14214","layer":"formal","project":"p8","title":"MPSTensor.IsCyclicSectorDecomp.offDiag_shift","kind":"theorem","summary":"∀ d D m : Nat [inst : NeZero D] [inst_1 : NeZero m] A : MPSTensor d D, MPSTensor.IsPeriodic m A…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCyclicSectorDecomp.offDiag_shift","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14215","layer":"formal","project":"p8","title":"MPSTensor.IsCyclicSectorDecompWith","kind":"def","summary":"d D m : Nat → [NeZero D] → [NeZero m] → MPSTensor d D → dim : Fin m → Nat → ((k : Fin m) → MPST…","labels":[],"detail_key":"p8","name":"MPSTensor.IsCyclicSectorDecompWith","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14216","layer":"formal","project":"p8","title":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_of_isPeriodic","kind":"theorem","summary":"∀ d D : Nat [inst : NeZero D] (A : MPSTensor d D) m : Nat [inst_1 : NeZero m], MPSTensor.IsPeri…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_cyclic_sector_decomp_after_blocking_of_isPeriodic","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14217","layer":"formal","project":"p8","title":"MPSTensor.exists_cyclic_sector_decomp_with_letter_after_blocking_of_isPeriodic","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D) m : Nat (hP : MPSTensor.IsPeriodic m A), Exists fun…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_cyclic_sector_decomp_with_letter_after_blocking_of_isPeriodic","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14218","layer":"formal","project":"p8","title":"MPSTensor.exists_cyclic_sector_decomp_with_letter_and_isometry_after_blocking_of_isPeriod…","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A : MPSTensor d D) m : Nat (hP : MPSTensor.IsPeriodic m A), Exists fun…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_cyclic_sector_decomp_with_letter_and_isometry_after_blocking_of_isPeriodic","module":"TNLean.MPS.Periodic.Overlap.SelfOverlapSetup"},{"id":"n14219","layer":"formal","project":"p8","title":"MPSTensor.exists_isSpectrallyPeriodic_of_irreducible_of_spectralRadius_one","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, Kraus.IsIrreducibleFamily A → Eq (spectralRadius Complex ((Modul…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_isSpectrallyPeriodic_of_irreducible_of_spectralRadius_one","module":"TNLean.MPS.Periodic.PeriodExistence"},{"id":"n14220","layer":"formal","project":"p8","title":"MPSTensor.PeriodicProjectiveRigidity","kind":"def","summary":"d D : Nat → G : Type u_1 → [inst : Group G] → MPSTensor d D → MonoidHom G (Matrix (Fin d) (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.PeriodicProjectiveRigidity","module":"TNLean.MPS.Periodic.ProjectiveRep"},{"id":"n14221","layer":"formal","project":"p8","title":"MPSTensor.cor_4_1_projective_rep","kind":"theorem","summary":"∀ d D : Nat G : Type u_1 [inst : Group G] (A : MPSTensor d D) (U : MonoidHom G (Matrix (Fin d)…","labels":[],"detail_key":"p8","name":"MPSTensor.cor_4_1_projective_rep","module":"TNLean.MPS.Periodic.ProjectiveRep"},{"id":"n14222","layer":"formal","project":"p8","title":"MPSTensor.cor_4_1_projective_rep_cocycle","kind":"theorem","summary":"∀ d D : Nat G : Type u_1 [inst : Group G] (A : MPSTensor d D) (U : MonoidHom G (Matrix (Fin d)…","labels":[],"detail_key":"p8","name":"MPSTensor.cor_4_1_projective_rep_cocycle","module":"TNLean.MPS.Periodic.ProjectiveRep"},{"id":"n14223","layer":"formal","project":"p8","title":"MPSTensor.projectiveRep_of_symmetry","kind":"theorem","summary":"∀ d D : Nat G : Type u_1 [inst : Group G] (A : MPSTensor d D) (U : 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(hStruct.leftVirtualBon…","labels":[],"detail_key":"p8","name":"MPSTensor.AppendixBStructuralData.leftVirtualBondProjection_comp_right","module":"TNLean.MPS.RFP.AppendixBVirtualBondSupport"},{"id":"n14336","layer":"formal","project":"p8","title":"MPSTensor.AppendixBStructuralData.rightVirtualBondProjection","kind":"def","summary":"d D : Nat → A : MPSTensor d D → A.AppendixBStructuralData → LinearMap (RingHom.id Complex) ((Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.AppendixBStructuralData.rightVirtualBondProjection","module":"TNLean.MPS.RFP.AppendixBVirtualBondSupport"},{"id":"n14337","layer":"formal","project":"p8","title":"MPSTensor.AppendixBStructuralData.twoSiteBondInsertion","kind":"def","summary":"d D : Nat → A : MPSTensor d D → A.AppendixBStructuralData → LinearMap (RingHom.id Complex) 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(MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isBNT_basisDirectSum","module":"TNLean.MPS.RFP.BNTDirectSumBasis"},{"id":"n14340","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isCPSVBasisOfNormalTensors_basisDirectSum","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isCPSVBasisOfNormalTensors_basisDirectSum","module":"TNLean.MPS.RFP.BNTDirectSumBasis"},{"id":"n14341","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isCPSVCanonicalForm_basisDirectSum","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isCPSVCanonicalForm_basisDirectSum","module":"TNLean.MPS.RFP.BNTDirectSumBasis"},{"id":"n14342","layer":"formal","project":"p8","title":"MPSTensor.blockDiagonal'_transferSum_toBlock","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)) (X : Matrix (Sigma fun k…","labels":[],"detail_key":"p8","name":"MPSTensor.blockDiagonal'_transferSum_toBlock","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14343","layer":"formal","project":"p8","title":"MPSTensor.blockTransferSum_blockTransferSum_eq_smul","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)) (c : Complex), Eq ((Kraus…","labels":[],"detail_key":"p8","name":"MPSTensor.blockTransferSum_blockTransferSum_eq_smul","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14344","layer":"formal","project":"p8","title":"MPSTensor.blockTransferSum_idempotent_of_pairwise_mixedMapLM","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)), (∀ (j j' : Fin r), IsIde…","labels":[],"detail_key":"p8","name":"MPSTensor.blockTransferSum_idempotent_of_pairwise_mixedMapLM","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14345","layer":"formal","project":"p8","title":"MPSTensor.isBNTLocallyOrthogonal_of_isTransferIdempotent_directSum","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)) [∀ (k : Fin r), NeZero (d…","labels":[],"detail_key":"p8","name":"MPSTensor.isBNTLocallyOrthogonal_of_isTransferIdempotent_directSum","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14346","layer":"formal","project":"p8","title":"MPSTensor.isPositiveGapBNTZCL_of_isTransferIdempotent_directSum","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)) [∀ (k : Fin r), NeZero (d…","labels":[],"detail_key":"p8","name":"MPSTensor.isPositiveGapBNTZCL_of_isTransferIdempotent_directSum","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14347","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_directSumTensor_iff_pairwise_mixedMapLM_isIdempotentElem","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (B : (k : Fin r) → MPSTensor d (d…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_directSumTensor_iff_pairwise_mixedMapLM_isIdempotentElem","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14348","layer":"formal","project":"p8","title":"MPSTensor.phases_eq_of_isTransferIdempotent_directSum_scaled_self","kind":"theorem","summary":"∀ d r D : Nat [NeZero D] (A : MPSTensor d D), A.IsTransferIdempotent → Ne (Kraus.transferMap A)…","labels":[],"detail_key":"p8","name":"MPSTensor.phases_eq_of_isTransferIdempotent_directSum_scaled_self","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14349","layer":"formal","project":"p8","title":"MPSTensor.transferMap_directSumTensor_reindex","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)) (Y : Matrix (Sigma fun k…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_directSumTensor_reindex","module":"TNLean.MPS.RFP.BNTOrthogonality"},{"id":"n14350","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_counterexample_to_unrestricted_zcl_iff_rfp","kind":"theorem","summary":"And (MPSTensor.IsBNTCanonicalForm (MPSTensor.SectorBNT.Examples.halvedDecomp MPSTensor.scalarUn…","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_counterexample_to_unrestricted_zcl_iff_rfp","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14351","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_eq_halvedDecomp_toTensor","kind":"theorem","summary":"Eq MPSTensor.halvedWeightTensor (MPSTensor.SectorBNT.Examples.halvedDecomp MPSTensor.scalarUnit…","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_eq_halvedDecomp_toTensor","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14352","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_isPhysicalBNTZCL","kind":"theorem","summary":"MPSTensor.halvedWeightTensor.IsPhysicalBNTZCL fun x => MPSTensor.scalarUnitTensor","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_isPhysicalBNTZCL","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14353","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_mpv","kind":"theorem","summary":"∀ N : Nat (σ : Fin N → Fin 1), Eq (MPSTensor.halvedWeightTensor.mpv σ) (HAdd.hAdd 1 (HPow.hPow…","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_mpv","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14354","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_not_isTransferIdempotent","kind":"theorem","summary":"Not MPSTensor.halvedWeightTensor.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_not_isTransferIdempotent","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14355","layer":"formal","project":"p8","title":"MPSTensor.halvedWeightTensor_sameMPV₂_halvedDecomp","kind":"theorem","summary":"MPSTensor.halvedWeightTensor.SameMPV₂ (MPSTensor.SectorBNT.Examples.halvedDecomp MPSTensor.scal…","labels":[],"detail_key":"p8","name":"MPSTensor.halvedWeightTensor_sameMPV₂_halvedDecomp","module":"TNLean.MPS.RFP.BNTWeightCounterexample"},{"id":"n14356","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_card_loop","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData), (Exists fun c => Exists fun N₀ => ∀…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_card_loop","module":"TNLean.MPS.RFP.BeigiEventuallyConstantSectorGraph"},{"id":"n14357","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_card_loop_of_has…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d (F : P.toTensor.BeigiSectorGraphData), MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_card_loop_of_hasBNTSectorData","module":"TNLean.MPS.RFP.BeigiEventuallyConstantSectorGraph"},{"id":"n14358","layer":"formal","project":"p8","title":"MPSTensor.HasBNTSectorData.eventually_parentHamiltonian_groundSpace_finrank_eq","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.HasBNTSectorData P → P.toTensor.HasNNC…","labels":[],"detail_key":"p8","name":"MPSTensor.HasBNTSectorData.eventually_parentHamiltonian_groundSpace_finrank_eq","module":"TNLean.MPS.RFP.BeigiGroundSpaceDimension"},{"id":"n14359","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_of_hasNNCPHGroundSpaces","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_of_hasNNCPHGroundSpaces","module":"TNLean.MPS.RFP.BeigiLoopBNTIdentification"},{"id":"n14360","layer":"formal","project":"p8","title":"MPSTensor.nncph_implies_rfp","kind":"theorem","summary":"∀ d D : Nat (B : MPSTensor d D) [NeZero D] r : Nat dim : Fin r → Nat (A : (j : Fin r) → MPSTens…","labels":[],"detail_key":"p8","name":"MPSTensor.nncph_implies_rfp","module":"TNLean.MPS.RFP.BeigiLoopBNTIdentification"},{"id":"n14361","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondNorm","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondNorm","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14362","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondNorm_eq_frobeniusNorm","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondNorm_eq_frobeniusNorm","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14363","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondNorm_ne_zero","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondNorm_ne_zero","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14364","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondNorm_pos","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondNorm_pos","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14365","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mpv_normalizedMinimalLoopTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mpv_normalizedMinimalLoopTensor","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14366","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopCoordinateTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14367","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopCoordinateTensor_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopCoordinateTensor_isTransferIdempotent","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14368","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14369","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isInjective","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14370","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isNormal","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isNormal","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14371","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_isTransferIdempotent","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14372","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_mixedMapLM_eq_zero_of_ne","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) l m : FiniteWeightedDigraph.Loop F.e…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.normalizedMinimalLoopTensor_mixedMapLM_eq_zero_of_ne","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14373","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14374","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor_comp_self","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.transferMap_minimalLoopCoordinateTensor_comp_self","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14375","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.transferMap_normalizedMinimalLoopCoordinateTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.transferMap_normalizedMinimalLoopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopFixedPoint"},{"id":"n14376","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_isInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_isInjective","module":"TNLean.MPS.RFP.BeigiLoopInjectivity"},{"id":"n14377","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor_isInjective","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor_isInjective","module":"TNLean.MPS.RFP.BeigiLoopInjectivity"},{"id":"n14378","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor_isNormal","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor_isNormal","module":"TNLean.MPS.RFP.BeigiLoopInjectivity"},{"id":"n14379","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.LoopSchmidtFactorization","kind":"inductive","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.LoopSchmidtFactorization","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14380","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopSchmidtFactorization","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopSchmidtFactorization","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14381","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopSchmidtRank","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopSchmidtRank","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14382","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopSchmidtRank_pos","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopSchmidtRank_pos","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14383","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14384","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_apply","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_apply","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14385","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_ne","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopCoordinateTensor_sector_ne","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14386","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.minimalLoopTensor","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14387","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mpv_minimalLoopCoordinateTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mpv_minimalLoopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14388","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mpv_minimalLoopTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mpv_minimalLoopTensor","module":"TNLean.MPS.RFP.BeigiLoopSchmidtSupport"},{"id":"n14389","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondVector","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondVector","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14390","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondVector_mem","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondVector_mem","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14391","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopBondVector_ne_zero","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopBondVector_ne_zero","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14392","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopCoordinateTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14393","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14394","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct_inner_eq_zero_of_ne","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) l m : FiniteWeightedDigraph.Loop F.e…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct_inner_eq_zero_of_ne","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14395","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct_ne_zero","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopCyclicProduct_ne_zero","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14396","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductState","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductState","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14397","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductStateES","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → FiniteWeightedDigraph.Loop F.edg…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductStateES","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14398","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductStateES_inner_eq_zero_of_ne","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) l m : FiniteWeightedDigraph.Loop F.e…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductStateES_inner_eq_zero_of_ne","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14399","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductStateES_linearIndependent","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) N : Nat [inst : NeZero N], LinearInd…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductStateES_linearIndependent","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14400","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductStateES_ne_zero","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductStateES_ne_zero","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14401","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductState_mem_ker_parentHamiltonian","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductState_mem_ker_parentHamiltonian","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14402","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductState_mem_parentHamiltonianGroundSpaceES","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductState_mem_parentHamiltonianGroundSpaceES","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14403","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopProductState_ne_zero","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopProductState_ne_zero","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14404","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.loopTensor","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (l : FiniteWeightedDigraph.Loop…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.loopTensor","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14405","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mpv_loopCoordinateTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mpv_loopCoordinateTensor","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14406","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mpv_loopTensor","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) (l : FiniteWeightedDigraph.Loop F.ed…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mpv_loopTensor","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14407","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData), (Exists fun c => Exists fun N₀ => ∀…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14408","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES_…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d (F : P.toTensor.BeigiSectorGraphData), MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.span_loopProductStateES_eq_parentHamiltonianGroundSpaceES_of_hasBNTSectorData","module":"TNLean.MPS.RFP.BeigiLoopTensor"},{"id":"n14409","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData","kind":"inductive","summary":"d D : Nat → MPSTensor d D → Type","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14410","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.IsEdge","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → Fin F.sectorCount → Fin F.sector…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.IsEdge","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14411","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.OrderedCycle","kind":"def","summary":"d D : Nat → A : MPSTensor d D → A.BeigiSectorGraphData → (N : Nat) → [NeZero N] → Type","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.OrderedCycle","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14412","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.edgeGroundSpace","kind":"def","summary":"d D : Nat → A : MPSTensor d D → (F : A.BeigiSectorGraphData) → (a b : Fin F.sectorCount) → Subm…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.edgeGroundSpace","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14413","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.mem_ker_parentHamiltonian_of_groundBondProduct_mulVec_eq_s…","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) N : Nat [NeZero N] (hN : LE.le 2 N)…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.mem_ker_parentHamiltonian_of_groundBondProduct_mulVec_eq_self","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14414","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) N : Nat (hN : LE.le 2 N), Eq (Module…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14415","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_two","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData), Eq (Module.finrank Complex (Subtype…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.parentHamiltonianGroundSpaceES_finrank_eq_two","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14416","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.singleKrausMap_groundBondProduct","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData) N : Nat [NeZero N] (hN : LE.le 2 N),…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.singleKrausMap_groundBondProduct","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14417","layer":"formal","project":"p8","title":"MPSTensor.BeigiSectorGraphData.unitary_mul_conjTranspose","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (F : A.BeigiSectorGraphData), Eq (HMul.hMul F.unitary F.unitary.c…","labels":[],"detail_key":"p8","name":"MPSTensor.BeigiSectorGraphData.unitary_mul_conjTranspose","module":"TNLean.MPS.RFP.BeigiSectorGraph"},{"id":"n14418","layer":"formal","project":"p8","title":"MPSTensor.IsNNCPH.beigiSectorGraphData","kind":"def","summary":"d D : Nat → A : MPSTensor d D → A.IsNNCPH 3 → A.BeigiSectorGraphData","labels":[],"detail_key":"p8","name":"MPSTensor.IsNNCPH.beigiSectorGraphData","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14419","layer":"formal","project":"p8","title":"MPSTensor.IsNNCPH.nonempty_beigiSectorGraphData","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNNCPH 3 → Nonempty A.BeigiSectorGraphData","labels":[],"detail_key":"p8","name":"MPSTensor.IsNNCPH.nonempty_beigiSectorGraphData","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14420","layer":"formal","project":"p8","title":"MPSTensor.IsNNCPH.twoSiteParentGroundProjectorMatrix_overlappingLifts_commute","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNNCPH 3 → Eq (HMul.hMul A.twoSiteParentGroundProjectorMatrix…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNNCPH.twoSiteParentGroundProjectorMatrix_overlappingLifts_commute","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14421","layer":"formal","project":"p8","title":"MPSTensor.twoSiteParentGroundProjectorMatrix_isStarProjection","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), IsStarProjection A.twoSiteParentGroundProjectorMatrix","labels":[],"detail_key":"p8","name":"MPSTensor.twoSiteParentGroundProjectorMatrix_isStarProjection","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14422","layer":"formal","project":"p8","title":"MPSTensor.twoSiteParentGroundProjectorMatrix_posSemidef","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), A.twoSiteParentGroundProjectorMatrix.PosSemidef","labels":[],"detail_key":"p8","name":"MPSTensor.twoSiteParentGroundProjectorMatrix_posSemidef","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14423","layer":"formal","project":"p8","title":"MPSTensor.twoSiteParentInteractionMatrix_isStarProjection","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), IsStarProjection A.twoSiteParentInteractionMatrix","labels":[],"detail_key":"p8","name":"MPSTensor.twoSiteParentInteractionMatrix_isStarProjection","module":"TNLean.MPS.RFP.BeigiSectorGraphConstruction"},{"id":"n14424","layer":"formal","project":"p8","title":"MPSTensor.IsNNCPH.exists_unitary_blockActions_of_twoSiteParentInteractionMatrix","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNNCPH 3 → Exists fun K => Exists fun dl => Exists fun dr =>…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNNCPH.exists_unitary_blockActions_of_twoSiteParentInteractionMatrix","module":"TNLean.MPS.RFP.BeigiSpatialDecomposition"},{"id":"n14425","layer":"formal","project":"p8","title":"MPSTensor.IsNNCPH.twoSiteParentInteractionMatrix_isHermitian_and_overlappingLifts_commute","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNNCPH 3 → And A.twoSiteParentInteractionMatrix.IsHermitian (…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNNCPH.twoSiteParentInteractionMatrix_isHermitian_and_overlappingLifts_commute","module":"TNLean.MPS.RFP.BeigiSpatialDecomposition"},{"id":"n14426","layer":"formal","project":"p8","title":"MPSTensor.twoSiteParentInteractionMatrix","kind":"def","summary":"d D : Nat → MPSTensor d D → Matrix (Prod (Fin d) (Fin d)) (Prod (Fin d) (Fin d)) Complex","labels":[],"detail_key":"p8","name":"MPSTensor.twoSiteParentInteractionMatrix","module":"TNLean.MPS.RFP.BeigiSpatialDecomposition"},{"id":"n14427","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor","kind":"def","summary":"MPSTensor (HMul.hMul 2 2) 2","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14428","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor_adjacent_twoPointExpectation","kind":"theorem","summary":"Eq (MPSTensor.bellPairChainTensor.physicalTwoPointExpectation 1 1 MPSTensor.bellPairChainZFirst…","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor_adjacent_twoPointExpectation","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14429","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor_isNormalTensor","kind":"theorem","summary":"MPSTensor.bellPairChainTensor.IsNormalTensor","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor_isNormalTensor","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14430","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor_isTransferIdempotent","kind":"theorem","summary":"MPSTensor.bellPairChainTensor.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor_isTransferIdempotent","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14431","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor_not_isPhysicalCID","kind":"theorem","summary":"Not MPSTensor.bellPairChainTensor.IsPhysicalCID","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor_not_isPhysicalCID","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14432","layer":"formal","project":"p8","title":"MPSTensor.bellPairChainTensor_shifted_twoPointExpectation","kind":"theorem","summary":"Eq (MPSTensor.bellPairChainTensor.physicalTwoPointExpectation 1 1 MPSTensor.bellPairChainZFirst…","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChainTensor_shifted_twoPointExpectation","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14433","layer":"formal","project":"p8","title":"MPSTensor.bellPairChain_isNormalTensor_isTransferIdempotent_and_not_isPhysicalCID","kind":"theorem","summary":"And MPSTensor.bellPairChainTensor.IsNormalTensor (And MPSTensor.bellPairChainTensor.IsTransferI…","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChain_isNormalTensor_isTransferIdempotent_and_not_isPhysicalCID","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14434","layer":"formal","project":"p8","title":"MPSTensor.bellPairChain_isTransferIdempotent_and_not_isPhysicalCID","kind":"theorem","summary":"And MPSTensor.bellPairChainTensor.IsTransferIdempotent (Not MPSTensor.bellPairChainTensor.IsPhy…","labels":[],"detail_key":"p8","name":"MPSTensor.bellPairChain_isTransferIdempotent_and_not_isPhysicalCID","module":"TNLean.MPS.RFP.BellPairCIDObstruction"},{"id":"n14435","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34PlusTensor","kind":"def","summary":"MPSTensor 2 1","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34PlusTensor","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14436","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34Tensor","kind":"def","summary":"MPSTensor 2 2","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34Tensor","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14437","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34ZeroTensor","kind":"def","summary":"MPSTensor 2 1","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34ZeroTensor","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14438","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34_blocked_overlap","kind":"theorem","summary":"∀ (n : Nat), Eq (MPSTensor.cpsvExample34ZeroTensor.mpvOverlap MPSTensor.cpsvExample34PlusTensor…","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34_blocked_overlap","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14439","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34_isPhysicalCID","kind":"theorem","summary":"MPSTensor.cpsvExample34Tensor.IsPhysicalCID","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34_isPhysicalCID","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14440","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34_mpv","kind":"theorem","summary":"∀ N : Nat, LT.lt 0 N → ∀ (σ : Fin N → Fin 2), Eq (MPSTensor.cpsvExample34Tensor.mpv σ) (HAdd.hA…","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34_mpv","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14441","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34_not_hasPhysicalBlockingIsometry","kind":"theorem","summary":"Not MPSTensor.cpsvExample34Tensor.HasPhysicalBlockingIsometry","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34_not_hasPhysicalBlockingIsometry","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14442","layer":"formal","project":"p8","title":"MPSTensor.cpsvExample34_not_isTransferIdempotent","kind":"theorem","summary":"Not MPSTensor.cpsvExample34Tensor.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.cpsvExample34_not_isTransferIdempotent","module":"TNLean.MPS.RFP.CPSVCIDNotRFPExample"},{"id":"n14443","layer":"formal","project":"p8","title":"MPSTensor.CPSVCanonicalFormData.weight_norm_one_and_block_rfp","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D (data : A.CPSVCanonicalFormData), A.IsTransferIdempotent → ∀ (k :…","labels":[],"detail_key":"p8","name":"MPSTensor.CPSVCanonicalFormData.weight_norm_one_and_block_rfp","module":"TNLean.MPS.RFP.CPSVCanonicalForm"},{"id":"n14444","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_coisometry_reconstruction_iff","kind":"theorem","summary":"∀ d D R : Nat (A : MPSTensor d D) (B : MPSTensor d R) (U : Matrix (Fin R) (Fin D) Complex), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_coisometry_reconstruction_iff","module":"TNLean.MPS.RFP.CPSVCanonicalForm"},{"id":"n14445","layer":"formal","project":"p8","title":"MPSTensor.norm_eq_one_and_isTransferIdempotent_of_isNormalTensor_smul","kind":"theorem","summary":"∀ d R : Nat (A : MPSTensor d R), A.IsNormalTensor → ∀ (c : Complex), Ne c 0 → (MPSTensor.IsTran…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_eq_one_and_isTransferIdempotent_of_isNormalTensor_smul","module":"TNLean.MPS.RFP.CPSVCanonicalForm"},{"id":"n14446","layer":"formal","project":"p8","title":"MPSTensor.rg_flow_converges_of_cf","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dim k)), MP…","labels":[],"detail_key":"p8","name":"MPSTensor.rg_flow_converges_of_cf","module":"TNLean.MPS.RFP.Convergence"},{"id":"n14447","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.complement_comm","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.complement_comm","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14448","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.hamiltonian_eq","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.hamiltonian_eq","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14449","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.mem_ground_iff","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.mem_ground_iff","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14450","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.pAX_comp_pK","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.pAX_comp_pK","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14451","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.pK_comp_pAX","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.pK_comp_pAX","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14452","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.pK_comp_pXB","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.pK_comp_pXB","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14453","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.pK_idem","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.pK_idem","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14454","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.pXB_comp_pK","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.pXB_comp_pK","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14455","layer":"formal","project":"p8","title":"Decorrelation.HasCommutingParentHam.reverse_product","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P_K : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"Decorrelation.HasCommutingParentHam.reverse_product","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14456","layer":"formal","project":"p8","title":"LinearMap.frustration_free_ham_eq","kind":"theorem","summary":"∀ E : Type u_1 [inst : AddCommGroup E] [inst_1 : Module Complex E] P Q : LinearMap (RingHom.id…","labels":[],"detail_key":"p8","name":"LinearMap.frustration_free_ham_eq","module":"TNLean.MPS.RFP.Decorrelation"},{"id":"n14457","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.hasPhysicalBlockingIsometry","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → A.HasPhysicalBlockingIsometry → B.HasPhysical…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.hasPhysicalBlockingIsometry","module":"TNLean.MPS.RFP.Defs"},{"id":"n14458","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.hasPhysicalBlockingIsometry_iff","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → Iff A.HasPhysicalBlockingIsometry B.HasPhysic…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.hasPhysicalBlockingIsometry_iff","module":"TNLean.MPS.RFP.Defs"},{"id":"n14459","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.isTransferIdempotent_iff","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D, A.GaugeEquiv B → Iff A.IsTransferIdempotent B.IsTransferIdempo…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.isTransferIdempotent_iff","module":"TNLean.MPS.RFP.Defs"},{"id":"n14460","layer":"formal","project":"p8","title":"MPSTensor.HasPhysicalBlockingIsometry","kind":"def","summary":"d D : Nat → MPSTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.HasPhysicalBlockingIsometry","module":"TNLean.MPS.RFP.Defs"},{"id":"n14461","layer":"formal","project":"p8","title":"MPSTensor.IsTransferIdempotent","kind":"def","summary":"d D : Nat → MPSTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.IsTransferIdempotent","module":"TNLean.MPS.RFP.Defs"},{"id":"n14462","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_iff_hasPhysicalBlockingIsometry","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.IsTransferIdempotent A.HasPhysicalBlockingIsometry","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_iff_hasPhysicalBlockingIsometry","module":"TNLean.MPS.RFP.Defs"},{"id":"n14463","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_iff_kraus_isometry","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.IsTransferIdempotent (Exists fun V => And (Eq (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_iff_kraus_isometry","module":"TNLean.MPS.RFP.Defs"},{"id":"n14464","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_of_kraus_isometry","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (V : Matrix (Prod (Fin d) (Fin d)) (Fin d) Complex), Eq (HMul.h…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_of_kraus_isometry","module":"TNLean.MPS.RFP.Defs"},{"id":"n14465","layer":"formal","project":"p8","title":"MPSTensor.AppearsAsRenormalizationFlowLimit","kind":"def","summary":"d D : Nat → MPSTensor d D → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.AppearsAsRenormalizationFlowLimit","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14466","layer":"formal","project":"p8","title":"MPSTensor.IsTransferIdempotent.appearsAsRenormalizationFlowLimit","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsTransferIdempotent → A.AppearsAsRenormalizationFlowLimit","labels":[],"detail_key":"p8","name":"MPSTensor.IsTransferIdempotent.appearsAsRenormalizationFlowLimit","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14467","layer":"formal","project":"p8","title":"MPSTensor.IsTransferIdempotent.of_appearsAsRenormalizationFlowLimit","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.AppearsAsRenormalizationFlowLimit → A.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.IsTransferIdempotent.of_appearsAsRenormalizationFlowLimit","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14468","layer":"formal","project":"p8","title":"MPSTensor.appearsAsRenormalizationFlowLimit_iff_hasPhysicalBlockingIsometry","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.AppearsAsRenormalizationFlowLimit A.HasPhysicalBlockingI…","labels":[],"detail_key":"p8","name":"MPSTensor.appearsAsRenormalizationFlowLimit_iff_hasPhysicalBlockingIsometry","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14469","layer":"formal","project":"p8","title":"MPSTensor.appearsAsRenormalizationFlowLimit_iff_isTransferIdempotent","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D), Iff A.AppearsAsRenormalizationFlowLimit A.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.appearsAsRenormalizationFlowLimit_iff_isTransferIdempotent","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14470","layer":"formal","project":"p8","title":"isIdempotentElem_of_tendsto_pow_two_pow","kind":"theorem","summary":"∀ R : Type u_1 [inst : TopologicalSpace R] [inst_1 : Monoid R] [ContinuousMul R] [T2Space R] (x…","labels":[],"detail_key":"p8","name":"isIdempotentElem_of_tendsto_pow_two_pow","module":"TNLean.MPS.RFP.FlowLimit"},{"id":"n14471","layer":"formal","project":"p8","title":"MPOTensor.IsOneLetterRFPViaTSUpToVirtualGauge","kind":"def","summary":"D : Nat → MPOTensor 1 D → Prop","labels":[],"detail_key":"p8","name":"MPOTensor.IsOneLetterRFPViaTSUpToVirtualGauge","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14472","layer":"formal","project":"p8","title":"MPSTensor.IsBlockedGaugePhaseFixedPoint","kind":"def","summary":"D : Nat → MPSTensor 1 D → 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data.IsWeightNormalized","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseDoubledTensor_exists_weightNormalized","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14476","layer":"formal","project":"p8","title":"MPSTensor.cubePhaseDoubledTensor_isCPSVCanonicalForm","kind":"theorem","summary":"(doubledTensor MPSTensor.cubePhaseTensor).toMPSTensor.IsCPSVCanonicalForm","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseDoubledTensor_isCPSVCanonicalForm","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14477","layer":"formal","project":"p8","title":"MPSTensor.cubePhaseDoubledTensor_isMPDO","kind":"theorem","summary":"(doubledTensor MPSTensor.cubePhaseTensor).IsMPDO","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseDoubledTensor_isMPDO","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14478","layer":"formal","project":"p8","title":"MPSTensor.cubePhaseTensor_isPureOneLetterRFPViaTSUpToVirtualGauge","kind":"theorem","summary":"MPSTensor.cubePhaseTensor.IsPureOneLetterRFPViaTSUpToVirtualGauge","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseTensor_isPureOneLetterRFPViaTSUpToVirtualGauge","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14479","layer":"formal","project":"p8","title":"MPSTensor.cubePhaseTensor_not_isTransferIdempotent","kind":"theorem","summary":"Not MPSTensor.cubePhaseTensor.IsTransferIdempotent","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseTensor_not_isTransferIdempotent","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14480","layer":"formal","project":"p8","title":"MPSTensor.cubeSwapGauge","kind":"def","summary":"Matrix.GeneralLinearGroup (Fin (Finset.univ.sum fun k => MPSTensor.cubePhaseBondDim k)) Complex","labels":[],"detail_key":"p8","name":"MPSTensor.cubeSwapGauge","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14481","layer":"formal","project":"p8","title":"MPSTensor.cube_phase_blocked_gauge_identity","kind":"theorem","summary":"∀ (i : Fin 1), Eq (MPSTensor.cubePhaseTensor i) (HSMul.hSMul MPSTensor.primitiveCubeRoot (HMul.…","labels":[],"detail_key":"p8","name":"MPSTensor.cube_phase_blocked_gauge_identity","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14482","layer":"formal","project":"p8","title":"MPSTensor.doubledVirtualGauge","kind":"def","summary":"D : Nat → Matrix.GeneralLinearGroup (Fin D) Complex → Matrix.GeneralLinearGroup (Fin (HMul.hMul…","labels":[],"detail_key":"p8","name":"MPSTensor.doubledVirtualGauge","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14483","layer":"formal","project":"p8","title":"MPSTensor.isPureOneLetterRFPViaTSUpToVirtualGauge_of_isBlockedGaugePhaseFixedPoint","kind":"theorem","summary":"∀ D : Nat (A : MPSTensor 1 D), A.IsBlockedGaugePhaseFixedPoint → A.IsPureOneLetterRFPViaTSUpToV…","labels":[],"detail_key":"p8","name":"MPSTensor.isPureOneLetterRFPViaTSUpToVirtualGauge_of_isBlockedGaugePhaseFixedPoint","module":"TNLean.MPS.RFP.GaugeBlockingCounterexample"},{"id":"n14484","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isPositiveGapBNTZCL_basisDirectSum_iff_hasNNCPHGroundSpaces","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → Iff ((MPSTensor…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isPositiveGapBNTZCL_basisDirectSum_iff_hasNNCPHGroundSpaces","module":"TNLean.MPS.RFP.MainMPSConditional"},{"id":"n14485","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isPositiveGapBNTZCL_implies_hasNNCPHGroundSpaces_basisDirect…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isPositiveGapBNTZCL_implies_hasNNCPHGroundSpaces_basisDirectSum","module":"TNLean.MPS.RFP.MainMPSConditional"},{"id":"n14486","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_iff_hasNNCPHGroundSpaces","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → Iff (MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_iff_hasNNCPHGroundSpaces","module":"TNLean.MPS.RFP.MainMPSConditional"},{"id":"n14487","layer":"formal","project":"p8","title":"MPSTensor.rfp_hasParentHamiltonianGroundSpaceSpanning_single","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) [NeZero D], A.IsTransferIdempotent → A.IsNormalTensor → A.HasPa…","labels":[],"detail_key":"p8","name":"MPSTensor.rfp_hasParentHamiltonianGroundSpaceSpanning_single","module":"TNLean.MPS.RFP.NNCPHGroundSpace"},{"id":"n14488","layer":"formal","project":"p8","title":"MPSTensor.rfp_implies_hasNNCPHGroundSpaces_single","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) [NeZero D], A.IsTransferIdempotent → A.IsNormalTensor → A.HasNN…","labels":[],"detail_key":"p8","name":"MPSTensor.rfp_implies_hasNNCPHGroundSpaces_single","module":"TNLean.MPS.RFP.NNCPHGroundSpace"},{"id":"n14489","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.rfp_implies_hasNNCPHGroundSpaces_basisDirectSum","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.rfp_implies_hasNNCPHGroundSpaces_basisDirectSum","module":"TNLean.MPS.RFP.NNCPHGroundSpacesMultiSector"},{"id":"n14490","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_isPositiveGapBNTZCL","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_isPositiveGapBNTZCL","module":"TNLean.MPS.RFP.NNCPHMultiSector"},{"id":"n14491","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.rfp_hasParentHamiltonianGroundSpaceSpanning_basisDirectSum","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.rfp_hasParentHamiltonianGroundSpaceSpanning_basisDirectSum","module":"TNLean.MPS.RFP.NNCPHMultiSector"},{"id":"n14492","layer":"formal","project":"p8","title":"MPSTensor.IsNormalTensor.isTransferIdempotent_iff_isIsometryCanonicalForm","kind":"theorem","summary":"∀ d D : Nat A : MPSTensor d D, A.IsNormalTensor → Iff A.IsTransferIdempotent A.IsIsometryCanoni…","labels":[],"detail_key":"p8","name":"MPSTensor.IsNormalTensor.isTransferIdempotent_iff_isIsometryCanonicalForm","module":"TNLean.MPS.RFP.NormalIsometryCharacterization"},{"id":"n14493","layer":"formal","project":"p8","title":"MPSTensor.leftPairLift_toMatrix_reindex","kind":"theorem","summary":"∀ d : Nat (Q : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace…","labels":[],"detail_key":"p8","name":"MPSTensor.leftPairLift_toMatrix_reindex","module":"TNLean.MPS.RFP.PairLiftCoordinates"},{"id":"n14494","layer":"formal","project":"p8","title":"MPSTensor.rightPairLift_toMatrix_reindex","kind":"theorem","summary":"∀ d : Nat (Q : LinearMap (RingHom.id Complex) (MPSTensor.NSiteSpace d 2) (MPSTensor.NSiteSpace…","labels":[],"detail_key":"p8","name":"MPSTensor.rightPairLift_toMatrix_reindex","module":"TNLean.MPS.RFP.PairLiftCoordinates"},{"id":"n14495","layer":"formal","project":"p8","title":"MPSTensor.cubePhaseTensor_not_tendsto_dyadic_transferMap","kind":"theorem","summary":"And MPSTensor.cubePhaseTensor.IsCPSVCanonicalForm (And MPSTensor.cubePhaseCanonicalData.IsWeigh…","labels":[],"detail_key":"p8","name":"MPSTensor.cubePhaseTensor_not_tendsto_dyadic_transferMap","module":"TNLean.MPS.RFP.PhaseOscillation"},{"id":"n14496","layer":"formal","project":"p8","title":"MPSTensor.primitiveCubeRoot_isPrimitiveRoot","kind":"theorem","summary":"IsPrimitiveRoot MPSTensor.primitiveCubeRoot 3","labels":[],"detail_key":"p8","name":"MPSTensor.primitiveCubeRoot_isPrimitiveRoot","module":"TNLean.MPS.RFP.PhaseOscillation"},{"id":"n14497","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.exists_basis_physicalObservableTransfer_rankOne_le_three_tot…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → Exists fun L =>…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.exists_basis_physicalObservableTransfer_rankOne_le_three_totalDim_pow_five","module":"TNLean.MPS.RFP.PhysicalObservableRealization"},{"id":"n14498","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_matrixUnit","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k) L : Nat, MPSTensor.WordTupl…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_matrixUnit","module":"TNLean.MPS.RFP.PhysicalObservableRealization"},{"id":"n14499","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_rankOne","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k) L : Nat, MPSTensor.WordTupl…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_rankOne","module":"TNLean.MPS.RFP.PhysicalObservableRealization"},{"id":"n14500","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_sectorSupported","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k) L : Nat, MPSTensor.WordTupl…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.exists_physicalObservableTransfer_directSum_sectorSupported","module":"TNLean.MPS.RFP.PhysicalObservableRealization"},{"id":"n14501","layer":"formal","project":"p8","title":"MPSTensor.WordTupleSpanTop.exists_simultaneous_left_inverse","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat A : (k : Fin r) → MPSTensor d (dim k) L : Nat, MPSTensor.WordTupl…","labels":[],"detail_key":"p8","name":"MPSTensor.WordTupleSpanTop.exists_simultaneous_left_inverse","module":"TNLean.MPS.RFP.PhysicalObservableRealization"},{"id":"n14502","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.rfp_implies_nncph_basisDirectSum","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.rfp_implies_nncph_basisDirectSum","module":"TNLean.MPS.RFP.ResidualFamilyCommutation"},{"id":"n14503","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.exists_residualFamilyAppendixBData","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → (MPSTensor.dire…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.exists_residualFamilyAppendixBData","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14504","layer":"formal","project":"p8","title":"MPSTensor.IsResidualIsometryFamily.residualFamilyPhysicalIsometryMatrix_isometry","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat U : (j : Fin r) → MPSTensor d (dim j), MPSTensor.IsResidualIsomet…","labels":[],"detail_key":"p8","name":"MPSTensor.IsResidualIsometryFamily.residualFamilyPhysicalIsometryMatrix_isometry","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14505","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData","kind":"inductive","summary":"d r : Nat → dim : Fin r → Nat → ((j : Fin r) → MPSTensor d (dim j)) → Type","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14506","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.TwoSiteBoundary","kind":"def","summary":"r : Nat → (Fin r → Nat) → Type","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.TwoSiteBoundary","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14507","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → B : (j : Fin r) → MPSTensor d (dim j) → MPSTensor.ResidualFamil…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14508","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat B : (j : Fin r) → MPSTensor d (dim j) (h : MPSTensor.ResidualFami…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14509","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range_eq_directSum_groundSpace","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat B : (j : Fin r) → MPSTensor d (dim j) (h : MPSTensor.ResidualFami…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondEmbedding_range_eq_directSum_groundSpace","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14510","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((j : Fin r) → MPSTensor d (dim j)) → LinearMap (RingHom.id Com…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14511","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_eq_complement","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (j : Fin r) → MPSTensor d (dim j)), Eq (MPSTensor.ResidualFa…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_eq_complement","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14512","layer":"formal","project":"p8","title":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_range","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat B : (j : Fin r) → MPSTensor d (dim j) (h : MPSTensor.ResidualFami…","labels":[],"detail_key":"p8","name":"MPSTensor.ResidualFamilyAppendixBData.twoSiteBondSupportProjection_range","module":"TNLean.MPS.RFP.ResidualFamilySupport"},{"id":"n14513","layer":"formal","project":"p8","title":"MPSTensor.residualFamilyPhysicalIsometryMatrix","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((j : Fin r) → MPSTensor d (dim j)) → Matrix (Fin d) 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(MPSTensor.…","labels":[],"detail_key":"p8","name":"MPSTensor.IsBNTCanonicalForm.isTransferIdempotent_basisDirectSum_iff","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14516","layer":"formal","project":"p8","title":"MPSTensor.IsResidualIsometryFamily","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → ((j : Fin r) → MPSTensor d (dim j)) → Prop","labels":[],"detail_key":"p8","name":"MPSTensor.IsResidualIsometryFamily","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14517","layer":"formal","project":"p8","title":"MPSTensor.exists_residualIsometryFamily_of_isTransferIdempotent_directSum","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (B : (k : Fin r) → MPSTensor d (d…","labels":[],"detail_key":"p8","name":"MPSTensor.exists_residualIsometryFamily_of_isTransferIdempotent_directSum","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14518","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_block_of_isTransferIdempotent_directSum","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (B : (k : Fin r) → MPSTensor d (d…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_block_of_isTransferIdempotent_directSum","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14519","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_directSumTensor_iff","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat [∀ (k : Fin r), NeZero (dim k)] (B : (k : Fin r) → MPSTensor d (d…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_directSumTensor_iff","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14520","layer":"formal","project":"p8","title":"MPSTensor.isTransferIdempotent_directSumTensor_of_isIsometryCanonicalForm","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (B : (k : Fin r) → MPSTensor d (dim k)), (∀ (k : Fin r), (B k).Is…","labels":[],"detail_key":"p8","name":"MPSTensor.isTransferIdempotent_directSumTensor_of_isIsometryCanonicalForm","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14521","layer":"formal","project":"p8","title":"MPSTensor.mixedMapLM_cast_eq_zero_iff","kind":"theorem","summary":"∀ d D₁ D₁' D₂ D₂' : Nat (h₁ : Eq D₁ D₁') (h₂ : Eq D₂ D₂') (A : MPSTensor d D₁) (B : MPSTensor 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U.GaugePhaseEquiv A → V.GaugePhaseEq…","labels":[],"detail_key":"p8","name":"MPSTensor.mixedMapLM_eq_zero_of_gaugePhaseEquiv","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14525","layer":"formal","project":"p8","title":"MPSTensor.residual_isometry_entry_of_mixedMapLM_eq_zero","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (U : MPSTensor d D₁) (V : MPSTensor d D₂), Eq (Kraus.mixedMapLM U V) 0 → ∀ (α β…","labels":[],"detail_key":"p8","name":"MPSTensor.residual_isometry_entry_of_mixedMapLM_eq_zero","module":"TNLean.MPS.RFP.ResidualIsometry"},{"id":"n14526","layer":"formal","project":"p8","title":"MPSTensor.IsResidualIsometryFamily.wordEntryFamily_one_gram","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat U : (j : Fin r) → MPSTensor d (dim j), 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[NeZero D], Kraus.IsNormal A → A.IsTransferIdempotent → Eq (Fin…","labels":[],"detail_key":"p8","name":"MPSTensor.rfp_nt_structural_full_unit_pair","module":"TNLean.MPS.RFP.StructuralFull"},{"id":"n14538","layer":"formal","project":"p8","title":"MPSTensor.unitPairIsometry_transfer","kind":"theorem","summary":"∀ d D : Nat (U : MPSTensor d D), (∀ (p q : Prod (Fin D) (Fin D)), Eq (Finset.univ.sum fun i =>…","labels":[],"detail_key":"p8","name":"MPSTensor.unitPairIsometry_transfer","module":"TNLean.MPS.RFP.StructuralFull"},{"id":"n14539","layer":"formal","project":"p8","title":"MPSTensor.IsBNTCanonicalForm.exists_basis_physicalObservables_expectation_eq_trace_mul_tr…","kind":"theorem","summary":"∀ d : Nat P : MPSTensor.SectorDecomposition d, MPSTensor.IsBNTCanonicalForm P → ∀ (j : Fin 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(Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.blockInclusion","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14572","layer":"formal","project":"p8","title":"MPSTensor.blockInclusion_apply","kind":"theorem","summary":"∀ r : Nat (dim : Fin r → Nat) (k : Fin r) (x : Fin (Finset.univ.sum fun j => dim j)) (y : Fin (…","labels":[],"detail_key":"p8","name":"MPSTensor.blockInclusion_apply","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14573","layer":"formal","project":"p8","title":"MPSTensor.blockInclusion_conjTranspose_mul_eq_zero","kind":"theorem","summary":"∀ r : Nat (dim : Fin r → Nat) k l : Fin r, Ne k l → Eq (HMul.hMul (MPSTensor.blockInclusion dim…","labels":[],"detail_key":"p8","name":"MPSTensor.blockInclusion_conjTranspose_mul_eq_zero","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14574","layer":"formal","project":"p8","title":"MPSTensor.blockInclusion_conjTranspose_mul_self","kind":"theorem","summary":"∀ r : Nat (dim : Fin r → Nat) (k : Fin r), Eq (HMul.hMul (MPSTensor.blockInclusion dim k).conjT…","labels":[],"detail_key":"p8","name":"MPSTensor.blockInclusion_conjTranspose_mul_self","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14575","layer":"formal","project":"p8","title":"MPSTensor.blockIndexCoordinateEquiv","kind":"def","summary":"r : Nat → ι : Type u_1 → (dim : Fin r → Nat) → (e : Equiv ι (Fin r)) → Equiv (Sigma fun k => Fi…","labels":[],"detail_key":"p8","name":"MPSTensor.blockIndexCoordinateEquiv","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14576","layer":"formal","project":"p8","title":"MPSTensor.evalWord_toTensorFromBlocks_eq_reindex_blockDiagonal","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dim k)) (w…","labels":[],"detail_key":"p8","name":"MPSTensor.evalWord_toTensorFromBlocks_eq_reindex_blockDiagonal","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14577","layer":"formal","project":"p8","title":"MPSTensor.mpv_toTensorFromBlocks_eq_sum","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dim k)) N :…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_toTensorFromBlocks_eq_sum","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14578","layer":"formal","project":"p8","title":"MPSTensor.mpv_toTensorFromBlocks_reindex","kind":"theorem","summary":"∀ d r r' : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dim k))…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_toTensorFromBlocks_reindex","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14579","layer":"formal","project":"p8","title":"MPSTensor.sameMPV₂Pos_of_coisometry_reconstruction","kind":"theorem","summary":"∀ s d n : Nat (T : MPSTensor s d) (B : MPSTensor s n) (U : Matrix (Fin n) (Fin d) Complex), Eq…","labels":[],"detail_key":"p8","name":"MPSTensor.sameMPV₂Pos_of_coisometry_reconstruction","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14580","layer":"formal","project":"p8","title":"MPSTensor.toTensorFromBlocks","kind":"def","summary":"d r : Nat → dim : Fin r → Nat → (Fin r → Complex) → ((k : Fin r) → MPSTensor d (dim k)) → MPSTe…","labels":[],"detail_key":"p8","name":"MPSTensor.toTensorFromBlocks","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14581","layer":"formal","project":"p8","title":"MPSTensor.toTensorFromBlocks_eq_reindex_blockDiagonal_equiv","kind":"theorem","summary":"∀ d r : Nat ι : Type u_1 [inst : DecidableEq ι] dim : Fin r → Nat (μ : Fin r → Complex) (A : (k…","labels":[],"detail_key":"p8","name":"MPSTensor.toTensorFromBlocks_eq_reindex_blockDiagonal_equiv","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14582","layer":"formal","project":"p8","title":"MPSTensor.toTensorFromBlocks_eq_reindex_of_equiv","kind":"theorem","summary":"∀ d r r' : Nat dim : Fin r → Nat dim' : Fin r' → Nat (μ : Fin r → Complex) (A : (k : Fin r) → M…","labels":[],"detail_key":"p8","name":"MPSTensor.toTensorFromBlocks_eq_reindex_of_equiv","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14583","layer":"formal","project":"p8","title":"MPSTensor.toTensorFromBlocks_mul_blockInclusion","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A : (k : Fin r) → MPSTensor d (dim k)) (k…","labels":[],"detail_key":"p8","name":"MPSTensor.toTensorFromBlocks_mul_blockInclusion","module":"TNLean.MPS.SharedInfra.BlockAssembly"},{"id":"n14584","layer":"formal","project":"p8","title":"MPSTensor.blockDiagonalGL","kind":"def","summary":"r : Nat → dim : Fin r → Nat → ((k : Fin r) → Matrix.GeneralLinearGroup (Fin (dim k)) Complex) →…","labels":[],"detail_key":"p8","name":"MPSTensor.blockDiagonalGL","module":"TNLean.MPS.SharedInfra.BlockGauge"},{"id":"n14585","layer":"formal","project":"p8","title":"MPSTensor.globalGaugeOfBlocks","kind":"def","summary":"r : Nat → dim : Fin r → Nat → ((k : Fin r) → Matrix.GeneralLinearGroup (Fin (dim k)) Complex) →…","labels":[],"detail_key":"p8","name":"MPSTensor.globalGaugeOfBlocks","module":"TNLean.MPS.SharedInfra.BlockGauge"},{"id":"n14586","layer":"formal","project":"p8","title":"MPSTensor.globalGaugeOfBlocks_unitaryGL_mem","kind":"theorem","summary":"∀ r : Nat dim : Fin r → Nat (U : (k : Fin r) → Subtype fun x => Membership.mem (Matrix.unitaryG…","labels":[],"detail_key":"p8","name":"MPSTensor.globalGaugeOfBlocks_unitaryGL_mem","module":"TNLean.MPS.SharedInfra.BlockGauge"},{"id":"n14587","layer":"formal","project":"p8","title":"MPSTensor.matched_block_gauge","kind":"def","summary":"d : Nat → Q : MPSTensor.SectorDecomposition d → ((k : Fin Q.basisCount) → Matrix.GeneralLinearG…","labels":[],"detail_key":"p8","name":"MPSTensor.matched_block_gauge","module":"TNLean.MPS.SharedInfra.BlockGauge"},{"id":"n14588","layer":"formal","project":"p8","title":"MPSTensor.toTensorFromBlocks_eq_globalGaugeOfBlocks_conj","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (μ : Fin r → Complex) (A B : (k : Fin r) → MPSTensor d (dim k)) (…","labels":[],"detail_key":"p8","name":"MPSTensor.toTensorFromBlocks_eq_globalGaugeOfBlocks_conj","module":"TNLean.MPS.SharedInfra.BlockGauge"},{"id":"n14589","layer":"formal","project":"p8","title":"MPSTensor.trace_evalWord_gauge_mul","kind":"theorem","summary":"∀ d D : Nat A B : MPSTensor d D (G : Matrix.GeneralLinearGroup (Fin D) Complex), (∀ (i : Fin d)…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_evalWord_gauge_mul","module":"TNLean.MPS.SharedInfra.BoundaryDecomposition"},{"id":"n14590","layer":"formal","project":"p8","title":"MPSTensor.trace_evalWord_gauge_toTensorFromBlocks_mul","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (weight : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_evalWord_gauge_toTensorFromBlocks_mul","module":"TNLean.MPS.SharedInfra.BoundaryDecomposition"},{"id":"n14591","layer":"formal","project":"p8","title":"MPSTensor.trace_evalWord_toTensorFromBlocks_mul","kind":"theorem","summary":"∀ d r : Nat dim : Fin r → Nat (weight : Fin r → Complex) (A : (j : Fin r) → MPSTensor d (dim j)…","labels":[],"detail_key":"p8","name":"MPSTensor.trace_evalWord_toTensorFromBlocks_mul","module":"TNLean.MPS.SharedInfra.BoundaryDecomposition"},{"id":"n14592","layer":"formal","project":"p8","title":"MPSTensor.GaugeEquiv.of_coisometry_reconstruction","kind":"theorem","summary":"∀ s n D : Nat A B : MPSTensor s D C E : MPSTensor s n (UA UB : Matrix (Fin n) (Fin D) Complex),…","labels":[],"detail_key":"p8","name":"MPSTensor.GaugeEquiv.of_coisometry_reconstruction","module":"TNLean.MPS.SharedInfra.CoisometryGauge"},{"id":"n14593","layer":"formal","project":"p8","title":"MPSTensor.unitaryGL","kind":"def","summary":"n : Type u_1 → [inst : Fintype n] → [inst_1 : DecidableEq n] → (Subtype fun x => Membership.mem…","labels":[],"detail_key":"p8","name":"MPSTensor.unitaryGL","module":"TNLean.MPS.SharedInfra.CoisometryGauge"},{"id":"n14594","layer":"formal","project":"p8","title":"MPSTensor.dim_eq_of_mpvOverlap_norm_tendsto_one_of_irreducible_TP","kind":"theorem","summary":"∀ d D₁ D₂ : Nat [NeZero D₁] [NeZero D₂] (A : MPSTensor d D₁) (B : MPSTensor d D₂), Kraus.IsIrre…","labels":[],"detail_key":"p8","name":"MPSTensor.dim_eq_of_mpvOverlap_norm_tendsto_one_of_irreducible_TP","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14595","layer":"formal","project":"p8","title":"MPSTensor.gaugePhaseEquiv_cast_compose_via_centre","kind":"theorem","summary":"∀ d D D₁ D₂ : Nat (h₁ : Eq D D₁) (h₂ : Eq D D₂) A : MPSTensor d D B : MPSTensor d D₁ C : MPSTen…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhaseEquiv_cast_compose_via_centre","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14596","layer":"formal","project":"p8","title":"MPSTensor.gaugePhaseEquiv_of_overlap_norm_tendsto_one_of_irreducible_TP","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Kraus.IsIrreducibleFamily A → Kraus.IsIrreducible…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhaseEquiv_of_overlap_norm_tendsto_one_of_irreducible_TP","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14597","layer":"formal","project":"p8","title":"MPSTensor.gaugePhaseEquiv_swap_cast","kind":"theorem","summary":"∀ d D₁ D₂ : Nat (h : Eq D₁ D₂) A : MPSTensor d D₁ B : MPSTensor d D₂, (cast ⋯ B).GaugePhaseEqui…","labels":[],"detail_key":"p8","name":"MPSTensor.gaugePhaseEquiv_swap_cast","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14598","layer":"formal","project":"p8","title":"MPSTensor.mixedTransferSpectralRadius_ge_one_of_mpvOverlap_norm_tendsto_one","kind":"theorem","summary":"∀ d D : Nat [NeZero D] (A B : MPSTensor d D), Filter.Tendsto (fun N => norm (A.mpvOverlap B N))…","labels":[],"detail_key":"p8","name":"MPSTensor.mixedTransferSpectralRadius_ge_one_of_mpvOverlap_norm_tendsto_one","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14599","layer":"formal","project":"p8","title":"MPSTensor.mpv_eq_pow_mul_of_gaugePhase","kind":"theorem","summary":"∀ d D : Nat (A B : MPSTensor d D) (X : Matrix.GeneralLinearGroup (Fin D) Complex) (ζ : Complex)…","labels":[],"detail_key":"p8","name":"MPSTensor.mpv_eq_pow_mul_of_gaugePhase","module":"TNLean.MPS.SharedInfra.GaugePhase"},{"id":"n14600","layer":"formal","project":"p8","title":"MPSTensor.kappa_norm_eq_one_of_leftCanonical_smul","kind":"theorem","summary":"∀ d D : Nat [NeZero D] κ ξ : Complex, Eq (norm ξ) 1 → ∀ (A B : MPSTensor d D), A.IsLeftCanonica…","labels":[],"detail_key":"p8","name":"MPSTensor.kappa_norm_eq_one_of_leftCanonical_smul","module":"TNLean.MPS.SharedInfra.Scaling"},{"id":"n14601","layer":"formal","project":"p8","title":"MPSTensor.mpv_smul","kind":"theorem","summary":"∀ d D : Nat (c : 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(MPSTensor.IsLeft…","labels":[],"detail_key":"p8","name":"MPSTensor.norm_eq_one_of_leftCanonical_smul","module":"TNLean.MPS.SharedInfra.Scaling"},{"id":"n14604","layer":"formal","project":"p8","title":"MPSTensor.transferMap_smul","kind":"theorem","summary":"∀ d D : Nat (c : Complex) (A : MPSTensor d D) (X : Matrix (Fin D) (Fin D) Complex), Eq ((Kraus.…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_smul","module":"TNLean.MPS.SharedInfra.Scaling"},{"id":"n14605","layer":"formal","project":"p8","title":"MPSTensor.transferMap_smul_eq_of_norm_eq_one","kind":"theorem","summary":"∀ d D : Nat (A : MPSTensor d D) (c : Complex), Eq (norm c) 1 → Eq (Kraus.transferMap fun i => H…","labels":[],"detail_key":"p8","name":"MPSTensor.transferMap_smul_eq_of_norm_eq_one","module":"TNLean.MPS.SharedInfra.Scaling"},{"id":"n14606","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatBlockInclusion","kind":"def","summary":"d : Nat → (P : MPSTensor.SectorDecomposition d) → (s : Fin P.totalCopies) → Matrix (Fin P.total…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatBlockInclusion","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14607","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatBlockInclusion_conjTranspose_mul_toTensor","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (s : Fin P.totalCopies) (i : Fin d), Eq (HMul.h…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatBlockInclusion_conjTranspose_mul_toTensor","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14608","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatBlockInclusion_isometry","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (s : Fin P.totalCopies), Eq (HMul.hMul (P.flatB…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatBlockInclusion_isometry","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14609","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatBlockInclusion_mul_conjTranspose","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (s : Fin P.totalCopies), Eq (HMul.hMul (P.flatB…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatBlockInclusion_mul_conjTranspose","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14610","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatWeight_smul_flatBasis_eq_compression","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (s : Fin P.totalCopies) (i : Fin d), Eq (HSMul.…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatWeight_smul_flatBasis_eq_compression","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14611","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.toTensor_mul_flatBlockInclusion","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (s : Fin P.totalCopies) (i : Fin d), Eq (HMul.h…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.toTensor_mul_flatBlockInclusion","module":"TNLean.MPS.SharedInfra.SectorCompression"},{"id":"n14612","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14613","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.basisCount_le_totalDim","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d), (∀ (j : Fin P.basisCount), LT.lt 0 (P.basisDim…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.basisCount_le_totalDim","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14614","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.basisDim_le_totalDim","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (j : Fin P.basisCount), LE.le (P.basisDim j) P.…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.basisDim_le_totalDim","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14615","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_not_eventually_zero","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (j : Fin P.basisCount), Not (Filter.Eventually…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_not_eventually_zero","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14616","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.coeff_powWeights","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (m n : Nat) (j : Fin P.basisCount), Eq ((P.powW…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.coeff_powWeights","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14617","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatBasis_flatIndexEquiv_heq","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (jq : Sigma fun j => Fin (P.copies j)), HEq (P.…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatBasis_flatIndexEquiv_heq","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14618","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatDim_flatIndexEquiv","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (jq : Sigma fun j => Fin (P.copies j)), Eq (P.f…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatDim_flatIndexEquiv","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14619","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.flatWeight_flatIndexEquiv","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) (jq : Sigma fun j => Fin (P.copies j)), Eq (P.f…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.flatWeight_flatIndexEquiv","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14620","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.markedTensor","kind":"def","summary":"d e : Nat → (P : MPSTensor.SectorDecomposition d) → ((j : Fin P.basisCount) → MPSTensor e (P.ba…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.markedTensor","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14621","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.mpv_toTensor_eq_sum_coeff","kind":"theorem","summary":"∀ d : Nat (P : MPSTensor.SectorDecomposition d) N : Nat (σ : Fin N → Fin d), Eq (P.toTensor.mpv…","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.mpv_toTensor_eq_sum_coeff","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14622","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.powWeights","kind":"def","summary":"d : Nat → MPSTensor.SectorDecomposition d → Nat → MPSTensor.SectorDecomposition d","labels":[],"detail_key":"p8","name":"MPSTensor.SectorDecomposition.powWeights","module":"TNLean.MPS.SharedInfra.SectorDecomposition"},{"id":"n14623","layer":"formal","project":"p8","title":"MPSTensor.SectorDecomposition.scaleWeights","kind":"def","summary":"d : Nat → MPSTensor.SectorDecomposition d → (c : Complex) → Ne c 0 → 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atibleLocalAutomorphism.algebraicEquiv_symm_apply","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15756","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.algebraicEquiv_symm_isometry","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.algebraicEquiv_symm_isometry","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15757","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.forwardHom","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.forwardHom","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15758","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.forwardHom_localObservable","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.forwardHom_localObservable","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15759","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.inverseHom","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.inverseHom","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15760","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.inverseHom_localObservable","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.inverseHom_localObservable","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15761","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.isQCA_quasiLocalEquiv","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.isQCA_quasiLocalEquiv","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15762","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.norm_algebraicEquiv","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.norm_algebraicEquiv","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15763","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.norm_algebraicEquiv_symm","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.norm_algebraicEquiv_symm","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15764","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.propagatesWithin_quasiLocalEquiv","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.propagatesWithin_quasiLocalEquiv","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15765","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv","kind":"def","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15766","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_algebraicToQuasiLocal","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_algebraicToQuasiLocal","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15767","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_quasiLocalObservable","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_quasiLocalObservable","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15768","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_symm_algebraicToQuasiLocal","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_symm_algebraicToQuasiLocal","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15769","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_symm_quasiLocalObservable","kind":"lemma","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.quasiLocalEquiv_symm_quasiLocalObservable","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15770","layer":"formal","project":"p8","title":"SpinChain.CompatibleLocalAutomorphism.translationCovariant_quasiLocalEquiv","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.CompatibleLocalAutomorphism.translationCovariant_quasiLocalEquiv","module":"TNLean.QCA.CompatibleLocalAutomorphism"},{"id":"n15771","layer":"formal","project":"p8","title":"SpinChain.QuasiLocalSupportedIn.commute_of_disjoint","kind":"theorem","summary":"","labels":[],"detail_key":"p8","name":"SpinChain.QuasiLocalSupportedIn.commute_of_disjoint","module":"TNLean.QCA.DisjointSupport"},{"id":"n15772","layer":"form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a group action of G on X, we can define an induced group action of G on Y^X by: g \\cdot f…","labels":["prop:MulActionColorings"],"detail_key":"p9"},{"id":"n16073","layer":"informal","project":"p9","title":"(1 \\cdot f)(x) = f(1^-1 \\cdot x) = f(1 \\cdot x) = f(x), \\\\ (g \\cdot (h \\cdot f))(x) = f(h…","kind":"proof","summary":"(1 \\cdot f)(x) = f(1^-1 \\cdot x) = f(1 \\cdot x) = f(x), \\\\ (g \\cdot (h \\cdot f))(x) = f(h^-1 \\c…","labels":[],"detail_key":"p9"},{"id":"n16074","layer":"informal","project":"p9","title":"def:numDistinctColorings","kind":"definition","summary":"The \\emphnumber of distinct colorings is defined as |Y^X/G|.","labels":["def:numDistinctColorings"],"detail_key":"p9"},{"id":"n16075","layer":"informal","project":"p9","title":"def:CyclesOfGroup","kind":"definition","summary":"Given g \\in G, the set of \\emphcycles of g is defined as X / \\sim_g, where \\sim_g is the equiva…","labels":["def:CyclesOfGroup"],"detail_key":"p9"},{"id":"n16076","layer":"informal","project":"p9","title":"prop:f-mem-fixedBy-iff-forall-eq-to-eq","kind":"proposition","summary":"For any g \\in G: f \\in (Y^X)^g \\iff \\forall x_1, x_2 \\in X: (x_1 \\sim_g x_2 \\implies f(x_1) = f…","labels":["prop:f-mem-fixedBy-iff-forall-eq-to-eq"],"detail_key":"p9"},{"id":"n16077","layer":"informal","project":"p9","title":"f \\in (Y^X)^g &\\iff g \\cdot f = f, \\\\ &\\iff \\forall x \\in X: (g \\cdot f)(x) = f(x), \\\\ &\\…","kind":"proof","summary":"f \\in (Y^X)^g &\\iff g \\cdot f = f, \\\\ &\\iff \\forall x \\in X: (g \\cdot f)(x) = f(x), \\\\ &\\iff \\f…","labels":[],"detail_key":"p9"},{"id":"n16078","layer":"informal","project":"p9","title":"prop:equiv-of-fixedBy-coloring-of-cycle-coloring","kind":"proposition","summary":"Let [x] denote the equivalence class of x in X/\\sim_g.\\\\ Let \\varphi : (Y^X)^g \\to Y^X/\\sim_g b…","labels":["prop:equiv-of-fixedBy-coloring-of-cycle-coloring"],"detail_key":"p9"},{"id":"n16079","layer":"informal","project":"p9","title":"\\varphi is well-defined because, by Proposition~\\refprop:f-mem-fixedBy-iff-forall-eq-to-e…","kind":"proof","summary":"\\varphi is well-defined because, by Proposition~\\refprop:f-mem-fixedBy-iff-forall-eq-to-eq, f \\…","labels":[],"detail_key":"p9"},{"id":"n16080","layer":"informal","project":"p9","title":"prop:forall-card-pow-numCyclesOfGroup-eq-card-fixedBy","kind":"proposition","summary":"\\forall g \\in G: |(Y^X)^g| = |Y|^c(g)","labels":["prop:forall-card-pow-numCyclesOfGroup-eq-card-fixedBy"],"detail_key":"p9"},{"id":"n16081","layer":"informal","project":"p9","title":"The equality |(Y^X)^g| = |Y^X/\\sim_g| follows from the bijection in Proposition~\\refprop:…","kind":"proof","summary":"The equality |(Y^X)^g| = |Y^X/\\sim_g| follows from the bijection in Proposition~\\refprop:equiv-…","labels":[],"detail_key":"p9"},{"id":"n16082","layer":"informal","project":"p9","title":"prop:numDistinctColorings-eq-sum-card-pow-numCyclesOfGroup-div-mul-card-group","kind":"theorem","summary":"\\emph(Pólya's enumeration theorem) The number of distinct colorings of X with colors in Y under…","labels":["prop:numDistinctColorings-eq-sum-card-pow-numCyclesOfGroup-div-mul-card-group"],"detail_key":"p9"},{"id":"n16083","layer":"informal","project":"p9","title":"We use \\emphBurnside's lemma on colorings to get: |Y^X/G| \\cdot |G| = \\sum_g \\in G |(Y^X)…","kind":"proof","summary":"We use \\emphBurnside's lemma on colorings to get: |Y^X/G| \\cdot |G| = \\sum_g \\in G |(Y^X)^g|. U…","labels":[],"detail_key":"p9"},{"id":"n16084","layer":"informal","project":"p9","title":"prop:numDistinctColorings-mul-card-group-eq-sum-numGroupOfNumCycles-mul-card-pow","kind":"theorem","summary":"|Y^X/G| \\cdot |G| = \\sum_i \\in [|X| + 1] |\\g \\in G \\mid c(g) = i\\| \\cdot |Y|^i.","labels":["prop:numDistinctColorings-mul-card-group-eq-sum-numGroupOfNumCycles-mul-card-pow"],"detail_key":"p9"},{"id":"n16085","layer":"informal","project":"p9","title":"For any g \\in G we have 0 \\leq c(g) \\leq |X|. Then: |Y^X/G| \\cdot |G| = \\sum_g \\in G |Y|^…","kind":"proof","summary":"For any g \\in G we have 0 \\leq c(g) \\leq |X|. Then: |Y^X/G| \\cdot |G| = \\sum_g \\in G |Y|^c(g) =…","labels":[],"detail_key":"p9"},{"id":"n16086","layer":"informal","project":"p9","title":"def:numCyclesOfPerm","kind":"definition","summary":"Given a permutation \\pi \\in S_X, the set of \\emphcycles of \\pi is defined as X / \\sim_\\pi, wher…","labels":["def:numCyclesOfPerm"],"detail_key":"p9"},{"id":"n16087","layer":"informal","project":"p9","title":"prop:numCyclesOfPerm-eq-card","kind":"proposition","summary":"For a permutation \\pi \\in S_X and a set S containing exactly one element from each cycle of \\pi…","labels":["prop:numCyclesOfPerm-eq-card"],"detail_key":"p9"},{"id":"n16088","layer":"informal","project":"p9","title":"We construct a bijection \\varphi : X / \\sim_\\pi \\to S defined by: \\varphi([x]) &= \\textth…","kind":"proof","summary":"We construct a bijection \\varphi : X / \\sim_\\pi \\to S defined by: \\varphi([x]) &= \\textthe uniq…","labels":[],"detail_key":"p9"},{"id":"n16089","layer":"informal","project":"p9","title":"prop:exists-perm-pow","kind":"proposition","summary":"Let \\pi be a permutation on a finite set X, and x \\in X. Then there exists some n \\in N such th…","labels":["prop:exists-perm-pow"],"detail_key":"p9"},{"id":"n16090","layer":"informal","project":"p9","title":"Since X is finite, the cycle containing x has finite length. We set n to be the length of…","kind":"proof","summary":"Since X is finite, the cycle containing x has finite length. We set n to be the length of this…","labels":[],"detail_key":"p9"},{"id":"n16091","layer":"informal","project":"p9","title":"prop:perm-pow-reduce","kind":"proposition","summary":"Let \\pi be a permutation such that \\pi^n(x) = x for some n \\in N. Then for any m, r \\in N, we h…","labels":["prop:perm-pow-reduce"],"detail_key":"p9"},{"id":"n16092","layer":"informal","project":"p9","title":"We prove this by induction on m.\\\\ If m = 0, the equation follows immediately.\\\\ If m > 0…","kind":"proof","summary":"We prove this by induction on m.\\\\ If m = 0, the equation follows immediately.\\\\ If m > 0, then…","labels":[],"detail_key":"p9"},{"id":"n16093","layer":"informal","project":"p9","title":"prop:forall-exists-lt-perm-pow-eq-perm-pow","kind":"proposition","summary":"Let \\pi be a permutation such that \\pi^n + 1(x) = x for some n \\in N. Then for any k \\in Z ther…","labels":["prop:forall-exists-lt-perm-pow-eq-perm-pow"],"detail_key":"p9"},{"id":"n16094","layer":"informal","project":"p9","title":"We use m = k \\bmod (n + 1). We will apply Proposition~\\refprop:perm-pow-reduce, which req…","kind":"proof","summary":"We use m = k \\bmod (n + 1). We will apply Proposition~\\refprop:perm-pow-reduce, which requires…","labels":[],"detail_key":"p9"},{"id":"n16095","layer":"informal","project":"p9","title":"def:visitCycle","kind":"definition","summary":"Let n \\in N, \\pi \\in S_n, x, y \\in [n] and v be a boolean array indexed by [n].\\\\ The function…","labels":["def:visitCycle"],"detail_key":"p9"},{"id":"n16096","layer":"informal","project":"p9","title":"prop:visitCycleAux-get!-true-of-get!-true","kind":"proposition","summary":"The function \\emphvisitCycleAux given v, (\\exists m \\in N, \\pi^m + 1(x) = y) returns an array t…","labels":["prop:visitCycleAux-get!-true-of-get!-true"],"detail_key":"p9"},{"id":"n16097","layer":"informal","project":"p9","title":"We prove this by induction on m.\\\\ If m = 0, then \\pi(x) = y is reached immediately and t…","kind":"proof","summary":"We prove this by induction on m.\\\\ If m = 0, then \\pi(x) = y is reached immediately and the fun…","labels":[],"detail_key":"p9"},{"id":"n16098","layer":"informal","project":"p9","title":"prop:visitCycleAux-spec","kind":"proposition","summary":"Let m \\in N be the smallest number satisfying \\pi^m + 1(x) = y. Then \\emphvisitCycleAux given v…","labels":["prop:visitCycleAux-spec"],"detail_key":"p9"},{"id":"n16099","layer":"informal","project":"p9","title":"This is proven by induction on m.\\\\ If m = 0, then \\pi(x) = y is reached immediately and…","kind":"proof","summary":"This is proven by induction on m.\\\\ If m = 0, then \\pi(x) = y is reached immediately and the fu…","labels":[],"detail_key":"p9"},{"id":"n16100","layer":"informal","project":"p9","title":"prop:visitCycle-spec","kind":"proposition","summary":"The function \\emphvisitCycle given \\pi, x, v returns an array identical to v, except that it is…","labels":["prop:visitCycle-spec"],"detail_key":"p9"},{"id":"n16101","layer":"informal","project":"p9","title":"Let m \\in N be the smallest number satisfying \\pi^m + 1(x) = x.\\\\ \\emphvisitCycle calls \\…","kind":"proof","summary":"Let m \\in N be the smallest number satisfying \\pi^m + 1(x) = x.\\\\ \\emphvisitCycle calls \\emphvi…","labels":[],"detail_key":"p9"},{"id":"n16102","layer":"informal","project":"p9","title":"def:computeNumCyclesOfPerm","kind":"definition","summary":"Let n, i, c \\in N, \\pi \\in S_n and v be some boolean array indexed by [n].\\\\ The function \\emph…","labels":["def:computeNumCyclesOfPerm"],"detail_key":"p9"},{"id":"n16103","layer":"informal","project":"p9","title":"prop:computeNumCyclesOfPermAux-exists-representatives","kind":"proposition","summary":"Let \\pi \\in S_n, v be a boolean array that is set to \\top at exactly those indices that are in…","labels":["prop:computeNumCyclesOfPermAux-exists-representatives"],"detail_key":"p9"},{"id":"n16104","layer":"informal","project":"p9","title":"This is proven by induction on i.\\\\ If i = 0 then the function returns c, which is alread…","kind":"proof","summary":"This is proven by induction on i.\\\\ If i = 0 then the function returns c, which is already the…","labels":[],"detail_key":"p9"},{"id":"n16105","layer":"informal","project":"p9","title":"prop:computeNumCyclesOfPerm-exists-representatives","kind":"proposition","summary":"The function \\emphcomputeNumCyclesOfPerm given \\pi computes the cardinality of some set of repr…","labels":["prop:computeNumCyclesOfPerm-exists-representatives"],"detail_key":"p9"},{"id":"n16106","layer":"informal","project":"p9","title":"The function \\emphcomputeNumCyclesOfPerm given \\pi returns the result of \\emphcomputeNumC…","kind":"proof","summary":"The function \\emphcomputeNumCyclesOfPerm given \\pi returns the result of \\emphcomputeNumCyclesO…","labels":[],"detail_key":"p9"},{"id":"n16107","layer":"informal","project":"p9","title":"prop:computeNumCyclesOfPerm-eq-numCyclesOfPerm","kind":"proposition","summary":"The function \\emphcomputeNumCyclesOfPerm given \\pi computes the number of cycles of \\pi.","labels":["prop:computeNumCyclesOfPerm-eq-numCyclesOfPerm"],"detail_key":"p9"},{"id":"n16108","layer":"informal","project":"p9","title":"By Proposition~\\refprop:computeNumCyclesOfPerm-exists-representatives, \\emphcomputeNumCyc…","kind":"proof","summary":"By Proposition~\\refprop:computeNumCyclesOfPerm-exists-representatives, \\emphcomputeNumCyclesOfP…","labels":[],"detail_key":"p9"},{"id":"n16109","layer":"informal","project":"p9","title":"prop:MulActionFin","kind":"proposition","summary":"Given a group action of G on X, we can define an induced group action of G on [|X|] with: g \\cd…","labels":["prop:MulActionFin"],"detail_key":"p9"},{"id":"n16110","layer":"informal","project":"p9","title":"1 \\cdot i = \\phi(1 \\cdot \\phi^-1(i)) = \\phi(\\phi^-1(1^-1 \\cdot i)) = i, \\\\ g \\cdot (h \\cd…","kind":"proof","summary":"1 \\cdot i = \\phi(1 \\cdot \\phi^-1(i)) = \\phi(\\phi^-1(1^-1 \\cdot i)) = i, \\\\ g \\cdot (h \\cdot i)…","labels":[],"detail_key":"p9"},{"id":"n16111","layer":"informal","project":"p9","title":"prop:numDistinctColorings-eq-numDistinctColorings-of-Fin","kind":"proposition","summary":"The number of distinct colorings of X with colors in Y under the group action of G on X is equa…","labels":["prop:numDistinctColorings-eq-numDistinctColorings-of-Fin"],"detail_key":"p9"},{"id":"n16112","layer":"informal","project":"p9","title":"We define the bijection: \\varphi : Y^X/G \\to Y^[|X|]/G, \\quad [f] \\mapsto [f \\circ \\phi^-…","kind":"proof","summary":"We define the bijection: \\varphi : Y^X/G \\to Y^[|X|]/G, \\quad [f] \\mapsto [f \\circ \\phi^-1] wit…","labels":[],"detail_key":"p9"},{"id":"n16113","layer":"informal","project":"p9","title":"prop:computeNumDistinctColorings","kind":"proposition","summary":"The function \\emphcomputeNumDistinctColorings computes the number of distinct colorings of X wi…","labels":["prop:computeNumDistinctColorings"],"detail_key":"p9"},{"id":"n16114","layer":"informal","project":"p9","title":"By Proposition~\\refprop:numDistinctColorings-eq-numDistinctColorings-of-Fin, we rewrite |…","kind":"proof","summary":"By Proposition~\\refprop:numDistinctColorings-eq-numDistinctColorings-of-Fin, we rewrite |Y^X/G|…","labels":[],"detail_key":"p9"},{"id":"n16115","layer":"informal","project":"p9","title":"def:numDistinctColoringsOfTrivialGroup","kind":"proposition","summary":"|Y^X / \\1\\| = |Y|^|X|.","labels":["def:numDistinctColoringsOfTrivialGroup"],"detail_key":"p9"},{"id":"n16116","layer":"informal","project":"p9","title":"We define a bijection \\varphi : Y^X/\\1\\ \\to Y^X by: \\varphi([f]) = f, \\quad \\varphi^-1(f)…","kind":"proof","summary":"We define a bijection \\varphi : Y^X/\\1\\ \\to Y^X by: \\varphi([f]) = f, \\quad \\varphi^-1(f) = [f]…","labels":[],"detail_key":"p9"},{"id":"n16117","layer":"informal","project":"p9","title":"def:numDistinctColoringsOfNecklace","kind":"definition","summary":"For n \\geq 1, the \\emphnumber of distinct colorings of a necklace with n beads and m colors is…","labels":["def:numDistinctColoringsOfNecklace"],"detail_key":"p9"},{"id":"n16118","layer":"informal","project":"p9","title":"prop:computeNumDistinctColoringsOfNecklace-eq-numDistinctColoringsOfNecklace","kind":"proposition","summary":"The number of distinct colorings of a necklace can be computed using the \\emphcomputeNumDistinc…","labels":["prop:computeNumDistinctColoringsOfNecklace-eq-numDistinctColoringsOfNecklace"],"detail_key":"p9"},{"id":"n16119","layer":"informal","project":"p9","title":"By Proposition~\\refprop:computeNumDistinctColorings.","kind":"proof","summary":"By Proposition~\\refprop:computeNumDistinctColorings.","labels":[],"detail_key":"p9"},{"id":"n16120","layer":"informal","project":"p9","title":"def:MulActionBracelet","kind":"definition","summary":"The dihedral group D_2n acts on Z_n with: r_i \\cdot x &= i + x, \\\\ sr_i \\cdot x &= n - 1 - (i +…","labels":["def:MulActionBracelet"],"detail_key":"p9"},{"id":"n16121","layer":"informal","project":"p9","title":"def:numDistinctColoringsOfBracelet","kind":"definition","summary":"For n \\geq 1, the \\emphnumber of distinct colorings of a bracelet with n beads and m colors is…","labels":["def:numDistinctColoringsOfBracelet"],"detail_key":"p9"},{"id":"n16122","layer":"informal","project":"p9","title":"prop:computeNumDistinctColoringsOfBracelet-eq-numDistinctColoringsOfBracelet","kind":"proposition","summary":"The number of distinct colorings of a bracelet can be computed using the \\emphcomputeNumDistinc…","labels":["prop:computeNumDistinctColoringsOfBracelet-eq-numDistinctColoringsOfBracelet"],"detail_key":"p9"},{"id":"n16123","layer":"informal","project":"p9","title":"By Proposition~\\refprop:computeNumDistinctColorings.","kind":"proof","summary":"By Proposition~\\refprop:computeNumDistinctColorings.","labels":[],"detail_key":"p9"},{"id":"n16124","layer":"informal","project":"p9","title":"def:rotationalSymmetriesOfCube","kind":"definition","summary":"We define two fundamental rotations of the cube: \\item r = (0\\ 1\\ 2\\ 3): A rotation of the cube…","labels":["def:rotationalSymmetriesOfCube"],"detail_key":"p9"},{"id":"n16125","layer":"informal","project":"p9","title":"prop:pow-add-val-eq-of-pow-eq-one","kind":"proposition","summary":"Let M be a monoid. If m \\in M satisfies m^n = 1, then for any i, j \\in [n], we have: m^(i + j)…","labels":["prop:pow-add-val-eq-of-pow-eq-one"],"detail_key":"p9"},{"id":"n16126","layer":"informal","project":"p9","title":"If i + j < n, the result follows immediately. Otherwise, since n \\leq i + j < 2n, we have…","kind":"proof","summary":"If i + j < n, the result follows immediately. Otherwise, since n \\leq i + j < 2n, we have: m^i…","labels":[],"detail_key":"p9"},{"id":"n16127","layer":"informal","project":"p9","title":"prop:mul_mem_rotationalSymmetriesOfCube","kind":"proposition","summary":"For all x \\in S, we have: r \\cdot x \\in S, \\quad s \\cdot x \\in S.","labels":["prop:mul_mem_rotationalSymmetriesOfCube"],"detail_key":"p9"},{"id":"n16128","layer":"informal","project":"p9","title":"For each x \\in S, we verify that the result remains in S.","kind":"proof","summary":"For each x \\in S, we verify that the result remains in S.","labels":[],"detail_key":"p9"},{"id":"n16129","layer":"informal","project":"p9","title":"prop:s-pow-r-pow-s-pow-mul-mem-rotationalSymmetriesOfCube","kind":"proposition","summary":"For all x \\in S and n, m, k \\in N, we have: s^n \\cdot r^m \\cdot s^k \\cdot x \\in S.","labels":["prop:s-pow-r-pow-s-pow-mul-mem-rotationalSymmetriesOfCube"],"detail_key":"p9"},{"id":"n16130","layer":"informal","project":"p9","title":"By induction on k. If k = 0, we perform another induction on m. If also m = 0, we perform…","kind":"proof","summary":"By induction on k. If k = 0, we perform another induction on m. If also m = 0, we perform anoth…","labels":[],"detail_key":"p9"},{"id":"n16131","layer":"informal","project":"p9","title":"prop:RotationalSymmetriesOfCubeSubgroup","kind":"proposition","summary":"S is a subgroup of S_6.","labels":["prop:RotationalSymmetriesOfCubeSubgroup"],"detail_key":"p9"},{"id":"n16132","layer":"informal","project":"p9","title":"\\item x_1, x_2 \\in S \\implies x_1 \\cdot x_2 = s^n \\cdot r^m \\cdot s^k \\cdot x_1 \\in S \\it…","kind":"proof","summary":"\\item x_1, x_2 \\in S \\implies x_1 \\cdot x_2 = s^n \\cdot r^m \\cdot s^k \\cdot x_1 \\in S \\item 1 =…","labels":[],"detail_key":"p9"},{"id":"n16133","layer":"informal","project":"p9","title":"def:numDistinctColoringsOfCube","kind":"definition","summary":"The \\emphnumber of distinct colorings of a cube with m colors is given by: |[m]^[6] / S|.","labels":["def:numDistinctColoringsOfCube"],"detail_key":"p9"},{"id":"n16134","layer":"informal","project":"p9","title":"prop:computeNumDistinctColoringsOfCube-eq-numDistinctColoringsOfCube","kind":"proposition","summary":"The number of distinct colorings of a cube can be computed using the \\emphcomputeNumDistinctCol…","labels":["prop:computeNumDistinctColoringsOfCube-eq-numDistinctColoringsOfCube"],"detail_key":"p9"},{"id":"n16135","layer":"informal","project":"p9","title":"By Proposition~\\refprop:computeNumDistinctColorings.","kind":"proof","summary":"By Proposition~\\refprop:computeNumDistinctColorings.","labels":[],"detail_key":"p9"},{"id":"n16136","layer":"informal","project":"p9","title":"def:numDistinctColoringsOfPerm","kind":"definition","summary":"The number of distinct colorings of \\emphn unordered, indistinguishable objects with m colors i…","labels":["def:numDistinctColoringsOfPerm"],"detail_key":"p9"},{"id":"n16137","layer":"informal","project":"p9","title":"def:numWeakCompositions","kind":"definition","summary":"The number of weak compositions of n into m parts is: 1 & \\textif n = m = 0, \\\\ 0 & \\textif m =…","labels":["def:numWeakCompositions"],"detail_key":"p9"},{"id":"n16138","layer":"informal","project":"p9","title":"def:colorings-contract-expand","kind":"definition","summary":"We define functions that contract and expand the domain and codomain of colorings: \\item \\emphc…","labels":["def:colorings-contract-expand"],"detail_key":"p9"},{"id":"n16139","layer":"informal","project":"p9","title":"prop:perm-perm-val-lt-of-perm-val-eq","kind":"proposition","summary":"Let \\pi \\in S_n be such that \\pi(i) = n for some i < n. Then \\pi(\\pi(i)) < n.","labels":["prop:perm-perm-val-lt-of-perm-val-eq"],"detail_key":"p9"},{"id":"n16140","layer":"informal","project":"p9","title":"Assume, for a contradiction, that \\pi(\\pi(i)) \\geq n. Then \\pi(\\pi(i)) = n = \\pi(i) \\impl…","kind":"proof","summary":"Assume, for a contradiction, that \\pi(\\pi(i)) \\geq n. Then \\pi(\\pi(i)) = n = \\pi(i) \\implies \\p…","labels":[],"detail_key":"p9"},{"id":"n16141","layer":"informal","project":"p9","title":"def:perm-contract-expand","kind":"definition","summary":"We define functions that contract and expand permutations: \\item \\emphpermContract: Contracts \\…","labels":["def:perm-contract-expand"],"detail_key":"p9"},{"id":"n16142","layer":"informal","project":"p9","title":"prop:numDistinctColoringsOfPerm-succ-succ","kind":"proposition","summary":"A recurrence formula for the number of distinct colorings of n + 1 unordered, indistinguishable…","labels":["prop:numDistinctColoringsOfPerm-succ-succ"],"detail_key":"p9"},{"id":"n16143","layer":"informal","project":"p9","title":"We construct a bijection \\varphi : \\left([m]^[n + 1]/S_n + 1\\right) \\cup \\left([m + 1]^[n…","kind":"proof","summary":"We construct a bijection \\varphi : \\left([m]^[n + 1]/S_n + 1\\right) \\cup \\left([m + 1]^[n]/S_n\\…","labels":[],"detail_key":"p9"},{"id":"n16144","layer":"informal","project":"p9","title":"prop:numDistinctColoringsOfPerm-eq-numWeakCompositions","kind":"proposition","summary":"The number of distinct colorings of n unordered, indistinguishable objects with m colors is equ…","labels":["prop:numDistinctColoringsOfPerm-eq-numWeakCompositions"],"detail_key":"p9"},{"id":"n16145","layer":"informal","project":"p9","title":"We use induction on m + n. \\\\ If m + n = 0, then n = m = 0, and we have |\\emptyset^\\empty…","kind":"proof","summary":"We use induction on m + n. \\\\ If m + n = 0, then n = m = 0, and we have |\\emptyset^\\emptyset /…","labels":[],"detail_key":"p9"},{"id":"n16146","layer":"informal","project":"p9","title":"def:stirlingFirstKind","kind":"definition","summary":"The \\hrefhttps://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind\\textttStirling number…","labels":["def:stirlingFirstKind"],"detail_key":"p9"},{"id":"n16147","layer":"informal","project":"p9","title":"prop:sum-stirlingFirstKind-mul-pow-eq-choose-mul-factorial","kind":"proposition","summary":"For any n, m \\in N with m > 0, we have \\sum_k = 0^n s(n, k) m^k = n! \\binomn + m - 1m - 1. Addi…","labels":["prop:sum-stirlingFirstKind-mul-pow-eq-choose-mul-factorial"],"detail_key":"p9"},{"id":"n16148","layer":"informal","project":"p9","title":"We apply \\emphPólya's enumeration theorem (Proposition~\\refprop:numDistinctColorings-mul-…","kind":"proof","summary":"We apply \\emphPólya's enumeration theorem (Proposition~\\refprop:numDistinctColorings-mul-card-g…","labels":[],"detail_key":"p9"},{"id":"n16149","layer":"formal","project":"p9","title":"CyclesOfGroupElements.CyclesOfGroup","kind":"def","summary":"(X : Type u) → G : Type v → [inst : Group G] → [inst : MulAction G X] → G → Type u","labels":[],"detail_key":"p9","name":"CyclesOfGroupElements.CyclesOfGroup","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16150","layer":"formal","project":"p9","title":"CyclesOfGroupElements.numCyclesOfGroup","kind":"def","summary":"(X : Type u) → G : Type v → [inst : Group G] → [inst_1 : MulAction G X] → (g : G) → [inst : Fin…","labels":[],"detail_key":"p9","name":"CyclesOfGroupElements.numCyclesOfGroup","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16151","layer":"formal","project":"p9","title":"CyclesOfGroupElements.numGroupOfNumCycles","kind":"def","summary":"(X : Type u) → (G : Type v) → [inst : Group G] → [inst_1 : MulAction G X] → [inst_2 : Fintype G…","labels":[],"detail_key":"p9","name":"CyclesOfGroupElements.numGroupOfNumCycles","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16152","layer":"formal","project":"p9","title":"DistinctColorings.MulActionColorings","kind":"def","summary":"(X : Type u) → (Y : Type v) → (G : Type w) → [inst : Group G] → [inst_1 : MulAction G X] → MulA…","labels":[],"detail_key":"p9","name":"DistinctColorings.MulActionColorings","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16153","layer":"formal","project":"p9","title":"DistinctColorings.numDistinctColorings","kind":"def","summary":"(X : Type u) → (Y : Type v) → (G : Type w) → [inst : Group G] → [inst_1 : MulAction G X] → [ins…","labels":[],"detail_key":"p9","name":"DistinctColorings.numDistinctColorings","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16154","layer":"formal","project":"p9","title":"Theorem.cycle_coloring_of_fixedBy_coloring","kind":"def","summary":"X : Type u → Y : Type v → G : Type w → [inst : Group G] → [inst_1 : MulAction G X] → (g : G) →…","labels":[],"detail_key":"p9","name":"Theorem.cycle_coloring_of_fixedBy_coloring","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16155","layer":"formal","project":"p9","title":"Theorem.equiv_of_fixedBy_coloring_of_cycle_coloring","kind":"def","summary":"X : Type u → Y : Type v → G : Type w → [inst : Group G] → [inst_1 : MulAction G X] → (g : G) →…","labels":[],"detail_key":"p9","name":"Theorem.equiv_of_fixedBy_coloring_of_cycle_coloring","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16156","layer":"formal","project":"p9","title":"Theorem.f_mem_fixedBy_iff_forall_eq_to_eq","kind":"theorem","summary":"∀ X : Type u Y : Type v G : Type w [inst : Group G] [inst_1 : MulAction G X] (g : G) (f : X → Y…","labels":[],"detail_key":"p9","name":"Theorem.f_mem_fixedBy_iff_forall_eq_to_eq","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16157","layer":"formal","project":"p9","title":"Theorem.fixedBy_coloring_of_cycle_coloring","kind":"def","summary":"X : Type u → Y : Type v → G : Type w → [inst : Group G] → [inst_1 : MulAction G X] → (g : G) →…","labels":[],"detail_key":"p9","name":"Theorem.fixedBy_coloring_of_cycle_coloring","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16158","layer":"formal","project":"p9","title":"Theorem.forall_card_pow_numCyclesOfGroup_eq_card_fixedBy","kind":"theorem","summary":"∀ X : Type u Y : Type v G : Type w [inst : Group G] [inst_1 : MulAction G X] [inst_2 : Fintype…","labels":[],"detail_key":"p9","name":"Theorem.forall_card_pow_numCyclesOfGroup_eq_card_fixedBy","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16159","layer":"formal","project":"p9","title":"Theorem.numDistinctColorings_eq_sum_card_pow_numCyclesOfGroup_div_card_group","kind":"theorem","summary":"∀ (X : Type u) (Y : Type v) (G : Type w) [inst : Group G] [inst_1 : MulAction G X] [inst_2 : Fi…","labels":[],"detail_key":"p9","name":"Theorem.numDistinctColorings_eq_sum_card_pow_numCyclesOfGroup_div_card_group","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16160","layer":"formal","project":"p9","title":"Theorem.numDistinctColorings_mul_card_group_eq_sum_card_pow_numCyclesOfGroup","kind":"theorem","summary":"∀ (X : Type u) (Y : Type v) (G : Type w) [inst : Group G] [inst_1 : MulAction G X] [inst_2 : Fi…","labels":[],"detail_key":"p9","name":"Theorem.numDistinctColorings_mul_card_group_eq_sum_card_pow_numCyclesOfGroup","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16161","layer":"formal","project":"p9","title":"Theorem.numDistinctColorings_mul_card_group_eq_sum_numGroupOfNumCycles_mul_card_pow","kind":"theorem","summary":"∀ (X : Type u) (Y : Type v) (G : Type w) [inst : Group G] [inst_1 : MulAction G X] [inst_2 : Fi…","labels":[],"detail_key":"p9","name":"Theorem.numDistinctColorings_mul_card_group_eq_sum_numGroupOfNumCycles_mul_card_pow","module":"PolyaEnumerationTheorem.Basic"},{"id":"n16162","layer":"formal","project":"p9","title":"ComputationNumberOfCycles.computeNumCyclesOfPerm","kind":"def","summary":"n : Nat → Equiv.Perm (Fin n) → 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Th…","labels":["lem:isTightMeasureSet_iff_basis"],"detail_key":"p10"},{"id":"n16232","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16233","layer":"informal","project":"p10","title":"Prokhorov's theorem","kind":"theorem","summary":"[Prokhorov's theorem] \\notready Let E be a complete separable metric space and let S \\subseteq…","labels":["thm:prokhorov"],"detail_key":"p10"},{"id":"n16234","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16235","layer":"informal","project":"p10","title":"lem:relatively_compact_of_tight","kind":"lemma","summary":"Let E be a metric space and let S \\subseteq P(E) be tight. Then the closure of S is compact.","labels":["lem:relatively_compact_of_tight"],"detail_key":"p10"},{"id":"n16236","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16237","layer":"informal","project":"p10","title":"def:separates_points","kind":"definition","summary":"\\mathlibok A set F of functions E \\to F separates points in E if for all x, y \\in E with x \\ne…","labels":["def:separates_points"],"detail_key":"p10"},{"id":"n16238","layer":"informal","project":"p10","title":"lem:bounded_continuous_separating","kind":"lemma","summary":"\\mathlibok In a Borel space E, if two probability measures \\mu, \\nu on E are such that \\mu[f] =…","labels":["lem:bounded_continuous_separating"],"detail_key":"p10"},{"id":"n16239","layer":"informal","project":"p10","title":"The Mathlib lemma \\textttMeasureTheory.FiniteMeasure.ext\\_of\\_forall\\_lintegral\\_eq shows…","kind":"proof","summary":"The Mathlib lemma \\textttMeasureTheory.FiniteMeasure.ext\\_of\\_forall\\_lintegral\\_eq shows that…","labels":[],"detail_key":"p10"},{"id":"n16240","layer":"informal","project":"p10","title":"lem:exp_character","kind":"lemma","summary":"\\mathlibok x \\mapsto (t \\mapsto \\exp(i \\langle t, x \\rangle)) is a monoid homomorphism from (E,…","labels":["lem:exp_character"],"detail_key":"p10"},{"id":"n16241","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16242","layer":"informal","project":"p10","title":"lem:starSubalgebra_expPoly","kind":"lemma","summary":"\\mathlibok The functions of the form x \\mapsto \\sum_k=1^n a_k e^i\\langle t_k, x\\rangle for n \\i…","labels":["lem:starSubalgebra_expPoly"],"detail_key":"p10"},{"id":"n16243","layer":"informal","project":"p10","title":"The monoid homomorphism can be lifted to a homomorphism of algebras from \\textttAddMonoid…","kind":"proof","summary":"The monoid homomorphism can be lifted to a homomorphism of algebras from \\textttAddMonoidAlgebr…","labels":[],"detail_key":"p10"},{"id":"n16244","layer":"informal","project":"p10","title":"lem:separates_points_expPoly","kind":"lemma","summary":"\\mathlibok The star-subalgebra M separates points.","labels":["lem:separates_points_expPoly"],"detail_key":"p10"},{"id":"n16245","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16246","layer":"informal","project":"p10","title":"lem:integral_restrict_compact","kind":"lemma","summary":"\\mathlibok Let \\mu be a finite measure on a Borel space E and let K be a measurable compact set…","labels":["lem:integral_restrict_compact"],"detail_key":"p10"},{"id":"n16247","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16248","layer":"informal","project":"p10","title":"lem:introduce_exponential","kind":"lemma","summary":"\\mathlibok For any f \\in C_b(E, R) and a probability measure \\mu, \\mu[f] = \\lim_\\varepsilon \\to…","labels":["lem:introduce_exponential"],"detail_key":"p10"},{"id":"n16249","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16250","layer":"informal","project":"p10","title":"lem:dist_integral_mulExpNegMulSq_comp_le","kind":"lemma","summary":"\\mathlibok Let \\mu, \\nu be two finite measures in a standard Borel space E. Let A be a subalgeb…","labels":["lem:dist_integral_mulExpNegMulSq_comp_le"],"detail_key":"p10"},{"id":"n16251","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16252","layer":"informal","project":"p10","title":"Subalgebras separating points","kind":"theorem","summary":"[Subalgebras separating points] \\mathlibok Let E be a complete separable pseudo-metric space. L…","labels":["thm:separating_starSubalgebra"],"detail_key":"p10"},{"id":"n16253","layer":"informal","project":"p10","title":"Let \\mu and \\nu be two probability measures such that \\mu[f] = \\nu[f] for all f \\in A. We…","kind":"proof","summary":"Let \\mu and \\nu be two probability measures such that \\mu[f] = \\nu[f] for all f \\in A. We want…","labels":[],"detail_key":"p10"},{"id":"n16254","layer":"informal","project":"p10","title":"lem:cvg_of_separating","kind":"lemma","summary":"Let A be a subalgebra of C_b(E, C) that separates points. Let \\mu, \\mu_1, \\mu_2, \\ldots be prob…","labels":["lem:cvg_of_separating"],"detail_key":"p10"},{"id":"n16255","layer":"informal","project":"p10","title":"To show convergence to \\mu, it suffices to show that every subsequence has a subsequence…","kind":"proof","summary":"To show convergence to \\mu, it suffices to show that every subsequence has a subsequence that c…","labels":[],"detail_key":"p10"},{"id":"n16256","layer":"informal","project":"p10","title":"Characteristic function","kind":"definition","summary":"[Characteristic function] \\mathlibok Let \\mu be a measure on a real inner product space E. The…","labels":["def:charFun"],"detail_key":"p10"},{"id":"n16257","layer":"informal","project":"p10","title":"lem:charFun_bounded","kind":"lemma","summary":"\\mathlibok For all t, \\Vert\\hat\\mu(t)\\Vert \\le 1.","labels":["lem:charFun_bounded"],"detail_key":"p10"},{"id":"n16258","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16259","layer":"informal","project":"p10","title":"lem:charFun_contDiff","kind":"lemma","summary":"If \\mu[|X|^n] < \\infty then the characteristic function of \\mu is continuously differentiable o…","labels":["lem:charFun_contDiff"],"detail_key":"p10"},{"id":"n16260","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16261","layer":"informal","project":"p10","title":"lem:charFun_continuous","kind":"lemma","summary":"The characteristic function is a continuous function.","labels":["lem:charFun_continuous"],"detail_key":"p10"},{"id":"n16262","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16263","layer":"informal","project":"p10","title":"lem:charFun_neg","kind":"lemma","summary":"\\mathlibok \\hat\\mu(-t) = \\overline\\hat\\mu(t).","labels":["lem:charFun_neg"],"detail_key":"p10"},{"id":"n16264","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16265","layer":"informal","project":"p10","title":"lem:charFun_smul","kind":"lemma","summary":"\\mathlibok For a \\in R and X a random variable with law \\mu, the characteristic function of a X…","labels":["lem:charFun_smul"],"detail_key":"p10"},{"id":"n16266","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16267","layer":"informal","project":"p10","title":"lem:charFun_add_of_indep","kind":"lemma","summary":"\\mathlibok If two random variables X, Y : \\Omega \\to S with laws \\mu and \\nu are independent, t…","labels":["lem:charFun_add_of_indep"],"detail_key":"p10"},{"id":"n16268","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16269","layer":"informal","project":"p10","title":"lem:integral_exp_I","kind":"lemma","summary":"\\mathlibok \\int_-r^r e^i t \\, d t = 2 \\sin r \\: .","labels":["lem:integral_exp_I"],"detail_key":"p10"},{"id":"n16270","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16271","layer":"informal","project":"p10","title":"lem:integral_charFun","kind":"lemma","summary":"\\mathlibok \\int_-r^r \\hat\\mu(t) dt = 2 r \\int \\frac\\sin(r x)r x \\, d\\mu(x)","labels":["lem:integral_charFun"],"detail_key":"p10"},{"id":"n16272","layer":"informal","project":"p10","title":"We will use Fubini's theorem to exchange the order of integration. \\int_-r^r \\hat\\mu(t) \\…","kind":"proof","summary":"We will use Fubini's theorem to exchange the order of integration. \\int_-r^r \\hat\\mu(t) \\, dt &…","labels":[],"detail_key":"p10"},{"id":"n16273","layer":"informal","project":"p10","title":"lem:charFun_bound_large","kind":"lemma","summary":"\\mathlibok For \\mu a probability measure on R and r > 0, \\mu \\left\\x \\mid |x| > r\\right\\ &\\le \\…","labels":["lem:charFun_bound_large"],"detail_key":"p10"},{"id":"n16274","layer":"informal","project":"p10","title":"We will use Lemma~\\reflem:integral_charFun with r replaced by 2/r. \\fracr2 \\int_-2/r^2/r…","kind":"proof","summary":"We will use Lemma~\\reflem:integral_charFun with r replaced by 2/r. \\fracr2 \\int_-2/r^2/r (1 - \\…","labels":[],"detail_key":"p10"},{"id":"n16275","layer":"informal","project":"p10","title":"lem:charFun_bound_inner","kind":"lemma","summary":"\\mathlibok For \\mu a probability measure on an inner product space E and r > 0, a \\in E, \\mu \\l…","labels":["lem:charFun_bound_inner"],"detail_key":"p10"},{"id":"n16276","layer":"informal","project":"p10","title":"TODO (see code)","kind":"proof","summary":"TODO (see code)","labels":[],"detail_key":"p10"},{"id":"n16277","layer":"informal","project":"p10","title":"lem:tendsto_M_of_tendsto_charFun","kind":"lemma","summary":"Let \\mu, \\mu_1, \\mu_2, \\ldots be probability measures on R^d with characteristic functions \\hat…","labels":["lem:tendsto_M_of_tendsto_charFun"],"detail_key":"p10"},{"id":"n16278","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16279","layer":"informal","project":"p10","title":"lem:ext_charFun","kind":"lemma","summary":"\\mathlibok Two finite measures on a complete separable metric space are equal iff they have the…","labels":["lem:ext_charFun"],"detail_key":"p10"},{"id":"n16280","layer":"informal","project":"p10","title":"See Mathlib PR \\#19783. The sub-algebra M of exponential polynomials is separating by Lem…","kind":"proof","summary":"See Mathlib PR \\#19783. The sub-algebra M of exponential polynomials is separating by Lemma~\\re…","labels":[],"detail_key":"p10"},{"id":"n16281","layer":"informal","project":"p10","title":"lem:tight_of_tendsto_charFun","kind":"lemma","summary":"Let (\\mu_n)_n \\in N be measures on R^d with characteristic functions (\\hat\\mu_n). If \\hat\\mu_n…","labels":["lem:tight_of_tendsto_charFun"],"detail_key":"p10"},{"id":"n16282","layer":"informal","project":"p10","title":"By Lemma~\\reflem:charFun_bound_inner and dominated convergence, for all a \\in R^d and r >…","kind":"proof","summary":"By Lemma~\\reflem:charFun_bound_inner and dominated convergence, for all a \\in R^d and r > 0, \\l…","labels":[],"detail_key":"p10"},{"id":"n16283","layer":"informal","project":"p10","title":"Convergence of characteristic functions and weak convergence of measures","kind":"theorem","summary":"[Convergence of characteristic functions and weak convergence of measures] Let \\mu, \\mu_1, \\mu_…","labels":["thm:charFun_tendsto_iff_measure_tendsto"],"detail_key":"p10"},{"id":"n16284","layer":"informal","project":"p10","title":"For all t, x \\mapsto e^i \\langle t, x \\rangle is a bounded continuous function. Hence by…","kind":"proof","summary":"For all t, x \\mapsto e^i \\langle t, x \\rangle is a bounded continuous function. Hence by the de…","labels":[],"detail_key":"p10"},{"id":"n16285","layer":"informal","project":"p10","title":"def:gaussianReal","kind":"definition","summary":"\\mathlibok A measure on R is Gaussian if it is equal to N(m, \\sigma^2) for some m \\in R and \\si…","labels":["def:gaussianReal"],"detail_key":"p10"},{"id":"n16286","layer":"informal","project":"p10","title":"lem:isProbabilityMeasure_gaussianReal","kind":"lemma","summary":"\\mathlibok A real Gaussian measure is a probability measure.","labels":["lem:isProbabilityMeasure_gaussianReal"],"detail_key":"p10"},{"id":"n16287","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16288","layer":"informal","project":"p10","title":"lem:gaussian_charFun","kind":"lemma","summary":"\\mathlibok The Gaussian distribution N(m, \\sigma^2) has characteristic function \\phi(t) = e^itm…","labels":["lem:gaussian_charFun"],"detail_key":"p10"},{"id":"n16289","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16290","layer":"informal","project":"p10","title":"lem:add_gaussianReal","kind":"lemma","summary":"\\mathlibok The sum of two independent real Gaussian random variables is Gaussian.","labels":["lem:add_gaussianReal"],"detail_key":"p10"},{"id":"n16291","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16292","layer":"informal","project":"p10","title":"def:isGaussian","kind":"definition","summary":"\\mathlibok A Borel measure \\mu on a separable Banach space E is said to be Gaussian if for all…","labels":["def:isGaussian"],"detail_key":"p10"},{"id":"n16293","layer":"informal","project":"p10","title":"lem:isProbabilityMeasure_isGaussian","kind":"lemma","summary":"\\mathlibok A Gaussian measure is a probability measure.","labels":["lem:isProbabilityMeasure_isGaussian"],"detail_key":"p10"},{"id":"n16294","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16295","layer":"informal","project":"p10","title":"lem:isGaussian_gaussianReal","kind":"lemma","summary":"\\mathlibok The real Gaussian measures are Gaussian measures in the sense of Definition~\\refdef:…","labels":["lem:isGaussian_gaussianReal"],"detail_key":"p10"},{"id":"n16296","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16297","layer":"informal","project":"p10","title":"lem:isGaussian_add","kind":"lemma","summary":"\\mathlibok The sum of two independent Gaussian random variables is Gaussian.","labels":["lem:isGaussian_add"],"detail_key":"p10"},{"id":"n16298","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16299","layer":"informal","project":"p10","title":"lem:stdGaussian_finiteDimensional","kind":"lemma","summary":"Let E be a finite dimensional real inner product space and let b_1, \\ldots, b_d be an orthonorm…","labels":["lem:stdGaussian_finiteDimensional"],"detail_key":"p10"},{"id":"n16300","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16301","layer":"informal","project":"p10","title":"lem:deriv_charFun","kind":"lemma","summary":"Let X be a real random variable with characteristic function \\phi with E[\\vert X \\vert^n] < \\in…","labels":["lem:deriv_charFun"],"detail_key":"p10"},{"id":"n16302","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16303","layer":"informal","project":"p10","title":"Peano form of Taylor's theorem","kind":"lemma","summary":"[Peano form of Taylor's theorem] \\mathlibok For a n-times continuously differentiable function…","labels":["lem:taylor_peano"],"detail_key":"p10"},{"id":"n16304","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16305","layer":"informal","project":"p10","title":"lem:iIndepFun_iff_pi_map_eq_map","kind":"lemma","summary":"\\mathlibok A finite collection of random variables are independent iff their joint law is the p…","labels":["lem:iIndepFun_iff_pi_map_eq_map"],"detail_key":"p10"},{"id":"n16306","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16307","layer":"informal","project":"p10","title":"lem:charFun_taylor","kind":"lemma","summary":"Let X be a real random variable with characteristic function \\phi, with E[\\vert X \\vert^n] < \\i…","labels":["lem:charFun_taylor"],"detail_key":"p10"},{"id":"n16308","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16309","layer":"informal","project":"p10","title":"lem:tendsto_pow_exp_of_isLittleO","kind":"lemma","summary":"For t\\inC, \\lim_n\\to\\infty(1+t/n+o(1/n))^n=\\exp(t) (where the little-o term may be complex).","labels":["lem:tendsto_pow_exp_of_isLittleO"],"detail_key":"p10"},{"id":"n16310","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16311","layer":"informal","project":"p10","title":"Central limit theorem","kind":"theorem","summary":"[Central limit theorem] Let X_1, X_2, \\ldots be i.i.d. real random variables with mean 0 and va…","labels":["thm:clt"],"detail_key":"p10"},{"id":"n16312","layer":"informal","project":"p10","title":"Let S_n = \\frac1\\sqrtn\\sum_k=1^n X_k. Let \\phi be the characteristic function of X_k. By…","kind":"proof","summary":"Let S_n = \\frac1\\sqrtn\\sum_k=1^n X_k. Let \\phi be the characteristic function of X_k. By Lemma~…","labels":[],"detail_key":"p10"},{"id":"n16313","layer":"informal","project":"p10","title":"Cramér-Wold","kind":"theorem","summary":"[Cramér-Wold] Let X, X_1, X_2, \\ldots be random variables in R^d. Then X_n \\xrightarrowd X iff…","labels":["thm:cramer_wold"],"detail_key":"p10"},{"id":"n16314","layer":"informal","project":"p10","title":"By Theorem~\\refthm:charFun_tendsto_iff_measure_tendsto, the convergence in distribution o…","kind":"proof","summary":"By Theorem~\\refthm:charFun_tendsto_iff_measure_tendsto, the convergence in distribution of the…","labels":[],"detail_key":"p10"},{"id":"n16315","layer":"informal","project":"p10","title":"Central limit theorem in R^d","kind":"theorem","summary":"[Central limit theorem in R^d] Let X_1, X_2, \\ldots be i.i.d. random variables in R^d with mean…","labels":["thm:multivariate_clt"],"detail_key":"p10"},{"id":"n16316","layer":"informal","project":"p10","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p10"},{"id":"n16317","layer":"informal","project":"p11","title":"Characteristic function","kind":"definition","summary":"[Characteristic function] \\mathlibok The characteristic function of a measure \\mu on a normed s…","labels":["def:charFunDual"],"detail_key":"p11"},{"id":"n16318","layer":"informal","project":"p11","title":"thm:ext_of_charFunDual","kind":"theorem","summary":"\\mathlibok In a separable Banach space, if two finite measures have same characteristic functio…","labels":["thm:ext_of_charFunDual"],"detail_key":"p11"},{"id":"n16319","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16320","layer":"informal","project":"p11","title":"Characteristic function","kind":"definition","summary":"[Characteristic function] \\mathlibok The characteristic function of a measure \\mu on an inner p…","labels":["def:charFun"],"detail_key":"p11"},{"id":"n16321","layer":"informal","project":"p11","title":"thm:ext_of_charFun","kind":"theorem","summary":"\\mathlibok In a separable Hilbert space, if two finite measures have same characteristic functi…","labels":["thm:ext_of_charFun"],"detail_key":"p11"},{"id":"n16322","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16323","layer":"informal","project":"p11","title":"lem:charFun_map_eq_charFunDual_smul","kind":"lemma","summary":"\\mathlibok Let \\mu be a measure on F and let L \\in F^*. Then \\widehatL_*\\mu(x) &= \\hat\\mu(x \\cd…","labels":["lem:charFun_map_eq_charFunDual_smul"],"detail_key":"p11"},{"id":"n16324","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16325","layer":"informal","project":"p11","title":"lem:charFunDual_map","kind":"lemma","summary":"\\mathlibok Let \\mu be a measure on a normed space E and let L be a continuous linear map from E…","labels":["lem:charFunDual_map"],"detail_key":"p11"},{"id":"n16326","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16327","layer":"informal","project":"p11","title":"Covariance","kind":"definition","summary":"[Covariance] \\mathlibok The covariance bilinear form of a measure \\mu on F with finite second m…","labels":["def:covarianceBilin"],"detail_key":"p11"},{"id":"n16328","layer":"informal","project":"p11","title":"lem:covarianceBilin_same_eq_variance","kind":"lemma","summary":"\\mathlibok For \\mu a measure on F with finite second moment and L \\in F^*, C_\\mu(L, L) = V_\\mu[…","labels":["lem:covarianceBilin_same_eq_variance"],"detail_key":"p11"},{"id":"n16329","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16330","layer":"informal","project":"p11","title":"Covariance in a Hilbert space","kind":"definition","summary":"[Covariance in a Hilbert space] \\mathlibok The covariance bilinear form of a finite measure \\mu…","labels":["def:covInnerBilin"],"detail_key":"p11"},{"id":"n16331","layer":"informal","project":"p11","title":"lem:covInnerBilin_map","kind":"lemma","summary":"\\mathlibok Let E and F be two Hilbert spaces with F finite dimensional, \\mu a finite measure on…","labels":["lem:covInnerBilin_map"],"detail_key":"p11"},{"id":"n16332","layer":"informal","project":"p11","title":"C'_L_*\\mu(u, v) &= (L_*\\mu)\\left[\\langle u, x - m_L_*\\mu\\rangle \\langle x - m_L_*\\mu, v \\…","kind":"proof","summary":"C'_L_*\\mu(u, v) &= (L_*\\mu)\\left[\\langle u, x - m_L_*\\mu\\rangle \\langle x - m_L_*\\mu, v \\rangle…","labels":[],"detail_key":"p11"},{"id":"n16333","layer":"informal","project":"p11","title":"Covariance matrix","kind":"definition","summary":"[Covariance matrix] The covariance matrix of a finite measure \\mu with finite second moment on…","labels":["def:covMatrix"],"detail_key":"p11"},{"id":"n16334","layer":"informal","project":"p11","title":"lem:covMatrix_map","kind":"lemma","summary":"Let E and F be two finite dimensional inner product spaces, \\mu a measure on E with finite seco…","labels":["lem:covMatrix_map"],"detail_key":"p11"},{"id":"n16335","layer":"informal","project":"p11","title":"On the left-hand side we have \\langle e_i, \\Sigma_L_*\\mu e_j\\rangle = C'_L_*\\mu(e_i, e_j)…","kind":"proof","summary":"On the left-hand side we have \\langle e_i, \\Sigma_L_*\\mu e_j\\rangle = C'_L_*\\mu(e_i, e_j) = C'_…","labels":[],"detail_key":"p11"},{"id":"n16336","layer":"informal","project":"p11","title":"Law of a stochastic process","kind":"definition","summary":"[Law of a stochastic process] The law of a stochastic process X is the measure on the measurabl…","labels":["def:processLaw"],"detail_key":"p11"},{"id":"n16337","layer":"informal","project":"p11","title":"Modification","kind":"definition","summary":"[Modification] We say that a stochastic process Y is a \\emphmodification of another stochastic…","labels":["def:modification"],"detail_key":"p11"},{"id":"n16338","layer":"informal","project":"p11","title":"Indistinguishable","kind":"definition","summary":"[Indistinguishable] We say that a stochastic processes Y is a \\emphindistinguishable from X if…","labels":["def:indistinguishable"],"detail_key":"p11"},{"id":"n16339","layer":"informal","project":"p11","title":"lem:Indistinguishable.Modification","kind":"lemma","summary":"If Y is indistinguishable from X, then Y is a modification of X.","labels":["lem:Indistinguishable.Modification"],"detail_key":"p11"},{"id":"n16340","layer":"informal","project":"p11","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p11"},{"id":"n16341","layer":"informal","project":"p11","title":"lem:map_eq_of_modification","kind":"lemma","summary":"\\mathlibok Let X, Y : T \\to \\Omega \\to E be two stochastic processes that are modifications of…","labels":["lem:map_eq_of_modification"],"detail_key":"p11"},{"id":"n16342","layer":"informal","project":"p11","title":"By the modification property, almost surely X_t_i = Y_t_i for all i \\in [n]. Thus the fun…","kind":"proof","summary":"By the modification property, almost surely X_t_i = Y_t_i for all i \\in [n]. Thus the function…","labels":[],"detail_key":"p11"},{"id":"n16343","layer":"informal","project":"p11","title":"lem:map_eq_iff","kind":"lemma","summary":"\\mathlibok Let X, Y : T \\to \\Omega \\to E be two stochastic processes. Then X and Y have same fi…","labels":["lem:map_eq_iff"],"detail_key":"p11"},{"id":"n16344","layer":"informal","project":"p11","title":"TODO: consider the \\pi-system of cylinder sets.","kind":"proof","summary":"TODO: consider the \\pi-system of cylinder sets.","labels":[],"detail_key":"p11"},{"id":"n16345","layer":"informal","project":"p11","title":"lem:indistinguishable_of_modification_of_continuous","kind":"lemma","summary":"Let T and E be topological spaces and suppose that T is separable Hausdorff. Let X, Y : T \\to \\…","labels":["lem:indistinguishable_of_modification_of_continuous"],"detail_key":"p11"},{"id":"n16346","layer":"informal","project":"p11","title":"Since T is separable, it has a countable dense subset D. Since D is countable, (\\forall t…","kind":"proof","summary":"Since T is separable, it has a countable dense subset D. Since D is countable, (\\forall t \\in D…","labels":[],"detail_key":"p11"},{"id":"n16347","layer":"informal","project":"p11","title":"Real Gaussian measure","kind":"definition","summary":"[Real Gaussian measure] \\mathlibok The real Gaussian measure with mean \\mu \\in R and variance \\…","labels":["def:gaussianReal"],"detail_key":"p11"},{"id":"n16348","layer":"informal","project":"p11","title":"lem:charFun_gaussianReal","kind":"lemma","summary":"\\mathlibok The characteristic function of a real Gaussian measure with mean \\mu and variance \\s…","labels":["lem:charFun_gaussianReal"],"detail_key":"p11"},{"id":"n16349","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16350","layer":"informal","project":"p11","title":"lem:centralMoment_two_mul_gaussianReal","kind":"lemma","summary":"The central moment of order 2n of a real Gaussian measure N(\\mu, \\sigma^2) is given by E[(X - \\…","labels":["lem:centralMoment_two_mul_gaussianReal"],"detail_key":"p11"},{"id":"n16351","layer":"informal","project":"p11","title":"E[(X - \\mu)^2n] &= \\int_-\\infty^\\infty (x - \\mu)^2n \\frac1\\sqrt2 \\pi \\sigma^2 e^-\\frac(x…","kind":"proof","summary":"E[(X - \\mu)^2n] &= \\int_-\\infty^\\infty (x - \\mu)^2n \\frac1\\sqrt2 \\pi \\sigma^2 e^-\\frac(x - \\mu)…","labels":[],"detail_key":"p11"},{"id":"n16352","layer":"informal","project":"p11","title":"Gaussian measure","kind":"definition","summary":"[Gaussian measure] \\mathlibok A measure \\mu on F is Gaussian if for every continuous linear for…","labels":["def:IsGaussian"],"detail_key":"p11"},{"id":"n16353","layer":"informal","project":"p11","title":"lem:IsGaussian.IsProbabilityMeasure","kind":"lemma","summary":"\\mathlibok A Gaussian measure is a probability measure.","labels":["lem:IsGaussian.IsProbabilityMeasure"],"detail_key":"p11"},{"id":"n16354","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16355","layer":"informal","project":"p11","title":"thm:isGaussian_iff_charFunDual_eq","kind":"theorem","summary":"\\mathlibok A finite measure \\mu on F is Gaussian if and only if for every continuous linear for…","labels":["thm:isGaussian_iff_charFunDual_eq"],"detail_key":"p11"},{"id":"n16356","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16357","layer":"informal","project":"p11","title":"lem:isGaussian_map","kind":"lemma","summary":"\\mathlibok Let F, G be two Banach spaces, let \\mu be a Gaussian measure on F and let T : F \\to…","labels":["lem:isGaussian_map"],"detail_key":"p11"},{"id":"n16358","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16359","layer":"informal","project":"p11","title":"lem:isGaussian_add_const","kind":"lemma","summary":"Let \\mu be a Gaussian measure on F and let c \\in F. Then the measure \\mu translated by c (the m…","labels":["lem:isGaussian_add_const"],"detail_key":"p11"},{"id":"n16360","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16361","layer":"informal","project":"p11","title":"lem:isGaussian_conv","kind":"lemma","summary":"\\mathlibok The convolution of two Gaussian measures is a Gaussian measure.","labels":["lem:isGaussian_conv"],"detail_key":"p11"},{"id":"n16362","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16363","layer":"informal","project":"p11","title":"thm:exists_integrable_exp_sq_of_map_rotation_eq_self","kind":"theorem","summary":"\\mathlibok Let \\mu be a finite measure on F such that \\mu \\times \\mu is invariant under the rot…","labels":["thm:exists_integrable_exp_sq_of_map_rotation_eq_self"],"detail_key":"p11"},{"id":"n16364","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16365","layer":"informal","project":"p11","title":"lem:IsGaussian.map_rotation_eq_self","kind":"lemma","summary":"\\mathlibok For a centered Gaussian measure \\mu, \\mu \\times \\mu is invariant by rotation.","labels":["lem:IsGaussian.map_rotation_eq_self"],"detail_key":"p11"},{"id":"n16366","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16367","layer":"informal","project":"p11","title":"Fernique's theorem","kind":"theorem","summary":"[Fernique's theorem] \\mathlibok For a Gaussian measure, there exists C > 0 such that the functi…","labels":["thm:IsGaussian.exists_integrable_exp_sq"],"detail_key":"p11"},{"id":"n16368","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16369","layer":"informal","project":"p11","title":"lem:IsGaussian.memLp_id","kind":"lemma","summary":"\\mathlibok A Gaussian measure \\mu has finite moments of all orders. In particular, there is a w…","labels":["lem:IsGaussian.memLp_id"],"detail_key":"p11"},{"id":"n16370","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16371","layer":"informal","project":"p11","title":"lem:isGaussian_iff_charFun_eq","kind":"lemma","summary":"A finite measure \\mu on a Hilbert space E is Gaussian if and only if for every t \\in E, the cha…","labels":["lem:isGaussian_iff_charFun_eq"],"detail_key":"p11"},{"id":"n16372","layer":"informal","project":"p11","title":"By Theorem~\\refthm:isGaussian_iff_charFunDual_eq, \\mu is Gaussian iff for every continuou…","kind":"proof","summary":"By Theorem~\\refthm:isGaussian_iff_charFunDual_eq, \\mu is Gaussian iff for every continuous line…","labels":[],"detail_key":"p11"},{"id":"n16373","layer":"informal","project":"p11","title":"lem:IsGaussian.charFun_eq","kind":"lemma","summary":"The characteristic function of a Gaussian measure \\mu on E is given by \\hat\\mu(t) = \\exp\\left(i…","labels":["lem:IsGaussian.charFun_eq"],"detail_key":"p11"},{"id":"n16374","layer":"informal","project":"p11","title":"By Lemma~\\reflem:isGaussian_iff_charFun_eq, for every t \\in E, \\hat\\mu(t) = \\exp\\left(i \\…","kind":"proof","summary":"By Lemma~\\reflem:isGaussian_iff_charFun_eq, for every t \\in E, \\hat\\mu(t) = \\exp\\left(i \\mu[\\la…","labels":[],"detail_key":"p11"},{"id":"n16375","layer":"informal","project":"p11","title":"lem:isGaussian_iff_gaussian_charFun","kind":"lemma","summary":"A finite measure \\mu on E is Gaussian if and only if there exists m \\in E and C positive semide…","labels":["lem:isGaussian_iff_gaussian_charFun"],"detail_key":"p11"},{"id":"n16376","layer":"informal","project":"p11","title":"Lemma~\\reflem:IsGaussian.charFun_eq states that the characteristic function of a Gaussian…","kind":"proof","summary":"Lemma~\\reflem:IsGaussian.charFun_eq states that the characteristic function of a Gaussian measu…","labels":[],"detail_key":"p11"},{"id":"n16377","layer":"informal","project":"p11","title":"lem:IsGaussian.ext_iff","kind":"lemma","summary":"Two Gaussian measures \\mu and \\nu on a separable Hilbert space are equal if and only if they ha…","labels":["lem:IsGaussian.ext_iff"],"detail_key":"p11"},{"id":"n16378","layer":"informal","project":"p11","title":"The forward direction is immediate. For the converse direction, it is enough to show that…","kind":"proof","summary":"The forward direction is immediate. For the converse direction, it is enough to show that \\mu a…","labels":[],"detail_key":"p11"},{"id":"n16379","layer":"informal","project":"p11","title":"Standard Gaussian measure","kind":"definition","summary":"[Standard Gaussian measure] Let (e_1, \\ldots, e_d) be an orthonormal basis of E and let \\mu be…","labels":["def:stdGaussian"],"detail_key":"p11"},{"id":"n16380","layer":"informal","project":"p11","title":"lem:integral_eval_pi","kind":"lemma","summary":"\\mathlibok For \\mu_1, \\ldots, \\mu_d probability measures on R and f : R \\to R integrable with r…","labels":["lem:integral_eval_pi"],"detail_key":"p11"},{"id":"n16381","layer":"informal","project":"p11","title":"As f is integrable, we can use Fubini theorem to obtain that \\int f(x_i) \\, d(\\mu_1 \\time…","kind":"proof","summary":"As f is integrable, we can use Fubini theorem to obtain that \\int f(x_i) \\, d(\\mu_1 \\times \\ldo…","labels":[],"detail_key":"p11"},{"id":"n16382","layer":"informal","project":"p11","title":"lem:isCentered_stdGaussian","kind":"lemma","summary":"The standard Gaussian measure on E is centered, i.e., \\mu[L] = 0 for every L \\in E^*.","labels":["lem:isCentered_stdGaussian"],"detail_key":"p11"},{"id":"n16383","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16384","layer":"informal","project":"p11","title":"lem:isProbabilityMeasure_stdGaussian","kind":"lemma","summary":"The standard Gaussian measure is a probability measure.","labels":["lem:isProbabilityMeasure_stdGaussian"],"detail_key":"p11"},{"id":"n16385","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16386","layer":"informal","project":"p11","title":"lem:charFun_stdGaussian","kind":"lemma","summary":"The characteristic function of the standard Gaussian measure on E is given by \\hat\\mu(t) = \\exp…","labels":["lem:charFun_stdGaussian"],"detail_key":"p11"},{"id":"n16387","layer":"informal","project":"p11","title":"Denote by \\nu the standard Gaussian measure on R. This is a straightforward computation:…","kind":"proof","summary":"Denote by \\nu the standard Gaussian measure on R. This is a straightforward computation: \\hat\\m…","labels":[],"detail_key":"p11"},{"id":"n16388","layer":"informal","project":"p11","title":"lem:isGaussian_stdGaussian","kind":"lemma","summary":"The standard Gaussian measure on E is a Gaussian measure.","labels":["lem:isGaussian_stdGaussian"],"detail_key":"p11"},{"id":"n16389","layer":"informal","project":"p11","title":"Since the standard Gaussian is a probability measure (hence finite), we can apply Lemma~\\…","kind":"proof","summary":"Since the standard Gaussian is a probability measure (hence finite), we can apply Lemma~\\reflem…","labels":[],"detail_key":"p11"},{"id":"n16390","layer":"informal","project":"p11","title":"lem:integral_id_stdGaussian","kind":"lemma","summary":"The mean of the standard Gaussian measure is 0.","labels":["lem:integral_id_stdGaussian"],"detail_key":"p11"},{"id":"n16391","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16392","layer":"informal","project":"p11","title":"lem:covMatrix_stdGaussian","kind":"lemma","summary":"The covariance matrix of the standard Gaussian measure is the identity matrix.","labels":["lem:covMatrix_stdGaussian"],"detail_key":"p11"},{"id":"n16393","layer":"informal","project":"p11","title":"From Lemma~\\reflem:charFun_stdGaussian, we know that for all t \\in R, \\hat\\mu(t) = \\exp\\l…","kind":"proof","summary":"From Lemma~\\reflem:charFun_stdGaussian, we know that for all t \\in R, \\hat\\mu(t) = \\exp\\left(-\\…","labels":[],"detail_key":"p11"},{"id":"n16394","layer":"informal","project":"p11","title":"Multivariate Gaussian","kind":"definition","summary":"[Multivariate Gaussian] The multivariate Gaussian measure on R^d with mean m \\in R^d and covari…","labels":["def:multivariateGaussian"],"detail_key":"p11"},{"id":"n16395","layer":"informal","project":"p11","title":"lem:integral_id_multivariateGaussian","kind":"lemma","summary":"The mean of the multivariate Gaussian measure N(m, \\Sigma) is m.","labels":["lem:integral_id_multivariateGaussian"],"detail_key":"p11"},{"id":"n16396","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16397","layer":"informal","project":"p11","title":"lem:covMatrix_multivariateGaussian","kind":"lemma","summary":"The covariance matrix of the multivariate Gaussian measure N(m, \\Sigma) is \\Sigma.","labels":["lem:covMatrix_multivariateGaussian"],"detail_key":"p11"},{"id":"n16398","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16399","layer":"informal","project":"p11","title":"lem:isGaussian_multivariateGaussian","kind":"lemma","summary":"A multivariate Gaussian measure is a Gaussian measure.","labels":["lem:isGaussian_multivariateGaussian"],"detail_key":"p11"},{"id":"n16400","layer":"informal","project":"p11","title":"The multivariate Gaussian measure is the pushforward of the standard Gaussian measure by…","kind":"proof","summary":"The multivariate Gaussian measure is the pushforward of the standard Gaussian measure by an aff…","labels":[],"detail_key":"p11"},{"id":"n16401","layer":"informal","project":"p11","title":"thm:charFun_multivariateGaussian","kind":"theorem","summary":"The characteristic function of a multivariate Gaussian measure N(m, \\Sigma) is given by \\hat\\mu…","labels":["thm:charFun_multivariateGaussian"],"detail_key":"p11"},{"id":"n16402","layer":"informal","project":"p11","title":"Since the multivariate Gaussian measure is a Gaussian measure, we can apply Lemma~\\reflem…","kind":"proof","summary":"Since the multivariate Gaussian measure is a Gaussian measure, we can apply Lemma~\\reflem:IsGau…","labels":[],"detail_key":"p11"},{"id":"n16403","layer":"informal","project":"p11","title":"Gaussian process","kind":"definition","summary":"[Gaussian process] \\mathlibok A process X : T \\to \\Omega \\to E is Gaussian if for every finite…","labels":["def:IsGaussianProcess"],"detail_key":"p11"},{"id":"n16404","layer":"informal","project":"p11","title":"lem:isGaussianProcess_of_modification","kind":"lemma","summary":"\\mathlibok Let X, Y : T \\to \\Omega \\to E be two stochastic processes that are modifications of…","labels":["lem:isGaussianProcess_of_modification"],"detail_key":"p11"},{"id":"n16405","layer":"informal","project":"p11","title":"Being a Gaussian process is defined in terms of the distribution of finite-dimensional ra…","kind":"proof","summary":"Being a Gaussian process is defined in terms of the distribution of finite-dimensional random v…","labels":[],"detail_key":"p11"},{"id":"n16406","layer":"informal","project":"p11","title":"Projective family","kind":"definition","summary":"[Projective family] \\mathlibok A family of measures P indexed by finite sets of T is projective…","labels":["def:IsProjectiveMeasureFamily"],"detail_key":"p11"},{"id":"n16407","layer":"informal","project":"p11","title":"Projective limit","kind":"definition","summary":"[Projective limit] \\mathlibok A measure \\mu on E^T is the projective limit of a projective fami…","labels":["def:IsProjectiveLimit"],"detail_key":"p11"},{"id":"n16408","layer":"informal","project":"p11","title":"Kolmogorov extension theorem","kind":"theorem","summary":"[Kolmogorov extension theorem] Let X be a Polish space, equipped with the Borel \\sigma-algebra,…","labels":["thm:kolmogorovExtension"],"detail_key":"p11"},{"id":"n16409","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16410","layer":"informal","project":"p11","title":"Gram matrix","kind":"definition","summary":"[Gram matrix] \\mathlibok Let v_1, \\ldots, v_n be vectors in an inner product space E. The Gram…","labels":["def:gramMatrix"],"detail_key":"p11"},{"id":"n16411","layer":"informal","project":"p11","title":"lem:posSemidef_gramMatrix","kind":"lemma","summary":"\\mathlibok A gram matrix is positive semidefinite.","labels":["lem:posSemidef_gramMatrix"],"detail_key":"p11"},{"id":"n16412","layer":"informal","project":"p11","title":"Symmetry is obvious from the definition. Let x \\in E. Then \\langle x, G x \\rangle &= \\sum…","kind":"proof","summary":"Symmetry is obvious from the definition. Let x \\in E. Then \\langle x, G x \\rangle &= \\sum_i,j x…","labels":[],"detail_key":"p11"},{"id":"n16413","layer":"informal","project":"p11","title":"lem:C_eq_gramMatrix","kind":"lemma","summary":"Let I = \\t_1, \\ldots, t_n\\ be a finite subset of R_+. For i \\le n, let v_i = I_[0, t_i] be the…","labels":["lem:C_eq_gramMatrix"],"detail_key":"p11"},{"id":"n16414","layer":"informal","project":"p11","title":"By definition of the inner product in L^2(R), \\langle v_i, v_j \\rangle &= \\int_R I_[0, t_…","kind":"proof","summary":"By definition of the inner product in L^2(R), \\langle v_i, v_j \\rangle &= \\int_R I_[0, t_i](x)…","labels":[],"detail_key":"p11"},{"id":"n16415","layer":"informal","project":"p11","title":"lem:posSemidef_brownianCov","kind":"lemma","summary":"For I = \\t_1, \\ldots, t_n\\ a finite subset of R_+, let C \\in R^n \\times n be the matrix C_ij =…","labels":["lem:posSemidef_brownianCov"],"detail_key":"p11"},{"id":"n16416","layer":"informal","project":"p11","title":"C is a Gram matrix by Lemma~\\reflem:C_eq_gramMatrix. By Lemma~\\reflem:posSemidef_gramMatr…","kind":"proof","summary":"C is a Gram matrix by Lemma~\\reflem:C_eq_gramMatrix. By Lemma~\\reflem:posSemidef_gramMatrix, it…","labels":[],"detail_key":"p11"},{"id":"n16417","layer":"informal","project":"p11","title":"Projective family of the Brownian motion","kind":"definition","summary":"[Projective family of the Brownian motion] For I = \\t_1, \\ldots, t_n\\ a finite subset of R_+, l…","labels":["def:gaussianProjectiveFamily"],"detail_key":"p11"},{"id":"n16418","layer":"informal","project":"p11","title":"lem:isProjectiveMeasureFamily_gaussianProjectiveFamily","kind":"lemma","summary":"The projective family of the Brownian motion is a projective family of measures.","labels":["lem:isProjectiveMeasureFamily_gaussianProjectiveFamily"],"detail_key":"p11"},{"id":"n16419","layer":"informal","project":"p11","title":"Let J \\subseteq I be finite subsets of R_+. We need to show that the restriction from R^I…","kind":"proof","summary":"Let J \\subseteq I be finite subsets of R_+. We need to show that the restriction from R^I to R^…","labels":[],"detail_key":"p11"},{"id":"n16420","layer":"informal","project":"p11","title":"def:gaussianLimit","kind":"definition","summary":"We denote by P_B the projective limit of the projective family of the Brownian motion given by…","labels":["def:gaussianLimit"],"detail_key":"p11"},{"id":"n16421","layer":"informal","project":"p11","title":"\\varepsilon-cover","kind":"definition","summary":"[\\varepsilon-cover] \\mathlibok A set C \\subseteq E is an \\varepsilon-cover of a set A \\subseteq…","labels":["def:IsCover"],"detail_key":"p11"},{"id":"n16422","layer":"informal","project":"p11","title":"External covering number","kind":"definition","summary":"[External covering number] \\mathlibok The external covering number of a set A \\subseteq E for \\…","labels":["def:externalCoveringNumber"],"detail_key":"p11"},{"id":"n16423","layer":"informal","project":"p11","title":"Internal covering number","kind":"definition","summary":"[Internal covering number] \\mathlibok The internal covering number of a set A \\subseteq E for \\…","labels":["def:internalCoveringNumber"],"detail_key":"p11"},{"id":"n16424","layer":"informal","project":"p11","title":"Separated set","kind":"definition","summary":"[Separated set] \\mathlibok A set c \\subseteq E is \\varepsilon-separated if for all x, y \\in c,…","labels":["def:IsSeparated"],"detail_key":"p11"},{"id":"n16425","layer":"informal","project":"p11","title":"Packing number","kind":"definition","summary":"[Packing number] \\mathlibok The packing number of a set A \\subseteq E for \\varepsilon > 0 is th…","labels":["def:packingNumber"],"detail_key":"p11"},{"id":"n16426","layer":"informal","project":"p11","title":"lem:externalCoveringNumber_le_internalCoveringNumber","kind":"lemma","summary":"\\mathlibok N^ext_\\varepsilon(A) \\le N^int_\\varepsilon(A).","labels":["lem:externalCoveringNumber_le_internalCoveringNumber"],"detail_key":"p11"},{"id":"n16427","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16428","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_le_packingNumber","kind":"lemma","summary":"\\mathlibok N^int_\\varepsilon(A) \\le P_\\varepsilon(A).","labels":["lem:internalCoveringNumber_le_packingNumber"],"detail_key":"p11"},{"id":"n16429","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16430","layer":"informal","project":"p11","title":"lem:packingNumber_two_le_externalCoveringNumber","kind":"lemma","summary":"\\mathlibok P_2\\varepsilon(A) \\le N^ext_\\varepsilon(A).","labels":["lem:packingNumber_two_le_externalCoveringNumber"],"detail_key":"p11"},{"id":"n16431","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16432","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_eq_one_of_diam_le","kind":"lemma","summary":"\\mathlibok If diam(A) \\le \\varepsilon and A is nonempty, then N^int_\\varepsilon(A) = 1.","labels":["lem:internalCoveringNumber_eq_one_of_diam_le"],"detail_key":"p11"},{"id":"n16433","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16434","layer":"informal","project":"p11","title":"lem:externalCoveringNumber_mono","kind":"lemma","summary":"\\mathlibok For B \\subseteq A, N^ext_\\varepsilon(B) \\le N^ext_\\varepsilon(A).","labels":["lem:externalCoveringNumber_mono"],"detail_key":"p11"},{"id":"n16435","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16436","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_subset_le","kind":"lemma","summary":"\\mathlibok For B \\subseteq A, N^int_\\varepsilon(B) \\le N^int_\\varepsilon/2(A).","labels":["lem:internalCoveringNumber_subset_le"],"detail_key":"p11"},{"id":"n16437","layer":"informal","project":"p11","title":"N^int_\\varepsilon(B) &\\le P_\\varepsilon(B) \\le N^ext_\\varepsilon/2(B) \\le N^ext_\\varepsil…","kind":"proof","summary":"N^int_\\varepsilon(B) &\\le P_\\varepsilon(B) \\le N^ext_\\varepsilon/2(B) \\le N^ext_\\varepsilon/2(A…","labels":[],"detail_key":"p11"},{"id":"n16438","layer":"informal","project":"p11","title":"lem:volume_le_of_isCover","kind":"lemma","summary":"Let A \\subseteq E and C \\subseteq E be a finite \\varepsilon-cover of A. Denote by V(A) the volu…","labels":["lem:volume_le_of_isCover"],"detail_key":"p11"},{"id":"n16439","layer":"informal","project":"p11","title":"Since C is a cover of A, A is a subset of the union of the closed balls B_\\varepsilon(c)…","kind":"proof","summary":"Since C is a cover of A, A is a subset of the union of the closed balls B_\\varepsilon(c) for c…","labels":[],"detail_key":"p11"},{"id":"n16440","layer":"informal","project":"p11","title":"lem:volume_le_externalCoveringNumber_mul","kind":"lemma","summary":"If 0 < \\varepsilon then V(A) \\le N^ext_\\varepsilon(A) V(B_\\varepsilon).","labels":["lem:volume_le_externalCoveringNumber_mul"],"detail_key":"p11"},{"id":"n16441","layer":"informal","project":"p11","title":"If A has no \\varepsilon-cover, then N^ext_\\varepsilon(A) = \\infty, and because 0 < \\varep…","kind":"proof","summary":"If A has no \\varepsilon-cover, then N^ext_\\varepsilon(A) = \\infty, and because 0 < \\varepsilon,…","labels":[],"detail_key":"p11"},{"id":"n16442","layer":"informal","project":"p11","title":"lem:le_volume_of_isSeparated","kind":"lemma","summary":"Let A \\subseteq E and let S \\subseteq A be an \\varepsilon-separated set. Then \\vert S \\vert V(B…","labels":["lem:le_volume_of_isSeparated"],"detail_key":"p11"},{"id":"n16443","layer":"informal","project":"p11","title":"Since S is \\varepsilon-separated, the closed balls B_\\varepsilon/2(s) for s \\in S are pai…","kind":"proof","summary":"Since S is \\varepsilon-separated, the closed balls B_\\varepsilon/2(s) for s \\in S are pairwise…","labels":[],"detail_key":"p11"},{"id":"n16444","layer":"informal","project":"p11","title":"lem:packingNumber_mul_le_volume","kind":"lemma","summary":"P_\\varepsilon(A) V(B_\\varepsilon/2) \\le V(A + B_\\varepsilon/2).","labels":["lem:packingNumber_mul_le_volume"],"detail_key":"p11"},{"id":"n16445","layer":"informal","project":"p11","title":"Use Lemma~\\reflem:le_volume_of_isSeparated with S an \\varepsilon-separated set of maximal…","kind":"proof","summary":"Use Lemma~\\reflem:le_volume_of_isSeparated with S an \\varepsilon-separated set of maximal cardi…","labels":[],"detail_key":"p11"},{"id":"n16446","layer":"informal","project":"p11","title":"lem:volume_div_le_internalCoveringNumber","kind":"lemma","summary":"If 0 < \\varepsilon then \\fracV(A)V(B_\\varepsilon) \\le N^int_\\varepsilon(A).","labels":["lem:volume_div_le_internalCoveringNumber"],"detail_key":"p11"},{"id":"n16447","layer":"informal","project":"p11","title":"We have \\fracV(A)V(B_\\varepsilon) \\le N^ext_\\varepsilon(A) by Lemma~\\reflem:volume_le_ext…","kind":"proof","summary":"We have \\fracV(A)V(B_\\varepsilon) \\le N^ext_\\varepsilon(A) by Lemma~\\reflem:volume_le_externalC…","labels":[],"detail_key":"p11"},{"id":"n16448","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_le_volume_div","kind":"lemma","summary":"If 0 < \\varepsilon < \\infty then N^int_\\varepsilon(A) \\le \\fracV(A + B_\\varepsilon/2)V(B_\\varep…","labels":["lem:internalCoveringNumber_le_volume_div"],"detail_key":"p11"},{"id":"n16449","layer":"informal","project":"p11","title":"We have N^int_\\varepsilon(A) \\le P_\\varepsilon(A) by Lemma~\\reflem:internalCoveringNumber…","kind":"proof","summary":"We have N^int_\\varepsilon(A) \\le P_\\varepsilon(A) by Lemma~\\reflem:internalCoveringNumber_le_pa…","labels":[],"detail_key":"p11"},{"id":"n16450","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_closedBall_ge","kind":"lemma","summary":"N_\\varepsilon^int(B_1) \\ge \\frac1\\varepsilon^d.","labels":["lem:internalCoveringNumber_closedBall_ge"],"detail_key":"p11"},{"id":"n16451","layer":"informal","project":"p11","title":"By Lemma~\\reflem:volume_div_le_internalCoveringNumber, N^int_\\varepsilon(B_1) &\\ge \\fracV…","kind":"proof","summary":"By Lemma~\\reflem:volume_div_le_internalCoveringNumber, N^int_\\varepsilon(B_1) &\\ge \\fracV(B_1)V…","labels":[],"detail_key":"p11"},{"id":"n16452","layer":"informal","project":"p11","title":"lem:internalCoveringNumber_closedBall_le","kind":"lemma","summary":"N_\\varepsilon^int(B_1) \\le \\left(\\frac2\\varepsilon + 1\\right)^d.","labels":["lem:internalCoveringNumber_closedBall_le"],"detail_key":"p11"},{"id":"n16453","layer":"informal","project":"p11","title":"By Lemma~\\reflem:internalCoveringNumber_le_volume_div, N^int_\\varepsilon(B_1) &\\le \\fracV…","kind":"proof","summary":"By Lemma~\\reflem:internalCoveringNumber_le_volume_div, N^int_\\varepsilon(B_1) &\\le \\fracV(B_1 +…","labels":[],"detail_key":"p11"},{"id":"n16454","layer":"informal","project":"p11","title":"Bounded internal covering number","kind":"definition","summary":"[Bounded internal covering number] Let diam(A) be the diameter of A \\subseteq E, i.e. diam(A) =…","labels":["def:HasBoundedInternalCoveringNumber"],"detail_key":"p11"},{"id":"n16455","layer":"informal","project":"p11","title":"lem:hasBoundedInternalCoveringNumber_unitInterval","kind":"lemma","summary":"The unit interval I = [0, 1] \\subseteq R has bounded internal covering number with constant 1 a…","labels":["lem:hasBoundedInternalCoveringNumber_unitInterval"],"detail_key":"p11"},{"id":"n16456","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16457","layer":"informal","project":"p11","title":"lem:hasBoundedInternalCoveringNumber_subset","kind":"lemma","summary":"If A has bounded internal covering number with constant c>0 and exponent d>0, then for all B \\s…","labels":["lem:hasBoundedInternalCoveringNumber_subset"],"detail_key":"p11"},{"id":"n16458","layer":"informal","project":"p11","title":"N^int_\\varepsilon(B) &\\le N^int_\\varepsilon/2(A) \\le c (\\varepsilon/2)^-d = 2^d c \\vareps…","kind":"proof","summary":"N^int_\\varepsilon(B) &\\le N^int_\\varepsilon/2(A) \\le c (\\varepsilon/2)^-d = 2^d c \\varepsilon^-…","labels":[],"detail_key":"p11"},{"id":"n16459","layer":"informal","project":"p11","title":"def:nearestPt","kind":"definition","summary":"Let S be a finite set of E and x \\in E. We denote by \\pi(x, S) the point in S which is closest…","labels":["def:nearestPt"],"detail_key":"p11"},{"id":"n16460","layer":"informal","project":"p11","title":"lem:dist_nearestPt_le","kind":"lemma","summary":"Let S be a finite set of E and x \\in E. Then for all y \\in S, d_E(x, \\pi(x, S)) \\le d_E(x, y).","labels":["lem:dist_nearestPt_le"],"detail_key":"p11"},{"id":"n16461","layer":"informal","project":"p11","title":"By definition.","kind":"proof","summary":"By definition.","labels":[],"detail_key":"p11"},{"id":"n16462","layer":"informal","project":"p11","title":"lem:dist_nearestPt_of_isCover","kind":"lemma","summary":"Let C_\\varepsilon be a finite \\varepsilon-cover of A \\subseteq E (assuming such a finite cover…","labels":["lem:dist_nearestPt_of_isCover"],"detail_key":"p11"},{"id":"n16463","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16464","layer":"informal","project":"p11","title":"Chaining sequence","kind":"definition","summary":"[Chaining sequence] Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite…","labels":["def:chainingSequence"],"detail_key":"p11"},{"id":"n16465","layer":"informal","project":"p11","title":"lem:chainingSequence_mem","kind":"lemma","summary":"Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\varepsilon_n-cover…","labels":["lem:chainingSequence_mem"],"detail_key":"p11"},{"id":"n16466","layer":"informal","project":"p11","title":"By definition.","kind":"proof","summary":"By definition.","labels":[],"detail_key":"p11"},{"id":"n16467","layer":"informal","project":"p11","title":"lem:dist_chainingSequence_add_one","kind":"lemma","summary":"Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\varepsilon_n-cover…","labels":["lem:dist_chainingSequence_add_one"],"detail_key":"p11"},{"id":"n16468","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:dist_nearestPt_of_isCover with S = C_i and x = \\barx_i+1.","kind":"proof","summary":"Apply Lemma~\\reflem:dist_nearestPt_of_isCover with S = C_i and x = \\barx_i+1.","labels":[],"detail_key":"p11"},{"id":"n16469","layer":"informal","project":"p11","title":"lem:dist_chainingSequence_le_sum","kind":"lemma","summary":"Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\varepsilon_n-cover…","labels":["lem:dist_chainingSequence_le_sum"],"detail_key":"p11"},{"id":"n16470","layer":"informal","project":"p11","title":"By the triangle inequality and Lemma~\\reflem:dist_chainingSequence_add_one, d_E(\\barx_m,…","kind":"proof","summary":"By the triangle inequality and Lemma~\\reflem:dist_chainingSequence_add_one, d_E(\\barx_m, x) \\le…","labels":[],"detail_key":"p11"},{"id":"n16471","layer":"informal","project":"p11","title":"lem:dist_chainingSequence_le","kind":"lemma","summary":"Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\varepsilon_n-cover…","labels":["lem:dist_chainingSequence_le"],"detail_key":"p11"},{"id":"n16472","layer":"informal","project":"p11","title":"Triangle inequality and Lemma~\\reflem:dist_chainingSequence_le_sum.","kind":"proof","summary":"Triangle inequality and Lemma~\\reflem:dist_chainingSequence_le_sum.","labels":[],"detail_key":"p11"},{"id":"n16473","layer":"informal","project":"p11","title":"cor:dist_chainingSequence_pow_two_le","kind":"corollary","summary":"For \\varepsilon_n = \\varepsilon_0 2^-n, with the hypothesis of Lemma~\\reflem:dist_chainingSeque…","labels":["cor:dist_chainingSequence_pow_two_le"],"detail_key":"p11"},{"id":"n16474","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16475","layer":"informal","project":"p11","title":"def:logSizeRadius","kind":"definition","summary":"\\mathlibok Let V be a finite subset of a metric space and let t \\in V and a > 1, c > 0. Let the…","labels":["def:logSizeRadius"],"detail_key":"p11"},{"id":"n16476","layer":"informal","project":"p11","title":"lem:card_logSizeRadius_ge","kind":"lemma","summary":"\\mathlibok a^r_V,t-1 \\le \\vert B_V(t, (r_V,t-1)c) \\vert~.","labels":["lem:card_logSizeRadius_ge"],"detail_key":"p11"},{"id":"n16477","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16478","layer":"informal","project":"p11","title":"lem:card_logSizeRadius_le","kind":"lemma","summary":"\\mathlibok \\vert B_V(t, r_V,tc) \\vert \\le a^r_V,t~.","labels":["lem:card_logSizeRadius_le"],"detail_key":"p11"},{"id":"n16479","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16480","layer":"informal","project":"p11","title":"Log-size ball sequence","kind":"definition","summary":"[Log-size ball sequence] \\mathlibok Let (T,d_T) be a metric space and let J \\subseteq T be fini…","labels":["def:logSizeBallSequence"],"detail_key":"p11"},{"id":"n16481","layer":"informal","project":"p11","title":"lem:logSizeRadius_logSizeBallSequence_le","kind":"lemma","summary":"\\mathlibok The radius of a log-size ball sequence (V_i, t_i, r_i)_i \\in N for (J, a, c, n) sati…","labels":["lem:logSizeRadius_logSizeBallSequence_le"],"detail_key":"p11"},{"id":"n16482","layer":"informal","project":"p11","title":"Since |J| \\le a^n, we have \\vert B_V_i(t_i, n c) \\vert \\le \\vert J \\vert \\le a^n.","kind":"proof","summary":"Since |J| \\le a^n, we have \\vert B_V_i(t_i, n c) \\vert \\le \\vert J \\vert \\le a^n.","labels":[],"detail_key":"p11"},{"id":"n16483","layer":"informal","project":"p11","title":"lem:logSizeBallSequence_V_anti","kind":"lemma","summary":"\\mathlibok The sets V_i of a log-size ball sequence (V_i, t_i, r_i)_i \\in N are a decreasing se…","labels":["lem:logSizeBallSequence_V_anti"],"detail_key":"p11"},{"id":"n16484","layer":"informal","project":"p11","title":"V_i+1 = V_i \\setminus B_V_i(t_i, (r_i - 1)c) hence V_i+1 \\subseteq V_i.","kind":"proof","summary":"V_i+1 = V_i \\setminus B_V_i(t_i, (r_i - 1)c) hence V_i+1 \\subseteq V_i.","labels":[],"detail_key":"p11"},{"id":"n16485","layer":"informal","project":"p11","title":"lem:logSizeBallSequence_eq_zero","kind":"lemma","summary":"\\mathlibok For any log-size ball sequence (V_i, t_i, r_i)_i \\in N for (J, a, c, n), for all k \\…","labels":["lem:logSizeBallSequence_eq_zero"],"detail_key":"p11"},{"id":"n16486","layer":"informal","project":"p11","title":"V_i+1 = V_i \\setminus B_V_i(t_i, (r_i - 1)c) and since t_i \\in B_V_i(t_i, (r_i - 1)c), we…","kind":"proof","summary":"V_i+1 = V_i \\setminus B_V_i(t_i, (r_i - 1)c) and since t_i \\in B_V_i(t_i, (r_i - 1)c), we have…","labels":[],"detail_key":"p11"},{"id":"n16487","layer":"informal","project":"p11","title":"lem:logSizeBallSequence_disjoint_B","kind":"lemma","summary":"\\mathlibok For i \\ne j, the balls B_V_i(t, (r_i-1)c) and B_V_j(t_j, (r_j-1)c) of a log-size bal…","labels":["lem:logSizeBallSequence_disjoint_B"],"detail_key":"p11"},{"id":"n16488","layer":"informal","project":"p11","title":"Assume w.l.o.g. that i < j. Then B_V_j(t_j, (r_j-1)c) \\subseteq V_j \\subseteq V_i+1. It s…","kind":"proof","summary":"Assume w.l.o.g. that i < j. Then B_V_j(t_j, (r_j-1)c) \\subseteq V_j \\subseteq V_i+1. It suffice…","labels":[],"detail_key":"p11"},{"id":"n16489","layer":"informal","project":"p11","title":"def:pairSet","kind":"definition","summary":"\\mathlibok Let (V_i, t_i, r_i)_i \\in N be a log-size ball sequence for (J, a, c, n). For i \\in…","labels":["def:pairSet"],"detail_key":"p11"},{"id":"n16490","layer":"informal","project":"p11","title":"lem:card_pairSet_le","kind":"lemma","summary":"\\mathlibok The cardinal of the pair set K of a log-size ball sequence for (J, a, c, n) satisfie…","labels":["lem:card_pairSet_le"],"detail_key":"p11"},{"id":"n16491","layer":"informal","project":"p11","title":"Using Lemma~\\reflem:card_logSizeRadius_le, the cardinal of K is bounded by \\vert K \\vert…","kind":"proof","summary":"Using Lemma~\\reflem:card_logSizeRadius_le, the cardinal of K is bounded by \\vert K \\vert &\\le \\…","labels":[],"detail_key":"p11"},{"id":"n16492","layer":"informal","project":"p11","title":"lem:dist_le_of_mem_pairSet","kind":"lemma","summary":"\\mathlibok Let (s, t) be a pair in the pair set K of a log-size ball sequence for (J, a, c, n).…","labels":["lem:dist_le_of_mem_pairSet"],"detail_key":"p11"},{"id":"n16493","layer":"informal","project":"p11","title":"A pair (t, s) \\in K is of the form (t_i, s) for s \\in B_V(t_i, r_i c) and satisfies d_T(t…","kind":"proof","summary":"A pair (t, s) \\in K is of the form (t_i, s) for s \\in B_V(t_i, r_i c) and satisfies d_T(t_i, s)…","labels":[],"detail_key":"p11"},{"id":"n16494","layer":"informal","project":"p11","title":"lem:sup_dist_le_two_mul_sup_dist_pairSet","kind":"lemma","summary":"\\mathlibok Let K be the pair set of a log-size ball sequence (V_i, t_i, r_i)_i \\in N for (J, a,…","labels":["lem:sup_dist_le_two_mul_sup_dist_pairSet"],"detail_key":"p11"},{"id":"n16495","layer":"informal","project":"p11","title":"Let (s, t) \\in J^2 such that d_T(s, t) \\le c. Then there exists a largest \\ell \\in N such…","kind":"proof","summary":"Let (s, t) \\in J^2 such that d_T(s, t) \\le c. Then there exists a largest \\ell \\in N such that…","labels":[],"detail_key":"p11"},{"id":"n16496","layer":"informal","project":"p11","title":"lem:pair_reduction","kind":"lemma","summary":"\\mathlibok Let (T,d_T) be a metric space. Let J \\subseteq T be finite, a > 1, c>0 and n \\in \\1,…","labels":["lem:pair_reduction","eq:chain1","eq:chain2","eq:chain3"],"detail_key":"p11"},{"id":"n16497","layer":"informal","project":"p11","title":"Let (V_i, t_i, r_i)_i \\in N be a log-size ball sequence for (J, a, c, n). We show that it…","kind":"proof","summary":"Let (V_i, t_i, r_i)_i \\in N be a log-size ball sequence for (J, a, c, n). We show that its pair…","labels":[],"detail_key":"p11"},{"id":"n16498","layer":"informal","project":"p11","title":"Kolmogorov condition","kind":"definition","summary":"[Kolmogorov condition] \\mathlibok Let X : T \\to \\Omega \\to E be a stochastic process, where (T,…","labels":["def:IsKolmogorovProcess"],"detail_key":"p11"},{"id":"n16499","layer":"informal","project":"p11","title":"lem:IsKolmogorovProcess.edist_eq_zero","kind":"lemma","summary":"\\mathlibok If X : T \\to \\Omega \\to E is a process that satisfies the Kolmogorov condition for e…","labels":["lem:IsKolmogorovProcess.edist_eq_zero"],"detail_key":"p11"},{"id":"n16500","layer":"informal","project":"p11","title":"It suffices to show that d_E(X_s, X_t)^p = 0 almost everywhere, which is in turn implied…","kind":"proof","summary":"It suffices to show that d_E(X_s, X_t)^p = 0 almost everywhere, which is in turn implied by E[d…","labels":[],"detail_key":"p11"},{"id":"n16501","layer":"informal","project":"p11","title":"lem:IsKolmogorovProcess.lintegral_sup_rpow_edist_eq_zero","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:IsKolmogorovProcess.lintegral_sup_rpow_edist_eq_zero"],"detail_key":"p11"},{"id":"n16502","layer":"informal","project":"p11","title":"Since T' is countable, we get from Lemma~\\reflem:IsKolmogorovProcess.edist_eq_zero that a…","kind":"proof","summary":"Since T' is countable, we get from Lemma~\\reflem:IsKolmogorovProcess.edist_eq_zero that almost…","labels":[],"detail_key":"p11"},{"id":"n16503","layer":"informal","project":"p11","title":"lem:IsKolmogorovProcess.aemeasurable","kind":"lemma","summary":"\\mathlibok If X : T \\to \\Omega \\to E is a function that satisfies the Kolmogorov condition, the…","labels":["lem:IsKolmogorovProcess.aemeasurable"],"detail_key":"p11"},{"id":"n16504","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16505","layer":"informal","project":"p11","title":"lem:aemeasurable_pair_of_aemeasurable","kind":"lemma","summary":"If E is separable and X : T \\to \\Omega \\to E is a process such that X_t is P-a.e. measurable fo…","labels":["lem:aemeasurable_pair_of_aemeasurable"],"detail_key":"p11"},{"id":"n16506","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16507","layer":"informal","project":"p11","title":"lem:IsKolmogorovProcess.aemeasurable_edist","kind":"lemma","summary":"\\mathlibok If X : T \\to \\Omega \\to E is a process that satisfies the Kolmogorov condition, then…","labels":["lem:IsKolmogorovProcess.aemeasurable_edist"],"detail_key":"p11"},{"id":"n16508","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16509","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_le_card_mul_rpow","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_le_card_mul_rpow"],"detail_key":"p11"},{"id":"n16510","layer":"informal","project":"p11","title":"E\\left[\\sup_(s,t) \\in C d_E(X_s, X_t)^p \\right] &\\le E\\left[\\sum_(s,t) \\in C d_E(X_s, X_t…","kind":"proof","summary":"E\\left[\\sup_(s,t) \\in C d_E(X_s, X_t)^p \\right] &\\le E\\left[\\sum_(s,t) \\in C d_E(X_s, X_t)^p \\r…","labels":[],"detail_key":"p11"},{"id":"n16511","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_of_dist_le","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_of_dist_le"],"detail_key":"p11"},{"id":"n16512","layer":"informal","project":"p11","title":"By Lemma~\\reflem:pair_reduction, there exists K \\subseteq J^2 such that |K| & \\le a |J| \\…","kind":"proof","summary":"By Lemma~\\reflem:pair_reduction, there exists K \\subseteq J^2 such that |K| & \\le a |J| \\:, \\\\…","labels":[],"detail_key":"p11"},{"id":"n16513","layer":"informal","project":"p11","title":"lem:scale_change","kind":"lemma","summary":"Let X : T \\to E. Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\v…","labels":["lem:scale_change"],"detail_key":"p11"},{"id":"n16514","layer":"informal","project":"p11","title":"By the triangle inequality, d_E(X_s, X_t) &\\le d_E(X_s, X_\\bars_m) + d(X_\\bars_m, X_\\bart…","kind":"proof","summary":"By the triangle inequality, d_E(X_s, X_t) &\\le d_E(X_s, X_\\bars_m) + d(X_\\bars_m, X_\\bart_m) +…","labels":[],"detail_key":"p11"},{"id":"n16515","layer":"informal","project":"p11","title":"cor:scale_change_rpow","kind":"corollary","summary":"Let X : T \\to E. Let (\\varepsilon_n)_n \\in N be a sequence of positive numbers, C_n a finite \\v…","labels":["cor:scale_change_rpow"],"detail_key":"p11"},{"id":"n16516","layer":"informal","project":"p11","title":"This is Lemma~\\reflem:scale_change, together with the fact that for a, b \\ge 0, (a + b)^p…","kind":"proof","summary":"This is Lemma~\\reflem:scale_change, together with the fact that for a, b \\ge 0, (a + b)^p \\le (…","labels":[],"detail_key":"p11"},{"id":"n16517","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_cover_of_dist_le","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_cover_of_dist_le"],"detail_key":"p11"},{"id":"n16518","layer":"informal","project":"p11","title":"Let \\barr = 1 + \\log_2 N^int_\\varepsilon(J). Then \\vert C \\vert = N^int_\\varepsilon(J) \\l…","kind":"proof","summary":"Let \\barr = 1 + \\log_2 N^int_\\varepsilon(J). Then \\vert C \\vert = N^int_\\varepsilon(J) \\le 2^\\b…","labels":[],"detail_key":"p11"},{"id":"n16519","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_cover_rescale","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_cover_rescale"],"detail_key":"p11"},{"id":"n16520","layer":"informal","project":"p11","title":"By definition of m, \\delta \\le \\varepsilon_0 2^-m+2. For s, t \\in C_k with d_T(s, t) \\le…","kind":"proof","summary":"By definition of m, \\delta \\le \\varepsilon_0 2^-m+2. For s, t \\in C_k with d_T(s, t) \\le \\delta…","labels":[],"detail_key":"p11"},{"id":"n16521","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_succ","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_succ"],"detail_key":"p11"},{"id":"n16522","layer":"informal","project":"p11","title":"E\\left[\\sup_t \\in C_k d_E(X_\\bart_j, X_\\bart_j+1)^p \\right] &\\le E\\left[\\sup_u \\in C_j+1…","kind":"proof","summary":"E\\left[\\sup_t \\in C_k d_E(X_\\bart_j, X_\\bart_j+1)^p \\right] &\\le E\\left[\\sup_u \\in C_j+1 d_E(X_…","labels":[],"detail_key":"p11"},{"id":"n16523","layer":"informal","project":"p11","title":"lem:integral_sup_dist_le_sum_rpow","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a stochastic process. Let (\\varepsilon_n)_n \\in N be a sequence o…","labels":["lem:integral_sup_dist_le_sum_rpow"],"detail_key":"p11"},{"id":"n16524","layer":"informal","project":"p11","title":"By the triangle inequality, \\sup_t \\in C_k d_E(X_t, X_\\bart_m)^p &\\le \\sup_t \\in C_k \\lef…","kind":"proof","summary":"By the triangle inequality, \\sup_t \\in C_k d_E(X_t, X_\\bart_m)^p &\\le \\sup_t \\in C_k \\left( \\su…","labels":[],"detail_key":"p11"},{"id":"n16525","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_le_sum","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_le_sum"],"detail_key":"p11"},{"id":"n16526","layer":"informal","project":"p11","title":"Put together Lemma~\\reflem:integral_sup_rpow_dist_succ and Lemma~\\reflem:integral_sup_dis…","kind":"proof","summary":"Put together Lemma~\\reflem:integral_sup_rpow_dist_succ and Lemma~\\reflem:integral_sup_dist_le_s…","labels":[],"detail_key":"p11"},{"id":"n16527","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_le_of_minimal_cover","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_le_of_minimal_cover"],"detail_key":"p11"},{"id":"n16528","layer":"informal","project":"p11","title":"By Lemma~\\reflem:integral_sup_rpow_dist_le_sum, we have E \\left[\\sup_t \\in C_k d_E(X_t, X…","kind":"proof","summary":"By Lemma~\\reflem:integral_sup_rpow_dist_le_sum, we have E \\left[\\sup_t \\in C_k d_E(X_t, X_\\bart…","labels":[],"detail_key":"p11"},{"id":"n16529","layer":"informal","project":"p11","title":"cor:integral_sup_rpow_dist_le_of_minimal_cover_two","kind":"corollary","summary":"Under the assumptions of Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover, for \\varepsi…","labels":["cor:integral_sup_rpow_dist_le_of_minimal_cover_two"],"detail_key":"p11"},{"id":"n16530","layer":"informal","project":"p11","title":"Applying first Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover, we get E \\left[\\…","kind":"proof","summary":"Applying first Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover, we get E \\left[\\sup_t…","labels":[],"detail_key":"p11"},{"id":"n16531","layer":"informal","project":"p11","title":"lem:integral_sup_dist_le_sum_rpow_of_le_one","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a stochastic process. Let (\\varepsilon_n)_n \\in N be a sequence o…","labels":["lem:integral_sup_dist_le_sum_rpow_of_le_one"],"detail_key":"p11"},{"id":"n16532","layer":"informal","project":"p11","title":"For 0 < p \\le 1, the power function is sub-additive, i.e. for a, b \\ge 0, (a + b)^p \\le a…","kind":"proof","summary":"For 0 < p \\le 1, the power function is sub-additive, i.e. for a, b \\ge 0, (a + b)^p \\le a^p + b…","labels":[],"detail_key":"p11"},{"id":"n16533","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_le_sum_of_le_one","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_le_sum_of_le_one"],"detail_key":"p11"},{"id":"n16534","layer":"informal","project":"p11","title":"Put together Lemma~\\reflem:integral_sup_rpow_dist_succ and Lemma~\\reflem:integral_sup_dis…","kind":"proof","summary":"Put together Lemma~\\reflem:integral_sup_rpow_dist_succ and Lemma~\\reflem:integral_sup_dist_le_s…","labels":[],"detail_key":"p11"},{"id":"n16535","layer":"informal","project":"p11","title":"lem:integral_sup_rpow_dist_le_of_minimal_cover_of_le_one","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:integral_sup_rpow_dist_le_of_minimal_cover_of_le_one"],"detail_key":"p11"},{"id":"n16536","layer":"informal","project":"p11","title":"By Lemma~\\reflem:integral_sup_rpow_dist_le_sum_of_le_one, we have E\\left[\\sup_t \\in C_k d…","kind":"proof","summary":"By Lemma~\\reflem:integral_sup_rpow_dist_le_sum_of_le_one, we have E\\left[\\sup_t \\in C_k d_E(X_t…","labels":[],"detail_key":"p11"},{"id":"n16537","layer":"informal","project":"p11","title":"cor:integral_sup_rpow_dist_le_of_minimal_cover_two_of_le_one","kind":"corollary","summary":"Under the assumptions of Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover_of_le_one, fo…","labels":["cor:integral_sup_rpow_dist_le_of_minimal_cover_two_of_le_one"],"detail_key":"p11"},{"id":"n16538","layer":"informal","project":"p11","title":"Applying first Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover_of_le_one, we get…","kind":"proof","summary":"Applying first Lemma~\\reflem:integral_sup_rpow_dist_le_of_minimal_cover_of_le_one, we get E \\le…","labels":[],"detail_key":"p11"},{"id":"n16539","layer":"informal","project":"p11","title":"def:Cp","kind":"definition","summary":"C_p = \\max\\left\\\\frac1\\left( 2^(q -d)/p - 1\\right)^p, \\frac1\\left( 2^(q -d) - 1\\right) \\right\\…","labels":["def:Cp"],"detail_key":"p11"},{"id":"n16540","layer":"informal","project":"p11","title":"lem:second_term_bound","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:second_term_bound"],"detail_key":"p11"},{"id":"n16541","layer":"informal","project":"p11","title":"This is the max of the two bounds obtained p \\ge 1 and p \\le 1.","kind":"proof","summary":"This is the max of the two bounds obtained p \\ge 1 and p \\le 1.","labels":[],"detail_key":"p11"},{"id":"n16542","layer":"informal","project":"p11","title":"lem:lintegral_sup_cover_eq_of_lt_iInf_dist","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process that satisfies the Kolmogorov condition for exponents (…","labels":["lem:lintegral_sup_cover_eq_of_lt_iInf_dist"],"detail_key":"p11"},{"id":"n16543","layer":"informal","project":"p11","title":"First, remark that C is actually a 0-cover of J. For s, t \\in J, let s', t' \\in C be such…","kind":"proof","summary":"First, remark that C is actually a 0-cover of J. For s, t \\in J, let s', t' \\in C be such that…","labels":[],"detail_key":"p11"},{"id":"n16544","layer":"informal","project":"p11","title":"thm:finite_set_bound_of_dist_le_of_diam_le","kind":"theorem","summary":"Suppose that T is a finite set with bounded internal covering number with constant c_1>0 and ex…","labels":["thm:finite_set_bound_of_dist_le_of_diam_le"],"detail_key":"p11"},{"id":"n16545","layer":"informal","project":"p11","title":"Let \\varepsilon_0 = diam(T). For all n \\in N, let C_n a finite \\varepsilon_n-cover of T w…","kind":"proof","summary":"Let \\varepsilon_0 = diam(T). For all n \\in N, let C_n a finite \\varepsilon_n-cover of T with C_…","labels":[],"detail_key":"p11"},{"id":"n16546","layer":"informal","project":"p11","title":"thm:finite_set_bound_of_dist_le_of_le_diam","kind":"theorem","summary":"Suppose that T is a finite set with bounded internal covering number with constant c_1>0 and ex…","labels":["thm:finite_set_bound_of_dist_le_of_le_diam"],"detail_key":"p11"},{"id":"n16547","layer":"informal","project":"p11","title":"Let \\varepsilon_0 = diam(T). For all n \\in N, let C_n a finite \\varepsilon_n-cover of T w…","kind":"proof","summary":"Let \\varepsilon_0 = diam(T). For all n \\in N, let C_n a finite \\varepsilon_n-cover of T with C_…","labels":[],"detail_key":"p11"},{"id":"n16548","layer":"informal","project":"p11","title":"cor:finite_set_bound_of_dist_le_of_le_diam_bis","kind":"corollary","summary":"With the same assumptions and notations as in Theorem~\\refthm:finite_set_bound_of_dist_le_of_le…","labels":["cor:finite_set_bound_of_dist_le_of_le_diam_bis"],"detail_key":"p11"},{"id":"n16549","layer":"informal","project":"p11","title":"We apply Theorem~\\refthm:finite_set_bound_of_dist_le_of_le_diam and then remark that for…","kind":"proof","summary":"We apply Theorem~\\refthm:finite_set_bound_of_dist_le_of_le_diam and then remark that for \\delta…","labels":[],"detail_key":"p11"},{"id":"n16550","layer":"informal","project":"p11","title":"cor:finite_set_bound_of_dist_le","kind":"corollary","summary":"Suppose that T is a finite set with bounded internal covering number with constant c_1>0 and ex…","labels":["cor:finite_set_bound_of_dist_le"],"detail_key":"p11"},{"id":"n16551","layer":"informal","project":"p11","title":"We combine Corollary~\\refcor:finite_set_bound_of_dist_le_of_le_diam_bis and Theorem~\\reft…","kind":"proof","summary":"We combine Corollary~\\refcor:finite_set_bound_of_dist_le_of_le_diam_bis and Theorem~\\refthm:fin…","labels":[],"detail_key":"p11"},{"id":"n16552","layer":"informal","project":"p11","title":"lem:integral_div_dist_le_sum_integral_dist_le","kind":"lemma","summary":"Let J \\subseteq T be a finite set and suppose that T has finite diameter. For k \\in N, let \\eta…","labels":["lem:integral_div_dist_le_sum_integral_dist_le"],"detail_key":"p11"},{"id":"n16553","layer":"informal","project":"p11","title":"We introduce for each k \\in N the set of pairs (s, t) such that \\eta_k < d_T(s, t) \\le 2…","kind":"proof","summary":"We introduce for each k \\in N the set of pairs (s, t) such that \\eta_k < d_T(s, t) \\le 2 \\eta_k…","labels":[],"detail_key":"p11"},{"id":"n16554","layer":"informal","project":"p11","title":"def:L","kind":"definition","summary":"We introduce the constant L(T, c_1, d, p, q, \\beta) &= 2^2p+5q+1 c_1 (diam(T)+1)^q-d \\\\&\\quad \\…","labels":["def:L"],"detail_key":"p11"},{"id":"n16555","layer":"informal","project":"p11","title":"lem:L_lt_top","kind":"lemma","summary":"For diam(T) < \\infty, p> 0, q > d > 0 and \\beta \\in (0, (q-d)/p), the constant L(T, c_1, d, p,…","labels":["lem:L_lt_top"],"detail_key":"p11"},{"id":"n16556","layer":"informal","project":"p11","title":"Let a_k = 2^2p+5q+1 M c_1 (diam(T)+1)^q-d 2^k (\\beta p - (q-d)) \\left(4^d \\left(\\max\\left…","kind":"proof","summary":"Let a_k = 2^2p+5q+1 M c_1 (diam(T)+1)^q-d 2^k (\\beta p - (q-d)) \\left(4^d \\left(\\max\\left\\0, \\l…","labels":[],"detail_key":"p11"},{"id":"n16557","layer":"informal","project":"p11","title":"lem:finite_set_bound","kind":"lemma","summary":"Suppose that J \\subseteq T is a finite set and that T has bounded internal covering number with…","labels":["lem:finite_set_bound"],"detail_key":"p11"},{"id":"n16558","layer":"informal","project":"p11","title":"Since J \\subseteq T, J has bounded internal covering number with constant 2^d c_1 and exp…","kind":"proof","summary":"Since J \\subseteq T, J has bounded internal covering number with constant 2^d c_1 and exponent…","labels":[],"detail_key":"p11"},{"id":"n16559","layer":"informal","project":"p11","title":"thm:countable_set_bound","kind":"theorem","summary":"Suppose that T has bounded internal covering number with constant c_1>0 and exponent d > 0. Let…","labels":["thm:countable_set_bound"],"detail_key":"p11"},{"id":"n16560","layer":"informal","project":"p11","title":"Build a monotone sequence of finite sets T_n \\subseteq T', use Lemma~\\reflem:finite_set_b…","kind":"proof","summary":"Build a monotone sequence of finite sets T_n \\subseteq T', use Lemma~\\reflem:finite_set_bound t…","labels":[],"detail_key":"p11"},{"id":"n16561","layer":"informal","project":"p11","title":"cor:countable_set_bound_of_le","kind":"corollary","summary":"Under the same assumptions as in Theorem~\\refthm:countable_set_bound, for every countable subse…","labels":["cor:countable_set_bound_of_le"],"detail_key":"p11"},{"id":"n16562","layer":"informal","project":"p11","title":"Immediately follows from Theorem~\\refthm:countable_set_bound.","kind":"proof","summary":"Immediately follows from Theorem~\\refthm:countable_set_bound.","labels":[],"detail_key":"p11"},{"id":"n16563","layer":"informal","project":"p11","title":"lem:holder_modification_single","kind":"lemma","summary":"Under the assumptions of Theorem~\\refthm:countable_set_bound, for E a complete space and \\beta…","labels":["lem:holder_modification_single"],"detail_key":"p11"},{"id":"n16564","layer":"informal","project":"p11","title":"Let T' be a countable dense subset of T. Let A be the event \\left\\\\sup_s, t \\in T';\\: s \\…","kind":"proof","summary":"Let T' be a countable dense subset of T. Let A be the event \\left\\\\sup_s, t \\in T';\\: s \\ne t \\…","labels":[],"detail_key":"p11"},{"id":"n16565","layer":"informal","project":"p11","title":"thm:holder_modification","kind":"theorem","summary":"Under the assumptions of Theorem~\\refthm:countable_set_bound, for E a complete space, there exi…","labels":["thm:holder_modification"],"detail_key":"p11"},{"id":"n16566","layer":"informal","project":"p11","title":"Let (\\beta_n) be an increasing sequence of numbers in (0, (q - d)/p) such that \\beta_n \\t…","kind":"proof","summary":"Let (\\beta_n) be an increasing sequence of numbers in (0, (q - d)/p) such that \\beta_n \\to (q -…","labels":[],"detail_key":"p11"},{"id":"n16567","layer":"informal","project":"p11","title":"Cover with bounded covering numbers","kind":"definition","summary":"[Cover with bounded covering numbers] A set T is said to have a cover with bounded covering num…","labels":["def:HasBoundedCoveringNumberCover"],"detail_key":"p11"},{"id":"n16568","layer":"informal","project":"p11","title":"lem:hasBoundedCoveringNumberCover_nnreal","kind":"lemma","summary":"R_+ has a cover with bounded covering numbers for the sets T_n = [0,n), constants c_n = n and e…","labels":["lem:hasBoundedCoveringNumberCover_nnreal"],"detail_key":"p11"},{"id":"n16569","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16570","layer":"informal","project":"p11","title":"thm:localized_holder_modification","kind":"theorem","summary":"Let T be a metric space with a cover (T_n) with bounded covering numbers with constants c_n and…","labels":["thm:localized_holder_modification"],"detail_key":"p11"},{"id":"n16571","layer":"informal","project":"p11","title":"For each n, by Theorem~\\refthm:holder_modification there is a modification Y_n of X seen…","kind":"proof","summary":"For each n, by Theorem~\\refthm:holder_modification there is a modification Y_n of X seen as a p…","labels":[],"detail_key":"p11"},{"id":"n16572","layer":"informal","project":"p11","title":"thm:localized_holder_modification_sup","kind":"theorem","summary":"Let T be a metric space with a cover (T_n) with bounded covering numbers with constants c_n and…","labels":["thm:localized_holder_modification_sup"],"detail_key":"p11"},{"id":"n16573","layer":"informal","project":"p11","title":"For each n, by Theorem~\\refthm:localized_holder_modification there is a modification Y_n…","kind":"proof","summary":"For each n, by Theorem~\\refthm:localized_holder_modification there is a modification Y_n of X s…","labels":[],"detail_key":"p11"},{"id":"n16574","layer":"informal","project":"p11","title":"pre-Brownian process","kind":"definition","summary":"[pre-Brownian process] Let \\Omega = R^R_+ and consider the probability space (\\Omega, P_B) (whe…","labels":["def:preBrownian"],"detail_key":"p11"},{"id":"n16575","layer":"informal","project":"p11","title":"lem:isGaussianProcess_preBrownian","kind":"lemma","summary":"The pre-Brownian process X of Definition~\\refdef:preBrownian is a Gaussian process.","labels":["lem:isGaussianProcess_preBrownian"],"detail_key":"p11"},{"id":"n16576","layer":"informal","project":"p11","title":"For any t_1, \\ldots, t_n \\in R_+, the distribution of (X_t_1, \\ldots, X_t_n) is given by…","kind":"proof","summary":"For any t_1, \\ldots, t_n \\in R_+, the distribution of (X_t_1, \\ldots, X_t_n) is given by a fini…","labels":[],"detail_key":"p11"},{"id":"n16577","layer":"informal","project":"p11","title":"lem:hasLaw_preBrownian_sub","kind":"lemma","summary":"Let X be the pre-Brownian process of Definition~\\refdef:preBrownian. Then, for all s, t \\in R_+…","labels":["lem:hasLaw_preBrownian_sub"],"detail_key":"p11"},{"id":"n16578","layer":"informal","project":"p11","title":"The map L : R^2 &\\to R \\\\ (x_1, x_2) &\\mapsto x_2 - x_1 is a continuous linear map, and (…","kind":"proof","summary":"The map L : R^2 &\\to R \\\\ (x_1, x_2) &\\mapsto x_2 - x_1 is a continuous linear map, and (X_s, X…","labels":[],"detail_key":"p11"},{"id":"n16579","layer":"informal","project":"p11","title":"lem:isKolmogorovProcess_preBrownian","kind":"lemma","summary":"The pre-Brownian process X of Definition~\\refdef:preBrownian satisfies the Kolmogorov condition…","labels":["lem:isKolmogorovProcess_preBrownian"],"detail_key":"p11"},{"id":"n16580","layer":"informal","project":"p11","title":"X_t - X_s is a Gaussian random variable with mean 0 and variance |t - s| (Lemma~\\reflem:h…","kind":"proof","summary":"X_t - X_s is a Gaussian random variable with mean 0 and variance |t - s| (Lemma~\\reflem:hasLaw_…","labels":[],"detail_key":"p11"},{"id":"n16581","layer":"informal","project":"p11","title":"Brownian motion","kind":"definition","summary":"[Brownian motion] By Theorem~\\refthm:localized_holder_modification_sup, there exists a modifica…","labels":["def:brownian"],"detail_key":"p11"},{"id":"n16582","layer":"informal","project":"p11","title":"lem:isGaussianProcess_brownian","kind":"lemma","summary":"The Brownian motion is a Gaussian process.","labels":["lem:isGaussianProcess_brownian"],"detail_key":"p11"},{"id":"n16583","layer":"informal","project":"p11","title":"The pre-Brownian process is a Gaussian process by Lemma~\\reflem:isGaussianProcess_preBrow…","kind":"proof","summary":"The pre-Brownian process is a Gaussian process by Lemma~\\reflem:isGaussianProcess_preBrownian.…","labels":[],"detail_key":"p11"},{"id":"n16584","layer":"informal","project":"p11","title":"lem:isHolderWith_brownian","kind":"lemma","summary":"The paths of the Brownian motion are locally Hölder continuous of all orders \\gamma \\in (0, 1/2…","labels":["lem:isHolderWith_brownian"],"detail_key":"p11"},{"id":"n16585","layer":"informal","project":"p11","title":"Consider the cover of R_+ given by T_n := [0, n + 1). By Lemma~\\reflem:hasBoundedCovering…","kind":"proof","summary":"Consider the cover of R_+ given by T_n := [0, n + 1). By Lemma~\\reflem:hasBoundedCoveringNumber…","labels":[],"detail_key":"p11"},{"id":"n16586","layer":"informal","project":"p11","title":"lem:continuous_brownian","kind":"lemma","summary":"The paths of the Brownian motion are continuous.","labels":["lem:continuous_brownian"],"detail_key":"p11"},{"id":"n16587","layer":"informal","project":"p11","title":"The paths are 1/4-Hölder continuous by Lemma~\\reflem:isHolderWith_brownian because 0 < 1/…","kind":"proof","summary":"The paths are 1/4-Hölder continuous by Lemma~\\reflem:isHolderWith_brownian because 0 < 1/4 < 1/…","labels":[],"detail_key":"p11"},{"id":"n16588","layer":"informal","project":"p11","title":"lem:hasLaw_brownian_eval","kind":"lemma","summary":"For t \\in R_+, the law of B_t (the Brownian motion at time t) is the real Gaussian measure N(0,…","labels":["lem:hasLaw_brownian_eval"],"detail_key":"p11"},{"id":"n16589","layer":"informal","project":"p11","title":"The Brownian motion B is a modification of the pre-Brownian process, and therefore has sa…","kind":"proof","summary":"The Brownian motion B is a modification of the pre-Brownian process, and therefore has same fin…","labels":[],"detail_key":"p11"},{"id":"n16590","layer":"informal","project":"p11","title":"lem:hasLaw_brownian_sub","kind":"lemma","summary":"For s, t \\in R_+, the law of B_t - B_s is the real Gaussian measure N(0,\\vert t - s \\vert).","labels":["lem:hasLaw_brownian_sub"],"detail_key":"p11"},{"id":"n16591","layer":"informal","project":"p11","title":"The Brownian motion B is a modification of the pre-Brownian process, and therefore has sa…","kind":"proof","summary":"The Brownian motion B is a modification of the pre-Brownian process, and therefore has same fin…","labels":[],"detail_key":"p11"},{"id":"n16592","layer":"informal","project":"p11","title":"def:HasIndepIncrements","kind":"definition","summary":"We say that a stochastic process X : T \\to \\Omega \\to E has independent increments if for all t…","labels":["def:HasIndepIncrements"],"detail_key":"p11"},{"id":"n16593","layer":"informal","project":"p11","title":"lem:hasIndepIncrements_brownian","kind":"lemma","summary":"The Brownian motion has independent increments.","labels":["lem:hasIndepIncrements_brownian"],"detail_key":"p11"},{"id":"n16594","layer":"informal","project":"p11","title":"Let t_1 \\le t_2 \\le \\ldots \\le t_n be n times in R_+. Then (B_t_2 - B_t_1, B_t_3 - B_t_2,…","kind":"proof","summary":"Let t_1 \\le t_2 \\le \\ldots \\le t_n be n times in R_+. Then (B_t_2 - B_t_1, B_t_3 - B_t_2, \\ldot…","labels":[],"detail_key":"p11"},{"id":"n16595","layer":"informal","project":"p11","title":"Auxiliary Wiener measure","kind":"definition","summary":"[Auxiliary Wiener measure] The pushforward of the measure P_B of Definition~\\refdef:gaussianLim…","labels":["def:wienerMeasureAux"],"detail_key":"p11"},{"id":"n16596","layer":"informal","project":"p11","title":"thm:ContinuousMap.borel_eq_iSup_comap_eval","kind":"theorem","summary":"The Borel sigma-algebra on C(R_+, R) coming from the compact-open topology is equal to the smal…","labels":["thm:ContinuousMap.borel_eq_iSup_comap_eval"],"detail_key":"p11"},{"id":"n16597","layer":"informal","project":"p11","title":"We prove that this holds for C(X, Y) as long as X and Y are second-countable topological…","kind":"proof","summary":"We prove that this holds for C(X, Y) as long as X and Y are second-countable topological spaces…","labels":[],"detail_key":"p11"},{"id":"n16598","layer":"informal","project":"p11","title":"def:MeasurableEquiv.continuousMap","kind":"definition","summary":"The identity is a measurable equivalence between the continuous functions of R^R_+ with the sub…","labels":["def:MeasurableEquiv.continuousMap"],"detail_key":"p11"},{"id":"n16599","layer":"informal","project":"p11","title":"Wiener measure","kind":"definition","summary":"[Wiener measure] The Wiener measure on C(R_+, R) with the Borel sigma-algebra is the map of the…","labels":["def:wienerMeasure"],"detail_key":"p11"},{"id":"n16600","layer":"informal","project":"p11","title":"Filtration","kind":"definition","summary":"[Filtration] \\mathlibok A filtration on a measurable space (\\Omega, A) with measure P indexed b…","labels":["def:filtration"],"detail_key":"p11"},{"id":"n16601","layer":"informal","project":"p11","title":"def:adapted","kind":"definition","summary":"\\mathlibok A process X : T \\to \\Omega \\to E is said to be adapted with respect to a filtration…","labels":["def:adapted"],"detail_key":"p11"},{"id":"n16602","layer":"informal","project":"p11","title":"Progressively measurable","kind":"definition","summary":"[Progressively measurable] \\mathlibok A stochastic process X is said to be progressively measur…","labels":["def:IsStronglyProgressive"],"detail_key":"p11"},{"id":"n16603","layer":"informal","project":"p11","title":"Right continuous function","kind":"definition","summary":"[Right continuous function] Let T be equipped with a partial order and f : T \\to E be a functio…","labels":["def:RightContinuous"],"detail_key":"p11"},{"id":"n16604","layer":"informal","project":"p11","title":"Cadlag function","kind":"definition","summary":"[Cadlag function] A function f : T \\to E is cadlag if it is right continuous and the limit \\lim…","labels":["def:IsCadlag"],"detail_key":"p11"},{"id":"n16605","layer":"informal","project":"p11","title":"lem:Adapted.isStronglyProgressive_of_rightContinuous","kind":"lemma","summary":"Suppose T is a linearly ordered set equipped with the order topology. If T is second countable…","labels":["lem:Adapted.isStronglyProgressive_of_rightContinuous"],"detail_key":"p11"},{"id":"n16606","layer":"informal","project":"p11","title":"Fixing t \\in T, let S be a union of a dense set of (-\\infty,t] and the set of points isol…","kind":"proof","summary":"Fixing t \\in T, let S be a union of a dense set of (-\\infty,t] and the set of points isolated f…","labels":[],"detail_key":"p11"},{"id":"n16607","layer":"informal","project":"p11","title":"Martingale","kind":"definition","summary":"[Martingale] \\mathlibok Let F be a filtration on a measurable space \\Omega with measure P index…","labels":["def:Martingale"],"detail_key":"p11"},{"id":"n16608","layer":"informal","project":"p11","title":"Submartingale","kind":"definition","summary":"[Submartingale] \\mathlibok Let F be a filtration on a measurable space \\Omega with measure P in…","labels":["def:Submartingale"],"detail_key":"p11"},{"id":"n16609","layer":"informal","project":"p11","title":"lem:condExp_sub_nonneg","kind":"lemma","summary":"\\mathlibok Let X be a real-valued submartingale with respect to a filtration F. Then for all i…","labels":["lem:condExp_sub_nonneg"],"detail_key":"p11"},{"id":"n16610","layer":"informal","project":"p11","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p11"},{"id":"n16611","layer":"informal","project":"p11","title":"lem:Submartingale.integrable_stoppedValue","kind":"lemma","summary":"\\mathlibok Let X be a submartingale. Then for all bounded stopping times \\tau, the stopped valu…","labels":["lem:Submartingale.integrable_stoppedValue"],"detail_key":"p11"},{"id":"n16612","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16613","layer":"informal","project":"p11","title":"lem:Martingale.congr","kind":"lemma","summary":"If X is a martingale and Y is an adapted modification of X, then Y is a martingale.","labels":["lem:Martingale.congr"],"detail_key":"p11"},{"id":"n16614","layer":"informal","project":"p11","title":"Let i \\le j in T. We want to show that P[Y_j \\mid F_i] = Y_i almost surely. It suffices t…","kind":"proof","summary":"Let i \\le j in T. We want to show that P[Y_j \\mid F_i] = Y_i almost surely. It suffices to show…","labels":[],"detail_key":"p11"},{"id":"n16615","layer":"informal","project":"p11","title":"lem:Submartingale.congr","kind":"lemma","summary":"If X is a submartingale and Y is an adapted modification of X, then Y is a submartingale.","labels":["lem:Submartingale.congr"],"detail_key":"p11"},{"id":"n16616","layer":"informal","project":"p11","title":"Let i \\le j in T. We want to show that P[Y_j \\mid F_i] \\ge Y_i almost surely. It suffices…","kind":"proof","summary":"Let i \\le j in T. We want to show that P[Y_j \\mid F_i] \\ge Y_i almost surely. It suffices to sh…","labels":[],"detail_key":"p11"},{"id":"n16617","layer":"informal","project":"p11","title":"Jensen's inequality for the conditional expectation","kind":"lemma","summary":"[Jensen's inequality for the conditional expectation] \\mathlibok Let X : \\Omega \\to E be an int…","labels":["lem:conditional_jensen"],"detail_key":"p11"},{"id":"n16618","layer":"informal","project":"p11","title":"Done in a Mathlib PR for finite measures: \\hrefhttps://github.com/leanprover-community/ma…","kind":"proof","summary":"Done in a Mathlib PR for finite measures: \\hrefhttps://github.com/leanprover-community/mathlib4…","labels":[],"detail_key":"p11"},{"id":"n16619","layer":"informal","project":"p11","title":"cor:norm_condExp_le","kind":"corollary","summary":"Let X : \\Omega \\to E be an integrable random variable with values in a normed space E. Then, fo…","labels":["cor:norm_condExp_le"],"detail_key":"p11"},{"id":"n16620","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16621","layer":"informal","project":"p11","title":"lem:Martingale.submartingale_convex_comp","kind":"lemma","summary":"Let X : T \\rightarrow \\Omega\\rightarrow E a martingale with values in a normed space E. Let \\ph…","labels":["lem:Martingale.submartingale_convex_comp"],"detail_key":"p11"},{"id":"n16622","layer":"informal","project":"p11","title":"By the conditional Jensen inequality (Lemma~\\reflem:conditional_jensen), \\phi(X_t) = \\phi…","kind":"proof","summary":"By the conditional Jensen inequality (Lemma~\\reflem:conditional_jensen), \\phi(X_t) = \\phi\\left(…","labels":[],"detail_key":"p11"},{"id":"n16623","layer":"informal","project":"p11","title":"cor:Martingale.submartingale_norm","kind":"corollary","summary":"Let X : T \\rightarrow \\Omega \\rightarrow E a martingale with values in a normed space E. Then \\…","labels":["cor:Martingale.submartingale_norm"],"detail_key":"p11"},{"id":"n16624","layer":"informal","project":"p11","title":"Same proof as Lemma~\\reflem:Martingale.submartingale_convex_comp, specialized to \\phi = \\…","kind":"proof","summary":"Same proof as Lemma~\\reflem:Martingale.submartingale_convex_comp, specialized to \\phi = \\Vert \\…","labels":[],"detail_key":"p11"},{"id":"n16625","layer":"informal","project":"p11","title":"lem:convex_of_submg_is_submg","kind":"lemma","summary":"Let X : T \\rightarrow \\Omega \\rightarrow E a sub-martingale. Let \\phi:E \\rightarrow R convex, c…","labels":["lem:convex_of_submg_is_submg"],"detail_key":"p11"},{"id":"n16626","layer":"informal","project":"p11","title":"By Jensen and the fact that \\phi is increasing \\phi(X_t) \\leq \\phi\\left( E[X_T\\ |\\ F_t] \\…","kind":"proof","summary":"By Jensen and the fact that \\phi is increasing \\phi(X_t) \\leq \\phi\\left( E[X_T\\ |\\ F_t] \\right)…","labels":[],"detail_key":"p11"},{"id":"n16627","layer":"informal","project":"p11","title":"Stopping time","kind":"definition","summary":"[Stopping time] \\mathlibok A stopping time with respect to some filtration F indexed by T is a…","labels":["def:IsStoppingTime"],"detail_key":"p11"},{"id":"n16628","layer":"informal","project":"p11","title":"\\sigma-algebra generated by a stopping time","kind":"definition","summary":"[\\sigma-algebra generated by a stopping time] \\mathlibok Given a stopping time \\tau on a time i…","labels":["def:StoppingTimeGen"],"detail_key":"p11"},{"id":"n16629","layer":"informal","project":"p11","title":"lem:StoppingTimeGenMono","kind":"lemma","summary":"\\mathlibok Let \\tau, \\sigma be stopping times such that \\tau \\le \\sigma. Then, F_\\tau \\subseteq…","labels":["lem:StoppingTimeGenMono"],"detail_key":"p11"},{"id":"n16630","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16631","layer":"informal","project":"p11","title":"Stopped process","kind":"definition","summary":"[Stopped process] \\mathlibok Let X : T \\to \\Omega \\to E be a stochastic process and let \\tau :…","labels":["def:stoppedProcess"],"detail_key":"p11"},{"id":"n16632","layer":"informal","project":"p11","title":"lem:Submartingale.stoppedProcess","kind":"lemma","summary":"\\mathlibok Let X : N \\to \\Omega \\to R be a sub-martingale and \\tau a stopping time with respect…","labels":["lem:Submartingale.stoppedProcess"],"detail_key":"p11"},{"id":"n16633","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16634","layer":"informal","project":"p11","title":"Hitting time","kind":"definition","summary":"[Hitting time] \\mathlibok For X : T \\to \\Omega \\to E a stochastic process, B a subset of E and…","labels":["def:hittingAfter"],"detail_key":"p11"},{"id":"n16635","layer":"informal","project":"p11","title":"def:hittingBtwn","kind":"definition","summary":"\\mathlibok For X : T \\to \\Omega \\to E a stochastic process, B a subset of E and t_0, t_1 \\in T,…","labels":["def:hittingBtwn"],"detail_key":"p11"},{"id":"n16636","layer":"informal","project":"p11","title":"lem:isStoppingTime_hittingBtwn","kind":"lemma","summary":"\\mathlibok The hitting time \\tau_B, t_0, t_1 of a measurable set by an adapted process on a dis…","labels":["lem:isStoppingTime_hittingBtwn"],"detail_key":"p11"},{"id":"n16637","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16638","layer":"informal","project":"p11","title":"Left continuation","kind":"definition","summary":"[Left continuation] Assume that T is a partial order. For F a filtration indexed by T and t \\in…","labels":["def:leftLimitFiltration"],"detail_key":"p11"},{"id":"n16639","layer":"informal","project":"p11","title":"Right continuation","kind":"definition","summary":"[Right continuation] Assume that T is a partial order. For F a filtration indexed by T and t \\i…","labels":["def:rightLimitFiltration"],"detail_key":"p11"},{"id":"n16640","layer":"informal","project":"p11","title":"lem:filtrationRightCont","kind":"lemma","summary":"The right continuation of F is a filtration.","labels":["lem:filtrationRightCont"],"detail_key":"p11"},{"id":"n16641","layer":"informal","project":"p11","title":"We endow T with the order topology. Let us first prove that the right continuation is non…","kind":"proof","summary":"We endow T with the order topology. Let us first prove that the right continuation is nondecrea…","labels":[],"detail_key":"p11"},{"id":"n16642","layer":"informal","project":"p11","title":"lem:rightContDef","kind":"lemma","summary":"Suppose T is a topological space with the topology being the order topology. For t \\in T, the r…","labels":["lem:rightContDef"],"detail_key":"p11"},{"id":"n16643","layer":"informal","project":"p11","title":"This follows from Definition~\\refdef:rightLimitFiltration because the topology on T agree…","kind":"proof","summary":"This follows from Definition~\\refdef:rightLimitFiltration because the topology on T agrees with…","labels":[],"detail_key":"p11"},{"id":"n16644","layer":"informal","project":"p11","title":"lem:rightContIsolated","kind":"lemma","summary":"Suppose T is a topological space with the topology being the order topology. Assume that t \\in…","labels":["lem:rightContIsolated"],"detail_key":"p11"},{"id":"n16645","layer":"informal","project":"p11","title":"This is a direct consequence of Lemma~\\reflem:rightContDef.","kind":"proof","summary":"This is a direct consequence of Lemma~\\reflem:rightContDef.","labels":[],"detail_key":"p11"},{"id":"n16646","layer":"informal","project":"p11","title":"lem:rightContSuccOrder","kind":"lemma","summary":"Assume that T is a linear order with successor. This means that for any t, there is an element…","labels":["lem:rightContSuccOrder"],"detail_key":"p11"},{"id":"n16647","layer":"informal","project":"p11","title":"Endow T with the order topology. In a linear order with successor equipped with the order…","kind":"proof","summary":"Endow T with the order topology. In a linear order with successor equipped with the order topol…","labels":[],"detail_key":"p11"},{"id":"n16648","layer":"informal","project":"p11","title":"lem:rightContIsMax","kind":"lemma","summary":"If t \\in T is maximal, F_t+ = F_t.","labels":["lem:rightContIsMax"],"detail_key":"p11"},{"id":"n16649","layer":"informal","project":"p11","title":"Endow T with the order topology. As t is maximal, it is isolated on the right for this to…","kind":"proof","summary":"Endow T with the order topology. As t is maximal, it is isolated on the right for this topology…","labels":[],"detail_key":"p11"},{"id":"n16650","layer":"informal","project":"p11","title":"lem:rightContExistsGt","kind":"lemma","summary":"If T is a linear order and there exists u > t such that (t, u) = \\emptyset, then F_t+ = F_t.","labels":["lem:rightContExistsGt"],"detail_key":"p11"},{"id":"n16651","layer":"informal","project":"p11","title":"Endow T with the order topology. The hypothesis implies that t is isolated on the right i…","kind":"proof","summary":"Endow T with the order topology. The hypothesis implies that t is isolated on the right in this…","labels":[],"detail_key":"p11"},{"id":"n16652","layer":"informal","project":"p11","title":"lem:rightContNeBot","kind":"lemma","summary":"Suppose T is a topological space with the topology being the order topology. Assume that t \\in…","labels":["lem:rightContNeBot"],"detail_key":"p11"},{"id":"n16653","layer":"informal","project":"p11","title":"This is a direct consequence of Lemma~\\reflem:rightContDef.","kind":"proof","summary":"This is a direct consequence of Lemma~\\reflem:rightContDef.","labels":[],"detail_key":"p11"},{"id":"n16654","layer":"informal","project":"p11","title":"lem:rightContNotIsMax","kind":"lemma","summary":"Assume that T is a densely ordered linear order, meaning that for all s < t, there exists u suc…","labels":["lem:rightContNotIsMax"],"detail_key":"p11"},{"id":"n16655","layer":"informal","project":"p11","title":"Endow T with the order topology. In a densely ordered linear order, a point which is not…","kind":"proof","summary":"Endow T with the order topology. In a densely ordered linear order, a point which is not maxima…","labels":[],"detail_key":"p11"},{"id":"n16656","layer":"informal","project":"p11","title":"lem:rightContEq","kind":"lemma","summary":"If T is a densely ordered linear order with no maximal element, then forall t \\in T we have F_t…","labels":["lem:rightContEq"],"detail_key":"p11"},{"id":"n16657","layer":"informal","project":"p11","title":"For all t, t is not maximal, so we can conclude by Lemma~\\reflem:rightContNotIsMax.","kind":"proof","summary":"For all t, t is not maximal, so we can conclude by Lemma~\\reflem:rightContNotIsMax.","labels":[],"detail_key":"p11"},{"id":"n16658","layer":"informal","project":"p11","title":"lem:leRightCont","kind":"lemma","summary":"The filtration F is contained in its right continuation.","labels":["lem:leRightCont"],"detail_key":"p11"},{"id":"n16659","layer":"informal","project":"p11","title":"Endow T with the order topology, and consider t \\in T. Using Lemma~\\reflem:rightContDef,…","kind":"proof","summary":"Endow T with the order topology, and consider t \\in T. Using Lemma~\\reflem:rightContDef, we spl…","labels":[],"detail_key":"p11"},{"id":"n16660","layer":"informal","project":"p11","title":"lem:rightContSelf","kind":"lemma","summary":"The right continuation of the right continuation of F is equal to the right continuation of F.","labels":["lem:rightContSelf"],"detail_key":"p11"},{"id":"n16661","layer":"informal","project":"p11","title":"Let t \\in T. From Lemma~\\reflem:leRightCont, we already now that F_t+ \\subseteq F_t++. En…","kind":"proof","summary":"Let t \\in T. From Lemma~\\reflem:leRightCont, we already now that F_t+ \\subseteq F_t++. Endow T…","labels":[],"detail_key":"p11"},{"id":"n16662","layer":"informal","project":"p11","title":"Basic properties of the right continuation","kind":"lemma","summary":"[Basic properties of the right continuation] Fake lemma for the dependency graph. Import this t…","labels":["lem:rightLimitFiltration_basic"],"detail_key":"p11"},{"id":"n16663","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16664","layer":"informal","project":"p11","title":"Right-continuous filtration","kind":"definition","summary":"[Right-continuous filtration] \\mathlibok We say that the filtration is \\emphright-continuous if…","labels":["def:rightContinuous"],"detail_key":"p11"},{"id":"n16665","layer":"informal","project":"p11","title":"lem:eqRightCont","kind":"lemma","summary":"\\mathlibok If F is right-continuous, then for all t \\in T, F_t = F_t+.","labels":["lem:eqRightCont"],"detail_key":"p11"},{"id":"n16666","layer":"informal","project":"p11","title":"This is a direct consequence of Definition~\\refdef:rightContinuous and Lemma~\\reflem:leRi…","kind":"proof","summary":"This is a direct consequence of Definition~\\refdef:rightContinuous and Lemma~\\reflem:leRightCon…","labels":[],"detail_key":"p11"},{"id":"n16667","layer":"informal","project":"p11","title":"lem:rightContinuousRightCont","kind":"lemma","summary":"\\mathlibok The right continuation of F is right-continuous.","labels":["lem:rightContinuousRightCont"],"detail_key":"p11"},{"id":"n16668","layer":"informal","project":"p11","title":"This follows immediately from Lemma~\\reflem:rightContSelf.","kind":"proof","summary":"This follows immediately from Lemma~\\reflem:rightContSelf.","labels":[],"detail_key":"p11"},{"id":"n16669","layer":"informal","project":"p11","title":"lem:rightContinuousMeasurableSet","kind":"lemma","summary":"If F is right-continuous, then for all t \\in T, any set A \\subseteq \\Omega which is F_t-measura…","labels":["lem:rightContinuousMeasurableSet"],"detail_key":"p11"},{"id":"n16670","layer":"informal","project":"p11","title":"This is a direct consequence of Definition~\\refdef:rightContinuous.","kind":"proof","summary":"This is a direct consequence of Definition~\\refdef:rightContinuous.","labels":[],"detail_key":"p11"},{"id":"n16671","layer":"informal","project":"p11","title":"Basic properties of right continuous filtrations","kind":"lemma","summary":"[Basic properties of right continuous filtrations] Fake lemma for the dependency graph. Import…","labels":["lem:rightContinuous_basic"],"detail_key":"p11"},{"id":"n16672","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16673","layer":"informal","project":"p11","title":"Usual conditions","kind":"definition","summary":"[Usual conditions] We say that a filtered probability space (\\Omega, F, P) satisfies the usual…","labels":["def:hasUsualConditions"],"detail_key":"p11"},{"id":"n16674","layer":"informal","project":"p11","title":"Predictable \\sigma-algebra","kind":"definition","summary":"[Predictable \\sigma-algebra] \\mathlibok Let F be a filtration on a measurable space indexed \\Om…","labels":["def:predictableMeasurableSpace"],"detail_key":"p11"},{"id":"n16675","layer":"informal","project":"p11","title":"Predictable process","kind":"definition","summary":"[Predictable process] \\mathlibok A process X : T \\to \\Omega \\to E is said to be predictable wit…","labels":["def:predictable"],"detail_key":"p11"},{"id":"n16676","layer":"informal","project":"p11","title":"lem:Predictable.progressive","kind":"lemma","summary":"\\mathlibok A predictable process is progressively measurable.","labels":["lem:Predictable.progressive"],"detail_key":"p11"},{"id":"n16677","layer":"informal","project":"p11","title":"Let X : T \\times \\Omega \\to E be a predictable process, we will show that it is progressi…","kind":"proof","summary":"Let X : T \\times \\Omega \\to E be a predictable process, we will show that it is progressively m…","labels":[],"detail_key":"p11"},{"id":"n16678","layer":"informal","project":"p11","title":"lem:predictable_Ioc_prod","kind":"lemma","summary":"\\mathlibok Sets of the form (s, t] \\times A for any A \\in F_s is measurable with respect to the…","labels":["lem:predictable_Ioc_prod"],"detail_key":"p11"},{"id":"n16679","layer":"informal","project":"p11","title":"For t \\le s, the set in question is empty and thusly, trivially measurable. On the other…","kind":"proof","summary":"For t \\le s, the set in question is empty and thusly, trivially measurable. On the other hand,…","labels":[],"detail_key":"p11"},{"id":"n16680","layer":"informal","project":"p11","title":"lem:predictable_nat_iff","kind":"lemma","summary":"\\mathlibok Let X : N \\to \\Omega \\to E be a stochastic process and let F be a filtration indexed…","labels":["lem:predictable_nat_iff"],"detail_key":"p11"},{"id":"n16681","layer":"informal","project":"p11","title":"Suppose first that X is predictable. Straightaway, X_0 is F_0-measurable as predictable i…","kind":"proof","summary":"Suppose first that X is predictable. Straightaway, X_0 is F_0-measurable as predictable implies…","labels":[],"detail_key":"p11"},{"id":"n16682","layer":"informal","project":"p11","title":"lem:Adapted.isPredictable_of_leftContinuous","kind":"lemma","summary":"An adapted process with left-continuous paths is predictable.","labels":["lem:Adapted.isPredictable_of_leftContinuous"],"detail_key":"p11"},{"id":"n16683","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16684","layer":"informal","project":"p11","title":"lem:condExpUI","kind":"lemma","summary":"If (X_i)_i \\in \\iota is a family of (probabilistically) uniformly integrable functions and (F_j…","labels":["lem:condExpUI"],"detail_key":"p11"},{"id":"n16685","layer":"informal","project":"p11","title":"Since (X_i)_i \\in \\iota is uniformly integrable, it is uniformly bounded in L^1, thus so…","kind":"proof","summary":"Since (X_i)_i \\in \\iota is uniformly integrable, it is uniformly bounded in L^1, thus so is (P[…","labels":[],"detail_key":"p11"},{"id":"n16686","layer":"informal","project":"p11","title":"lem:uniformIntegrable_of_bounded","kind":"lemma","summary":"Let (X_i)_i \\in \\iota be a family of random variables that is bounded in L^p, for p > q \\ge 1.…","labels":["lem:uniformIntegrable_of_bounded"],"detail_key":"p11"},{"id":"n16687","layer":"informal","project":"p11","title":"For any measurable set s we have by Hölder inequality that \\|X_t I_s\\|_q \\le \\|X_t\\|_p P(…","kind":"proof","summary":"For any measurable set s we have by Hölder inequality that \\|X_t I_s\\|_q \\le \\|X_t\\|_p P(s)^1/q…","labels":[],"detail_key":"p11"},{"id":"n16688","layer":"informal","project":"p11","title":"lem:uniformIntegrable_stoppedValue_martingale","kind":"lemma","summary":"Let X be a martingale on a discrete index set and let (\\tau_k)_k \\in N be a sequence of stoppin…","labels":["lem:uniformIntegrable_stoppedValue_martingale"],"detail_key":"p11"},{"id":"n16689","layer":"informal","project":"p11","title":"By optional sampling (Lemma~\\reflem:optionalSampling_discrete), we have that for each k,…","kind":"proof","summary":"By optional sampling (Lemma~\\reflem:optionalSampling_discrete), we have that for each k, X_\\tau…","labels":[],"detail_key":"p11"},{"id":"n16690","layer":"informal","project":"p11","title":"lem:uniformIntegrable_stoppedValue_martingale_of_countable_range","kind":"lemma","summary":"Let X be a martingale and let (\\tau_k)_k \\in N be a sequence of stopping times that are uniform…","labels":["lem:uniformIntegrable_stoppedValue_martingale_of_countable_range"],"detail_key":"p11"},{"id":"n16691","layer":"informal","project":"p11","title":"Same proof as in Lemma~\\reflem:uniformIntegrable_stoppedValue_martingale.","kind":"proof","summary":"Same proof as in Lemma~\\reflem:uniformIntegrable_stoppedValue_martingale.","labels":[],"detail_key":"p11"},{"id":"n16692","layer":"informal","project":"p11","title":"lem:uniformIntegrableAdd","kind":"lemma","summary":"Let (X_t)_t \\in T and (Y_t)_t \\in T be two families of uniformly integrable random variables. T…","labels":["lem:uniformIntegrableAdd"],"detail_key":"p11"},{"id":"n16693","layer":"informal","project":"p11","title":"The familie","kind":"proof","summary":"The familie","labels":[],"detail_key":"p11"},{"id":"n16694","layer":"informal","project":"p11","title":"lem:uniformIntegrableDominated","kind":"lemma","summary":"Let (X_s)_s \\in S be a family of random variables and (Y_t)_t \\in T be a family of uniformly in…","labels":["lem:uniformIntegrableDominated"],"detail_key":"p11"},{"id":"n16695","layer":"informal","project":"p11","title":"Let \\epsilon > 0. The family Y is uniformly integrable, thus there exists C \\ge 0 such th…","kind":"proof","summary":"Let \\epsilon > 0. The family Y is uniformly integrable, thus there exists C \\ge 0 such that for…","labels":[],"detail_key":"p11"},{"id":"n16696","layer":"informal","project":"p11","title":"lem:uniformIntegrableDominatedSingleton","kind":"lemma","summary":"Let (X_t)_t \\in T be a family of random variables and Y be a real random variable in L^p. If fo…","labels":["lem:uniformIntegrableDominatedSingleton"],"detail_key":"p11"},{"id":"n16697","layer":"informal","project":"p11","title":"Because Y is in L^p, we deduce that \\Y\\ is uniformly integrable. The conclusion then foll…","kind":"proof","summary":"Because Y is in L^p, we deduce that \\Y\\ is uniformly integrable. The conclusion then follows fr…","labels":[],"detail_key":"p11"},{"id":"n16698","layer":"informal","project":"p11","title":"lem:uniformIntegrableNorm","kind":"lemma","summary":"If (X_t)_t \\in T is a family of uniformly integrable random variables, then so is (\\|X_t\\|)_t \\…","labels":["lem:uniformIntegrableNorm"],"detail_key":"p11"},{"id":"n16699","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:uniformIntegrableDominated with Y := X.","kind":"proof","summary":"Apply Lemma~\\reflem:uniformIntegrableDominated with Y := X.","labels":[],"detail_key":"p11"},{"id":"n16700","layer":"informal","project":"p11","title":"lem:uniformIntegrableIffNorm","kind":"lemma","summary":"Let (X_t)_t \\in T be a family of uniformly integrable random variables. It is uniformly integra…","labels":["lem:uniformIntegrableIffNorm"],"detail_key":"p11"},{"id":"n16701","layer":"informal","project":"p11","title":"The forward direction is Lemma~\\reflem:uniformIntegrableNorm. The converse direction foll…","kind":"proof","summary":"The forward direction is Lemma~\\reflem:uniformIntegrableNorm. The converse direction follows fr…","labels":[],"detail_key":"p11"},{"id":"n16702","layer":"informal","project":"p11","title":"lem:uniformIntegrableComp","kind":"lemma","summary":"If (X_t)_t \\in T is uniformly integrable and \\phi : S \\to T, then (X_\\phi(s))_s \\in S is unifor…","labels":["lem:uniformIntegrableComp"],"detail_key":"p11"},{"id":"n16703","layer":"informal","project":"p11","title":"This is immediate from the definition.","kind":"proof","summary":"This is immediate from the definition.","labels":[],"detail_key":"p11"},{"id":"n16704","layer":"informal","project":"p11","title":"lem:uniformIntegrable_stoppedValue_submartingale","kind":"lemma","summary":"Let X be a submartingale on a discrete index set and let (\\tau_k)_k \\in N be a sequence of stop…","labels":["lem:uniformIntegrable_stoppedValue_submartingale"],"detail_key":"p11"},{"id":"n16705","layer":"informal","project":"p11","title":"Use Doob decomposition to write X_n = M_n + A_n, where M (Definition~\\refdef:martingalePa…","kind":"proof","summary":"Use Doob decomposition to write X_n = M_n + A_n, where M (Definition~\\refdef:martingalePart) is…","labels":[],"detail_key":"p11"},{"id":"n16706","layer":"informal","project":"p11","title":"lem:memLp_of_tendstoInMeasure","kind":"lemma","summary":"Let (X_n)_n \\in N be a sequence of p-uniformly integrable stochastic processes and suppose X_n…","labels":["lem:memLp_of_tendstoInMeasure"],"detail_key":"p11"},{"id":"n16707","layer":"informal","project":"p11","title":"Since X_n \\to X in probability, it has a subsequence (X_n_k) \\subseteq (X_n) which conver…","kind":"proof","summary":"Since X_n \\to X in probability, it has a subsequence (X_n_k) \\subseteq (X_n) which converges to…","labels":[],"detail_key":"p11"},{"id":"n16708","layer":"informal","project":"p11","title":"lem:uniformIntegrable_of_tendsto_ae","kind":"lemma","summary":"Let (X_t)_t \\in T be a family of p-uniformly integrable stochastic processes. Then the family o…","labels":["lem:uniformIntegrable_of_tendsto_ae"],"detail_key":"p11"},{"id":"n16709","layer":"informal","project":"p11","title":"Let \\epsilon > 0. There exists \\delta > 0 such that for all t\\in T and all measurable set…","kind":"proof","summary":"Let \\epsilon > 0. There exists \\delta > 0 such that for all t\\in T and all measurable set S suc…","labels":[],"detail_key":"p11"},{"id":"n16710","layer":"informal","project":"p11","title":"lem:uniformIntegrable_of_tendstoInMeasure","kind":"lemma","summary":"Let (X_t)_t \\in T be a family of p-uniformly integrable stochastic processes. Then the family o…","labels":["lem:uniformIntegrable_of_tendstoInMeasure"],"detail_key":"p11"},{"id":"n16711","layer":"informal","project":"p11","title":"A subfamily of a p-uniformly integrable family is p-uniformly integrable. As convergence…","kind":"proof","summary":"A subfamily of a p-uniformly integrable family is p-uniformly integrable. As convergence in pro…","labels":[],"detail_key":"p11"},{"id":"n16712","layer":"informal","project":"p11","title":"Vitali convergence theorem","kind":"lemma","summary":"[Vitali convergence theorem] \\mathlibok A sequence of functions converges in L^1 if and only if…","labels":["lem:vitali"],"detail_key":"p11"},{"id":"n16713","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16714","layer":"informal","project":"p11","title":"Ordered Monoid","kind":"definition","summary":"[Ordered Monoid] \\mathlibok Let (M, +) be a commutative monoid that is also a partial order. It…","labels":["def:isOrderedAddMonoid"],"detail_key":"p11"},{"id":"n16715","layer":"informal","project":"p11","title":"Ordered Module","kind":"definition","summary":"[Ordered Module] \\mathlibok Let \\alpha, \\beta be preorders with 0 elements and such that there…","labels":["def:isOrderedModule"],"detail_key":"p11"},{"id":"n16716","layer":"informal","project":"p11","title":"Order-closed topology","kind":"definition","summary":"[Order-closed topology] \\mathlibok Let X be a topological space that is also a preorder. The sp…","labels":["def:orderClosedTopology"],"detail_key":"p11"},{"id":"n16717","layer":"informal","project":"p11","title":"Optional sampling (discrete time)","kind":"lemma","summary":"[Optional sampling (discrete time)] \\mathlibok Let X be a discrete time martingale with respect…","labels":["lem:optionalSampling_discrete"],"detail_key":"p11"},{"id":"n16718","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16719","layer":"informal","project":"p11","title":"lem:optionalSampling_discrete_submartingale","kind":"lemma","summary":"Let X be a discrete time submartingale with respect to the filtration F taking values in a real…","labels":["lem:optionalSampling_discrete_submartingale"],"detail_key":"p11"},{"id":"n16720","layer":"informal","project":"p11","title":"Use Doob decomposition to write X_n = M_n + A_n, where M (Definition~\\refdef:martingalePa…","kind":"proof","summary":"Use Doob decomposition to write X_n = M_n + A_n, where M (Definition~\\refdef:martingalePart) is…","labels":[],"detail_key":"p11"},{"id":"n16721","layer":"informal","project":"p11","title":"lem:optionalSampling_discrete_supermartingale","kind":"lemma","summary":"Let X be a discrete time supermartingale with respect to the filtration F taking values in a re…","labels":["lem:optionalSampling_discrete_supermartingale"],"detail_key":"p11"},{"id":"n16722","layer":"informal","project":"p11","title":"We know that -X is a submartingale, so from Lemma~\\reflem:optionalSampling_discrete_subma…","kind":"proof","summary":"We know that -X is a submartingale, so from Lemma~\\reflem:optionalSampling_discrete_submartinga…","labels":[],"detail_key":"p11"},{"id":"n16723","layer":"informal","project":"p11","title":"Discrete approximation sequence","kind":"definition","summary":"[Discrete approximation sequence] Given a stopping time \\(\\tau : \\Omega \\to T \\cup \\\\infty\\\\),…","labels":["def:approxSeq"],"detail_key":"p11"},{"id":"n16724","layer":"informal","project":"p11","title":"Approximable time index","kind":"definition","summary":"[Approximable time index] A time index set \\(T\\) is said to be approximable if for any stopping…","labels":["def:approximableTimeIndex"],"detail_key":"p11"},{"id":"n16725","layer":"informal","project":"p11","title":"lem:tendsto_stoppedValue_discreteApproxSequence","kind":"lemma","summary":"Given a right continuous process \\(X\\) and a discrete approximation sequence \\((\\tau_n)\\) of th…","labels":["lem:tendsto_stoppedValue_discreteApproxSequence"],"detail_key":"p11"},{"id":"n16726","layer":"informal","project":"p11","title":"This follows directly as \\(X\\) is right continuous and \\(\\tau_n \\downarrow \\tau\\) a.s.","kind":"proof","summary":"This follows directly as \\(X\\) is right continuous and \\(\\tau_n \\downarrow \\tau\\) a.s.","labels":[],"detail_key":"p11"},{"id":"n16727","layer":"informal","project":"p11","title":"lem:discreteApproxSequence_of","kind":"lemma","summary":"Let \\(\\tau\\) be a stopping time bounded by \\(t \\in T\\) and \\((\\tau_n)\\) be a discrete approxima…","labels":["lem:discreteApproxSequence_of"],"detail_key":"p11"},{"id":"n16728","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16729","layer":"informal","project":"p11","title":"lem:uniformIntegrable_stoppedValue_discreteApproxSequence","kind":"lemma","summary":"Let \\(\\tau\\) be a stopping time bounded by \\(t \\in T\\) and \\((\\tau_n)\\) be a discrete approxima…","labels":["lem:uniformIntegrable_stoppedValue_discreteApproxSequence"],"detail_key":"p11"},{"id":"n16730","layer":"informal","project":"p11","title":"Follows directly by Lemma~\\reflem:uniformIntegrable_stoppedValue_martingale_of_countable_…","kind":"proof","summary":"Follows directly by Lemma~\\reflem:uniformIntegrable_stoppedValue_martingale_of_countable_range…","labels":[],"detail_key":"p11"},{"id":"n16731","layer":"informal","project":"p11","title":"lem:tendsto_eLpNorm_stoppedValue_discreteApproxSequence","kind":"lemma","summary":"Let \\(\\tau\\) be a stopping time bounded by \\(t \\in T\\) and \\((\\tau_n)\\) be a discrete approxima…","labels":["lem:tendsto_eLpNorm_stoppedValue_discreteApproxSequence"],"detail_key":"p11"},{"id":"n16732","layer":"informal","project":"p11","title":"By Lemma~\\reflem:tendsto_stoppedValue_discreteApproxSequence, as \\(X\\) is right continuou…","kind":"proof","summary":"By Lemma~\\reflem:tendsto_stoppedValue_discreteApproxSequence, as \\(X\\) is right continuous we h…","labels":[],"detail_key":"p11"},{"id":"n16733","layer":"informal","project":"p11","title":"lem:stoppingTime_approximation","kind":"lemma","summary":"T = R_+ is an approximable time index. In particular, for any stopping time \\tau on \\overlineR_…","labels":["lem:stoppingTime_approximation"],"detail_key":"p11"},{"id":"n16734","layer":"informal","project":"p11","title":"Clearly \\tau_n \\downarrow \\tau as n \\to \\infty and so it remains to show that each \\tau_n…","kind":"proof","summary":"Clearly \\tau_n \\downarrow \\tau as n \\to \\infty and so it remains to show that each \\tau_n is a…","labels":[],"detail_key":"p11"},{"id":"n16735","layer":"informal","project":"p11","title":"lem:stoppingTime_approximationNat","kind":"lemma","summary":"T = N is an approximable time index.","labels":["lem:stoppingTime_approximationNat"],"detail_key":"p11"},{"id":"n16736","layer":"informal","project":"p11","title":"Immediate as we can take \\(\\tau_n = \\tau\\) for all \\(n\\).","kind":"proof","summary":"Immediate as we can take \\(\\tau_n = \\tau\\) for all \\(n\\).","labels":[],"detail_key":"p11"},{"id":"n16737","layer":"informal","project":"p11","title":"Optional sampling (continuous time)","kind":"lemma","summary":"[Optional sampling (continuous time)] Let X be a right-continuous F-martingale on an approximab…","labels":["lem:optionalSampling"],"detail_key":"p11"},{"id":"n16738","layer":"informal","project":"p11","title":"Fixing A \\in F_\\sigma, we need to show that P[X_\\tau I_A] = P[X_\\sigma \\wedge \\tau I_A].…","kind":"proof","summary":"Fixing A \\in F_\\sigma, we need to show that P[X_\\tau I_A] = P[X_\\sigma \\wedge \\tau I_A]. Let (\\…","labels":[],"detail_key":"p11"},{"id":"n16739","layer":"informal","project":"p11","title":"lem:optionalSamplingSubmartingale","kind":"lemma","summary":"Let X be a right-continuous F-submartingale on an approximable time index. Then, for any stoppi…","labels":["lem:optionalSamplingSubmartingale"],"detail_key":"p11"},{"id":"n16740","layer":"informal","project":"p11","title":"Fixing A \\in F_\\sigma, we need to show that P[X_\\tau I_A] \\le P[X_\\sigma \\wedge \\tau I_A]…","kind":"proof","summary":"Fixing A \\in F_\\sigma, we need to show that P[X_\\tau I_A] \\le P[X_\\sigma \\wedge \\tau I_A]. Let…","labels":[],"detail_key":"p11"},{"id":"n16741","layer":"informal","project":"p11","title":"def:limitProcess","kind":"definition","summary":"\\mathlibok Let X : T \\to \\Omega \\to E be a stochastic process, let F be a filtration on \\Omega…","labels":["def:limitProcess"],"detail_key":"p11"},{"id":"n16742","layer":"informal","project":"p11","title":"thm:tendsto_limitProcess_of_cadlag","kind":"theorem","summary":"Let X be an uniformly integrable cadlag martingale with respect to the filtration F. Then there…","labels":["thm:tendsto_limitProcess_of_cadlag"],"detail_key":"p11"},{"id":"n16743","layer":"informal","project":"p11","title":"lem:limitProcess_ae_eq","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be a process such that there exists g : \\Omega \\to E that is a.e. st…","labels":["lem:limitProcess_ae_eq"],"detail_key":"p11"},{"id":"n16744","layer":"informal","project":"p11","title":"Because there exist a function that is a.e. strongly measurable with respect to \\bigsqcup…","kind":"proof","summary":"Because there exist a function that is a.e. strongly measurable with respect to \\bigsqcup t, F_…","labels":[],"detail_key":"p11"},{"id":"n16745","layer":"informal","project":"p11","title":"lem:limitProcess_congr","kind":"lemma","summary":"Let X and Y be two indistinguishable processes. Then X_\\infty and Y_\\infty are a.e. equal.","labels":["lem:limitProcess_congr"],"detail_key":"p11"},{"id":"n16746","layer":"informal","project":"p11","title":"If there exists g : \\Omega \\to E that is a.e. strongly measurable with respect to \\bigsqc…","kind":"proof","summary":"If there exists g : \\Omega \\to E that is a.e. strongly measurable with respect to \\bigsqcup t,…","labels":[],"detail_key":"p11"},{"id":"n16747","layer":"informal","project":"p11","title":"lem:limitProcess_const","kind":"lemma","summary":"Let c \\in E and X be the constant process equal to c. Then X_\\infty is a.e. equal to c.","labels":["lem:limitProcess_const"],"detail_key":"p11"},{"id":"n16748","layer":"informal","project":"p11","title":"This follows from Lemma~\\reflem:limitProcess_ae_eq as a constant function converges to a…","kind":"proof","summary":"This follows from Lemma~\\reflem:limitProcess_ae_eq as a constant function converges to a consta…","labels":[],"detail_key":"p11"},{"id":"n16749","layer":"informal","project":"p11","title":"lem:limitProcess_smul","kind":"lemma","summary":"If X is a stochastic process and c \\in R, then (c \\cdot X)_\\infty is a.e. equal to c \\cdot X_\\i…","labels":["lem:limitProcess_smul"],"detail_key":"p11"},{"id":"n16750","layer":"informal","project":"p11","title":"If c = 0","kind":"proof","summary":"If c = 0","labels":[],"detail_key":"p11"},{"id":"n16751","layer":"informal","project":"p11","title":"lem:limitProcess_neg","kind":"lemma","summary":"If X is a stochastic process, then (-X)_\\infty is a.e. equal to -X_\\infty.","labels":["lem:limitProcess_neg"],"detail_key":"p11"},{"id":"n16752","layer":"informal","project":"p11","title":"If there","kind":"proof","summary":"If there","labels":[],"detail_key":"p11"},{"id":"n16753","layer":"informal","project":"p11","title":"thm:condExp_limitProcess","kind":"theorem","summary":"If X is a càdlàg and uniformly integrable martingale, then for any t, almost surely, P[X_\\infty…","labels":["thm:condExp_limitProcess"],"detail_key":"p11"},{"id":"n16754","layer":"informal","project":"p11","title":"thm:condExp_limitProcess_stopped","kind":"theorem","summary":"If X is a càdlàg and uniformly integrable martingale, then for any stopping time \\tau, almost s…","labels":["thm:condExp_limitProcess_stopped"],"detail_key":"p11"},{"id":"n16755","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16756","layer":"informal","project":"p11","title":"lem:limitProcess_stoppedProcess","kind":"lemma","summary":"Let M be a càdlàg uniformly integrable martingale and \\tau be a stopping time. Then almost sure…","labels":["lem:limitProcess_stoppedProcess"],"detail_key":"p11"},{"id":"n16757","layer":"informal","project":"p11","title":"We apply Lemma~\\reflem:limitProcess_ae_eq. Because M is right-continuous and adapted, it…","kind":"proof","summary":"We apply Lemma~\\reflem:limitProcess_ae_eq. Because M is right-continuous and adapted, it is pro…","labels":[],"detail_key":"p11"},{"id":"n16758","layer":"informal","project":"p11","title":"Doob's maximal inequality for N","kind":"lemma","summary":"[Doob's maximal inequality for N] \\mathlibok Let X : N \\rightarrow \\Omega \\rightarrow R be a no…","labels":["lem:maximal_ineq"],"detail_key":"p11"},{"id":"n16759","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16760","layer":"informal","project":"p11","title":"Doob's maximal Inequality for countable","kind":"lemma","summary":"[Doob's maximal Inequality for countable] Let X : I \\rightarrow \\Omega \\rightarrow R be a non-n…","labels":["lem:doob_countable"],"detail_key":"p11"},{"id":"n16761","layer":"informal","project":"p11","title":"For any finite subset J \\subset I with M \\in J, we have by Lemma~\\reflem:maximal_ineq P\\l…","kind":"proof","summary":"For any finite subset J \\subset I with M \\in J, we have by Lemma~\\reflem:maximal_ineq P\\left( \\…","labels":[],"detail_key":"p11"},{"id":"n16762","layer":"informal","project":"p11","title":"Doob Lp Inequality for countable","kind":"lemma","summary":"[Doob Lp Inequality for countable] Let X : I \\rightarrow \\Omega \\rightarrow R be a non-negative…","labels":["lem:doob_Lp_countable"],"detail_key":"p11"},{"id":"n16763","layer":"informal","project":"p11","title":"E\\left[ \\sup_i \\le MX_i^p \\right] = p \\int_0^\\infty P\\left( \\sup_i \\le MX_i \\geq \\lambda…","kind":"proof","summary":"E\\left[ \\sup_i \\le MX_i^p \\right] = p \\int_0^\\infty P\\left( \\sup_i \\le MX_i \\geq \\lambda \\right…","labels":[],"detail_key":"p11"},{"id":"n16764","layer":"informal","project":"p11","title":"Doob Inequality","kind":"theorem","summary":"[Doob Inequality] Let X: R_+ \\to \\Omega \\to R be a right-continuous non-negative sub-martingale…","labels":["thm:doob_ineq"],"detail_key":"p11"},{"id":"n16765","layer":"informal","project":"p11","title":"Since X is right-continuous and [0,T] is a compact interval, we have that \\sup_t\\in[0,T]X…","kind":"proof","summary":"Since X is right-continuous and [0,T] is a compact interval, we have that \\sup_t\\in[0,T]X_t = \\…","labels":[],"detail_key":"p11"},{"id":"n16766","layer":"informal","project":"p11","title":"Doob Inequality for normed spaces","kind":"corollary","summary":"[Doob Inequality for normed spaces] Let X:R_+ \\to \\Omega \\to E be a right-continuous martingale…","labels":["cor:doob_ineq_norm"],"detail_key":"p11"},{"id":"n16767","layer":"informal","project":"p11","title":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale.…","kind":"proof","summary":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale. Then a…","labels":[],"detail_key":"p11"},{"id":"n16768","layer":"informal","project":"p11","title":"Doob's Lp inequality in R","kind":"theorem","summary":"[Doob's Lp inequality in R] Let X:R \\rightarrow \\Omega \\rightarrow R be a right-continuous non-…","labels":["thm:doob_lp"],"detail_key":"p11"},{"id":"n16769","layer":"informal","project":"p11","title":"Since X is right-continuous and [0,T] is a compact interval, we have that \\sup_t\\in[0,T]X…","kind":"proof","summary":"Since X is right-continuous and [0,T] is a compact interval, we have that \\sup_t\\in[0,T]X_t = \\…","labels":[],"detail_key":"p11"},{"id":"n16770","layer":"informal","project":"p11","title":"Doob's Lp inequality for normed spaces","kind":"corollary","summary":"[Doob's Lp inequality for normed spaces] Let X : R \\rightarrow \\Omega\\rightarrow E be a right-c…","labels":["cor:doob_lp_norm"],"detail_key":"p11"},{"id":"n16771","layer":"informal","project":"p11","title":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale.…","kind":"proof","summary":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale. Then a…","labels":[],"detail_key":"p11"},{"id":"n16772","layer":"informal","project":"p11","title":"cor:doob_lp_norm_top","kind":"corollary","summary":"Let X : R \\rightarrow \\Omega\\rightarrow E be a right-continuous martingale with values in a nor…","labels":["cor:doob_lp_norm_top"],"detail_key":"p11"},{"id":"n16773","layer":"informal","project":"p11","title":"Take limits on both sides in Corollary~\\refcor:doob_lp_norm.","kind":"proof","summary":"Take limits on both sides in Corollary~\\refcor:doob_lp_norm.","labels":[],"detail_key":"p11"},{"id":"n16774","layer":"informal","project":"p11","title":"Stopped Submartingale","kind":"lemma","summary":"[Stopped Submartingale] Let X:R_+ \\to \\Omega \\to R be a cadlag submartingale and \\tau a stoppin…","labels":["lem:Submartingale.stoppedProcess_of_cadlag"],"detail_key":"p11"},{"id":"n16775","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16776","layer":"informal","project":"p11","title":"Doob Inequality for stopping times","kind":"lemma","summary":"[Doob Inequality for stopping times] Let X:R\\times\\Omega\\rightarrow R be a right-continuous non…","labels":["lem:doob_ineq_stop"],"detail_key":"p11"},{"id":"n16777","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16778","layer":"informal","project":"p11","title":"Doob Inequality for stopping times in normed spaces","kind":"corollary","summary":"[Doob Inequality for stopping times in normed spaces] Let X:R\\times\\Omega\\rightarrow E be a rig…","labels":["cor:doob_ineq_stop"],"detail_key":"p11"},{"id":"n16779","layer":"informal","project":"p11","title":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale.…","kind":"proof","summary":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale. Then a…","labels":[],"detail_key":"p11"},{"id":"n16780","layer":"informal","project":"p11","title":"Doob's Lp Inequality for stopping times","kind":"lemma","summary":"[Doob's Lp Inequality for stopping times] Let X:R\\times\\Omega\\rightarrow R be a right-continuou…","labels":["lem:doob_ineq_stop_exp_val"],"detail_key":"p11"},{"id":"n16781","layer":"informal","project":"p11","title":"8.1.3 Pascucci.","kind":"proof","summary":"8.1.3 Pascucci.","labels":[],"detail_key":"p11"},{"id":"n16782","layer":"informal","project":"p11","title":"Doob's Lp Inequality for stopping times in normed spaces","kind":"corollary","summary":"[Doob's Lp Inequality for stopping times in normed spaces] Let X:R\\times\\Omega\\rightarrow E be…","labels":["cor:doob_ineq_stop_exp_val"],"detail_key":"p11"},{"id":"n16783","layer":"informal","project":"p11","title":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale.…","kind":"proof","summary":"By Corollary~\\refcor:Martingale.submartingale_norm, \\lVert X \\rVert is a sub-martingale. Then a…","labels":[],"detail_key":"p11"},{"id":"n16784","layer":"informal","project":"p11","title":"def:image2_prod","kind":"definition","summary":"\\mathlibok The product of two pavings S and T is the set of sets of the form s \\times t where s…","labels":["def:image2_prod"],"detail_key":"p11"},{"id":"n16785","layer":"informal","project":"p11","title":"def:supClosure","kind":"definition","summary":"\\mathlibok For a paving S, we denote by S^\\cup f the set of finite unions of sets in S.","labels":["def:supClosure"],"detail_key":"p11"},{"id":"n16786","layer":"informal","project":"p11","title":"def:infClosure","kind":"definition","summary":"\\mathlibok For a paving S, we denote by S^\\cap f the set of finite intersections of sets in S.","labels":["def:infClosure"],"detail_key":"p11"},{"id":"n16787","layer":"informal","project":"p11","title":"def:countableSupClosure","kind":"definition","summary":"We denote by S_\\sigma the set of countable unions of sets in S.","labels":["def:countableSupClosure"],"detail_key":"p11"},{"id":"n16788","layer":"informal","project":"p11","title":"def:countableInfClosure","kind":"definition","summary":"We denote by S_\\delta the set of countable intersections of sets in S.","labels":["def:countableInfClosure"],"detail_key":"p11"},{"id":"n16789","layer":"informal","project":"p11","title":"lem:InfClosed.mem_countableInfClosure_iff","kind":"lemma","summary":"If a paving S is closed under pairwise intersection, then a set s is in S_\\delta if and only if…","labels":["lem:InfClosed.mem_countableInfClosure_iff"],"detail_key":"p11"},{"id":"n16790","layer":"informal","project":"p11","title":"The ``monotone decreasing'' condition is the only non-trivial part of the statement: with…","kind":"proof","summary":"The ``monotone decreasing'' condition is the only non-trivial part of the statement: without it…","labels":[],"detail_key":"p11"},{"id":"n16791","layer":"informal","project":"p11","title":"lem:SupClosed.mem_countableSupClosure_iff","kind":"lemma","summary":"If a paving S is closed under pairwise union, then a set s is in S_\\sigma if and only if there…","labels":["lem:SupClosed.mem_countableSupClosure_iff"],"detail_key":"p11"},{"id":"n16792","layer":"informal","project":"p11","title":"Similar to the proof of Lemma~\\reflem:InfClosed.mem_countableInfClosure_iff.","kind":"proof","summary":"Similar to the proof of Lemma~\\reflem:InfClosed.mem_countableInfClosure_iff.","labels":[],"detail_key":"p11"},{"id":"n16793","layer":"informal","project":"p11","title":"def:IsCompactSystem","kind":"definition","summary":"\\mathlibok A set of sets K is a compact system if for every countable family (C_n)_n \\in N of s…","labels":["def:IsCompactSystem"],"detail_key":"p11"},{"id":"n16794","layer":"informal","project":"p11","title":"lem:isCompactSystem_isCompact_isClosed","kind":"lemma","summary":"The family of compact closed sets of a topological space is a compact system.","labels":["lem:isCompactSystem_isCompact_isClosed"],"detail_key":"p11"},{"id":"n16795","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16796","layer":"informal","project":"p11","title":"lem:isCompactSystem_isCompact","kind":"lemma","summary":"The family of compact sets of a T2 topological space is a compact system.","labels":["lem:isCompactSystem_isCompact"],"detail_key":"p11"},{"id":"n16797","layer":"informal","project":"p11","title":"The T2 assumption ensures that compact sets are closed, so the family of compact sets is…","kind":"proof","summary":"The T2 assumption ensures that compact sets are closed, so the family of compact sets is the fa…","labels":[],"detail_key":"p11"},{"id":"n16798","layer":"informal","project":"p11","title":"lem:IsCompactSystem.mono","kind":"lemma","summary":"If S \\subseteq T and T is a compact system, then S is a compact system.","labels":["lem:IsCompactSystem.mono"],"detail_key":"p11"},{"id":"n16799","layer":"informal","project":"p11","title":"Any sequence of sets in S is a sequence of sets in T, so if the intersection of the seque…","kind":"proof","summary":"Any sequence of sets in S is a sequence of sets in T, so if the intersection of the sequence is…","labels":[],"detail_key":"p11"},{"id":"n16800","layer":"informal","project":"p11","title":"lem:isCompactSystem_Icc","kind":"lemma","summary":"The family of closed compact intervals of R is a compact system.","labels":["lem:isCompactSystem_Icc"],"detail_key":"p11"},{"id":"n16801","layer":"informal","project":"p11","title":"The family of closed compact intervals of R is a family of compact sets, so it is a compa…","kind":"proof","summary":"The family of closed compact intervals of R is a family of compact sets, so it is a compact sys…","labels":[],"detail_key":"p11"},{"id":"n16802","layer":"informal","project":"p11","title":"lem:IsCompactSystem.pi","kind":"lemma","summary":"The product of a countable family of compact systems is a compact system.","labels":["lem:IsCompactSystem.pi"],"detail_key":"p11"},{"id":"n16803","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16804","layer":"informal","project":"p11","title":"lem:IsCompactSystem.sigma","kind":"lemma","summary":"The sum of a countable family of compact systems is a compact system.","labels":["lem:IsCompactSystem.sigma"],"detail_key":"p11"},{"id":"n16805","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16806","layer":"informal","project":"p11","title":"lem:IsCompactSystem.image2_prod","kind":"lemma","summary":"The product of two compact systems S \\times T is a compact system.","labels":["lem:IsCompactSystem.image2_prod"],"detail_key":"p11"},{"id":"n16807","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16808","layer":"informal","project":"p11","title":"lem:IsCompactSystem.supClosure","kind":"lemma","summary":"If S is a compact system, then S^\\cup f is a compact system.","labels":["lem:IsCompactSystem.supClosure"],"detail_key":"p11"},{"id":"n16809","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16810","layer":"informal","project":"p11","title":"lem:IsCompactSystem.countableInfClosure","kind":"lemma","summary":"If S is a compact system, then S_\\delta is a compact system.","labels":["lem:IsCompactSystem.countableInfClosure"],"detail_key":"p11"},{"id":"n16811","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16812","layer":"informal","project":"p11","title":"lem:IsCompactSystem.infClosure","kind":"lemma","summary":"If S is a compact system, then S^\\cap f is a compact system.","labels":["lem:IsCompactSystem.infClosure"],"detail_key":"p11"},{"id":"n16813","layer":"informal","project":"p11","title":"S_\\delta is a compact system by Lemma~\\reflem:IsCompactSystem.countableInfClosure, and S^…","kind":"proof","summary":"S_\\delta is a compact system by Lemma~\\reflem:IsCompactSystem.countableInfClosure, and S^\\cap f…","labels":[],"detail_key":"p11"},{"id":"n16814","layer":"informal","project":"p11","title":"thm:fst_iInter_of_supClosure_image2_prod_of_antitone","kind":"theorem","summary":"Let S be a paving of X and T be a compact system of K. Let (B_n)_n \\in N be a monotone decreasi…","labels":["thm:fst_iInter_of_supClosure_image2_prod_of_antitone"],"detail_key":"p11"},{"id":"n16815","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16816","layer":"informal","project":"p11","title":"Analytic set","kind":"definition","summary":"[Analytic set] Let F be a paving of a type X. A set s of X is said to be F-analytic if there ex…","labels":["def:IsPavingAnalytic"],"detail_key":"p11"},{"id":"n16817","layer":"informal","project":"p11","title":"lem:isPavingAnalytic_of_prop","kind":"lemma","summary":"Every set in F is F-analytic: F \\subseteq A(F).","labels":["lem:isPavingAnalytic_of_prop"],"detail_key":"p11"},{"id":"n16818","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16819","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.mono","kind":"lemma","summary":"If F \\subseteq G and s is F-analytic, then s is G-analytic: A(F) \\subseteq A(G).","labels":["lem:IsPavingAnalytic.mono"],"detail_key":"p11"},{"id":"n16820","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16821","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.exists_mem_countableSupClosure_superset","kind":"lemma","summary":"If s is F-analytic, then there exists a set t in F_\\sigma such that s \\subseteq t.","labels":["lem:IsPavingAnalytic.exists_mem_countableSupClosure_superset"],"detail_key":"p11"},{"id":"n16822","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16823","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.iUnion","kind":"lemma","summary":"A countable union of F-analytic sets is F-analytic.","labels":["lem:IsPavingAnalytic.iUnion"],"detail_key":"p11"},{"id":"n16824","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16825","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.iInter","kind":"lemma","summary":"A countable intersection of F-analytic sets is F-analytic.","labels":["lem:IsPavingAnalytic.iInter"],"detail_key":"p11"},{"id":"n16826","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16827","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.fst","kind":"lemma","summary":"Let F be a set of sets of X and K be a compact system of K, with \\emptyset \\in K. If a set s of…","labels":["lem:IsPavingAnalytic.fst"],"detail_key":"p11"},{"id":"n16828","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16829","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.prod_left","kind":"lemma","summary":"If t \\in T and s is F-analytic, then t \\times s is (T \\times F)-analytic: T \\times A(F) \\subset…","labels":["lem:IsPavingAnalytic.prod_left"],"detail_key":"p11"},{"id":"n16830","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16831","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.prod","kind":"lemma","summary":"If s is F-analytic and t is G-analytic, then s \\times t is (F \\times G)-analytic: A(F) \\times A…","labels":["lem:IsPavingAnalytic.prod"],"detail_key":"p11"},{"id":"n16832","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16833","layer":"informal","project":"p11","title":"lem:isPavingAnalytic_isPavingAnalytic","kind":"lemma","summary":"A(A(F)) = A(F).","labels":["lem:isPavingAnalytic_isPavingAnalytic"],"detail_key":"p11"},{"id":"n16834","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16835","layer":"informal","project":"p11","title":"lem:isPavingAnalytic_of_measurableSet_generateFrom","kind":"lemma","summary":"Let F be a paving with \\emptyset \\in F and let \\sigma(F) be the \\sigma-algebra generated by F.…","labels":["lem:isPavingAnalytic_of_measurableSet_generateFrom"],"detail_key":"p11"},{"id":"n16836","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16837","layer":"informal","project":"p11","title":"lem:MeasurableSet.isPavingAnalytic_Icc_real","kind":"lemma","summary":"A measurable set of R is I_R-analytic. That is, M_R \\subseteq A(I_R).","labels":["lem:MeasurableSet.isPavingAnalytic_Icc_real"],"detail_key":"p11"},{"id":"n16838","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16839","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic_measurableSet_iff_isPavingAnalytic_Icc","kind":"lemma","summary":"A set of R is M_R-analytic if and only if it is I_R-analytic.","labels":["lem:IsPavingAnalytic_measurableSet_iff_isPavingAnalytic_Icc"],"detail_key":"p11"},{"id":"n16840","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16841","layer":"informal","project":"p11","title":"lem:MeasurableSet.isPavingAnalytic_image2_prod","kind":"lemma","summary":"Let X be a measurable space and s be a measurable set of X \\times R. Then s is (M_X \\times I_R)…","labels":["lem:MeasurableSet.isPavingAnalytic_image2_prod"],"detail_key":"p11"},{"id":"n16842","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16843","layer":"informal","project":"p11","title":"lem:isPavingAnalytic_fst_of_image2_prod_measurableSet_Icc","kind":"lemma","summary":"Let X be a measurable space and s be a set of X \\times R which is (M_X \\times I_R)-analytic. Th…","labels":["lem:isPavingAnalytic_fst_of_image2_prod_measurableSet_Icc"],"detail_key":"p11"},{"id":"n16844","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16845","layer":"informal","project":"p11","title":"lem:MeasurableSet.isPavingAnalytic_fst","kind":"lemma","summary":"Let X be a measurable space and s be a measurable set of X \\times R. Then the projection of s o…","labels":["lem:MeasurableSet.isPavingAnalytic_fst"],"detail_key":"p11"},{"id":"n16846","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16847","layer":"informal","project":"p11","title":"def:IsMeasurableAnalytic","kind":"definition","summary":"We say that a set s of a measurable space X is measurably analytic for a measurable space K if…","labels":["def:IsMeasurableAnalytic"],"detail_key":"p11"},{"id":"n16848","layer":"informal","project":"p11","title":"lem:IsMeasurableAnalyticFor.isMeasurableAnalytic","kind":"lemma","summary":"Let K be a standard Borel space. If a set s of X is measurably analytic for K, then it is measu…","labels":["lem:IsMeasurableAnalyticFor.isMeasurableAnalytic"],"detail_key":"p11"},{"id":"n16849","layer":"informal","project":"p11","title":"There is a measurable embedding of K into R.","kind":"proof","summary":"There is a measurable embedding of K into R.","labels":[],"detail_key":"p11"},{"id":"n16850","layer":"informal","project":"p11","title":"lem:IsMeasurableAnalytic.isPavingAnalytic","kind":"lemma","summary":"If a set s of X is measurably analytic, then it is M_X-analytic.","labels":["lem:IsMeasurableAnalytic.isPavingAnalytic"],"detail_key":"p11"},{"id":"n16851","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16852","layer":"informal","project":"p11","title":"def:Capacity","kind":"definition","summary":"Let F be a set of sets of a type X. An F-capacity is a function I from the sets of X to R_+,\\in…","labels":["def:Capacity"],"detail_key":"p11"},{"id":"n16853","layer":"informal","project":"p11","title":"lem:Capacity.comp_fst_aux","kind":"lemma","summary":"Let F be a set of sets of X and let I be an F-capacity. Suppose that \\emptyset \\in F and F is c…","labels":["lem:Capacity.comp_fst_aux"],"detail_key":"p11"},{"id":"n16854","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16855","layer":"informal","project":"p11","title":"def:Capacity.comp_fst","kind":"definition","summary":"Let F be a set of sets of X and let I be an F-capacity. Suppose that \\emptyset \\in F and F is c…","labels":["def:Capacity.comp_fst"],"detail_key":"p11"},{"id":"n16856","layer":"informal","project":"p11","title":"def:IsCapacitable","kind":"definition","summary":"Let F be a set of sets of X and let I be an F-capacity. A set s of X is said to be I-capacitabl…","labels":["def:IsCapacitable"],"detail_key":"p11"},{"id":"n16857","layer":"informal","project":"p11","title":"lem:isCapacitable_of_prop","kind":"lemma","summary":"For I an F-capacity, every set in F is I-capacitable.","labels":["lem:isCapacitable_of_prop"],"detail_key":"p11"},{"id":"n16858","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16859","layer":"informal","project":"p11","title":"lem:isCapacitable_mem_countableInfClosure_countableSupClosure","kind":"lemma","summary":"Suppose that F contains the empty set and is closed under pairwise intersections and unions, an…","labels":["lem:isCapacitable_mem_countableInfClosure_countableSupClosure"],"detail_key":"p11"},{"id":"n16860","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16861","layer":"informal","project":"p11","title":"lem:mem_countableInfClosure_fst","kind":"lemma","summary":"Suppose that F contains the empty set and is closed under pairwise intersections and unions, an…","labels":["lem:mem_countableInfClosure_fst"],"detail_key":"p11"},{"id":"n16862","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16863","layer":"informal","project":"p11","title":"lem:IsCapacitable.fst","kind":"lemma","summary":"Let F be a set of sets of X and let I be an F-capacity. Suppose that F contains the empty set a…","labels":["lem:IsCapacitable.fst"],"detail_key":"p11"},{"id":"n16864","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16865","layer":"informal","project":"p11","title":"Choquet's capacitability theorem","kind":"theorem","summary":"[Choquet's capacitability theorem] Let F be a set of sets of X and let I be an F-capacity. Supp…","labels":["thm:IsPavingAnalytic.isCapacitable"],"detail_key":"p11"},{"id":"n16866","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16867","layer":"informal","project":"p11","title":"def:Measure.capacity","kind":"definition","summary":"Every finite measure defines a capacity by I(s) = \\mu(s) (where \\mu(s) is the value of the oute…","labels":["def:Measure.capacity"],"detail_key":"p11"},{"id":"n16868","layer":"informal","project":"p11","title":"lem:isCapacitable_measure_iff","kind":"lemma","summary":"A set is capacitable with respect to the capacity defined by a finite measure if and only if it…","labels":["lem:isCapacitable_measure_iff"],"detail_key":"p11"},{"id":"n16869","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16870","layer":"informal","project":"p11","title":"lem:IsPavingAnalytic.nullMeasurableSet","kind":"lemma","summary":"An M_X-analytic set of a measurable space X is universally measurable: it is null-measurable fo…","labels":["lem:IsPavingAnalytic.nullMeasurableSet"],"detail_key":"p11"},{"id":"n16871","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16872","layer":"informal","project":"p11","title":"lem:IsMeasurableAnalytic.nullMeasurableSet","kind":"lemma","summary":"A measurably analytic set of a measurable space X is universally measurable: it is null-measura…","labels":["lem:IsMeasurableAnalytic.nullMeasurableSet"],"detail_key":"p11"},{"id":"n16873","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16874","layer":"informal","project":"p11","title":"Measurable projection","kind":"theorem","summary":"[Measurable projection] Let X and Y be measurable spaces, with Y standard Borel. Then the proje…","labels":["thm:MeasurableSet.nullMeasurableSet_fst"],"detail_key":"p11"},{"id":"n16875","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16876","layer":"informal","project":"p11","title":"Monotone class","kind":"definition","summary":"[Monotone class] Let M be a collection of subsets of a set X. We say that M is a monotone class…","labels":["def:monotone_class"],"detail_key":"p11"},{"id":"n16877","layer":"informal","project":"p11","title":"Monotone class theorem","kind":"theorem","summary":"[Monotone class theorem] Let \\(G\\) be an algebra of subsets of a set \\(X\\). Then the monotone c…","labels":["thm:monotone_class"],"detail_key":"p11"},{"id":"n16878","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16879","layer":"informal","project":"p11","title":"Progressively measurable set","kind":"definition","summary":"[Progressively measurable set] A subset of [0, \\infty) \\times \\Omega is progressively measurabl…","labels":["def:progr_meas_set"],"detail_key":"p11"},{"id":"n16880","layer":"informal","project":"p11","title":"Debut of a set","kind":"definition","summary":"[Debut of a set] Let E \\subseteq [0, \\infty) \\times \\Omega , define D_E = \\inf\\left\\lbrace t \\g…","labels":["def:debut_set"],"detail_key":"p11"},{"id":"n16881","layer":"informal","project":"p11","title":"thm:debut_of_progr_meas_is_stop_time","kind":"theorem","summary":"If E is a progressively measurable set and the filtration satisfies the usual conditions, then…","labels":["thm:debut_of_progr_meas_is_stop_time"],"detail_key":"p11"},{"id":"n16882","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16883","layer":"informal","project":"p11","title":"thm:hitting_is_stopping_time","kind":"theorem","summary":"If X : T \\to \\Omega \\to E is a progressively measurable process with respect to a filtration sa…","labels":["thm:hitting_is_stopping_time"],"detail_key":"p11"},{"id":"n16884","layer":"informal","project":"p11","title":"Since B is a Borel subset of S and X is progressively measurable, then 1_B(X_t) is also p…","kind":"proof","summary":"Since B is a Borel subset of S and X is progressively measurable, then 1_B(X_t) is also progres…","labels":[],"detail_key":"p11"},{"id":"n16885","layer":"informal","project":"p11","title":"def:leastGE","kind":"definition","summary":"\\mathlibok Let T be a conditionally complete linear order with a bottom element and let R be a…","labels":["def:leastGE"],"detail_key":"p11"},{"id":"n16886","layer":"informal","project":"p11","title":"lem:isStoppingTime_leastGE","kind":"lemma","summary":"Let T be a conditionally complete linear order with a bottom element, which is a Polish space f…","labels":["lem:isStoppingTime_leastGE"],"detail_key":"p11"},{"id":"n16887","layer":"informal","project":"p11","title":"This is a direct application of Theorem~\\refthm:hitting_is_stopping_time with the set B =…","kind":"proof","summary":"This is a direct application of Theorem~\\refthm:hitting_is_stopping_time with the set B = [a, +…","labels":[],"detail_key":"p11"},{"id":"n16888","layer":"informal","project":"p11","title":"cor:isStoppingTime_leastGE_of_rightContinuous","kind":"corollary","summary":"If X : ι \\to Ω \\to ℝ is a right-continuous and adapted process with respect to a filtration sat…","labels":["cor:isStoppingTime_leastGE_of_rightContinuous"],"detail_key":"p11"},{"id":"n16889","layer":"informal","project":"p11","title":"This follows from Lemma~\\reflem:isStoppingTime_leastGE since X is progressively measurabl…","kind":"proof","summary":"This follows from Lemma~\\reflem:isStoppingTime_leastGE since X is progressively measurable by L…","labels":[],"detail_key":"p11"},{"id":"n16890","layer":"informal","project":"p11","title":"def:leastGT","kind":"definition","summary":"Let T be a conditionally complete linear order with a bottom element and let R be a preorder. F…","labels":["def:leastGT"],"detail_key":"p11"},{"id":"n16891","layer":"informal","project":"p11","title":"lem:isStoppingTime_leastGT","kind":"lemma","summary":"Let T be a conditionally complete linear order with a bottom element, which is a Polish space f…","labels":["lem:isStoppingTime_leastGT"],"detail_key":"p11"},{"id":"n16892","layer":"informal","project":"p11","title":"This is a direct application of Theorem~\\refthm:hitting_is_stopping_time with the set B =…","kind":"proof","summary":"This is a direct application of Theorem~\\refthm:hitting_is_stopping_time with the set B = (a, +…","labels":[],"detail_key":"p11"},{"id":"n16893","layer":"informal","project":"p11","title":"lem:leastGT_lt_iff","kind":"lemma","summary":"For t \\in T, \\tau_X > a < t if and only if there exists s < t such that X_s > a.","labels":["lem:leastGT_lt_iff"],"detail_key":"p11"},{"id":"n16894","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16895","layer":"informal","project":"p11","title":"def:preLocalizingSequence","kind":"definition","summary":"A pre-localizing sequence is a sequence of stopping times (\\tau_n)_n \\in N such that \\tau_n \\to…","labels":["def:preLocalizingSequence"],"detail_key":"p11"},{"id":"n16896","layer":"informal","project":"p11","title":"Localizing sequence","kind":"definition","summary":"[Localizing sequence] A localizing sequence is a sequence of stopping times (\\tau_n)_n \\in N su…","labels":["def:localizingSequence"],"detail_key":"p11"},{"id":"n16897","layer":"informal","project":"p11","title":"lem:localizingSequence_const_top","kind":"lemma","summary":"The constant sequence \\tau_n = \\infty is a localizing sequence.","labels":["lem:localizingSequence_const_top"],"detail_key":"p11"},{"id":"n16898","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16899","layer":"informal","project":"p11","title":"lem:localizingSequence_min","kind":"lemma","summary":"Let (\\sigma_n), (\\tau_n) be localizing sequences. Then (\\sigma_n \\wedge \\tau_n) is a localizing…","labels":["lem:localizingSequence_min"],"detail_key":"p11"},{"id":"n16900","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16901","layer":"informal","project":"p11","title":"lem:isLocalizingSequence_of_isPreLocalizingSequence","kind":"lemma","summary":"If (\\tau_n)_n \\in N is a pre-localizing sequence, then the sequence defined by \\tau'_n = \\inf_m…","labels":["lem:isLocalizingSequence_of_isPreLocalizingSequence"],"detail_key":"p11"},{"id":"n16902","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16903","layer":"informal","project":"p11","title":"lem:isPreLocalizingSequence_of_isLocalizingSequence","kind":"lemma","summary":"Let (\\tau_n)_n \\in N be a localizing sequence and let (\\sigma_n,k)_k \\in N be a localizing sequ…","labels":["lem:isPreLocalizingSequence_of_isLocalizingSequence"],"detail_key":"p11"},{"id":"n16904","layer":"informal","project":"p11","title":"For each n, since \\sigma_n,k \\to \\infty a.s. as k \\to \\infty, we may choose k_n \\in N suc…","kind":"proof","summary":"For each n, since \\sigma_n,k \\to \\infty a.s. as k \\to \\infty, we may choose k_n \\in N such that…","labels":[],"detail_key":"p11"},{"id":"n16905","layer":"informal","project":"p11","title":"lem:isLocalizingSequence_ae","kind":"lemma","summary":"Let P be a predicate on paths and suppose X is a stochastic process satisfying P a.s. Then, def…","labels":["lem:isLocalizingSequence_ae"],"detail_key":"p11"},{"id":"n16906","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16907","layer":"informal","project":"p11","title":"Local property","kind":"definition","summary":"[Local property] \\mathlibok Let P be a class of stochastic processes (or equivalently a predica…","labels":["def:locally"],"detail_key":"p11"},{"id":"n16908","layer":"informal","project":"p11","title":"lem:implies_locally","kind":"lemma","summary":"\\mathlibok For any class of processes P, we have P \\subseteq P_loc.","labels":["lem:implies_locally"],"detail_key":"p11"},{"id":"n16909","layer":"informal","project":"p11","title":"Take \\tau_n = \\infty for all n.","kind":"proof","summary":"Take \\tau_n = \\infty for all n.","labels":[],"detail_key":"p11"},{"id":"n16910","layer":"informal","project":"p11","title":"lem:locally_mono","kind":"lemma","summary":"\\mathlibok If P \\subseteq Q then P_loc \\subseteq Q_loc.","labels":["lem:locally_mono"],"detail_key":"p11"},{"id":"n16911","layer":"informal","project":"p11","title":"Let X \\in P_loc. Then there exists a localizing sequence (\\tau_n)_n \\in N such that for a…","kind":"proof","summary":"Let X \\in P_loc. Then there exists a localizing sequence (\\tau_n)_n \\in N such that for all n \\…","labels":[],"detail_key":"p11"},{"id":"n16912","layer":"informal","project":"p11","title":"def:stable","kind":"definition","summary":"\\mathlibok A class of stochastic processes P is stable if whenever X is in P, then for any stop…","labels":["def:stable"],"detail_key":"p11"},{"id":"n16913","layer":"informal","project":"p11","title":"lem:isStable_locally","kind":"lemma","summary":"\\mathlibok If P is a stable class of processes, then P_loc is also stable.","labels":["lem:isStable_locally"],"detail_key":"p11"},{"id":"n16914","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16915","layer":"informal","project":"p11","title":"lem:locally_inter","kind":"lemma","summary":"\\mathlibok If P, Q are stable classes of processes then (P\\cap Q)_loc = P_loc\\cap Q_loc.","labels":["lem:locally_inter"],"detail_key":"p11"},{"id":"n16916","layer":"informal","project":"p11","title":"The forward direction is trivial so we only provide proof for the reverse. Suppose that X…","kind":"proof","summary":"The forward direction is trivial so we only provide proof for the reverse. Suppose that X \\in P…","labels":[],"detail_key":"p11"},{"id":"n16917","layer":"informal","project":"p11","title":"lem:locally_of_isPreLocalizingSequence","kind":"lemma","summary":"\\mathlibok Let P be a stable class of processes and let (\\tau_n)_n \\in N be a pre-localizing se…","labels":["lem:locally_of_isPreLocalizingSequence"],"detail_key":"p11"},{"id":"n16918","layer":"informal","project":"p11","title":"Using the localizing sequence defined by Lemma~\\reflem:isLocalizingSequence_of_isPreLocal…","kind":"proof","summary":"Using the localizing sequence defined by Lemma~\\reflem:isLocalizingSequence_of_isPreLocalizingS…","labels":[],"detail_key":"p11"},{"id":"n16919","layer":"informal","project":"p11","title":"lem:locally_locally","kind":"lemma","summary":"\\mathlibok Suppose that the filtration is right-continuous. For any stable class of processes P…","labels":["lem:locally_locally"],"detail_key":"p11"},{"id":"n16920","layer":"informal","project":"p11","title":"(P_loc)_loc \\supseteq P_loc by Lemma~\\reflem:isStable_locally so we only prove the revers…","kind":"proof","summary":"(P_loc)_loc \\supseteq P_loc by Lemma~\\reflem:isStable_locally so we only prove the reverse incl…","labels":[],"detail_key":"p11"},{"id":"n16921","layer":"informal","project":"p11","title":"Local implication from global implication","kind":"lemma","summary":"[Local implication from global implication] \\mathlibok Suppose that the filtration is right-con…","labels":["lem:local_induction"],"detail_key":"p11"},{"id":"n16922","layer":"informal","project":"p11","title":"Since X \\in P_loc, then X \\in (Q_loc)_loc by assumption and Lemma~\\reflem:locally_mono. B…","kind":"proof","summary":"Since X \\in P_loc, then X \\in (Q_loc)_loc by assumption and Lemma~\\reflem:locally_mono. By Lemm…","labels":[],"detail_key":"p11"},{"id":"n16923","layer":"informal","project":"p11","title":"lem:locally_of_ae","kind":"lemma","summary":"If P be a predicate on paths such that the constant path 0 satisfies P and X is a stochastic pr…","labels":["lem:locally_of_ae"],"detail_key":"p11"},{"id":"n16924","layer":"informal","project":"p11","title":"Follows directly by using the localizing sequence defined in Lemma~\\reflem:isLocalizingSe…","kind":"proof","summary":"Follows directly by using the localizing sequence defined in Lemma~\\reflem:isLocalizingSequence…","labels":[],"detail_key":"p11"},{"id":"n16925","layer":"informal","project":"p11","title":"lem:locally_rightContinuous","kind":"lemma","summary":"A stochastic process X is locally right continuous if and only if it is right continuous almost…","labels":["lem:locally_rightContinuous"],"detail_key":"p11"},{"id":"n16926","layer":"informal","project":"p11","title":"If X is a.s. right continuous, then it is locally right continuous by Lemma~\\reflem:local…","kind":"proof","summary":"If X is a.s. right continuous, then it is locally right continuous by Lemma~\\reflem:locally_of_…","labels":[],"detail_key":"p11"},{"id":"n16927","layer":"informal","project":"p11","title":"lem:locally_leftLimit","kind":"lemma","summary":"A stochastic process X has left limits locally if and only if it has left limits almost surely.","labels":["lem:locally_leftLimit"],"detail_key":"p11"},{"id":"n16928","layer":"informal","project":"p11","title":"Same proof as in Lemma~\\reflem:locally_rightContinuous.","kind":"proof","summary":"Same proof as in Lemma~\\reflem:locally_rightContinuous.","labels":[],"detail_key":"p11"},{"id":"n16929","layer":"informal","project":"p11","title":"lem:locally_isCadlag","kind":"lemma","summary":"A stochastic process X is locally cadlag if and only if it is cadlag almost surely.","labels":["lem:locally_isCadlag"],"detail_key":"p11"},{"id":"n16930","layer":"informal","project":"p11","title":"The forward direction follows from Lemmas~\\reflem:locally_rightContinuous and \\reflem:loc…","kind":"proof","summary":"The forward direction follows from Lemmas~\\reflem:locally_rightContinuous and \\reflem:locally_l…","labels":[],"detail_key":"p11"},{"id":"n16931","layer":"informal","project":"p11","title":"lem:isStable_rightContinuous","kind":"lemma","summary":"The class of right continuous processes is stable.","labels":["lem:isStable_rightContinuous"],"detail_key":"p11"},{"id":"n16932","layer":"informal","project":"p11","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p11"},{"id":"n16933","layer":"informal","project":"p11","title":"lem:isStable_left_limit","kind":"lemma","summary":"The class of processes with left limits is stable.","labels":["lem:isStable_left_limit"],"detail_key":"p11"},{"id":"n16934","layer":"informal","project":"p11","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p11"},{"id":"n16935","layer":"informal","project":"p11","title":"lem:isStable_isCadlag","kind":"lemma","summary":"The class of cadlag processes is stable.","labels":["lem:isStable_isCadlag"],"detail_key":"p11"},{"id":"n16936","layer":"informal","project":"p11","title":"Follows from Lemmas~\\reflem:isStable_rightContinuous and \\reflem:isStable_left_limit.","kind":"proof","summary":"Follows from Lemmas~\\reflem:isStable_rightContinuous and \\reflem:isStable_left_limit.","labels":[],"detail_key":"p11"},{"id":"n16937","layer":"informal","project":"p11","title":"lem:isStable_isStronglyProgressive","kind":"lemma","summary":"The class of progressively measurable processes is stable.","labels":["lem:isStable_isStronglyProgressive"],"detail_key":"p11"},{"id":"n16938","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16939","layer":"informal","project":"p11","title":"Local martingale","kind":"definition","summary":"[Local martingale] We say a stochastic process (M_t)_t \\in T is a local martingale if it is loc…","labels":["def:IsLocalMartingale"],"detail_key":"p11"},{"id":"n16940","layer":"informal","project":"p11","title":"def:IsLocalSubmartingale","kind":"definition","summary":"A stochastic process is a local submartingale if it is locally a cadlag submartingale in the se…","labels":["def:IsLocalSubmartingale"],"detail_key":"p11"},{"id":"n16941","layer":"informal","project":"p11","title":"lem:Martingale.IsLocalMartingale","kind":"lemma","summary":"Every cadlag martingale is a local martingale.","labels":["lem:Martingale.IsLocalMartingale"],"detail_key":"p11"},{"id":"n16942","layer":"informal","project":"p11","title":"This follows from Lemma \\reflem:implies_locally.","kind":"proof","summary":"This follows from Lemma \\reflem:implies_locally.","labels":[],"detail_key":"p11"},{"id":"n16943","layer":"informal","project":"p11","title":"lem:stable_IsMartingale","kind":"lemma","summary":"The class of cadlag martingales is stable. That is, if M is a cadlag martingale and \\tau is a s…","labels":["lem:stable_IsMartingale"],"detail_key":"p11"},{"id":"n16944","layer":"informal","project":"p11","title":"Clearly, the stopped process M^\\tauI_\\tau > 0 is cadlag and it remains to show that it is…","kind":"proof","summary":"Clearly, the stopped process M^\\tauI_\\tau > 0 is cadlag and it remains to show that it is a mar…","labels":[],"detail_key":"p11"},{"id":"n16945","layer":"informal","project":"p11","title":"lem:stable_IsSubmartingale","kind":"lemma","summary":"The class of cadlag submartingales is stable. That is, if M is a cadlag submartingale and \\tau…","labels":["lem:stable_IsSubmartingale"],"detail_key":"p11"},{"id":"n16946","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16947","layer":"informal","project":"p11","title":"thm:IsLocalMartingale.eq_zero_of_finiteVariation","kind":"theorem","summary":"Let M be a continuous local martingale with M_0 = 0. If M is also a finite variation process, t…","labels":["thm:IsLocalMartingale.eq_zero_of_finiteVariation"],"detail_key":"p11"},{"id":"n16948","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16949","layer":"informal","project":"p11","title":"Integrable supremum","kind":"definition","summary":"[Integrable supremum] We say that a stochastic process is integrable if the map (t,\\omega) \\map…","labels":["def:HasIntegrableSup"],"detail_key":"p11"},{"id":"n16950","layer":"informal","project":"p11","title":"Locally integrable supremum","kind":"definition","summary":"[Locally integrable supremum] A process has locally integrable supremum if it is locally a proc…","labels":["def:locallyIntegrableSup"],"detail_key":"p11"},{"id":"n16951","layer":"informal","project":"p11","title":"Doob-Meyer class, class D","kind":"definition","summary":"[Doob-Meyer class, class D] A stochastic process (X_t) is of class D (or in the Doob-Meyer clas…","labels":["def:classD"],"detail_key":"p11"},{"id":"n16952","layer":"informal","project":"p11","title":"Class DL","kind":"definition","summary":"[Class DL] A stochastic process (X_t) is of class DL if it is progressively measurable and for…","labels":["def:classDL"],"detail_key":"p11"},{"id":"n16953","layer":"informal","project":"p11","title":"lem:classDLOfClassD","kind":"lemma","summary":"A stochastic process of class D is of class DL.","labels":["lem:classDLOfClassD"],"detail_key":"p11"},{"id":"n16954","layer":"informal","project":"p11","title":"This follows from the definitions and Lemma~\\reflem:uniformIntegrableComp.","kind":"proof","summary":"This follows from the definitions and Lemma~\\reflem:uniformIntegrableComp.","labels":[],"detail_key":"p11"},{"id":"n16955","layer":"informal","project":"p11","title":"lem:ClassD.uniformIntegrable","kind":"lemma","summary":"A stochastic process of class D is uniformly integrable.","labels":["lem:ClassD.uniformIntegrable"],"detail_key":"p11"},{"id":"n16956","layer":"informal","project":"p11","title":"Use the Class D property with the constant stopping times \\tau_n = n for all n and apply…","kind":"proof","summary":"Use the Class D property with the constant stopping times \\tau_n = n for all n and apply Lemma~…","labels":[],"detail_key":"p11"},{"id":"n16957","layer":"informal","project":"p11","title":"lem:classDL_iff_norm","kind":"lemma","summary":"A progressively measurable process is of class DL iff its norm is of class DL.","labels":["lem:classDL_iff_norm"],"detail_key":"p11"},{"id":"n16958","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16959","layer":"informal","project":"p11","title":"lem:classD_of_uniformIntegrable_bounded_stoppingTime","kind":"lemma","summary":"If a stochastic process is progressively measurable and the set \\X_\\tau \\mid \\tau \\text is a st…","labels":["lem:classD_of_uniformIntegrable_bounded_stoppingTime"],"detail_key":"p11"},{"id":"n16960","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16961","layer":"informal","project":"p11","title":"lem:classD_iff_norm","kind":"lemma","summary":"A progressively measurable process is of class D iff its norm is of class D.","labels":["lem:classD_iff_norm"],"detail_key":"p11"},{"id":"n16962","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16963","layer":"informal","project":"p11","title":"lem:Submartingale.classDL","kind":"lemma","summary":"Every nonnegative right-continuous submartingale is of class DL.","labels":["lem:Submartingale.classDL"],"detail_key":"p11"},{"id":"n16964","layer":"informal","project":"p11","title":"Let t \\in T and \\tau \\le t be a stopping time. By Lemma~\\reflem:optionalSamplingSubmartin…","kind":"proof","summary":"Let t \\in T and \\tau \\le t be a stopping time. By Lemma~\\reflem:optionalSamplingSubmartingale a…","labels":[],"detail_key":"p11"},{"id":"n16965","layer":"informal","project":"p11","title":"lem:Submartingale.classD_iff_uniformIntegrable","kind":"lemma","summary":"A nonnegative right-continuous submartingale is of class D if and only if it is uniformly integ…","labels":["lem:Submartingale.classD_iff_uniformIntegrable"],"detail_key":"p11"},{"id":"n16966","layer":"informal","project":"p11","title":"Assume that X is uniformly integrable. Just like what we did in the proof of Lemma~\\refle…","kind":"proof","summary":"Assume that X is uniformly integrable. Just like what we did in the proof of Lemma~\\reflem:Subm…","labels":[],"detail_key":"p11"},{"id":"n16967","layer":"informal","project":"p11","title":"lem:Martingale.classDL","kind":"lemma","summary":"Every càdlàg martingale is of class DL.","labels":["lem:Martingale.classDL"],"detail_key":"p11"},{"id":"n16968","layer":"informal","project":"p11","title":"Let X be càdlàg martingale. By Lemma~\\reflem:Martingale.submartingale_convex_comp, (|X_t|…","kind":"proof","summary":"Let X be càdlàg martingale. By Lemma~\\reflem:Martingale.submartingale_convex_comp, (|X_t|)_t \\i…","labels":[],"detail_key":"p11"},{"id":"n16969","layer":"informal","project":"p11","title":"lem:Martingale.classD_iff_uniformIntegrable","kind":"lemma","summary":"A right-continuous martingale is of class D if and only if it is uniformly integrable.","labels":["lem:Martingale.classD_iff_uniformIntegrable"],"detail_key":"p11"},{"id":"n16970","layer":"informal","project":"p11","title":"Applying Lemma~\\reflem:classD_iff_norm and Lemma~\\reflem:uniformIntegrableIffNorm, it suf…","kind":"proof","summary":"Applying Lemma~\\reflem:classD_iff_norm and Lemma~\\reflem:uniformIntegrableIffNorm, it suffices…","labels":[],"detail_key":"p11"},{"id":"n16971","layer":"informal","project":"p11","title":"lem:isStable_stronglyMeasurable_uncurry","kind":"lemma","summary":"The class of processes for which the induced map (t,\\omega)\\mapsto X_t(\\omega) is strongly meas…","labels":["lem:isStable_stronglyMeasurable_uncurry"],"detail_key":"p11"},{"id":"n16972","layer":"informal","project":"p11","title":"The stopped process at \\tau is obtained by precomposition with (t, \\omega) \\mapsto (\\min…","kind":"proof","summary":"The stopped process at \\tau is obtained by precomposition with (t, \\omega) \\mapsto (\\min (\\tau(…","labels":[],"detail_key":"p11"},{"id":"n16973","layer":"informal","project":"p11","title":"lem:isStable_hasIntegrableSup","kind":"lemma","summary":"The class of process with integrable supremum is stable.","labels":["lem:isStable_hasIntegrableSup"],"detail_key":"p11"},{"id":"n16974","layer":"informal","project":"p11","title":"Let X be a process with integrable supremum and \\tau be a stopping time. Let t \\in T. The…","kind":"proof","summary":"Let X be a process with integrable supremum and \\tau be a stopping time. Let t \\in T. Then (X^\\…","labels":[],"detail_key":"p11"},{"id":"n16975","layer":"informal","project":"p11","title":"lem:isStable_hasLocallyIntegrableSup","kind":"lemma","summary":"The class of process with locally integrable supremum is stable.","labels":["lem:isStable_hasLocallyIntegrableSup"],"detail_key":"p11"},{"id":"n16976","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:isStable_hasIntegrableSup and Lemma~\\reflem:isStable_locally.","kind":"proof","summary":"Apply Lemma~\\reflem:isStable_hasIntegrableSup and Lemma~\\reflem:isStable_locally.","labels":[],"detail_key":"p11"},{"id":"n16977","layer":"informal","project":"p11","title":"lem:isStable_classD","kind":"lemma","summary":"The class D is stable.","labels":["lem:isStable_classD"],"detail_key":"p11"},{"id":"n16978","layer":"informal","project":"p11","title":"Let X be","kind":"proof","summary":"Let X be","labels":[],"detail_key":"p11"},{"id":"n16979","layer":"informal","project":"p11","title":"lem:isStable_classDL","kind":"lemma","summary":"The class DL is stable.","labels":["lem:isStable_classDL"],"detail_key":"p11"},{"id":"n16980","layer":"informal","project":"p11","title":"Let X be","kind":"proof","summary":"Let X be","labels":[],"detail_key":"p11"},{"id":"n16981","layer":"informal","project":"p11","title":"lem:Integrable.classDL","kind":"lemma","summary":"Let X be a progressively measurable stochastic process with integrable supremum (Definition~\\re…","labels":["lem:Integrable.classDL"],"detail_key":"p11"},{"id":"n16982","layer":"informal","project":"p11","title":"Let t \\in T. For every stopping time \\tau with \\tau \\le t, we have \\|X_\\tau\\| \\le X^*_t.…","kind":"proof","summary":"Let t \\in T. For every stopping time \\tau with \\tau \\le t, we have \\|X_\\tau\\| \\le X^*_t. Measur…","labels":[],"detail_key":"p11"},{"id":"n16983","layer":"informal","project":"p11","title":"lem:IsStronglyProgressive.hasStronglyMeasurableSup","kind":"lemma","summary":"If the filtration satisfies the usual conditions, a progressively measurable process has a supr…","labels":["lem:IsStronglyProgressive.hasStronglyMeasurableSup"],"detail_key":"p11"},{"id":"n16984","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n16985","layer":"informal","project":"p11","title":"lem:HasLocallyIntegrableSup.locally_classDL","kind":"lemma","summary":"Let X be a progressively measurable stochastic process with locally integrable supremum. Then X…","labels":["lem:HasLocallyIntegrableSup.locally_classDL"],"detail_key":"p11"},{"id":"n16986","layer":"informal","project":"p11","title":"Combine Lemma~\\reflem:Integrable.classDL and Lemma~\\reflem:locally_mono. Use Lemma~\\refle…","kind":"proof","summary":"Combine Lemma~\\reflem:Integrable.classDL and Lemma~\\reflem:locally_mono. Use Lemma~\\reflem:isSt…","labels":[],"detail_key":"p11"},{"id":"n16987","layer":"informal","project":"p11","title":"lem:ClassDL.locally_classD","kind":"lemma","summary":"If X is of class DL then it is locally of class D.","labels":["lem:ClassDL.locally_classD"],"detail_key":"p11"},{"id":"n16988","layer":"informal","project":"p11","title":"Take \\tau_n := n. Then \\X^\\tau_n_\\sigma \\mid \\sigma \\text is a finite stopping time\\ & =…","kind":"proof","summary":"Take \\tau_n := n. Then \\X^\\tau_n_\\sigma \\mid \\sigma \\text is a finite stopping time\\ & = \\X_\\si…","labels":[],"detail_key":"p11"},{"id":"n16989","layer":"informal","project":"p11","title":"lem:locally_classD_of_locally_classDL","kind":"lemma","summary":"If the filtration is right-continuous and X is locally of class DL then it is locally of class…","labels":["lem:locally_classD_of_locally_classDL"],"detail_key":"p11"},{"id":"n16990","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:local_induction using Lemma~\\reflem:ClassDL.locally_classD and Lemma~…","kind":"proof","summary":"Apply Lemma~\\reflem:local_induction using Lemma~\\reflem:ClassDL.locally_classD and Lemma~\\refle…","labels":[],"detail_key":"p11"},{"id":"n16991","layer":"informal","project":"p11","title":"lem:isBounded_image_of_isCadlag_of_isCompact","kind":"lemma","summary":"Assume T is a linear order endowed with a topology making it first countable and E is a pseudo-…","labels":["lem:isBounded_image_of_isCadlag_of_isCompact"],"detail_key":"p11"},{"id":"n16992","layer":"informal","project":"p11","title":"Let K \\sub","kind":"proof","summary":"Let K \\sub","labels":[],"detail_key":"p11"},{"id":"n16993","layer":"informal","project":"p11","title":"lem:isLocalizingSequence_leastGE","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact, and that the filtratio…","labels":["lem:isLocalizingSequence_leastGE"],"detail_key":"p11"},{"id":"n16994","layer":"informal","project":"p11","title":"By Corollar","kind":"proof","summary":"By Corollar","labels":[],"detail_key":"p11"},{"id":"n16995","layer":"informal","project":"p11","title":"lem:sup_stoppedProcess_le","kind":"lemma","summary":"For Y a stochastic process, let Y^*_t = \\sup_s \\le t \\Vert Y_s \\Vert. Let X be a stochastic pro…","labels":["lem:sup_stoppedProcess_le"],"detail_key":"p11"},{"id":"n16996","layer":"informal","project":"p11","title":"If \\tau > t, then for all s \\le t, \\|X_s\\| \\le n, and thus (X^\\tau)^*_t = \\sup_s \\le t \\|…","kind":"proof","summary":"If \\tau > t, then for all s \\le t, \\|X_s\\| \\le n, and thus (X^\\tau)^*_t = \\sup_s \\le t \\|X_\\tau…","labels":[],"detail_key":"p11"},{"id":"n16997","layer":"informal","project":"p11","title":"lem:ClassDL.hasLocallyIntegrableSup","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact, and that the filtratio…","labels":["lem:ClassDL.hasLocallyIntegrableSup"],"detail_key":"p11"},{"id":"n16998","layer":"informal","project":"p11","title":"Set \\tau_n","kind":"proof","summary":"Set \\tau_n","labels":[],"detail_key":"p11"},{"id":"n16999","layer":"informal","project":"p11","title":"lem:hasLocallyIntegrableSup_of_locally_classDL","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact. Assume that the filtra…","labels":["lem:hasLocallyIntegrableSup_of_locally_classDL"],"detail_key":"p11"},{"id":"n17000","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:local_induction using Lemma~\\reflem:ClassDL.hasLocallyIntegrableSup a…","kind":"proof","summary":"Apply Lemma~\\reflem:local_induction using Lemma~\\reflem:ClassDL.hasLocallyIntegrableSup and Lem…","labels":[],"detail_key":"p11"},{"id":"n17001","layer":"informal","project":"p11","title":"lem:locally_classDL_iff_locallyIntegrableSup","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact, and that the filtratio…","labels":["lem:locally_classDL_iff_locallyIntegrableSup"],"detail_key":"p11"},{"id":"n17002","layer":"informal","project":"p11","title":"The two directions are proved in Lemmas~\\reflem:hasLocallyIntegrableSup_of_locally_classD…","kind":"proof","summary":"The two directions are proved in Lemmas~\\reflem:hasLocallyIntegrableSup_of_locally_classDL and…","labels":[],"detail_key":"p11"},{"id":"n17003","layer":"informal","project":"p11","title":"lem:locally_classD_iff_locally_classDL","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact, and that the filtratio…","labels":["lem:locally_classD_iff_locally_classDL"],"detail_key":"p11"},{"id":"n17004","layer":"informal","project":"p11","title":"The forward direction follows from Lemma~\\reflem:classDLOfClassD along with Lemma~\\reflem…","kind":"proof","summary":"The forward direction follows from Lemma~\\reflem:classDLOfClassD along with Lemma~\\reflem:local…","labels":[],"detail_key":"p11"},{"id":"n17005","layer":"informal","project":"p11","title":"lem:locally_classD_iff_locallyIntegrableSup","kind":"lemma","summary":"Assume T has a bottom element and that its closed intervals are compact, and that the filtratio…","labels":["lem:locally_classD_iff_locallyIntegrableSup"],"detail_key":"p11"},{"id":"n17006","layer":"informal","project":"p11","title":"This follows from Lemmas~\\reflem:locally_classD_iff_locally_classDL and \\reflem:locally_c…","kind":"proof","summary":"This follows from Lemmas~\\reflem:locally_classD_iff_locally_classDL and \\reflem:locally_classDL…","labels":[],"detail_key":"p11"},{"id":"n17007","layer":"informal","project":"p11","title":"lem:Submartingale.locallyIntegrableSup","kind":"lemma","summary":"Every cadlag submartingale for a right-continuous filtration has locally integrable supremum.","labels":["lem:Submartingale.locallyIntegrableSup"],"detail_key":"p11"},{"id":"n17008","layer":"informal","project":"p11","title":"Define the stopping times \\sigma_n = \\inf\\t \\mid \\Vert X_t \\Vert \\ge n\\ and set \\tau_n =…","kind":"proof","summary":"Define the stopping times \\sigma_n = \\inf\\t \\mid \\Vert X_t \\Vert \\ge n\\ and set \\tau_n = \\sigma…","labels":[],"detail_key":"p11"},{"id":"n17009","layer":"informal","project":"p11","title":"lem:Submartingale.locally_classD","kind":"lemma","summary":"Every cadlag submartingale is locally of class D.","labels":["lem:Submartingale.locally_classD"],"detail_key":"p11"},{"id":"n17010","layer":"informal","project":"p11","title":"By Lemma~\\reflem:locally_classD_iff_locally_classDL, it suffices to show that every cadla…","kind":"proof","summary":"By Lemma~\\reflem:locally_classD_iff_locally_classDL, it suffices to show that every cadlag subm…","labels":[],"detail_key":"p11"},{"id":"n17011","layer":"informal","project":"p11","title":"lem:IsLocalSubmartingale.locally_classD","kind":"lemma","summary":"Every local submartingale is locally of class D.","labels":["lem:IsLocalSubmartingale.locally_classD"],"detail_key":"p11"},{"id":"n17012","layer":"informal","project":"p11","title":"By Lemma~\\reflem:local_induction, it suffices to show that if X is a submartingale then i…","kind":"proof","summary":"By Lemma~\\reflem:local_induction, it suffices to show that if X is a submartingale then it is l…","labels":[],"detail_key":"p11"},{"id":"n17013","layer":"informal","project":"p11","title":"lem:IsLocalMartingale.locally_classD","kind":"lemma","summary":"\\notready Every local martingale is locally of class D.","labels":["lem:IsLocalMartingale.locally_classD"],"detail_key":"p11"},{"id":"n17014","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17015","layer":"informal","project":"p11","title":"lem:IsLocalMartingale.martingale_iff_classDL","kind":"lemma","summary":"\\notready A local martingale is a cadlag martingale if and only if it is of class DL.","labels":["lem:IsLocalMartingale.martingale_iff_classDL"],"detail_key":"p11"},{"id":"n17016","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17017","layer":"informal","project":"p11","title":"lem:IsLocalSubmartingale.submartingale_iff_classDL_of_nonnegative","kind":"lemma","summary":"\\notready A nonnegative local submartingale is a cadlag submartingale if and only if it is of c…","labels":["lem:IsLocalSubmartingale.submartingale_iff_classDL_of_nonnegative"],"detail_key":"p11"},{"id":"n17018","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17019","layer":"informal","project":"p11","title":"Semi-ring of sets","kind":"definition","summary":"[Semi-ring of sets] \\mathlibok A \\emphsemi-ring of sets C is a family of sets containing \\empty…","labels":["def:semiRingOfSets"],"detail_key":"p11"},{"id":"n17020","layer":"informal","project":"p11","title":"lem:semiRingOfSets_of_intervals","kind":"lemma","summary":"\\mathlibok The family of intervals of the form (s, t] with s < t in a linear order T is a semi-…","labels":["lem:semiRingOfSets_of_intervals"],"detail_key":"p11"},{"id":"n17021","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17022","layer":"informal","project":"p11","title":"lem:semiRingOfSets_disjoint_of_union","kind":"lemma","summary":"\\mathlibok Suppose that C is a semi-ring of sets, and let A_k \\in C, k \\in \\1, ..., n\\~. Then t…","labels":["lem:semiRingOfSets_disjoint_of_union"],"detail_key":"p11"},{"id":"n17023","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17024","layer":"informal","project":"p11","title":"cor:semiRingOfSets_subtractUnion","kind":"corollary","summary":"\\mathlibok Suppose that C is a semi-ring of sets, and let A_k \\in C, k \\in \\1, ..., n\\, and B\\i…","labels":["cor:semiRingOfSets_subtractUnion"],"detail_key":"p11"},{"id":"n17025","layer":"informal","project":"p11","title":"This is easily done by induction on n, thanks to the Lemma~\\reflem:semiRingOfSets_disjoin…","kind":"proof","summary":"This is easily done by induction on n, thanks to the Lemma~\\reflem:semiRingOfSets_disjoint_of_u…","labels":[],"detail_key":"p11"},{"id":"n17026","layer":"informal","project":"p11","title":"lem:semiRingOfSets_disjoint_of_union_finer","kind":"lemma","summary":"The family of pairwise disjoint sets B_j in the Lemma~\\reflem:semiRingOfSets_disjoint_of_union…","labels":["lem:semiRingOfSets_disjoint_of_union_finer"],"detail_key":"p11"},{"id":"n17027","layer":"informal","project":"p11","title":"TODO: The construction in Lean, MeasureTheory.IsSetSemiring.disjointOfUnion_props lacks t…","kind":"proof","summary":"TODO: The construction in Lean, MeasureTheory.IsSetSemiring.disjointOfUnion_props lacks this pr…","labels":[],"detail_key":"p11"},{"id":"n17028","layer":"informal","project":"p11","title":"Ring of sets","kind":"definition","summary":"[Ring of sets] \\mathlibok A \\emphring of sets C is a family of sets containing \\emptyset, stabl…","labels":["def:ringOfSets"],"detail_key":"p11"},{"id":"n17029","layer":"informal","project":"p11","title":"lem:setRing_of_finiteUnions","kind":"lemma","summary":"\\mathlibok If C is a semi-ring of sets, then finite unions of elements of C form a ring of sets…","labels":["lem:setRing_of_finiteUnions"],"detail_key":"p11"},{"id":"n17030","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17031","layer":"informal","project":"p11","title":"cor:semiRingOfSets_subtract_ringElem","kind":"corollary","summary":"Suppose that D is a ring of sets, generated by the semi-ring C~. If A\\in D and B\\in C, then the…","labels":["cor:semiRingOfSets_subtract_ringElem"],"detail_key":"p11"},{"id":"n17032","layer":"informal","project":"p11","title":"This is just another wording of the Corollary~\\refcor:semiRingOfSets_subtractUnion, thank…","kind":"proof","summary":"This is just another wording of the Corollary~\\refcor:semiRingOfSets_subtractUnion, thanks to t…","labels":[],"detail_key":"p11"},{"id":"n17033","layer":"informal","project":"p11","title":"Predictable rectangles","kind":"definition","summary":"[Predictable rectangles] The class of \\emphpredictable rectangles on T \\times \\Omega is R_T :=…","labels":["def:predictableRectangles"],"detail_key":"p11"},{"id":"n17034","layer":"informal","project":"p11","title":"lem:predictableRectangles_isSetSemiring","kind":"lemma","summary":"The class R_T of predictable rectangles is a semi-ring of sets.","labels":["lem:predictableRectangles_isSetSemiring"],"detail_key":"p11"},{"id":"n17035","layer":"informal","project":"p11","title":"Take two elements A, B \\in R_T~. If only one of A, B is of the form \\0\\ \\times F, they ar…","kind":"proof","summary":"Take two elements A, B \\in R_T~. If only one of A, B is of the form \\0\\ \\times F, they are disj…","labels":[],"detail_key":"p11"},{"id":"n17036","layer":"informal","project":"p11","title":"Elementary predictable set","kind":"definition","summary":"[Elementary predictable set] A set A \\subseteq T \\times \\Omega is an \\emphelementary predictabl…","labels":["def:elementaryPredictableSet"],"detail_key":"p11"},{"id":"n17037","layer":"informal","project":"p11","title":"lem:predictableRectangles_generate_elementaryPredictableSets","kind":"lemma","summary":"Every elementary predictable set (Definition~\\refdef:elementaryPredictableSet) is a finite unio…","labels":["lem:predictableRectangles_generate_elementaryPredictableSets"],"detail_key":"p11"},{"id":"n17038","layer":"informal","project":"p11","title":"This follows immediately from the definitions.","kind":"proof","summary":"This follows immediately from the definitions.","labels":[],"detail_key":"p11"},{"id":"n17039","layer":"informal","project":"p11","title":"lem:elementaryPredictableSets_isSetRing_of","kind":"lemma","summary":"The family of elementary predictable sets (Definition~\\refdef:elementaryPredictableSet) is a ri…","labels":["lem:elementaryPredictableSets_isSetRing_of"],"detail_key":"p11"},{"id":"n17040","layer":"informal","project":"p11","title":"Since elementary predictable sets are finite unions of predictable rectangles (Lemma~\\ref…","kind":"proof","summary":"Since elementary predictable sets are finite unions of predictable rectangles (Lemma~\\reflem:pr…","labels":[],"detail_key":"p11"},{"id":"n17041","layer":"informal","project":"p11","title":"lem:elementaryPredictableSets_asDisjointUnions","kind":"lemma","summary":"Any elementary predictable set (Definition~\\refdef:elementaryPredictableSet) is a disjoint unio…","labels":["lem:elementaryPredictableSets_asDisjointUnions"],"detail_key":"p11"},{"id":"n17042","layer":"informal","project":"p11","title":"Since elementary predictable sets are finite unions of predictable rectangles (Lemma~\\ref…","kind":"proof","summary":"Since elementary predictable sets are finite unions of predictable rectangles (Lemma~\\reflem:pr…","labels":[],"detail_key":"p11"},{"id":"n17043","layer":"informal","project":"p11","title":"lem:predictableSet_elementaryPredictableSet","kind":"lemma","summary":"An elementary predictable set is measurable with respect to the predictable \\sigma-algebra.","labels":["lem:predictableSet_elementaryPredictableSet"],"detail_key":"p11"},{"id":"n17044","layer":"informal","project":"p11","title":"It is a union of sets of the form 0 \\times B with B \\in F_0 or of the form (s, t] \\times…","kind":"proof","summary":"It is a union of sets of the form 0 \\times B with B \\in F_0 or of the form (s, t] \\times B with…","labels":[],"detail_key":"p11"},{"id":"n17045","layer":"informal","project":"p11","title":"Simple process","kind":"definition","summary":"[Simple process] Let (s_k < t_k)_k \\in \\1, ..., n\\ be points in a linear order T with a bottom…","labels":["def:simpleProcess"],"detail_key":"p11"},{"id":"n17046","layer":"informal","project":"p11","title":"lem:simpleProcess_admitsDisjointIntervals","kind":"lemma","summary":"The definition of simple process does not change if we require the intervals (s_k, t_k] to be d…","labels":["lem:simpleProcess_admitsDisjointIntervals"],"detail_key":"p11"},{"id":"n17047","layer":"informal","project":"p11","title":"Fix a simple process V. The union of its time intervals is A=\\bigcup_k=1^n A_k with A_k =…","kind":"proof","summary":"Fix a simple process V. The union of its time intervals is A=\\bigcup_k=1^n A_k with A_k = (s_k,…","labels":[],"detail_key":"p11"},{"id":"n17048","layer":"informal","project":"p11","title":"lem:predictable_simpleProcess","kind":"lemma","summary":"A simple process is predictable.","labels":["lem:predictable_simpleProcess"],"detail_key":"p11"},{"id":"n17049","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17050","layer":"informal","project":"p11","title":"lem:iSup_comap_simpleProcess","kind":"lemma","summary":"Real simple processes generate the predictable \\sigma-algebra, i.e., the predictable \\sigma-alg…","labels":["lem:iSup_comap_simpleProcess"],"detail_key":"p11"},{"id":"n17051","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17052","layer":"informal","project":"p11","title":"lem:elementaryPredictableSet_iff_indicator","kind":"lemma","summary":"A set A \\subseteq T \\times \\Omega is an elementary predictable set if and only if the indicator…","labels":["lem:elementaryPredictableSet_iff_indicator"],"detail_key":"p11"},{"id":"n17053","layer":"informal","project":"p11","title":"Suppose that A is an elementary predictable set; by Lemma~\\reflem:elementaryPredictableSe…","kind":"proof","summary":"Suppose that A is an elementary predictable set; by Lemma~\\reflem:elementaryPredictableSets_asD…","labels":[],"detail_key":"p11"},{"id":"n17054","layer":"informal","project":"p11","title":"lem:addCommGroup_simpleProcess","kind":"lemma","summary":"The simple processes E_T, F form an additive commutative group.","labels":["lem:addCommGroup_simpleProcess"],"detail_key":"p11"},{"id":"n17055","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17056","layer":"informal","project":"p11","title":"lem:module_simpleProcess","kind":"lemma","summary":"The simple processes E_T, F form a module over the scalars R.","labels":["lem:module_simpleProcess"],"detail_key":"p11"},{"id":"n17057","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17058","layer":"informal","project":"p11","title":"Elementary stochastic integral","kind":"definition","summary":"[Elementary stochastic integral] Let V \\in E_T, E be a simple process and let X be a stochastic…","labels":["def:elemStochIntegralBilin"],"detail_key":"p11"},{"id":"n17059","layer":"informal","project":"p11","title":"lem:simpleProcess_map2","kind":"lemma","summary":"For V, W two simple processes with values in E and F respectively, the process defined by B(V,…","labels":["lem:simpleProcess_map2"],"detail_key":"p11"},{"id":"n17060","layer":"informal","project":"p11","title":"It has values B(\\eta_i, \\xi_j) on the intervals (s_i, t_i] \\cap (u_j, v_j] if V and W are…","kind":"proof","summary":"It has values B(\\eta_i, \\xi_j) on the intervals (s_i, t_i] \\cap (u_j, v_j] if V and W are defin…","labels":[],"detail_key":"p11"},{"id":"n17061","layer":"informal","project":"p11","title":"def:elemStochIntegralLinear","kind":"definition","summary":"Let E and F be normed real vector spaces. We call \\emphlinear elementary stochastic integral th…","labels":["def:elemStochIntegralLinear"],"detail_key":"p11"},{"id":"n17062","layer":"informal","project":"p11","title":"def:elemStochIntegral","kind":"definition","summary":"We call \\emphscalar elementary stochastic integral the general elementary stochastic integral o…","labels":["def:elemStochIntegral"],"detail_key":"p11"},{"id":"n17063","layer":"informal","project":"p11","title":"lem:elemStochIntegral_linear","kind":"lemma","summary":"The elementary stochastic integrals \\bullet_B, \\bullet_L and \\bullet_R are linear in both argum…","labels":["lem:elemStochIntegral_linear"],"detail_key":"p11"},{"id":"n17064","layer":"informal","project":"p11","title":"(In Lean, this is split into several lemmas about each argument and add/sub/smul.) Immedi…","kind":"proof","summary":"(In Lean, this is split into several lemmas about each argument and add/sub/smul.) Immediate fr…","labels":[],"detail_key":"p11"},{"id":"n17065","layer":"informal","project":"p11","title":"lem:elemStochIntegralBilin_zero","kind":"lemma","summary":"(V \\bullet_B X)_0 = 0 for every simple process V and stochastic process X.","labels":["lem:elemStochIntegralBilin_zero"],"detail_key":"p11"},{"id":"n17066","layer":"informal","project":"p11","title":"(V \\bullet_B X)_0 &= \\sum_k=1^n B (\\eta_k, X^0_t_k - X^0_s_k) \\\\ &= \\sum_k=1^n B (\\eta_k,…","kind":"proof","summary":"(V \\bullet_B X)_0 &= \\sum_k=1^n B (\\eta_k, X^0_t_k - X^0_s_k) \\\\ &= \\sum_k=1^n B (\\eta_k, X_0 -…","labels":[],"detail_key":"p11"},{"id":"n17067","layer":"informal","project":"p11","title":"lem:elemStochIntegralBilin_const","kind":"lemma","summary":"Let X_c be a constant process (equal to the same random variable for all times). Then for every…","labels":["lem:elemStochIntegralBilin_const"],"detail_key":"p11"},{"id":"n17068","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17069","layer":"informal","project":"p11","title":"lem:elemStochIntegralBilin_assoc","kind":"lemma","summary":"Let E, F, G, H, I, J be normed real vector spaces. Let B_1 : E \\times H \\to I, B_2 : F \\times G…","labels":["lem:elemStochIntegralBilin_assoc"],"detail_key":"p11"},{"id":"n17070","layer":"informal","project":"p11","title":"Unfold definitions, use linearity.","kind":"proof","summary":"Unfold definitions, use linearity.","labels":[],"detail_key":"p11"},{"id":"n17071","layer":"informal","project":"p11","title":"cor:elemStochIntegral_assoc_real_bilin","kind":"corollary","summary":"Let E, F, G be normed real vector spaces. Let B : E \\times F \\to G be a continuous bilinear map…","labels":["cor:elemStochIntegral_assoc_real_bilin"],"detail_key":"p11"},{"id":"n17072","layer":"informal","project":"p11","title":"Use Lemma~\\reflem:elemStochIntegralBilin_assoc.","kind":"proof","summary":"Use Lemma~\\reflem:elemStochIntegralBilin_assoc.","labels":[],"detail_key":"p11"},{"id":"n17073","layer":"informal","project":"p11","title":"lem:elemStochIntegral_assoc","kind":"lemma","summary":"V \\bullet_R (W \\bullet_R X) = (V \\times W) \\bullet_R X for real-valued simple processes V, W an…","labels":["lem:elemStochIntegral_assoc"],"detail_key":"p11"},{"id":"n17074","layer":"informal","project":"p11","title":"Apply Corollary~\\refcor:elemStochIntegral_assoc_real_bilin with B the scalar multiplicati…","kind":"proof","summary":"Apply Corollary~\\refcor:elemStochIntegral_assoc_real_bilin with B the scalar multiplication.","labels":[],"detail_key":"p11"},{"id":"n17075","layer":"informal","project":"p11","title":"lem:elemStochIntegralLinear_assoc","kind":"lemma","summary":"For simple process W with values in L(E, F), simple process V with values in L(F, G), and stoch…","labels":["lem:elemStochIntegralLinear_assoc"],"detail_key":"p11"},{"id":"n17076","layer":"informal","project":"p11","title":"Use Lemma~\\reflem:elemStochIntegralBilin_assoc with B_1, B_2, B_4 equal to the evaluation…","kind":"proof","summary":"Use Lemma~\\reflem:elemStochIntegralBilin_assoc with B_1, B_2, B_4 equal to the evaluation maps…","labels":[],"detail_key":"p11"},{"id":"n17077","layer":"informal","project":"p11","title":"lem:cadlag_elemStochIntegralBilin","kind":"lemma","summary":"For V \\in E_T, F, X : T \\to \\Omega \\to E a càdlàg process and a continuous bilinear map B: E \\t…","labels":["lem:cadlag_elemStochIntegralBilin"],"detail_key":"p11"},{"id":"n17078","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17079","layer":"informal","project":"p11","title":"lem:elemStochIntegral_stoppedProcess","kind":"lemma","summary":"Let X be a stochastic process, V a simple process and τ a stopping time. Then (V \\bullet_B X)^τ…","labels":["lem:elemStochIntegral_stoppedProcess"],"detail_key":"p11"},{"id":"n17080","layer":"informal","project":"p11","title":"(V \\bullet_B X)^\\tau_t &= (V \\bullet_B X)_t \\wedge \\tau \\\\ &= \\sum_k=1^n B (\\eta_k, X^t \\…","kind":"proof","summary":"(V \\bullet_B X)^\\tau_t &= (V \\bullet_B X)_t \\wedge \\tau \\\\ &= \\sum_k=1^n B (\\eta_k, X^t \\wedge…","labels":[],"detail_key":"p11"},{"id":"n17081","layer":"informal","project":"p11","title":"lem:stoppedProcess_eq_elemStochIntegral","kind":"lemma","summary":"\\notready Let X be a stochastic process and τ be a stopping time taking finitely many values. T…","labels":["lem:stoppedProcess_eq_elemStochIntegral"],"detail_key":"p11"},{"id":"n17082","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17083","layer":"informal","project":"p11","title":"lem:elemStochIntegral_stoppedIntegrator","kind":"lemma","summary":"Let X be a stochastic process, V a real-valued simple process, and τ a stopping time taking fin…","labels":["lem:elemStochIntegral_stoppedIntegrator"],"detail_key":"p11"},{"id":"n17084","layer":"informal","project":"p11","title":"Replace, in the formula (Lemma~\\reflem:elemStochIntegral_stoppedProcess): \\[ (V \\bullet_R…","kind":"proof","summary":"Replace, in the formula (Lemma~\\reflem:elemStochIntegral_stoppedProcess): \\[ (V \\bullet_R X)^τ…","labels":[],"detail_key":"p11"},{"id":"n17085","layer":"informal","project":"p11","title":"lem:tendsto_elemStochIntegral_limitProcess","kind":"lemma","summary":"For V a simple process and X a stochastic process, (V \\bullet_B X)_t tends to its limit process…","labels":["lem:tendsto_elemStochIntegral_limitProcess"],"detail_key":"p11"},{"id":"n17086","layer":"informal","project":"p11","title":"That process is eventually constant.","kind":"proof","summary":"That process is eventually constant.","labels":[],"detail_key":"p11"},{"id":"n17087","layer":"informal","project":"p11","title":"lem:limitProcess_elemStochIntegral","kind":"lemma","summary":"(V \\bullet_B X)_\\infty &= \\sum_k=1^n B (\\eta_k, X_t_k - X_s_k) \\: .","labels":["lem:limitProcess_elemStochIntegral"],"detail_key":"p11"},{"id":"n17088","layer":"informal","project":"p11","title":"Immediate from the definition.","kind":"proof","summary":"Immediate from the definition.","labels":[],"detail_key":"p11"},{"id":"n17089","layer":"informal","project":"p11","title":"lem:submartingale_iff_integral_elemStochIntegral_nonneg","kind":"lemma","summary":"An adapted integrable process X is a submartingale if and only if for every bounded simple proc…","labels":["lem:submartingale_iff_integral_elemStochIntegral_nonneg"],"detail_key":"p11"},{"id":"n17090","layer":"informal","project":"p11","title":"First suppose that X is a submartingale. The simple process V can be written as V_t = \\et…","kind":"proof","summary":"First suppose that X is a submartingale. The simple process V can be written as V_t = \\eta_0 1_…","labels":[],"detail_key":"p11"},{"id":"n17091","layer":"informal","project":"p11","title":"lem:martingale_iff_integral_elemStochIntegral_eq_zero","kind":"lemma","summary":"An adapted integrable process X is a martingale if and only if for every bounded simple process…","labels":["lem:martingale_iff_integral_elemStochIntegral_eq_zero"],"detail_key":"p11"},{"id":"n17092","layer":"informal","project":"p11","title":"There might not be an order on the type E of values of X, so we cannot just say that X is…","kind":"proof","summary":"There might not be an order on the type E of values of X, so we cannot just say that X is a mar…","labels":[],"detail_key":"p11"},{"id":"n17093","layer":"informal","project":"p11","title":"cor:Submartingale.integral_elemStochIntegral_le","kind":"corollary","summary":"Let X be a submartingale and A be an elementary predictable set. Then for all t \\in T, E[(1_A \\…","labels":["cor:Submartingale.integral_elemStochIntegral_le"],"detail_key":"p11"},{"id":"n17094","layer":"informal","project":"p11","title":"Let A be an elementary predictable set and let t \\in T. E[(1_A \\bullet_R X)_t] &= E[X_t -…","kind":"proof","summary":"Let A be an elementary predictable set and let t \\in T. E[(1_A \\bullet_R X)_t] &= E[X_t - X_0]…","labels":[],"detail_key":"p11"},{"id":"n17095","layer":"informal","project":"p11","title":"lem:Submartingale.elemStochIntegral","kind":"lemma","summary":"Let X be a submartingale and V be a nonnegative bounded simple process. Then the elementary sto…","labels":["lem:Submartingale.elemStochIntegral"],"detail_key":"p11"},{"id":"n17096","layer":"informal","project":"p11","title":"We use Lemma~\\reflem:submartingale_iff_integral_elemStochIntegral_nonneg and have to show…","kind":"proof","summary":"We use Lemma~\\reflem:submartingale_iff_integral_elemStochIntegral_nonneg and have to show that…","labels":[],"detail_key":"p11"},{"id":"n17097","layer":"informal","project":"p11","title":"lem:Martingale.elemStochIntegral","kind":"lemma","summary":"Let X be a martingale and V be a bounded simple process. Then the elementary stochastic integra…","labels":["lem:Martingale.elemStochIntegral"],"detail_key":"p11"},{"id":"n17098","layer":"informal","project":"p11","title":"We use Lemma~\\reflem:martingale_iff_integral_elemStochIntegral_eq_zero and have to show t…","kind":"proof","summary":"We use Lemma~\\reflem:martingale_iff_integral_elemStochIntegral_eq_zero and have to show that fo…","labels":[],"detail_key":"p11"},{"id":"n17099","layer":"informal","project":"p11","title":"Stochastic intervals","kind":"definition","summary":"[Stochastic intervals] Let T be a time domain and let \\sigma, \\tau be functions \\Omega \\to T \\c…","labels":["def:stochasticInterval"],"detail_key":"p11"},{"id":"n17100","layer":"informal","project":"p11","title":"lem:predictable_stochasticInterval","kind":"lemma","summary":"If \\sigma and \\tau are stopping times, then the stochastic interval ]\\!]\\sigma, \\tau]\\!] is a p…","labels":["lem:predictable_stochasticInterval"],"detail_key":"p11"},{"id":"n17101","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17102","layer":"informal","project":"p11","title":"lem:elementaryPredictableSet_stochasticInterval","kind":"lemma","summary":"If \\sigma and \\tau are stopping times on N, with \\tau bounded, then the stochastic interval ]\\!…","labels":["lem:elementaryPredictableSet_stochasticInterval"],"detail_key":"p11"},{"id":"n17103","layer":"informal","project":"p11","title":"Let n be an upper bound for \\tau. To see that the stochastic interval is an elementary pr…","kind":"proof","summary":"Let n be an upper bound for \\tau. To see that the stochastic interval is an elementary predicta…","labels":[],"detail_key":"p11"},{"id":"n17104","layer":"informal","project":"p11","title":"def:lowerCrossingTimeAux","kind":"definition","summary":"\\mathlibok Let X : T \\to \\Omega \\to R be a stochastic process, t \\in T, and a, b \\in R. The aux…","labels":["def:lowerCrossingTimeAux"],"detail_key":"p11"},{"id":"n17105","layer":"informal","project":"p11","title":"Upper crossing time","kind":"definition","summary":"[Upper crossing time] \\mathlibok Let X : T \\to \\Omega \\to R be a stochastic process, t \\in T, a…","labels":["def:upperCrossingTime"],"detail_key":"p11"},{"id":"n17106","layer":"informal","project":"p11","title":"Lower crossing time","kind":"definition","summary":"[Lower crossing time] \\mathlibok Let X : T \\to \\Omega \\to R be a stochastic process, t \\in T, a…","labels":["def:lowerCrossingTime"],"detail_key":"p11"},{"id":"n17107","layer":"informal","project":"p11","title":"Upcrossings before time t","kind":"definition","summary":"[Upcrossings before time t] \\mathlibok Let X : T \\to \\Omega \\to R be a stochastic process, t \\i…","labels":["def:upcrossingsBefore"],"detail_key":"p11"},{"id":"n17108","layer":"informal","project":"p11","title":"Upcrossings","kind":"definition","summary":"[Upcrossings] \\mathlibok Let X : T \\to \\Omega \\to R be a stochastic process and a, b \\in R. The…","labels":["def:upcrossings"],"detail_key":"p11"},{"id":"n17109","layer":"informal","project":"p11","title":"lem:tendsto_of_no_upcrossings","kind":"lemma","summary":"\\mathlibok Let u : \\beta \\to \\alpha, for \\alpha a densely ordered, conditionally complete linea…","labels":["lem:tendsto_of_no_upcrossings"],"detail_key":"p11"},{"id":"n17110","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17111","layer":"informal","project":"p11","title":"lem:isStoppingTime_upperCrossingTime","kind":"lemma","summary":"For an adapted process X : N \\to \\Omega \\to R indexed by the natural numbers, the upper crossin…","labels":["lem:isStoppingTime_upperCrossingTime"],"detail_key":"p11"},{"id":"n17112","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17113","layer":"informal","project":"p11","title":"lem:isStoppingTime_lowerCrossingTime","kind":"lemma","summary":"For an adapted process X : N \\to \\Omega \\to R indexed by the natural numbers, the lower crossin…","labels":["lem:isStoppingTime_lowerCrossingTime"],"detail_key":"p11"},{"id":"n17114","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17115","layer":"informal","project":"p11","title":"def:upcrossingPredictableSet","kind":"definition","summary":"Let X : N \\to \\Omega \\to R be an adapted process on the natural numbers. The upcrossing predict…","labels":["def:upcrossingPredictableSet"],"detail_key":"p11"},{"id":"n17116","layer":"informal","project":"p11","title":"lem:predictable_upcrossingPredictableSet","kind":"lemma","summary":"The upcrossing predictable set A_N of X : N \\to \\Omega \\to R before time N is a predictable set.","labels":["lem:predictable_upcrossingPredictableSet"],"detail_key":"p11"},{"id":"n17117","layer":"informal","project":"p11","title":"By Lemmas \\reflem:isStoppingTime_lowerCrossingTime and \\reflem:isStoppingTime_upperCrossi…","kind":"proof","summary":"By Lemmas \\reflem:isStoppingTime_lowerCrossingTime and \\reflem:isStoppingTime_upperCrossingTime…","labels":[],"detail_key":"p11"},{"id":"n17118","layer":"informal","project":"p11","title":"lem:elementaryPredictableSet_upcrossingPredictableSet","kind":"lemma","summary":"The upcrossing predictable set A_N of X : N \\to \\Omega \\to R before time N is an elementary pre…","labels":["lem:elementaryPredictableSet_upcrossingPredictableSet"],"detail_key":"p11"},{"id":"n17119","layer":"informal","project":"p11","title":"By Lemmas \\reflem:isStoppingTime_lowerCrossingTime and \\reflem:isStoppingTime_upperCrossi…","kind":"proof","summary":"By Lemmas \\reflem:isStoppingTime_lowerCrossingTime and \\reflem:isStoppingTime_upperCrossingTime…","labels":[],"detail_key":"p11"},{"id":"n17120","layer":"informal","project":"p11","title":"lem:upcrossing_simpleProcess_le_nat","kind":"lemma","summary":"Let X : N \\to \\Omega \\to R be an adapted process on the natural numbers. Then for all a < b in…","labels":["lem:upcrossing_simpleProcess_le_nat"],"detail_key":"p11"},{"id":"n17121","layer":"informal","project":"p11","title":"(1_A_t \\bullet X)_t &= \\sum_k=0^t (X_\\tau^k+1_a, b, t \\wedge t - X_\\sigma^k_a, b, t \\wedg…","kind":"proof","summary":"(1_A_t \\bullet X)_t &= \\sum_k=0^t (X_\\tau^k+1_a, b, t \\wedge t - X_\\sigma^k_a, b, t \\wedge t) \\…","labels":[],"detail_key":"p11"},{"id":"n17122","layer":"informal","project":"p11","title":"lem:upcrossing_simpleProcess_le","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be an adapted process on a finite time domain T. Then for all a < b…","labels":["lem:upcrossing_simpleProcess_le"],"detail_key":"p11"},{"id":"n17123","layer":"informal","project":"p11","title":"Go from T to a subset of N and use Lemma~\\reflem:upcrossing_simpleProcess_le_nat?","kind":"proof","summary":"Go from T to a subset of N and use Lemma~\\reflem:upcrossing_simpleProcess_le_nat?","labels":[],"detail_key":"p11"},{"id":"n17124","layer":"informal","project":"p11","title":"thm:exists_rightContinuous_modification_of_bounded_elemStochIntegral","kind":"theorem","summary":"Let T be a separable time domain, and let X : T \\to \\Omega \\to R be an adapted stochastic proce…","labels":["thm:exists_rightContinuous_modification_of_bounded_elemStochIntegral"],"detail_key":"p11"},{"id":"n17125","layer":"informal","project":"p11","title":"Let D be a countable dense subset of T (e.g.\\ the rationals in T) and let t \\in T and let…","kind":"proof","summary":"Let D be a countable dense subset of T (e.g.\\ the rationals in T) and let t \\in T and let S be…","labels":[],"detail_key":"p11"},{"id":"n17126","layer":"informal","project":"p11","title":"cor:exists_caldag_modification","kind":"corollary","summary":"Let T be a separable time domain, and let X : T \\to \\Omega \\to R be an adapted stochastic proce…","labels":["cor:exists_caldag_modification"],"detail_key":"p11"},{"id":"n17127","layer":"informal","project":"p11","title":"Let Y be the modification of X given by Theorem~\\refthm:exists_rightContinuous_modificati…","kind":"proof","summary":"Let Y be the modification of X given by Theorem~\\refthm:exists_rightContinuous_modification_of_…","labels":[],"detail_key":"p11"},{"id":"n17128","layer":"informal","project":"p11","title":"lem:Submartingale.integral_elemStochIntegral_bounded","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a submartingale. Then for every t \\in T, the set \\E[(1_A \\bullet…","labels":["lem:Submartingale.integral_elemStochIntegral_bounded"],"detail_key":"p11"},{"id":"n17129","layer":"informal","project":"p11","title":"Since X is a submartingale, for any elementary predictable set A, we have (Corollary~\\ref…","kind":"proof","summary":"Since X is a submartingale, for any elementary predictable set A, we have (Corollary~\\refcor:Su…","labels":[],"detail_key":"p11"},{"id":"n17130","layer":"informal","project":"p11","title":"lem:Submartingale.exists_cadlag_modification_of_rightContinuous","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a submartingale which is right-continuous in probability. Then X…","labels":["lem:Submartingale.exists_cadlag_modification_of_rightContinuous"],"detail_key":"p11"},{"id":"n17131","layer":"informal","project":"p11","title":"We apply Corollary~\\refcor:exists_caldag_modification, in which the boundedness condition…","kind":"proof","summary":"We apply Corollary~\\refcor:exists_caldag_modification, in which the boundedness condition of Th…","labels":[],"detail_key":"p11"},{"id":"n17132","layer":"informal","project":"p11","title":"lem:Submartingale.exists_modifications_limits","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a submartingale. Then X has a modification Y such that for all t…","labels":["lem:Submartingale.exists_modifications_limits"],"detail_key":"p11"},{"id":"n17133","layer":"informal","project":"p11","title":"Apply Theorem~\\refthm:exists_rightContinuous_modification_of_bounded_elemStochIntegral, i…","kind":"proof","summary":"Apply Theorem~\\refthm:exists_rightContinuous_modification_of_bounded_elemStochIntegral, in whic…","labels":[],"detail_key":"p11"},{"id":"n17134","layer":"informal","project":"p11","title":"lem:Submartingale.uniformIntegrable_of_antitone_of_ge","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a submartingale and let t_n be a decreasing sequence in T which i…","labels":["lem:Submartingale.uniformIntegrable_of_antitone_of_ge"],"detail_key":"p11"},{"id":"n17135","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17136","layer":"informal","project":"p11","title":"lem:Submartingale.le_condExp_of_tendsto","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a submartingale and t \\in T. Let t_n be a decreasing sequence in…","labels":["lem:Submartingale.le_condExp_of_tendsto"],"detail_key":"p11"},{"id":"n17137","layer":"informal","project":"p11","title":"By the submartingale property, for all n we have that X_t \\le P[X_t_n \\mid F_t] almost su…","kind":"proof","summary":"By the submartingale property, for all n we have that X_t \\le P[X_t_n \\mid F_t] almost surely.…","labels":[],"detail_key":"p11"},{"id":"n17138","layer":"informal","project":"p11","title":"thm:Submartingale.exists_cadlag_modification_iff_rightContinuous","kind":"theorem","summary":"Let X : T \\to \\Omega \\to R be a submartingale with respect to a right-continuous filtration. Th…","labels":["thm:Submartingale.exists_cadlag_modification_iff_rightContinuous"],"detail_key":"p11"},{"id":"n17139","layer":"informal","project":"p11","title":"Let Y be the modification of X given by Lemma~\\reflem:Submartingale.exists_modifications_…","kind":"proof","summary":"Let Y be the modification of X given by Lemma~\\reflem:Submartingale.exists_modifications_limits…","labels":[],"detail_key":"p11"},{"id":"n17140","layer":"informal","project":"p11","title":"thm:Martingale.exists_cadlag_modification","kind":"theorem","summary":"Let X : T \\to \\Omega \\to R be a martingale with respect to a right-continuous filtration. Then…","labels":["thm:Martingale.exists_cadlag_modification"],"detail_key":"p11"},{"id":"n17141","layer":"informal","project":"p11","title":"X is in particular a submartingale, and for all t \\in T, we have that E[X_t] = E[X_0] by…","kind":"proof","summary":"X is in particular a submartingale, and for all t \\in T, we have that E[X_t] = E[X_0] by the ma…","labels":[],"detail_key":"p11"},{"id":"n17142","layer":"informal","project":"p11","title":"lem:exists_cadlag_mod_of_nonneg_submg","kind":"lemma","summary":"Let the filtered probability space satisfy the usual conditions. Then every nonnegative submart…","labels":["lem:exists_cadlag_mod_of_nonneg_submg"],"detail_key":"p11"},{"id":"n17143","layer":"informal","project":"p11","title":"See 8.2.3 of Pascucci.","kind":"proof","summary":"See 8.2.3 of Pascucci.","labels":[],"detail_key":"p11"},{"id":"n17144","layer":"informal","project":"p11","title":"lem:exists_cadlag_mod_of_local_mg","kind":"lemma","summary":"Let the filtered probability space satisfy the usual conditions. Then every local martingale X…","labels":["lem:exists_cadlag_mod_of_local_mg"],"detail_key":"p11"},{"id":"n17145","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17146","layer":"informal","project":"p11","title":"lem:komlos_convex","kind":"lemma","summary":"Let (f_n)_n\\inN be a sequence in a vector space E and \\phi : E \\to R_+ be a function such that…","labels":["lem:komlos_convex"],"detail_key":"p11"},{"id":"n17147","layer":"informal","project":"p11","title":"Let B be the bound of (\\phi(f_n))_n\\inN. Then for all n\\inN and g\\in convex(f_n,f_n+1,\\cd…","kind":"proof","summary":"Let B be the bound of (\\phi(f_n))_n\\inN. Then for all n\\inN and g\\in convex(f_n,f_n+1,\\cdots) w…","labels":[],"detail_key":"p11"},{"id":"n17148","layer":"informal","project":"p11","title":"lem:komlos_norm","kind":"lemma","summary":"Let H be a Hilbert space and (f_n)_n\\inN a bounded sequence in H. Then there exist functions g_…","labels":["lem:komlos_norm"],"detail_key":"p11"},{"id":"n17149","layer":"informal","project":"p11","title":"Consider \\phi : H \\to R_+ defined by \\phi(f) = \\|f\\|_2^2, which is convex. Then Lemma \\re…","kind":"proof","summary":"Consider \\phi : H \\to R_+ defined by \\phi(f) = \\|f\\|_2^2, which is convex. Then Lemma \\reflem:k…","labels":[],"detail_key":"p11"},{"id":"n17150","layer":"informal","project":"p11","title":"lem:convex_of_converg_seq_is_converg","kind":"lemma","summary":"Let (x_n)_n \\in N be a sequence in a real vector space converging to x. Let C((x_n)) be the set…","labels":["lem:convex_of_converg_seq_is_converg"],"detail_key":"p11"},{"id":"n17151","layer":"informal","project":"p11","title":"Let \\varepsilon>0. By convergence of x_n, there exists \\barn such that for all n \\ge \\bar…","kind":"proof","summary":"Let \\varepsilon>0. By convergence of x_n, there exists \\barn such that for all n \\ge \\barn, \\Ve…","labels":[],"detail_key":"p11"},{"id":"n17152","layer":"informal","project":"p11","title":"def:convex_weights_product","kind":"definition","summary":"If (a_m)_m \\in N are convex weights and (b^n_m)_n,m \\in N is such that for all n, the (b^n_m) a…","labels":["def:convex_weights_product"],"detail_key":"p11"},{"id":"n17153","layer":"informal","project":"p11","title":"lem:komlos_convex_weights","kind":"lemma","summary":"Let E be","labels":["lem:komlos_convex_weights"],"detail_key":"p11"},{"id":"n17154","layer":"informal","project":"p11","title":"First by Lemma~\\reflem:komlos_norm applied to (x_n^(1))_n\\inN in the Hilbert space E, the…","kind":"proof","summary":"First by Lemma~\\reflem:komlos_norm applied to (x_n^(1))_n\\inN in the Hilbert space E, there exi…","labels":[],"detail_key":"p11"},{"id":"n17155","layer":"informal","project":"p11","title":"lem:komlos_convex_weights_tendsto","kind":"lemma","summary":"Let E be a Hilbert space and for i \\in N, let (x_n^(i))_n \\in N be a bounded sequence in E. Let…","labels":["lem:komlos_convex_weights_tendsto"],"detail_key":"p11"},{"id":"n17156","layer":"informal","project":"p11","title":"Let i \\in N. By Lemma~\\reflem:convex_of_converg_seq_is_converg, there is uniform converge…","kind":"proof","summary":"Let i \\in N. By Lemma~\\reflem:convex_of_converg_seq_is_converg, there is uniform convergence ov…","labels":[],"detail_key":"p11"},{"id":"n17157","layer":"informal","project":"p11","title":"lem:komlos_convex_weights_diagonal","kind":"lemma","summary":"Let E be a Hilbert space and for i \\in N, let (x_n^(i))_n \\in N be a bounded sequence in E. The…","labels":["lem:komlos_convex_weights_diagonal"],"detail_key":"p11"},{"id":"n17158","layer":"informal","project":"p11","title":"Let (\\lambda^k,n_\\cdot)_k, n \\in N be convex weights satisfying the conclusion of Lemma~\\…","kind":"proof","summary":"Let (\\lambda^k,n_\\cdot)_k, n \\in N be convex weights satisfying the conclusion of Lemma~\\reflem…","labels":[],"detail_key":"p11"},{"id":"n17159","layer":"informal","project":"p11","title":"lem:komlos_convex_aux","kind":"lemma","summary":"Let E be a Hilbert space and let (f_n)_n \\in N be a sequence in \\Omega \\to E. For i \\in N, set…","labels":["lem:komlos_convex_aux"],"detail_key":"p11"},{"id":"n17160","layer":"informal","project":"p11","title":"Use Lemma~\\reflem:komlos_convex_weights_diagonal in the Hilbert space L^2(E) with the seq…","kind":"proof","summary":"Use Lemma~\\reflem:komlos_convex_weights_diagonal in the Hilbert space L^2(E) with the sequence…","labels":[],"detail_key":"p11"},{"id":"n17161","layer":"informal","project":"p11","title":"Komlòs Lemma","kind":"lemma","summary":"[Komlòs Lemma] Let (f_n)_n\\inN be a uniformly integrable sequence of functions \\Omega \\to E, fo…","labels":["lem:komlos"],"detail_key":"p11"},{"id":"n17162","layer":"informal","project":"p11","title":"For i,n\\inN set f_n^(i):=f_n 1_(|f_n|\\leq i) such that f_n^(i)\\in L^2. Using \\reflem:koml…","kind":"proof","summary":"For i,n\\inN set f_n^(i):=f_n 1_(|f_n|\\leq i) such that f_n^(i)\\in L^2. Using \\reflem:komlos_con…","labels":[],"detail_key":"p11"},{"id":"n17163","layer":"informal","project":"p11","title":"Komlòs lemma - nonnegative, a.e. convergence","kind":"lemma","summary":"[Komlòs lemma - nonnegative, a.e. convergence] Let (f_n)_n\\inN be a sequence of random variable…","labels":["lem:komlos_ennreal"],"detail_key":"p11"},{"id":"n17164","layer":"informal","project":"p11","title":"Let \\phi : (\\Omega \\to [0, \\infty]) \\to [0, \\infty] be defined by \\phi(X) = E[e^-X]. Then…","kind":"proof","summary":"Let \\phi : (\\Omega \\to [0, \\infty]) \\to [0, \\infty] be defined by \\phi(X) = E[e^-X]. Then \\phi…","labels":[],"detail_key":"p11"},{"id":"n17165","layer":"informal","project":"p11","title":"Predictable part","kind":"definition","summary":"[Predictable part] \\mathlibok Let X : N \\to \\Omega \\to E be a process indexed by N, for E a Ban…","labels":["def:predictablePart"],"detail_key":"p11"},{"id":"n17166","layer":"informal","project":"p11","title":"Martingale part","kind":"definition","summary":"[Martingale part] \\mathlibok Let X : N \\to \\Omega \\to E be a process indexed by N, for E a Bana…","labels":["def:martingalePart"],"detail_key":"p11"},{"id":"n17167","layer":"informal","project":"p11","title":"lem:predictablePart_zero","kind":"lemma","summary":"\\mathlibok We have A_0 = 0.","labels":["lem:predictablePart_zero"],"detail_key":"p11"},{"id":"n17168","layer":"informal","project":"p11","title":"By definition.","kind":"proof","summary":"By definition.","labels":[],"detail_key":"p11"},{"id":"n17169","layer":"informal","project":"p11","title":"lem:martingalePart_zero","kind":"lemma","summary":"\\mathlibok M_0 = X_0.","labels":["lem:martingalePart_zero"],"detail_key":"p11"},{"id":"n17170","layer":"informal","project":"p11","title":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_zero, A…","kind":"proof","summary":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_zero, A_0 = 0…","labels":[],"detail_key":"p11"},{"id":"n17171","layer":"informal","project":"p11","title":"lem:predictablePart_add_one","kind":"lemma","summary":"\\mathlibok For any integer n \\ge 0, A_n+1 = A_n + E[X_n+1 - X_n \\mid F_n].","labels":["lem:predictablePart_add_one"],"detail_key":"p11"},{"id":"n17172","layer":"informal","project":"p11","title":"Let n \\in N. Then A_n+1 & = \\sum_k=0^n E[X_k+1-X_k \\mid F_k] \\\\ & = \\sum_k=0^n-1 E[X_k+1-…","kind":"proof","summary":"Let n \\in N. Then A_n+1 & = \\sum_k=0^n E[X_k+1-X_k \\mid F_k] \\\\ & = \\sum_k=0^n-1 E[X_k+1-X_k \\m…","labels":[],"detail_key":"p11"},{"id":"n17173","layer":"informal","project":"p11","title":"lem:martingalePart_add_one","kind":"lemma","summary":"M_n+1 = M_n + X_n+1 - X_n - E[X_n+1 - X_n \\mid F_n].","labels":["lem:martingalePart_add_one"],"detail_key":"p11"},{"id":"n17174","layer":"informal","project":"p11","title":"Using Lemma~\\reflem:predictablePart_add_one, we have for n \\in N, M_n+1 & = X_n+1 - A_n+1…","kind":"proof","summary":"Using Lemma~\\reflem:predictablePart_add_one, we have for n \\in N, M_n+1 & = X_n+1 - A_n+1 \\\\ &…","labels":[],"detail_key":"p11"},{"id":"n17175","layer":"informal","project":"p11","title":"lem:Martingale.predictablePart_eq_zero","kind":"lemma","summary":"\\mathlibok If X is a martingale, then A = 0 almost surely.","labels":["lem:Martingale.predictablePart_eq_zero"],"detail_key":"p11"},{"id":"n17176","layer":"informal","project":"p11","title":"By the martingale property, each conditional expectation in the definition of A is zero.","kind":"proof","summary":"By the martingale property, each conditional expectation in the definition of A is zero.","labels":[],"detail_key":"p11"},{"id":"n17177","layer":"informal","project":"p11","title":"lem:isStronglyPredictable.predictablePart_eq","kind":"lemma","summary":"\\mathlibok If X is predictable, then A = X - X_0 almost surely.","labels":["lem:isStronglyPredictable.predictablePart_eq"],"detail_key":"p11"},{"id":"n17178","layer":"informal","project":"p11","title":"Since X is predictable, for all n \\in N, X_n+1 is F_n-measurable and thus E[X_n+1 - X_n \\…","kind":"proof","summary":"Since X is predictable, for all n \\in N, X_n+1 is F_n-measurable and thus E[X_n+1 - X_n \\mid F_…","labels":[],"detail_key":"p11"},{"id":"n17179","layer":"informal","project":"p11","title":"lem:martingalePart_eq_zero","kind":"lemma","summary":"\\mathlibok If X is predictable, then M = X_0 almost surely.","labels":["lem:martingalePart_eq_zero"],"detail_key":"p11"},{"id":"n17180","layer":"informal","project":"p11","title":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:isStronglyPredictable.p…","kind":"proof","summary":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:isStronglyPredictable.predict…","labels":[],"detail_key":"p11"},{"id":"n17181","layer":"informal","project":"p11","title":"lem:Martingale.martingalePart_eq","kind":"lemma","summary":"\\mathlibok If X is a martingale, then M = X almost surely.","labels":["lem:Martingale.martingalePart_eq"],"detail_key":"p11"},{"id":"n17182","layer":"informal","project":"p11","title":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:Martingale.predictableP…","kind":"proof","summary":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:Martingale.predictablePart_eq…","labels":[],"detail_key":"p11"},{"id":"n17183","layer":"informal","project":"p11","title":"lem:predictablePart_smul","kind":"lemma","summary":"\\mathlibok For any scalar c, the predictable part of c X is c A.","labels":["lem:predictablePart_smul"],"detail_key":"p11"},{"id":"n17184","layer":"informal","project":"p11","title":"Linearity of the conditional expectation.","kind":"proof","summary":"Linearity of the conditional expectation.","labels":[],"detail_key":"p11"},{"id":"n17185","layer":"informal","project":"p11","title":"lem:martingalePart_smul","kind":"lemma","summary":"\\mathlibok For any scalar c, the martingale part of c X is c M.","labels":["lem:martingalePart_smul"],"detail_key":"p11"},{"id":"n17186","layer":"informal","project":"p11","title":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_smul, t…","kind":"proof","summary":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_smul, the pre…","labels":[],"detail_key":"p11"},{"id":"n17187","layer":"informal","project":"p11","title":"lem:predictablePart_add","kind":"lemma","summary":"\\mathlibok The predictable part of X + Y is the sum of the predictable part of X and the predic…","labels":["lem:predictablePart_add"],"detail_key":"p11"},{"id":"n17188","layer":"informal","project":"p11","title":"Linearity of the conditional expectation.","kind":"proof","summary":"Linearity of the conditional expectation.","labels":[],"detail_key":"p11"},{"id":"n17189","layer":"informal","project":"p11","title":"lem:martingalePart_add","kind":"lemma","summary":"\\mathlibok The martingale part of X + Y is the sum of the martingale part of X and the martinga…","labels":["lem:martingalePart_add"],"detail_key":"p11"},{"id":"n17190","layer":"informal","project":"p11","title":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_add, th…","kind":"proof","summary":"By definition of the martingale part, M = X - A. By Lemma~\\reflem:predictablePart_add, the pred…","labels":[],"detail_key":"p11"},{"id":"n17191","layer":"informal","project":"p11","title":"lem:adapted_predictablePart","kind":"lemma","summary":"\\mathlibok The predictable part A is adapted to the filtration (F_n+1)_n \\in N.","labels":["lem:adapted_predictablePart"],"detail_key":"p11"},{"id":"n17192","layer":"informal","project":"p11","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p11"},{"id":"n17193","layer":"informal","project":"p11","title":"lem:predictable_predictablePart","kind":"lemma","summary":"\\mathlibok The predictable part of a process is predictable.","labels":["lem:predictable_predictablePart"],"detail_key":"p11"},{"id":"n17194","layer":"informal","project":"p11","title":"By Lemma~\\reflem:predictable_nat_iff, the process A is predictable if A_0 is F_0-measurab…","kind":"proof","summary":"By Lemma~\\reflem:predictable_nat_iff, the process A is predictable if A_0 is F_0-measurable and…","labels":[],"detail_key":"p11"},{"id":"n17195","layer":"informal","project":"p11","title":"lem:martingale_martingalePart","kind":"lemma","summary":"\\mathlibok Suppose that the filtration is \\sigma-finite and that X is adapted, with X_n integra…","labels":["lem:martingale_martingalePart"],"detail_key":"p11"},{"id":"n17196","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17197","layer":"informal","project":"p11","title":"lem:Submartingale.monotone_predictablePart","kind":"lemma","summary":"\\mathlibok The predictable part of a real-valued submartingale is an almost surely nondecreasin…","labels":["lem:Submartingale.monotone_predictablePart"],"detail_key":"p11"},{"id":"n17198","layer":"informal","project":"p11","title":"Let X be a submartingale and let A be its predictable part. Then for all n \\geq 0, from L…","kind":"proof","summary":"Let X be a submartingale and let A be its predictable part. Then for all n \\geq 0, from Lemma~\\…","labels":[],"detail_key":"p11"},{"id":"n17199","layer":"informal","project":"p11","title":"lem:Submartingale.predictablePart_nonneg","kind":"lemma","summary":"\\mathlibok The predictable part of a real-valued submartingale is almost surely nonnegative.","labels":["lem:Submartingale.predictablePart_nonneg"],"detail_key":"p11"},{"id":"n17200","layer":"informal","project":"p11","title":"By Lemma~\\reflem:Submartingale.monotone_predictablePart, the predictable part A is almost…","kind":"proof","summary":"By Lemma~\\reflem:Submartingale.monotone_predictablePart, the predictable part A is almost surel…","labels":[],"detail_key":"p11"},{"id":"n17201","layer":"informal","project":"p11","title":"lem:centering_basic","kind":"lemma","summary":"\\mathlibok Fake lemma: import this lemma when using the basic properties of the Doob decomposit…","labels":["lem:centering_basic"],"detail_key":"p11"},{"id":"n17202","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17203","layer":"informal","project":"p11","title":"lem:IsStoppingTime_leastGT_predictablePart","kind":"lemma","summary":"Let X be a real adapted process and let A be its predictable part. Let c \\in R. The hitting tim…","labels":["lem:IsStoppingTime_leastGT_predictablePart"],"detail_key":"p11"},{"id":"n17204","layer":"informal","project":"p11","title":"Since A_n is predictable, A_n + 1 is adapted. The hitting time of an adapted process is a…","kind":"proof","summary":"Since A_n is predictable, A_n + 1 is adapted. The hitting time of an adapted process is a stopp…","labels":[],"detail_key":"p11"},{"id":"n17205","layer":"informal","project":"p11","title":"lem:leastGT_predictablePart_le","kind":"lemma","summary":"Let X be a real adapted process and let A be its predictable part. Let c \\in R. Then A_\\tau_A_n…","labels":["lem:leastGT_predictablePart_le"],"detail_key":"p11"},{"id":"n17206","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17207","layer":"informal","project":"p11","title":"lem:leastGT_predictablePart_sub_ge","kind":"lemma","summary":"Let X be a real adapted process and let A be its predictable part. Let a, b \\in R with a \\le b.…","labels":["lem:leastGT_predictablePart_sub_ge"],"detail_key":"p11"},{"id":"n17208","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17209","layer":"informal","project":"p11","title":"lem:uniformIntegrable_predictablePart_aux1","kind":"lemma","summary":"Let T \\in N and let X be an adapted process with X_n integrable for all n and such that X_T = 0…","labels":["lem:uniformIntegrable_predictablePart_aux1"],"detail_key":"p11"},{"id":"n17210","layer":"informal","project":"p11","title":"By definition and since X_T = 0, M_T = X_T - A_T = - A_T. Since M is a martingale it foll…","kind":"proof","summary":"By definition and since X_T = 0, M_T = X_T - A_T = - A_T. Since M is a martingale it follows by…","labels":[],"detail_key":"p11"},{"id":"n17211","layer":"informal","project":"p11","title":"lem:predictablePart_gt_iff_leastGT_lt","kind":"lemma","summary":"Suppose that X is a submartingale and let T \\ge 1, c \\in R. Then A_T > c \\iff \\tau^T(c) < T.","labels":["lem:predictablePart_gt_iff_leastGT_lt"],"detail_key":"p11"},{"id":"n17212","layer":"informal","project":"p11","title":"By Lemma~\\reflem:leastGT_lt_iff, \\tau^T(c) < T \\iff \\exists s < T, A_s+1 > c, which is eq…","kind":"proof","summary":"By Lemma~\\reflem:leastGT_lt_iff, \\tau^T(c) < T \\iff \\exists s < T, A_s+1 > c, which is equivale…","labels":[],"detail_key":"p11"},{"id":"n17213","layer":"informal","project":"p11","title":"lem:uniformIntegrable_predictablePart_aux2","kind":"lemma","summary":"Suppose that X is a submartingale with X_T = 0. Then E\\left[A_T I\\A_T > c\\ \\right] \\le c P(\\tau…","labels":["lem:uniformIntegrable_predictablePart_aux2"],"detail_key":"p11"},{"id":"n17214","layer":"informal","project":"p11","title":"By Lemma~\\reflem:predictablePart_gt_iff_leastGT_lt, E\\left[A_T I\\A_T > c\\ \\right] = E\\lef…","kind":"proof","summary":"By Lemma~\\reflem:predictablePart_gt_iff_leastGT_lt, E\\left[A_T I\\A_T > c\\ \\right] = E\\left[A_T…","labels":[],"detail_key":"p11"},{"id":"n17215","layer":"informal","project":"p11","title":"lem:uniformIntegrable_predictablePart_aux3","kind":"lemma","summary":"Suppose that X is a submartingale with X_T = 0. Then P(\\tau^T(c) < T) &\\le - \\frac2c \\int_\\tau^…","labels":["lem:uniformIntegrable_predictablePart_aux3"],"detail_key":"p11"},{"id":"n17216","layer":"informal","project":"p11","title":"Notice that \\\\tau^T(c)<T\\\\subseteq \\\\tau^T(c/2)<T\\, thus \\int_\\tau^T(c/2)<T -X_\\tau^T(c/2…","kind":"proof","summary":"Notice that \\\\tau^T(c)<T\\\\subseteq \\\\tau^T(c/2)<T\\, thus \\int_\\tau^T(c/2)<T -X_\\tau^T(c/2)dP &=…","labels":[],"detail_key":"p11"},{"id":"n17217","layer":"informal","project":"p11","title":"lem:uniformIntegrable_predictablePart_aux4","kind":"lemma","summary":"Suppose that X is a submartingale with X_T = 0. Then E\\left[A_T I\\A_T > c\\ \\right] \\le - 2 \\int…","labels":["lem:uniformIntegrable_predictablePart_aux4"],"detail_key":"p11"},{"id":"n17218","layer":"informal","project":"p11","title":"Put together the bounds of Lemma~\\reflem:uniformIntegrable_predictablePart_aux2 and Lemma…","kind":"proof","summary":"Put together the bounds of Lemma~\\reflem:uniformIntegrable_predictablePart_aux2 and Lemma~\\refl…","labels":[],"detail_key":"p11"},{"id":"n17219","layer":"informal","project":"p11","title":"lem:uniformIntegrable_predictablePart_aux5","kind":"lemma","summary":"Suppose that X is a submartingale with X_T = 0. Then P(\\tau^T(c) < T) \\le -\\fracE[X_0]c \\: .","labels":["lem:uniformIntegrable_predictablePart_aux5"],"detail_key":"p11"},{"id":"n17220","layer":"informal","project":"p11","title":"Starting with Lemma~\\reflem:predictablePart_gt_iff_leastGT_lt, P(\\tau^T(c)<T) =P(A_T>c) \\…","kind":"proof","summary":"Starting with Lemma~\\reflem:predictablePart_gt_iff_leastGT_lt, P(\\tau^T(c)<T) =P(A_T>c) \\stackr…","labels":[],"detail_key":"p11"},{"id":"n17221","layer":"informal","project":"p11","title":"Dyadics","kind":"definition","summary":"[Dyadics] For T>0, let D_n^T = \\left\\lbrace \\frack2^nT \\mid k=0,\\cdots 2^n\\right\\rbrace be the…","labels":["def:dyadics"],"detail_key":"p11"},{"id":"n17222","layer":"informal","project":"p11","title":"S, A, M","kind":"definition","summary":"[S, A, M] For n \\in N, the restriction of S to D_n^T is a discrete time submartingale S^n : N \\…","labels":["def:SAM"],"detail_key":"p11"},{"id":"n17223","layer":"informal","project":"p11","title":"lem:A_uniform_integrable","kind":"lemma","summary":"The sequence (A^n_2^n)_n\\inN is uniformly integrable (bounded in L^1 norm).","labels":["lem:A_uniform_integrable"],"detail_key":"p11"},{"id":"n17224","layer":"informal","project":"p11","title":"WLOG S^n_2^n = S_T=0 and S_t\\leq 0 (else consider S_t-E\\left[S_T\\vertF_t\\right]). We writ…","kind":"proof","summary":"WLOG S^n_2^n = S_T=0 and S_t\\leq 0 (else consider S_t-E\\left[S_T\\vertF_t\\right]). We write \\tau…","labels":[],"detail_key":"p11"},{"id":"n17225","layer":"informal","project":"p11","title":"lem:M_uniform_integrable","kind":"lemma","summary":"The sequence (M^n_2^n)_n\\inN is uniformly integrable (bounded in L^1 norm).","labels":["lem:M_uniform_integrable"],"detail_key":"p11"},{"id":"n17226","layer":"informal","project":"p11","title":"M^n_2^n = S_2^n - A^n_2^n, also S is of class D hence uniformly integrable and A^n_2^n is…","kind":"proof","summary":"M^n_2^n = S_2^n - A^n_2^n, also S is of class D hence uniformly integrable and A^n_2^n is unifo…","labels":[],"detail_key":"p11"},{"id":"n17227","layer":"informal","project":"p11","title":"lem:M_n_cadlag_mg","kind":"lemma","summary":"The martingale on [0, T] defined by t \\mapsto E[M^n_2^n\\vertF_t] admits a modification which is…","labels":["lem:M_n_cadlag_mg"],"detail_key":"p11"},{"id":"n17228","layer":"informal","project":"p11","title":"By theorem \\refthm:Martingale.exists_cadlag_modification","kind":"proof","summary":"By theorem \\refthm:Martingale.exists_cadlag_modification","labels":[],"detail_key":"p11"},{"id":"n17229","layer":"informal","project":"p11","title":"def:barM","kind":"definition","summary":"For t\\in[0,T] let \\overlineM^n_t be the cadlag modification of t \\mapsto E[M^n_2^n \\mid F_t] fr…","labels":["def:barM"],"detail_key":"p11"},{"id":"n17230","layer":"informal","project":"p11","title":"lem:M_cal_converges_L1","kind":"lemma","summary":"There exists an M : \\Omega \\to R and convex weights \\lambda^n_n,\\cdots,\\lambda^n_N_n such that…","labels":["lem:M_cal_converges_L1"],"detail_key":"p11"},{"id":"n17231","layer":"informal","project":"p11","title":"By lemma \\reflem:M_uniform_integrable (M^n_T)_n\\inN is uniformly bounded in L^1, thus by…","kind":"proof","summary":"By lemma \\reflem:M_uniform_integrable (M^n_T)_n\\inN is uniformly bounded in L^1, thus by lemma…","labels":[],"detail_key":"p11"},{"id":"n17232","layer":"informal","project":"p11","title":"def:calM","kind":"definition","summary":"For t\\in[0,T] let M^n_t=\\lambda^n_n \\overlineM^n_t+\\cdots +\\lambda^n_N_n \\overlineM^N_n_t be th…","labels":["def:calM"],"detail_key":"p11"},{"id":"n17233","layer":"informal","project":"p11","title":"lem:M_cal_cadlag","kind":"lemma","summary":"M^n is cadlag.","labels":["lem:M_cal_cadlag"],"detail_key":"p11"},{"id":"n17234","layer":"informal","project":"p11","title":"By construction and \\reflem:M_n_cadlag_mg","kind":"proof","summary":"By construction and \\reflem:M_n_cadlag_mg","labels":[],"detail_key":"p11"},{"id":"n17235","layer":"informal","project":"p11","title":"def:M","kind":"definition","summary":"Let M_t be a cadlag modification of E[M \\mid F_t], obtained by applying Theorem~\\refthm:Marting…","labels":["def:M"],"detail_key":"p11"},{"id":"n17236","layer":"informal","project":"p11","title":"lem:M1_komlos","kind":"lemma","summary":"For every t\\in[0,T] we have M^n_t\\stackrelL^1\\rightarrowM_t.","labels":["lem:M1_komlos"],"detail_key":"p11"},{"id":"n17237","layer":"informal","project":"p11","title":"equation_DM_e7","kind":"proof","summary":"By Jensen's inequality, the tower lemma and lemma \\reflem:M_cal_converges_L1 \\nonumber E[|M^n_t…","labels":["equation_DM_e7"],"detail_key":"p11"},{"id":"n17238","layer":"informal","project":"p11","title":"def:barA","kind":"definition","summary":"For t\\in[0,T] let \\overlineA^n_t be process defined by \\overlineA^n_s := \\sum_m < 2^n A^n_m+1 1…","labels":["def:barA"],"detail_key":"p11"},{"id":"n17239","layer":"informal","project":"p11","title":"lem:barA_eq","kind":"lemma","summary":"For s \\in D^T_n, \\overlineA^n_s = A^n_m where m is such that s = m2^-nT.","labels":["lem:barA_eq"],"detail_key":"p11"},{"id":"n17240","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17241","layer":"informal","project":"p11","title":"lem:leftContinuous_barA","kind":"lemma","summary":"\\overlineA^n is left continuous.","labels":["lem:leftContinuous_barA"],"detail_key":"p11"},{"id":"n17242","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17243","layer":"informal","project":"p11","title":"lem:predictable_barA","kind":"lemma","summary":"\\overlineA^n is predictable.","labels":["lem:predictable_barA"],"detail_key":"p11"},{"id":"n17244","layer":"informal","project":"p11","title":"Since \\overlineA^n is left continuous and adapted, it is predictable (Lemma~\\reflem:Adapt…","kind":"proof","summary":"Since \\overlineA^n is left continuous and adapted, it is predictable (Lemma~\\reflem:Adapted.isP…","labels":[],"detail_key":"p11"},{"id":"n17245","layer":"informal","project":"p11","title":"def:calA","kind":"definition","summary":"Let A^n := \\lambda^n_n \\overlineA^n+\\cdots +\\lambda^n_N_n\\overlineA^N_n, in which the \\lambda^n…","labels":["def:calA"],"detail_key":"p11"},{"id":"n17246","layer":"informal","project":"p11","title":"lem:leftContinuous_calA","kind":"lemma","summary":"A^n is left continuous.","labels":["lem:leftContinuous_calA"],"detail_key":"p11"},{"id":"n17247","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17248","layer":"informal","project":"p11","title":"lem:predictable_calA","kind":"lemma","summary":"A^n is predictable.","labels":["lem:predictable_calA"],"detail_key":"p11"},{"id":"n17249","layer":"informal","project":"p11","title":"Either use that it is left continuous and adapted, or that it is a convex combination of…","kind":"proof","summary":"Either use that it is left continuous and adapted, or that it is a convex combination of predic…","labels":[],"detail_key":"p11"},{"id":"n17250","layer":"informal","project":"p11","title":"lem:calA_mono","kind":"lemma","summary":"A^n is non-decreasing on [0,T].","labels":["lem:calA_mono"],"detail_key":"p11"},{"id":"n17251","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17252","layer":"informal","project":"p11","title":"def:A","kind":"definition","summary":"Let A_t = S_t - M_t for t \\in [0, T].","labels":["def:A"],"detail_key":"p11"},{"id":"n17253","layer":"informal","project":"p11","title":"lem:cadlag_A","kind":"lemma","summary":"A is cadlag.","labels":["lem:cadlag_A"],"detail_key":"p11"},{"id":"n17254","layer":"informal","project":"p11","title":"Since S and M are cadlag, their difference A = S - M is also cadlag.","kind":"proof","summary":"Since S and M are cadlag, their difference A = S - M is also cadlag.","labels":[],"detail_key":"p11"},{"id":"n17255","layer":"informal","project":"p11","title":"lem:tendsto_calA_A_L1","kind":"lemma","summary":"For every t\\in D^T, A^n_t converges to A_t in L^1.","labels":["lem:tendsto_calA_A_L1"],"detail_key":"p11"},{"id":"n17256","layer":"informal","project":"p11","title":"By Lemma \\reflem:M1_komlos, for all t\\inD^T, A^n_t = S_t - M^n_t \\stackrelL^1\\rightarrow…","kind":"proof","summary":"By Lemma \\reflem:M1_komlos, for all t\\inD^T, A^n_t = S_t - M^n_t \\stackrelL^1\\rightarrow S_t-M_…","labels":[],"detail_key":"p11"},{"id":"n17257","layer":"informal","project":"p11","title":"lem:tendsto_calA_A_ae","kind":"lemma","summary":"For every t\\in D^T, there exists a subsequence (k_n) such that A^k_n_t converges to A_t almost…","labels":["lem:tendsto_calA_A_ae"],"detail_key":"p11"},{"id":"n17258","layer":"informal","project":"p11","title":"By Lemma \\reflem:tendsto_calA_A_L1, we have convergence in L^1 for every t\\in D^T, which…","kind":"proof","summary":"By Lemma \\reflem:tendsto_calA_A_L1, we have convergence in L^1 for every t\\in D^T, which implie…","labels":[],"detail_key":"p11"},{"id":"n17259","layer":"informal","project":"p11","title":"lem:A_mono_dyadics","kind":"lemma","summary":"A is almost surely non-decreasing on D^T.","labels":["lem:A_mono_dyadics"],"detail_key":"p11"},{"id":"n17260","layer":"informal","project":"p11","title":"Since D^T is countable, to prove that A is almost surely non-decreasing on D^T it is enou…","kind":"proof","summary":"Since D^T is countable, to prove that A is almost surely non-decreasing on D^T it is enough to…","labels":[],"detail_key":"p11"},{"id":"n17261","layer":"informal","project":"p11","title":"lem:A_increasing","kind":"lemma","summary":"(A_t)_t\\in[0,T] is almost surely non-decreasing.","labels":["lem:A_increasing"],"detail_key":"p11"},{"id":"n17262","layer":"informal","project":"p11","title":"A is almost surely non-decreasing on D^T by Lemma~\\reflem:A_mono_dyadics. Since A is cadl…","kind":"proof","summary":"A is almost surely non-decreasing on D^T by Lemma~\\reflem:A_mono_dyadics. Since A is cadlag (th…","labels":[],"detail_key":"p11"},{"id":"n17263","layer":"informal","project":"p11","title":"lem:jump_A","kind":"lemma","summary":"For q,k \\in N, let \\tau_q, k be the q-th time that the process A_t has a jump higher than 1/(k+…","labels":["lem:jump_A"],"detail_key":"p11"},{"id":"n17264","layer":"informal","project":"p11","title":"Since A is non-decreasing it has finitely many jumps of size higher than 1/(k+1) for each…","kind":"proof","summary":"Since A is non-decreasing it has finitely many jumps of size higher than 1/(k+1) for each k. Th…","labels":[],"detail_key":"p11"},{"id":"n17265","layer":"informal","project":"p11","title":"lem:incr_fun_lim_right_cont_limsup_ineq","kind":"lemma","summary":"If f_n, f : [0, 1] \\rightarrow R are increasing functions such that f is right continuous and \\…","labels":["lem:incr_fun_lim_right_cont_limsup_ineq"],"detail_key":"p11"},{"id":"n17266","layer":"informal","project":"p11","title":"Let t\\in[0,T] and s\\inD^T such that t<s. We have \\limsup_n f_n(t)\\leq \\limsup_n f_n(s)=f(…","kind":"proof","summary":"Let t\\in[0,T] and s\\inD^T such that t<s. We have \\limsup_n f_n(t)\\leq \\limsup_n f_n(s)=f(s). Si…","labels":[],"detail_key":"p11"},{"id":"n17267","layer":"informal","project":"p11","title":"lem:incr_fun_lim_right_cont_lim_eq","kind":"lemma","summary":"If f_n, f : [0, 1] \\rightarrow R are increasing functions such that f is right continuous and \\…","labels":["lem:incr_fun_lim_right_cont_lim_eq"],"detail_key":"p11"},{"id":"n17268","layer":"informal","project":"p11","title":"By lemma \\reflem:incr_fun_lim_right_cont_limsup_ineq it is enough to show that \\liminf_n…","kind":"proof","summary":"By lemma \\reflem:incr_fun_lim_right_cont_limsup_ineq it is enough to show that \\liminf_n f_n(t)…","labels":[],"detail_key":"p11"},{"id":"n17269","layer":"informal","project":"p11","title":"lem:lim_Exp_A_n_tau_is_Exp_A_tau","kind":"lemma","summary":"Let \\tau be an (F_t)_t\\in[0,T] stopping time. We have \\lim_nE[A^n_\\tau]=E[A_\\tau].","labels":["lem:lim_Exp_A_n_tau_is_Exp_A_tau"],"detail_key":"p11"},{"id":"n17270","layer":"informal","project":"p11","title":"Let \\sigma_n:=\\inf\\left(t\\inD^T_n\\vert t>\\tau\\right). By construction of A^n we have A^n_…","kind":"proof","summary":"Let \\sigma_n:=\\inf\\left(t\\inD^T_n\\vert t>\\tau\\right). By construction of A^n we have A^n_\\tau=A…","labels":[],"detail_key":"p11"},{"id":"n17271","layer":"informal","project":"p11","title":"lem:limsup_A_n_tau_is_A_tau_ae","kind":"lemma","summary":"Let \\tau be an (F_t)_t\\in[0,T] stopping time. We have \\limsup_n A_\\tau^n = A_\\tau.","labels":["lem:limsup_A_n_tau_is_A_tau_ae"],"detail_key":"p11"},{"id":"n17272","layer":"informal","project":"p11","title":"Firstly we notice that \\liminf_n E[A_\\tau^n] \\leq \\limsup_n E [A_\\tau^n ] \\leq E[\\limsup_…","kind":"proof","summary":"Firstly we notice that \\liminf_n E[A_\\tau^n] \\leq \\limsup_n E [A_\\tau^n ] \\leq E[\\limsup_n A_\\t…","labels":[],"detail_key":"p11"},{"id":"n17273","layer":"informal","project":"p11","title":"Doob-Meyer decomposition","kind":"theorem","summary":"[Doob-Meyer decomposition] Let S = (S_t )_0\\leq t\\leq T be a cadlag submartingale of class D. T…","labels":["thm:Doob_Meyer"],"detail_key":"p11"},{"id":"n17274","layer":"informal","project":"p11","title":"By construction M is a cadlag martingale and A_0=0 and by lemma \\reflem:A_increasing A is…","kind":"proof","summary":"By construction M is a cadlag martingale and A_0=0 and by lemma \\reflem:A_increasing A is incre…","labels":[],"detail_key":"p11"},{"id":"n17275","layer":"informal","project":"p11","title":"Doob-Meyer decomposition","kind":"theorem","summary":"[Doob-Meyer decomposition] An adapted process X is a cadlag local submartingale iff X = M + A w…","labels":["thm:local_doobMeyer"],"detail_key":"p11"},{"id":"n17276","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17277","layer":"informal","project":"p11","title":"lem:submartingale_iff_integrable","kind":"lemma","summary":"An adapted increasing process A is a submartingale iff it is integrable.","labels":["lem:submartingale_iff_integrable"],"detail_key":"p11"},{"id":"n17278","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17279","layer":"informal","project":"p11","title":"lem:local_doobMeyer_increasing","kind":"lemma","summary":"An adapted locally integrable increasing process A can be written as a sum of a local martingal…","labels":["lem:local_doobMeyer_increasing"],"detail_key":"p11"},{"id":"n17280","layer":"informal","project":"p11","title":"A locally integrable increasing process is a local submartingale, thus we can apply Theor…","kind":"proof","summary":"A locally integrable increasing process is a local submartingale, thus we can apply Theorem~\\re…","labels":[],"detail_key":"p11"},{"id":"n17281","layer":"informal","project":"p11","title":"Jumps of a process","kind":"definition","summary":"[Jumps of a process] The jumps of a process X : T \\to \\Omega \\to E is the process \\Delta X : T…","labels":["def:jump"],"detail_key":"p11"},{"id":"n17282","layer":"informal","project":"p11","title":"Jump part","kind":"definition","summary":"[Jump part] The jump part (or purely discontinuous part) of a process X : T \\to \\Omega \\to E is…","labels":["def:jumpPart"],"detail_key":"p11"},{"id":"n17283","layer":"informal","project":"p11","title":"Continuous part","kind":"definition","summary":"[Continuous part] The continuous part of a process X : T \\to \\Omega \\to E is the process X^c :…","labels":["def:continuousPart"],"detail_key":"p11"},{"id":"n17284","layer":"informal","project":"p11","title":"Large jumps","kind":"definition","summary":"[Large jumps] The large jump part of a process X : T \\to \\Omega \\to E at level \\varepsilon is t…","labels":["def:largeJump"],"detail_key":"p11"},{"id":"n17285","layer":"informal","project":"p11","title":"Extended variation","kind":"definition","summary":"[Extended variation] \\mathlibok The (extended-real-valued) variation of a function f : T \\to E…","labels":["def:eVariationOn"],"detail_key":"p11"},{"id":"n17286","layer":"informal","project":"p11","title":"Bounded variation","kind":"definition","summary":"[Bounded variation] \\mathlibok A function f : T \\to E is of bounded variation on a set s if its…","labels":["def:BoundedVariationOn"],"detail_key":"p11"},{"id":"n17287","layer":"informal","project":"p11","title":"Locally bounded variation","kind":"definition","summary":"[Locally bounded variation] \\mathlibok A function f : T \\to E is of locally bounded variation o…","labels":["def:LocallyBoundedVariationOn"],"detail_key":"p11"},{"id":"n17288","layer":"informal","project":"p11","title":"lem:LocallyBoundedVariationOn.countable_not_continuous_at","kind":"lemma","summary":"The set of points of discontinuity of a function of locally bounded variation is at most counta…","labels":["lem:LocallyBoundedVariationOn.countable_not_continuous_at"],"detail_key":"p11"},{"id":"n17289","layer":"informal","project":"p11","title":"Mathlib has the result for monotone functions and has the fact that a function of locally…","kind":"proof","summary":"Mathlib has the result for monotone functions and has the fact that a function of locally bound…","labels":[],"detail_key":"p11"},{"id":"n17290","layer":"informal","project":"p11","title":"Finite variation process, V","kind":"definition","summary":"[Finite variation process, V] A process X : T \\to \\Omega \\to E is of finite variation if it is…","labels":["def:HasFiniteVariation"],"detail_key":"p11"},{"id":"n17291","layer":"informal","project":"p11","title":"lem:HasFiniteVariation.isCadlag","kind":"lemma","summary":"A finite variation process is càdlàg.","labels":["lem:HasFiniteVariation.isCadlag"],"detail_key":"p11"},{"id":"n17292","layer":"informal","project":"p11","title":"Right-continuity is by definition. Left limits exist because a function of locally bounde…","kind":"proof","summary":"Right-continuity is by definition. Left limits exist because a function of locally bounded vari…","labels":[],"detail_key":"p11"},{"id":"n17293","layer":"informal","project":"p11","title":"def:variationProcess","kind":"definition","summary":"The variation of a process X : T \\to \\Omega \\to E is the process V_X : T \\to \\Omega \\to R defin…","labels":["def:variationProcess"],"detail_key":"p11"},{"id":"n17294","layer":"informal","project":"p11","title":"lem:monotone_variationProcess","kind":"lemma","summary":"The variation process V_X of a process X is non-decreasing.","labels":["lem:monotone_variationProcess"],"detail_key":"p11"},{"id":"n17295","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17296","layer":"informal","project":"p11","title":"lem:rightContinuous_variationProcess","kind":"lemma","summary":"The variation process V_X of a right-continuous process X is right-continuous.","labels":["lem:rightContinuous_variationProcess"],"detail_key":"p11"},{"id":"n17297","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17298","layer":"informal","project":"p11","title":"Integrable variation, A","kind":"definition","summary":"[Integrable variation, A] We say that a stochastic process X has integrable variation if its va…","labels":["def:HasIntegrableVariation"],"detail_key":"p11"},{"id":"n17299","layer":"informal","project":"p11","title":"lem:HasIntegrableVariation.eq_sub_monotone","kind":"lemma","summary":"A process with integrable variation can be written as the difference of two non-decreasing adap…","labels":["lem:HasIntegrableVariation.eq_sub_monotone"],"detail_key":"p11"},{"id":"n17300","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17301","layer":"informal","project":"p11","title":"lem:local_doobMeyer_integrable_variation","kind":"lemma","summary":"A process in A_loc (locally integrable variation) can be written as a sum of a local martingale…","labels":["lem:local_doobMeyer_integrable_variation"],"detail_key":"p11"},{"id":"n17302","layer":"informal","project":"p11","title":"We write the process as a difference of two non-decreasing locally integrable processes,…","kind":"proof","summary":"We write the process as a difference of two non-decreasing locally integrable processes, and th…","labels":[],"detail_key":"p11"},{"id":"n17303","layer":"informal","project":"p11","title":"lem:martingalePart_jumpPart","kind":"lemma","summary":"Let M \\in M \\cap A be a martingale with integrable variation. Then the jump part A = \\sum_0 < s…","labels":["lem:martingalePart_jumpPart"],"detail_key":"p11"},{"id":"n17304","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17305","layer":"informal","project":"p11","title":"lem:Martingale.continuous_of_isStronglyPredictable_of_uniformIntegrable","kind":"lemma","summary":"A predictable uniformly integrable martingale is continuous.","labels":["lem:Martingale.continuous_of_isStronglyPredictable_of_uniformIntegrable"],"detail_key":"p11"},{"id":"n17306","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17307","layer":"informal","project":"p11","title":"Square integrable martingales","kind":"definition","summary":"[Square integrable martingales] Let T be a linear order with bottom element 0, on which we have…","labels":["def:IsSquareIntegrable"],"detail_key":"p11"},{"id":"n17308","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.uniformIntegrable","kind":"lemma","summary":"A square integrable martingale is uniformly integrable.","labels":["lem:IsSquareIntegrable.uniformIntegrable"],"detail_key":"p11"},{"id":"n17309","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:uniformIntegrable_of_bounded.","kind":"proof","summary":"Apply Lemma~\\reflem:uniformIntegrable_of_bounded.","labels":[],"detail_key":"p11"},{"id":"n17310","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.module","kind":"lemma","summary":"If M and N are square integrable martingales and a \\in R, then M + N and a M are square integra…","labels":["lem:IsSquareIntegrable.module"],"detail_key":"p11"},{"id":"n17311","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17312","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.submartingale_sq","kind":"lemma","summary":"If M is a square integrable martingale, then \\Vert M \\Vert^2 is a submartingale.","labels":["lem:IsSquareIntegrable.submartingale_sq"],"detail_key":"p11"},{"id":"n17313","layer":"informal","project":"p11","title":"Apply Lemma~\\reflem:Martingale.submartingale_convex_comp with the convex function f: x \\m…","kind":"proof","summary":"Apply Lemma~\\reflem:Martingale.submartingale_convex_comp with the convex function f: x \\mapsto…","labels":[],"detail_key":"p11"},{"id":"n17314","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.eLpNorm_two_mono","kind":"lemma","summary":"For M a square integrable martingale, the function t \\mapsto \\Vert M_t \\Vert_L^2 is non-decreas…","labels":["lem:IsSquareIntegrable.eLpNorm_two_mono"],"detail_key":"p11"},{"id":"n17315","layer":"informal","project":"p11","title":"By Lemma~\\reflem:IsSquareIntegrable.submartingale_sq, \\Vert M_t \\Vert^2 is a submartingal…","kind":"proof","summary":"By Lemma~\\reflem:IsSquareIntegrable.submartingale_sq, \\Vert M_t \\Vert^2 is a submartingale. Thu…","labels":[],"detail_key":"p11"},{"id":"n17316","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.tendsto_limitProcess","kind":"lemma","summary":"For M a square integrable martingale, we have M_t \\to M_\\infty almost surely and in L^2 as t \\t…","labels":["lem:IsSquareIntegrable.tendsto_limitProcess"],"detail_key":"p11"},{"id":"n17317","layer":"informal","project":"p11","title":"TODO: use a martingale convergence theorem. Check whether Theorem~\\refthm:tendsto_limitPr…","kind":"proof","summary":"TODO: use a martingale convergence theorem. Check whether Theorem~\\refthm:tendsto_limitProcess_…","labels":[],"detail_key":"p11"},{"id":"n17318","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.limitProcess_add","kind":"lemma","summary":"For M and N two square integrable martingales, (M + N)_\\infty and M_\\infty + N_\\infty are a.e.…","labels":["lem:IsSquareIntegrable.limitProcess_add"],"detail_key":"p11"},{"id":"n17319","layer":"informal","project":"p11","title":"Lemma~\\reflem:limitProcess_ae_eq with g := M_\\infty + N_\\infty.","kind":"proof","summary":"Lemma~\\reflem:limitProcess_ae_eq with g := M_\\infty + N_\\infty.","labels":[],"detail_key":"p11"},{"id":"n17320","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.limitProcess_sub","kind":"lemma","summary":"For M and N two square integrable martingales, (M - N)_\\infty and M_\\infty - N_\\infty are a.e.…","labels":["lem:IsSquareIntegrable.limitProcess_sub"],"detail_key":"p11"},{"id":"n17321","layer":"informal","project":"p11","title":"Lemma~\\reflem:limitProcess_ae_eq with g := M_\\infty - N_\\infty.","kind":"proof","summary":"Lemma~\\reflem:limitProcess_ae_eq with g := M_\\infty - N_\\infty.","labels":[],"detail_key":"p11"},{"id":"n17322","layer":"informal","project":"p11","title":"lem:condExp_limitProcess","kind":"lemma","summary":"For M a square integrable martingale and t \\in T, we have that P[X_\\infty | F_t] is a.e. equal…","labels":["lem:condExp_limitProcess"],"detail_key":"p11"},{"id":"n17323","layer":"informal","project":"p11","title":"Apply Theorem~\\refthm:condExp_limitProcess.","kind":"proof","summary":"Apply Theorem~\\refthm:condExp_limitProcess.","labels":[],"detail_key":"p11"},{"id":"n17324","layer":"informal","project":"p11","title":"lem:condExp_limitProcess_stopped","kind":"lemma","summary":"For M a square integrable martingale and \\tau a stopping time, we have that P[X_\\infty | F_\\tau…","labels":["lem:condExp_limitProcess_stopped"],"detail_key":"p11"},{"id":"n17325","layer":"informal","project":"p11","title":"Apply Theorem~\\refthm:condExp_limitProcess_stopped.","kind":"proof","summary":"Apply Theorem~\\refthm:condExp_limitProcess_stopped.","labels":[],"detail_key":"p11"},{"id":"n17326","layer":"informal","project":"p11","title":"lem:iSup_eLpNorm_le_eLpNorm_limitProcess","kind":"lemma","summary":"For M a càdlàg and uniformly integrable martingale, \\sup_t \\in T \\Vert M_t \\Vert_2 \\le \\Vert M_…","labels":["lem:iSup_eLpNorm_le_eLpNorm_limitProcess"],"detail_key":"p11"},{"id":"n17327","layer":"informal","project":"p11","title":"By Lemma~\\reflem:condExp_limitProcess, We have that P[M_\\infty | F_t] = M t almost surely…","kind":"proof","summary":"By Lemma~\\reflem:condExp_limitProcess, We have that P[M_\\infty | F_t] = M t almost surely, thus…","labels":[],"detail_key":"p11"},{"id":"n17328","layer":"informal","project":"p11","title":"lem:isSquareIntegrable_of_limitProcess","kind":"lemma","summary":"A càdlàg and uniformly integrable martingale M is square integrable if and only if M_\\infty is…","labels":["lem:isSquareIntegrable_of_limitProcess"],"detail_key":"p11"},{"id":"n17329","layer":"informal","project":"p11","title":"It is a càdlàg martingale by hypothesis, and the bound follows from Lemma~\\reflem:IsSquar…","kind":"proof","summary":"It is a càdlàg martingale by hypothesis, and the bound follows from Lemma~\\reflem:IsSquareInteg…","labels":[],"detail_key":"p11"},{"id":"n17330","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.stoppedProcess","kind":"lemma","summary":"If M is a square integrable martingale and \\tau is a stopping time, then M^\\tau is a square int…","labels":["lem:IsSquareIntegrable.stoppedProcess"],"detail_key":"p11"},{"id":"n17331","layer":"informal","project":"p11","title":"We apply Lemma~\\reflem:isSquareIntegrable_of_limitProcess. First, M^\\tau is a martingale.…","kind":"proof","summary":"We apply Lemma~\\reflem:isSquareIntegrable_of_limitProcess. First, M^\\tau is a martingale. Moreo…","labels":[],"detail_key":"p11"},{"id":"n17332","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.sup_eLpNorm_eq_eLpNorm_limitProcess","kind":"lemma","summary":"For M a square integrable martingale, \\sup_t \\in T \\Vert M_t \\Vert_L^2 &= \\Vert M_\\infty \\Vert_…","labels":["lem:IsSquareIntegrable.sup_eLpNorm_eq_eLpNorm_limitProcess"],"detail_key":"p11"},{"id":"n17333","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17334","layer":"informal","project":"p11","title":"lem:tendsto_ae_condExp","kind":"lemma","summary":"For f : \\Omega \\to E, we have almost surely that \\lim_t \\to +\\infty P[f | F_t] = P\\left[f | \\bi…","labels":["lem:tendsto_ae_condExp"],"detail_key":"p11"},{"id":"n17335","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17336","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.integral_iSup_norm_rpow_rpow_inv_le_limitProcess","kind":"lemma","summary":"If M is a square integrable random variable, then \\left(\\int (\\sup_t \\|X_t(\\omega)\\|)^2\\right)^…","labels":["lem:IsSquareIntegrable.integral_iSup_norm_rpow_rpow_inv_le_limitProcess"],"detail_key":"p11"},{"id":"n17337","layer":"informal","project":"p11","title":"Combine Corollary~\\refcor:doob_lp_norm_top, the fact that suprema commute with power, and…","kind":"proof","summary":"Combine Corollary~\\refcor:doob_lp_norm_top, the fact that suprema commute with power, and Lemma…","labels":[],"detail_key":"p11"},{"id":"n17338","layer":"informal","project":"p11","title":"lem:memLp_two_stoppedValue","kind":"lemma","summary":"If M is a square integrable martingale and \\tau is a stopping time, then M_\\tau is in L^2.","labels":["lem:memLp_two_stoppedValue"],"detail_key":"p11"},{"id":"n17339","layer":"informal","project":"p11","title":"This follows from the fact that M_\\tau = M^\\tau_\\infty by Lemma~\\reflem:limitProcess_stop…","kind":"proof","summary":"This follows from the fact that M_\\tau = M^\\tau_\\infty by Lemma~\\reflem:limitProcess_stoppedPro…","labels":[],"detail_key":"p11"},{"id":"n17340","layer":"informal","project":"p11","title":"def:IsPurelyDiscontinuous","kind":"definition","summary":"A stochastic process X is purely discontinuous if it is a square integrable martingale such tha…","labels":["def:IsPurelyDiscontinuous"],"detail_key":"p11"},{"id":"n17341","layer":"informal","project":"p11","title":"def:L2Martingales","kind":"definition","summary":"We denote by M^2(E) or simply M^2 the space of equivalence classes with respect to indistinguis…","labels":["def:L2Martingales"],"detail_key":"p11"},{"id":"n17342","layer":"informal","project":"p11","title":"lem:L2Martingales.module","kind":"lemma","summary":"The space M^2(E) is a real vector space.","labels":["lem:L2Martingales.module"],"detail_key":"p11"},{"id":"n17343","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17344","layer":"informal","project":"p11","title":"def:L2Martingales.norm","kind":"definition","summary":"We define a norm on M^2 by \\Vert M \\Vert = \\Vert M_\\infty \\Vert_L^2 \\: .","labels":["def:L2Martingales.norm"],"detail_key":"p11"},{"id":"n17345","layer":"informal","project":"p11","title":"lem:L2Martingales.norm_eq_zero","kind":"lemma","summary":"For M \\in M^2(E), \\Vert M \\Vert = 0 if and only if M = 0.","labels":["lem:L2Martingales.norm_eq_zero"],"detail_key":"p11"},{"id":"n17346","layer":"informal","project":"p11","title":"By Lemma~\\reflem:IsSquareIntegrable.sup_eLpNorm_eq_eLpNorm_limitProcess, \\Vert M \\Vert =…","kind":"proof","summary":"By Lemma~\\reflem:IsSquareIntegrable.sup_eLpNorm_eq_eLpNorm_limitProcess, \\Vert M \\Vert = 0 if a…","labels":[],"detail_key":"p11"},{"id":"n17347","layer":"informal","project":"p11","title":"def:L2Martingales.inner","kind":"definition","summary":"We define an inner product on M^2 by \\langle M, N \\rangle_M^2 = E[M_\\infty N_\\infty] \\: .","labels":["def:L2Martingales.inner"],"detail_key":"p11"},{"id":"n17348","layer":"informal","project":"p11","title":"thm:hilbertSpace_L2Martingales","kind":"theorem","summary":"If T is separable then the space M^2 is a Hilbert space.","labels":["thm:hilbertSpace_L2Martingales"],"detail_key":"p11"},{"id":"n17349","layer":"informal","project":"p11","title":"We show that the map \\phi : M^2 & \\to \\left\\f \\in L^2(E, P) | f \\text is \\left(\\bigsqcup_…","kind":"proof","summary":"We show that the map \\phi : M^2 & \\to \\left\\f \\in L^2(E, P) | f \\text is \\left(\\bigsqcup_t, F_t…","labels":[],"detail_key":"p11"},{"id":"n17350","layer":"informal","project":"p11","title":"lem:eLpNorm_elemStochIntegralBilin_le","kind":"lemma","summary":"For V \\in E_T, F bounded by a constant D, M \\in M^2(E) and a continuous bilinear map B: E \\time…","labels":["lem:eLpNorm_elemStochIntegralBilin_le"],"detail_key":"p11"},{"id":"n17351","layer":"informal","project":"p11","title":"Let C be a bound on \\Vert M_t \\Vert_L^2 for all t \\in T~. Let (s_k < t_k)_k \\in \\1, ...,…","kind":"proof","summary":"Let C be a bound on \\Vert M_t \\Vert_L^2 for all t \\in T~. Let (s_k < t_k)_k \\in \\1, ..., n\\ and…","labels":[],"detail_key":"p11"},{"id":"n17352","layer":"informal","project":"p11","title":"lem:isSquareIntegrable_elemStochIntegralBilin","kind":"lemma","summary":"For V \\in E_T, F, M \\in M^2(E) and a continuous bilinear map B: E \\times F \\to G, the elementar…","labels":["lem:isSquareIntegrable_elemStochIntegralBilin"],"detail_key":"p11"},{"id":"n17353","layer":"informal","project":"p11","title":"By Lemma~\\reflem:cadlag_elemStochIntegralBilin, V \\bullet_B M is càdlàg, and we know that…","kind":"proof","summary":"By Lemma~\\reflem:cadlag_elemStochIntegralBilin, V \\bullet_B M is càdlàg, and we know that it is…","labels":[],"detail_key":"p11"},{"id":"n17354","layer":"informal","project":"p11","title":"def:continuousSquareIntegrable","kind":"definition","summary":"We denote by M^2,c the submodule of M^2 made of continuous square integrable martingales.","labels":["def:continuousSquareIntegrable"],"detail_key":"p11"},{"id":"n17355","layer":"informal","project":"p11","title":"lem:exists_subsequence_ae_tendsto_uniformly","kind":"lemma","summary":"Let (M^(n)) be a sequence of square integrable martingales and N be a square integrable marting…","labels":["lem:exists_subsequence_ae_tendsto_uniformly"],"detail_key":"p11"},{"id":"n17356","layer":"informal","project":"p11","title":"We can find \\phi increasing such that \\|M^(\\phi(n))_\\infty - N_\\infty\\|_2 \\le 1/2^n. Then…","kind":"proof","summary":"We can find \\phi increasing such that \\|M^(\\phi(n))_\\infty - N_\\infty\\|_2 \\le 1/2^n. Then, P[\\s…","labels":[],"detail_key":"p11"},{"id":"n17357","layer":"informal","project":"p11","title":"lem:isClosed_continuousSquareIntegrable","kind":"lemma","summary":"The submodule M^2,c is closed in the Hilbert space M^2.","labels":["lem:isClosed_continuousSquareIntegrable"],"detail_key":"p11"},{"id":"n17358","layer":"informal","project":"p11","title":"Take M^(n)","kind":"proof","summary":"Take M^(n)","labels":[],"detail_key":"p11"},{"id":"n17359","layer":"informal","project":"p11","title":"def:continuousPartSquareIntegrable","kind":"definition","summary":"If M is a square integrable martingale, we define its \\emphcontinuous part as the orthogonal pr…","labels":["def:continuousPartSquareIntegrable"],"detail_key":"p11"},{"id":"n17360","layer":"informal","project":"p11","title":"lem:continuous_continuousPart","kind":"lemma","summary":"The continuous part of a square integrable martingale is continuous.","labels":["lem:continuous_continuousPart"],"detail_key":"p11"},{"id":"n17361","layer":"informal","project":"p11","title":"By definition of the orthogonal projection.","kind":"proof","summary":"By definition of the orthogonal projection.","labels":[],"detail_key":"p11"},{"id":"n17362","layer":"informal","project":"p11","title":"lem:isSquareIntegrable_continuousPart","kind":"lemma","summary":"The continuous part of a square integrable martingale is a square integrable martingale.","labels":["lem:isSquareIntegrable_continuousPart"],"detail_key":"p11"},{"id":"n17363","layer":"informal","project":"p11","title":"By definition of the orthogonal projection.","kind":"proof","summary":"By definition of the orthogonal projection.","labels":[],"detail_key":"p11"},{"id":"n17364","layer":"informal","project":"p11","title":"def:discontinuousPart","kind":"definition","summary":"If M is a square integrable martingale, we define its \\emphdiscontinuous part by M^d := M - M^c.","labels":["def:discontinuousPart"],"detail_key":"p11"},{"id":"n17365","layer":"informal","project":"p11","title":"lem:isSquareIntegrable_discontinuousPart","kind":"lemma","summary":"The discontinuous part of a square integrable martingale is a square integrable martingale.","labels":["lem:isSquareIntegrable_discontinuousPart"],"detail_key":"p11"},{"id":"n17366","layer":"informal","project":"p11","title":"By definition.","kind":"proof","summary":"By definition.","labels":[],"detail_key":"p11"},{"id":"n17367","layer":"informal","project":"p11","title":"def:discontinuousSquareIntegrable","kind":"definition","summary":"We denote by M^2,d the orthogonal of M^2,c and call its elements purely discontinuous square in…","labels":["def:discontinuousSquareIntegrable"],"detail_key":"p11"},{"id":"n17368","layer":"informal","project":"p11","title":"lem:mem_discontinuousSquareIntegrable","kind":"lemma","summary":"If M is a purely discontinuous martingale in the sense of Definition~\\refdef:IsPurelyDiscontinu…","labels":["lem:mem_discontinuousSquareIntegrable"],"detail_key":"p11"},{"id":"n17369","layer":"informal","project":"p11","title":"By definition of the orthogonal projection.","kind":"proof","summary":"By definition of the orthogonal projection.","labels":[],"detail_key":"p11"},{"id":"n17370","layer":"informal","project":"p11","title":"lem:isPurelyDiscontinuous_discontinuousPart","kind":"lemma","summary":"The discontinuous part of square integrable martingale is purely discontinuous.","labels":["lem:isPurelyDiscontinuous_discontinuousPart"],"detail_key":"p11"},{"id":"n17371","layer":"informal","project":"p11","title":"By definition of the orthogonal projection.","kind":"proof","summary":"By definition of the orthogonal projection.","labels":[],"detail_key":"p11"},{"id":"n17372","layer":"informal","project":"p11","title":"lem:unique_continuousPart","kind":"lemma","summary":"If M is a square integrable martingale, then the decomposition M = M^c + M^d as the sum of a co…","labels":["lem:unique_continuousPart"],"detail_key":"p11"},{"id":"n17373","layer":"informal","project":"p11","title":"A closed subspace is the complement of its orthogonal subspace.","kind":"proof","summary":"A closed subspace is the complement of its orthogonal subspace.","labels":[],"detail_key":"p11"},{"id":"n17374","layer":"informal","project":"p11","title":"lem:IsPurelyDiscontinuous.stoppedProcess","kind":"lemma","summary":"If M is a purely discontinuous martingale and \\tau is a stopping time, then M^\\tau is a purely…","labels":["lem:IsPurelyDiscontinuous.stoppedProcess"],"detail_key":"p11"},{"id":"n17375","layer":"informal","project":"p11","title":"By Lemma~\\reflem:IsSquareIntegrable.stoppedProcess M^\\tau is a square integrable martinga…","kind":"proof","summary":"By Lemma~\\reflem:IsSquareIntegrable.stoppedProcess M^\\tau is a square integrable martingale. Le…","labels":[],"detail_key":"p11"},{"id":"n17376","layer":"informal","project":"p11","title":"lem:continuousPart_stoppedProcess","kind":"lemma","summary":"If M is a square integrable martingale and \\tau is a stopping time, then (M^\\tau)^c = (M^c)^\\ta…","labels":["lem:continuousPart_stoppedProcess"],"detail_key":"p11"},{"id":"n17377","layer":"informal","project":"p11","title":"We can write M^\\tau = (M^c + M^d)^\\tau = (M^c)^\\tau + (M^d)^\\tau. We conclude by Lemma~\\r…","kind":"proof","summary":"We can write M^\\tau = (M^c + M^d)^\\tau = (M^c)^\\tau + (M^d)^\\tau. We conclude by Lemma~\\reflem:…","labels":[],"detail_key":"p11"},{"id":"n17378","layer":"informal","project":"p11","title":"lem:inner_elemStochIntegral","kind":"lemma","summary":"For V \\in E_T, R and M, N \\in M^2, we have \\langle V \\bullet_R M, N \\rangle_M^2 &= V \\bullet_R…","labels":["lem:inner_elemStochIntegral"],"detail_key":"p11"},{"id":"n17379","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17380","layer":"informal","project":"p11","title":"Locally square-integrable martingales","kind":"definition","summary":"[Locally square-integrable martingales] A process is locally square-integrable if it locally sa…","labels":["def:IsLocallySquareIntegrable"],"detail_key":"p11"},{"id":"n17381","layer":"informal","project":"p11","title":"lem:IsSquareIntegrable.isLocallySquareIntegrable","kind":"lemma","summary":"Every square-integrable martingale is locally square-integrable: M^2 \\subseteq M^2_loc~.","labels":["lem:IsSquareIntegrable.isLocallySquareIntegrable"],"detail_key":"p11"},{"id":"n17382","layer":"informal","project":"p11","title":"This follows from Lemma~\\reflem:implies_locally.","kind":"proof","summary":"This follows from Lemma~\\reflem:implies_locally.","labels":[],"detail_key":"p11"},{"id":"n17383","layer":"informal","project":"p11","title":"lem:IsLocallySquareIntegrable.isLocalSubmartingale_sq_norm","kind":"lemma","summary":"If M \\in M^2_loc, then \\Vert M \\Vert^2 is a càdlàg local submartingale.","labels":["lem:IsLocallySquareIntegrable.isLocalSubmartingale_sq_norm"],"detail_key":"p11"},{"id":"n17384","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17385","layer":"informal","project":"p11","title":"lem:IsLocalMartingale.isLocallySquareIntegrable_of_continuous","kind":"lemma","summary":"A continuous local martingale is locally square-integrable: M^c_loc \\subseteq M^2_loc~.","labels":["lem:IsLocalMartingale.isLocallySquareIntegrable_of_continuous"],"detail_key":"p11"},{"id":"n17386","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17387","layer":"informal","project":"p11","title":"Predictable quadratic variation","kind":"definition","summary":"[Predictable quadratic variation] For M \\in M^2_loc with càdlàg paths, the predictable quadrati…","labels":["def:quadraticVariation"],"detail_key":"p11"},{"id":"n17388","layer":"informal","project":"p11","title":"lem:predictable_quadraticVariation","kind":"lemma","summary":"The predictable quadratic variation \\langle M \\rangle of M \\in M^2_loc is a predictable process.","labels":["lem:predictable_quadraticVariation"],"detail_key":"p11"},{"id":"n17389","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17390","layer":"informal","project":"p11","title":"lem:cadlag_quadraticVariation","kind":"lemma","summary":"The predictable quadratic variation \\langle M \\rangle of M \\in M^2_loc is càdlàg.","labels":["lem:cadlag_quadraticVariation"],"detail_key":"p11"},{"id":"n17391","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17392","layer":"informal","project":"p11","title":"lem:locallyIntegrable_quadraticVariation","kind":"lemma","summary":"The predictable quadratic variation \\langle M \\rangle of M \\in M^2_loc is locally integrable.","labels":["lem:locallyIntegrable_quadraticVariation"],"detail_key":"p11"},{"id":"n17393","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17394","layer":"informal","project":"p11","title":"lem:quadraticVariation_zero","kind":"lemma","summary":"\\langle M \\rangle_0 = 0~.","labels":["lem:quadraticVariation_zero"],"detail_key":"p11"},{"id":"n17395","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17396","layer":"informal","project":"p11","title":"lem:monotone_quadraticVariation","kind":"lemma","summary":"The predictable quadratic variation \\langle M \\rangle of M \\in M^2_loc is non-decreasing.","labels":["lem:monotone_quadraticVariation"],"detail_key":"p11"},{"id":"n17397","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17398","layer":"informal","project":"p11","title":"lem:local_martingale_sub_quadraticVariation","kind":"lemma","summary":"For M \\in M^2_loc, the process \\Vert M_t \\Vert^2 - \\langle M \\rangle_t is a local martingale.","labels":["lem:local_martingale_sub_quadraticVariation"],"detail_key":"p11"},{"id":"n17399","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17400","layer":"informal","project":"p11","title":"Predictable covariation","kind":"definition","summary":"[Predictable covariation] For M, N \\in M^2_loc, the predictable covariation \\langle M, N \\rangl…","labels":["def:covariation"],"detail_key":"p11"},{"id":"n17401","layer":"informal","project":"p11","title":"lem:predictable_covariation","kind":"lemma","summary":"The predictable covariation \\langle M, N \\rangle of M, N \\in M^2_loc is a predictable process.","labels":["lem:predictable_covariation"],"detail_key":"p11"},{"id":"n17402","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17403","layer":"informal","project":"p11","title":"lem:cadlag_covariation","kind":"lemma","summary":"The predictable covariation \\langle M, N \\rangle of M, N \\in M^2_loc is càdlàg.","labels":["lem:cadlag_covariation"],"detail_key":"p11"},{"id":"n17404","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17405","layer":"informal","project":"p11","title":"lem:covariation_zero","kind":"lemma","summary":"\\langle M, N \\rangle_0 = 0~.","labels":["lem:covariation_zero"],"detail_key":"p11"},{"id":"n17406","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17407","layer":"informal","project":"p11","title":"lem:local_martingale_sub_covariation","kind":"lemma","summary":"For M, N \\in M^2_loc, the process \\langle M_t, N_t \\rangle_E - \\langle M, N \\rangle_t is a loca…","labels":["lem:local_martingale_sub_covariation"],"detail_key":"p11"},{"id":"n17408","layer":"informal","project":"p11","title":"&\\langle M_t, N_t \\rangle_E - \\langle M, N \\rangle_t \\\\ &= \\frac14\\left( \\left(\\Vert M_t…","kind":"proof","summary":"&\\langle M_t, N_t \\rangle_E - \\langle M, N \\rangle_t \\\\ &= \\frac14\\left( \\left(\\Vert M_t + N_t…","labels":[],"detail_key":"p11"},{"id":"n17409","layer":"informal","project":"p11","title":"lem:covariation_eq_inner","kind":"lemma","summary":"Let M and N be square integrable martingales. Then E\\left[\\langle M,N \\rangle_\\infty\\right] = \\…","labels":["lem:covariation_eq_inner"],"detail_key":"p11"},{"id":"n17410","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17411","layer":"informal","project":"p11","title":"lem:quadraticVariation_brownian","kind":"lemma","summary":"Let B be a standard Brownian motion. Then the quadratic variation of B is given by \\langle B \\r…","labels":["lem:quadraticVariation_brownian"],"detail_key":"p11"},{"id":"n17412","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17413","layer":"informal","project":"p11","title":"lem:IsLocalMartingale.locally_hasIntegrableVariation_largeJumps","kind":"lemma","summary":"The large jump process of a local martingale is a process with locally integrable variation (it…","labels":["lem:IsLocalMartingale.locally_hasIntegrableVariation_largeJumps"],"detail_key":"p11"},{"id":"n17414","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17415","layer":"informal","project":"p11","title":"thm:local_martingale_decomposition","kind":"theorem","summary":"Let M be a local martingale. Then for any \\varepsilon > 0, M can be decomposed as M = M_0 + U +…","labels":["thm:local_martingale_decomposition"],"detail_key":"p11"},{"id":"n17416","layer":"informal","project":"p11","title":"See \\citehe2019semimartingale, 7.17","kind":"proof","summary":"See \\citehe2019semimartingale, 7.17","labels":[],"detail_key":"p11"},{"id":"n17417","layer":"informal","project":"p11","title":"Semimartingale","kind":"definition","summary":"[Semimartingale] A process X is a semimartingale if it can be decomposed as X = M + A, where M…","labels":["def:IsSemimartingale"],"detail_key":"p11"},{"id":"n17418","layer":"informal","project":"p11","title":"lem:IsSemimartingale.decomposition","kind":"lemma","summary":"A semimartingale X can be decomposed as X = M + A, where M is a locally bounded martingale with…","labels":["lem:IsSemimartingale.decomposition"],"detail_key":"p11"},{"id":"n17419","layer":"informal","project":"p11","title":"Decompose X = M + A as in Definition~\\refdef:IsSemimartingale. Then decompose M as in The…","kind":"proof","summary":"Decompose X = M + A as in Definition~\\refdef:IsSemimartingale. Then decompose M as in Theorem~\\…","labels":[],"detail_key":"p11"},{"id":"n17420","layer":"informal","project":"p11","title":"Continuous martingale part","kind":"definition","summary":"[Continuous martingale part] TODO. Denoted by X^c.","labels":["def:IsSemimartingale.continuousMartingalePart"],"detail_key":"p11"},{"id":"n17421","layer":"informal","project":"p11","title":"Quadratic covariation of semimartingales","kind":"definition","summary":"[Quadratic covariation of semimartingales] Let X and Y be semimartingales. Their quadratic cova…","labels":["def:quadCovariation"],"detail_key":"p11"},{"id":"n17422","layer":"informal","project":"p11","title":"Quadratic variation of a semimartingale","kind":"definition","summary":"[Quadratic variation of a semimartingale] The quadratic variation of a semimartingale X is defi…","labels":["def:quadVariation"],"detail_key":"p11"},{"id":"n17423","layer":"informal","project":"p11","title":"Special semimartingale","kind":"definition","summary":"[Special semimartingale] A semimartingale X is a special semimartingale if it can be decomposed…","labels":["def:IsSpecialSemimartingale"],"detail_key":"p11"},{"id":"n17424","layer":"informal","project":"p11","title":"lem:IsSpecialSemimartingale.decomposition","kind":"theorem","summary":"A special semimartingale X can be decomposed as X = M + A, where M is a local martingale and A…","labels":["lem:IsSpecialSemimartingale.decomposition"],"detail_key":"p11"},{"id":"n17425","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17426","layer":"informal","project":"p11","title":"Additive content","kind":"definition","summary":"[Additive content] \\mathlibok Let E be a commutative monoid and A be a family of sets. An \\emph…","labels":["def:addContent"],"detail_key":"p11"},{"id":"n17427","layer":"informal","project":"p11","title":"lem:addContent_extension","kind":"lemma","summary":"\\mathlibok Let \\lambda be an additive content on a semi-ring of sets C~. Then \\lambda admits a…","labels":["lem:addContent_extension"],"detail_key":"p11"},{"id":"n17428","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17429","layer":"informal","project":"p11","title":"Content associated with a process","kind":"definition","summary":"[Content associated with a process] Let X : T \\to \\Omega \\to E, where E is a Banach space, be a…","labels":["def:contentAssociatedWithProcess"],"detail_key":"p11"},{"id":"n17430","layer":"informal","project":"p11","title":"lem:contentAssociatedWithProcess_wellDefined","kind":"lemma","summary":"The content \\lambda_X associated with X is uniquely defined.","labels":["lem:contentAssociatedWithProcess_wellDefined"],"detail_key":"p11"},{"id":"n17431","layer":"informal","project":"p11","title":"The uniqueness of \\lambda_X is guaranteed by Lemma~\\reflem:addContent_extension, since th…","kind":"proof","summary":"The uniqueness of \\lambda_X is guaranteed by Lemma~\\reflem:addContent_extension, since the pred…","labels":[],"detail_key":"p11"},{"id":"n17432","layer":"informal","project":"p11","title":"Partition of a set","kind":"definition","summary":"[Partition of a set] \\mathlibok An \\emphpartition of a set A is a family of pairwise disjoint s…","labels":["def:partitionOfSet"],"detail_key":"p11"},{"id":"n17433","layer":"informal","project":"p11","title":"Bounded variation of a content","kind":"definition","summary":"[Bounded variation of a content] An additive content \\lambda on a ring A of sets, with values i…","labels":["def:contentBoundedVariation"],"detail_key":"p11"},{"id":"n17434","layer":"informal","project":"p11","title":"lem:contentBoundedVariation_equivOnGenerator","kind":"lemma","summary":"We can equivalently take supremum over C-partitions of A in the above definition, when A is gen…","labels":["lem:contentBoundedVariation_equivOnGenerator"],"detail_key":"p11"},{"id":"n17435","layer":"informal","project":"p11","title":"This follows (by triangle inequality and additivity of the content) from the Definition~\\…","kind":"proof","summary":"This follows (by triangle inequality and additivity of the content) from the Definition~\\refdef…","labels":[],"detail_key":"p11"},{"id":"n17436","layer":"informal","project":"p11","title":"lem:contentEquivElementaryIntegral","kind":"lemma","summary":"The content \\lambda_X associated with a process X satisfies, for every elementary predictable s…","labels":["lem:contentEquivElementaryIntegral"],"detail_key":"p11"},{"id":"n17437","layer":"informal","project":"p11","title":"By Lemma~\\reflem:elementaryPredictableSet_iff_indicator, the process 1_A is a simple one,…","kind":"proof","summary":"By Lemma~\\reflem:elementaryPredictableSet_iff_indicator, the process 1_A is a simple one, (1_A)…","labels":[],"detail_key":"p11"},{"id":"n17438","layer":"informal","project":"p11","title":"Quasi-martingale","kind":"definition","summary":"[Quasi-martingale] Let E be a Banach space. An adapted integrable process X : T \\to \\Omega \\to…","labels":["def:Quasimartingale"],"detail_key":"p11"},{"id":"n17439","layer":"informal","project":"p11","title":"lem:realContentBoundedVariation_equivalent_definition","kind":"lemma","summary":"The content \\lambda_X associated with a real-valued process X has bounded variation on (0, t] \\…","labels":["lem:realContentBoundedVariation_equivalent_definition"],"detail_key":"p11"},{"id":"n17440","layer":"informal","project":"p11","title":"The above boundedness condition, \\[ \\sup \\Big\\ |E[(1_A \\bullet X)_t]| \\:\\Big|\\: A \\text e…","kind":"proof","summary":"The above boundedness condition, \\[ \\sup \\Big\\ |E[(1_A \\bullet X)_t]| \\:\\Big|\\: A \\text element…","labels":[],"detail_key":"p11"},{"id":"n17441","layer":"informal","project":"p11","title":"def:StieltjesFunction.measure","kind":"definition","summary":"\\mathlibok Let f : T \\to R be a right-continuous monotone function on a conditionally complete…","labels":["def:StieltjesFunction.measure"],"detail_key":"p11"},{"id":"n17442","layer":"informal","project":"p11","title":"lem:StieltjesFunction.measurable_measure","kind":"lemma","summary":"Let X : T \\to \\Omega \\to R be a right-continuous adapted process which is monotone in the time…","labels":["lem:StieltjesFunction.measurable_measure"],"detail_key":"p11"},{"id":"n17443","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17444","layer":"informal","project":"p11","title":"def:BoundedVariationOn.vectorMeasure","kind":"definition","summary":"\\mathlibok Let f : T \\to E be a function of bounded variation on a suitable order T, with E a c…","labels":["def:BoundedVariationOn.vectorMeasure"],"detail_key":"p11"},{"id":"n17445","layer":"informal","project":"p11","title":"lem:BoundedVariationOn.measurable_vectorMeasure","kind":"lemma","summary":"Let X : T \\to \\Omega \\to E be an adapted process which is of bounded variation in the time vari…","labels":["lem:BoundedVariationOn.measurable_vectorMeasure"],"detail_key":"p11"},{"id":"n17446","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17447","layer":"informal","project":"p11","title":"lem:eVariationOn_eq_totalVariation","kind":"lemma","summary":"Denoting by \\vert df \\vert the total variation of the measure df~, V_f([a, b]) = \\vert df \\vert…","labels":["lem:eVariationOn_eq_totalVariation"],"detail_key":"p11"},{"id":"n17448","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17449","layer":"informal","project":"p11","title":"lem:BoundedVariationOn.vectorMeasure_of_monotone","kind":"lemma","summary":"If f : T \\to R is a right-continuous monotone function, then the measure df coincides with the…","labels":["lem:BoundedVariationOn.vectorMeasure_of_monotone"],"detail_key":"p11"},{"id":"n17450","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17451","layer":"informal","project":"p11","title":"lem:Filtration.predictable_le_prod","kind":"lemma","summary":"The predictable \\sigma-algebra on T \\times \\Omega is a sub-\\sigma-algebra of the product \\sigma…","labels":["lem:Filtration.predictable_le_prod"],"detail_key":"p11"},{"id":"n17452","layer":"informal","project":"p11","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p11"},{"id":"n17453","layer":"informal","project":"p11","title":"L2 space of predictable processes","kind":"definition","summary":"[L2 space of predictable processes] Let \\mu be a measure on T and P a measure on \\Omega. We den…","labels":["def:L2Predictable"],"detail_key":"p11"},{"id":"n17454","layer":"informal","project":"p11","title":"lem:L2Predictable.inner_eq","kind":"lemma","summary":"For X, Y \\in L^2(\\mu, P), we have \\langle X, Y \\rangle_L^2(\\mu, P) = P\\left[ \\int_0^\\infty \\lan…","labels":["lem:L2Predictable.inner_eq"],"detail_key":"p11"},{"id":"n17455","layer":"informal","project":"p11","title":"The inner product in L^2 spaces is defined as the integral of the pointwise inner product…","kind":"proof","summary":"The inner product in L^2 spaces is defined as the integral of the pointwise inner products. \\la…","labels":[],"detail_key":"p11"},{"id":"n17456","layer":"informal","project":"p11","title":"lem:simpleProcess_mem_L2Predictable","kind":"lemma","summary":"Any simple process in E_T, E is in L^2(\\mu, P)~. (Lean remark: we mean that the uncurried versi…","labels":["lem:simpleProcess_mem_L2Predictable"],"detail_key":"p11"},{"id":"n17457","layer":"informal","project":"p11","title":"A simple process is bounded by definition.","kind":"proof","summary":"A simple process is bounded by definition.","labels":[],"detail_key":"p11"},{"id":"n17458","layer":"informal","project":"p11","title":"lem:L2Predictable.sq_norm_simpleProcess","kind":"lemma","summary":"Let V \\in E_T, E and \\mu a measure on T~. Let f be a right-continuous non-decreasing function s…","labels":["lem:L2Predictable.sq_norm_simpleProcess"],"detail_key":"p11"},{"id":"n17459","layer":"informal","project":"p11","title":"TODO: this proof (and the result of the lemma) assumes that the intervals defining the si…","kind":"proof","summary":"TODO: this proof (and the result of the lemma) assumes that the intervals defining the simple p…","labels":[],"detail_key":"p11"},{"id":"n17460","layer":"informal","project":"p11","title":"L2 space with respect to a square integrable martingale","kind":"definition","summary":"[L2 space with respect to a square integrable martingale] Let M be a square integrable martinga…","labels":["def:L2M"],"detail_key":"p11"},{"id":"n17461","layer":"informal","project":"p11","title":"lem:sq_norm_elemStochIntegral","kind":"lemma","summary":"For V \\in E and M \\in M^2, then V \\bullet M \\in M^2 (by Lemma~\\reflem:isSquareIntegrable_elemSt…","labels":["lem:sq_norm_elemStochIntegral"],"detail_key":"p11"},{"id":"n17462","layer":"informal","project":"p11","title":"There are two steps to the proof. First, in order to make sense of \\Vert V \\Vert_L^2(M),…","kind":"proof","summary":"There are two steps to the proof. First, in order to make sense of \\Vert V \\Vert_L^2(M), we def…","labels":[],"detail_key":"p11"},{"id":"n17463","layer":"informal","project":"p11","title":"lem:integral_process_eq_zero","kind":"lemma","summary":"Let X\\in L^2(M) such that \\int_0^t X_s \\: d\\langle M \\rangle_s = 0 for all t \\ge 0 a.s.. 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Type u_2 E : Type u_3 mΩ : MeasurableSpace Ω P : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.Filtration.limitProcess_congr","module":"BrownianMotion.Auxiliary.LimitProcess"},{"id":"n17487","layer":"formal","project":"p11","title":"MeasureTheory.Filtration.limitProcess_neg","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 mΩ : MeasurableSpace Ω P : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.Filtration.limitProcess_neg","module":"BrownianMotion.Auxiliary.LimitProcess"},{"id":"n17488","layer":"formal","project":"p11","title":"MeasureTheory.Filtration.limitProcess_smul","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 mΩ : MeasurableSpace Ω P : MeasureTheory.Measure Ω 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Ω)","labels":[],"detail_key":"p11","name":"ProbabilityTheory.stochIco","module":"BrownianMotion.StochasticIntegral.StochasticInterval"},{"id":"n17769","layer":"formal","project":"p11","title":"ProbabilityTheory.stochIoc","kind":"def","summary":"ι : Type u_1 → Ω : Type u_2 → [Preorder ι] → (Ω → WithTop ι) → (Ω → WithTop ι) → Set (Prod ι Ω)","labels":[],"detail_key":"p11","name":"ProbabilityTheory.stochIoc","module":"BrownianMotion.StochasticIntegral.StochasticInterval"},{"id":"n17770","layer":"formal","project":"p11","title":"ProbabilityTheory.stochIoc.exists_elementaryPredictableSet","kind":"theorem","summary":"∀ Ω : Type u_2 mΩ : MeasurableSpace Ω 𝓕 : MeasureTheory.Filtration Nat mΩ σ τ : Ω → ENat, Measu…","labels":[],"detail_key":"p11","name":"ProbabilityTheory.stochIoc.exists_elementaryPredictableSet","module":"BrownianMotion.StochasticIntegral.StochasticInterval"},{"id":"n17771","layer":"formal","project":"p11","title":"ProbabilityTheory.stochIoo","kind":"def","summary":"ι : Type u_1 → Ω : Type u_2 → [Preorder ι] → (Ω → WithTop ι) → (Ω → WithTop ι) → Set (Prod ι Ω)","labels":[],"detail_key":"p11","name":"ProbabilityTheory.stochIoo","module":"BrownianMotion.StochasticIntegral.StochasticInterval"},{"id":"n17772","layer":"formal","project":"p11","title":"MeasureTheory.Filtration.limitProcess_stoppedProcess","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 [inst : LinearOrder ι] mΩ : MeasurableSpace Ω P : Meas…","labels":[],"detail_key":"p11","name":"MeasureTheory.Filtration.limitProcess_stoppedProcess","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17773","layer":"formal","project":"p11","title":"MeasureTheory.Martingale.ae_tendsto_limitProcess","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 [inst : LinearOrder ι] mΩ : MeasurableSpace Ω P : Meas…","labels":[],"detail_key":"p11","name":"MeasureTheory.Martingale.ae_tendsto_limitProcess","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17774","layer":"formal","project":"p11","title":"MeasureTheory.Martingale.condExp_limitProcess_ae_eq","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 [inst : LinearOrder ι] mΩ : MeasurableSpace Ω P : Meas…","labels":[],"detail_key":"p11","name":"MeasureTheory.Martingale.condExp_limitProcess_ae_eq","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17775","layer":"formal","project":"p11","title":"MeasureTheory.Martingale.condExp_limitProcess_ae_eq'","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 [inst : LinearOrder ι] mΩ : MeasurableSpace Ω P : Meas…","labels":[],"detail_key":"p11","name":"MeasureTheory.Martingale.condExp_limitProcess_ae_eq'","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17776","layer":"formal","project":"p11","title":"MeasureTheory.Martingale.uniformIntegrable_stoppedValue","kind":"theorem","summary":"∀ Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω ι : Type u_6 [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.Martingale.uniformIntegrable_stoppedValue","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17777","layer":"formal","project":"p11","title":"MeasureTheory.Martingale.uniformIntegrable_stoppedValue_of_countable_range","kind":"theorem","summary":"∀ Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω ι : Type u_6 [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.Martingale.uniformIntegrable_stoppedValue_of_countable_range","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17778","layer":"formal","project":"p11","title":"MeasureTheory.Submartingale.uniformIntegrable_stoppedValue","kind":"theorem","summary":"∀ Ω : Type u_3 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω ι : Type u_6 [inst : LinearOr…","labels":[],"detail_key":"p11","name":"MeasureTheory.Submartingale.uniformIntegrable_stoppedValue","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17779","layer":"formal","project":"p11","title":"MeasureTheory.UniformIntegrable.add","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.UniformIntegrable.add","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17780","layer":"formal","project":"p11","title":"MeasureTheory.UniformIntegrable.comp","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω κ :…","labels":[],"detail_key":"p11","name":"MeasureTheory.UniformIntegrable.comp","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17781","layer":"formal","project":"p11","title":"MeasureTheory.UniformIntegrable.condExp'","kind":"theorem","summary":"∀ ι : Type u_1 κ : Type u_2 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.…","labels":[],"detail_key":"p11","name":"MeasureTheory.UniformIntegrable.condExp'","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17782","layer":"formal","project":"p11","title":"MeasureTheory.UniformIntegrable.norm","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.UniformIntegrable.norm","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17783","layer":"formal","project":"p11","title":"MeasureTheory.iSup_eLpNorm_le_eLpNorm_limitProcess","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_2 E : Type u_3 [inst : LinearOrder ι] mΩ : MeasurableSpace Ω P : Meas…","labels":[],"detail_key":"p11","name":"MeasureTheory.iSup_eLpNorm_le_eLpNorm_limitProcess","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17784","layer":"formal","project":"p11","title":"MeasureTheory.uniformIntegrable_iff_norm","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.uniformIntegrable_iff_norm","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17785","layer":"formal","project":"p11","title":"MeasureTheory.uniformIntegrable_of_dominated","kind":"theorem","summary":"∀ ι : Type u_1 κ : Type u_2 Ω : Type u_3 E : Type u_4 F : Type u_5 mΩ : MeasurableSpace Ω μ : M…","labels":[],"detail_key":"p11","name":"MeasureTheory.uniformIntegrable_of_dominated","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17786","layer":"formal","project":"p11","title":"MeasureTheory.uniformIntegrable_of_dominated_singleton","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω [in…","labels":[],"detail_key":"p11","name":"MeasureTheory.uniformIntegrable_of_dominated_singleton","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17787","layer":"formal","project":"p11","title":"MeasureTheory.uniformIntegrable_of_eLpNorm_le","kind":"theorem","summary":"∀ ι : Type u_1 Ω : Type u_3 E : Type u_4 mΩ : MeasurableSpace Ω μ : MeasureTheory.Measure Ω X :…","labels":[],"detail_key":"p11","name":"MeasureTheory.uniformIntegrable_of_eLpNorm_le","module":"BrownianMotion.StochasticIntegral.UniformIntegrable"},{"id":"n17788","layer":"informal","project":"p12","title":"Risk","kind":"definition","summary":"[Risk] The risk of an estimator \\haty on the estimation problem (P, y, \\ell') at \\theta \\in \\Th…","labels":["def:risk"],"detail_key":"p12"},{"id":"n17789","layer":"informal","project":"p12","title":"Bayesian risk","kind":"definition","summary":"[Bayesian risk] The Bayesian risk of an estimator \\haty on (P, y, \\ell') for a prior \\pi \\in M(…","labels":["def:bayesianRisk"],"detail_key":"p12"},{"id":"n17790","layer":"informal","project":"p12","title":"Bayes risk","kind":"definition","summary":"[Bayes risk] The Bayes risk of (P, y, \\ell') for prior \\pi \\in M(\\Theta) is R^P_\\pi = \\inf_\\hat…","labels":["def:bayesRisk"],"detail_key":"p12"},{"id":"n17791","layer":"informal","project":"p12","title":"Bayes estimator","kind":"definition","summary":"[Bayes estimator] An estimator \\haty is said to be a Bayes estimator for a prior \\pi \\in P(\\The…","labels":["def:bayesEstimator"],"detail_key":"p12"},{"id":"n17792","layer":"informal","project":"p12","title":"Minimax risk","kind":"definition","summary":"[Minimax risk] The minimax risk of (P, y, \\ell') is R^* = \\inf_\\haty : X \\rightsquigarrow Z \\su…","labels":["def:minimaxRisk"],"detail_key":"p12"},{"id":"n17793","layer":"informal","project":"p12","title":"lem:bayesRisk_le_minimaxRisk","kind":"lemma","summary":"R_B^* \\le R^*.","labels":["lem:bayesRisk_le_minimaxRisk"],"detail_key":"p12"},{"id":"n17794","layer":"informal","project":"p12","title":"For any \\pi \\in P( X) and any estimator, \\pi\\left[\\hat\\mu_\\theta\\left[\\ell'(y(\\theta), \\h…","kind":"proof","summary":"For any \\pi \\in P( X) and any estimator, \\pi\\left[\\hat\\mu_\\theta\\left[\\ell'(y(\\theta), \\haty(\\t…","labels":[],"detail_key":"p12"},{"id":"n17795","layer":"informal","project":"p12","title":"lem:bayesRisk_le_const","kind":"lemma","summary":"The Bayes risk of a prior \\pi \\in M(\\Theta) on (P, y, \\ell') with P a Markov kernel satisfies R…","labels":["lem:bayesRisk_le_const"],"detail_key":"p12"},{"id":"n17796","layer":"informal","project":"p12","title":"The infimum over all Markov kernels in the definition of the Bayes risk of \\pi is bounded…","kind":"proof","summary":"The infimum over all Markov kernels in the definition of the Bayes risk of \\pi is bounded from…","labels":[],"detail_key":"p12"},{"id":"n17797","layer":"informal","project":"p12","title":"lem:bayesRisk_const","kind":"lemma","summary":"The Bayes risk of a prior \\pi \\in M(\\Theta) on (P, y, \\ell') with P a constant Markov kernel is…","labels":["lem:bayesRisk_const"],"detail_key":"p12"},{"id":"n17798","layer":"informal","project":"p12","title":"Let \\xi be the measure such that P(\\theta) = \\xi for all \\theta. R^P_\\pi &= \\inf_\\haty(\\p…","kind":"proof","summary":"Let \\xi be the measure such that P(\\theta) = \\xi for all \\theta. R^P_\\pi &= \\inf_\\haty(\\pi \\tim…","labels":[],"detail_key":"p12"},{"id":"n17799","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"theorem","summary":"[Data-processing inequality] For P : \\Theta \\rightsquigarrow X and \\kappa : X \\rightsquigarrow…","labels":["thm:data_proc_bayesRisk"],"detail_key":"p12"},{"id":"n17800","layer":"informal","project":"p12","title":"The risk R^\\kappa \\circ P_\\pi is R^\\kappa \\circ P_\\pi &= \\inf_\\haty : X' \\rightsquigarrow…","kind":"proof","summary":"The risk R^\\kappa \\circ P_\\pi is R^\\kappa \\circ P_\\pi &= \\inf_\\haty : X' \\rightsquigarrow Z (\\p…","labels":[],"detail_key":"p12"},{"id":"n17801","layer":"informal","project":"p12","title":"lem:bayesRisk_compProd_le_fst","kind":"lemma","summary":"For P : \\Theta \\rightsquigarrow X and \\kappa : \\Theta \\times X \\rightsquigarrow X' a Markov ker…","labels":["lem:bayesRisk_compProd_le_fst"],"detail_key":"p12"},{"id":"n17802","layer":"informal","project":"p12","title":"Use Theorem~\\refthm:data_proc_bayesRisk: P is the composition of P \\otimes \\kappa and the…","kind":"proof","summary":"Use Theorem~\\refthm:data_proc_bayesRisk: P is the composition of P \\otimes \\kappa and the deter…","labels":[],"detail_key":"p12"},{"id":"n17803","layer":"informal","project":"p12","title":"lem:bayesRisk_compProd_le_snd","kind":"lemma","summary":"For P :","labels":["lem:bayesRisk_compProd_le_snd"],"detail_key":"p12"},{"id":"n17804","layer":"informal","project":"p12","title":"Use Theorem~\\refthm:data_proc_bayesRisk: (P \\otimes \\kappa)_ X' is the composition of P \\…","kind":"proof","summary":"Use Theorem~\\refthm:data_proc_bayesRisk: (P \\otimes \\kappa)_ X' is the composition of P \\otimes…","labels":[],"detail_key":"p12"},{"id":"n17805","layer":"informal","project":"p12","title":"lem:bayesRisk_concave","kind":"lemma","summary":"The Bayes risk R_\\pi^P is concave in P : \\Theta \\rightsquigarrow X~.","labels":["lem:bayesRisk_concave"],"detail_key":"p12"},{"id":"n17806","layer":"informal","project":"p12","title":"The infimum of a sum is larger than the sum of the infimums: R_\\pi^\\lambda P_1 + (1 - \\la…","kind":"proof","summary":"The infimum of a sum is larger than the sum of the infimums: R_\\pi^\\lambda P_1 + (1 - \\lambda)P…","labels":[],"detail_key":"p12"},{"id":"n17807","layer":"informal","project":"p12","title":"lem:bayesianRisk_bayesInv","kind":"lemma","summary":"The Bayesian risk of a Markov kernel \\haty : X \\rightsquigarrow Z with respect to a prior \\pi \\…","labels":["lem:bayesianRisk_bayesInv"],"detail_key":"p12"},{"id":"n17808","layer":"informal","project":"p12","title":"Use the main property of the Bayesian inverse. (P_\\pi^\\dagger \\times \\haty) \\circ P \\circ…","kind":"proof","summary":"Use the main property of the Bayesian inverse. (P_\\pi^\\dagger \\times \\haty) \\circ P \\circ \\pi &…","labels":[],"detail_key":"p12"},{"id":"n17809","layer":"informal","project":"p12","title":"lem:bayesianRisk_ge_inf_bayesInv","kind":"lemma","summary":"The Bayesian risk of a Markov kernel \\haty : X \\rightsquigarrow Z with respect to a prior \\pi \\…","labels":["lem:bayesianRisk_ge_inf_bayesInv"],"detail_key":"p12"},{"id":"n17810","layer":"informal","project":"p12","title":"Starting from the equality of Lemma~\\reflem:bayesianRisk_bayesInv, we get R^P_\\pi(\\haty)…","kind":"proof","summary":"Starting from the equality of Lemma~\\reflem:bayesianRisk_bayesInv, we get R^P_\\pi(\\haty) &= ((P…","labels":[],"detail_key":"p12"},{"id":"n17811","layer":"informal","project":"p12","title":"Generalized Bayes estimator","kind":"definition","summary":"[Generalized Bayes estimator] The generalized Bayes estimator for prior \\pi \\in P(\\Theta) on (P…","labels":["def:genBayesEstimator"],"detail_key":"p12"},{"id":"n17812","layer":"informal","project":"p12","title":"lem:bayesianRisk_genBayesEstimator","kind":"lemma","summary":"The Bayesian risk of the generalized Bayes estimator \\haty_B is R^P_\\pi(\\haty_B) = (P \\circ \\pi…","labels":["lem:bayesianRisk_genBayesEstimator"],"detail_key":"p12"},{"id":"n17813","layer":"informal","project":"p12","title":"Start from the equality of Lemma~\\reflem:bayesianRisk_bayesInv and use the definition of…","kind":"proof","summary":"Start from the equality of Lemma~\\reflem:bayesianRisk_bayesInv and use the definition of the ge…","labels":[],"detail_key":"p12"},{"id":"n17814","layer":"informal","project":"p12","title":"thm:isBayesEstimator_genBayesEstimator","kind":"theorem","summary":"When the generalized Bayes estimator is well defined, it is a Bayes estimator. The value of the…","labels":["thm:isBayesEstimator_genBayesEstimator"],"detail_key":"p12"},{"id":"n17815","layer":"informal","project":"p12","title":"By Lemma~\\reflem:bayesianRisk_ge_inf_bayesInv, the Bayesian risk of the generalized Bayes…","kind":"proof","summary":"By Lemma~\\reflem:bayesianRisk_ge_inf_bayesInv, the Bayesian risk of the generalized Bayes estim…","labels":[],"detail_key":"p12"},{"id":"n17816","layer":"informal","project":"p12","title":"lem:bayesRisk_eq_rnDeriv","kind":"lemma","summary":"When the generalized Bayes estimator is well defined, the Bayes risk with respect to the prior…","labels":["lem:bayesRisk_eq_rnDeriv"],"detail_key":"p12"},{"id":"n17817","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17818","layer":"informal","project":"p12","title":"lem:bayesRisk_binary_eq_sub_bayesInv","kind":"lemma","summary":"When the generalized Bayes estimator is well defined, the Bayes risk with respect to the prior…","labels":["lem:bayesRisk_binary_eq_sub_bayesInv"],"detail_key":"p12"},{"id":"n17819","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17820","layer":"informal","project":"p12","title":"lem:bayesRisk_binary_eq_sub","kind":"lemma","summary":"When the generalized Bayes estimator is well defined, the Bayes risk with respect to the prior…","labels":["lem:bayesRisk_binary_eq_sub"],"detail_key":"p12"},{"id":"n17821","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17822","layer":"informal","project":"p12","title":"lem:genBayesEstimator_binary","kind":"lemma","summary":"The generalized Bayes estimator for prior \\pi \\in P(\\Theta) on the estimation problem defined b…","labels":["lem:genBayesEstimator_binary"],"detail_key":"p12"},{"id":"n17823","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17824","layer":"informal","project":"p12","title":"lem:bayesRisk_binary_le_sub_prod","kind":"lemma","summary":"Suppose that \\Theta is finite and let \\xi \\in P(\\Theta). The Bayes risk with respect to the pri…","labels":["lem:bayesRisk_binary_le_sub_prod"],"detail_key":"p12"},{"id":"n17825","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17826","layer":"informal","project":"p12","title":"def:riskIncrease","kind":"definition","summary":"The Bayes risk increase I^P_\\pi(\\kappa) of a kernel \\kappa : X \\rightsquigarrow X' with respect…","labels":["def:riskIncrease"],"detail_key":"p12"},{"id":"n17827","layer":"informal","project":"p12","title":"lem:riskIncrease_nonneg","kind":"lemma","summary":"For \\kappa a Markov kernel, I^P_\\pi(\\kappa) \\ge 0~.","labels":["lem:riskIncrease_nonneg"],"detail_key":"p12"},{"id":"n17828","layer":"informal","project":"p12","title":"Use Theorem~\\refthm:data_proc_bayesRisk.","kind":"proof","summary":"Use Theorem~\\refthm:data_proc_bayesRisk.","labels":[],"detail_key":"p12"},{"id":"n17829","layer":"informal","project":"p12","title":"lem:riskIncrease_comp","kind":"lemma","summary":"For \\kappa : X \\rightsquigarrow X' and \\eta : X' \\rightsquigarrow X'' two Markov kernels, I^P_\\…","labels":["lem:riskIncrease_comp"],"detail_key":"p12"},{"id":"n17830","layer":"informal","project":"p12","title":"I^P_\\pi(\\kappa) + I^\\kappa \\circ P_\\pi(\\eta) &= R^\\kappa \\circ P_\\pi - R^P_\\pi + R^\\eta \\…","kind":"proof","summary":"I^P_\\pi(\\kappa) + I^\\kappa \\circ P_\\pi(\\eta) &= R^\\kappa \\circ P_\\pi - R^P_\\pi + R^\\eta \\circ \\…","labels":[],"detail_key":"p12"},{"id":"n17831","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"lemma","summary":"[Data-processing inequality] For any measurable space X, let d_ X : X \\rightsquigarrow * be the…","labels":["lem:riskIncrease_comp_del"],"detail_key":"p12"},{"id":"n17832","layer":"informal","project":"p12","title":"By Lemma~\\reflem:riskIncrease_comp, then Lemma~\\reflem:riskIncrease_nonneg, I_\\pi^P(d_ X'…","kind":"proof","summary":"By Lemma~\\reflem:riskIncrease_comp, then Lemma~\\reflem:riskIncrease_nonneg, I_\\pi^P(d_ X' \\circ…","labels":[],"detail_key":"p12"},{"id":"n17833","layer":"informal","project":"p12","title":"lem:bayesInv_binary","kind":"lemma","summary":"The Bayesian inverse of a kernel P : \\0,1\\ \\rightsquigarrow X with respect to a prior \\xi \\in M…","labels":["lem:bayesInv_binary"],"detail_key":"p12"},{"id":"n17834","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17835","layer":"informal","project":"p12","title":"lem:bayesRisk_binary","kind":"lemma","summary":"For \\Theta = \\0,1\\, the Bayes risk of a prior \\xi \\in M(\\0,1\\) is R^P_\\xi = (P \\circ \\xi)\\left[…","labels":["lem:bayesRisk_binary"],"detail_key":"p12"},{"id":"n17836","layer":"informal","project":"p12","title":"Use Theorem~\\refthm:isBayesEstimator_genBayesEstimator, with the value of P_\\xi^\\dagger g…","kind":"proof","summary":"Use Theorem~\\refthm:isBayesEstimator_genBayesEstimator, with the value of P_\\xi^\\dagger given b…","labels":[],"detail_key":"p12"},{"id":"n17837","layer":"informal","project":"p12","title":"def:bayesBinaryRisk","kind":"definition","summary":"The Bayes binary risk between measures \\mu and \\nu with respect to prior \\xi \\in M(\\0,1\\), deno…","labels":["def:bayesBinaryRisk"],"detail_key":"p12"},{"id":"n17838","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_mul","kind":"lemma","summary":"For all a, b > 0, B_\\xi(\\mu, \\nu) = B_(a \\xi_0, b \\xi_1)(a^-1 \\mu, b^-1 \\nu)~.","labels":["lem:bayesBinaryRisk_mul"],"detail_key":"p12"},{"id":"n17839","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17840","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_one_one","kind":"lemma","summary":"B_\\xi(\\mu, \\nu) = B_(1,1)(\\xi_0\\mu, \\xi_1\\nu)~.","labels":["lem:bayesBinaryRisk_one_one"],"detail_key":"p12"},{"id":"n17841","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17842","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"theorem","summary":"[Data-processing inequality] For \\mu, \\nu \\in M( X) and \\kappa : X \\rightsquigarrow Y a Markov…","labels":["thm:data_proc_bayesBinaryRisk"],"detail_key":"p12"},{"id":"n17843","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:data_proc_bayesRisk.","kind":"proof","summary":"Apply Theorem~\\refthm:data_proc_bayesRisk.","labels":[],"detail_key":"p12"},{"id":"n17844","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_self","kind":"lemma","summary":"For \\mu \\in M( X), B_\\xi(\\mu, \\mu) = \\min\\\\xi_0, \\xi_1\\ \\mu( X)~.","labels":["lem:bayesBinaryRisk_self"],"detail_key":"p12"},{"id":"n17845","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17846","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_nonneg","kind":"lemma","summary":"For all measures \\mu, \\nu, B_\\xi(\\mu, \\nu) \\ge 0~.","labels":["lem:bayesBinaryRisk_nonneg"],"detail_key":"p12"},{"id":"n17847","layer":"informal","project":"p12","title":"It is an infimum of non-negative values.","kind":"proof","summary":"It is an infimum of non-negative values.","labels":[],"detail_key":"p12"},{"id":"n17848","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_le","kind":"lemma","summary":"For all measures \\mu, \\nu, B_\\xi(\\mu, \\nu) \\le \\min\\\\xi_0 \\mu( X), \\xi_1 \\nu( X)\\~.","labels":["lem:bayesBinaryRisk_le"],"detail_key":"p12"},{"id":"n17849","layer":"informal","project":"p12","title":"Let d_ X : X \\rightsquigarrow * be the discard kernel and \\delta_* be the only probabilit…","kind":"proof","summary":"Let d_ X : X \\rightsquigarrow * be the discard kernel and \\delta_* be the only probability meas…","labels":[],"detail_key":"p12"},{"id":"n17850","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_symm","kind":"lemma","summary":"For \\mu, \\nu \\in M( X) and \\xi \\in M(\\0,1\\), B_\\xi(\\mu, \\nu) = B_\\xi_\\leftrightarrow(\\nu, \\mu)…","labels":["lem:bayesBinaryRisk_symm"],"detail_key":"p12"},{"id":"n17851","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17852","layer":"informal","project":"p12","title":"lem:genBayesEstimator_bayesBinaryRisk","kind":"lemma","summary":"The generalized Bayes estimator for the Bayes binary risk with prior \\xi \\in M(\\0,1\\) is x \\map…","labels":["lem:genBayesEstimator_bayesBinaryRisk"],"detail_key":"p12"},{"id":"n17853","layer":"informal","project":"p12","title":"The generalized Bayes estimator is defined by x \\mapsto \\arg\\min_z P_\\xi^\\dagger(x)\\left[…","kind":"proof","summary":"The generalized Bayes estimator is defined by x \\mapsto \\arg\\min_z P_\\xi^\\dagger(x)\\left[\\theta…","labels":[],"detail_key":"p12"},{"id":"n17854","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_eq_event","kind":"lemma","summary":"B_\\xi(\\mu, \\nu) = \\inf_E \\text event \\left( \\xi_0 \\mu(E) + \\xi_1 \\nu(E^c) \\right) \\: .","labels":["lem:bayesBinaryRisk_eq_event"],"detail_key":"p12"},{"id":"n17855","layer":"informal","project":"p12","title":"By definition, B_\\xi(\\mu, \\nu) = \\inf_\\haty : X \\rightsquigarrow \\0,1\\\\left(\\xi_0 (\\haty…","kind":"proof","summary":"By definition, B_\\xi(\\mu, \\nu) = \\inf_\\haty : X \\rightsquigarrow \\0,1\\\\left(\\xi_0 (\\haty \\circ…","labels":[],"detail_key":"p12"},{"id":"n17856","layer":"informal","project":"p12","title":"thm:bayesBinaryRisk_eq","kind":"theorem","summary":"The Bayes risk of simple binary hypothesis testing for prior \\xi \\in M(\\0,1\\) is B_\\xi(\\mu, \\nu…","labels":["thm:bayesBinaryRisk_eq"],"detail_key":"p12"},{"id":"n17857","layer":"informal","project":"p12","title":"By Theorem~\\refthm:isBayesEstimator_genBayesEstimator, the generalized Bayes estimator is…","kind":"proof","summary":"By Theorem~\\refthm:isBayesEstimator_genBayesEstimator, the generalized Bayes estimator is a Bay…","labels":[],"detail_key":"p12"},{"id":"n17858","layer":"informal","project":"p12","title":"cor:bayesBinaryRisk_eq_abs","kind":"corollary","summary":"B_\\xi(\\mu, \\nu) = \\frac12\\left((P \\circ \\xi)( X) - (P \\circ \\xi)\\left[x \\mapsto \\left\\vert \\xi_…","labels":["cor:bayesBinaryRisk_eq_abs"],"detail_key":"p12"},{"id":"n17859","layer":"informal","project":"p12","title":"Use Theorem~\\refthm:bayesBinaryRisk_eq and the equality \\min\\a,b\\ = \\frac12(a + b - \\vert…","kind":"proof","summary":"Use Theorem~\\refthm:bayesBinaryRisk_eq and the equality \\min\\a,b\\ = \\frac12(a + b - \\vert a - b…","labels":[],"detail_key":"p12"},{"id":"n17860","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_bernoulli","kind":"lemma","summary":"Let \\haty_B be the generalized Bayes estimator for simple binary hypothesis testing. The distri…","labels":["lem:bayesBinaryRisk_bernoulli"],"detail_key":"p12"},{"id":"n17861","layer":"informal","project":"p12","title":"\\ell takes values in \\0,1\\ so the law has to be Bernoulli. It has mean B_\\pi(\\mu, \\nu) be…","kind":"proof","summary":"\\ell takes values in \\0,1\\ so the law has to be Bernoulli. It has mean B_\\pi(\\mu, \\nu) because…","labels":[],"detail_key":"p12"},{"id":"n17862","layer":"informal","project":"p12","title":"Dummy lemma: bayesBinaryRisk properties","kind":"lemma","summary":"[Dummy lemma: bayesBinaryRisk properties] Dummy node to summarize properties of the Bayes binar…","labels":["lem:bayesBinaryRisk_properties"],"detail_key":"p12"},{"id":"n17863","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17864","layer":"informal","project":"p12","title":"lem:bayesRisk_mono_prod","kind":"lemma","summary":"If n \\le m then R_\\pi^P^\\otimes n \\ge R_\\pi^P^\\otimes m.","labels":["lem:bayesRisk_mono_prod"],"detail_key":"p12"},{"id":"n17865","layer":"informal","project":"p12","title":"Use the data-processing inequality, Theorem~\\refthm:data_proc_bayesRisk, for the determin…","kind":"proof","summary":"Use the data-processing inequality, Theorem~\\refthm:data_proc_bayesRisk, for the deterministic…","labels":[],"detail_key":"p12"},{"id":"n17866","layer":"informal","project":"p12","title":"def:priorSampleComplexity","kind":"definition","summary":"The sample complexity of Bayesian estimation with respect to a prior \\pi \\in M(\\Theta) at risk…","labels":["def:priorSampleComplexity"],"detail_key":"p12"},{"id":"n17867","layer":"informal","project":"p12","title":"Divergence","kind":"definition","summary":"[Divergence] A divergence between measures is a function D which for any measurable space X and…","labels":["def:div"],"detail_key":"p12"},{"id":"n17868","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"definition","summary":"[Data-processing inequality] A divergence D is said to satisfy the data-processing inequality (…","labels":["def:dpi"],"detail_key":"p12"},{"id":"n17869","layer":"informal","project":"p12","title":"Second marginal","kind":"lemma","summary":"[Second marginal] Let D be a divergence that satisfies the DPI. Let \\mu, \\nu \\in M( X) and let…","labels":["lem:div_comp_le_div_compProd"],"detail_key":"p12"},{"id":"n17870","layer":"informal","project":"p12","title":"Use the DPI for the deterministic kernel of the function (x,y) \\mapsto y.","kind":"proof","summary":"Use the DPI for the deterministic kernel of the function (x,y) \\mapsto y.","labels":[],"detail_key":"p12"},{"id":"n17871","layer":"informal","project":"p12","title":"First marginal","kind":"lemma","summary":"[First marginal] Let D be a divergence that satisfies the DPI. Let \\mu, \\nu \\in M( X) and let \\…","labels":["lem:div_le_div_compProd"],"detail_key":"p12"},{"id":"n17872","layer":"informal","project":"p12","title":"Use the DPI for the deterministic kernel of the function (x,y) \\mapsto x.","kind":"proof","summary":"Use the DPI for the deterministic kernel of the function (x,y) \\mapsto x.","labels":[],"detail_key":"p12"},{"id":"n17873","layer":"informal","project":"p12","title":"lem:div_compProd_right","kind":"lemma","summary":"Let D be a divergence that satisfies the DPI. Let \\mu, \\nu \\in M( X) and let \\kappa : X \\rights…","labels":["lem:div_compProd_right"],"detail_key":"p12"},{"id":"n17874","layer":"informal","project":"p12","title":"Lemma~\\reflem:div_le_div_compProd gives one inequality. The other inequality is the DPI:…","kind":"proof","summary":"Lemma~\\reflem:div_le_div_compProd gives one inequality. The other inequality is the DPI: \\mu \\o…","labels":[],"detail_key":"p12"},{"id":"n17875","layer":"informal","project":"p12","title":"Conditional divergence","kind":"definition","summary":"[Conditional divergence] Let D be a divergence. The conditional divergence of kernels \\kappa, \\…","labels":["def:condDiv"],"detail_key":"p12"},{"id":"n17876","layer":"informal","project":"p12","title":"Conditioning increases divergence","kind":"lemma","summary":"[Conditioning increases divergence] Let D be a divergence that satisfies the DPI and for which…","labels":["lem:div_comp_le_div_compProd_right"],"detail_key":"p12"},{"id":"n17877","layer":"informal","project":"p12","title":"This is a special case of Lemma~\\reflem:div_comp_le_div_compProd.","kind":"proof","summary":"This is a special case of Lemma~\\reflem:div_comp_le_div_compProd.","labels":[],"detail_key":"p12"},{"id":"n17878","layer":"informal","project":"p12","title":"def:statInfo","kind":"definition","summary":"The statistical information between measures \\mu and \\nu with respect to prior \\xi \\in M(\\0,1\\)…","labels":["def:statInfo"],"detail_key":"p12"},{"id":"n17879","layer":"informal","project":"p12","title":"lem:statInfo_self","kind":"lemma","summary":"For \\mu \\in M( X), I_\\xi(\\mu, \\mu) = 0~.","labels":["lem:statInfo_self"],"detail_key":"p12"},{"id":"n17880","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:bayesBinaryRisk_self.","kind":"proof","summary":"Use Lemma~\\reflem:bayesBinaryRisk_self.","labels":[],"detail_key":"p12"},{"id":"n17881","layer":"informal","project":"p12","title":"lem:statInfo_nonneg","kind":"lemma","summary":"For \\mu, \\nu \\in M( X), I_\\xi(\\mu, \\nu) \\ge 0~.","labels":["lem:statInfo_nonneg"],"detail_key":"p12"},{"id":"n17882","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:bayesBinaryRisk_le.","kind":"proof","summary":"Use Lemma~\\reflem:bayesBinaryRisk_le.","labels":[],"detail_key":"p12"},{"id":"n17883","layer":"informal","project":"p12","title":"lem:statInfo_le","kind":"lemma","summary":"For \\mu, \\nu \\in M( X), I_\\xi(\\mu, \\nu) \\le \\min\\\\xi_0 \\mu( X), \\xi_1 \\nu( X)\\~.","labels":["lem:statInfo_le"],"detail_key":"p12"},{"id":"n17884","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:bayesBinaryRisk_nonneg.","kind":"proof","summary":"Use Lemma~\\reflem:bayesBinaryRisk_nonneg.","labels":[],"detail_key":"p12"},{"id":"n17885","layer":"informal","project":"p12","title":"lem:statInfo_symm","kind":"lemma","summary":"For \\mu, \\nu \\in M( X) and \\xi \\in M(\\0,1\\), I_\\xi(\\mu, \\nu) = I_\\xi_\\leftrightarrow(\\nu, \\mu)~.","labels":["lem:statInfo_symm"],"detail_key":"p12"},{"id":"n17886","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:bayesBinaryRisk_symm.","kind":"proof","summary":"Use Lemma~\\reflem:bayesBinaryRisk_symm.","labels":[],"detail_key":"p12"},{"id":"n17887","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"theorem","summary":"[Data-processing inequality] For \\mu, \\nu \\in M( X), \\xi \\in M(\\0,1\\) and \\kappa : X \\rightsqui…","labels":["thm:data_proc_statInfo"],"detail_key":"p12"},{"id":"n17888","layer":"informal","project":"p12","title":"Since \\kappa is a Markov kernel, I_\\xi(\\kappa \\circ \\mu, \\kappa \\circ \\nu) &= \\min\\\\xi_0(…","kind":"proof","summary":"Since \\kappa is a Markov kernel, I_\\xi(\\kappa \\circ \\mu, \\kappa \\circ \\nu) &= \\min\\\\xi_0(\\kappa…","labels":[],"detail_key":"p12"},{"id":"n17889","layer":"informal","project":"p12","title":"cor:statInfo_data_proc_event","kind":"corollary","summary":"Let \\mu, \\nu be two measures on X, \\xi \\in M(\\0,1\\) and let E be an event on X. Let \\mu_E and \\…","labels":["cor:statInfo_data_proc_event"],"detail_key":"p12"},{"id":"n17890","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:data_proc_statInfo to the deterministic kernel X \\rightsquigarrow \\…","kind":"proof","summary":"Apply Theorem~\\refthm:data_proc_statInfo to the deterministic kernel X \\rightsquigarrow \\0,1\\ g…","labels":[],"detail_key":"p12"},{"id":"n17891","layer":"informal","project":"p12","title":"lem:statInfo_eq_sub_min","kind":"lemma","summary":"For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\), for any measure \\zeta with \\mu \\ll \\zeta and…","labels":["lem:statInfo_eq_sub_min"],"detail_key":"p12"},{"id":"n17892","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:bayesBinaryRisk_eq to get that result for \\zeta = P \\circ \\xi. Then…","kind":"proof","summary":"Apply Theorem~\\refthm:bayesBinaryRisk_eq to get that result for \\zeta = P \\circ \\xi. Then for o…","labels":[],"detail_key":"p12"},{"id":"n17893","layer":"informal","project":"p12","title":"lem:statInfo_eq_integral_abs_sub","kind":"lemma","summary":"For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\), for any measure \\zeta with \\mu \\ll \\zeta and…","labels":["lem:statInfo_eq_integral_abs_sub"],"detail_key":"p12"},{"id":"n17894","layer":"informal","project":"p12","title":"Apply Corollary~\\refcor:bayesBinaryRisk_eq_abs and write \\min\\\\xi_0\\mu( X), \\xi_1\\nu( X)\\…","kind":"proof","summary":"Apply Corollary~\\refcor:bayesBinaryRisk_eq_abs and write \\min\\\\xi_0\\mu( X), \\xi_1\\nu( X)\\ = \\fr…","labels":[],"detail_key":"p12"},{"id":"n17895","layer":"informal","project":"p12","title":"Integral form of the statistical information","kind":"theorem","summary":"[Integral form of the statistical information] For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\…","labels":["thm:statInfo_eq_integral"],"detail_key":"p12"},{"id":"n17896","layer":"informal","project":"p12","title":"By Theorem~\\refthm:bayesBinaryRisk_eq, I_\\xi(\\mu, \\nu) &= \\min\\\\xi_0\\mu( X), \\xi_1\\nu( X)…","kind":"proof","summary":"By Theorem~\\refthm:bayesBinaryRisk_eq, I_\\xi(\\mu, \\nu) &= \\min\\\\xi_0\\mu( X), \\xi_1\\nu( X)\\ - (P…","labels":[],"detail_key":"p12"},{"id":"n17897","layer":"informal","project":"p12","title":"cor:statInfo_eq_integral_abs","kind":"corollary","summary":"For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\), I_\\xi(\\mu, \\nu) &= -\\frac12 \\left\\vert\\xi_0…","labels":["cor:statInfo_eq_integral_abs"],"detail_key":"p12"},{"id":"n17898","layer":"informal","project":"p12","title":"Start from Theorem~\\refthm:statInfo_eq_integral, then use \\max\\a,b\\ = \\frac12\\left( a + b…","kind":"proof","summary":"Start from Theorem~\\refthm:statInfo_eq_integral, then use \\max\\a,b\\ = \\frac12\\left( a + b + \\ve…","labels":[],"detail_key":"p12"},{"id":"n17899","layer":"informal","project":"p12","title":"lem:statInfo_eq_sub_inf_event","kind":"lemma","summary":"For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\), I_\\xi(\\mu, \\nu) &= \\min\\\\xi_0 \\mu( X), \\xi_1…","labels":["lem:statInfo_eq_sub_inf_event"],"detail_key":"p12"},{"id":"n17900","layer":"informal","project":"p12","title":"This is a direct application of Lemma~\\reflem:bayesBinaryRisk_eq_event.","kind":"proof","summary":"This is a direct application of Lemma~\\reflem:bayesBinaryRisk_eq_event.","labels":[],"detail_key":"p12"},{"id":"n17901","layer":"informal","project":"p12","title":"lem:statInfo_eq_sup_event","kind":"lemma","summary":"For finite measures \\mu, \\nu and \\xi \\in M(\\0,1\\), I_\\xi(\\mu, \\nu) &= - \\max\\0, \\xi_1 \\nu( X) -…","labels":["lem:statInfo_eq_sup_event"],"detail_key":"p12"},{"id":"n17902","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17903","layer":"informal","project":"p12","title":"Dummy lemma: statInfo properties","kind":"lemma","summary":"[Dummy lemma: statInfo properties] Dummy node to summarize properties of the statistical inform…","labels":["lem:statInfo_properties"],"detail_key":"p12"},{"id":"n17904","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17905","layer":"informal","project":"p12","title":"def:deGrootInfo","kind":"definition","summary":"The DeGroot statistical information between finite measures \\mu and \\nu for \\pi \\in [0,1] is I_…","labels":["def:deGrootInfo"],"detail_key":"p12"},{"id":"n17906","layer":"informal","project":"p12","title":"def:eGamma","kind":"definition","summary":"The E_\\gamma or hockey-stick divergence between finite measures \\mu and \\nu for \\gamma \\in (0,+…","labels":["def:eGamma"],"detail_key":"p12"},{"id":"n17907","layer":"informal","project":"p12","title":"def:TV","kind":"definition","summary":"The total variation distance between finite measures \\mu and \\nu is \\TV(\\mu, \\nu) = I_(1,1)(\\mu…","labels":["def:TV"],"detail_key":"p12"},{"id":"n17908","layer":"informal","project":"p12","title":"lem:tv_self","kind":"lemma","summary":"For \\mu \\in M( X), \\TV(\\mu, \\mu) = 0~.","labels":["lem:tv_self"],"detail_key":"p12"},{"id":"n17909","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:statInfo_self.","kind":"proof","summary":"Use Lemma~\\reflem:statInfo_self.","labels":[],"detail_key":"p12"},{"id":"n17910","layer":"informal","project":"p12","title":"lem:tv_nonneg","kind":"lemma","summary":"For \\mu, \\nu \\in M( X), \\TV(\\mu, \\nu) \\ge 0~.","labels":["lem:tv_nonneg"],"detail_key":"p12"},{"id":"n17911","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:statInfo_nonneg.","kind":"proof","summary":"Use Lemma~\\reflem:statInfo_nonneg.","labels":[],"detail_key":"p12"},{"id":"n17912","layer":"informal","project":"p12","title":"lem:tv_le","kind":"lemma","summary":"For \\mu, \\nu \\in M( X), \\TV(\\mu, \\nu) \\le \\min\\\\mu( X), \\nu( X)\\~.","labels":["lem:tv_le"],"detail_key":"p12"},{"id":"n17913","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:statInfo_le.","kind":"proof","summary":"Use Lemma~\\reflem:statInfo_le.","labels":[],"detail_key":"p12"},{"id":"n17914","layer":"informal","project":"p12","title":"Data-processing inequality","kind":"theorem","summary":"[Data-processing inequality] For \\mu, \\nu \\in M( X) and \\kappa : X \\rightsquigarrow Y a Markov…","labels":["thm:tv_data_proc"],"detail_key":"p12"},{"id":"n17915","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:data_proc_statInfo.","kind":"proof","summary":"Apply Theorem~\\refthm:data_proc_statInfo.","labels":[],"detail_key":"p12"},{"id":"n17916","layer":"informal","project":"p12","title":"Dummy lemma: TV properties","kind":"lemma","summary":"[Dummy lemma: TV properties] Dummy node to summarize properties of the total variation distance.","labels":["lem:tv_properties"],"detail_key":"p12"},{"id":"n17917","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17918","layer":"informal","project":"p12","title":"lem:tv_eq_sub_min","kind":"lemma","summary":"For finite measures \\mu, \\nu, for any measure \\zeta with \\mu \\ll \\zeta and \\nu \\ll \\zeta~, \\TV(…","labels":["lem:tv_eq_sub_min"],"detail_key":"p12"},{"id":"n17919","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:statInfo_eq_sub_min.","kind":"proof","summary":"Apply Lemma~\\reflem:statInfo_eq_sub_min.","labels":[],"detail_key":"p12"},{"id":"n17920","layer":"informal","project":"p12","title":"lem:tv_eq_integral_abs_sub","kind":"lemma","summary":"For finite measures \\mu, \\nu, for any measure \\zeta with \\mu \\ll \\zeta and \\nu \\ll \\zeta~, \\TV(…","labels":["lem:tv_eq_integral_abs_sub"],"detail_key":"p12"},{"id":"n17921","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:statInfo_eq_integral_abs_sub.","kind":"proof","summary":"Apply Lemma~\\reflem:statInfo_eq_integral_abs_sub.","labels":[],"detail_key":"p12"},{"id":"n17922","layer":"informal","project":"p12","title":"Integral form of the total variation distance","kind":"lemma","summary":"[Integral form of the total variation distance] For finite measures \\mu, \\nu, \\TV(\\mu, \\nu) &=…","labels":["lem:tv_eq_integral"],"detail_key":"p12"},{"id":"n17923","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:statInfo_eq_integral.","kind":"proof","summary":"Apply Theorem~\\refthm:statInfo_eq_integral.","labels":[],"detail_key":"p12"},{"id":"n17924","layer":"informal","project":"p12","title":"lem:tv_eq_integral_abs","kind":"lemma","summary":"For finite measures \\mu, \\nu, \\TV(\\mu, \\nu) &= -\\frac12 \\left\\vert \\mu( X) - \\nu( X)\\right\\vert…","labels":["lem:tv_eq_integral_abs"],"detail_key":"p12"},{"id":"n17925","layer":"informal","project":"p12","title":"Apply Corollary~\\refcor:statInfo_eq_integral_abs.","kind":"proof","summary":"Apply Corollary~\\refcor:statInfo_eq_integral_abs.","labels":[],"detail_key":"p12"},{"id":"n17926","layer":"informal","project":"p12","title":"lem:tv_eq_sub_inf_event","kind":"lemma","summary":"For finite measures \\mu, \\nu, \\TV(\\mu, \\nu) &= \\min\\\\mu( X), \\nu( X)\\ - \\inf_E \\text event \\lef…","labels":["lem:tv_eq_sub_inf_event"],"detail_key":"p12"},{"id":"n17927","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:statInfo_eq_sub_inf_event.","kind":"proof","summary":"Apply Lemma~\\reflem:statInfo_eq_sub_inf_event.","labels":[],"detail_key":"p12"},{"id":"n17928","layer":"informal","project":"p12","title":"thm:tv_eq_sup_sub_measure","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X with \\mu( X) \\leq \\nu( X).\\\\ Then TV(\\mu, \\nu) = \\sup_…","labels":["thm:tv_eq_sup_sub_measure"],"detail_key":"p12"},{"id":"n17929","layer":"informal","project":"p12","title":"Starting from Lemma~\\reflem:tv_eq_sub_inf_event, \\TV(\\mu, \\nu) &= \\min\\\\mu( X), \\nu( X)\\…","kind":"proof","summary":"Starting from Lemma~\\reflem:tv_eq_sub_inf_event, \\TV(\\mu, \\nu) &= \\min\\\\mu( X), \\nu( X)\\ - \\inf…","labels":[],"detail_key":"p12"},{"id":"n17930","layer":"informal","project":"p12","title":"lem:tv_eq_sup_aux","kind":"lemma","summary":"Let F = \\f : X \\to R \\mid \\Vert f \\Vert_\\infty \\le 1\\. Then for \\mu, \\nu finite measures with \\…","labels":["lem:tv_eq_sup_aux"],"detail_key":"p12"},{"id":"n17931","layer":"informal","project":"p12","title":"Let p,q be the respective densities of \\mu, \\nu with respect to \\xi=\\mu+\\nu. For any f \\i…","kind":"proof","summary":"Let p,q be the respective densities of \\mu, \\nu with respect to \\xi=\\mu+\\nu. For any f \\in F, \\…","labels":[],"detail_key":"p12"},{"id":"n17932","layer":"informal","project":"p12","title":"thm:tv_eq_sup_sub_integral","kind":"theorem","summary":"Let F = \\f : X \\to R \\mid \\Vert f \\Vert_\\infty \\le 1\\. Then for \\mu, \\nu finite measures with \\…","labels":["thm:tv_eq_sup_sub_integral"],"detail_key":"p12"},{"id":"n17933","layer":"informal","project":"p12","title":"Lemma~\\reflem:tv_eq_sup_aux gives \\TV(\\mu, \\nu) \\ge \\frac12 \\sup_f \\in F \\left( \\mu[f] -…","kind":"proof","summary":"Lemma~\\reflem:tv_eq_sup_aux gives \\TV(\\mu, \\nu) \\ge \\frac12 \\sup_f \\in F \\left( \\mu[f] - \\nu[f]…","labels":[],"detail_key":"p12"},{"id":"n17934","layer":"informal","project":"p12","title":"lem:tv_data_proc_event","kind":"lemma","summary":"Let \\mu, \\nu be two measures on X and let E be an event. Let \\mu_E and \\nu_E be the two Bernoul…","labels":["lem:tv_data_proc_event"],"detail_key":"p12"},{"id":"n17935","layer":"informal","project":"p12","title":"Apply Corollary~\\refcor:statInfo_data_proc_event.","kind":"proof","summary":"Apply Corollary~\\refcor:statInfo_data_proc_event.","labels":[],"detail_key":"p12"},{"id":"n17936","layer":"informal","project":"p12","title":"f-divergence","kind":"definition","summary":"[f-divergence] Let f : R \\to R and let \\mu, \\nu be two measures on a measurable space X. The f-…","labels":["def:fDiv"],"detail_key":"p12"},{"id":"n17937","layer":"informal","project":"p12","title":"lem:fDiv_ne_top_iff","kind":"lemma","summary":"For \\mu and \\nu two finite measures, D_f(\\mu, \\nu) is finite if and only if x \\mapsto f\\left(\\f…","labels":["lem:fDiv_ne_top_iff"],"detail_key":"p12"},{"id":"n17938","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17939","layer":"informal","project":"p12","title":"lem:fDiv_const","kind":"lemma","summary":"For \\nu a finite measure, for all a \\in R, D_x \\mapsto a(\\mu, \\nu) = a \\nu( X).","labels":["lem:fDiv_const"],"detail_key":"p12"},{"id":"n17940","layer":"informal","project":"p12","title":"Compute the integral.","kind":"proof","summary":"Compute the integral.","labels":[],"detail_key":"p12"},{"id":"n17941","layer":"informal","project":"p12","title":"lem:fDiv_self","kind":"lemma","summary":"If f(1) = 0 then D_f(\\mu, \\mu) = 0.","labels":["lem:fDiv_self"],"detail_key":"p12"},{"id":"n17942","layer":"informal","project":"p12","title":"\\fracd \\mud \\mu(x) = 1 almost everywhere and f(1) = 0.","kind":"proof","summary":"\\fracd \\mud \\mu(x) = 1 almost everywhere and f(1) = 0.","labels":[],"detail_key":"p12"},{"id":"n17943","layer":"informal","project":"p12","title":"lem:fDiv_eq_zero_iff","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. Assume f'(\\infty) = + \\infty and f(1) = 0. Then D_f(\\…","labels":["lem:fDiv_eq_zero_iff"],"detail_key":"p12"},{"id":"n17944","layer":"informal","project":"p12","title":"If \\mu = \\nu then D_f(\\mu, \\nu) = 0 by Lemma~\\reflem:fDiv_self. For the other direction u…","kind":"proof","summary":"If \\mu = \\nu then D_f(\\mu, \\nu) = 0 by Lemma~\\reflem:fDiv_self. For the other direction use the…","labels":[],"detail_key":"p12"},{"id":"n17945","layer":"informal","project":"p12","title":"lem:fDiv_id","kind":"lemma","summary":"D_x \\mapsto x(\\mu, \\nu) = \\mu( X).","labels":["lem:fDiv_id"],"detail_key":"p12"},{"id":"n17946","layer":"informal","project":"p12","title":"Compute the integral: its value is (\\fracd\\mud\\nu\\cdot \\nu)( X). Then D_x\\mapsto x(\\mu, \\…","kind":"proof","summary":"Compute the integral: its value is (\\fracd\\mud\\nu\\cdot \\nu)( X). Then D_x\\mapsto x(\\mu, \\nu) =…","labels":[],"detail_key":"p12"},{"id":"n17947","layer":"informal","project":"p12","title":"lem:fDiv_mul","kind":"lemma","summary":"For all a \\ge 0, D_a f(\\mu, \\nu) = a D_f(\\mu, \\nu).","labels":["lem:fDiv_mul"],"detail_key":"p12"},{"id":"n17948","layer":"informal","project":"p12","title":"Linearity of the integral.","kind":"proof","summary":"Linearity of the integral.","labels":[],"detail_key":"p12"},{"id":"n17949","layer":"informal","project":"p12","title":"lem:fDiv_add","kind":"lemma","summary":"D_f + g(\\mu, \\nu) = D_f(\\mu, \\nu) + D_g(\\mu, \\nu).","labels":["lem:fDiv_add"],"detail_key":"p12"},{"id":"n17950","layer":"informal","project":"p12","title":"Linearity of the integral.","kind":"proof","summary":"Linearity of the integral.","labels":[],"detail_key":"p12"},{"id":"n17951","layer":"informal","project":"p12","title":"lem:fDiv_add_linear","kind":"lemma","summary":"For finite measures \\mu and \\nu with \\mu( X) = \\nu( X), for all a \\in R, D_f + a(x - 1)(\\mu, \\n…","labels":["lem:fDiv_add_linear"],"detail_key":"p12"},{"id":"n17952","layer":"informal","project":"p12","title":"Linearity (Lemmas~\\reflem:fDiv_add and~\\reflem:fDiv_mul), then Lemma~\\reflem:fDiv_const a…","kind":"proof","summary":"Linearity (Lemmas~\\reflem:fDiv_add and~\\reflem:fDiv_mul), then Lemma~\\reflem:fDiv_const and~\\re…","labels":[],"detail_key":"p12"},{"id":"n17953","layer":"informal","project":"p12","title":"lem:fDiv_map_measurableEmbedding","kind":"lemma","summary":"Let \\mu and \\nu be two measures on X and let g : X \\to Y be a measurable embedding. Then D_f(g_…","labels":["lem:fDiv_map_measurableEmbedding"],"detail_key":"p12"},{"id":"n17954","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17955","layer":"informal","project":"p12","title":"lem:fDiv_symm","kind":"lemma","summary":"D_f(\\mu, \\nu) = D_x \\mapsto xf(1/x)(\\nu, \\mu)~.","labels":["lem:fDiv_symm"],"detail_key":"p12"},{"id":"n17956","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17957","layer":"informal","project":"p12","title":"lem:fDiv_add_smul_left","kind":"lemma","summary":"(\\mu, \\nu) \\mapsto D_f(a \\mu + b \\nu, \\nu) is an f-divergence for the function x \\mapsto f(ax +…","labels":["lem:fDiv_add_smul_left"],"detail_key":"p12"},{"id":"n17958","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17959","layer":"informal","project":"p12","title":"lem:fDiv_add_smul_right","kind":"lemma","summary":"(\\mu, \\nu) \\mapsto D_f(\\mu, a \\mu + b \\nu) is an f-divergence for the function x \\mapsto (ax+b)…","labels":["lem:fDiv_add_smul_right"],"detail_key":"p12"},{"id":"n17960","layer":"informal","project":"p12","title":"We use Lemma~\\reflem:fDiv_symm, then Lemma~\\reflem:fDiv_add_smul_left and finally Lemma~\\…","kind":"proof","summary":"We use Lemma~\\reflem:fDiv_symm, then Lemma~\\reflem:fDiv_add_smul_left and finally Lemma~\\reflem…","labels":[],"detail_key":"p12"},{"id":"n17961","layer":"informal","project":"p12","title":"lem:fDiv_add_smul_right'","kind":"lemma","summary":"(\\mu, \\nu) \\mapsto D_f(\\nu, a \\mu + b \\nu) is an f-divergence for the function x \\mapsto (ax+b)…","labels":["lem:fDiv_add_smul_right'"],"detail_key":"p12"},{"id":"n17962","layer":"informal","project":"p12","title":"We use Lemma~\\reflem:fDiv_add_smul_right, then Lemma~\\reflem:fDiv_symm. D_f(\\nu, a \\mu +…","kind":"proof","summary":"We use Lemma~\\reflem:fDiv_add_smul_right, then Lemma~\\reflem:fDiv_symm. D_f(\\nu, a \\mu + b \\nu)…","labels":[],"detail_key":"p12"},{"id":"n17963","layer":"informal","project":"p12","title":"lem:fDiv_absolutelyContinuous_add_mutuallySingular","kind":"lemma","summary":"Let \\mu_1, \\mu_2 and \\nu be finite measures on X, with \\mu_1 \\ll \\nu and \\mu_2 \\perp \\nu. Then…","labels":["lem:fDiv_absolutelyContinuous_add_mutuallySingular"],"detail_key":"p12"},{"id":"n17964","layer":"informal","project":"p12","title":"\\fracd(\\mu_1 + \\mu_2)d \\nu = \\fracd \\mu_1d \\nu a.e. and (\\mu_1 + \\mu_2)_\\perp \\nu = \\mu_2.","kind":"proof","summary":"\\fracd(\\mu_1 + \\mu_2)d \\nu = \\fracd \\mu_1d \\nu a.e. and (\\mu_1 + \\mu_2)_\\perp \\nu = \\mu_2.","labels":[],"detail_key":"p12"},{"id":"n17965","layer":"informal","project":"p12","title":"Superseded by Lemma~\\reflem:fDiv_add_measure_le","kind":"lemma","summary":"[Superseded by Lemma~\\reflem:fDiv_add_measure_le] Let \\mu_1, \\mu_2, \\nu be three finite measure…","labels":["lem:fDiv_add_measure_le_of_ac"],"detail_key":"p12"},{"id":"n17966","layer":"informal","project":"p12","title":"D_f(\\mu_1 + \\mu_2, \\nu) &= \\int_x f \\left( \\fracd \\mu_1d\\nu(x) + \\fracd\\mu_2d\\nu(x) \\righ…","kind":"proof","summary":"D_f(\\mu_1 + \\mu_2, \\nu) &= \\int_x f \\left( \\fracd \\mu_1d\\nu(x) + \\fracd\\mu_2d\\nu(x) \\right) \\pa…","labels":[],"detail_key":"p12"},{"id":"n17967","layer":"informal","project":"p12","title":"lem:fDiv_add_measure_le","kind":"lemma","summary":"Let \\mu_1, \\mu_2, \\nu be three finite measures on X. Then D_f(\\mu_1 + \\mu_2, \\nu) \\le D_f(\\mu_1…","labels":["lem:fDiv_add_measure_le"],"detail_key":"p12"},{"id":"n17968","layer":"informal","project":"p12","title":"From Lemma~\\reflem:fDiv_absolutelyContinuous_add_mutuallySingular, then Lemma~\\reflem:fDi…","kind":"proof","summary":"From Lemma~\\reflem:fDiv_absolutelyContinuous_add_mutuallySingular, then Lemma~\\reflem:fDiv_add_…","labels":[],"detail_key":"p12"},{"id":"n17969","layer":"informal","project":"p12","title":"lem:fDiv_eq_add_withDensity_derivAtTop","kind":"lemma","summary":"Let \\mu and \\nu be two finite measures on X. Then D_f(\\mu, \\nu) = D_f(\\fracd\\mud\\nu\\cdot \\nu, \\…","labels":["lem:fDiv_eq_add_withDensity_derivAtTop"],"detail_key":"p12"},{"id":"n17970","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:fDiv_absolutelyContinuous_add_mutuallySingular to \\fracd\\mud\\nu\\cdot…","kind":"proof","summary":"Apply Lemma~\\reflem:fDiv_absolutelyContinuous_add_mutuallySingular to \\fracd\\mud\\nu\\cdot \\nu an…","labels":[],"detail_key":"p12"},{"id":"n17971","layer":"informal","project":"p12","title":"Superseded by Lemma~\\reflem:le_fDiv","kind":"lemma","summary":"[Superseded by Lemma~\\reflem:le_fDiv] Let \\mu be a finite measure and \\nu be a probability meas…","labels":["lem:le_fDiv_of_ac"],"detail_key":"p12"},{"id":"n17972","layer":"informal","project":"p12","title":"Since \\mu \\ll \\nu the f-divergence is only the integral part. Then by Jensen's inequality…","kind":"proof","summary":"Since \\mu \\ll \\nu the f-divergence is only the integral part. Then by Jensen's inequality, D_f(…","labels":[],"detail_key":"p12"},{"id":"n17973","layer":"informal","project":"p12","title":"lem:le_fDiv","kind":"lemma","summary":"Let \\mu be a finite measure and \\nu be a probability measure on the same space X. Then f(\\mu( X…","labels":["lem:le_fDiv"],"detail_key":"p12"},{"id":"n17974","layer":"informal","project":"p12","title":"By convexity, then Lemma~\\reflem:le_fDiv_of_ac and finally Lemma~\\reflem:fDiv_eq_add_with…","kind":"proof","summary":"By convexity, then Lemma~\\reflem:le_fDiv_of_ac and finally Lemma~\\reflem:fDiv_eq_add_withDensit…","labels":[],"detail_key":"p12"},{"id":"n17975","layer":"informal","project":"p12","title":"lem:fDiv_nonneg","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. If f(1) = 0 then D_f(\\mu, \\nu) \\ge 0.","labels":["lem:fDiv_nonneg"],"detail_key":"p12"},{"id":"n17976","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:le_fDiv and use f(\\mu( X)) = f(1) = 0.","kind":"proof","summary":"Apply Lemma~\\reflem:le_fDiv and use f(\\mu( X)) = f(1) = 0.","labels":[],"detail_key":"p12"},{"id":"n17977","layer":"informal","project":"p12","title":"Dummy lemma: fDiv properties","kind":"lemma","summary":"[Dummy lemma: fDiv properties] Dummy node to summarize properties of f-divergences.","labels":["lem:fDiv_properties"],"detail_key":"p12"},{"id":"n17978","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17979","layer":"informal","project":"p12","title":"Conditional f-divergence","kind":"definition","summary":"[Conditional f-divergence] Let f : R \\to R, \\mu a measure on X and \\kappa, \\eta : X \\rightsquig…","labels":["def:condFDiv"],"detail_key":"p12"},{"id":"n17980","layer":"informal","project":"p12","title":"lem:condFDiv_nonneg","kind":"lemma","summary":"Let \\mu be a measure on X and \\kappa, \\eta : X \\rightsquigarrow Y two Markov kernels. If f(1) =…","labels":["lem:condFDiv_nonneg"],"detail_key":"p12"},{"id":"n17981","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:fDiv_nonneg.","kind":"proof","summary":"Apply Lemma~\\reflem:fDiv_nonneg.","labels":[],"detail_key":"p12"},{"id":"n17982","layer":"informal","project":"p12","title":"lem:condFDiv_const","kind":"lemma","summary":"Let \\mu, \\nu be measures on X, where \\mu is finite, and let \\xi be a finite measure on Y. Then…","labels":["lem:condFDiv_const"],"detail_key":"p12"},{"id":"n17983","layer":"informal","project":"p12","title":"D_f(x \\mapsto \\mu, x \\mapsto \\nu \\mid \\xi) = \\xi\\left[x \\mapsto D_f(\\mu, \\nu)\\right] = D_…","kind":"proof","summary":"D_f(x \\mapsto \\mu, x \\mapsto \\nu \\mid \\xi) = \\xi\\left[x \\mapsto D_f(\\mu, \\nu)\\right] = D_f(\\mu,…","labels":[],"detail_key":"p12"},{"id":"n17984","layer":"informal","project":"p12","title":"lem:integrable_fDiv_compProd_iff","kind":"lemma","summary":"Let \\mu be a finite measure on X and \\kappa, \\eta : X \\rightsquigarrow Y be two finite kernels…","labels":["lem:integrable_fDiv_compProd_iff"],"detail_key":"p12"},{"id":"n17985","layer":"informal","project":"p12","title":"Since x \\mapsto f \\left(\\fracd(\\mu \\otimes \\kappa)d(\\mu \\otimes \\eta) x \\right) is measur…","kind":"proof","summary":"Since x \\mapsto f \\left(\\fracd(\\mu \\otimes \\kappa)d(\\mu \\otimes \\eta) x \\right) is measurable,…","labels":[],"detail_key":"p12"},{"id":"n17986","layer":"informal","project":"p12","title":"lem:condFDiv_ne_top_iff","kind":"lemma","summary":"Let \\mu be a finite measure on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two finite kern…","labels":["lem:condFDiv_ne_top_iff"],"detail_key":"p12"},{"id":"n17987","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17988","layer":"informal","project":"p12","title":"lem:fDiv_compProd_ne_top_iff","kind":"lemma","summary":"Let \\mu be a finite measure on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two finite kern…","labels":["lem:fDiv_compProd_ne_top_iff"],"detail_key":"p12"},{"id":"n17989","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17990","layer":"informal","project":"p12","title":"lem:fDiv_compProd_left","kind":"lemma","summary":"Let \\mu be a finite measure on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two finite kern…","labels":["lem:fDiv_compProd_left"],"detail_key":"p12"},{"id":"n17991","layer":"informal","project":"p12","title":"By Lemma~\\reflem:fDiv_compProd_ne_top_iff, the conditions on which the two divergences ar…","kind":"proof","summary":"By Lemma~\\reflem:fDiv_compProd_ne_top_iff, the conditions on which the two divergences are fini…","labels":[],"detail_key":"p12"},{"id":"n17992","layer":"informal","project":"p12","title":"Dummy lemma: condFDiv properties","kind":"lemma","summary":"[Dummy lemma: condFDiv properties] Dummy node to summarize properties of conditional f-divergen…","labels":["lem:condFDiv_properties"],"detail_key":"p12"},{"id":"n17993","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17994","layer":"informal","project":"p12","title":"Curvature measure","kind":"definition","summary":"[Curvature measure] Let f: R \\to R be a convex function. Then its right derivative f'_+(x) \\col…","labels":["def:curvatureMeasure"],"detail_key":"p12"},{"id":"n17995","layer":"informal","project":"p12","title":"lem:curvatureMeasure_mul","kind":"lemma","summary":"For a \\ge 0 and f: R \\to R a convex function, the curvature measure of af is \\gamma_af = a \\gam…","labels":["lem:curvatureMeasure_mul"],"detail_key":"p12"},{"id":"n17996","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17997","layer":"informal","project":"p12","title":"lem:curvatureMeasure_add","kind":"lemma","summary":"For f,g: R \\to R two convex functions, the curvature measure of f+g is \\gamma_f+g = \\gamma_f +…","labels":["lem:curvatureMeasure_add"],"detail_key":"p12"},{"id":"n17998","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n17999","layer":"informal","project":"p12","title":"thm:integration_by_parts","kind":"theorem","summary":"\\notready If f and g are two Stieltjes functions with associated measures \\mu_f and \\mu_g and f…","labels":["thm:integration_by_parts"],"detail_key":"p12"},{"id":"n18000","layer":"informal","project":"p12","title":"\\notready","kind":"proof","summary":"\\notready","labels":[],"detail_key":"p12"},{"id":"n18001","layer":"informal","project":"p12","title":"lem:convex_taylor","kind":"lemma","summary":"For f: R \\to R a convex function and x,y \\in R, f(y) - f(x) - (y - x)f'_+(x) &= \\int_z \\in (x,y…","labels":["lem:convex_taylor"],"detail_key":"p12"},{"id":"n18002","layer":"informal","project":"p12","title":"Let \\Lambda be the Lebesgue measure and let x < y. Since f has right derivative f'_+ in (…","kind":"proof","summary":"Let \\Lambda be the Lebesgue measure and let x < y. Since f has right derivative f'_+ in (x,y) a…","labels":[],"detail_key":"p12"},{"id":"n18003","layer":"informal","project":"p12","title":"def:statInfoFun","kind":"definition","summary":"For a,b \\in (0, +\\infty) let \\phi_a,b : R \\to R be the function defined by \\phi_a,b(x) &= \\max\\…","labels":["def:statInfoFun"],"detail_key":"p12"},{"id":"n18004","layer":"informal","project":"p12","title":"cor:convex_taylor_statInfoFun","kind":"corollary","summary":"For f: R \\to R a convex function, for all x \\in R~, f(x) &= f(1) + f'_+(1) (x - 1) + \\int_y \\ph…","labels":["cor:convex_taylor_statInfoFun"],"detail_key":"p12"},{"id":"n18005","layer":"informal","project":"p12","title":"By Lemma~\\reflem:convex_taylor, for x > 1, f(x) - f(1) - f'_+(1) (x - 1) &= \\int_y \\in (1…","kind":"proof","summary":"By Lemma~\\reflem:convex_taylor, for x > 1, f(x) - f(1) - f'_+(1) (x - 1) &= \\int_y \\in (1, x] (…","labels":[],"detail_key":"p12"},{"id":"n18006","layer":"informal","project":"p12","title":"lem:curvatureMeasure_statInfoFun","kind":"lemma","summary":"The curvature measure of the function \\phi_a,b is \\gamma_\\phi_a,b = a\\delta_b/a~, where \\delta_…","labels":["lem:curvatureMeasure_statInfoFun"],"detail_key":"p12"},{"id":"n18007","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18008","layer":"informal","project":"p12","title":"thm:fDiv_eq_integral_eGamma","kind":"theorem","summary":"For two finite measures \\mu, \\nu \\in M( X), D_f(\\mu, \\nu) = f(1) \\nu( X) + f'_+(1)(\\mu( X) - \\n…","labels":["thm:fDiv_eq_integral_eGamma"],"detail_key":"p12"},{"id":"n18009","layer":"informal","project":"p12","title":"We prove the result for f(1) = 0 and f'_+(1) = 0 for simplicity of exposition. From Corol…","kind":"proof","summary":"We prove the result for f(1) = 0 and f'_+(1) = 0 for simplicity of exposition. From Corollary~\\…","labels":[],"detail_key":"p12"},{"id":"n18010","layer":"informal","project":"p12","title":"\\citeliese2006divergences,liese2012phi","kind":"theorem","summary":"[\\citeliese2006divergences,liese2012phi] D_f(\\mu, \\nu) = f(1) \\nu( X) + f'_+(1)(\\mu( X) - \\nu(…","labels":["thm:fDiv_eq_integral"],"detail_key":"p12"},{"id":"n18011","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18012","layer":"informal","project":"p12","title":"lem:DPI_of_DPI_ac","kind":"lemma","summary":"Let \\kappa : X \\rightsquigarrow Y be a Markov kernel and \\nu \\in M( X) be a finite measure. Sup…","labels":["lem:DPI_of_DPI_ac"],"detail_key":"p12"},{"id":"n18013","layer":"informal","project":"p12","title":"We use \\kappa \\circ \\mu = \\kappa \\circ (\\fracd\\mud\\nu\\cdot \\nu) + \\kappa \\circ \\mu_\\perp…","kind":"proof","summary":"We use \\kappa \\circ \\mu = \\kappa \\circ (\\fracd\\mud\\nu\\cdot \\nu) + \\kappa \\circ \\mu_\\perp \\nu~.…","labels":[],"detail_key":"p12"},{"id":"n18014","layer":"informal","project":"p12","title":"Composition-product with a kernel","kind":"lemma","summary":"[Composition-product with a kernel] Let \\mu, \\nu be two measures on X and let \\kappa : X \\right…","labels":["thm:fDiv_compProd_right_1"],"detail_key":"p12"},{"id":"n18015","layer":"informal","project":"p12","title":"By Lemma~\\refcor:rnDeriv_compProd_left and the fact that (\\mu \\otimes \\kappa)_\\perp \\nu \\…","kind":"proof","summary":"By Lemma~\\refcor:rnDeriv_compProd_left and the fact that (\\mu \\otimes \\kappa)_\\perp \\nu \\otimes…","labels":[],"detail_key":"p12"},{"id":"n18016","layer":"informal","project":"p12","title":"cor:fDiv_prod_right_1","kind":"corollary","summary":"Let \\mu, \\nu be two measures on X and let \\xi be a measure on Y. Then D_f(\\mu \\times \\xi, \\nu \\…","labels":["cor:fDiv_prod_right_1"],"detail_key":"p12"},{"id":"n18017","layer":"informal","project":"p12","title":"Apply Lemma~\\refthm:fDiv_compProd_right_1 with \\kappa the constant kernel with value \\xi.","kind":"proof","summary":"Apply Lemma~\\refthm:fDiv_compProd_right_1 with \\kappa the constant kernel with value \\xi.","labels":[],"detail_key":"p12"},{"id":"n18018","layer":"informal","project":"p12","title":"thm:fDiv_map_le","kind":"theorem","summary":"Let \\mu, \\nu \\in M ( X) be two finite measures and let g : X \\to Y be a measurable function. Th…","labels":["thm:fDiv_map_le"],"detail_key":"p12"},{"id":"n18019","layer":"informal","project":"p12","title":"By Lemma~\\reflem:DPI_of_DPI_ac, it is sufficient to prove the inequality with the additio…","kind":"proof","summary":"By Lemma~\\reflem:DPI_of_DPI_ac, it is sufficient to prove the inequality with the additional hy…","labels":[],"detail_key":"p12"},{"id":"n18020","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\mu, \\nu be two finite measures on X and let \\kappa : X \\rightsquigarrow…","labels":["thm:fDiv_data_proc_3"],"detail_key":"p12"},{"id":"n18021","layer":"informal","project":"p12","title":"Let \\pi_Y : X \\times Y \\to Y be the projection \\pi_Y((x,y)) = y. Then \\kappa \\circ \\mu =…","kind":"proof","summary":"Let \\pi_Y : X \\times Y \\to Y be the projection \\pi_Y((x,y)) = y. Then \\kappa \\circ \\mu = \\pi_Y*…","labels":[],"detail_key":"p12"},{"id":"n18022","layer":"informal","project":"p12","title":"thm:fDiv_le_compProd_1","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two Mar…","labels":["thm:fDiv_le_compProd_1"],"detail_key":"p12"},{"id":"n18023","layer":"informal","project":"p12","title":"TODO: the following proof assumes \\mu \\ll \\nu and \\kappa(x) \\ll \\eta(x) for \\nu-almost al…","kind":"proof","summary":"TODO: the following proof assumes \\mu \\ll \\nu and \\kappa(x) \\ll \\eta(x) for \\nu-almost all x. D…","labels":[],"detail_key":"p12"},{"id":"n18024","layer":"informal","project":"p12","title":"Marginals","kind":"theorem","summary":"[Marginals] Let \\mu and \\nu be two measures on X \\times Y where Y is standard Borel, and let \\m…","labels":["thm:fDiv_fst_le_1"],"detail_key":"p12"},{"id":"n18025","layer":"informal","project":"p12","title":"We introduce conditional kernels and write D(\\mu, \\nu) = D(\\mu_X \\otimes \\mu_Y|X, \\nu_X \\…","kind":"proof","summary":"We introduce conditional kernels and write D(\\mu, \\nu) = D(\\mu_X \\otimes \\mu_Y|X, \\nu_X \\otimes…","labels":[],"detail_key":"p12"},{"id":"n18026","layer":"informal","project":"p12","title":"lem:fDiv_comp_le_compProd_1","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on a standard Borel space X and let \\kappa, \\eta : X \\right…","labels":["lem:fDiv_comp_le_compProd_1"],"detail_key":"p12"},{"id":"n18027","layer":"informal","project":"p12","title":"By definition, \\kappa \\circ \\mu is the marginal of \\mu \\otimes \\kappa (a measure on X \\ti…","kind":"proof","summary":"By definition, \\kappa \\circ \\mu is the marginal of \\mu \\otimes \\kappa (a measure on X \\times Y)…","labels":[],"detail_key":"p12"},{"id":"n18028","layer":"informal","project":"p12","title":"Conditioning increases f-divergence","kind":"theorem","summary":"[Conditioning increases f-divergence] Let \\mu be a measure on a standard Borel space X and let…","labels":["thm:fDiv_comp_le_condFDiv_1"],"detail_key":"p12"},{"id":"n18029","layer":"informal","project":"p12","title":"By Lemma~\\reflem:fDiv_comp_le_compProd_1, D_f(\\kappa \\circ \\mu, \\eta \\circ \\mu) \\le D_f(\\…","kind":"proof","summary":"By Lemma~\\reflem:fDiv_comp_le_compProd_1, D_f(\\kappa \\circ \\mu, \\eta \\circ \\mu) \\le D_f(\\mu \\ot…","labels":[],"detail_key":"p12"},{"id":"n18030","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\mu, \\nu be two measures on X and let \\kappa : X \\rightsquigarrow Y be a…","labels":["thm:fDiv_data_proc_1"],"detail_key":"p12"},{"id":"n18031","layer":"informal","project":"p12","title":"By Lemma~\\reflem:fDiv_comp_le_compProd_1, D_f(\\kappa \\circ \\mu, \\kappa \\circ \\nu) \\le D_f…","kind":"proof","summary":"By Lemma~\\reflem:fDiv_comp_le_compProd_1, D_f(\\kappa \\circ \\mu, \\kappa \\circ \\nu) \\le D_f(\\mu \\…","labels":[],"detail_key":"p12"},{"id":"n18032","layer":"informal","project":"p12","title":"lem:fDiv_statInfoFun_eq","kind":"lemma","summary":"Let a,b \\in [0, +\\infty) and let \\mu, \\nu be two measures on X. D_\\phi_a,b(\\mu, \\nu) &= \\textsi…","labels":["lem:fDiv_statInfoFun_eq"],"detail_key":"p12"},{"id":"n18033","layer":"informal","project":"p12","title":"If a \\le b, D_\\phi_a,b(\\mu, \\nu) &= \\nu\\left[ \\max\\0, a \\fracd\\mud\\nu - b\\ \\right] + a \\m…","kind":"proof","summary":"If a \\le b, D_\\phi_a,b(\\mu, \\nu) &= \\nu\\left[ \\max\\0, a \\fracd\\mud\\nu - b\\ \\right] + a \\mu_\\per…","labels":[],"detail_key":"p12"},{"id":"n18034","layer":"informal","project":"p12","title":"cor:fDiv_statInfoFun_eq_statInfo","kind":"corollary","summary":"Let a,b \\in [0, +\\infty) and let \\mu, \\nu be two measures on X. D_\\phi_a,b(\\mu, \\nu) = I_(a,b)(…","labels":["cor:fDiv_statInfoFun_eq_statInfo"],"detail_key":"p12"},{"id":"n18035","layer":"informal","project":"p12","title":"Combine Lemma~\\reflem:fDiv_statInfoFun_eq and Corollary~\\refcor:statInfo_eq_integral_abs.","kind":"proof","summary":"Combine Lemma~\\reflem:fDiv_statInfoFun_eq and Corollary~\\refcor:statInfo_eq_integral_abs.","labels":[],"detail_key":"p12"},{"id":"n18036","layer":"informal","project":"p12","title":"lem:fDiv_phi_data_proc","kind":"lemma","summary":"Let a,b \\in [0, +\\infty). Let \\mu, \\nu be two finite measures on X and let \\kappa : X \\rightsqu…","labels":["lem:fDiv_phi_data_proc"],"detail_key":"p12"},{"id":"n18037","layer":"informal","project":"p12","title":"By Corollary~\\refcor:fDiv_statInfoFun_eq_statInfo, D_\\phi_a,b(\\mu, \\nu) = I_(a,b)(\\mu, \\n…","kind":"proof","summary":"By Corollary~\\refcor:fDiv_statInfoFun_eq_statInfo, D_\\phi_a,b(\\mu, \\nu) = I_(a,b)(\\mu, \\nu) + \\…","labels":[],"detail_key":"p12"},{"id":"n18038","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\mu, \\nu be two finite measures on X and let \\kappa : X \\rightsquigarrow…","labels":["thm:fDiv_data_proc_2"],"detail_key":"p12"},{"id":"n18039","layer":"informal","project":"p12","title":"By Theorem~\\refthm:fDiv_eq_integral_eGamma, D_f(\\mu, \\nu) = f(1) \\nu( X) + f'_+(1) (\\mu(…","kind":"proof","summary":"By Theorem~\\refthm:fDiv_eq_integral_eGamma, D_f(\\mu, \\nu) = f(1) \\nu( X) + f'_+(1) (\\mu( X) - \\…","labels":[],"detail_key":"p12"},{"id":"n18040","layer":"informal","project":"p12","title":"thm:fDiv_trim_le","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X and let A be a sub-\\sigma-algebra of X. Then D_f(\\mu_|…","labels":["thm:fDiv_trim_le"],"detail_key":"p12"},{"id":"n18041","layer":"informal","project":"p12","title":"The measure \\mu_| A is equal to the map of \\mu by the identity, seen as a function to X w…","kind":"proof","summary":"The measure \\mu_| A is equal to the map of \\mu by the identity, seen as a function to X with th…","labels":[],"detail_key":"p12"},{"id":"n18042","layer":"informal","project":"p12","title":"thm:iSup_fDiv_trim","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X. Then \\sup_ A \\text finite D_f(\\mu_| A, \\nu_| A) = D_f…","labels":["thm:iSup_fDiv_trim"],"detail_key":"p12"},{"id":"n18043","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18044","layer":"informal","project":"p12","title":"lem:fDiv_map_eq_fDiv_trim_of_ac","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) be finite measures with \\mu \\ll \\nu and let g : X \\to Y be a measurable…","labels":["lem:fDiv_map_eq_fDiv_trim_of_ac"],"detail_key":"p12"},{"id":"n18045","layer":"informal","project":"p12","title":"Using Lemma~\\reflem:rnDeriv_map_eq_rnDeriv_trim, D_f(g_* \\mu, g_* \\nu) &= \\int_y f \\left(…","kind":"proof","summary":"Using Lemma~\\reflem:rnDeriv_map_eq_rnDeriv_trim, D_f(g_* \\mu, g_* \\nu) &= \\int_y f \\left( \\frac…","labels":[],"detail_key":"p12"},{"id":"n18046","layer":"informal","project":"p12","title":"thm:fDiv_le_compProd_2","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two Mar…","labels":["thm:fDiv_le_compProd_2"],"detail_key":"p12"},{"id":"n18047","layer":"informal","project":"p12","title":"Since \\kappa is a Markov kernel, \\mu is the composition of \\mu \\otimes \\kappa and the det…","kind":"proof","summary":"Since \\kappa is a Markov kernel, \\mu is the composition of \\mu \\otimes \\kappa and the determini…","labels":[],"detail_key":"p12"},{"id":"n18048","layer":"informal","project":"p12","title":"thm:fDiv_comp_le_compProd_2","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two fin…","labels":["thm:fDiv_comp_le_compProd_2"],"detail_key":"p12"},{"id":"n18049","layer":"informal","project":"p12","title":"\\kappa \\circ \\mu is the composition of \\mu \\otimes \\kappa and the deterministic kernel fo…","kind":"proof","summary":"\\kappa \\circ \\mu is the composition of \\mu \\otimes \\kappa and the deterministic kernel for the…","labels":[],"detail_key":"p12"},{"id":"n18050","layer":"informal","project":"p12","title":"Conditioning increases f-divergence","kind":"theorem","summary":"[Conditioning increases f-divergence] Let \\mu be a finite measure X and let \\kappa, \\eta : X \\r…","labels":["thm:fDiv_comp_le_condFDiv_2"],"detail_key":"p12"},{"id":"n18051","layer":"informal","project":"p12","title":"This is a particular case of Theorem~\\refthm:fDiv_comp_le_compProd_2.","kind":"proof","summary":"This is a particular case of Theorem~\\refthm:fDiv_comp_le_compProd_2.","labels":[],"detail_key":"p12"},{"id":"n18052","layer":"informal","project":"p12","title":"Marginals","kind":"theorem","summary":"[Marginals] Let \\mu and \\nu be two measures on X \\times Y, and let \\mu_X, \\nu_X be their margin…","labels":["thm:fDiv_fst_le_2"],"detail_key":"p12"},{"id":"n18053","layer":"informal","project":"p12","title":"The measure \\mu_X is the composition of \\mu and the deterministic kernel for the function…","kind":"proof","summary":"The measure \\mu_X is the composition of \\mu and the deterministic kernel for the function (x,y)…","labels":[],"detail_key":"p12"},{"id":"n18054","layer":"informal","project":"p12","title":"lem:fDiv_compProd_prod_eq","kind":"lemma","summary":"Let \\mu, \\nu be two measures on X and let \\kappa : X \\rightsquigarrow ( X \\times Y) be a Markov…","labels":["lem:fDiv_compProd_prod_eq"],"detail_key":"p12"},{"id":"n18055","layer":"informal","project":"p12","title":"D_f(\\kappa \\circ \\mu, \\kappa \\circ \\nu) \\le D_f(\\mu, \\nu) by Theorem~\\refthm:fDiv_data_pr…","kind":"proof","summary":"D_f(\\kappa \\circ \\mu, \\kappa \\circ \\nu) \\le D_f(\\mu, \\nu) by Theorem~\\refthm:fDiv_data_proc_2.…","labels":[],"detail_key":"p12"},{"id":"n18056","layer":"informal","project":"p12","title":"cor:data_proc_event","kind":"corollary","summary":"Let \\mu, \\nu be two measures on X and let E be an event. Then D_f(\\mu, \\nu) \\ge d_f(\\mu(E), \\nu…","labels":["cor:data_proc_event"],"detail_key":"p12"},{"id":"n18057","layer":"informal","project":"p12","title":"Use the deterministic kernel \\kappa : X \\rightsquigarrow \\0, 1\\ with \\kappa(x) = \\delta_1…","kind":"proof","summary":"Use the deterministic kernel \\kappa : X \\rightsquigarrow \\0, 1\\ with \\kappa(x) = \\delta_1 I\\x \\…","labels":[],"detail_key":"p12"},{"id":"n18058","layer":"informal","project":"p12","title":"lem:fDiv_bernoulli_convex","kind":"lemma","summary":"For all y \\in [0,1], x \\mapsto d_f(x, y) is convex and attains a minimum at x = y.","labels":["lem:fDiv_bernoulli_convex"],"detail_key":"p12"},{"id":"n18059","layer":"informal","project":"p12","title":"d_f(x, y) &= y f(\\fracxy) + (1 - y) f(\\frac1 - x1 - y) \\: , \\\\ (x \\mapsto d_f(x, y))'_+(x…","kind":"proof","summary":"d_f(x, y) &= y f(\\fracxy) + (1 - y) f(\\frac1 - x1 - y) \\: , \\\\ (x \\mapsto d_f(x, y))'_+(x) &= f…","labels":[],"detail_key":"p12"},{"id":"n18060","layer":"informal","project":"p12","title":"lem:fDiv_bounded_ge_fDiv_mean","kind":"lemma","summary":"Let \\mu, \\nu \\in P([0,1]). Then D_f(\\mu, \\nu) \\ge d_f(\\mu[X], \\nu[X]) \\: .","labels":["lem:fDiv_bounded_ge_fDiv_mean"],"detail_key":"p12"},{"id":"n18061","layer":"informal","project":"p12","title":"Let u be the uniform distribution on [0,1]. Then D_f(\\mu, \\nu) = D_f(\\mu \\times u, \\nu \\t…","kind":"proof","summary":"Let u be the uniform distribution on [0,1]. Then D_f(\\mu, \\nu) = D_f(\\mu \\times u, \\nu \\times u…","labels":[],"detail_key":"p12"},{"id":"n18062","layer":"informal","project":"p12","title":"cor:fDiv_ge_fDiv_mean_comp","kind":"corollary","summary":"Let \\mu, \\nu \\in P( X) and let \\kappa : X \\rightsquigarrow [0,1]. Then D_f(\\mu, \\nu) \\ge d_f((\\…","labels":["cor:fDiv_ge_fDiv_mean_comp"],"detail_key":"p12"},{"id":"n18063","layer":"informal","project":"p12","title":"First, by the data-processing inequality (Theorem~\\refthm:fDiv_data_proc_2), D_f(\\mu, \\nu…","kind":"proof","summary":"First, by the data-processing inequality (Theorem~\\refthm:fDiv_data_proc_2), D_f(\\mu, \\nu) \\ge…","labels":[],"detail_key":"p12"},{"id":"n18064","layer":"informal","project":"p12","title":"lem:fDiv_estimation_ge","kind":"lemma","summary":"Let \\pi, \\xi \\in P(\\Theta) and P, Q : \\Theta \\rightsquigarrow X. Suppose that the loss \\ell' ta…","labels":["lem:fDiv_estimation_ge"],"detail_key":"p12"},{"id":"n18065","layer":"informal","project":"p12","title":"Suppose first that R_\\pi^Q \\ge R_\\xi^P. Let \\haty_B be a Bayes estimator for the estimati…","kind":"proof","summary":"Suppose first that R_\\pi^Q \\ge R_\\xi^P. Let \\haty_B be a Bayes estimator for the estimation tas…","labels":[],"detail_key":"p12"},{"id":"n18066","layer":"informal","project":"p12","title":"Joint convexity","kind":"theorem","summary":"[Joint convexity] The function (\\mu, \\nu) \\mapsto D_f(\\mu, \\nu) is convex.","labels":["thm:fDiv_convex"],"detail_key":"p12"},{"id":"n18067","layer":"informal","project":"p12","title":"Let \\mu_0, \\mu_1, \\nu_0, \\nu_1 be four measures. Let \\lambda \\in [0,1]. Let \\xi be the pr…","kind":"proof","summary":"Let \\mu_0, \\mu_1, \\nu_0, \\nu_1 be four measures. Let \\lambda \\in [0,1]. Let \\xi be the probabil…","labels":[],"detail_key":"p12"},{"id":"n18068","layer":"informal","project":"p12","title":"lem:fDiv_mono_fun","kind":"lemma","summary":"If f \\le g, then D_f(\\mu, \\nu) \\le D_g(\\mu, \\nu).","labels":["lem:fDiv_mono_fun"],"detail_key":"p12"},{"id":"n18069","layer":"informal","project":"p12","title":"Monotonicity of the integral and of f \\mapsto f'(\\infty).","kind":"proof","summary":"Monotonicity of the integral and of f \\mapsto f'(\\infty).","labels":[],"detail_key":"p12"},{"id":"n18070","layer":"informal","project":"p12","title":"lem:fDiv_le_of_deriv2_le","kind":"lemma","summary":"If f(1) = 0, g(1) = 0, f'(1) = 0, g'(1) = 0, and both f and g have a second derivative, then D_…","labels":["lem:fDiv_le_of_deriv2_le"],"detail_key":"p12"},{"id":"n18071","layer":"informal","project":"p12","title":"By Taylor with integral remainder, if f'' \\le \\beta g'', f(x) = \\int_1^x f''(t) (x - t) d…","kind":"proof","summary":"By Taylor with integral remainder, if f'' \\le \\beta g'', f(x) = \\int_1^x f''(t) (x - t) dt \\le…","labels":[],"detail_key":"p12"},{"id":"n18072","layer":"informal","project":"p12","title":"\\citesason2016f","kind":"theorem","summary":"[\\citesason2016f] Suppose that f(1) = g(1) = 0 and that f'(1) = g'(1) = 0, and that g>0 on (0,1…","labels":["thm:fDiv_eq_sup_mul_fDiv"],"detail_key":"p12"},{"id":"n18073","layer":"informal","project":"p12","title":"TODO: proof done for measurable singletons? By Lemma~\\reflem:fDiv_mono_fun, \\sup_\\mu, \\nu…","kind":"proof","summary":"TODO: proof done for measurable singletons? By Lemma~\\reflem:fDiv_mono_fun, \\sup_\\mu, \\nu \\in P…","labels":[],"detail_key":"p12"},{"id":"n18074","layer":"informal","project":"p12","title":"thm:bh_fDiv","kind":"theorem","summary":"Let \\mu, \\nu \\in P( X). If f(1) = 0, D_f(\\mu, \\nu) &\\ge f(1 + \\TV(\\mu, \\nu)) + f(1 - \\TV(\\mu, \\…","labels":["thm:bh_fDiv"],"detail_key":"p12"},{"id":"n18075","layer":"informal","project":"p12","title":"Let V: x \\mapsto \\max\\left\\0, \\fracd\\mud\\nu(x) - 1\\right\\ and W: x \\mapsto \\max\\left\\0, 1…","kind":"proof","summary":"Let V: x \\mapsto \\max\\left\\0, \\fracd\\mud\\nu(x) - 1\\right\\ and W: x \\mapsto \\max\\left\\0, 1 - \\fr…","labels":[],"detail_key":"p12"},{"id":"n18076","layer":"informal","project":"p12","title":"Bretagnolle-Huber inequality","kind":"corollary","summary":"[Bretagnolle-Huber inequality] TODO: move this somewhere after the definition of KL. For \\mu, \\…","labels":["cor:bh_kl"],"detail_key":"p12"},{"id":"n18077","layer":"informal","project":"p12","title":"Use the second inequality of Theorem~\\refthm:bh_fDiv, for f: x \\mapsto x \\log x~.","kind":"proof","summary":"Use the second inequality of Theorem~\\refthm:bh_fDiv, for f: x \\mapsto x \\log x~.","labels":[],"detail_key":"p12"},{"id":"n18078","layer":"informal","project":"p12","title":"cor:cor:bh_hellingerAlpha","kind":"corollary","summary":"TODO: move this somewhere after the definition of \\He_\\alpha. Let \\mu, \\nu \\in P( X). For \\alph…","labels":["cor:cor:bh_hellingerAlpha"],"detail_key":"p12"},{"id":"n18079","layer":"informal","project":"p12","title":"Use the second inequality of Theorem~\\refthm:bh_fDiv, for f: x \\mapsto \\fracx^1+\\alpha -…","kind":"proof","summary":"Use the second inequality of Theorem~\\refthm:bh_fDiv, for f: x \\mapsto \\fracx^1+\\alpha - 1\\alph…","labels":[],"detail_key":"p12"},{"id":"n18080","layer":"informal","project":"p12","title":"lem:statInfo_eq_fDiv","kind":"lemma","summary":"On probability measures, the statistical information I_\\xi is an f-divergence for the function…","labels":["lem:statInfo_eq_fDiv"],"detail_key":"p12"},{"id":"n18081","layer":"informal","project":"p12","title":"This is a reformulation of Theorem~\\refthm:statInfo_eq_integral.","kind":"proof","summary":"This is a reformulation of Theorem~\\refthm:statInfo_eq_integral.","labels":[],"detail_key":"p12"},{"id":"n18082","layer":"informal","project":"p12","title":"lem:tv_eq_fDiv","kind":"lemma","summary":"On probability measures, the total variation distance \\TV is an f-divergence for the function x…","labels":["lem:tv_eq_fDiv"],"detail_key":"p12"},{"id":"n18083","layer":"informal","project":"p12","title":"This is a reformulation of Lemma~\\reflem:tv_eq_integral_abs.","kind":"proof","summary":"This is a reformulation of Lemma~\\reflem:tv_eq_integral_abs.","labels":[],"detail_key":"p12"},{"id":"n18084","layer":"informal","project":"p12","title":"Kullback-Leibler divergence","kind":"definition","summary":"[Kullback-Leibler divergence] Let \\mu, \\nu be two measures on X. The Kullback-Leibler divergenc…","labels":["def:KL"],"detail_key":"p12"},{"id":"n18085","layer":"informal","project":"p12","title":"lem:kl_eq_fDiv","kind":"lemma","summary":"\\KL(\\mu, \\nu) = D_f(\\mu, \\nu) for f: x \\mapsto x \\log x.","labels":["lem:kl_eq_fDiv"],"detail_key":"p12"},{"id":"n18086","layer":"informal","project":"p12","title":"Simple computation.","kind":"proof","summary":"Simple computation.","labels":[],"detail_key":"p12"},{"id":"n18087","layer":"informal","project":"p12","title":"Conditional Kullback-Leibler divergence","kind":"definition","summary":"[Conditional Kullback-Leibler divergence] Let \\mu be a measure on X and \\kappa, \\eta : X \\right…","labels":["def:condKL"],"detail_key":"p12"},{"id":"n18088","layer":"informal","project":"p12","title":"lem:condKL_eq_condFDiv","kind":"lemma","summary":"\\KL(\\kappa, \\eta \\mid \\mu) = D_f(\\kappa, \\eta \\mid \\mu) for f: x \\mapsto x \\log x.","labels":["lem:condKL_eq_condFDiv"],"detail_key":"p12"},{"id":"n18089","layer":"informal","project":"p12","title":"Simple computation.","kind":"proof","summary":"Simple computation.","labels":[],"detail_key":"p12"},{"id":"n18090","layer":"informal","project":"p12","title":"lem:kl_self","kind":"lemma","summary":"\\KL(\\mu, \\mu) = 0.","labels":["lem:kl_self"],"detail_key":"p12"},{"id":"n18091","layer":"informal","project":"p12","title":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv, then apply Lemma~\\reflem:fDiv_s…","kind":"proof","summary":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv, then apply Lemma~\\reflem:fDiv_self.","labels":[],"detail_key":"p12"},{"id":"n18092","layer":"informal","project":"p12","title":"lem:condKL_const","kind":"lemma","summary":"Let \\mu, \\nu be finite measures on X, \\xi be a finite measure on Y. Then \\KL(x \\mapsto \\mu, x \\…","labels":["lem:condKL_const"],"detail_key":"p12"},{"id":"n18093","layer":"informal","project":"p12","title":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv and Lemma~\\reflem:condKL_eq_cond…","kind":"proof","summary":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv and Lemma~\\reflem:condKL_eq_condFDiv,…","labels":[],"detail_key":"p12"},{"id":"n18094","layer":"informal","project":"p12","title":"Marginals","kind":"theorem","summary":"[Marginals] Let \\mu and \\nu be two measures on X \\times Y, and let \\mu_X, \\nu_X be their margin…","labels":["thm:kl_fst_le"],"detail_key":"p12"},{"id":"n18095","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:fDiv_fst_le_2.","kind":"proof","summary":"Apply Theorem~\\refthm:fDiv_fst_le_2.","labels":[],"detail_key":"p12"},{"id":"n18096","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\mu, \\nu be two measures on X and let \\kappa : X \\rightsquigarrow Y be a…","labels":["thm:kl_data_proc"],"detail_key":"p12"},{"id":"n18097","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:fDiv_data_proc_2.","kind":"proof","summary":"Apply Theorem~\\refthm:fDiv_data_proc_2.","labels":[],"detail_key":"p12"},{"id":"n18098","layer":"informal","project":"p12","title":"lem:kl_ge","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X. Then \\KL(\\mu, \\nu) \\ge \\mu( X) \\log \\frac\\mu( X)\\nu(…","labels":["lem:kl_ge"],"detail_key":"p12"},{"id":"n18099","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18100","layer":"informal","project":"p12","title":"Gibbs' inequality","kind":"lemma","summary":"[Gibbs' inequality] Let \\mu, \\nu be two probability measures. Then \\KL(\\mu, \\nu) \\ge 0.","labels":["lem:kl_nonneg"],"detail_key":"p12"},{"id":"n18101","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:kl_ge and use \\mu( X) = \\nu ( X).","kind":"proof","summary":"Apply Lemma~\\reflem:kl_ge and use \\mu( X) = \\nu ( X).","labels":[],"detail_key":"p12"},{"id":"n18102","layer":"informal","project":"p12","title":"Converse Gibbs' inequality","kind":"lemma","summary":"[Converse Gibbs' inequality] Let \\mu, \\nu be two probability measures. Then \\KL(\\mu, \\nu) = 0 i…","labels":["lem:kl_eq_zero_iff"],"detail_key":"p12"},{"id":"n18103","layer":"informal","project":"p12","title":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv, then apply Lemma~\\reflem:fDiv_e…","kind":"proof","summary":"KL is an f-divergence thanks to Lemma~\\reflem:kl_eq_fDiv, then apply Lemma~\\reflem:fDiv_eq_zero…","labels":[],"detail_key":"p12"},{"id":"n18104","layer":"informal","project":"p12","title":"lem:kl_convex","kind":"lemma","summary":"(\\mu, \\nu) \\mapsto \\KL(\\mu, \\nu) is convex.","labels":["lem:kl_convex"],"detail_key":"p12"},{"id":"n18105","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:fDiv_convex","kind":"proof","summary":"Apply Theorem~\\refthm:fDiv_convex","labels":[],"detail_key":"p12"},{"id":"n18106","layer":"informal","project":"p12","title":"lem:integrable_llr_compProd_iff","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X and \\kappa, \\eta : X \\rightsquigarrow Y be two Markov…","labels":["lem:integrable_llr_compProd_iff",":1",":2",":3"],"detail_key":"p12"},{"id":"n18107","layer":"informal","project":"p12","title":":i","kind":"proof","summary":"Note that from the hypothesis it easily follows that \\mu \\ll \\nu and \\mu-a.e.\\ \\kappa(x) \\ll \\e…","labels":[":i",":ii"],"detail_key":"p12"},{"id":"n18108","layer":"informal","project":"p12","title":"thm:kl_compProd_aux","kind":"theorem","summary":"Let \\mu, \\nu be two finite measures on X and \\kappa, \\eta : X \\rightsquigarrow Y two Markov ker…","labels":["thm:kl_compProd_aux"],"detail_key":"p12"},{"id":"n18109","layer":"informal","project":"p12","title":"By Lemma~\\reflem:ac_compProd_iff, \\mu \\otimes \\kappa \\ll \\nu \\otimes \\eta \\iff \\left( \\mu…","kind":"proof","summary":"By Lemma~\\reflem:ac_compProd_iff, \\mu \\otimes \\kappa \\ll \\nu \\otimes \\eta \\iff \\left( \\mu \\ll \\…","labels":[],"detail_key":"p12"},{"id":"n18110","layer":"informal","project":"p12","title":"Chain rule, kernel version","kind":"theorem","summary":"[Chain rule, kernel version] Let \\mu, \\nu be two finite measures on X and \\kappa, \\eta : X \\rig…","labels":["thm:kl_compProd"],"detail_key":"p12"},{"id":"n18111","layer":"informal","project":"p12","title":"Handle the case where \\mu \\otimes \\kappa is not absolutely continuous with respect to \\nu…","kind":"proof","summary":"Handle the case where \\mu \\otimes \\kappa is not absolutely continuous with respect to \\nu \\otim…","labels":[],"detail_key":"p12"},{"id":"n18112","layer":"informal","project":"p12","title":"Chain rule, with Bayesian inverse","kind":"theorem","summary":"[Chain rule, with Bayesian inverse] Let \\mu, \\nu be two finite measures on X and \\kappa, \\eta :…","labels":["thm:kl_compProd_bayesInv"],"detail_key":"p12"},{"id":"n18113","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18114","layer":"informal","project":"p12","title":"Chain rule, product version","kind":"theorem","summary":"[Chain rule, product version] Let \\mu, \\nu be two finite measures on X \\times Y, where Y is sta…","labels":["thm:kl_fst_add_condKL"],"detail_key":"p12"},{"id":"n18115","layer":"informal","project":"p12","title":"Write \\mu = \\mu_X \\otimes \\mu_Y|X and \\nu = \\nu_X \\otimes \\nu_Y|X, then use Theorem~\\reft…","kind":"proof","summary":"Write \\mu = \\mu_X \\otimes \\mu_Y|X and \\nu = \\nu_X \\otimes \\nu_Y|X, then use Theorem~\\refthm:kl_…","labels":[],"detail_key":"p12"},{"id":"n18116","layer":"informal","project":"p12","title":"lem:kl_prod_two'","kind":"lemma","summary":"Let \\mu_1, \\nu_1 be finite measures on X and \\mu_2, \\nu_2 probability measures on Y. Then \\KL(\\…","labels":["lem:kl_prod_two'"],"detail_key":"p12"},{"id":"n18117","layer":"informal","project":"p12","title":"Write \\mu_1 \\times \\mu_2 and \\nu_1 \\times \\nu_2 as composition products of a measure and…","kind":"proof","summary":"Write \\mu_1 \\times \\mu_2 and \\nu_1 \\times \\nu_2 as composition products of a measure and a cons…","labels":[],"detail_key":"p12"},{"id":"n18118","layer":"informal","project":"p12","title":"Tensorization","kind":"theorem","summary":"[Tensorization] For \\mu_1 a probability measure on X, \\nu_1 a finite measure on X and \\mu_2, \\n…","labels":["thm:kl_prod_two"],"detail_key":"p12"},{"id":"n18119","layer":"informal","project":"p12","title":"This is a particular case of Lemma~\\reflem:kl_prod_two'.","kind":"proof","summary":"This is a particular case of Lemma~\\reflem:kl_prod_two'.","labels":[],"detail_key":"p12"},{"id":"n18120","layer":"informal","project":"p12","title":"Tensorization - finite product","kind":"theorem","summary":"[Tensorization - finite product] Let I be a finite index set. Let (\\mu_i)_i \\in I, (\\nu_i)_i \\i…","labels":["thm:kl_pi"],"detail_key":"p12"},{"id":"n18121","layer":"informal","project":"p12","title":"Induction on the size of I, using Theorem~\\refthm:kl_prod_two.","kind":"proof","summary":"Induction on the size of I, using Theorem~\\refthm:kl_prod_two.","labels":[],"detail_key":"p12"},{"id":"n18122","layer":"informal","project":"p12","title":"lem:kl_chain_rule_cond_event","kind":"lemma","summary":"Let \\mu, \\nu be two measures and E an event. Let \\mu_|E be the measure defined by \\mu_|E(A) = \\…","labels":["lem:kl_chain_rule_cond_event"],"detail_key":"p12"},{"id":"n18123","layer":"informal","project":"p12","title":"Let \\mu_E be the Bernoulli distribution with \\mu_E(\\1\\) = \\mu(E), and define \\nu_E simila…","kind":"proof","summary":"Let \\mu_E be the Bernoulli distribution with \\mu_E(\\1\\) = \\mu(E), and define \\nu_E similarly. L…","labels":[],"detail_key":"p12"},{"id":"n18124","layer":"informal","project":"p12","title":"lem:expectation_llr_event","kind":"lemma","summary":"Let \\mu, \\nu be two measures and E an event. Then \\mu(E)\\log\\frac\\mu(E)\\nu(E) \\le \\mu\\left[I(E)…","labels":["lem:expectation_llr_event"],"detail_key":"p12"},{"id":"n18125","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:kl_ge to the measures \\mu and \\nu restricted to E. \\mu\\left[I(E)\\log…","kind":"proof","summary":"Apply Lemma~\\reflem:kl_ge to the measures \\mu and \\nu restricted to E. \\mu\\left[I(E)\\log \\fracd…","labels":[],"detail_key":"p12"},{"id":"n18126","layer":"informal","project":"p12","title":"Dummy lemma: KL properties","kind":"lemma","summary":"[Dummy lemma: KL properties] Dummy node to summarize properties of the Kullback-Leibler diverge…","labels":["lem:kl_properties"],"detail_key":"p12"},{"id":"n18127","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18128","layer":"informal","project":"p12","title":"Hellinger \\alpha-divergence","kind":"definition","summary":"[Hellinger \\alpha-divergence] Let \\mu, \\nu be two measures on X. The Hellinger divergence of or…","labels":["def:hellingerAlpha"],"detail_key":"p12"},{"id":"n18129","layer":"informal","project":"p12","title":"lem:hellingerAlpha_ne_top_of_lt_one","kind":"lemma","summary":"For \\alpha \\in [0, 1) and finite measures \\mu, \\nu, \\He_\\alpha(\\mu, \\nu) < \\infty.","labels":["lem:hellingerAlpha_ne_top_of_lt_one"],"detail_key":"p12"},{"id":"n18130","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18131","layer":"informal","project":"p12","title":"lem:hellingerAlpha_eq_integral","kind":"lemma","summary":"For \\alpha \\in (0,1)\\cup(1, \\infty), \\mu a finite measure and \\nu a probability measure, if \\He…","labels":["lem:hellingerAlpha_eq_integral"],"detail_key":"p12"},{"id":"n18132","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18133","layer":"informal","project":"p12","title":"lem:integral_rpow_rnDeriv","kind":"lemma","summary":"For \\alpha \\in (0,1)\\cup(1, \\infty), \\mu, \\nu two sigma-finite measures, \\int_x \\left(\\fracd \\m…","labels":["lem:integral_rpow_rnDeriv"],"detail_key":"p12"},{"id":"n18134","layer":"informal","project":"p12","title":"\\int_x \\left(\\fracd \\mud \\nu(x)\\right)^\\alpha \\partial \\nu &= \\int_x \\left(\\fracd \\mud (\\…","kind":"proof","summary":"\\int_x \\left(\\fracd \\mud \\nu(x)\\right)^\\alpha \\partial \\nu &= \\int_x \\left(\\fracd \\mud (\\mu + \\…","labels":[],"detail_key":"p12"},{"id":"n18135","layer":"informal","project":"p12","title":"lem:hellingerAlpha_symm","kind":"lemma","summary":"For \\alpha \\in (0, 1) and finite measures \\mu, \\nu with \\mu( X) = \\nu( X), (1 - \\alpha) \\He_\\al…","labels":["lem:hellingerAlpha_symm"],"detail_key":"p12"},{"id":"n18136","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:integral_rpow_rnDeriv.","kind":"proof","summary":"Use Lemma~\\reflem:integral_rpow_rnDeriv.","labels":[],"detail_key":"p12"},{"id":"n18137","layer":"informal","project":"p12","title":"Conditional Hellinger \\alpha-divergence","kind":"definition","summary":"[Conditional Hellinger \\alpha-divergence] Let \\mu be a measure on X and \\kappa, \\eta : X \\right…","labels":["def:condHellingerAlpha"],"detail_key":"p12"},{"id":"n18138","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\alpha > 0, \\mu, \\nu be two finite measures on X and let \\kappa : X \\righ…","labels":["thm:hellingerAlpha_data_proc"],"detail_key":"p12"},{"id":"n18139","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:fDiv_data_proc_2.","kind":"proof","summary":"Apply Theorem~\\refthm:fDiv_data_proc_2.","labels":[],"detail_key":"p12"},{"id":"n18140","layer":"informal","project":"p12","title":"lem:hellingerAlpha_nonneg","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. Then \\He_\\alpha(\\mu, \\nu) \\ge 0.","labels":["lem:hellingerAlpha_nonneg"],"detail_key":"p12"},{"id":"n18141","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:fDiv_nonneg.","kind":"proof","summary":"Apply Lemma~\\reflem:fDiv_nonneg.","labels":[],"detail_key":"p12"},{"id":"n18142","layer":"informal","project":"p12","title":"lem:hellingerAlpha_convex","kind":"lemma","summary":"(\\mu, \\nu) \\mapsto \\He_\\alpha(\\mu, \\nu) is convex.","labels":["lem:hellingerAlpha_convex"],"detail_key":"p12"},{"id":"n18143","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:fDiv_convex","kind":"proof","summary":"Apply Theorem~\\refthm:fDiv_convex","labels":[],"detail_key":"p12"},{"id":"n18144","layer":"informal","project":"p12","title":"Dummy lemma: hellingerAlpha properties","kind":"lemma","summary":"[Dummy lemma: hellingerAlpha properties] Dummy node to summarize properties of the Hellinger \\a…","labels":["lem:hellingerAlpha_properties"],"detail_key":"p12"},{"id":"n18145","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18146","layer":"informal","project":"p12","title":"Rényi divergence","kind":"definition","summary":"[Rényi divergence] Let \\mu, \\nu be two measures on X. The Rényi divergence of order \\alpha \\in…","labels":["def:Renyi"],"detail_key":"p12"},{"id":"n18147","layer":"informal","project":"p12","title":"lem:renyiDiv_zero","kind":"lemma","summary":"For \\mu a sigma-finite measure and \\nu a finite measure \\[R_0(\\mu, \\nu) = - \\log(\\nu\\x \\mid \\fr…","labels":["lem:renyiDiv_zero"],"detail_key":"p12"},{"id":"n18148","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18149","layer":"informal","project":"p12","title":"lem:renyiDiv_eq_top_iff_mutuallySingular_of_lt_one","kind":"lemma","summary":"For \\alpha \\in [0, 1) and finite measures \\mu, \\nu, \\[R_\\alpha(\\mu, \\nu) = \\infty \\iff \\mu \\per…","labels":["lem:renyiDiv_eq_top_iff_mutuallySingular_of_lt_one"],"detail_key":"p12"},{"id":"n18150","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18151","layer":"informal","project":"p12","title":"lem:renyi_eq_log_integral","kind":"lemma","summary":"For \\alpha \\in (0,1)\\cup(1, \\infty) and finite measures \\mu, \\nu, if \\left(\\fracd \\mud \\nu\\righ…","labels":["lem:renyi_eq_log_integral"],"detail_key":"p12"},{"id":"n18152","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18153","layer":"informal","project":"p12","title":"lem:renyi_eq_log_integral'","kind":"lemma","summary":"For \\alpha \\in (0,1)\\cup(1, \\infty) and finite measures \\mu, \\nu, if \\left(\\fracd \\mud \\nu\\righ…","labels":["lem:renyi_eq_log_integral'"],"detail_key":"p12"},{"id":"n18154","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18155","layer":"informal","project":"p12","title":"lem:renyi_symm","kind":"lemma","summary":"For \\alpha \\in (0, 1) and finite measures \\mu, \\nu with \\mu( X) = \\nu( X), \\[(1 - \\alpha) R_\\al…","labels":["lem:renyi_symm"],"detail_key":"p12"},{"id":"n18156","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:hellingerAlpha_symm.","kind":"proof","summary":"Use Lemma~\\reflem:hellingerAlpha_symm.","labels":[],"detail_key":"p12"},{"id":"n18157","layer":"informal","project":"p12","title":"lem:renyi_cgf","kind":"lemma","summary":"The cumulant generating function of \\log\\fracd\\mud\\nu under \\nu is \\alpha \\mapsto (\\alpha - 1)…","labels":["lem:renyi_cgf"],"detail_key":"p12"},{"id":"n18158","layer":"informal","project":"p12","title":"Unfold the definitions, using Lemma~\\reflem:renyi_eq_log_integral for the Rényi divergenc…","kind":"proof","summary":"Unfold the definitions, using Lemma~\\reflem:renyi_eq_log_integral for the Rényi divergence.","labels":[],"detail_key":"p12"},{"id":"n18159","layer":"informal","project":"p12","title":"lem:renyi_cgf_2","kind":"lemma","summary":"Set \\alpha > 0. If R_1+\\alpha(\\mu, \\nu) < \\infty, the cumulant generating function of \\log\\frac…","labels":["lem:renyi_cgf_2"],"detail_key":"p12"},{"id":"n18160","layer":"informal","project":"p12","title":"Unfold the definitions, using Lemma~\\reflem:renyi_eq_log_integral' for the Rényi divergen…","kind":"proof","summary":"Unfold the definitions, using Lemma~\\reflem:renyi_eq_log_integral' for the Rényi divergence.","labels":[],"detail_key":"p12"},{"id":"n18161","layer":"informal","project":"p12","title":"Conditional Rényi divergence","kind":"definition","summary":"[Conditional Rényi divergence] Let \\mu be a measure on X and \\kappa, \\eta : X \\rightsquigarrow…","labels":["def:condRenyi"],"detail_key":"p12"},{"id":"n18162","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] Let \\mu, \\nu be two finite measures on X and let \\kappa : X \\rightsquigarrow…","labels":["thm:renyi_data_proc"],"detail_key":"p12"},{"id":"n18163","layer":"informal","project":"p12","title":"The function x \\mapsto \\frac1\\alpha - 1\\log (1 + (\\alpha - 1)x) is non-decreasing and D_f…","kind":"proof","summary":"The function x \\mapsto \\frac1\\alpha - 1\\log (1 + (\\alpha - 1)x) is non-decreasing and D_f satis…","labels":[],"detail_key":"p12"},{"id":"n18164","layer":"informal","project":"p12","title":"lem:renyi_data_proc_event","kind":"lemma","summary":"Let \\mu, \\nu be two measures on X and let E be an event. Let \\mu_E and \\nu_E be the two Bernoul…","labels":["lem:renyi_data_proc_event"],"detail_key":"p12"},{"id":"n18165","layer":"informal","project":"p12","title":"By Corollary~\\refcor:data_proc_event, D_f(\\mu, \\nu) \\ge D_f(\\mu_E, \\nu_E), hence R_\\alpha…","kind":"proof","summary":"By Corollary~\\refcor:data_proc_event, D_f(\\mu, \\nu) \\ge D_f(\\mu_E, \\nu_E), hence R_\\alpha(\\mu,…","labels":[],"detail_key":"p12"},{"id":"n18166","layer":"informal","project":"p12","title":"lem:renyi_tendsto_renyi_zero","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures. R_0(\\mu, \\nu) = \\lim_\\alpha \\downarrow 0 R_\\alpha(\\mu, \\nu…","labels":["lem:renyi_tendsto_renyi_zero"],"detail_key":"p12"},{"id":"n18167","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18168","layer":"informal","project":"p12","title":"lem:renyi_tendsto_renyi_one","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures. R_1(\\mu, \\nu) = \\lim_\\alpha \\uparrow 1 R_\\alpha(\\mu, \\nu).","labels":["lem:renyi_tendsto_renyi_one"],"detail_key":"p12"},{"id":"n18169","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18170","layer":"informal","project":"p12","title":"lem:renyi_tendsto_renyi_one_above","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures such that there exists \\alpha > 1 with R_\\alpha(\\mu, \\nu) f…","labels":["lem:renyi_tendsto_renyi_one_above"],"detail_key":"p12"},{"id":"n18171","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18172","layer":"informal","project":"p12","title":"lem:renyi_monotone","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures. Then \\alpha \\mapsto R_\\alpha(\\mu, \\nu) is nondecreasing on…","labels":["lem:renyi_monotone"],"detail_key":"p12"},{"id":"n18173","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18174","layer":"informal","project":"p12","title":"lem:renyi_continuous","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures. Then \\alpha \\mapsto R_\\alpha(\\mu, \\nu) is continuous on [0…","labels":["lem:renyi_continuous"],"detail_key":"p12"},{"id":"n18175","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18176","layer":"informal","project":"p12","title":"def:renyiMeasure","kind":"definition","summary":"Let \\mu, \\nu be two measures on X and let \\alpha \\in (0, +\\infty) \\backslash \\1\\. Let p = \\frac…","labels":["def:renyiMeasure"],"detail_key":"p12"},{"id":"n18177","layer":"informal","project":"p12","title":"lem:renyiMeasure_ac","kind":"lemma","summary":"For \\alpha \\in (0,1), \\mu^(\\alpha, \\nu) \\ll \\mu and \\mu^(\\alpha, \\nu) \\ll \\nu.","labels":["lem:renyiMeasure_ac"],"detail_key":"p12"},{"id":"n18178","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18179","layer":"informal","project":"p12","title":"lem:rnDeriv_renyiMeasure","kind":"lemma","summary":"\\fracd \\mu^(\\alpha, \\nu)d \\nu = \\left(\\fracd\\mud\\nu\\right)^\\alpha e^-(\\alpha - 1) R_\\alpha(\\mu,…","labels":["lem:rnDeriv_renyiMeasure"],"detail_key":"p12"},{"id":"n18180","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18181","layer":"informal","project":"p12","title":"lem:kl_renyiMeasure_eq","kind":"lemma","summary":"Let \\mu, \\nu, \\xi be three measures on X and let \\alpha \\in (0, 1). Then \\KL(\\xi, \\mu^(\\alpha,…","labels":["lem:kl_renyiMeasure_eq"],"detail_key":"p12"},{"id":"n18182","layer":"informal","project":"p12","title":"Unfold definitions and compute?","kind":"proof","summary":"Unfold definitions and compute?","labels":[],"detail_key":"p12"},{"id":"n18183","layer":"informal","project":"p12","title":"cor:renyi_eq_add_kl","kind":"corollary","summary":"Let \\mu, \\nu, \\xi be three measures on X and let \\alpha \\in (0, 1). Then (1 - \\alpha) R_\\alpha(…","labels":["cor:renyi_eq_add_kl"],"detail_key":"p12"},{"id":"n18184","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:kl_renyiMeasure_eq to \\xi = \\mu^(\\alpha, \\nu).","kind":"proof","summary":"Apply Lemma~\\reflem:kl_renyiMeasure_eq to \\xi = \\mu^(\\alpha, \\nu).","labels":[],"detail_key":"p12"},{"id":"n18185","layer":"informal","project":"p12","title":"lem:renyi_eq_inf_add_kl","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures on X and let \\alpha \\in (0, 1). Then (1 - \\alpha) R_\\a…","labels":["lem:renyi_eq_inf_add_kl"],"detail_key":"p12"},{"id":"n18186","layer":"informal","project":"p12","title":"By Lemma~\\reflem:kl_renyiMeasure_eq and Lemma~\\reflem:kl_nonneg, for all \\xi, \\alpha \\KL(…","kind":"proof","summary":"By Lemma~\\reflem:kl_renyiMeasure_eq and Lemma~\\reflem:kl_nonneg, for all \\xi, \\alpha \\KL(\\xi, \\…","labels":[],"detail_key":"p12"},{"id":"n18187","layer":"informal","project":"p12","title":"lem:renyi_eq_inf_kl","kind":"lemma","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). Let \\pi_\\alpha = (\\alpha, 1 - \\alpha) \\in P(\\…","labels":["lem:renyi_eq_inf_kl"],"detail_key":"p12"},{"id":"n18188","layer":"informal","project":"p12","title":"\\KL\\left( \\pi_\\alpha \\times \\xi, \\pi_\\alpha \\otimes P \\right) &= \\alpha \\KL(\\xi, \\mu) + (…","kind":"proof","summary":"\\KL\\left( \\pi_\\alpha \\times \\xi, \\pi_\\alpha \\otimes P \\right) &= \\alpha \\KL(\\xi, \\mu) + (1 - \\a…","labels":[],"detail_key":"p12"},{"id":"n18189","layer":"informal","project":"p12","title":"Chain rule","kind":"theorem","summary":"[Chain rule] Let \\mu, \\nu be two measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be t…","labels":["thm:renyi_chain_rule"],"detail_key":"p12"},{"id":"n18190","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18191","layer":"informal","project":"p12","title":"Chain rule, with Bayesian inverse","kind":"theorem","summary":"[Chain rule, with Bayesian inverse] Let \\mu, \\nu be two finite measures on X and \\kappa, \\eta :…","labels":["thm:renyi_compProd_bayesInv"],"detail_key":"p12"},{"id":"n18192","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18193","layer":"informal","project":"p12","title":"cor:renyi_prod_two","kind":"corollary","summary":"Let \\mu, \\nu \\in P( X) and \\xi, \\lambda \\in P( Y). Then R_\\alpha(\\mu \\times \\xi, \\nu \\times \\la…","labels":["cor:renyi_prod_two"],"detail_key":"p12"},{"id":"n18194","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:renyi_chain_rule (the product of measures is a special case of \\oti…","kind":"proof","summary":"Apply Theorem~\\refthm:renyi_chain_rule (the product of measures is a special case of \\otimes).…","labels":[],"detail_key":"p12"},{"id":"n18195","layer":"informal","project":"p12","title":"Tensorization - finite product","kind":"theorem","summary":"[Tensorization - finite product] Let I be a finite index set. Let (\\mu_i)_i \\in I, (\\nu_i)_i \\i…","labels":["thm:renyi_prod"],"detail_key":"p12"},{"id":"n18196","layer":"informal","project":"p12","title":"Induction over the finite index set, using Corollary~\\refcor:renyi_prod_two.","kind":"proof","summary":"Induction over the finite index set, using Corollary~\\refcor:renyi_prod_two.","labels":[],"detail_key":"p12"},{"id":"n18197","layer":"informal","project":"p12","title":"lem:renyi_prod_n","kind":"corollary","summary":"Let \\mu, \\nu be two probability measures on X. Let n \\in N and write \\mu^\\otimes n for the prod…","labels":["lem:renyi_prod_n"],"detail_key":"p12"},{"id":"n18198","layer":"informal","project":"p12","title":"Apply Theorem~\\refthm:renyi_prod.","kind":"proof","summary":"Apply Theorem~\\refthm:renyi_prod.","labels":[],"detail_key":"p12"},{"id":"n18199","layer":"informal","project":"p12","title":"Tensorization - countable product","kind":"theorem","summary":"[Tensorization - countable product] Let I be a countable index set. Let (\\mu_i)_i \\in I, (\\nu_i…","labels":["thm:renyi_prod_countable"],"detail_key":"p12"},{"id":"n18200","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18201","layer":"informal","project":"p12","title":"Dummy lemma: Renyi properties","kind":"lemma","summary":"[Dummy lemma: Renyi properties] Dummy node to summarize properties of the Rényi divergence.","labels":["lem:renyi_properties"],"detail_key":"p12"},{"id":"n18202","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18203","layer":"informal","project":"p12","title":"Squared Hellinger distance","kind":"definition","summary":"[Squared Hellinger distance] Let \\mu, \\nu be two measures. The squared Hellinger distance betwe…","labels":["def:Hellinger"],"detail_key":"p12"},{"id":"n18204","layer":"informal","project":"p12","title":"Hellinger and Rényi","kind":"lemma","summary":"[Hellinger and Rényi] Let \\mu, \\nu be two probability measures. Then R_1/2(\\mu, \\nu) = -2\\log(1…","labels":["lem:renyi_half_eq_log_hellinger"],"detail_key":"p12"},{"id":"n18205","layer":"informal","project":"p12","title":"R_1/2(\\mu, \\nu) = -2 \\log (1 - \\frac12 D_f(\\nu, \\mu)) for f : x \\mapsto -2 (\\sqrtx - 1).…","kind":"proof","summary":"R_1/2(\\mu, \\nu) = -2 \\log (1 - \\frac12 D_f(\\nu, \\mu)) for f : x \\mapsto -2 (\\sqrtx - 1). Using…","labels":[],"detail_key":"p12"},{"id":"n18206","layer":"informal","project":"p12","title":"lem:hellinger_le_renyi","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. Then 2 \\Hsq(\\mu, \\nu) \\le R_1/2(\\mu, \\nu).","labels":["lem:hellinger_le_renyi"],"detail_key":"p12"},{"id":"n18207","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:renyi_half_eq_log_hellinger, then -\\log(1 - x) \\ge x.","kind":"proof","summary":"Use Lemma~\\reflem:renyi_half_eq_log_hellinger, then -\\log(1 - x) \\ge x.","labels":[],"detail_key":"p12"},{"id":"n18208","layer":"informal","project":"p12","title":"lem:hellinger_le_tv","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. Then \\Hsq(\\mu, \\nu) \\le \\TV(\\mu, \\nu).","labels":["lem:hellinger_le_tv"],"detail_key":"p12"},{"id":"n18209","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18210","layer":"informal","project":"p12","title":"lem:tv_le_hellinger","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures. Then \\TV(\\mu, \\nu) \\le \\sqrt\\Hsq(\\mu, \\nu)(2 - \\Hsq(\\…","labels":["lem:tv_le_hellinger"],"detail_key":"p12"},{"id":"n18211","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18212","layer":"informal","project":"p12","title":"cor:one_sub_hellinger_squared_le_one_sub_tv","kind":"corollary","summary":"Let \\mu, \\nu be two probability measures. Then \\frac12(1 - \\Hsq(\\mu, \\nu))^2 \\le 1 - \\TV(\\mu, \\…","labels":["cor:one_sub_hellinger_squared_le_one_sub_tv"],"detail_key":"p12"},{"id":"n18213","layer":"informal","project":"p12","title":"We use Lemma~\\reflem:tv_le_hellinger. 1 - \\TV(\\mu, \\nu) - \\frac12(1 - \\Hsq(\\mu, \\nu))^2 &…","kind":"proof","summary":"We use Lemma~\\reflem:tv_le_hellinger. 1 - \\TV(\\mu, \\nu) - \\frac12(1 - \\Hsq(\\mu, \\nu))^2 &= \\fra…","labels":[],"detail_key":"p12"},{"id":"n18214","layer":"informal","project":"p12","title":"cor:one_sub_hellinger_le_one_sub_tv","kind":"corollary","summary":"Let \\mu, \\nu be two probability measures. Then 1 - \\sqrt1 - (1 - \\Hsq(\\mu, \\nu))^2 \\le 1 - \\TV(…","labels":["cor:one_sub_hellinger_le_one_sub_tv"],"detail_key":"p12"},{"id":"n18215","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:tv_le_hellinger and~\\reflem:hellinger_le_tv.","kind":"proof","summary":"Use Lemma~\\reflem:tv_le_hellinger and~\\reflem:hellinger_le_tv.","labels":[],"detail_key":"p12"},{"id":"n18216","layer":"informal","project":"p12","title":"cor:one_sub_tv_bound_renyi","kind":"corollary","summary":"Let \\mu, \\nu be two probability measures on X and let n \\in N, and \\mu^\\otimes n, \\nu^\\otimes n…","labels":["cor:one_sub_tv_bound_renyi"],"detail_key":"p12"},{"id":"n18217","layer":"informal","project":"p12","title":"Use Corollary~\\refcor:one_sub_hellinger_le_one_sub_tv and Lemma~\\reflem:renyi_half_eq_log…","kind":"proof","summary":"Use Corollary~\\refcor:one_sub_hellinger_le_one_sub_tv and Lemma~\\reflem:renyi_half_eq_log_helli…","labels":[],"detail_key":"p12"},{"id":"n18218","layer":"informal","project":"p12","title":"Chernoff divergence","kind":"definition","summary":"[Chernoff divergence] The Chernoff divergence of order \\alpha > 0 between two measures \\mu and…","labels":["def:Chernoff"],"detail_key":"p12"},{"id":"n18219","layer":"informal","project":"p12","title":"lem:chernoff_eq_kl","kind":"lemma","summary":"C_1(\\mu, \\nu) = \\inf_\\xi \\in P( X)\\max\\\\KL(\\xi, \\mu), \\KL(\\xi, \\nu)\\ .","labels":["lem:chernoff_eq_kl"],"detail_key":"p12"},{"id":"n18220","layer":"informal","project":"p12","title":"This is R_1 = \\KL (by definition).","kind":"proof","summary":"This is R_1 = \\KL (by definition).","labels":[],"detail_key":"p12"},{"id":"n18221","layer":"informal","project":"p12","title":"Symmetry","kind":"lemma","summary":"[Symmetry] C_\\alpha(\\mu, \\nu) = C_\\alpha(\\nu, \\mu).","labels":["lem:chernoff_symm"],"detail_key":"p12"},{"id":"n18222","layer":"informal","project":"p12","title":"Immediate from the definition.","kind":"proof","summary":"Immediate from the definition.","labels":[],"detail_key":"p12"},{"id":"n18223","layer":"informal","project":"p12","title":"Monotonicity","kind":"lemma","summary":"[Monotonicity] The function \\alpha \\mapsto C_\\alpha(\\mu, \\nu) is monotone.","labels":["lem:chernoff_mono"],"detail_key":"p12"},{"id":"n18224","layer":"informal","project":"p12","title":"Consequence of Lemma~\\reflem:renyi_monotone.","kind":"proof","summary":"Consequence of Lemma~\\reflem:renyi_monotone.","labels":[],"detail_key":"p12"},{"id":"n18225","layer":"informal","project":"p12","title":"lem:chernoff_eq_max_renyi","kind":"lemma","summary":"C_1(\\mu, \\nu) = \\max_\\alpha\\in [0,1] (1 - \\alpha)R_\\alpha(\\mu, \\nu) \\: .","labels":["lem:chernoff_eq_max_renyi"],"detail_key":"p12"},{"id":"n18226","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18227","layer":"informal","project":"p12","title":"Jensen-Shannon divergence","kind":"definition","summary":"[Jensen-Shannon divergence] The Jensen-Shannon divergence indexed by \\alpha \\in (0,1) between t…","labels":["def:jensenShannon"],"detail_key":"p12"},{"id":"n18228","layer":"informal","project":"p12","title":"lem:jensenShannon_eq_kl","kind":"lemma","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). Let \\pi_\\alpha = (\\alpha, 1 - \\alpha) \\in P(\\…","labels":["lem:jensenShannon_eq_kl"],"detail_key":"p12"},{"id":"n18229","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18230","layer":"informal","project":"p12","title":"lem:jensenShannon_eq_fDiv","kind":"lemma","summary":"\\JS_\\alpha is an f-divergence for f(x) = \\alpha x \\log(x) - (\\alpha x + 1 - \\alpha) \\log (\\alph…","labels":["lem:jensenShannon_eq_fDiv"],"detail_key":"p12"},{"id":"n18231","layer":"informal","project":"p12","title":"(\\mu, \\nu) \\mapsto \\KL(\\mu, \\alpha \\mu + (1 - \\alpha) \\nu) is an f-divergence for the fun…","kind":"proof","summary":"(\\mu, \\nu) \\mapsto \\KL(\\mu, \\alpha \\mu + (1 - \\alpha) \\nu) is an f-divergence for the function…","labels":[],"detail_key":"p12"},{"id":"n18232","layer":"informal","project":"p12","title":"lem:jensenShannon_symm","kind":"lemma","summary":"For \\alpha \\in (0,1) and \\mu, \\nu \\in M( X), \\JS_\\alpha(\\mu, \\nu) = \\JS_1 - \\alpha(\\nu, \\mu) \\:…","labels":["lem:jensenShannon_symm"],"detail_key":"p12"},{"id":"n18233","layer":"informal","project":"p12","title":"Immediate from the definition.","kind":"proof","summary":"Immediate from the definition.","labels":[],"detail_key":"p12"},{"id":"n18234","layer":"informal","project":"p12","title":"lem:jensenShannon_eq_inf_add_kl","kind":"lemma","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). Then \\JS_\\alpha(\\mu, \\nu) = \\inf_\\xi \\in P( X…","labels":["lem:jensenShannon_eq_inf_add_kl"],"detail_key":"p12"},{"id":"n18235","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18236","layer":"informal","project":"p12","title":"lem:jensenShannon_eq_inf_kl","kind":"lemma","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). Let \\pi_\\alpha = (\\alpha, 1 - \\alpha) \\in P(\\…","labels":["lem:jensenShannon_eq_inf_kl"],"detail_key":"p12"},{"id":"n18237","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18238","layer":"informal","project":"p12","title":"lem:jensenShannon_prod_n","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures on X. Let n \\in N and write \\mu^\\otimes n for the prod…","labels":["lem:jensenShannon_prod_n"],"detail_key":"p12"},{"id":"n18239","layer":"informal","project":"p12","title":"By Lemma~\\reflem:jensenShannon_eq_inf_add_kl, \\JS_\\alpha(\\mu^\\otimes n, \\nu^\\otimes n) &=…","kind":"proof","summary":"By Lemma~\\reflem:jensenShannon_eq_inf_add_kl, \\JS_\\alpha(\\mu^\\otimes n, \\nu^\\otimes n) &= \\inf_…","labels":[],"detail_key":"p12"},{"id":"n18240","layer":"informal","project":"p12","title":"lem:jensenShannon_le_hellingerAlpha","kind":"lemma","summary":"For \\mu, \\nu \\in P( X) and \\alpha, \\lambda \\in (0,1)~, \\JS_\\alpha(\\mu, \\nu) \\le \\frac(1 - \\alph…","labels":["lem:jensenShannon_le_hellingerAlpha"],"detail_key":"p12"},{"id":"n18241","layer":"informal","project":"p12","title":"The main tool is Lemma~\\reflem:fDiv_le_of_deriv2_le. \\JS_\\alpha is an f-divergence with f…","kind":"proof","summary":"The main tool is Lemma~\\reflem:fDiv_le_of_deriv2_le. \\JS_\\alpha is an f-divergence with functio…","labels":[],"detail_key":"p12"},{"id":"n18242","layer":"informal","project":"p12","title":"cor:jensenShannon_le_hellingerAlpha_of_le_half","kind":"corollary","summary":"For \\mu, \\nu \\in P( X) and \\alpha \\in (0,1), \\lambda \\le 1/2~, \\JS_\\alpha(\\mu, \\nu) &\\le \\alpha…","labels":["cor:jensenShannon_le_hellingerAlpha_of_le_half"],"detail_key":"p12"},{"id":"n18243","layer":"informal","project":"p12","title":"By Lemma~\\reflem:jensenShannon_le_hellingerAlpha for \\alpha and 1 - \\lambda, \\JS_\\alpha(\\…","kind":"proof","summary":"By Lemma~\\reflem:jensenShannon_le_hellingerAlpha for \\alpha and 1 - \\lambda, \\JS_\\alpha(\\mu, \\n…","labels":[],"detail_key":"p12"},{"id":"n18244","layer":"informal","project":"p12","title":"def:mutualInfo","kind":"definition","summary":"The mutual information is, for \\rho \\in M( X \\times Y)~, I(\\rho) = \\KL(\\rho, \\rho_X \\times \\rho…","labels":["def:mutualInfo"],"detail_key":"p12"},{"id":"n18245","layer":"informal","project":"p12","title":"def:condMutualInfo","kind":"definition","summary":"Let \\kappa : Z \\rightsquigarrow X \\times Y. The conditional mutual information of \\kappa with r…","labels":["def:condMutualInfo"],"detail_key":"p12"},{"id":"n18246","layer":"informal","project":"p12","title":"lem:mutualInfo_eq_condKL","kind":"lemma","summary":"For \\mu \\in M( X) and \\kappa : X \\rightsquigarrow Y a Markov kernel, I(\\mu \\otimes \\kappa) = KL…","labels":["lem:mutualInfo_eq_condKL"],"detail_key":"p12"},{"id":"n18247","layer":"informal","project":"p12","title":"The first equality is the definition of I, together with (\\mu \\otimes \\kappa)_X = \\mu (si…","kind":"proof","summary":"The first equality is the definition of I, together with (\\mu \\otimes \\kappa)_X = \\mu (since \\k…","labels":[],"detail_key":"p12"},{"id":"n18248","layer":"informal","project":"p12","title":"lem:mutualInfo_symm","kind":"lemma","summary":"I(\\rho_\\leftrightarrow) = I(\\rho)~.","labels":["lem:mutualInfo_symm"],"detail_key":"p12"},{"id":"n18249","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18250","layer":"informal","project":"p12","title":"lem:mutualInfo_eq_jensenShannon","kind":"lemma","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). Let \\pi_\\alpha = (\\alpha, 1 - \\alpha) \\in P(\\…","labels":["lem:mutualInfo_eq_jensenShannon"],"detail_key":"p12"},{"id":"n18251","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:mutualInfo_eq_condKL and Lemma~\\reflem:jensenShannon_eq_kl.","kind":"proof","summary":"Use Lemma~\\reflem:mutualInfo_eq_condKL and Lemma~\\reflem:jensenShannon_eq_kl.","labels":[],"detail_key":"p12"},{"id":"n18252","layer":"informal","project":"p12","title":"Data-processing","kind":"theorem","summary":"[Data-processing] For \\rho \\in M( X \\times Y), \\kappa : X \\rightsquigarrow X' and \\eta : Y \\rig…","labels":["thm:mutualInfo_data_proc"],"detail_key":"p12"},{"id":"n18253","layer":"informal","project":"p12","title":"((\\kappa \\parallel \\eta) \\circ \\rho)_X' = \\kappa \\circ \\rho_X and ((\\kappa \\parallel \\eta…","kind":"proof","summary":"((\\kappa \\parallel \\eta) \\circ \\rho)_X' = \\kappa \\circ \\rho_X and ((\\kappa \\parallel \\eta) \\cir…","labels":[],"detail_key":"p12"},{"id":"n18254","layer":"informal","project":"p12","title":"cor:mutualInfo_compProd_le","kind":"corollary","summary":"For \\mu \\in X, \\kappa : X \\rightsquigarrow Y and \\eta : Y \\rightsquigarrow Z two Markov kernels…","labels":["cor:mutualInfo_compProd_le"],"detail_key":"p12"},{"id":"n18255","layer":"informal","project":"p12","title":"First, rewrite \\mu \\otimes (\\eta \\circ \\kappa) = (id \\parallel \\eta) \\circ (\\mu \\otimes \\…","kind":"proof","summary":"First, rewrite \\mu \\otimes (\\eta \\circ \\kappa) = (id \\parallel \\eta) \\circ (\\mu \\otimes \\kappa)…","labels":[],"detail_key":"p12"},{"id":"n18256","layer":"informal","project":"p12","title":"def:mutualInfoLeft","kind":"definition","summary":"Let D be a divergence between measures. The left D-mutual information for a measure \\mu \\in M(…","labels":["def:mutualInfoLeft"],"detail_key":"p12"},{"id":"n18257","layer":"informal","project":"p12","title":"def:mutualInfoRight","kind":"definition","summary":"Let D be a divergence between measures. The right D-mutual information for a measure \\mu \\in M(…","labels":["def:mutualInfoRight"],"detail_key":"p12"},{"id":"n18258","layer":"informal","project":"p12","title":"lem:mutualInfoLeft_eq_renyi","kind":"lemma","summary":"For \\mu \\in P(\\0,1\\) and \\kappa : \\0,1\\ \\rightsquigarrow Y, I_\\KL^L(\\mu, \\kappa) = (1 - \\mu_0)…","labels":["lem:mutualInfoLeft_eq_renyi"],"detail_key":"p12"},{"id":"n18259","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18260","layer":"informal","project":"p12","title":"lem:mutualInfoRight_eq_mutualInfo","kind":"lemma","summary":"For \\mu \\in P( X) and \\kappa : X \\rightsquigarrow Y, I_\\KL^R(\\mu, \\kappa) = I(\\mu \\otimes \\kapp…","labels":["lem:mutualInfoRight_eq_mutualInfo"],"detail_key":"p12"},{"id":"n18261","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18262","layer":"informal","project":"p12","title":"cor:mutualInfoRight_eq_jensenShannon","kind":"corollary","summary":"For \\mu \\in P(\\0,1\\) and \\kappa : \\0,1\\ \\rightsquigarrow Y, I_\\KL^R(\\mu, \\kappa) = \\JS_\\mu_0(\\k…","labels":["cor:mutualInfoRight_eq_jensenShannon"],"detail_key":"p12"},{"id":"n18263","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18264","layer":"informal","project":"p12","title":"thm:mutualInfoLeft_comp_le","kind":"theorem","summary":"If the divergence D satisfies the data-processing inequality, then for all \\mu \\in M( X), \\kapp…","labels":["thm:mutualInfoLeft_comp_le"],"detail_key":"p12"},{"id":"n18265","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18266","layer":"informal","project":"p12","title":"lem:mutualInfoLeft_estimation_ge","kind":"lemma","summary":"Let \\pi \\in P(\\Theta) and P : \\Theta \\rightsquigarrow X. Suppose that the loss \\ell' of an esti…","labels":["lem:mutualInfoLeft_estimation_ge"],"detail_key":"p12"},{"id":"n18267","layer":"informal","project":"p12","title":"The left mutual information I_D^L(\\pi, P) is an infimum: I_D^L(\\pi, P) = \\inf_\\xi \\in P(…","kind":"proof","summary":"The left mutual information I_D^L(\\pi, P) is an infimum: I_D^L(\\pi, P) = \\inf_\\xi \\in P( X) D(\\…","labels":[],"detail_key":"p12"},{"id":"n18268","layer":"informal","project":"p12","title":"\\citezhou2018non","kind":"lemma","summary":"[\\citezhou2018non] Let \\zeta be a measure such that \\mu \\ll \\zeta and \\nu \\ll \\zeta. Let p = \\f…","labels":["lem:bayesBinaryRisk_eq_exp_renyi_mul_integral"],"detail_key":"p12"},{"id":"n18269","layer":"informal","project":"p12","title":"B_\\xi(\\mu, \\nu) &= \\int_x \\min \\left\\\\xi_0 p(x), \\xi_1 q(x)\\right\\ \\partial\\zeta \\\\ &= \\i…","kind":"proof","summary":"B_\\xi(\\mu, \\nu) &= \\int_x \\min \\left\\\\xi_0 p(x), \\xi_1 q(x)\\right\\ \\partial\\zeta \\\\ &= \\int_x (…","labels":[],"detail_key":"p12"},{"id":"n18270","layer":"informal","project":"p12","title":"cor:bayesBinaryRisk_le_exp_renyi","kind":"corollary","summary":"For \\alpha \\in (0,1), B_\\xi(\\mu, \\nu) \\le e^-(1 - \\alpha) R_\\alpha(\\xi_0\\mu, \\xi_1\\nu) = \\xi_0^…","labels":["cor:bayesBinaryRisk_le_exp_renyi"],"detail_key":"p12"},{"id":"n18271","layer":"informal","project":"p12","title":"Use g_\\alpha(x) \\le 0 in Lemma~\\reflem:bayesBinaryRisk_eq_exp_renyi_mul_integral.","kind":"proof","summary":"Use g_\\alpha(x) \\le 0 in Lemma~\\reflem:bayesBinaryRisk_eq_exp_renyi_mul_integral.","labels":[],"detail_key":"p12"},{"id":"n18272","layer":"informal","project":"p12","title":"lem:bayesBinaryRisk_le_exp_chernoff","kind":"lemma","summary":"B_\\xi(\\mu, \\nu) \\le e^- C_1(\\xi_0\\mu, \\xi_1\\nu) \\: .","labels":["lem:bayesBinaryRisk_le_exp_chernoff"],"detail_key":"p12"},{"id":"n18273","layer":"informal","project":"p12","title":"Optimize over \\alpha in Corollary~\\refcor:bayesBinaryRisk_le_exp_renyi: the optimal value…","kind":"proof","summary":"Optimize over \\alpha in Corollary~\\refcor:bayesBinaryRisk_le_exp_renyi: the optimal value gives…","labels":[],"detail_key":"p12"},{"id":"n18274","layer":"informal","project":"p12","title":"lem:one_sub_tv_le_exp_chernoff","kind":"lemma","summary":"For probability measures, 1 - \\TV(\\mu, \\nu) \\le e^- C_1(\\mu, \\nu) \\: .","labels":["lem:one_sub_tv_le_exp_chernoff"],"detail_key":"p12"},{"id":"n18275","layer":"informal","project":"p12","title":"By definition, since \\mu and \\nu are probability measures, 1 - \\TV(\\mu, \\nu) = B_(1,1)(\\m…","kind":"proof","summary":"By definition, since \\mu and \\nu are probability measures, 1 - \\TV(\\mu, \\nu) = B_(1,1)(\\mu, \\nu…","labels":[],"detail_key":"p12"},{"id":"n18276","layer":"informal","project":"p12","title":"lem:kl_estimation_ge","kind":"lemma","summary":"Let \\pi, \\xi \\in P(\\Theta) and P, Q : \\Theta \\rightsquigarrow X. Suppose that the loss \\ell' ta…","labels":["lem:kl_estimation_ge"],"detail_key":"p12"},{"id":"n18277","layer":"informal","project":"p12","title":"This is Lemma~\\reflem:fDiv_estimation_ge specialized to the Kullback-Leibler divergence.","kind":"proof","summary":"This is Lemma~\\reflem:fDiv_estimation_ge specialized to the Kullback-Leibler divergence.","labels":[],"detail_key":"p12"},{"id":"n18278","layer":"informal","project":"p12","title":"cor:kl_estimation_ge_sub_kl'","kind":"corollary","summary":"Let \\pi, \\xi \\in P(\\Theta) and P, Q : \\Theta \\rightsquigarrow X. Suppose that the loss \\ell' ta…","labels":["cor:kl_estimation_ge_sub_kl'"],"detail_key":"p12"},{"id":"n18279","layer":"informal","project":"p12","title":"Use \\KL(\\pi \\otimes Q, \\xi \\otimes P) = \\KL(\\pi, \\xi) + \\KL(\\pi \\otimes Q, \\pi \\otimes P)…","kind":"proof","summary":"Use \\KL(\\pi \\otimes Q, \\xi \\otimes P) = \\KL(\\pi, \\xi) + \\KL(\\pi \\otimes Q, \\pi \\otimes P) in Le…","labels":[],"detail_key":"p12"},{"id":"n18280","layer":"informal","project":"p12","title":"cor:kl_estimation_ge_binary","kind":"corollary","summary":"Let \\alpha, \\beta \\in (0, 1). Let P, Q : \\0,1\\ \\rightsquigarrow X. We write \\pi_\\alpha for the…","labels":["cor:kl_estimation_ge_binary"],"detail_key":"p12"},{"id":"n18281","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:kl_estimation_ge.","kind":"proof","summary":"Apply Lemma~\\reflem:kl_estimation_ge.","labels":[],"detail_key":"p12"},{"id":"n18282","layer":"informal","project":"p12","title":"cor:kl_estimation_ge_sub_kl","kind":"corollary","summary":"Let \\mu, \\nu \\in P( X) and let \\alpha, \\beta \\in (0, 1). Let P, Q : \\0,1\\ \\rightsquigarrow X. W…","labels":["cor:kl_estimation_ge_sub_kl"],"detail_key":"p12"},{"id":"n18283","layer":"informal","project":"p12","title":"Apply Corollary~\\refcor:kl_estimation_ge_sub_kl'.","kind":"proof","summary":"Apply Corollary~\\refcor:kl_estimation_ge_sub_kl'.","labels":[],"detail_key":"p12"},{"id":"n18284","layer":"informal","project":"p12","title":"lem:kl_estimation_ge_prod","kind":"lemma","summary":"Let \\mu, \\nu, \\xi \\in P( X) and let \\alpha, \\beta \\in (0, 1). Let P : \\0,1\\ \\rightsquigarrow X…","labels":["lem:kl_estimation_ge_prod"],"detail_key":"p12"},{"id":"n18285","layer":"informal","project":"p12","title":"We apply Corollary~\\refcor:kl_estimation_ge_sub_kl for the constant kernel with value \\xi…","kind":"proof","summary":"We apply Corollary~\\refcor:kl_estimation_ge_sub_kl for the constant kernel with value \\xi and u…","labels":[],"detail_key":"p12"},{"id":"n18286","layer":"informal","project":"p12","title":"Fano's inequality, binary case","kind":"theorem","summary":"[Fano's inequality, binary case] Let \\mu, \\nu \\in P( X) and let \\alpha \\in (0, 1). h_2(\\alpha)…","labels":["thm:sub_bayesBinaryRisk_le_jensenShannon"],"detail_key":"p12"},{"id":"n18287","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:jensenShannon_eq_inf_kl and then Lemma~\\reflem:kl_estimation_ge_prod…","kind":"proof","summary":"Apply Lemma~\\reflem:jensenShannon_eq_inf_kl and then Lemma~\\reflem:kl_estimation_ge_prod with \\…","labels":[],"detail_key":"p12"},{"id":"n18288","layer":"informal","project":"p12","title":"thm:log_inv_bayesBinaryRisk_le_renyi","kind":"theorem","summary":"For \\alpha, \\beta \\in (0, 1/2), \\beta \\log\\frac\\alphaB_\\alpha(\\mu, \\nu) + (1 - \\beta) \\log\\frac…","labels":["thm:log_inv_bayesBinaryRisk_le_renyi"],"detail_key":"p12"},{"id":"n18289","layer":"informal","project":"p12","title":"By Lemma~\\reflem:renyi_eq_inf_kl and then Lemma~\\reflem:kl_estimation_ge_prod, (1 - \\beta…","kind":"proof","summary":"By Lemma~\\reflem:renyi_eq_inf_kl and then Lemma~\\reflem:kl_estimation_ge_prod, (1 - \\beta) R_\\b…","labels":[],"detail_key":"p12"},{"id":"n18290","layer":"informal","project":"p12","title":"lem:testing_bound_renyi_mean","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures on X and E an event. Let \\alpha \\in (0,1). Then \\mu(E)…","labels":["lem:testing_bound_renyi_mean"],"detail_key":"p12"},{"id":"n18291","layer":"informal","project":"p12","title":"Let \\mu_E and \\nu_E be the two Bernoulli distributions with respective means \\mu(E) and \\…","kind":"proof","summary":"Let \\mu_E and \\nu_E be the two Bernoulli distributions with respective means \\mu(E) and \\nu(E).…","labels":[],"detail_key":"p12"},{"id":"n18292","layer":"informal","project":"p12","title":"cor:testing_bound_hellinger","kind":"corollary","summary":"Let \\mu, \\nu be two probability measures on X and E an event. Then \\sqrt\\mu(E) + \\sqrt\\nu(E^c)…","labels":["cor:testing_bound_hellinger"],"detail_key":"p12"},{"id":"n18293","layer":"informal","project":"p12","title":"The inequality is an application of Lemma~\\reflem:testing_bound_renyi_mean for \\alpha = 1…","kind":"proof","summary":"The inequality is an application of Lemma~\\reflem:testing_bound_renyi_mean for \\alpha = 1/2. Th…","labels":[],"detail_key":"p12"},{"id":"n18294","layer":"informal","project":"p12","title":"Change of measure lemma","kind":"lemma","summary":"[Change of measure lemma] Let \\mu, \\nu be two measures on X with \\mu \\ll \\nu and let E be an ev…","labels":["lem:llr_change_measure"],"detail_key":"p12"},{"id":"n18295","layer":"informal","project":"p12","title":"\\nu(E) \\ge \\mu\\left[I(E) e^- \\log\\fracd \\mud \\nu \\right] &\\ge \\mu\\left[I\\left(E \\cap \\lef…","kind":"proof","summary":"\\nu(E) \\ge \\mu\\left[I(E) e^- \\log\\fracd \\mud \\nu \\right] &\\ge \\mu\\left[I\\left(E \\cap \\left\\\\log…","labels":[],"detail_key":"p12"},{"id":"n18296","layer":"informal","project":"p12","title":"Change of measure - functions","kind":"lemma","summary":"[Change of measure - functions] Let \\mu, \\nu be two measures on X with \\mu \\ll \\nu and let f :…","labels":["lem:llr_change_measure_fun"],"detail_key":"p12"},{"id":"n18297","layer":"informal","project":"p12","title":"\\nu[f] \\ge \\mu\\left[f e^- \\log\\fracd \\mud \\nu \\right] &\\ge \\mu\\left[f I\\left\\\\log\\fracd \\…","kind":"proof","summary":"\\nu[f] \\ge \\mu\\left[f e^- \\log\\fracd \\mud \\nu \\right] &\\ge \\mu\\left[f I\\left\\\\log\\fracd \\mud \\n…","labels":[],"detail_key":"p12"},{"id":"n18298","layer":"informal","project":"p12","title":"lem:change_measure_risk","kind":"lemma","summary":"Consider an estimation problem with loss \\ell' : Y \\times Z \\to [0,1]. Let \\pi, \\zeta \\in P(\\Th…","labels":["lem:change_measure_risk"],"detail_key":"p12"},{"id":"n18299","layer":"informal","project":"p12","title":"Let \\haty_B be a Bayes estimator for (P, y, \\ell'). If no such estimator exists, the proo…","kind":"proof","summary":"Let \\haty_B be a Bayes estimator for (P, y, \\ell'). If no such estimator exists, the proof can…","labels":[],"detail_key":"p12"},{"id":"n18300","layer":"informal","project":"p12","title":"lem:change_measure_risk_inf","kind":"lemma","summary":"Consider an estimation problem with loss \\ell' : Y \\times Z \\to [0,1]. Let \\pi, \\zeta \\in P(\\Th…","labels":["lem:change_measure_risk_inf"],"detail_key":"p12"},{"id":"n18301","layer":"informal","project":"p12","title":"For any \\xi, we apply Lemma~\\reflem:change_measure_risk to \\zeta \\times \\xi and \\pi \\otim…","kind":"proof","summary":"For any \\xi, we apply Lemma~\\reflem:change_measure_risk to \\zeta \\times \\xi and \\pi \\otimes P a…","labels":[],"detail_key":"p12"},{"id":"n18302","layer":"informal","project":"p12","title":"3 points change of measure","kind":"lemma","summary":"[3 points change of measure] Let \\mu, \\nu, \\xi \\in P( X) and let E be an event on X. Let \\beta_…","labels":["lem:llr_change_measure_add"],"detail_key":"p12"},{"id":"n18303","layer":"informal","project":"p12","title":"Two applications of Lemma~\\reflem:llr_change_measure, then sum them and use \\xi(E)+\\xi(E^…","kind":"proof","summary":"Two applications of Lemma~\\reflem:llr_change_measure, then sum them and use \\xi(E)+\\xi(E^c) = 1.","labels":[],"detail_key":"p12"},{"id":"n18304","layer":"informal","project":"p12","title":"Change of measure - mean","kind":"corollary","summary":"[Change of measure - mean] Let \\mu, \\nu be two measures on X and let E be an event on X. Let \\b…","labels":["cor:kl_change_measure"],"detail_key":"p12"},{"id":"n18305","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:llr_change_measure with the choice \\KL(\\mu, \\nu) + \\beta for \\beta.","kind":"proof","summary":"Use Lemma~\\reflem:llr_change_measure with the choice \\KL(\\mu, \\nu) + \\beta for \\beta.","labels":[],"detail_key":"p12"},{"id":"n18306","layer":"informal","project":"p12","title":"lem:change_measure_risk_mean","kind":"lemma","summary":"For \\alpha \\in (0,1), let \\pi_\\alpha \\in P(\\0,1\\) be the measure (\\alpha, 1 - \\alpha). Let \\alp…","labels":["lem:change_measure_risk_mean"],"detail_key":"p12"},{"id":"n18307","layer":"informal","project":"p12","title":"An application of Lemma~\\reflem:change_measure_risk_inf gives B_\\pi_\\alpha(\\mu, \\nu) e^\\b…","kind":"proof","summary":"An application of Lemma~\\reflem:change_measure_risk_inf gives B_\\pi_\\alpha(\\mu, \\nu) e^\\beta \\g…","labels":[],"detail_key":"p12"},{"id":"n18308","layer":"informal","project":"p12","title":"Change of measure - variance","kind":"lemma","summary":"[Change of measure - variance] Let \\mu, \\nu be two measures on X such that \\mu\\left[\\left(\\log\\…","labels":["lem:llr_change_measure_variance"],"detail_key":"p12"},{"id":"n18309","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:llr_change_measure with the choice \\KL(\\mu, \\nu) + \\sqrt\\Var_\\mu[\\log\\f…","kind":"proof","summary":"Use Lemma~\\reflem:llr_change_measure with the choice \\KL(\\mu, \\nu) + \\sqrt\\Var_\\mu[\\log\\fracd \\…","labels":[],"detail_key":"p12"},{"id":"n18310","layer":"informal","project":"p12","title":"Change of measure - c.g.f.","kind":"lemma","summary":"[Change of measure - c.g.f.] For \\mu, \\nu finite measures and \\alpha, \\beta > 0, \\mu\\left\\ \\log…","labels":["lem:renyi_chernoff_bound"],"detail_key":"p12"},{"id":"n18311","layer":"informal","project":"p12","title":"This is a Chernoff bound, using that the cumulant generating function of \\log\\fracd\\mud\\n…","kind":"proof","summary":"This is a Chernoff bound, using that the cumulant generating function of \\log\\fracd\\mud\\nu unde…","labels":[],"detail_key":"p12"},{"id":"n18312","layer":"informal","project":"p12","title":"lem:renyi_change_measure","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X and let E be an event on X. Let \\alpha,\\beta > 0. Then…","labels":["lem:renyi_change_measure"],"detail_key":"p12"},{"id":"n18313","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:llr_change_measure with the choice R_1+\\alpha(\\mu, \\nu) + \\beta for \\be…","kind":"proof","summary":"Use Lemma~\\reflem:llr_change_measure with the choice R_1+\\alpha(\\mu, \\nu) + \\beta for \\beta. Th…","labels":[],"detail_key":"p12"},{"id":"n18314","layer":"informal","project":"p12","title":"lem:change_measure_risk_cgf","kind":"lemma","summary":"For \\alpha \\in (0,1), let \\pi_\\alpha \\in P(\\0,1\\) be the measure (\\alpha, 1 - \\alpha). Let \\alp…","labels":["lem:change_measure_risk_cgf"],"detail_key":"p12"},{"id":"n18315","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18316","layer":"informal","project":"p12","title":"lem:change_measure_variance_add","kind":"lemma","summary":"Let \\mu, \\nu, \\xi be three probability measures on X and let E be an event on X. For \\beta > 0~…","labels":["lem:change_measure_variance_add"],"detail_key":"p12"},{"id":"n18317","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:llr_change_measure_add with the choices \\KL(\\xi, \\mu) + \\sqrt\\beta \\Var…","kind":"proof","summary":"Use Lemma~\\reflem:llr_change_measure_add with the choices \\KL(\\xi, \\mu) + \\sqrt\\beta \\Var_\\xi\\l…","labels":[],"detail_key":"p12"},{"id":"n18318","layer":"informal","project":"p12","title":"lem:renyi_change_measure_add","kind":"lemma","summary":"Let \\mu, \\nu, \\xi be three probability measures on X and let E be an event on X. Let \\alpha, \\b…","labels":["lem:renyi_change_measure_add"],"detail_key":"p12"},{"id":"n18319","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:llr_change_measure_add with the choices R_1+\\alpha(\\xi, \\mu) + \\beta an…","kind":"proof","summary":"Use Lemma~\\reflem:llr_change_measure_add with the choices R_1+\\alpha(\\xi, \\mu) + \\beta and R_1+…","labels":[],"detail_key":"p12"},{"id":"n18320","layer":"informal","project":"p12","title":"lem:testing_bound_renyi_one_add","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures on X and let E be an event on X. Let \\alpha > 0. Then…","labels":["lem:testing_bound_renyi_one_add"],"detail_key":"p12"},{"id":"n18321","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:renyi_change_measure_add with \\beta = \\log(4)/\\alpha and use that \\mu(E…","kind":"proof","summary":"Use Lemma~\\reflem:renyi_change_measure_add with \\beta = \\log(4)/\\alpha and use that \\mu(E) e^R_…","labels":[],"detail_key":"p12"},{"id":"n18322","layer":"informal","project":"p12","title":"lem:testing_bound_renyi_one_add_n","kind":"lemma","summary":"Let \\mu, \\nu be two probability measures on X, let n \\in N and let E be an event on X^n. For al…","labels":["lem:testing_bound_renyi_one_add_n"],"detail_key":"p12"},{"id":"n18323","layer":"informal","project":"p12","title":"Use Lemma~\\reflem:renyi_prod_n in Lemma~\\reflem:testing_bound_renyi_one_add. TODO: add a…","kind":"proof","summary":"Use Lemma~\\reflem:renyi_prod_n in Lemma~\\reflem:testing_bound_renyi_one_add. TODO: add a lemma…","labels":[],"detail_key":"p12"},{"id":"n18324","layer":"informal","project":"p12","title":"thm:testing_bound_chernoff","kind":"theorem","summary":"Let \\mu, \\nu be two probability measures on X and let (E_n)_n \\in N be events on X^n. For all \\…","labels":["thm:testing_bound_chernoff"],"detail_key":"p12"},{"id":"n18325","layer":"informal","project":"p12","title":"Let \\xi be a probability measure on X and \\beta > 0. By Corollary~\\refcor:kl_change_measu…","kind":"proof","summary":"Let \\xi be a probability measure on X and \\beta > 0. By Corollary~\\refcor:kl_change_measure, \\m…","labels":[],"detail_key":"p12"},{"id":"n18326","layer":"informal","project":"p12","title":"def:binaryPriorSampleComplexity","kind":"definition","summary":"The sample complexity of simple binary hypothesis testing with prior (\\pi, 1 - \\pi) \\in P(\\0, 1…","labels":["def:binaryPriorSampleComplexity"],"detail_key":"p12"},{"id":"n18327","layer":"informal","project":"p12","title":"lem:binaryPriorSampleComplexity_eq_zero","kind":"lemma","summary":"For \\delta \\ge \\min\\\\pi, 1 - \\pi\\, the sample complexity of simple binary hypothesis testing is…","labels":["lem:binaryPriorSampleComplexity_eq_zero"],"detail_key":"p12"},{"id":"n18328","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18329","layer":"informal","project":"p12","title":"lem:binaryPriorSampleComplexity_le_renyi","kind":"lemma","summary":"The sample complexity of simple binary hypothesis testing satisfies n(\\mu, \\nu, \\pi, \\delta) \\l…","labels":["lem:binaryPriorSampleComplexity_le_renyi"],"detail_key":"p12"},{"id":"n18330","layer":"informal","project":"p12","title":"It suffices to show that B_\\pi(\\mu^\\otimes n_0, \\nu^\\otimes n_0) \\le \\delta~. By Corollar…","kind":"proof","summary":"It suffices to show that B_\\pi(\\mu^\\otimes n_0, \\nu^\\otimes n_0) \\le \\delta~. By Corollary~\\ref…","labels":[],"detail_key":"p12"},{"id":"n18331","layer":"informal","project":"p12","title":"lem:binaryPriorSampleComplexity_ge_renyi","kind":"lemma","summary":"For \\delta \\le \\pi \\le 1/2, the sample complexity of simple binary hypothesis testing satisfies…","labels":["lem:binaryPriorSampleComplexity_ge_renyi"],"detail_key":"p12"},{"id":"n18332","layer":"informal","project":"p12","title":"By Theorem~\\refthm:log_inv_bayesBinaryRisk_le_renyi, \\log\\frac\\piB_\\pi(\\mu^\\otimes n, \\nu…","kind":"proof","summary":"By Theorem~\\refthm:log_inv_bayesBinaryRisk_le_renyi, \\log\\frac\\piB_\\pi(\\mu^\\otimes n, \\nu^\\otim…","labels":[],"detail_key":"p12"},{"id":"n18333","layer":"informal","project":"p12","title":"lem:binaryPriorSampleComplexity_ge_jensenShannon","kind":"lemma","summary":"For \\delta \\le \\min\\\\pi, 1 - \\pi\\, the sample complexity of simple binary hypothesis testing sa…","labels":["lem:binaryPriorSampleComplexity_ge_jensenShannon"],"detail_key":"p12"},{"id":"n18334","layer":"informal","project":"p12","title":"We start from Theorem~\\refthm:sub_bayesBinaryRisk_le_jensenShannon. \\JS_\\pi(\\mu^\\otimes n…","kind":"proof","summary":"We start from Theorem~\\refthm:sub_bayesBinaryRisk_le_jensenShannon. \\JS_\\pi(\\mu^\\otimes n, \\nu^…","labels":[],"detail_key":"p12"},{"id":"n18335","layer":"informal","project":"p12","title":"lem:llr_filtration_nat","kind":"lemma","summary":"For \\mu, \\nu \\in P( X) with \\mu \\ll \\nu and n \\in N, \\nu^\\otimes N_| F_n-almost surely, \\fracd…","labels":["lem:llr_filtration_nat"],"detail_key":"p12"},{"id":"n18336","layer":"informal","project":"p12","title":"By Lemma~\\reflem:rnDeriv_map_eq_rnDeriv_trim, \\fracd \\mu^\\otimes N_| F_nd \\nu^\\otimes N_|…","kind":"proof","summary":"By Lemma~\\reflem:rnDeriv_map_eq_rnDeriv_trim, \\fracd \\mu^\\otimes N_| F_nd \\nu^\\otimes N_| F_n(x…","labels":[],"detail_key":"p12"},{"id":"n18337","layer":"informal","project":"p12","title":"lem:llr_stopping_time","kind":"lemma","summary":"For \\mu, \\nu \\in P( X) with \\mu \\ll \\nu, \\nu_\\tau-almost surely, \\fracd \\mu_\\taud \\nu_\\tau(x) =…","labels":["lem:llr_stopping_time"],"detail_key":"p12"},{"id":"n18338","layer":"informal","project":"p12","title":"It suffices to show the equality of their integrals on F_\\tau-measurable sets. Let E be s…","kind":"proof","summary":"It suffices to show the equality of their integrals on F_\\tau-measurable sets. Let E be such a…","labels":[],"detail_key":"p12"},{"id":"n18339","layer":"informal","project":"p12","title":"lem:wald_equation","kind":"lemma","summary":"\\notready Let \\mu \\in P( X) and let (X_n)_n \\in N be \\mu-integrable real random variables. Let…","labels":["lem:wald_equation"],"detail_key":"p12"},{"id":"n18340","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18341","layer":"informal","project":"p12","title":"thm:kl_stopping_time","kind":"theorem","summary":"For \\mu, \\nu \\in P( X), if \\tau has finite expectation then \\KL(\\mu_\\tau, \\nu_\\tau) = \\mu[\\tau]…","labels":["thm:kl_stopping_time"],"detail_key":"p12"},{"id":"n18342","layer":"informal","project":"p12","title":"TODO: need Wald's first identity.","kind":"proof","summary":"TODO: need Wald's first identity.","labels":[],"detail_key":"p12"},{"id":"n18343","layer":"informal","project":"p12","title":"lem:renyiMeasure_stopping_time","kind":"lemma","summary":"For \\mu, \\nu two probability measures on X and \\alpha \\in (0,1), (\\mu^(\\alpha, \\nu))_\\tau = (\\m…","labels":["lem:renyiMeasure_stopping_time"],"detail_key":"p12"},{"id":"n18344","layer":"informal","project":"p12","title":"This is a guess, I did not check it. TODO","kind":"proof","summary":"This is a guess, I did not check it. TODO","labels":[],"detail_key":"p12"},{"id":"n18345","layer":"informal","project":"p12","title":"thm:renyi_stopping_time","kind":"theorem","summary":"For \\mu, \\nu two probability measures on X and \\alpha \\in (0,1), R_\\alpha(\\mu_\\tau, \\nu_\\tau) =…","labels":["thm:renyi_stopping_time"],"detail_key":"p12"},{"id":"n18346","layer":"informal","project":"p12","title":"From Corollary~\\refcor:renyi_eq_add_kl, (1 - \\alpha) R_\\alpha(\\mu, \\nu) = \\alpha \\KL(\\mu^…","kind":"proof","summary":"From Corollary~\\refcor:renyi_eq_add_kl, (1 - \\alpha) R_\\alpha(\\mu, \\nu) = \\alpha \\KL(\\mu^(\\alph…","labels":[],"detail_key":"p12"},{"id":"n18347","layer":"informal","project":"p12","title":"Measurable space of measures","kind":"definition","summary":"[Measurable space of measures] \\mathlibok Let B be the Borel \\sigma-algebra on R_+,\\infty. Let…","labels":["def:measure_measurableSpace"],"detail_key":"p12"},{"id":"n18348","layer":"informal","project":"p12","title":"lem:measurable_measure_fun","kind":"lemma","summary":"\\mathlibok Let X, Y be measurable spaces, and let f : X \\to M( Y) such that for all measurable…","labels":["lem:measurable_measure_fun"],"detail_key":"p12"},{"id":"n18349","layer":"informal","project":"p12","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p12"},{"id":"n18350","layer":"informal","project":"p12","title":"Kernel","kind":"definition","summary":"[Kernel] \\mathlibok Let X, Y be two measurable spaces. A probability transition kernel (or simp…","labels":["def:kernel"],"detail_key":"p12"},{"id":"n18351","layer":"informal","project":"p12","title":"Deterministic kernel","kind":"definition","summary":"[Deterministic kernel] \\mathlibok The deterministic kernel defined by a measurable function f :…","labels":["def:deterministic_kernel"],"detail_key":"p12"},{"id":"n18352","layer":"informal","project":"p12","title":"thm:kernel_ext","kind":"theorem","summary":"\\mathlibok Two kernels \\kappa, \\eta : X \\rightsquigarrow Y are equal iff for all measurable fun…","labels":["thm:kernel_ext"],"detail_key":"p12"},{"id":"n18353","layer":"informal","project":"p12","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p12"},{"id":"n18354","layer":"informal","project":"p12","title":"Finite kernel","kind":"definition","summary":"[Finite kernel] \\mathlibok A kernel \\kappa : X \\rightsquigarrow Y is said to be finite if there…","labels":["def:finite_kernel"],"detail_key":"p12"},{"id":"n18355","layer":"informal","project":"p12","title":"Markov kernel","kind":"definition","summary":"[Markov kernel] \\mathlibok A kernel \\kappa : X \\rightsquigarrow Y is said to be a Markov kernel…","labels":["def:markov_kernel"],"detail_key":"p12"},{"id":"n18356","layer":"informal","project":"p12","title":"s-finite kernel","kind":"definition","summary":"[s-finite kernel] \\mathlibok A kernel \\kappa : X \\rightsquigarrow Y is said to be a s-finite if…","labels":["def:sFinite_kernel"],"detail_key":"p12"},{"id":"n18357","layer":"informal","project":"p12","title":"Composition-product","kind":"definition","summary":"[Composition-product] \\mathlibok Let \\kappa : X \\rightsquigarrow Y and \\eta : ( X \\times Y) \\ri…","labels":["def:kernel_compProd"],"detail_key":"p12"},{"id":"n18358","layer":"informal","project":"p12","title":"Composition-product of a measure and a kernel","kind":"definition","summary":"[Composition-product of a measure and a kernel] \\mathlibok Let \\mu \\in M( X) be an s-finite mea…","labels":["def:measure_compProd"],"detail_key":"p12"},{"id":"n18359","layer":"informal","project":"p12","title":"Composition","kind":"definition","summary":"[Composition] \\mathlibok Let \\kappa : X \\rightsquigarrow Y and \\eta : Y \\rightsquigarrow Z be t…","labels":["def:kernel_comp"],"detail_key":"p12"},{"id":"n18360","layer":"informal","project":"p12","title":"Product","kind":"definition","summary":"[Product] \\mathlibok Let \\kappa : X \\rightsquigarrow Y and \\eta : X \\rightsquigarrow Z be two s…","labels":["def:kernel_prod"],"detail_key":"p12"},{"id":"n18361","layer":"informal","project":"p12","title":"Parallel product","kind":"definition","summary":"[Parallel product] Let \\kappa : X \\rightsquigarrow Y and \\eta : X' \\rightsquigarrow Y' be two s…","labels":["def:kernel_parallel_prod"],"detail_key":"p12"},{"id":"n18362","layer":"informal","project":"p12","title":"Blackwell sufficiency","kind":"definition","summary":"[Blackwell sufficiency] We define a partial order on kernels by the following. Let \\kappa : X \\…","labels":["def:blackwellOrder"],"detail_key":"p12"},{"id":"n18363","layer":"informal","project":"p12","title":"Disintegration in standard Borel spaces","kind":"theorem","summary":"[Disintegration in standard Borel spaces] \\mathlibok If either X is countable of Y has a counta…","labels":["thm:disintegration"],"detail_key":"p12"},{"id":"n18364","layer":"informal","project":"p12","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p12"},{"id":"n18365","layer":"informal","project":"p12","title":"Dummy lemma: kernel properties","kind":"lemma","summary":"[Dummy lemma: kernel properties] Dummy node to summarize kernel properties.","labels":["lem:kernel_properties"],"detail_key":"p12"},{"id":"n18366","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18367","layer":"informal","project":"p12","title":"def:bayesInv","kind":"definition","summary":"For \\mu \\in M( X) and \\kappa : X \\rightsquigarrow Y, a Bayesian inverse of \\kappa is a Markov k…","labels":["def:bayesInv"],"detail_key":"p12"},{"id":"n18368","layer":"informal","project":"p12","title":"Existence of the Bayesian inverse","kind":"lemma","summary":"[Existence of the Bayesian inverse] For X standard Borel, \\mu and \\kappa s-finite, the Bayesian…","labels":["lem:exists_bayesInv"],"detail_key":"p12"},{"id":"n18369","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18370","layer":"informal","project":"p12","title":"a.e.-uniqueness of the Bayesian inverse","kind":"lemma","summary":"[a.e.-uniqueness of the Bayesian inverse] Let \\mu \\in M( X) be a finite measure and let \\kappa…","labels":["lem:eq_bayesInv_of_compProd_eq"],"detail_key":"p12"},{"id":"n18371","layer":"informal","project":"p12","title":"TODO: uniqueness of the conditional kernel.","kind":"proof","summary":"TODO: uniqueness of the conditional kernel.","labels":[],"detail_key":"p12"},{"id":"n18372","layer":"informal","project":"p12","title":"lem:bayesInv_comp_self","kind":"lemma","summary":"For \\mu \\in M( X) s-finite, \\kappa : X \\rightsquigarrow Y a Markov kernel and \\kappa_\\mu^\\dagge…","labels":["lem:bayesInv_comp_self"],"detail_key":"p12"},{"id":"n18373","layer":"informal","project":"p12","title":"The measure \\kappa_\\mu^\\dagger \\circ (\\kappa \\circ \\mu) is the projection on X of (\\kappa…","kind":"proof","summary":"The measure \\kappa_\\mu^\\dagger \\circ (\\kappa \\circ \\mu) is the projection on X of (\\kappa \\circ…","labels":[],"detail_key":"p12"},{"id":"n18374","layer":"informal","project":"p12","title":"lem:bayesInv_self","kind":"lemma","summary":"For X and Y two standard Borel spaces, \\mu \\in M( X) s-finite and \\kappa : X \\rightsquigarrow Y…","labels":["lem:bayesInv_self"],"detail_key":"p12"},{"id":"n18375","layer":"informal","project":"p12","title":"By uniqueness of the disintegration, it suffices to show that (\\kappa \\circ \\mu) \\otimes…","kind":"proof","summary":"By uniqueness of the disintegration, it suffices to show that (\\kappa \\circ \\mu) \\otimes \\kappa…","labels":[],"detail_key":"p12"},{"id":"n18376","layer":"informal","project":"p12","title":"lem:bayesInv_id","kind":"lemma","summary":"Let \\mu \\in M ( X) and let \\textupid : X \\rightsquigarrow X be the identity kernel. Then \\textu…","labels":["lem:bayesInv_id"],"detail_key":"p12"},{"id":"n18377","layer":"informal","project":"p12","title":"It suffices to show that \\mu \\otimes \\textupid = ((\\textupid \\circ \\mu) \\otimes \\textupid…","kind":"proof","summary":"It suffices to show that \\mu \\otimes \\textupid = ((\\textupid \\circ \\mu) \\otimes \\textupid)_\\lef…","labels":[],"detail_key":"p12"},{"id":"n18378","layer":"informal","project":"p12","title":"lem:bayesInv_comp","kind":"lemma","summary":"Let \\mu \\in M( X), \\kappa : X \\rightsquigarrow Y and \\eta : Y \\rightsquigarrow Z. Then (\\eta \\c…","labels":["lem:bayesInv_comp"],"detail_key":"p12"},{"id":"n18379","layer":"informal","project":"p12","title":"It suffices to show that \\mu \\otimes (\\eta \\circ \\kappa) = ((\\eta \\circ \\kappa \\circ \\mu)…","kind":"proof","summary":"It suffices to show that \\mu \\otimes (\\eta \\circ \\kappa) = ((\\eta \\circ \\kappa \\circ \\mu) \\otim…","labels":[],"detail_key":"p12"},{"id":"n18380","layer":"informal","project":"p12","title":"lem:rnDeriv_bayesInv","kind":"lemma","summary":"Let \\mu \\in M( X) and \\kappa : X \\rightsquigarrow Y such that \\kappa_\\mu^\\dagger exists and suc…","labels":["lem:rnDeriv_bayesInv"],"detail_key":"p12"},{"id":"n18381","layer":"informal","project":"p12","title":"\\mu-almost all x and (\\kappa \\circ \\mu)-almost all y is the same as (\\mu \\times (\\kappa \\…","kind":"proof","summary":"\\mu-almost all x and (\\kappa \\circ \\mu)-almost all y is the same as (\\mu \\times (\\kappa \\circ \\…","labels":[],"detail_key":"p12"},{"id":"n18382","layer":"informal","project":"p12","title":"Dummy lemma: bayesInv properties","kind":"lemma","summary":"[Dummy lemma: bayesInv properties] Dummy node to summarize properties of the Bayesian inverse.","labels":["lem:bayesInv_properties"],"detail_key":"p12"},{"id":"n18383","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18384","layer":"informal","project":"p12","title":"def:kernel_rnDeriv","kind":"definition","summary":"Let \\kappa, \\","labels":["def:kernel_rnDeriv"],"detail_key":"p12"},{"id":"n18385","layer":"informal","project":"p12","title":"lem:rnDeriv_unique","kind":"lemma","summary":"Let \\kappa, \\eta : X \\rightsquigarrow Y be two finite kernels, with either X countable or Y cou…","labels":["lem:rnDeriv_unique"],"detail_key":"p12"},{"id":"n18386","layer":"informal","project":"p12","title":"Let f and \\xi be such that \\kappa = f \\cdot \\eta + \\xi with \\xi(x) \\perp \\eta(x) for all…","kind":"proof","summary":"Let f and \\xi be such that \\kappa = f \\cdot \\eta + \\xi with \\xi(x) \\perp \\eta(x) for all x. The…","labels":[],"detail_key":"p12"},{"id":"n18387","layer":"informal","project":"p12","title":"cor:rnDeriv_value","kind":"corollary","summary":"For all x \\in X, for \\eta(x)-almost all y \\in Y, \\fracd \\kappad \\eta(x, y) = \\fracd \\kappa(x)d…","labels":["cor:rnDeriv_value"],"detail_key":"p12"},{"id":"n18388","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:rnDeriv_unique.","kind":"proof","summary":"Apply Lemma~\\reflem:rnDeriv_unique.","labels":[],"detail_key":"p12"},{"id":"n18389","layer":"informal","project":"p12","title":"lem:ac_compProd_iff","kind":"lemma","summary":"Let \\mu, \\nu be two \\sigma-finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be…","labels":["lem:ac_compProd_iff"],"detail_key":"p12"},{"id":"n18390","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18391","layer":"informal","project":"p12","title":"lem:mutuallySingular_compProd","kind":"lemma","summary":"Let \\mu, \\nu be two \\sigma-finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be…","labels":["lem:mutuallySingular_compProd"],"detail_key":"p12"},{"id":"n18392","layer":"informal","project":"p12","title":"First, let's state two facts about mutually singular measures that we will use without pr…","kind":"proof","summary":"First, let's state two facts about mutually singular measures that we will use without proof: \\…","labels":[],"detail_key":"p12"},{"id":"n18393","layer":"informal","project":"p12","title":"lem:singularPart_compProd","kind":"lemma","summary":"Let \\mu, \\nu be two \\sigma-finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be…","labels":["lem:singularPart_compProd"],"detail_key":"p12"},{"id":"n18394","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18395","layer":"informal","project":"p12","title":"lem:rnDeriv_eq_ac_left","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two fin…","labels":["lem:rnDeriv_eq_ac_left"],"detail_key":"p12"},{"id":"n18396","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18397","layer":"informal","project":"p12","title":"cor:rnDeriv_compProd_left","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) and let \\kappa : X \\rightsquigarrow Y be a finite kernel. Then for (\\nu…","labels":["cor:rnDeriv_compProd_left"],"detail_key":"p12"},{"id":"n18398","layer":"informal","project":"p12","title":"We can suppose \\mu \\ll \\nu without loss of generality (by Lemma~\\reflem:rnDeriv_eq_ac_lef…","kind":"proof","summary":"We can suppose \\mu \\ll \\nu without loss of generality (by Lemma~\\reflem:rnDeriv_eq_ac_left). Th…","labels":[],"detail_key":"p12"},{"id":"n18399","layer":"informal","project":"p12","title":"lem:rnDeriv_chain","kind":"lemma","summary":"\\mathlibok Let \\mu, \\nu, \\xi be \\sigma-finite measures on X. \\item If \\mu \\ll \\nu then \\xi-almo…","labels":["lem:rnDeriv_chain"],"detail_key":"p12"},{"id":"n18400","layer":"informal","project":"p12","title":"\\mathlibok","kind":"proof","summary":"\\mathlibok","labels":[],"detail_key":"p12"},{"id":"n18401","layer":"informal","project":"p12","title":"Chain rule for Radon-Nikodym derivatives","kind":"theorem","summary":"[Chain rule for Radon-Nikodym derivatives] Let \\mu, \\nu be two finite measures on X and let \\ka…","labels":["thm:rnDeriv_chain_compProd"],"detail_key":"p12"},{"id":"n18402","layer":"informal","project":"p12","title":"By the first point of Lemma~\\reflem:rnDeriv_chain, (\\nu \\otimes \\eta)-almost surely, \\fra…","kind":"proof","summary":"By the first point of Lemma~\\reflem:rnDeriv_chain, (\\nu \\otimes \\eta)-almost surely, \\fracd(\\mu…","labels":[],"detail_key":"p12"},{"id":"n18403","layer":"informal","project":"p12","title":"lem:rnDeriv_eq_ac","kind":"lemma","summary":"Let \\mu, \\nu be two measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two finite ker…","labels":["lem:rnDeriv_eq_ac"],"detail_key":"p12"},{"id":"n18404","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18405","layer":"informal","project":"p12","title":"cor:rnDeriv_compProd_right","kind":"lemma","summary":"Let \\mu \\in M( X) be a finite measure and \\kappa, \\eta : X \\rightsquigarrow Y be two finite ker…","labels":["cor:rnDeriv_compProd_right"],"detail_key":"p12"},{"id":"n18406","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18407","layer":"informal","project":"p12","title":"lem:ae_rnDeriv_ne_zero","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) be two \\sigma-finite measures and let p be a predicate on X. If p is tru…","labels":["lem:ae_rnDeriv_ne_zero"],"detail_key":"p12"},{"id":"n18408","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18409","layer":"informal","project":"p12","title":"lem:rnDeriv_compProd","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X and let \\kappa, \\eta : X \\rightsquigarrow Y be two fin…","labels":["lem:rnDeriv_compProd"],"detail_key":"p12"},{"id":"n18410","layer":"informal","project":"p12","title":"First, by Theorem~\\refthm:rnDeriv_chain_compProd, for (\\nu \\otimes \\eta)-almost all (x,y)…","kind":"proof","summary":"First, by Theorem~\\refthm:rnDeriv_chain_compProd, for (\\nu \\otimes \\eta)-almost all (x,y), \\fra…","labels":[],"detail_key":"p12"},{"id":"n18411","layer":"informal","project":"p12","title":"lem:rnDeriv_map_eq_condexp","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) with \\mu \\ll \\nu, g : X \\to Y a measurable function and denote by g^* Y…","labels":["lem:rnDeriv_map_eq_condexp"],"detail_key":"p12"},{"id":"n18412","layer":"informal","project":"p12","title":"We show that the integrals of the two functions agree on all g^* Y-measurable sets. It su…","kind":"proof","summary":"We show that the integrals of the two functions agree on all g^* Y-measurable sets. It suffices…","labels":[],"detail_key":"p12"},{"id":"n18413","layer":"informal","project":"p12","title":"lem:rnDeriv_comp_eq_condexp","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) with \\mu \\ll \\nu and let \\kappa, \\eta : X \\rightsquigarrow Y be finite k…","labels":["lem:rnDeriv_comp_eq_condexp"],"detail_key":"p12"},{"id":"n18414","layer":"informal","project":"p12","title":"Let \\pi_Y : X \\times Y \\to Y be the projection \\pi_Y(x,y) = y. Remark that \\kappa \\circ \\…","kind":"proof","summary":"Let \\pi_Y : X \\times Y \\to Y be the projection \\pi_Y(x,y) = y. Remark that \\kappa \\circ \\mu = \\…","labels":[],"detail_key":"p12"},{"id":"n18415","layer":"informal","project":"p12","title":"\\citecsiszar1963informationstheoretische","kind":"lemma","summary":"[\\citecsiszar1963informationstheoretische] Let \\mu, \\nu \\in M( X) with \\mu \\ll \\nu and let \\kap…","labels":["lem:rnDeriv_comp_eq_condexp_right"],"detail_key":"p12"},{"id":"n18416","layer":"informal","project":"p12","title":"Apply Lemma~\\reflem:rnDeriv_comp_eq_condexp to get, (\\nu \\otimes \\kappa)-almost everywher…","kind":"proof","summary":"Apply Lemma~\\reflem:rnDeriv_comp_eq_condexp to get, (\\nu \\otimes \\kappa)-almost everywhere, \\fr…","labels":[],"detail_key":"p12"},{"id":"n18417","layer":"informal","project":"p12","title":"lem:rnDeriv_trim_of_ac","kind":"lemma","summary":"Let \\mu, \\nu be two finite measures on X with \\mu \\ll \\nu and let A be a sub-\\sigma-algebra of…","labels":["lem:rnDeriv_trim_of_ac"],"detail_key":"p12"},{"id":"n18418","layer":"informal","project":"p12","title":"The restriction \\mu_| A is the map of \\mu by the identity, seen as a function from X with…","kind":"proof","summary":"The restriction \\mu_| A is the map of \\mu by the identity, seen as a function from X with its \\…","labels":[],"detail_key":"p12"},{"id":"n18419","layer":"informal","project":"p12","title":"lem:rnDeriv_map_eq_rnDeriv_trim","kind":"lemma","summary":"Let \\mu, \\nu \\in M( X) with \\mu \\ll \\nu, g : X \\to Y a measurable function and denote by g^* Y…","labels":["lem:rnDeriv_map_eq_rnDeriv_trim"],"detail_key":"p12"},{"id":"n18420","layer":"informal","project":"p12","title":"Combine Lemma~\\reflem:rnDeriv_map_eq_condexp and Lemma~\\reflem:rnDeriv_trim_of_ac.","kind":"proof","summary":"Combine Lemma~\\reflem:rnDeriv_map_eq_condexp and Lemma~\\reflem:rnDeriv_trim_of_ac.","labels":[],"detail_key":"p12"},{"id":"n18421","layer":"informal","project":"p12","title":"def:derivAtTop","kind":"definition","summary":"We define f'(\\infty) := \\limsup_x \\to + \\infty f(x)/x. This can be equal to +\\infty (but not -\\…","labels":["def:derivAtTop"],"detail_key":"p12"},{"id":"n18422","layer":"informal","project":"p12","title":"lem:integrable_f_rnDeriv_of_derivAtTop_ne_top","kind":"lemma","summary":"If \\mu and \\nu are two finite measures and f'(\\infty) < \\infty, then x \\mapsto f\\left(\\fracd\\mu…","labels":["lem:integrable_f_rnDeriv_of_derivAtTop_ne_top"],"detail_key":"p12"},{"id":"n18423","layer":"informal","project":"p12","title":"By convexity and f(0) < \\infty, f'(\\infty)<\\infty, we can sandwich f between two affine f…","kind":"proof","summary":"By convexity and f(0) < \\infty, f'(\\infty)<\\infty, we can sandwich f between two affine functio…","labels":[],"detail_key":"p12"},{"id":"n18424","layer":"informal","project":"p12","title":"thm:condexp_jensen","kind":"theorem","summary":"\\notready For f convex, f\\left(\\mu [g \\mid m]\\right) \\le \\mu [f \\circ g \\mid m].","labels":["thm:condexp_jensen"],"detail_key":"p12"},{"id":"n18425","layer":"informal","project":"p12","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p12"},{"id":"n18426","layer":"formal","project":"p12","title":"ProbabilityTheory.integrable_f_rnDeriv_compProd_iff","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 mα : MeasurableSpace α mβ : MeasurableSpace β μ ν : MeasureTheory.M…","labels":[],"detail_key":"p12","name":"ProbabilityTheory.integrable_f_rnDeriv_compProd_iff","module":"TestingLowerBounds.CompProd"},{"id":"n18427","layer":"formal","project":"p12","title":"ConvexOn.convex_taylor","kind":"theorem","summary":"∀ f : Real → Real, ConvexOn Real Set.univ f → Continuous f → ∀ a b : Real, Eq (HSub.hSub (HSub.…","labels":[],"detail_key":"p12","name":"ConvexOn.convex_taylor","module":"TestingLowerBounds.CurvatureMeasure"},{"id":"n18428","layer":"formal","project":"p12","title":"ConvexOn.curvatureMeasure","kind":"def","summary":"(Real → Real) → MeasureTheory.Measure Real","labels":[],"detail_key":"p12","name":"ConvexOn.curvatureMeasure","module":"TestingLowerBounds.CurvatureMeasure"},{"id":"n18429","layer":"formal","project":"p12","title":"derivAtTop","kind":"def","summary":"(Real → Real) → EReal","labels":[],"detail_key":"p12","name":"derivAtTop","module":"TestingLowerBounds.DerivAtTop"},{"id":"n18430","layer":"formal","project":"p12","title":"ProbabilityTheory.chernoffDiv","kind":"def","summary":"α : Type u_1 → mα : MeasurableSpace α → Real → MeasureTheory.Measure α → MeasureTheory.Measure…","labels":[],"detail_key":"p12","name":"ProbabilityTheory.chernoffDiv","module":"TestingLowerBounds.Divergences.Chernoff"},{"id":"n18431","layer":"formal","project":"p12","title":"ProbabilityTheory.chernoffDiv_one","kind":"theorem","summary":"∀ α : Type u_1 mα : MeasurableSpace α (μ ν : MeasureTheory.Measure α), Eq (ProbabilityTheory.ch…","labels":[],"detail_key":"p12","name":"ProbabilityTheory.chernoffDiv_one","module":"TestingLowerBounds.Divergences.Chernoff"},{"id":"n18432","layer":"formal","project":"p12","title":"ProbabilityTheory.condHellingerDiv","kind":"def","summary":"α : Type u_1 → β : Type u_2 → mα : MeasurableSpace α → mβ : MeasurableSpace β → Real → Probabil…","labels":[],"detail_key":"p12","name":"ProbabilityTheory.condHellingerDiv","module":"TestingLowerBounds.Divergences.CondHellinger"},{"id":"n18433","layer":"formal","project":"p12","title":"ProbabilityTheory.condRenyiDiv","kind":"def","summary":"α : Type u_1 → β : Type u_2 → mα : MeasurableSpace α → mβ : MeasurableSpace β → Real → Probabil…","labels":[],"detail_key":"p12","name":"ProbabilityTheory.condRenyiDiv","module":"TestingLowerBounds.Divergences.CondRenyi"},{"id":"n18434","layer":"formal","project":"p12","title":"ProbabilityTheory.deGrootInfo","kind":"def","summary":"𝒳 : Type u_1 → m𝒳 : MeasurableSpace 𝒳 → 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Simply state the inv…","kind":"proof","summary":"This follows rather quickly from \\reflem:InvariantDistributionValue. Simply state the invariant…","labels":[],"detail_key":"p13"},{"id":"n18613","layer":"informal","project":"p13","title":"lem:ShiftedInvariantDistributionValue","kind":"lemma","summary":"Suppose we have an index n s.t. we know that \\lambda_i = 0 for all i < n, and furthermore that…","labels":["lem:ShiftedInvariantDistributionValue"],"detail_key":"p13"},{"id":"n18614","layer":"informal","project":"p13","title":"This will just be copying the proof from \\reflem:InvariantDistributionValue, but this tim…","kind":"proof","summary":"This will just be copying the proof from \\reflem:InvariantDistributionValue, but this time inst…","labels":[],"detail_key":"p13"},{"id":"n18615","layer":"informal","project":"p13","title":"lem:MaxUnreachableInvariance","kind":"lemma","summary":"If there exists finite A \\subsetneq N such that for all n \\in A the departure rate at state n i…","labels":["lem:MaxUnreachableInvariance"],"detail_key":"p13"},{"id":"n18616","layer":"informal","project":"p13","title":"We will need to do induction over i. base case i = 0:\\\\ \\qquad then use \\reflem:Unreachab…","kind":"proof","summary":"We will need to do induction over i. base case i = 0:\\\\ \\qquad then use \\reflem:UnreachableInva…","labels":[],"detail_key":"p13"},{"id":"n18617","layer":"informal","project":"p13","title":"lem:BeforeCutScaled","kind":"lemma","summary":"For any two scheduling policies whose departure rates differ at exactly one index n, \\forall i…","labels":["lem:BeforeCutScaled"],"detail_key":"p13"},{"id":"n18618","layer":"informal","project":"p13","title":"We will need to do a case distinction: either the rate is 0 or it is isn't.\\\\ If it is: A…","kind":"proof","summary":"We will need to do a case distinction: either the rate is 0 or it is isn't.\\\\ If it is: Apply p…","labels":[],"detail_key":"p13"},{"id":"n18619","layer":"informal","project":"p13","title":"lem:AfterCutScaled","kind":"lemma","summary":"For any two scheduling policies whose departure rates differ at exactly one index n, \\forall i…","labels":["lem:AfterCutScaled"],"detail_key":"p13"},{"id":"n18620","layer":"informal","project":"p13","title":"We will need to do a case distinction: either the rate is 0 or it is isn't.\\\\ If it is: A…","kind":"proof","summary":"We will need to do a case distinction: either the rate is 0 or it is isn't.\\\\ If it is: Apply t…","labels":[],"detail_key":"p13"},{"id":"n18621","layer":"informal","project":"p13","title":"lem:InvariantDistributionsSmaller","kind":"lemma","summary":"For any two scheduling policies P and Q whose departure rates differ at one index n, and P's de…","labels":["lem:InvariantDistributionsSmaller"],"detail_key":"p13"},{"id":"n18622","layer":"informal","project":"p13","title":"eqn","kind":"proof","summary":"Clearly we first use \\reflem:BeforeCutScaled, lem:AfterCutScaled We will need to do a few case…","labels":["eqn"],"detail_key":"p13"},{"id":"n18623","layer":"informal","project":"p13","title":"cor:NumberInTheSystemSmaller","kind":"corollary","summary":"In the previous situation E[N]^P \\leq E[N]^Q.","labels":["cor:NumberInTheSystemSmaller"],"detail_key":"p13"},{"id":"n18624","layer":"informal","project":"p13","title":"Because c \\leq 1 for i < n and C \\geq 1 for i \\geq n we find that: \\sum_i=0^n-1 i \\lambda…","kind":"proof","summary":"Because c \\leq 1 for i < n and C \\geq 1 for i \\geq n we find that: \\sum_i=0^n-1 i \\lambda_Q,i +…","labels":[],"detail_key":"p13"},{"id":"n18625","layer":"informal","project":"p13","title":"thm:theorem2.3","kind":"theorem","summary":"If we have two scheduling policies P and Q. And P's departurerates are always higher than or eq…","labels":["thm:theorem2.3"],"detail_key":"p13"},{"id":"n18626","layer":"informal","project":"p13","title":"We will create a set of intermediary policies between P and Q to prove what we wanted to…","kind":"proof","summary":"We will create a set of intermediary policies between P and Q to prove what we wanted to prove.…","labels":[],"detail_key":"p13"},{"id":"n18627","layer":"formal","project":"p13","title":"InvariantDistribution","kind":"def","summary":"RateMatrix → (Nat → Real) → Prop","labels":[],"detail_key":"p13","name":"InvariantDistribution","module":"BscThesisFormalisation.definitions"},{"id":"n18628","layer":"formal","project":"p13","title":"MeanResponseTime","kind":"def","summary":"(lambda : Nat → Real) → (Q : queue) → InvariantDistribution Q.Q lambda → Real","labels":[],"detail_key":"p13","name":"MeanResponseTime","module":"BscThesisFormalisation.definitions"},{"id":"n18629","layer":"formal","project":"p13","title":"Policy","kind":"def","summary":"(n : Nat) → (PiLp 1 fun x => Nat) → (Nat → Nat) → (Nat → Nat → PiLp 1 fun x => Real) → Prop","labels":[],"detail_key":"p13","name":"Policy","module":"BscThesisFormalisation.definitions"},{"id":"n18630","layer":"formal","project":"p13","title":"RateMatrix","kind":"inductive","summary":"Type","labels":[],"detail_key":"p13","name":"RateMatrix","module":"BscThesisFormalisation.definitions"},{"id":"n18631","layer":"formal","project":"p13","title":"SchedulePolicy","kind":"inductive","summary":"Type","labels":[],"detail_key":"p13","name":"SchedulePolicy","module":"BscThesisFormalisation.definitions"},{"id":"n18632","layer":"formal","project":"p13","title":"SpeedUpFunction","kind":"def","summary":"(n : Nat) → (PiLp 1 fun x => Real) → (PiLp 1 fun x => Nat) → ((PiLp 1 fun x => Real) → Real) →…","labels":[],"detail_key":"p13","name":"SpeedUpFunction","module":"BscThesisFormalisation.definitions"},{"id":"n18633","layer":"formal","project":"p13","title":"Sublinear","kind":"def","summary":"(n : Nat) → (PiLp 1 fun x => Real) → ((PiLp 1 fun x => Real) → Real) → Prop","labels":[],"detail_key":"p13","name":"Sublinear","module":"BscThesisFormalisation.definitions"},{"id":"n18634","layer":"formal","project":"p13","title":"coreSpace","kind":"def","summary":"(n : Nat) → (PiLp 1 fun x => Nat) → Set (PiLp 1 fun x => Real)","labels":[],"detail_key":"p13","name":"coreSpace","module":"BscThesisFormalisation.definitions"},{"id":"n18635","layer":"formal","project":"p13","title":"myConcave","kind":"def","summary":"(n : Nat) → (PiLp 1 fun x => Nat) → ((PiLp 1 fun x => Real) → Real) → Prop","labels":[],"detail_key":"p13","name":"myConcave","module":"BscThesisFormalisation.definitions"},{"id":"n18636","layer":"formal","project":"p13","title":"lemma2_3_1","kind":"theorem","summary":"∀ (P : RateMatrix) (lambdaP : Nat → Real), And (InvariantDistribution P lambdaP) (∀ (i : Nat),…","labels":[],"detail_key":"p13","name":"lemma2_3_1","module":"BscThesisFormalisation.identities"},{"id":"n18637","layer":"formal","project":"p13","title":"lemma2_3_3a","kind":"theorem","summary":"∀ (P : RateMatrix) (lambdaP : Nat → Real), InvariantDistribution P lambdaP → ∀ (n : Nat), Ne n…","labels":[],"detail_key":"p13","name":"lemma2_3_3a","module":"BscThesisFormalisation.identities"},{"id":"n18638","layer":"formal","project":"p13","title":"lemma2_1","kind":"theorem","summary":"∀ (n : Nat) (speedvec : PiLp 1 fun x => Real) (cN : PiLp 1 fun x => Nat) (cR : PiLp 1 fun x =>…","labels":[],"detail_key":"p13","name":"lemma2_1","module":"BscThesisFormalisation.lemma2_1"},{"id":"n18639","layer":"formal","project":"p13","title":"lemma2_3_3","kind":"theorem","summary":"∀ (P : RateMatrix) (lambdaP : Nat → Real), InvariantDistribution P lambdaP → ∀ (n : Nat) [inst…","labels":[],"detail_key":"p13","name":"lemma2_3_3","module":"BscThesisFormalisation.lemma2_3"},{"id":"n18640","layer":"formal","project":"p13","title":"EQUIOptimal","kind":"theorem","summary":"∀ (Que : queue) (Equi : MeanResponseTimePolicy), isEqui Equi.Q → ∀ (x : MeanResponseTimePolicy)…","labels":[],"detail_key":"p13","name":"EQUIOptimal","module":"BscThesisFormalisation.theorem2_2"},{"id":"n18641","layer":"informal","project":"p14","title":"Boolean hypercube","kind":"definition","summary":"[Boolean hypercube] The \\emphBoolean hypercube of dimension n is \\0,1\\^n, represented as the fu…","labels":["BooleanAnalysis.BoolCube"],"detail_key":"p14"},{"id":"n18642","layer":"informal","project":"p14","title":"Boolean function","kind":"definition","summary":"[Boolean function] A \\emphBoolean function of arity n is a function f : \\0,1\\^n \\to R.","labels":["BooleanAnalysis.BooleanFunc"],"detail_key":"p14"},{"id":"n18643","layer":"informal","project":"p14","title":"Uniform 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f,f\\rangle.","labels":["BooleanAnalysis.l2Norm"],"detail_key":"p14"},{"id":"n18647","layer":"informal","project":"p14","title":"Inner product is symmetric","kind":"lemma","summary":"[Inner product is symmetric] \\langle f, g\\rangle = \\langle g, f\\rangle.","labels":["BooleanAnalysis.innerProduct_comm"],"detail_key":"p14"},{"id":"n18648","layer":"informal","project":"p14","title":"Inner product is nonneg on the diagonal","kind":"lemma","summary":"[Inner product is nonneg on the diagonal] \\langle f, f\\rangle \\ge 0.","labels":["BooleanAnalysis.innerProduct_self_nonneg"],"detail_key":"p14"},{"id":"n18649","layer":"informal","project":"p14","title":"Inner product is linear in the first argument","kind":"lemma","summary":"[Inner product is linear in the first argument] \\langle f + g, h\\rangle = \\langle f, h\\rangle +…","labels":["BooleanAnalysis.innerProduct_add_left"],"detail_key":"p14"},{"id":"n18650","layer":"informal","project":"p14","title":"Sign 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=…","labels":["BooleanAnalysis.chiS_mul_of_disjoint"],"detail_key":"p14"},{"id":"n18657","layer":"informal","project":"p14","title":"Product of two characters","kind":"lemma","summary":"[Product of two characters] \\chi_S(x)\\cdot\\chi_T(x) = \\chi_S\\mathbin\\triangle T(x), where \\tria…","labels":["BooleanAnalysis.chiS_mul_chiS"],"detail_key":"p14"},{"id":"n18658","layer":"informal","project":"p14","title":"Character under global bit flip","kind":"lemma","summary":"[Character under global bit flip] Flipping all bits multiplies the character by (-1)^|S|: \\[ \\c…","labels":["BooleanAnalysis.chiS_neg"],"detail_key":"p14"},{"id":"n18659","layer":"informal","project":"p14","title":"Count of Walsh characters","kind":"lemma","summary":"[Count of Walsh characters] There are exactly 2^n Walsh characters, matching the dimension of L…","labels":["BooleanAnalysis.card_walsh_characters"],"detail_key":"p14"},{"id":"n18660","layer":"informal","project":"p14","title":"Fourier 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\\langl…","labels":["BooleanAnalysis.fourier_coeff_chi"],"detail_key":"p14"},{"id":"n18664","layer":"informal","project":"p14","title":"Self inner product of a character is 1","kind":"lemma","summary":"[Self inner product of a character is 1] \\langle \\chi_S, \\chi_S\\rangle = 1.","labels":["BooleanAnalysis.innerProduct_chi_self"],"detail_key":"p14"},{"id":"n18665","layer":"informal","project":"p14","title":"Parseval's identity","kind":"theorem","summary":"[Parseval's identity] The squared L^2 norm of f equals the sum of squared Fourier coefficients:…","labels":["BooleanAnalysis.parseval"],"detail_key":"p14"},{"id":"n18666","layer":"informal","project":"p14","title":"Bit flip","kind":"definition","summary":"[Bit flip] The point x^i\\in\\0,1\\^n obtained from x by flipping the i-th coordinate.","labels":["BooleanAnalysis.flipBit"],"detail_key":"p14"},{"id":"n18667","layer":"informal","project":"p14","title":"Influence","kind":"definition","summary":"[Influence] The \\emphinfluence of coordinate i on f measures how often flipping bit i changes t…","labels":["BooleanAnalysis.influence"],"detail_key":"p14"},{"id":"n18668","layer":"informal","project":"p14","title":"Total influence","kind":"definition","summary":"[Total influence] I[f] = \\sum_i=1^n Inf_i[f].","labels":["BooleanAnalysis.totalInfluence"],"detail_key":"p14"},{"id":"n18669","layer":"informal","project":"p14","title":"Influence of a Walsh character","kind":"lemma","summary":"[Influence of a Walsh character] Inf_i[\\chi_S] = [i\\in S].","labels":["BooleanAnalysis.influence_chi"],"detail_key":"p14"},{"id":"n18670","layer":"informal","project":"p14","title":"Influence via Fourier coefficients","kind":"theorem","summary":"[Influence via Fourier coefficients] \\[ Inf_i[f] \\;=\\; \\sum_\\substackS\\subseteq[n]\\\\ i\\in S \\ha…","labels":["BooleanAnalysis.influence_eq_sum_fourier"],"detail_key":"p14"},{"id":"n18671","layer":"informal","project":"p14","title":"Total influence via Fourier coefficients","kind":"theorem","summary":"[Total influence via Fourier coefficients] \\[ I[f] \\;=\\; \\sum_S\\subseteq[n] |S|\\,\\hat f(S)^2. \\]","labels":["BooleanAnalysis.totalInfluence_eq_sum_sq_deg"],"detail_key":"p14"},{"id":"n18672","layer":"informal","project":"p14","title":"Influence is at most 1","kind":"lemma","summary":"[Influence is at most 1] For any \\pm 1-valued function and any coordinate i, Inf_i[f] \\le 1.","labels":["BooleanAnalysis.influence_le_one"],"detail_key":"p14"},{"id":"n18673","layer":"informal","project":"p14","title":"Average influence lower bound","kind":"lemma","summary":"[Average influence lower bound] There exists a coordinate i with Inf_i[f] \\ge I[f]/n.","labels":["BooleanAnalysis.max_influence_lower_bound"],"detail_key":"p14"},{"id":"n18674","layer":"informal","project":"p14","title":"Noise operator","kind":"definition","summary":"[Noise operator] The \\emphnoise operator T_\\rho with parameter \\rho\\in[-1,1], defined via the F…","labels":["BooleanAnalysis.noiseOp"],"detail_key":"p14"},{"id":"n18675","layer":"informal","project":"p14","title":"Noise operator in the Fourier domain","kind":"theorem","summary":"[Noise operator in the Fourier domain] \\[ \\widehatT_\\rho f(S) \\;=\\; \\rho^|S|\\,\\hat f(S). \\]","labels":["BooleanAnalysis.noiseOp_fourier"],"detail_key":"p14"},{"id":"n18676","layer":"informal","project":"p14","title":"Stability formula","kind":"theorem","summary":"[Stability formula] \\[ \\langle f, T_\\rho f\\rangle \\;=\\; \\sum_S\\subseteq[n] \\rho^|S|\\,\\hat f(S)^…","labels":["BooleanAnalysis.stability_formula"],"detail_key":"p14"},{"id":"n18677","layer":"informal","project":"p14","title":"Dictator","kind":"definition","summary":"[Dictator] The \\emphdictator function for coordinate i outputs the sign of the i-th bit: dict_i…","labels":["BooleanAnalysis.dictator"],"detail_key":"p14"},{"id":"n18678","layer":"informal","project":"p14","title":"Dictator equals singleton 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This models an…","labels":["BooleanAnalysis.isOddFunc"],"detail_key":"p14"},{"id":"n18685","layer":"informal","project":"p14","title":"\\pm 1-valued function","kind":"definition","summary":"[\\pm 1-valued function] f is \\emph\\pm 1-valued if f(x)\\in\\-1,1\\ for all x.","labels":["BooleanAnalysis.isPmOne"],"detail_key":"p14"},{"id":"n18686","layer":"informal","project":"p14","title":"Odd functions have zero even-level Fourier coefficients","kind":"lemma","summary":"[Odd functions have zero even-level Fourier coefficients] If f is odd and |S| is even, then \\ha…","labels":["BooleanAnalysis.fourierCoeff_odd_even"],"detail_key":"p14"},{"id":"n18687","layer":"informal","project":"p14","title":"Self inner product of a \\pm 1 function","kind":"lemma","summary":"[Self inner product of a \\pm 1 function] If f is \\pm 1-valued then \\langle f, f\\rangle = 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g.","labels":["BooleanAnalysis.derivative_add"],"detail_key":"p14"},{"id":"n18702","layer":"informal","project":"p14","title":"Derivative commutes with scalars","kind":"lemma","summary":"[Derivative commutes with scalars] The discrete derivative commutes with scalar multiplication:…","labels":["BooleanAnalysis.derivative_smul"],"detail_key":"p14"},{"id":"n18703","layer":"informal","project":"p14","title":"Derivative as a linear map","kind":"definition","summary":"[Derivative as a linear map] The discrete derivative D_i packaged as an R-linear map on the spa…","labels":["BooleanAnalysis.derivativeLm"],"detail_key":"p14"},{"id":"n18704","layer":"informal","project":"p14","title":"Expectation operator","kind":"definition","summary":"[Expectation operator] The \\emphexpectation operator in direction i averages f over the two val…","labels":["BooleanAnalysis.expectationOperator"],"detail_key":"p14"},{"id":"n18705","layer":"informal","project":"p14","title":"Expectation operator is additive","kind":"lemma","summary":"[Expectation operator is additive] The expectation operator respects addition: E_i(f + g) = E_i…","labels":["BooleanAnalysis.expectation_add"],"detail_key":"p14"},{"id":"n18706","layer":"informal","project":"p14","title":"Expectation operator commutes with scalars","kind":"lemma","summary":"[Expectation operator commutes with scalars] The expectation operator commutes with scalar mult…","labels":["BooleanAnalysis.expectation_smul"],"detail_key":"p14"},{"id":"n18707","layer":"informal","project":"p14","title":"Expectation operator as a linear map","kind":"definition","summary":"[Expectation operator as a linear map] The expectation operator E_i packaged as an R-linear map…","labels":["BooleanAnalysis.expectationLm"],"detail_key":"p14"},{"id":"n18708","layer":"informal","project":"p14","title":"Noise operator is self-adjoint","kind":"lemma","summary":"[Noise operator is self-adjoint] The noise operator T_\\rho is self-adjoint with respect to the…","labels":["BooleanAnalysis.noiseOp_self_adjoint"],"detail_key":"p14"},{"id":"n18709","layer":"informal","project":"p14","title":"Unanimity","kind":"definition","summary":"[Unanimity] A social welfare function f:\\0,1\\^n\\toR is \\emphunanimous if f(\\textttfalse,\\dots,\\…","labels":["ArrowTheorem.unanimity"],"detail_key":"p14"},{"id":"n18710","layer":"informal","project":"p14","title":"Dictatorship","kind":"definition","summary":"[Dictatorship] f is a \\emphdictatorship if there exists a voter i such that f = dict_i, i.e.\\ s…","labels":["ArrowTheorem.isDictator"],"detail_key":"p14"},{"id":"n18711","layer":"informal","project":"p14","title":"Pairwise preference functions","kind":"definition","summary":"[Pairwise preference functions] abPref(k), bcPref(k), caPref(k) give the Boolean preference of…","labels":["ArrowTheorem.abPref","ArrowTheorem.bcPref","ArrowTheorem.caPref"],"detail_key":"p14"},{"id":"n18712","layer":"informal","project":"p14","title":"Profile","kind":"definition","summary":"[Profile] A \\emphprofile assigns each of the n voters one of the 6 orderings: p : Fin\\,n \\to Fi…","labels":["ArrowTheorem.Profile"],"detail_key":"p14"},{"id":"n18713","layer":"informal","project":"p14","title":"Vote vectors","kind":"definition","summary":"[Vote vectors] Given a profile p, the vote vectors abVotes(p), bcVotes(p), caVotes(p) \\in \\0,1\\…","labels":["ArrowTheorem.abVotes","ArrowTheorem.bcVotes","ArrowTheorem.caVotes"],"detail_key":"p14"},{"id":"n18714","layer":"informal","project":"p14","title":"Acyclicity","kind":"definition","summary":"[Acyclicity] A social welfare function f is \\emphacyclic if no profile of transitive voter orde…","labels":["ArrowTheorem.acyclic"],"detail_key":"p14"},{"id":"n18715","layer":"informal","project":"p14","title":"Pairwise sign sums","kind":"lemma","summary":"[Pairwise sign sums] Summing pairwise sign products over all 6 orderings: \\[ \\sum_k=0^5 s_ab(k)…","labels":["ArrowTheorem.sum_abPref_bcPref","ArrowTheorem.sum_bcPref_caPref","ArrowTheorem.sum_abPref_caPref"],"detail_key":"p14"},{"id":"n18716","layer":"informal","project":"p14","title":"Correlation function","kind":"definition","summary":"[Correlation function] The \\emphFourier correlation function of f is \\[ corr(f) \\;=\\; \\sum_S\\su…","labels":["ArrowTheorem.corrFunc"],"detail_key":"p14"},{"id":"n18717","layer":"informal","project":"p14","title":"Expected pairwise product equals correlation function","kind":"lemma","summary":"[Expected pairwise product equals correlation function] For voters with i.i.d.\\ uniform orderin…","labels":["ArrowTheorem.expected_product_eq_corrFunc"],"detail_key":"p14"},{"id":"n18718","layer":"informal","project":"p14","title":"Correlation function lower bound","kind":"lemma","summary":"[Correlation function lower bound] For any odd \\pm 1-valued function, \\[ corr(f) \\;\\ge\\; -1/3.…","labels":["ArrowTheorem.corrFunc_ge_neg_third"],"detail_key":"p14"},{"id":"n18719","layer":"informal","project":"p14","title":"Equality forces all weight on level 1","kind":"lemma","summary":"[Equality forces all weight on level 1] If corr(f) = -1/3, then \\hat f(S) = 0 for all S with |S…","labels":["ArrowTheorem.corrFunc_eq_neg_third_of_weight_one"],"detail_key":"p14"},{"id":"n18720","layer":"informal","project":"p14","title":"Acyclicity implies corr(f) = -1/3","kind":"lemma","summary":"[Acyclicity implies corr(f) = -1/3] If f is \\pm 1-valued and acyclic, then corr(f) = -1/3. \\emp…","labels":["ArrowTheorem.acyclic_implies_corrFunc"],"detail_key":"p14"},{"id":"n18721","layer":"informal","project":"p14","title":"Degree-1 unanimous \\pm 1 function is a dictator","kind":"lemma","summary":"[Degree-1 unanimous \\pm 1 function is a dictator] Suppose f is \\pm 1-valued, unanimous, and \\ha…","labels":["ArrowTheorem.degree_one_implies_dictator"],"detail_key":"p14"},{"id":"n18722","layer":"informal","project":"p14","title":"Arrow's Impossibility Theorem","kind":"theorem","summary":"[Arrow's Impossibility Theorem] Let f:\\0,1\\^n\\toR be a social welfare function that is odd (ant…","labels":["ArrowTheorem.arrow_theorem"],"detail_key":"p14"},{"id":"n18723","layer":"informal","project":"p14","title":"Sum of ab preference signs vanishes","kind":"lemma","summary":"[Sum of ab preference signs vanishes] Summing the signed a-vs-b preference over all six orderin…","labels":["ArrowTheorem.sum_abPref_sign"],"detail_key":"p14"},{"id":"n18724","layer":"informal","project":"p14","title":"Sum of bc preference signs vanishes","kind":"lemma","summary":"[Sum of bc preference signs vanishes] Summing the signed b-vs-c preference over all six orderin…","labels":["ArrowTheorem.sum_bcPref_sign"],"detail_key":"p14"},{"id":"n18725","layer":"informal","project":"p14","title":"Sum of ca preference signs vanishes","kind":"lemma","summary":"[Sum of ca preference signs vanishes] Summing the signed c-vs-a preference over all six orderin…","labels":["ArrowTheorem.sum_caPref_sign"],"detail_key":"p14"},{"id":"n18726","layer":"informal","project":"p14","title":"Product over a finset as an indicator product","kind":"lemma","summary":"[Product over a finset as an indicator product] For A\\subseteqFin\\,n and g:Fin\\,n\\toR, \\[ \\prod…","labels":["ArrowTheorem.prod_finset_eq_prod_univ_ite"],"detail_key":"p14"},{"id":"n18727","layer":"informal","project":"p14","title":"General profile kernel","kind":"lemma","summary":"[General profile kernel] Let xPref,yPref:Fin\\,6\\toBool be two preference assignments with balan…","labels":["ArrowTheorem.profile_kernel_gen"],"detail_key":"p14"},{"id":"n18728","layer":"informal","project":"p14","title":"ab--bc kernel","kind":"lemma","summary":"[ab--bc kernel] Specialization of the general kernel to the ab--bc vote pair: for all S,T\\subse…","labels":["ArrowTheorem.profile_inner_product_kernel"],"detail_key":"p14"},{"id":"n18729","layer":"informal","project":"p14","title":"bc--ca kernel","kind":"lemma","summary":"[bc--ca kernel] Specialization of the general kernel to the bc--ca vote pair: for all S,T\\subse…","labels":["ArrowTheorem.profile_kernel_bcca"],"detail_key":"p14"},{"id":"n18730","layer":"informal","project":"p14","title":"ab--ca kernel","kind":"lemma","summary":"[ab--ca kernel] Specialization of the general kernel to the ab--ca vote pair: for all S,T\\subse…","labels":["ArrowTheorem.profile_kernel_abca"],"detail_key":"p14"},{"id":"n18731","layer":"informal","project":"p14","title":"Expected product from a kernel","kind":"lemma","summary":"[Expected product from a kernel] Let f:\\0,1\\^n\\toR and let votes_1,votes_2 map each profile to…","labels":["ArrowTheorem.expected_product_helper"],"detail_key":"p14"},{"id":"n18732","layer":"informal","project":"p14","title":"Expected product for the bc--ca pair","kind":"lemma","summary":"[Expected product for the bc--ca pair] For any f:\\0,1\\^n\\toR, \\bigl(\\tfrac16\\bigr)^n\\sum_p f(bc…","labels":["ArrowTheorem.expected_product_bcca"],"detail_key":"p14"},{"id":"n18733","layer":"informal","project":"p14","title":"Expected product for the ab--ca pair","kind":"lemma","summary":"[Expected product for the ab--ca pair] For any f:\\0,1\\^n\\toR, \\bigl(\\tfrac16\\bigr)^n\\sum_p f(ab…","labels":["ArrowTheorem.expected_product_abca"],"detail_key":"p14"},{"id":"n18734","layer":"informal","project":"p14","title":"Switching Lemma -- Lean statement","kind":"theorem","summary":"[Switching Lemma -- Lean statement] Let n>0, let f:\\0,1\\^n\\to\\0,1\\ be a DNF formula, and fix w,…","labels":["thm:switching_lemma","SwitchingLemma2.switching_lemma"],"detail_key":"p14"},{"id":"n18735","layer":"informal","project":"p14","title":"Switching corollary","kind":"theorem","summary":"[Switching corollary] Under the same hypotheses as Theorem~\\refthm:switching_lemma, \\[ \\bigl|\\\\…","labels":["thm:switching_corollary","SwitchingLemma2.switching_corollary"],"detail_key":"p14"},{"id":"n18736","layer":"informal","project":"p14","title":"Codeword","kind":"definition","summary":"[Codeword] A codeword of length n over alphabet \\alpha is a function c : Fin\\,n \\to \\alpha.","labels":["ErrorCorrectingCodes.Codeword"],"detail_key":"p14"},{"id":"n18737","layer":"informal","project":"p14","title":"Pointwise operations","kind":"definition","summary":"[Pointwise operations] Pointwise addition, subtraction, and the all-zero codeword.","labels":["ErrorCorrectingCodes.Codeword.add","ErrorCorrectingCodes.Codeword.sub","ErrorCorrectingCodes.Codeword.zero"],"detail_key":"p14"},{"id":"n18738","layer":"informal","project":"p14","title":"Code","kind":"definition","summary":"[Code] A (block) code of length n over \\alpha is a finite set (\\textttFinset) of codewords.","labels":["ErrorCorrectingCodes.Codeword.Code"],"detail_key":"p14"},{"id":"n18739","layer":"informal","project":"p14","title":"Linear code via generator matrix","kind":"definition","summary":"[Linear code via generator matrix] A code C is linear with generator matrix G (shape n\\times m)…","labels":["ErrorCorrectingCodes.Codeword.Linear_Code"],"detail_key":"p14"},{"id":"n18740","layer":"informal","project":"p14","title":"Linear code, existential form","kind":"definition","summary":"[Linear code, existential form] Existential version: there exists some m and generator matrix G…","labels":["ErrorCorrectingCodes.Codeword.Linear_Code'"],"detail_key":"p14"},{"id":"n18741","layer":"informal","project":"p14","title":"q-ary entropy","kind":"definition","summary":"[q-ary entropy] \\[ H_q(p) \\;=\\; p\\log_q(q-1) - p\\log_q p - (1-p)\\log_q(1-p). \\]","labels":["ErrorCorrectingCodes.Codeword.qaryEntropy"],"detail_key":"p14"},{"id":"n18742","layer":"informal","project":"p14","title":"Hamming distance","kind":"definition","summary":"[Hamming distance] Hamming distance between two codewords, wrapping Mathlib's \\texttthammingDis…","labels":["ErrorCorrectingCodes.Codeword.hamming_distance"],"detail_key":"p14"},{"id":"n18743","layer":"informal","project":"p14","title":"Distance predicate for a code","kind":"definition","summary":"[Distance predicate for a code] \\textttdistance C d holds when d is the minimum Hamming distanc…","labels":["ErrorCorrectingCodes.Codeword.distance"],"detail_key":"p14"},{"id":"n18744","layer":"informal","project":"p14","title":"Rate","kind":"definition","summary":"[Rate] \\[ R(C) = \\frac\\log|C|n\\log|\\alpha|. \\]","labels":["ErrorCorrectingCodes.Codeword.rate"],"detail_key":"p14"},{"id":"n18745","layer":"informal","project":"p14","title":"Weight","kind":"definition","summary":"[Weight] The Hamming weight of c is d(c, 0).","labels":["ErrorCorrectingCodes.Codeword.weight"],"detail_key":"p14"},{"id":"n18746","layer":"informal","project":"p14","title":"Maximal size at distance","kind":"definition","summary":"[Maximal size at distance] \\textttmax\\_size n d A asserts existence of a code of size A and min…","labels":["ErrorCorrectingCodes.Codeword.max_size"],"detail_key":"p14"},{"id":"n18747","layer":"informal","project":"p14","title":"Hamming ball","kind":"definition","summary":"[Hamming ball] The Hamming ball of radius l around c: \\[ B_l(c) = \\c' : d(c',c) \\le l\\, \\] impl…","labels":["ErrorCorrectingCodes.Codeword.hamming_ball"],"detail_key":"p14"},{"id":"n18748","layer":"informal","project":"p14","title":"Distance is at most block length","kind":"lemma","summary":"[Distance is at most block length] If a code C has minimum distance d, then d \\le n.","labels":["ErrorCorrectingCodes.Codeword.dist_le_length"],"detail_key":"p14"},{"id":"n18749","layer":"informal","project":"p14","title":"Alphabet size positive","kind":"lemma","summary":"[Alphabet size positive] If q \\ge 2, then q > 0 as a real number.","labels":["ErrorCorrectingCodes.Codeword.natCast_pos_of_two_le"],"detail_key":"p14"},{"id":"n18750","layer":"informal","project":"p14","title":"Alphabet size exceeds one","kind":"lemma","summary":"[Alphabet size exceeds one] If q \\ge 2, then 1 < q as a real number.","labels":["ErrorCorrectingCodes.Codeword.natCast_one_lt_of_two_le"],"detail_key":"p14"},{"id":"n18751","layer":"informal","project":"p14","title":"q-1 positive","kind":"lemma","summary":"[q-1 positive] If q \\ge 2, then 0 < q - 1 as a real number.","labels":["ErrorCorrectingCodes.Codeword.natCast_sub_one_pos_of_two_le"],"detail_key":"p14"},{"id":"n18752","layer":"informal","project":"p14","title":"Alphabet size differs from one","kind":"lemma","summary":"[Alphabet size differs from one] If q \\ge 2, then q \\ne 1 as a real number.","labels":["ErrorCorrectingCodes.Codeword.natCast_ne_one_of_two_le"],"detail_key":"p14"},{"id":"n18753","layer":"informal","project":"p14","title":"1-p positive when p < 1","kind":"lemma","summary":"[1-p positive when p < 1] If p < 1, then 0 < 1 - p.","labels":["ErrorCorrectingCodes.Codeword.one_sub_pos_of_lt_one"],"detail_key":"p14"},{"id":"n18754","layer":"informal","project":"p14","title":"p(1-p) positive","kind":"lemma","summary":"[p(1-p) positive] If 0 < p and p < 1, then 0 < p(1 - p).","labels":["ErrorCorrectingCodes.Codeword.mul_one_sub_pos"],"detail_key":"p14"},{"id":"n18755","layer":"informal","project":"p14","title":"Below 1 - 1/q implies below one","kind":"lemma","summary":"[Below 1 - 1/q implies below one] If 0 < q and p \\le 1 - 1/q, then p < 1.","labels":["ErrorCorrectingCodes.Codeword.lt_one_of_le_one_sub_inv"],"detail_key":"p14"},{"id":"n18756","layer":"informal","project":"p14","title":"Singleton bound","kind":"theorem","summary":"[Singleton bound] Assuming \\alpha is nontrivial, a code C\\subseteq\\alpha^n with minimum distanc…","labels":["ErrorCorrectingCodes.Codeword.singleton_bound"],"detail_key":"p14"},{"id":"n18757","layer":"informal","project":"p14","title":"Cardinality of Hamming balls","kind":"theorem","summary":"[Cardinality of Hamming balls] For any center c and radius l, \\[ |B_l(c)| \\;=\\; \\sum_i=0^l \\bin…","labels":["ErrorCorrectingCodes.Codeword.hamming_ball_size"],"detail_key":"p14"},{"id":"n18758","layer":"informal","project":"p14","title":"Decoding balls around distinct codewords are disjoint","kind":"lemma","summary":"[Decoding balls around distinct codewords are disjoint] If C has minimum distance d>0, then for…","labels":["ErrorCorrectingCodes.Codeword.hamming_ball_non_intersect"],"detail_key":"p14"},{"id":"n18759","layer":"informal","project":"p14","title":"\\textttFinset.Disjoint formulation","kind":"lemma","summary":"[\\textttFinset.Disjoint formulation] Reformulation of the previous lemma as \\textttFinset.Disjo…","labels":["ErrorCorrectingCodes.Codeword.hamming_ball'_disjoint"],"detail_key":"p14"},{"id":"n18760","layer":"informal","project":"p14","title":"Hamming (sphere-packing) bound","kind":"theorem","summary":"[Hamming (sphere-packing) bound] Let q=|\\alpha|>1. If C\\subseteq\\alpha^n has minimum distance d…","labels":["ErrorCorrectingCodes.Codeword.hamming_bound"],"detail_key":"p14"},{"id":"n18761","layer":"informal","project":"p14","title":"Entropy upper bound on Hamming balls","kind":"theorem","summary":"[Entropy upper bound on Hamming balls] For 0<p\\le 1-1/q and q=|\\alpha|, \\[ |B_\\lfloor np\\rfloor…","labels":["ErrorCorrectingCodes.Codeword.hamming_ball_size_asymptotic_upper_bound"],"detail_key":"p14"},{"id":"n18762","layer":"informal","project":"p14","title":"Simplifying q^H_q(p)","kind":"lemma","summary":"[Simplifying q^H_q(p)] For q\\ge 2 and 0<p<1, \\[ q^H_q(p) \\;=\\; (q-1)^p\\,p^-p\\,(1-p)^-(1-p). \\]","labels":["ErrorCorrectingCodes.Codeword.q_pow_qary_entropy_simp"],"detail_key":"p14"},{"id":"n18763","layer":"informal","project":"p14","title":"Same identity, alternate exponentiation","kind":"lemma","summary":"[Same identity, alternate exponentiation] Variant using a different exponentiation operator.","labels":["ErrorCorrectingCodes.Codeword.q_pow_qary_entropy_simp'"],"detail_key":"p14"},{"id":"n18764","layer":"informal","project":"p14","title":"Square-root/floor inequality","kind":"lemma","summary":"[Square-root/floor inequality] For x\\ge 0, \\[ \\sqrtx - \\sqrt\\lfloor x\\rfloor \\;\\le\\; 1. \\]","labels":["ErrorCorrectingCodes.Codeword.sqrt_sub_sqrt_floor_le_one"],"detail_key":"p14"},{"id":"n18765","layer":"informal","project":"p14","title":"Asymptotic lower bound on binomial term","kind":"lemma","summary":"[Asymptotic lower bound on binomial term] For 0<p<1 and q\\ge 2, eventually \\[ \\tbinomn\\lfloor n…","labels":["ErrorCorrectingCodes.Codeword.binomial_coef_asymptotic_lower_bound'"],"detail_key":"p14"},{"id":"n18766","layer":"informal","project":"p14","title":"Positivity of q-ary entropy","kind":"theorem","summary":"[Positivity of q-ary entropy] For q=|\\alpha| and 0<p\\le 1-1/q, \\[ H_q(p) \\;>\\; 0. \\]","labels":["ErrorCorrectingCodes.Codeword.qary_entropy_pos"],"detail_key":"p14"},{"id":"n18767","layer":"informal","project":"p14","title":"Expansion of the q-ary entropy exponent","kind":"lemma","summary":"[Expansion of the q-ary entropy exponent] Algebraic rewrite of the q-ary entropy exponent: for…","labels":["ErrorCorrectingCodes.Codeword.qary_entropy_logb_expand"],"detail_key":"p14"},{"id":"n18768","layer":"informal","project":"p14","title":"AM--GM for two nonnegative reals","kind":"lemma","summary":"[AM--GM for two nonnegative reals] For nonnegative reals a,b\\ge 0, the geometric mean is at mos…","labels":["ErrorCorrectingCodes.Codeword.am_gm_sqrt_le_half_sum"],"detail_key":"p14"},{"id":"n18769","layer":"informal","project":"p14","title":"A Stirling square-root identity","kind":"lemma","summary":"[A Stirling square-root identity] For n\\ge 0, \\[ \\sqrt2\\pi n\\,\\cdot\\,\\sqrt\\pi/2\\,\\cdot\\,\\sqrtn…","labels":["ErrorCorrectingCodes.Codeword.sqrt_two_pi_mul_sqrt_pi_half"],"detail_key":"p14"},{"id":"n18770","layer":"informal","project":"p14","title":"Positivity of \\lfloor np\\rfloor for large n","kind":"lemma","summary":"[Positivity of \\lfloor np\\rfloor for large n] Let 0<p and 0<1-p, and set N_2 = \\lceil 2/(p(1-p)…","labels":["ErrorCorrectingCodes.Codeword.floor_np_pos"],"detail_key":"p14"},{"id":"n18771","layer":"informal","project":"p14","title":"Stirling-based ratio bound for the binomial term","kind":"lemma","summary":"[Stirling-based ratio bound for the binomial term] Let a,b,n\\inN with a,b>0, a+b=n, and let c>0…","labels":["ErrorCorrectingCodes.Codeword.stirling_comb_bound"],"detail_key":"p14"},{"id":"n18772","layer":"informal","project":"p14","title":"Distance equals minimum nonzero weight","kind":"theorem","summary":"[Distance equals minimum nonzero weight] If C is linear (witnessed by some generator matrix) an…","labels":["ErrorCorrectingCodes.Codeword.Linear_Code_dist_eq_min_weight"],"detail_key":"p14"},{"id":"n18773","layer":"informal","project":"p14","title":"Uniform distribution on vectors","kind":"definition","summary":"[Uniform distribution on vectors] The uniform probability mass function on \\alpha^n, encoded as…","labels":["ErrorCorrectingCodes.Codeword.uniform_vector_dist"],"detail_key":"p14"},{"id":"n18774","layer":"informal","project":"p14","title":"Finiteness of constrained matrix set","kind":"lemma","summary":"[Finiteness of constrained matrix set] For fixed x\\in\\alpha^k and v\\in\\alpha^n, the set \\G : Gx…","labels":["ErrorCorrectingCodes.Codeword.finite_matrix_dist"],"detail_key":"p14"},{"id":"n18775","layer":"informal","project":"p14","title":"Distribution induced by random matrices","kind":"definition","summary":"[Distribution induced by random matrices] \\mu_x(v) is the fraction of n\\times k matrices G sati…","labels":["ErrorCorrectingCodes.Codeword.matrix_dist"],"detail_key":"p14"},{"id":"n18776","layer":"informal","project":"p14","title":"Row extraction","kind":"definition","summary":"[Row extraction] Utility returning the i-th row of a matrix as a 1\\times k matrix.","labels":["ErrorCorrectingCodes.Codeword.get_matrix_row"],"detail_key":"p14"},{"id":"n18777","layer":"informal","project":"p14","title":"Uniformity of Gx for nonzero x","kind":"theorem","summary":"[Uniformity of Gx for nonzero x] If x\\neq 0 and k\\ge 1, then for a uniformly random n\\times k m…","labels":["ErrorCorrectingCodes.Codeword.uniformity_lemma"],"detail_key":"p14"},{"id":"n18778","layer":"informal","project":"p14","title":"Probability of low-weight codeword","kind":"theorem","summary":"[Probability of low-weight codeword] For any nonzero x\\in\\alpha^k, \\[ \\frac|\\G : wt(Gx) < d\\||\\…","labels":["ErrorCorrectingCodes.Codeword.prob_leq_ball_size"],"detail_key":"p14"},{"id":"n18779","layer":"informal","project":"p14","title":"Existence bound via union bound","kind":"theorem","summary":"[Existence bound via union bound] The number of n\\times k matrices that send some nonzero messa…","labels":["ErrorCorrectingCodes.Codeword.existence_bound"],"detail_key":"p14"},{"id":"n18780","layer":"informal","project":"p14","title":"Gilbert--Varshamov bound","kind":"theorem","summary":"[Gilbert--Varshamov bound] If k \\le n - \\lceil\\log_q|B_d-1(0)|\\rceil - 1, then there exists a g…","labels":["ErrorCorrectingCodes.Codeword.gv_bound"],"detail_key":"p14"},{"id":"n18781","layer":"informal","project":"p14","title":"List-decodable code","kind":"definition","summary":"[List-decodable code] A code C is (\\rho,L)-list-decodable if every Hamming ball of radius \\lflo…","labels":["ErrorCorrectingCodes.Codeword.list_decodable"],"detail_key":"p14"},{"id":"n18782","layer":"informal","project":"p14","title":"Existence of list-decodable codes","kind":"lemma","summary":"[Existence of list-decodable codes] If the ball volume V and code size M satisfy the counting i…","labels":["ErrorCorrectingCodes.Codeword.exists_listDecodable_code"],"detail_key":"p14"},{"id":"n18783","layer":"informal","project":"p14","title":"Binomial ratio bound","kind":"lemma","summary":"[Binomial ratio bound] For k\\le M\\le N, \\[ \\frac\\binomN-kM-k\\binomNM \\;\\le\\; \\Bigl(\\fracMN\\Bigr…","labels":["ErrorCorrectingCodes.Codeword.binom_ratio_bound"],"detail_key":"p14"},{"id":"n18784","layer":"informal","project":"p14","title":"Counting inequality for list decoding","kind":"lemma","summary":"[Counting inequality for list decoding] For the choice r = 1-H_q(\\rho)-1/L, M=\\lfloor q^rn\\rflo…","labels":["ErrorCorrectingCodes.Codeword.listDecoding_counting_ineq"],"detail_key":"p14"},{"id":"n18785","layer":"informal","project":"p14","title":"List-decoding capacity","kind":"theorem","summary":"[List-decoding capacity] For any q=|\\alpha|, 0<\\rho\\le 1-1/q, and L\\ge 1, there exist codes of…","labels":["ErrorCorrectingCodes.Codeword.list_decoding_capacity"],"detail_key":"p14"},{"id":"n18786","layer":"informal","project":"p14","title":"Maximum admissible code size","kind":"definition","summary":"[Maximum admissible code size] A_0(n,d,w) is the maximum size of an (n,d,w)-admissible code.","labels":["A0"],"detail_key":"p14"},{"id":"n18787","layer":"informal","project":"p14","title":"Johnson bound, radius form","kind":"theorem","summary":"[Johnson bound, radius form] If n > 0, 1 \\le d, 2d \\le n, and w \\le J_2(n,d), then \\[ A_0(n,d,w…","labels":["binary_johnson_bound_radius"],"detail_key":"p14"},{"id":"n18788","layer":"informal","project":"p14","title":"Binary vector type","kind":"definition","summary":"[Binary vector type] The type of binary words of length n, namely functions Fin\\,n \\to Bool.","labels":["CodingTheory.Johnson.BitVec"],"detail_key":"p14"},{"id":"n18789","layer":"informal","project":"p14","title":"Euclidean coordinate space","kind":"definition","summary":"[Euclidean coordinate space] Abbreviation for the n-dimensional real Euclidean space R^n with t…","labels":["CodingTheory.Johnson.Euc"],"detail_key":"p14"},{"id":"n18790","layer":"informal","project":"p14","title":"Hamming weight","kind":"definition","summary":"[Hamming weight] For a binary word x of length n, its weight is the number of coordinates where…","labels":["CodingTheory.Johnson.wt"],"detail_key":"p14"},{"id":"n18791","layer":"informal","project":"p14","title":"Hamming distance","kind":"definition","summary":"[Hamming distance] For binary words x,y of length n, the Hamming distance is the number of coor…","labels":["CodingTheory.Johnson.hdist"],"detail_key":"p14"},{"id":"n18792","layer":"informal","project":"p14","title":"\\pm 1 embedding","kind":"definition","summary":"[\\pm 1 embedding] The map sending a binary word x to the real vector in R^n whose i-th coordina…","labels":["CodingTheory.Johnson.pmOne"],"detail_key":"p14"},{"id":"n18793","layer":"informal","project":"p14","title":"All-ones vector","kind":"definition","summary":"[All-ones vector] The vector in R^n all of whose coordinates equal 1.","labels":["CodingTheory.Johnson.ones"],"detail_key":"p14"},{"id":"n18794","layer":"informal","project":"p14","title":"Shifted \\pm 1 vector","kind":"definition","summary":"[Shifted \\pm 1 vector] For a real parameter \\alpha and a binary word x, the vector \\hatx^\\alpha…","labels":["CodingTheory.Johnson.shifted"],"detail_key":"p14"},{"id":"n18795","layer":"informal","project":"p14","title":"Normalization","kind":"definition","summary":"[Normalization] The rescaling u \\mapsto \\lVert u\\rVert^-1 \\cdot u of a vector of R^n.","labels":["CodingTheory.Johnson.normalize"],"detail_key":"p14"},{"id":"n18796","layer":"informal","project":"p14","title":"Coordinates of the all-ones vector","kind":"lemma","summary":"[Coordinates of the all-ones vector] Every coordinate of 1 \\in R^n equals 1: for each i, 1_i =…","labels":["CodingTheory.Johnson.ones_apply"],"detail_key":"p14"},{"id":"n18797","layer":"informal","project":"p14","title":"\\pm 1 embedding at a false coordinate","kind":"lemma","summary":"[\\pm 1 embedding at a false coordinate] If x_i = \\textttfalse then pmOne(x)_i = 1.","labels":["CodingTheory.Johnson.pmOne_apply_false"],"detail_key":"p14"},{"id":"n18798","layer":"informal","project":"p14","title":"\\pm 1 embedding at a true coordinate","kind":"lemma","summary":"[\\pm 1 embedding at a true coordinate] If x_i = \\texttttrue then pmOne(x)_i = -1.","labels":["CodingTheory.Johnson.pmOne_apply_true"],"detail_key":"p14"},{"id":"n18799","layer":"informal","project":"p14","title":"Binary Johnson radius","kind":"definition","summary":"[Binary Johnson radius] The real quantity \\[ J_2(n,d) \\;=\\; \\fracn - \\sqrtn\\,(n - 2d)2. \\]","labels":["CodingTheory.Johnson.J2"],"detail_key":"p14"},{"id":"n18800","layer":"informal","project":"p14","title":"Shift parameter \\alpha","kind":"definition","summary":"[Shift parameter \\alpha] The real quantity \\[ \\alpha(n,d) \\;=\\; \\sqrt\\fracn - 2dn. \\]","labels":["CodingTheory.Johnson.alpha"],"detail_key":"p14"},{"id":"n18801","layer":"informal","project":"p14","title":"Admissible code","kind":"definition","summary":"[Admissible code] A finite set C of binary words of length n is (n,d,w)-admissible when any two…","labels":["CodingTheory.Johnson.AdmissibleCode"],"detail_key":"p14"},{"id":"n18802","layer":"informal","project":"p14","title":"Coordinatewise product of \\pm 1 embeddings","kind":"lemma","summary":"[Coordinatewise product of \\pm 1 embeddings] For all binary words x,y and every coordinate i, p…","labels":["CodingTheory.Johnson.coord_mul_pmOne"],"detail_key":"p14"},{"id":"n18803","layer":"informal","project":"p14","title":"Inner product of two \\pm 1 embeddings","kind":"lemma","summary":"[Inner product of two \\pm 1 embeddings] For all binary words x,y of length n, \\[ \\langle pmOne(…","labels":["CodingTheory.Johnson.inner_pmOne_pmOne"],"detail_key":"p14"},{"id":"n18804","layer":"informal","project":"p14","title":"Inner product with the all-ones vector","kind":"lemma","summary":"[Inner product with the all-ones vector] For every binary word x of length n, \\[ \\langle pmOne(…","labels":["CodingTheory.Johnson.inner_pmOne_ones"],"detail_key":"p14"},{"id":"n18805","layer":"informal","project":"p14","title":"Squared norm of the all-ones vector","kind":"lemma","summary":"[Squared norm of the all-ones vector] \\langle 1, 1\\rangle = n in R^n.","labels":["CodingTheory.Johnson.inner_ones_ones"],"detail_key":"p14"},{"id":"n18806","layer":"informal","project":"p14","title":"Bilinear expansion of the shifted inner product","kind":"lemma","summary":"[Bilinear expansion of the shifted inner product] For all \\alpha \\in R and binary words x,y, \\[…","labels":["CodingTheory.Johnson.inner_shifted_expand"],"detail_key":"p14"},{"id":"n18807","layer":"informal","project":"p14","title":"Upper bound on the shifted inner product","kind":"lemma","summary":"[Upper bound on the shifted inner product] If \\alpha \\ge 0, hdist(x,y) \\ge d, and wt(x), wt(y)…","labels":["CodingTheory.Johnson.inner_shifted_le_expr"],"detail_key":"p14"},{"id":"n18808","layer":"informal","project":"p14","title":"Nonnegativity of \\alpha","kind":"lemma","summary":"[Nonnegativity of \\alpha] For all n,d one has 0 \\le \\alpha(n,d).","labels":["CodingTheory.Johnson.alpha_nonneg"],"detail_key":"p14"},{"id":"n18809","layer":"informal","project":"p14","title":"Square of \\alpha","kind":"lemma","summary":"[Square of \\alpha] If n > 0 and 2d \\le n then \\alpha(n,d)^2 = (n - 2d)/n.","labels":["CodingTheory.Johnson.alpha_sq"],"detail_key":"p14"},{"id":"n18810","layer":"informal","project":"p14","title":"\\alpha is less than one","kind":"lemma","summary":"[\\alpha is less than one] If n > 0, d \\ge 1, and 2d \\le n, then \\alpha(n,d) < 1.","labels":["CodingTheory.Johnson.alpha_lt_one_of_hd1"],"detail_key":"p14"},{"id":"n18811","layer":"informal","project":"p14","title":"Johnson arithmetic inequality","kind":"lemma","summary":"[Johnson arithmetic inequality] If n > 0, 2d \\le n, and w \\le J_2(n,d), then for \\alpha = \\alph…","labels":["CodingTheory.Johnson.johnson_arith"],"detail_key":"p14"},{"id":"n18812","layer":"informal","project":"p14","title":"Projections keep non-positive inner products","kind":"lemma","summary":"[Projections keep non-positive inner products] Let V be a real inner product space, u \\in V a u…","labels":["CodingTheory.Johnson.inner_proj_le_zero"],"detail_key":"p14"},{"id":"n18813","layer":"informal","project":"p14","title":"Squared norm of an orthogonal component","kind":"lemma","summary":"[Squared norm of an orthogonal component] For a unit vector u and any x in a real inner product…","labels":["CodingTheory.Johnson.proj_norm_sq"],"detail_key":"p14"},{"id":"n18814","layer":"informal","project":"p14","title":"Nonvanishing of the orthogonal component","kind":"lemma","summary":"[Nonvanishing of the orthogonal component] If u and x are unit vectors with \\langle x,u\\rangle…","labels":["CodingTheory.Johnson.proj_nonzero"],"detail_key":"p14"},{"id":"n18815","layer":"informal","project":"p14","title":"Injectivity of the projection on a set of unit vectors","kind":"lemma","summary":"[Injectivity of the projection on a set of unit vectors] Let u be a unit vector and S a set of…","labels":["CodingTheory.Johnson.proj_inj_on"],"detail_key":"p14"},{"id":"n18816","layer":"informal","project":"p14","title":"Orthogonal projection off a unit vector","kind":"definition","summary":"[Orthogonal projection off a unit vector] For u,v in a real inner product space, orthProj(u,v)…","labels":["CodingTheory.Johnson.orthProj"],"detail_key":"p14"},{"id":"n18817","layer":"informal","project":"p14","title":"The projection lies in the orthogonal complement","kind":"lemma","summary":"[The projection lies in the orthogonal complement] If \\lVert u\\rVert = 1 then orthProj(u,v) \\in…","labels":["CodingTheory.Johnson.orthProj_mem_orthogonal"],"detail_key":"p14"},{"id":"n18818","layer":"informal","project":"p14","title":"The projection of a unit vector is nonzero","kind":"lemma","summary":"[The projection of a unit vector is nonzero] If u and v are unit vectors with v \\neq u and v \\n…","labels":["CodingTheory.Johnson.orthProj_ne_zero"],"detail_key":"p14"},{"id":"n18819","layer":"informal","project":"p14","title":"Projections preserve non-positive inner products","kind":"lemma","summary":"[Projections preserve non-positive inner products] Let \\lVert u\\rVert = 1 and let v,w satisfy \\…","labels":["CodingTheory.Johnson.orthProj_inner_nonpos"],"detail_key":"p14"},{"id":"n18820","layer":"informal","project":"p14","title":"Normalized projections of distinct vectors differ","kind":"lemma","summary":"[Normalized projections of distinct vectors differ] Let u,v,w be unit vectors with \\langle v,u\\…","labels":["CodingTheory.Johnson.normalized_orthProj_injective"],"detail_key":"p14"},{"id":"n18821","layer":"informal","project":"p14","title":"Normalized projection is a unit vector","kind":"lemma","summary":"[Normalized projection is a unit vector] If u,v are unit vectors with v \\neq u and v \\neq -u, t…","labels":["CodingTheory.Johnson.norm_normalized_orthProj"],"detail_key":"p14"},{"id":"n18822","layer":"informal","project":"p14","title":"Non-positive inner product of normalized projections","kind":"lemma","summary":"[Non-positive inner product of normalized projections] Under \\lVert u\\rVert = 1, \\langle v,w\\ra…","labels":["CodingTheory.Johnson.normalized_orthProj_inner_nonpos"],"detail_key":"p14"},{"id":"n18823","layer":"informal","project":"p14","title":"Rank of the orthogonal complement of a line","kind":"lemma","summary":"[Rank of the orthogonal complement of a line] For a unit vector u in a finite-dimensional real…","labels":["CodingTheory.Johnson.finrank_orthogonal_span_singleton"],"detail_key":"p14"},{"id":"n18824","layer":"informal","project":"p14","title":"Discarding u and -u costs at most two elements","kind":"lemma","summary":"[Discarding u and -u costs at most two elements] For a finite subset S of an additive group and…","labels":["CodingTheory.Johnson.card_filter_add_two"],"detail_key":"p14"},{"id":"n18825","layer":"informal","project":"p14","title":"Normalized projection as an element of the complement","kind":"definition","summary":"[Normalized projection as an element of the complement] Given a unit vector u, this packages th…","labels":["CodingTheory.Johnson.mkProj"],"detail_key":"p14"},{"id":"n18826","layer":"informal","project":"p14","title":"Underlying vector of \\textttmkProj","kind":"lemma","summary":"[Underlying vector of \\textttmkProj] The value in V underlying mkProj(u,v) is \\lVert orthProj(u…","labels":["CodingTheory.Johnson.mkProj_val"],"detail_key":"p14"},{"id":"n18827","layer":"informal","project":"p14","title":"Rankin bound in a finite-dimensional space","kind":"theorem","summary":"[Rankin bound in a finite-dimensional space] Let V be a finite-dimensional real inner product s…","labels":["CodingTheory.Johnson.rankin_bound_general"],"detail_key":"p14"},{"id":"n18828","layer":"informal","project":"p14","title":"Rankin bound in R^n","kind":"theorem","summary":"[Rankin bound in R^n] A finite set S of unit vectors in R^n with pairwise non-positive inner pr…","labels":["CodingTheory.Johnson.rankin_finset_bound"],"detail_key":"p14"},{"id":"n18829","layer":"informal","project":"p14","title":"Shifted vectors are nonzero for \\alpha < 1","kind":"lemma","summary":"[Shifted vectors are nonzero for \\alpha < 1] If n > 0 and 0 \\le \\alpha < 1, then \\hatx^\\alpha \\…","labels":["CodingTheory.Johnson.shifted_ne_zero_of_alpha_lt_one"],"detail_key":"p14"},{"id":"n18830","layer":"informal","project":"p14","title":"Johnson cardinality bound, parametric form","kind":"theorem","summary":"[Johnson cardinality bound, parametric form] Let n > 0 and let C be a finite set of binary word…","labels":["CodingTheory.Johnson.binary_johnson_card_bound_parametric"],"detail_key":"p14"},{"id":"n18831","layer":"informal","project":"p14","title":"Binary Johnson cardinality bound","kind":"theorem","summary":"[Binary Johnson cardinality bound] Assume n > 0, 1 \\le d and 2d \\le n. If C is a finite set of…","labels":["CodingTheory.Johnson.binary_johnson_card_bound"],"detail_key":"p14"},{"id":"n18832","layer":"informal","project":"p14","title":"Johnson bound for admissible codes","kind":"theorem","summary":"[Johnson bound for admissible codes] Assume n > 0, 1 \\le d and 2d \\le n. Every (n,d,w)-admissib…","labels":["CodingTheory.Johnson.binary_johnson_card_bound_of_admissible"],"detail_key":"p14"},{"id":"n18833","layer":"informal","project":"p14","title":"Uniform bound transfers to A_0","kind":"lemma","summary":"[Uniform bound transfers to A_0] If every (n,d,w)-admissible code C has \\left\\lvert C\\right\\rve…","labels":["A0_le_of_forall_le"],"detail_key":"p14"},{"id":"n18834","layer":"informal","project":"p14","title":"Symplectic vector space","kind":"definition","summary":"[Symplectic vector space] \\notready For a finite field F_q and n \\in N, the \\emphsymplectic spa…","labels":[],"detail_key":"p14"},{"id":"n18835","layer":"informal","project":"p14","title":"Isotropic subspace","kind":"definition","summary":"[Isotropic subspace] \\notready A subspace S \\le V is \\emphisotropic if \\omega(u,v)=0 for all u,…","labels":[],"detail_key":"p14"},{"id":"n18836","layer":"informal","project":"p14","title":"CSS validity","kind":"definition","summary":"[CSS validity] \\notready Classical codes C_Z, C_X \\le F_q^n satisfy the \\emphCSS condition if \\…","labels":[],"detail_key":"p14"},{"id":"n18837","layer":"informal","project":"p14","title":"CSS stabilizer","kind":"definition","summary":"[CSS stabilizer] \\notready Given C_Z \\perp C_X, the \\emphCSS stabilizer is the subspace of V ge…","labels":[],"detail_key":"p14"},{"id":"n18838","layer":"informal","project":"p14","title":"CSS validity iff isotropy","kind":"theorem","summary":"[CSS validity iff isotropy] \\notready The CSS stabilizer is isotropic if and only if the CSS co…","labels":[],"detail_key":"p14"},{"id":"n18839","layer":"informal","project":"p14","title":"CSS dimension formula","kind":"theorem","summary":"[CSS dimension formula] \\notready If C_Z \\perp C_X, then \\dim(cssStabilizer) = \\dim(C_Z) + \\dim…","labels":[],"detail_key":"p14"},{"id":"n18840","layer":"informal","project":"p14","title":"CSS code","kind":"definition","summary":"[CSS code] \\notready The \\emphCSS code Q(C_Z, C_X) encodes k = n - \\dim(C_Z) - \\dim(C_X) logica…","labels":[],"detail_key":"p14"},{"id":"n18841","layer":"informal","project":"p14","title":"Pauli basis","kind":"definition","summary":"[Pauli basis] The four Pauli basis elements \\I,X,Y,Z\\, represented as an inductive type.","labels":["PauliBasis"],"detail_key":"p14"},{"id":"n18842","layer":"informal","project":"p14","title":"Pauli string","kind":"definition","summary":"[Pauli string] A \\emphPauli string of length n is a function p : Fin\\,n \\to PauliBasis.","labels":["PauliString"],"detail_key":"p14"},{"id":"n18843","layer":"informal","project":"p14","title":"Support and weight","kind":"definition","summary":"[Support and weight] The \\emphsupport supp(p) \\subseteq Fin\\,n consists of coordinates where p(…","labels":["support","weight"],"detail_key":"p14"},{"id":"n18844","layer":"informal","project":"p14","title":"Pauli error set","kind":"definition","summary":"[Pauli error set] The set of all Pauli strings of weight at most t: \\[ E(n,t) = \\bigl\\p : Pauli…","labels":["PauliErrorsLe"],"detail_key":"p14"},{"id":"n18845","layer":"informal","project":"p14","title":"Cardinality of the Pauli error set","kind":"theorem","summary":"[Cardinality of the Pauli error set] \\[ |E(n,t)| \\;=\\; \\sum_i=0^t\\binomni\\,3^i. \\]","labels":["card_pauliErrorsLe"],"detail_key":"p14"},{"id":"n18846","layer":"informal","project":"p14","title":"Partition by weight j and exact support S: each of the \\binomnj supports of size j carrie…","kind":"proof","summary":"Partition by weight j and exact support S: each of the \\binomnj supports of size j carries 3^j…","labels":[],"detail_key":"p14"},{"id":"n18847","layer":"informal","project":"p14","title":"n-qubit Hilbert space","kind":"definition","summary":"[n-qubit Hilbert space] The n-qubit Hilbert space H_n = \\ell^2\\!\\bigl(\\0,1\\^n,C\\bigr), implemen…","labels":["Hn"],"detail_key":"p14"},{"id":"n18848","layer":"informal","project":"p14","title":"Dimension of H_n","kind":"lemma","summary":"[Dimension of H_n] \\dim_C(H_n) = 2^n.","labels":["finrank_Hn"],"detail_key":"p14"},{"id":"n18849","layer":"informal","project":"p14","title":"Pauli operator","kind":"definition","summary":"[Pauli operator] For p \\in PauliString\\,n, the associated Pauli operator \\hatp : H_n \\to H_n.","labels":["pauliOp"],"detail_key":"p14"},{"id":"n18850","layer":"informal","project":"p14","title":"Knill--Laflamme condition","kind":"definition","summary":"[Knill--Laflamme condition] A subspace C \\le H_n satisfies the \\emphKnill--Laflamme condition f…","labels":["KnillLaflamme"],"detail_key":"p14"},{"id":"n18851","layer":"informal","project":"p14","title":"Non-degenerate code","kind":"definition","summary":"[Non-degenerate code] A code is \\emphnon-degenerate if it satisfies the Knill--Laflamme conditi…","labels":["IsNondegenerate"],"detail_key":"p14"},{"id":"n18852","layer":"informal","project":"p14","title":"Error sphere","kind":"definition","summary":"[Error sphere] The \\empherror sphere ES(C,t) is the subspace \\bigvee_wt(p)\\le t \\hatp(C), i.e.\\…","labels":["ErrorSphere"],"detail_key":"p14"},{"id":"n18853","layer":"informal","project":"p14","title":"Error subspaces are pairwise orthogonal","kind":"lemma","summary":"[Error subspaces are pairwise orthogonal] If C is non-degenerate and E \\neq F both have weight…","labels":["error_subspaces_orthogonal"],"detail_key":"p14"},{"id":"n18854","layer":"informal","project":"p14","title":"Dimension of the error sphere","kind":"lemma","summary":"[Dimension of the error sphere] If C is non-degenerate, \\[ \\dim(ES(C,t)) \\;=\\; |E(n,t)|\\cdot\\di…","labels":["error_sphere_dimension"],"detail_key":"p14"},{"id":"n18855","layer":"informal","project":"p14","title":"Quantum Hamming bound","kind":"theorem","summary":"[Quantum Hamming bound] If C \\le H_n is non-degenerate, then \\[ \\left(\\sum_i=0^t\\binomni\\,3^i\\r…","labels":["quantum_hamming_bound"],"detail_key":"p14"},{"id":"n18856","layer":"informal","project":"p14","title":"Pauli I matrix","kind":"definition","summary":"[Pauli I matrix] The 2\\times 2 identity matrix over C, i.e.\\ the Pauli operator I.","labels":["sigmaI"],"detail_key":"p14"},{"id":"n18857","layer":"informal","project":"p14","title":"Pauli X matrix","kind":"definition","summary":"[Pauli X matrix] The bit-flip Pauli matrix X = 0 & 1 \\\\ 1 & 0 over C.","labels":["sigmaX"],"detail_key":"p14"},{"id":"n18858","layer":"informal","project":"p14","title":"Pauli Y matrix","kind":"definition","summary":"[Pauli Y matrix] The bit-phase-flip Pauli matrix Y = 0 & -i \\\\ i & 0 over C.","labels":["sigmaY"],"detail_key":"p14"},{"id":"n18859","layer":"informal","project":"p14","title":"Pauli Z matrix","kind":"definition","summary":"[Pauli Z matrix] The phase-flip Pauli matrix Z = 1 & 0 \\\\ 0 & -1 over C.","labels":["sigmaZ"],"detail_key":"p14"},{"id":"n18860","layer":"informal","project":"p14","title":"Matrix of a Pauli basis element","kind":"definition","summary":"[Matrix of a Pauli basis element] Maps each Pauli basis element \\I,X,Y,Z\\ to its corresponding…","labels":["PauliBasis.toMatrix"],"detail_key":"p14"},{"id":"n18861","layer":"informal","project":"p14","title":"Matrix of a Pauli string","kind":"definition","summary":"[Matrix of a Pauli string] For a Pauli string p on n qubits, the 2^n\\times 2^n matrix obtained…","labels":["pauliMatrix"],"detail_key":"p14"},{"id":"n18862","layer":"informal","project":"p14","title":"Embedding of a non-identity Pauli","kind":"definition","summary":"[Embedding of a non-identity Pauli] Embeds a non-identity Pauli (X, Y, or Z) from the three-ele…","labels":["PauliNZ.toBasis"],"detail_key":"p14"},{"id":"n18863","layer":"informal","project":"p14","title":"Pauli string with prescribed support","kind":"definition","summary":"[Pauli string with prescribed support] Given a finset S \\subseteq Fin\\,n and an assignment f :…","labels":["mkWithSupport"],"detail_key":"p14"},{"id":"n18864","layer":"informal","project":"p14","title":"Support of \\textttmkWithSupport","kind":"lemma","summary":"[Support of \\textttmkWithSupport] The Pauli string mkWithSupport\\,S\\,f has support exactly S.","labels":["support_mkWithSupport"],"detail_key":"p14"},{"id":"n18865","layer":"informal","project":"p14","title":"Pauli strings with exact support","kind":"definition","summary":"[Pauli strings with exact support] The finset of all Pauli strings on n qubits whose support eq…","labels":["pauliStringsExactSupport"],"detail_key":"p14"},{"id":"n18866","layer":"informal","project":"p14","title":"Count of Pauli strings with exact support","kind":"lemma","summary":"[Count of Pauli strings with exact support] The number of Pauli strings with support exactly S…","labels":["card_pauliStringsExactSupport"],"detail_key":"p14"},{"id":"n18867","layer":"informal","project":"p14","title":"Quantum code","kind":"definition","summary":"[Quantum code] An n-qubit quantum code is a subspace of the n-qubit Hilbert space H_n.","labels":["Code"],"detail_key":"p14"},{"id":"n18868","layer":"informal","project":"p14","title":"Adjoint of a Pauli operator","kind":"definition","summary":"[Adjoint of a Pauli operator] The adjoint (Hermitian conjugate) \\hatp^\\dagger : H_n \\to H_n of…","labels":["pauliOpAdjoint"],"detail_key":"p14"},{"id":"n18869","layer":"informal","project":"p14","title":"Projection onto a code","kind":"definition","summary":"[Projection onto a code] The orthogonal projection of H_n onto a subspace C, viewed as an endom…","labels":["codeProj"],"detail_key":"p14"},{"id":"n18870","layer":"informal","project":"p14","title":"Action of the code projection","kind":"lemma","summary":"[Action of the code projection] For any x \\in H_n, the value P_C\\,x coincides with the orthogon…","labels":["codeProj_apply"],"detail_key":"p14"},{"id":"n18871","layer":"informal","project":"p14","title":"Image of the code projection lies in C","kind":"lemma","summary":"[Image of the code projection lies in C] For any x \\in H_n, the projected vector P_C\\,x belongs…","labels":["codeProj_mem"],"detail_key":"p14"},{"id":"n18872","layer":"informal","project":"p14","title":"Projection fixes code vectors","kind":"lemma","summary":"[Projection fixes code vectors] If x \\in C, then P_C\\,x = x.","labels":["codeProj_eq_self_of_mem"],"detail_key":"p14"},{"id":"n18873","layer":"informal","project":"p14","title":"Idempotence of the code projection","kind":"lemma","summary":"[Idempotence of the code projection] The code projection is idempotent: P_C(P_C\\,x) = P_C\\,x fo…","labels":["codeProj_idempotent"],"detail_key":"p14"},{"id":"n18874","layer":"informal","project":"p14","title":"Quantum Hamming bound, raw form","kind":"theorem","summary":"[Quantum Hamming bound, raw form] For a non-degenerate [[n,k]] quantum code C \\le H_n correctin…","labels":["quantum_hamming_bound_raw"],"detail_key":"p14"},{"id":"n18875","layer":"informal","project":"p14","title":"Embedded non-identity Paulis are not I","kind":"lemma","summary":"[Embedded non-identity Paulis are not I] For every a \\in PauliNZ, its image a.toBasis in the Pa…","labels":["PauliNZ.toBasis_ne_I"],"detail_key":"p14"},{"id":"n18876","layer":"informal","project":"p14","title":"Symplectic form","kind":"definition","summary":"[Symplectic form] For u = (x,z), v = (x',z') \\in V = F_p^n \\times F_p^n, \\[ \\omega(u,v) \\;=\\; \\…","labels":["sym_form"],"detail_key":"p14"},{"id":"n18877","layer":"informal","project":"p14","title":"Bilinearity","kind":"lemma","summary":"[Bilinearity] \\omega is bilinear: additive and scalar-homogeneous in each argument.","labels":["sym_form_add_left","sym_form_add_right","sym_form_smul_left","sym_form_smul_right"],"detail_key":"p14"},{"id":"n18878","layer":"informal","project":"p14","title":"Antisymmetry","kind":"lemma","summary":"[Antisymmetry] \\omega(u,v) = -\\omega(v,u).","labels":["sym_form_swap"],"detail_key":"p14"},{"id":"n18879","layer":"informal","project":"p14","title":"Nondegeneracy","kind":"lemma","summary":"[Nondegeneracy] If \\omega(u,v) = 0 for all v, then u = 0.","labels":["sym_form_nondegenerate"],"detail_key":"p14"},{"id":"n18880","layer":"informal","project":"p14","title":"Bundled bilinear form","kind":"definition","summary":"[Bundled bilinear form] \\omega packaged as a \\textttLinearMap.BilinForm.","labels":["symB"],"detail_key":"p14"},{"id":"n18881","layer":"informal","project":"p14","title":"Support and weight","kind":"definition","summary":"[Support and weight] The \\emphsupport supp(v) \\subseteq Fin\\,n consists of coordinates where ei…","labels":["supp","wt"],"detail_key":"p14"},{"id":"n18882","layer":"informal","project":"p14","title":"Weight bound","kind":"lemma","summary":"[Weight bound] wt(v) \\le n for all v \\in V.","labels":["wt_le_n"],"detail_key":"p14"},{"id":"n18883","layer":"informal","project":"p14","title":"Support submodule","kind":"definition","summary":"[Support submodule] For C \\subseteq Fin\\,n, the \\emphsupport submodule V_C = \\v \\in V \\mid supp…","labels":["V_sub"],"detail_key":"p14"},{"id":"n18884","layer":"informal","project":"p14","title":"Dimension of support submodule","kind":"lemma","summary":"[Dimension of support submodule] \\dim_F_p(V_C) = 2|C|.","labels":["dim_V_sub"],"detail_key":"p14"},{"id":"n18885","layer":"informal","project":"p14","title":"Symplectic orthogonal complement","kind":"definition","summary":"[Symplectic orthogonal complement] For a submodule S \\le V, S^\\perp_\\omega = \\v \\in V \\mid \\for…","labels":["sym_orth"],"detail_key":"p14"},{"id":"n18886","layer":"informal","project":"p14","title":"Dimension of the symplectic orthogonal","kind":"lemma","summary":"[Dimension of the symplectic orthogonal] \\dim(S^\\perp_\\omega) = 2n - \\dim(S).","labels":["finrank_sym_orth"],"detail_key":"p14"},{"id":"n18887","layer":"informal","project":"p14","title":"Isotropic submodule","kind":"definition","summary":"[Isotropic submodule] S is \\emphisotropic if S \\le S^\\perp_\\omega, i.e.\\ \\omega(u,v) = 0 for al…","labels":["IsIsotropic"],"detail_key":"p14"},{"id":"n18888","layer":"informal","project":"p14","title":"Quantum code distance","kind":"definition","summary":"[Quantum code distance] d(S) = \\min\\wt(v) \\mid v \\in S^\\perp_\\omega \\setminus S,\\; wt(v) \\neq 0…","labels":["code_dist"],"detail_key":"p14"},{"id":"n18889","layer":"informal","project":"p14","title":"Distance is at most n","kind":"lemma","summary":"[Distance is at most n] d(S) \\le n.","labels":["code_dist_le_n"],"detail_key":"p14"},{"id":"n18890","layer":"informal","project":"p14","title":"Erasure correctability","kind":"definition","summary":"[Erasure correctability] An erasure set E \\subseteq Fin\\,n is \\emphcorrectable for S if every v…","labels":["correctable"],"detail_key":"p14"},{"id":"n18891","layer":"informal","project":"p14","title":"Distance implies correctability","kind":"lemma","summary":"[Distance implies correctability] If |E| < d(S), then E is correctable.","labels":["dist_implies_correctable"],"detail_key":"p14"},{"id":"n18892","layer":"informal","project":"p14","title":"Any v \\in S^\\perp_\\omega \\cap V_E outside S witnesses d(S) \\le wt(v), while v \\in V_E for…","kind":"proof","summary":"Any v \\in S^\\perp_\\omega \\cap V_E outside S witnesses d(S) \\le wt(v), while v \\in V_E forces su…","labels":[],"detail_key":"p14"},{"id":"n18893","layer":"informal","project":"p14","title":"Logical dimension","kind":"definition","summary":"[Logical dimension] k(S) = n - \\dim(S).","labels":["code_k"],"detail_key":"p14"},{"id":"n18894","layer":"informal","project":"p14","title":"Two disjoint correctable sets bound k","kind":"lemma","summary":"[Two disjoint correctable sets bound k] Let S be isotropic. If A,B \\subseteq Fin\\,n are disjoin…","labels":["two_disjoint_correctable_sets_bound_logical_dimension"],"detail_key":"p14"},{"id":"n18895","layer":"informal","project":"p14","title":"By the cleaning identity, g(S, univ \\setminus A) = 2(n - \\dim S) (\\textttg\\_complement\\_c…","kind":"proof","summary":"By the cleaning identity, g(S, univ \\setminus A) = 2(n - \\dim S) (\\textttg\\_complement\\_correct…","labels":[],"detail_key":"p14"},{"id":"n18896","layer":"informal","project":"p14","title":"Quantum Singleton bound","kind":"theorem","summary":"[Quantum Singleton bound] For any isotropic submodule S \\le V, \\[ k(S) + 2(d(S) - 1) \\;\\le\\; n.…","labels":["quantum_singleton_bound"],"detail_key":"p14"},{"id":"n18897","layer":"informal","project":"p14","title":"Choose disjoint erasure sets A, B of size d(S)-1 (\\textttexists\\_disjoint\\_finsets\\_card)…","kind":"proof","summary":"Choose disjoint erasure sets A, B of size d(S)-1 (\\textttexists\\_disjoint\\_finsets\\_card); both…","labels":[],"detail_key":"p14"},{"id":"n18898","layer":"informal","project":"p14","title":"Prime field F_p","kind":"definition","summary":"[Prime field F_p] The prime field GF(p), realized as Z/pZ via \\textttZMod.","labels":["F"],"detail_key":"p14"},{"id":"n18899","layer":"informal","project":"p14","title":"Symplectic vector space","kind":"definition","summary":"[Symplectic vector space] The ambient space V = F_p^n \\times F_p^n of pairs of coordinate vecto…","labels":["V"],"detail_key":"p14"},{"id":"n18900","layer":"informal","project":"p14","title":"Evaluation of the bundled form","kind":"lemma","summary":"[Evaluation of the bundled form] The bundled bilinear form symB agrees with \\omega: symB(x,y) =…","labels":["symB_apply"],"detail_key":"p14"},{"id":"n18901","layer":"informal","project":"p14","title":"Restriction to coordinates in C","kind":"definition","summary":"[Restriction to coordinates in C] For C \\subseteq Fin\\,n, the linear map V_C \\to (C \\to F_p) \\t…","labels":["restrictToC"],"detail_key":"p14"},{"id":"n18902","layer":"informal","project":"p14","title":"Extension from coordinates in C","kind":"definition","summary":"[Extension from coordinates in C] The linear map (C \\to F_p) \\times (C \\to F_p) \\to V_C that ex…","labels":["extendFromC"],"detail_key":"p14"},{"id":"n18903","layer":"informal","project":"p14","title":"restrictToC is a left inverse of extendFromC","kind":"lemma","summary":"[restrictToC is a left inverse of extendFromC] For every x, restrictToC_C(extendFromC_C(x)) = x.","labels":["restrictToC_extendFromC"],"detail_key":"p14"},{"id":"n18904","layer":"informal","project":"p14","title":"extendFromC is a left inverse of restrictToC","kind":"lemma","summary":"[extendFromC is a left inverse of restrictToC] For every x \\in V_C, extendFromC_C(restrictToC_C…","labels":["extendFromC_restrictToC"],"detail_key":"p14"},{"id":"n18905","layer":"informal","project":"p14","title":"Isomorphism V_C \\cong (F_p^C)^2","kind":"definition","summary":"[Isomorphism V_C \\cong (F_p^C)^2] The linear equivalence V_C \\simeq (C \\to F_p) \\times (C \\to F…","labels":["V_sub_iso"],"detail_key":"p14"},{"id":"n18906","layer":"informal","project":"p14","title":"Restriction map r_E","kind":"definition","summary":"[Restriction map r_E] The linear map r_E : V \\to V_E that zeroes out the coordinates of a vecto…","labels":["r_E"],"detail_key":"p14"},{"id":"n18907","layer":"informal","project":"p14","title":"Intersection S_M = S \\cap V_M","kind":"definition","summary":"[Intersection S_M = S \\cap V_M] For a submodule S \\le V and M \\subseteq Fin\\,n, the submodule S…","labels":["S_M"],"detail_key":"p14"},{"id":"n18908","layer":"informal","project":"p14","title":"Intersection S^\\perp_M = S^\\perp_\\omega \\cap V_M","kind":"definition","summary":"[Intersection S^\\perp_M = S^\\perp_\\omega \\cap V_M] The submodule S^\\perp_M = S^\\perp_\\omega \\ca…","labels":["S_perp_M"],"detail_key":"p14"},{"id":"n18909","layer":"informal","project":"p14","title":"Supportable logical operators count g(M)","kind":"definition","summary":"[Supportable logical operators count g(M)] g(S,M) = \\dim_F_p(S^\\perp_M) - \\dim_F_p(S_M).","labels":["g"],"detail_key":"p14"},{"id":"n18910","layer":"informal","project":"p14","title":"Kernel of r_E","kind":"lemma","summary":"[Kernel of r_E] \\ker(r_E) = V_univ \\setminus E, the subspace supported on the complement of E.","labels":["ker_r_E"],"detail_key":"p14"},{"id":"n18911","layer":"informal","project":"p14","title":"Complement of E","kind":"definition","summary":"[Complement of E] E^c = Fin\\,n \\setminus E, the complement finset of E.","labels":["E_c"],"detail_key":"p14"},{"id":"n18912","layer":"informal","project":"p14","title":"Complement as set difference","kind":"lemma","summary":"[Complement as set difference] E^c = univ \\setminus E.","labels":["E_c_eq"],"detail_key":"p14"},{"id":"n18913","layer":"informal","project":"p14","title":"Rank-nullity for the restriction of S to E","kind":"lemma","summary":"[Rank-nullity for the restriction of S to E] \\dim_F_p(r_E(S)) = \\dim_F_p(S) - \\dim_F_p(S \\cap V…","labels":["dim_map_r_E"],"detail_key":"p14"},{"id":"n18914","layer":"informal","project":"p14","title":"Symplectic form respects restriction","kind":"lemma","summary":"[Symplectic form respects restriction] If v \\in V_M, then \\omega(v,s) = \\omega(v, r_M(s)) for a…","labels":["sym_form_r_E"],"detail_key":"p14"},{"id":"n18915","layer":"informal","project":"p14","title":"Restriction as an endomorphism of V","kind":"definition","summary":"[Restriction as an endomorphism of V] The composition r_E^V : V \\to V of r_E with the inclusion…","labels":["r_E_V"],"detail_key":"p14"},{"id":"n18916","layer":"informal","project":"p14","title":"Restriction on the left argument","kind":"lemma","summary":"[Restriction on the left argument] If v \\in V_M, then \\omega(r_M^V(s), v) = \\omega(s, v) for al…","labels":["sym_form_r_E_left"],"detail_key":"p14"},{"id":"n18917","layer":"informal","project":"p14","title":"Non-degeneracy on V_M","kind":"lemma","summary":"[Non-degeneracy on V_M] If v \\in V_M and \\omega(v,w) = 0 for all w \\in V_M, then v = 0.","labels":["sym_form_nondegenerate_on_V_sub"],"detail_key":"p14"},{"id":"n18918","layer":"informal","project":"p14","title":"Restriction on the left argument, second form","kind":"lemma","summary":"[Restriction on the left argument, second form] If v \\in V_M, then \\omega(r_M^V(s), v) = \\omega…","labels":["sym_form_left_restrict"],"detail_key":"p14"},{"id":"n18919","layer":"informal","project":"p14","title":"Orthogonal intersection equals orthogonal of restricted image","kind":"lemma","summary":"[Orthogonal intersection equals orthogonal of restricted image] S^\\perp_\\omega \\cap V_M = (r_M^…","labels":["orth_inter_eq_orth_map"],"detail_key":"p14"},{"id":"n18920","layer":"informal","project":"p14","title":"Restricted bundled form","kind":"definition","summary":"[Restricted bundled form] The symplectic bilinear form symB restricted to the subspace V_M.","labels":["symB_sub"],"detail_key":"p14"},{"id":"n18921","layer":"informal","project":"p14","title":"Restricted symplectic form","kind":"definition","summary":"[Restricted symplectic form] An abbreviation for symB\\_sub, the symplectic form viewed as a bil…","labels":["sym_form_sub"],"detail_key":"p14"},{"id":"n18922","layer":"informal","project":"p14","title":"Evaluation of the restricted form","kind":"lemma","summary":"[Evaluation of the restricted form] For x,y \\in V_M, sym\\_form\\_sub_M(x,y) = \\omega(x,y) comput…","labels":["sym_form_sub_apply"],"detail_key":"p14"},{"id":"n18923","layer":"informal","project":"p14","title":"Non-degeneracy of the restricted form","kind":"lemma","summary":"[Non-degeneracy of the restricted form] The restricted symplectic form on V_M is non-degenerate.","labels":["sym_form_sub_nondegenerate"],"detail_key":"p14"},{"id":"n18924","layer":"informal","project":"p14","title":"Orthogonal intersection as an image","kind":"lemma","summary":"[Orthogonal intersection as an image] S^\\perp_\\omega \\cap V_M is the image under V_M \\hookright…","labels":["orth_inter_eq_orth_sub_image"],"detail_key":"p14"},{"id":"n18925","layer":"informal","project":"p14","title":"Reflexivity of the restricted form","kind":"lemma","summary":"[Reflexivity of the restricted form] The restricted symplectic form on V_M is reflexive.","labels":["sym_form_sub_isRefl"],"detail_key":"p14"},{"id":"n18926","layer":"informal","project":"p14","title":"Dimension of the orthogonal intersection","kind":"lemma","summary":"[Dimension of the orthogonal intersection] \\dim_F_p(S^\\perp_\\omega \\cap V_M) = 2|M| - \\dim_F_p(…","labels":["dim_orth_inter"],"detail_key":"p14"},{"id":"n18927","layer":"informal","project":"p14","title":"Expansion of g(M)","kind":"lemma","summary":"[Expansion of g(M)] For isotropic S, \\[ g(S,M) = 2|M| + \\dim(S_M^c) - \\dim(S) - \\dim(S_M). \\]","labels":["g_expansion"],"detail_key":"p14"},{"id":"n18928","layer":"informal","project":"p14","title":"Double complement","kind":"lemma","summary":"[Double complement] (M^c)^c = M.","labels":["E_c_E_c"],"detail_key":"p14"},{"id":"n18929","layer":"informal","project":"p14","title":"Sum of restricted dimensions is bounded","kind":"lemma","summary":"[Sum of restricted dimensions is bounded] \\dim_F_p(S_M) + \\dim_F_p(S_M^c) \\le \\dim_F_p(S).","labels":["dim_S_M_add_dim_S_M_c_le_dim_S"],"detail_key":"p14"},{"id":"n18930","layer":"informal","project":"p14","title":"Formula for g(M)","kind":"lemma","summary":"[Formula for g(M)] For isotropic S, \\[ g(S,M) = \\bigl(2|M| + \\dim(S_M^c)\\bigr) - \\bigl(\\dim(S)…","labels":["g_formula"],"detail_key":"p14"},{"id":"n18931","layer":"informal","project":"p14","title":"Additive identity for g(M)","kind":"lemma","summary":"[Additive identity for g(M)] For isotropic S, \\[ g(S,M) + \\dim(S_M) + \\dim(S) = 2|M| + \\dim(S_M…","labels":["g_add_dims"],"detail_key":"p14"},{"id":"n18932","layer":"informal","project":"p14","title":"Dimension inequality","kind":"lemma","summary":"[Dimension inequality] For isotropic S, \\dim(S) + \\dim(S_M) \\le 2|M| + \\dim(S_M^c).","labels":["dim_ineq_aux"],"detail_key":"p14"},{"id":"n18933","layer":"informal","project":"p14","title":"Cleaning dimension identity","kind":"lemma","summary":"[Cleaning dimension identity] For isotropic S, \\[ g(S,M) + g(S,M^c) = 2n - 2\\dim_F_p(S). \\]","labels":["cleaning_dimension_identity"],"detail_key":"p14"},{"id":"n18934","layer":"informal","project":"p14","title":"Cardinalities of a set and its complement","kind":"lemma","summary":"[Cardinalities of a set and its complement] |M| + |M^c| = n.","labels":["card_add_compl"],"detail_key":"p14"},{"id":"n18935","layer":"informal","project":"p14","title":"Correctable sets have g = 0","kind":"lemma","summary":"[Correctable sets have g = 0] If M is correctable for S, then g(S,M) = 0.","labels":["correctable_implies_g_zero"],"detail_key":"p14"},{"id":"n18936","layer":"informal","project":"p14","title":"Correctable complement has g(M^c) = 2k","kind":"lemma","summary":"[Correctable complement has g(M^c) = 2k] If S is isotropic and M is correctable, then g(S,M^c)…","labels":["g_complement_correctable"],"detail_key":"p14"},{"id":"n18937","layer":"informal","project":"p14","title":"Bound on g for a correctable part plus a set","kind":"lemma","summary":"[Bound on g for a correctable part plus a set] If B and C are disjoint and B is correctable for…","labels":["g_le_two_card_C"],"detail_key":"p14"},{"id":"n18938","layer":"informal","project":"p14","title":"Rank--nullity for the restriction map r_C : S^\\perp_\\omega \\cap V_B \\cup C \\to V_C: its k…","kind":"proof","summary":"Rank--nullity for the restriction map r_C : S^\\perp_\\omega \\cap V_B \\cup C \\to V_C: its kernel…","labels":[],"detail_key":"p14"},{"id":"n18939","layer":"informal","project":"p14","title":"Full support is the whole space","kind":"lemma","summary":"[Full support is the whole space] V_univ = \\top, the entire space V.","labels":["V_sub_univ_eq_top"],"detail_key":"p14"},{"id":"n18940","layer":"informal","project":"p14","title":"Dimension of V","kind":"lemma","summary":"[Dimension of V] \\dim_F_p(V) = 2n.","labels":["finrank_V"],"detail_key":"p14"},{"id":"n18941","layer":"informal","project":"p14","title":"Non-degeneracy of the bundled form","kind":"lemma","summary":"[Non-degeneracy of the bundled form] The bundled symplectic form symB is non-degenerate.","labels":["symB_nondegenerate"],"detail_key":"p14"},{"id":"n18942","layer":"informal","project":"p14","title":"Reflexivity of the bundled form","kind":"lemma","summary":"[Reflexivity of the bundled form] The bundled symplectic form symB is reflexive.","labels":["symB_isRefl"],"detail_key":"p14"},{"id":"n18943","layer":"informal","project":"p14","title":"Isotropic dimension bound","kind":"lemma","summary":"[Isotropic dimension bound] If S is isotropic, then \\dim_F_p(S) \\le n.","labels":["finrank_le_n_of_isotropic"],"detail_key":"p14"},{"id":"n18944","layer":"informal","project":"p14","title":"Zero logical dimension forces zero distance","kind":"lemma","summary":"[Zero logical dimension forces zero distance] If S is isotropic and k(S) = 0, then d(S) = 0.","labels":["code_dist_eq_zero_of_code_k_eq_zero"],"detail_key":"p14"},{"id":"n18945","layer":"informal","project":"p14","title":"Existence of disjoint finsets of given size","kind":"lemma","summary":"[Existence of disjoint finsets of given size] Whenever 2t \\le n, there exist disjoint finsets A…","labels":["exists_disjoint_finsets_card"],"detail_key":"p14"},{"id":"n18946","layer":"informal","project":"p14","title":"Merge of two gate arrays","kind":"definition","summary":"[Merge of two gate arrays] Given gate arrays g_1 : Fin\\,m_1 \\to \\alpha and g_2 : Fin\\,m_2 \\to \\…","labels":["LMN.mergeGates"],"detail_key":"p14"},{"id":"n18947","layer":"informal","project":"p14","title":"Left projection of merge","kind":"lemma","summary":"[Left projection of merge] For every i : Fin\\,m_1, evaluating the merged array at the left-embe…","labels":["LMN.mergeGates_castAdd"],"detail_key":"p14"},{"id":"n18948","layer":"informal","project":"p14","title":"Right projection of merge","kind":"lemma","summary":"[Right projection of merge] For every i : Fin\\,m_2, evaluating the merged array at the right-em…","labels":["LMN.mergeGates_natAdd"],"detail_key":"p14"},{"id":"n18949","layer":"informal","project":"p14","title":"Reindex into left half evaluates the original","kind":"lemma","summary":"[Reindex into left half evaluates the original] For a circuit c on m_1 gates and Boolean gate v…","labels":["LMN.reidx_eval_mergeGates_left"],"detail_key":"p14"},{"id":"n18950","layer":"informal","project":"p14","title":"Reindex into right half evaluates the original","kind":"lemma","summary":"[Reindex into right half evaluates the original] For a circuit c on m_2 gates and Boolean gate…","labels":["LMN.reidx_eval_mergeGates_right"],"detail_key":"p14"},{"id":"n18951","layer":"informal","project":"p14","title":"Width bound on the left part of a merge","kind":"lemma","summary":"[Width bound on the left part of a merge] If every g_1\\,k has width at most l, then for each i…","labels":["LMN.mergeGates_width_left"],"detail_key":"p14"},{"id":"n18952","layer":"informal","project":"p14","title":"Width bound on the right part of a merge","kind":"lemma","summary":"[Width bound on the right part of a merge] If every g_2\\,k has width at most l, then for each i…","labels":["LMN.mergeGates_width_right"],"detail_key":"p14"},{"id":"n18953","layer":"informal","project":"p14","title":"Width bound preserved under merging","kind":"lemma","summary":"[Width bound preserved under merging] If every gate of g_1 and every gate of g_2 has width at m…","labels":["LMN.mergeGates_width"],"detail_key":"p14"},{"id":"n18954","layer":"informal","project":"p14","title":"Variable injectivity preserved under merging","kind":"lemma","summary":"[Variable injectivity preserved under merging] If, in each DNF gate of g_1 and of g_2, any two…","labels":["LMN.mergeGates_varInj"],"detail_key":"p14"},{"id":"n18955","layer":"informal","project":"p14","title":"No-duplication preserved under merging","kind":"lemma","summary":"[No-duplication preserved under merging] If every term of every DNF gate of g_1 and of g_2 has…","labels":["LMN.mergeGates_nodup"],"detail_key":"p14"},{"id":"n18956","layer":"informal","project":"p14","title":"B-reasonable random variable","kind":"definition","summary":"[B-reasonable random variable] A real random variable X on a measurable space \\Omega with measu…","labels":["Bonami.IsBReasonable"],"detail_key":"p14"},{"id":"n18957","layer":"informal","project":"p14","title":"Tail bound for B-reasonable variables","kind":"lemma","summary":"[Tail bound for B-reasonable variables] If X is B-reasonable and not equivalent to 0 (its secon…","labels":["Bonami.b_reasonable_tail_bound"],"detail_key":"p14"},{"id":"n18958","layer":"informal","project":"p14","title":"B-reasonability from minimal probability","kind":"lemma","summary":"[B-reasonability from minimal probability] Let X be a discrete random variable on a finite spac…","labels":["Bonami.min_prob_b_reasonable"],"detail_key":"p14"},{"id":"n18959","layer":"informal","project":"p14","title":"Paley--Zygmund inequality","kind":"lemma","summary":"[Paley--Zygmund inequality] Let Z \\ge 0 be a nonnegative random variable with finite first and…","labels":["Bonami.paley_zygmund_ineq"],"detail_key":"p14"},{"id":"n18960","layer":"informal","project":"p14","title":"Anti-concentration for B-reasonable variables","kind":"lemma","summary":"[Anti-concentration for B-reasonable variables] If X is B-reasonable and not equivalent to 0, t…","labels":["Bonami.b_reasonable_anticon_zero"],"detail_key":"p14"},{"id":"n18961","layer":"informal","project":"p14","title":"Restriction of the last coordinate","kind":"definition","summary":"[Restriction of the last coordinate] Given a Boolean function f on n+1 variables and a bit b, t…","labels":["Bonami.restrictLast"],"detail_key":"p14"},{"id":"n18962","layer":"informal","project":"p14","title":"Average over the last coordinate","kind":"definition","summary":"[Average over the last coordinate] The average avgLast\\,f of f over its last coordinate is the…","labels":["Bonami.avgLast"],"detail_key":"p14"},{"id":"n18963","layer":"informal","project":"p14","title":"Half-difference over the last coordinate","kind":"definition","summary":"[Half-difference over the last coordinate] The half-difference diffLast\\,f of f over its last c…","labels":["Bonami.diffLast"],"detail_key":"p14"},{"id":"n18964","layer":"informal","project":"p14","title":"Restriction at false","kind":"lemma","summary":"[Restriction at false] For every x, the restriction of f fixing the last coordinate to false eq…","labels":["Bonami.restrictLast_false_eq"],"detail_key":"p14"},{"id":"n18965","layer":"informal","project":"p14","title":"Restriction at true","kind":"lemma","summary":"[Restriction at true] For every x, the restriction of f fixing the last coordinate to true equa…","labels":["Bonami.restrictLast_true_eq"],"detail_key":"p14"},{"id":"n18966","layer":"informal","project":"p14","title":"Sum over the hypercube splits on the last bit","kind":"lemma","summary":"[Sum over the hypercube splits on the last bit] For any \\varphi on the cube of dimension n+1, t…","labels":["Bonami.sum_boolCube_succ"],"detail_key":"p14"},{"id":"n18967","layer":"informal","project":"p14","title":"Uniform weight halves under one extra variable","kind":"lemma","summary":"[Uniform weight halves under one extra variable] The uniform point weight satisfies uniformWeig…","labels":["Bonami.uniformWeight_succ"],"detail_key":"p14"},{"id":"n18968","layer":"informal","project":"p14","title":"Fourier coefficient of the average","kind":"lemma","summary":"[Fourier coefficient of the average] For each S \\subseteq Fin\\,n, the Fourier coefficient of av…","labels":["Bonami.fourierCoeff_avgLast"],"detail_key":"p14"},{"id":"n18969","layer":"informal","project":"p14","title":"Fourier coefficient of the half-difference","kind":"lemma","summary":"[Fourier coefficient of the half-difference] For each S \\subseteq Fin\\,n, the Fourier coefficie…","labels":["Bonami.fourierCoeff_diffLast"],"detail_key":"p14"},{"id":"n18970","layer":"informal","project":"p14","title":"Expectation splits over the last coordinate","kind":"lemma","summary":"[Expectation splits over the last coordinate] The expectation of \\varphi on the cube of dimensi…","labels":["Bonami.expect_succ_eq"],"detail_key":"p14"},{"id":"n18971","layer":"informal","project":"p14","title":"Fourth-power algebraic identity","kind":"lemma","summary":"[Fourth-power algebraic identity] For all real a, b, (a+b)^4 + (a-b)^4 = 2\\,(a^4 + 6\\,a^2 b^2 +…","labels":["Bonami.fourth_pow_sum"],"detail_key":"p14"},{"id":"n18972","layer":"informal","project":"p14","title":"Second-power algebraic identity","kind":"lemma","summary":"[Second-power algebraic identity] For all real a, b, (a+b)^2 + (a-b)^2 = 2\\,(a^2 + b^2).","labels":["Bonami.second_pow_sum"],"detail_key":"p14"},{"id":"n18973","layer":"informal","project":"p14","title":"Fourth-moment decomposition","kind":"lemma","summary":"[Fourth-moment decomposition] Writing g = avgLast\\,f and h = diffLast\\,f, the fourth moment of…","labels":["Bonami.fourth_moment_decomp"],"detail_key":"p14"},{"id":"n18974","layer":"informal","project":"p14","title":"Second-moment decomposition","kind":"lemma","summary":"[Second-moment decomposition] Writing g = avgLast\\,f and h = diffLast\\,f, the second moment of…","labels":["Bonami.second_moment_decomp"],"detail_key":"p14"},{"id":"n18975","layer":"informal","project":"p14","title":"Cauchy--Schwarz for the expectation","kind":"lemma","summary":"[Cauchy--Schwarz for the expectation] For Boolean functions g, h on the cube, \\bigl(E[g^2 h^2]\\…","labels":["Bonami.expect_cs_sq"],"detail_key":"p14"},{"id":"n18976","layer":"informal","project":"p14","title":"Nonnegativity of E[f^2]","kind":"lemma","summary":"[Nonnegativity of E[f^2]] For every Boolean function f, 0 \\le E[f^2].","labels":["Bonami.expect_sq_nonneg"],"detail_key":"p14"},{"id":"n18977","layer":"informal","project":"p14","title":"Nonnegativity of E[g^2 h^2]","kind":"lemma","summary":"[Nonnegativity of E[g^2 h^2]] For Boolean functions g, h, 0 \\le E[g^2 h^2].","labels":["Bonami.expect_sq_nonneg_prod"],"detail_key":"p14"},{"id":"n18978","layer":"informal","project":"p14","title":"Nonnegativity of E[f^4]","kind":"lemma","summary":"[Nonnegativity of E[f^4]] For every Boolean function f, 0 \\le E[f^4].","labels":["Bonami.expect_fourth_nonneg"],"detail_key":"p14"},{"id":"n18979","layer":"informal","project":"p14","title":"Degree bound for the average","kind":"lemma","summary":"[Degree bound for the average] If f has degree at most k, then its average over the last coordi…","labels":["Bonami.degree_avgLast"],"detail_key":"p14"},{"id":"n18980","layer":"informal","project":"p14","title":"Degree bound for the half-difference","kind":"lemma","summary":"[Degree bound for the half-difference] If f has degree at most k, then its half-difference over…","labels":["Bonami.degree_diffLast"],"detail_key":"p14"},{"id":"n18981","layer":"informal","project":"p14","title":"Degree-zero functions are constant","kind":"lemma","summary":"[Degree-zero functions are constant] A Boolean function f of degree at most 0 is constant: f(x)…","labels":["Bonami.degree_zero_const"],"detail_key":"p14"},{"id":"n18982","layer":"informal","project":"p14","title":"Fourth moment of a constant function","kind":"lemma","summary":"[Fourth moment of a constant function] For a Boolean function f of degree at most 0 (hence cons…","labels":["Bonami.degree_zero_fourth_moment"],"detail_key":"p14"},{"id":"n18983","layer":"informal","project":"p14","title":"Algebraic inductive step for Bonami","kind":"lemma","summary":"[Algebraic inductive step for Bonami] The key algebraic inequality driving the inductive step o…","labels":["Bonami.bonami_algebra"],"detail_key":"p14"},{"id":"n18984","layer":"informal","project":"p14","title":"Bonami inequality in expectation form","kind":"lemma","summary":"[Bonami inequality in expectation form] The main Bonami inequality stated through the expectati…","labels":["Bonami.bonami_expect"],"detail_key":"p14"},{"id":"n18985","layer":"informal","project":"p14","title":"Moments equal expectations under the uniform law","kind":"lemma","summary":"[Moments equal expectations under the uniform law] If P is a probability measure on the cube as…","labels":["Bonami.moment_eq_expect"],"detail_key":"p14"},{"id":"n18986","layer":"informal","project":"p14","title":"Uniform measure on the hypercube","kind":"definition","summary":"[Uniform measure on the hypercube] The canonical uniform probability measure on the Boolean hyp…","labels":["Bonami.uniformMeasure"],"detail_key":"p14"},{"id":"n18987","layer":"informal","project":"p14","title":"Uniform measure of a point","kind":"lemma","summary":"[Uniform measure of a point] The uniform measure assigns each point x of the cube the combinato…","labels":["Bonami.uniformMeasure_apply"],"detail_key":"p14"},{"id":"n18988","layer":"informal","project":"p14","title":"Bonami lemma","kind":"lemma","summary":"[Bonami lemma] A Boolean function f of degree at most k is 9^k-reasonable under the uniform mea…","labels":["Bonami.bonami_lemma"],"detail_key":"p14"},{"id":"n18989","layer":"informal","project":"p14","title":"Enumeration of the one-bit cube","kind":"lemma","summary":"[Enumeration of the one-bit cube] The universal finite set of points of BoolCube\\,1 equals the…","labels":["OneBit.boolCube1_univ"],"detail_key":"p14"},{"id":"n18990","layer":"informal","project":"p14","title":"Enumeration of subsets of Fin\\,1","kind":"lemma","summary":"[Enumeration of subsets of Fin\\,1] The universal finite set of subsets of Fin\\,1 equals \\\\empty…","labels":["OneBit.finsetFin1_univ"],"detail_key":"p14"},{"id":"n18991","layer":"informal","project":"p14","title":"The two one-bit inputs differ","kind":"lemma","summary":"[The two one-bit inputs differ] The constant-\\mathitfalse and constant-\\mathittrue inputs on Fi…","labels":["OneBit.boolCube1_ne"],"detail_key":"p14"},{"id":"n18992","layer":"informal","project":"p14","title":"Empty set differs from \\0\\ in Fin\\,1","kind":"lemma","summary":"[Empty set differs from \\0\\ in Fin\\,1] The empty subset of Fin\\,1 is not equal to \\0\\.","labels":["OneBit.finsetFin1_ne"],"detail_key":"p14"},{"id":"n18993","layer":"informal","project":"p14","title":"One-bit value at \\mathitfalse","kind":"lemma","summary":"[One-bit value at \\mathitfalse] For a one-bit function f, its value on the all-\\mathitfalse inp…","labels":["OneBit.one_bit_val_false"],"detail_key":"p14"},{"id":"n18994","layer":"informal","project":"p14","title":"One-bit value at \\mathittrue","kind":"lemma","summary":"[One-bit value at \\mathittrue] For a one-bit function f, its value on the all-\\mathittrue input…","labels":["OneBit.one_bit_val_true"],"detail_key":"p14"},{"id":"n18995","layer":"informal","project":"p14","title":"Expected square of the noised one-bit function","kind":"lemma","summary":"[Expected square of the noised one-bit function] For a one-bit function f and noise rate \\rho,…","labels":["OneBit.expect_noiseOp_sq_one_bit"],"detail_key":"p14"},{"id":"n18996","layer":"informal","project":"p14","title":"Expected p-th power of |f| on one bit","kind":"lemma","summary":"[Expected p-th power of |f| on one bit] For a one-bit function f, writing a = \\widehatf(\\emptys…","labels":["OneBit.expect_abs_rpow_one_bit"],"detail_key":"p14"},{"id":"n18997","layer":"informal","project":"p14","title":"L^p norm monotonicity (power-mean inequality)","kind":"lemma","summary":"[L^p norm monotonicity (power-mean inequality)] For 1 \\le r \\le s and any f : BoolCube\\,n \\to R…","labels":["OneBit.lp_norm_mono"],"detail_key":"p14"},{"id":"n18998","layer":"informal","project":"p14","title":"Two-point inequality on the unit interval","kind":"theorem","summary":"[Two-point inequality on the unit interval] For 1 \\le p \\le 2 and 0 \\le b \\le 1, \\[ \\big(1 + (p…","labels":["OneBit.two_point_ineq_unit"],"detail_key":"p14"},{"id":"n18999","layer":"informal","project":"p14","title":"Two-point inequality: the a=0 case","kind":"lemma","summary":"[Two-point inequality: the a=0 case] For 1 \\le p \\le 2 one has (p-1)^p/2 \\le 1, which yields th…","labels":["OneBit.two_point_ineq_a_zero"],"detail_key":"p14"},{"id":"n19000","layer":"informal","project":"p14","title":"Noise operator does not increase the absolute-value L^2 contribution","kind":"lemma","summary":"[Noise operator does not increase the absolute-value L^2 contribution] For real a, b and 0 \\le…","labels":["OneBit.noise_l2_abs_mono"],"detail_key":"p14"},{"id":"n19001","layer":"informal","project":"p14","title":"Two-point inequality (full version)","kind":"theorem","summary":"[Two-point inequality (full version)] For 1 \\le p \\le 2, all a, b \\in R, and \\rho \\ge 0 with \\r…","labels":["OneBit.two_point_ineq"],"detail_key":"p14"},{"id":"n19002","layer":"informal","project":"p14","title":"One-bit (p,2)-hypercontractivity","kind":"theorem","summary":"[One-bit (p,2)-hypercontractivity] For f : BoolCube\\,1 \\to R, 1 \\le p \\le 2, and \\rho \\ge 0 wit…","labels":["OneBit.one_bit_p2_hypercontractivity"],"detail_key":"p14"},{"id":"n19003","layer":"informal","project":"p14","title":"Sign times value equals absolute value","kind":"lemma","summary":"[Sign times value equals absolute value] For all real x, sign(x)\\cdot x = |x|.","labels":["OneBit.sign_mul_self"],"detail_key":"p14"},{"id":"n19004","layer":"informal","project":"p14","title":"Absolute value of sign is one for nonzero input","kind":"lemma","summary":"[Absolute value of sign is one for nonzero input] For x \\neq 0, |sign(x)| = 1.","labels":["OneBit.abs_sign_eq_one"],"detail_key":"p14"},{"id":"n19005","layer":"informal","project":"p14","title":"Expectation of a nonnegative function is nonnegative","kind":"lemma","summary":"[Expectation of a nonnegative function is nonnegative] If f \\ge 0 pointwise, then E[f] \\ge 0.","labels":["OneBit.expect_nonneg_of_nonneg"],"detail_key":"p14"},{"id":"n19006","layer":"informal","project":"p14","title":"Expectation of a constant function","kind":"lemma","summary":"[Expectation of a constant function] The expectation over the cube of the constant function wit…","labels":["OneBit.expect_const_eq"],"detail_key":"p14"},{"id":"n19007","layer":"informal","project":"p14","title":"Cauchy--Schwarz for the Boolean inner product","kind":"lemma","summary":"[Cauchy--Schwarz for the Boolean inner product] For f, g : BoolCube\\,n \\to R, \\[ \\langle f, g\\r…","labels":["OneBit.cauchy_schwarz_bool"],"detail_key":"p14"},{"id":"n19008","layer":"informal","project":"p14","title":"H\\\"older sharpness","kind":"lemma","summary":"[H\\\"older sharpness] For H\\\"older conjugate exponents (p, q) and any function u, there exists a…","labels":["OneBit.holder_sharpness"],"detail_key":"p14"},{"id":"n19009","layer":"informal","project":"p14","title":"Noise operator duality","kind":"theorem","summary":"[Noise operator duality] If, for H\\\"older conjugate exponents p and p' with 1 \\le p, the noise…","labels":["OneBit.noise_operator_duality"],"detail_key":"p14"},{"id":"n19010","layer":"informal","project":"p14","title":"One-bit (2,q)-hypercontractivity","kind":"theorem","summary":"[One-bit (2,q)-hypercontractivity] For g : BoolCube\\,1 \\to R and q \\ge 2, \\[ \\lVert T_1/\\sqrtq-…","labels":["OneBit.one_bit_2q_hypercontractivity"],"detail_key":"p14"},{"id":"n19011","layer":"informal","project":"p14","title":"Character of a lifted set ignores the last coordinate","kind":"lemma","summary":"[Character of a lifted set ignores the last coordinate] For a subset S \\subseteq Fin\\,n, a poin…","labels":["SimpleHypercontractivity.chiS_snoc_castSucc"],"detail_key":"p14"},{"id":"n19012","layer":"informal","project":"p14","title":"Character of a lifted set with the last coordinate","kind":"lemma","summary":"[Character of a lifted set with the last coordinate] For S \\subseteq Fin\\,n, a point x, and a b…","labels":["SimpleHypercontractivity.chiS_snoc_with_last"],"detail_key":"p14"},{"id":"n19013","layer":"informal","project":"p14","title":"Partition of a sum over subsets of Fin(n+1)","kind":"lemma","summary":"[Partition of a sum over subsets of Fin(n+1)] For any \\varphi on subsets of Fin(n+1), the total…","labels":["SimpleHypercontractivity.finset_fin_succ_sum_partition"],"detail_key":"p14"},{"id":"n19014","layer":"informal","project":"p14","title":"Cardinality of a lifted set","kind":"lemma","summary":"[Cardinality of a lifted set] Lifting a set along castSucc preserves its cardinality: |castSucc…","labels":["SimpleHypercontractivity.card_image_castSucc"],"detail_key":"p14"},{"id":"n19015","layer":"informal","project":"p14","title":"Cardinality of a lifted set with the last coordinate","kind":"lemma","summary":"[Cardinality of a lifted set with the last coordinate] Adjoining the (necessarily new) last coo…","labels":["SimpleHypercontractivity.card_image_castSucc_union_last"],"detail_key":"p14"},{"id":"n19016","layer":"informal","project":"p14","title":"Noise operator decomposes along the last coordinate","kind":"lemma","summary":"[Noise operator decomposes along the last coordinate] For f : \\0,1\\^n+1 \\to R the noise operato…","labels":["SimpleHypercontractivity.noiseOp_snoc"],"detail_key":"p14"},{"id":"n19017","layer":"informal","project":"p14","title":"Fourth moment decomposition with the noise operator","kind":"lemma","summary":"[Fourth moment decomposition with the noise operator] For f : \\0,1\\^n+1 \\to R, writing g = avgL…","labels":["SimpleHypercontractivity.fourth_moment_noise_decomp"],"detail_key":"p14"},{"id":"n19018","layer":"informal","project":"p14","title":"(2,4)-Hypercontractivity (Bonami--Beckner)","kind":"theorem","summary":"[(2,4)-Hypercontractivity (Bonami--Beckner)] For any Boolean function f : \\0,1\\^n \\to R and noi…","labels":["SimpleHypercontractivity.hypercontractivity_2_4"],"detail_key":"p14"},{"id":"n19019","layer":"informal","project":"p14","title":"Hölder's inequality for p = 4/3, q = 4","kind":"lemma","summary":"[Hölder's inequality for p = 4/3, q = 4] For Boolean functions f, g, \\[ \\langle f, g \\rangle \\l…","labels":["SimpleHypercontractivity.innerProduct_le_L43_L4"],"detail_key":"p14"},{"id":"n19020","layer":"informal","project":"p14","title":"(4/3,2)-Hypercontractivity with \\rho = 1/\\sqrt3","kind":"theorem","summary":"[(4/3,2)-Hypercontractivity with \\rho = 1/\\sqrt3] Boolean functions are (4/3, 2)-hypercontracti…","labels":["SimpleHypercontractivity.hypercontractivity_4_div_3_2"],"detail_key":"p14"},{"id":"n19021","layer":"informal","project":"p14","title":"L^2 norm of T_\\rho f in Fourier space","kind":"lemma","summary":"[L^2 norm of T_\\rho f in Fourier space] The squared L^2 norm of T_\\rho f expands over the Fouri…","labels":["SimpleHypercontractivity.noise_l2_fourier"],"detail_key":"p14"},{"id":"n19022","layer":"informal","project":"p14","title":"Contractivity (q = 2 case)","kind":"theorem","summary":"[Contractivity (q = 2 case)] For \\rho^2 \\le 1 and any Boolean function f, \\[ E[(T_\\rho f)^2] \\l…","labels":["SimpleHypercontractivity.contractivity"],"detail_key":"p14"},{"id":"n19023","layer":"informal","project":"p14","title":"Binomial coefficient bound","kind":"lemma","summary":"[Binomial coefficient bound] For natural numbers k \\ge 1 and j \\le k, \\[ \\binom2k2j \\le \\binomk…","labels":["SimpleHypercontractivity.binom_coeff_ineq"],"detail_key":"p14"},{"id":"n19024","layer":"informal","project":"p14","title":"q-th moment decomposition along the last coordinate","kind":"lemma","summary":"[q-th moment decomposition along the last coordinate] For f : \\0,1\\^n+1 \\to R, writing g = avgL…","labels":["SimpleHypercontractivity.qth_moment_decomp"],"detail_key":"p14"},{"id":"n19025","layer":"informal","project":"p14","title":"q-th moment decomposition with the noise operator","kind":"lemma","summary":"[q-th moment decomposition with the noise operator] For f : \\0,1\\^n+1 \\to R, writing g = T_\\rho…","labels":["SimpleHypercontractivity.noise_qth_moment_decomp"],"detail_key":"p14"},{"id":"n19026","layer":"informal","project":"p14","title":"(2,2k)-Hypercontractivity (Bonami--Beckner)","kind":"theorem","summary":"[(2,2k)-Hypercontractivity (Bonami--Beckner)] For any Boolean function f : \\0,1\\^n \\to R, integ…","labels":["SimpleHypercontractivity.hypercontractivity_2_2k"],"detail_key":"p14"},{"id":"n19027","layer":"informal","project":"p14","title":"(2,q)-Hypercontractivity for even q","kind":"theorem","summary":"[(2,q)-Hypercontractivity for even q] For an even integer q \\ge 2 and noise parameter \\rho with…","labels":["SimpleHypercontractivity.hypercontractivity_2_q"],"detail_key":"p14"},{"id":"n19028","layer":"informal","project":"p14","title":"(2,2)-Hypercontractivity","kind":"theorem","summary":"[(2,2)-Hypercontractivity] For \\rho^2 \\le 1 and any Boolean function f, \\[ E[(T_\\rho f)^2] \\le…","labels":["SimpleHypercontractivity.hypercontractivity_2_2"],"detail_key":"p14"},{"id":"n19029","layer":"informal","project":"p14","title":"(2,6)-Hypercontractivity","kind":"theorem","summary":"[(2,6)-Hypercontractivity] For \\rho^2 \\le 1/5 and any Boolean function f, \\[ E[(T_\\rho f)^6] \\l…","labels":["SimpleHypercontractivity.hypercontractivity_2_6"],"detail_key":"p14"},{"id":"n19030","layer":"informal","project":"p14","title":"(2,2k)-Hypercontractivity in norm form","kind":"theorem","summary":"[(2,2k)-Hypercontractivity in norm form] For k \\ge 1 and noise parameter \\rho with \\rho^2 \\le 1…","labels":["SimpleHypercontractivity.hypercontractivity_2_2k_rpow"],"detail_key":"p14"},{"id":"n19031","layer":"informal","project":"p14","title":"Inner product equals expected square","kind":"lemma","summary":"[Inner product equals expected square] The self inner product of a Boolean function equals the…","labels":["SimpleHypercontractivity.innerProduct_eq_expect_sq"],"detail_key":"p14"},{"id":"n19032","layer":"informal","project":"p14","title":"Nonnegativity of the expected square of T_\\rho f","kind":"lemma","summary":"[Nonnegativity of the expected square of T_\\rho f] For any \\rho and Boolean function f, E[(T_\\r…","labels":["SimpleHypercontractivity.expect_sq_noiseOp_nonneg"],"detail_key":"p14"},{"id":"n19033","layer":"informal","project":"p14","title":"Nonnegativity of an expected absolute power","kind":"lemma","summary":"[Nonnegativity of an expected absolute power] For any real exponent p and Boolean function f, E…","labels":["SimpleHypercontractivity.expect_rpow_abs_nonneg"],"detail_key":"p14"},{"id":"n19034","layer":"informal","project":"p14","title":"Composition of noise operators","kind":"lemma","summary":"[Composition of noise operators] Composing two noise operators multiplies their parameters: T_\\…","labels":["SimpleHypercontractivity.noiseOp_compose"],"detail_key":"p14"},{"id":"n19035","layer":"informal","project":"p14","title":"(p,2)-Hypercontractivity via duality","kind":"theorem","summary":"[(p,2)-Hypercontractivity via duality] Let 1 < p, q \\ge 2 with 1/p + 1/q = 1. Given a Hölder in…","labels":["SimpleHypercontractivity.hypercontractivity_p_2_general"],"detail_key":"p14"},{"id":"n19036","layer":"informal","project":"p14","title":"Key algebraic inequality closing the recurrence","kind":"lemma","summary":"[Key algebraic inequality closing the recurrence] Let a, b, A, B, C, \\rho be reals with a \\ge 0…","labels":["SimpleHypercontractivity.hypercontractivity_algebra'"],"detail_key":"p14"},{"id":"n19037","layer":"informal","project":"p14","title":"Lower bound on logb_2(2s/\\varepsilon)","kind":"lemma","summary":"[Lower bound on logb_2(2s/\\varepsilon)] For s \\in N with 0 < s and \\varepsilon \\in R with 0 < \\…","labels":["LMN.logb_2s_div_eps_pos"],"detail_key":"p14"},{"id":"n19038","layer":"informal","project":"p14","title":"Monotonicity of logb_2(2s/\\varepsilon) in s","kind":"lemma","summary":"[Monotonicity of logb_2(2s/\\varepsilon) in s] For s \\in N with 0 < s and \\varepsilon \\in R with…","labels":["LMN.logb_2_div_eps_le_l"],"detail_key":"p14"},{"id":"n19039","layer":"informal","project":"p14","title":"Bound s \\cdot 2^-l \\le \\varepsilon/2","kind":"lemma","summary":"[Bound s \\cdot 2^-l \\le \\varepsilon/2] For s \\in N with 0 < s and \\varepsilon \\in R with 0 < \\v…","labels":["LMN.size_times_two_pow_neg_l_le"],"detail_key":"p14"},{"id":"n19040","layer":"informal","project":"p14","title":"Value of 2^-logb_2(2/\\varepsilon)","kind":"lemma","summary":"[Value of 2^-logb_2(2/\\varepsilon)] For \\varepsilon \\in R with 0 < \\varepsilon, \\[ 2^-\\log_2(2/…","labels":["LMN.two_pow_neg_logb_2_div_eps"],"detail_key":"p14"},{"id":"n19041","layer":"informal","project":"p14","title":"Iterative circuit reduction bound with Chernoff tails","kind":"lemma","summary":"[Iterative circuit reduction bound with Chernoff tails] Let c be a circuit on n variables with…","labels":["LMN.iterative_reduction_bound"],"detail_key":"p14"},{"id":"n19042","layer":"informal","project":"p14","title":"O'Donnell Lemma 4.28, with tails","kind":"theorem","summary":"[O'Donnell Lemma 4.28, with tails] Let c be a circuit on n variables with n > 0, depth at most…","labels":["LMN.odonnell_lemma_4_28"],"detail_key":"p14"},{"id":"n19043","layer":"informal","project":"p14","title":"Restrictions with a fixed number of free variables","kind":"definition","summary":"[Restrictions with a fixed number of free variables] For natural numbers n and k, this is the f…","labels":["BernoulliCost.fixedSizeRestrs"],"detail_key":"p14"},{"id":"n19044","layer":"informal","project":"p14","title":"Fixed-size restriction probability R_k","kind":"definition","summary":"[Fixed-size restriction probability R_k] The probability of an event under the fixed-size restr…","labels":["BernoulliCost.fixedSizeRestrProb"],"detail_key":"p14"},{"id":"n19045","layer":"informal","project":"p14","title":"Binomial probability mass function","kind":"definition","summary":"[Binomial probability mass function] The binomial probability mass function, \\Pr[Bin(n,p) = k]…","labels":["BernoulliCost.binomialPMF"],"detail_key":"p14"},{"id":"n19046","layer":"informal","project":"p14","title":"Fixed-size restriction probability is nonnegative","kind":"lemma","summary":"[Fixed-size restriction probability is nonnegative] For any event, the fixed-size restriction p…","labels":["BernoulliCost.fixedSizeRestrProb_nonneg"],"detail_key":"p14"},{"id":"n19047","layer":"informal","project":"p14","title":"Fixed-size restriction probability is at most one","kind":"lemma","summary":"[Fixed-size restriction probability is at most one] For any event, the fixed-size restriction p…","labels":["BernoulliCost.fixedSizeRestrProb_le_one"],"detail_key":"p14"},{"id":"n19048","layer":"informal","project":"p14","title":"Binomial PMF is nonnegative","kind":"lemma","summary":"[Binomial PMF is nonnegative] If 0 \\le p \\le 1, then 0 \\le binomialPMF\\,n\\,p\\,k for every k.","labels":["BernoulliCost.binomialPMF_nonneg"],"detail_key":"p14"},{"id":"n19049","layer":"informal","project":"p14","title":"Binomial PMF sums to one","kind":"lemma","summary":"[Binomial PMF sums to one] If 0 \\le p \\le 1, then the binomial PMF sums to 1 over k ranging in…","labels":["BernoulliCost.binomialPMF_sum_one"],"detail_key":"p14"},{"id":"n19050","layer":"informal","project":"p14","title":"Decomposition of Bernoulli probability by free-variable count","kind":"lemma","summary":"[Decomposition of Bernoulli probability by free-variable count] For 0 \\le p \\le 1 and any event…","labels":["BernoulliCost.bernoulli_decompose"],"detail_key":"p14"},{"id":"n19051","layer":"informal","project":"p14","title":"Chernoff upper-tail bound for the binomial distribution","kind":"lemma","summary":"[Chernoff upper-tail bound for the binomial distribution] For 0 < p \\le 1, the upper tail of th…","labels":["BernoulliCost.chernoff_binomial_upper_tail"],"detail_key":"p14"},{"id":"n19052","layer":"informal","project":"p14","title":"Bernoulli restriction cost, exact version","kind":"theorem","summary":"[Bernoulli restriction cost, exact version] Let n > 0, 0 < p \\le 1, and w, s > 0. If the event…","labels":["BernoulliCost.bernoulli_restriction_cost"],"detail_key":"p14"},{"id":"n19053","layer":"informal","project":"p14","title":"Eventual smallness of the exponential tail","kind":"lemma","summary":"[Eventual smallness of the exponential tail] For any p > 0 and any \\varepsilon > 0, there exist…","labels":["BernoulliCost.exp_neg_eventually_small"],"detail_key":"p14"},{"id":"n19054","layer":"informal","project":"p14","title":"Bernoulli restriction cost, asymptotic version","kind":"theorem","summary":"[Bernoulli restriction cost, asymptotic version] Let 0 < p \\le 1, w, s > 0, and \\varepsilon > 0…","labels":["BernoulliCost.bernoulli_restriction_asymptotic"],"detail_key":"p14"},{"id":"n19055","layer":"informal","project":"p14","title":"Concatenation of a list of lists","kind":"definition","summary":"[Concatenation of a list of lists] Flattens a list of lists [\\,\\ell_1, \\ell_2, \\dots\\,] into th…","labels":["LMN.listConcat"],"detail_key":"p14"},{"id":"n19056","layer":"informal","project":"p14","title":"Bounded width iff all terms bounded","kind":"lemma","summary":"[Bounded width iff all terms bounded] For a list of terms ts, the folded maximum of their width…","labels":["LMN.width_le_iff_forall"],"detail_key":"p14"},{"id":"n19057","layer":"informal","project":"p14","title":"CNF concatenation preserves width","kind":"lemma","summary":"[CNF concatenation preserves width] If CNFs \\psi_1, \\dots, \\psi_s each have width at most \\ell,…","labels":["LMN.cnf_concat_width_le"],"detail_key":"p14"},{"id":"n19058","layer":"informal","project":"p14","title":"CNF concatenation evaluates as conjunction","kind":"lemma","summary":"[CNF concatenation evaluates as conjunction] For any input x, the value of the concatenated CNF…","labels":["LMN.cnf_concat_eval"],"detail_key":"p14"},{"id":"n19059","layer":"informal","project":"p14","title":"DNF concatenation preserves width","kind":"lemma","summary":"[DNF concatenation preserves width] If DNFs \\varphi_1, \\dots, \\varphi_s each have width at most…","labels":["LMN.dnf_concat_width_le"],"detail_key":"p14"},{"id":"n19060","layer":"informal","project":"p14","title":"DNF concatenation evaluates as disjunction","kind":"lemma","summary":"[DNF concatenation evaluates as disjunction] For any input x, the value of the concatenated DNF…","labels":["LMN.dnf_concat_eval"],"detail_key":"p14"},{"id":"n19061","layer":"informal","project":"p14","title":"Circuit compression for CNFs under AND","kind":"theorem","summary":"[Circuit compression for CNFs under AND] Let an AND gate have a list of children, each of which…","labels":["LMN.compression_and_of_cnfs"],"detail_key":"p14"},{"id":"n19062","layer":"informal","project":"p14","title":"Circuit compression for DNFs under OR","kind":"theorem","summary":"[Circuit compression for DNFs under OR] Dual to the AND case: if an OR gate has children each e…","labels":["LMN.compression_or_of_dnfs"],"detail_key":"p14"},{"id":"n19063","layer":"informal","project":"p14","title":"One-step reduction failure bound","kind":"theorem","summary":"[One-step reduction failure bound] For s_2 layer-2 DNF gates of width at most w (with the usual…","labels":["LMN.one_step_reduction_failure_bound"],"detail_key":"p14"},{"id":"n19064","layer":"informal","project":"p14","title":"One-step decision-tree depth bound","kind":"theorem","summary":"[One-step decision-tree depth bound] Under a Bernoulli restriction with p \\le 1/(40w) on s_2 wi…","labels":["LMN.one_step_dtDepth_bound"],"detail_key":"p14"},{"id":"n19065","layer":"informal","project":"p14","title":"Complement probability sums to one","kind":"lemma","summary":"[Complement probability sums to one] For a Bernoulli restriction probability with parameter 0 \\…","labels":["LMN.bernoulliRestrProb_complement"],"detail_key":"p14"},{"id":"n19066","layer":"informal","project":"p14","title":"One-step reduction with compression","kind":"theorem","summary":"[One-step reduction with compression] Combining Steps 6 and 7: after a Bernoulli restriction wi…","labels":["LMN.one_step_reduction_with_compression"],"detail_key":"p14"},{"id":"n19067","layer":"informal","project":"p14","title":"Restriction respects pointwise equality","kind":"lemma","summary":"[Restriction respects pointwise equality] If f,g:(Fin\\,n\\to\\0,1\\)\\to\\0,1\\ agree pointwise, i.e.…","labels":["LMN.restrictFn_ext'"],"detail_key":"p14"},{"id":"n19068","layer":"informal","project":"p14","title":"Bernoulli restriction probability depends only on the function","kind":"lemma","summary":"[Bernoulli restriction probability depends only on the function] If f and g agree pointwise, th…","labels":["LMN.bernoulliRestrProb_congr_fn"],"detail_key":"p14"},{"id":"n19069","layer":"informal","project":"p14","title":"De-duplicate literals by variable","kind":"definition","summary":"[De-duplicate literals by variable] Given a term t (a list of literals), dedupTermVar\\,t keeps…","labels":["LMN.dedupTermVar"],"detail_key":"p14"},{"id":"n19070","layer":"informal","project":"p14","title":"Contradictory term test","kind":"definition","summary":"[Contradictory term test] termHasContradiction\\,t returns \\texttttrue iff the term t contains t…","labels":["LMN.termHasContradiction"],"detail_key":"p14"},{"id":"n19071","layer":"informal","project":"p14","title":"Clean a DNF","kind":"definition","summary":"[Clean a DNF] cleanDNF\\,d first discards every contradictory term of the DNF d and then applies…","labels":["LMN.cleanDNF"],"detail_key":"p14"},{"id":"n19072","layer":"informal","project":"p14","title":"Clean a CNF","kind":"definition","summary":"[Clean a CNF] cleanCNF\\,c filters out the contradictory (tautological) clauses of the CNF c and…","labels":["LMN.cleanCNF"],"detail_key":"p14"},{"id":"n19073","layer":"informal","project":"p14","title":"De-duplication yields no duplicate literals","kind":"lemma","summary":"[De-duplication yields no duplicate literals] For every term t, the de-duplicated term dedupTer…","labels":["LMN.dedupTermVar_nodup"],"detail_key":"p14"},{"id":"n19074","layer":"informal","project":"p14","title":"De-duplication is variable-injective","kind":"lemma","summary":"[De-duplication is variable-injective] Within dedupTermVar\\,t the variable of a literal determi…","labels":["LMN.dedupTermVar_var_inj"],"detail_key":"p14"},{"id":"n19075","layer":"informal","project":"p14","title":"De-duplication does not increase width","kind":"lemma","summary":"[De-duplication does not increase width] The de-duplicated term is no longer than the original:…","labels":["LMN.dedupTermVar_width_le"],"detail_key":"p14"},{"id":"n19076","layer":"informal","project":"p14","title":"Contradictory terms evaluate to false","kind":"lemma","summary":"[Contradictory terms evaluate to false] If termHasContradiction\\,t = \\texttttrue, then for ever…","labels":["LMN.contradiction_term_eval_false"],"detail_key":"p14"},{"id":"n19077","layer":"informal","project":"p14","title":"De-duplication preserves term evaluation","kind":"lemma","summary":"[De-duplication preserves term evaluation] For a non-contradictory term t (i.e.\\ termHasContrad…","labels":["LMN.dedupTermVar_preserves_term_eval"],"detail_key":"p14"},{"id":"n19078","layer":"informal","project":"p14","title":"cleanDNF preserves evaluation","kind":"lemma","summary":"[cleanDNF preserves evaluation] For every DNF d and input x, the cleaned formula computes the s…","labels":["LMN.cleanDNF_eval"],"detail_key":"p14"},{"id":"n19079","layer":"informal","project":"p14","title":"Contradictory clauses evaluate to true","kind":"lemma","summary":"[Contradictory clauses evaluate to true] If termHasContradiction\\,t = \\texttttrue, then viewed…","labels":["LMN.contradiction_clause_eval_true"],"detail_key":"p14"},{"id":"n19080","layer":"informal","project":"p14","title":"De-duplication preserves clause evaluation","kind":"lemma","summary":"[De-duplication preserves clause evaluation] For a non-contradictory clause t, de-duplication p…","labels":["LMN.dedupTermVar_preserves_clause_eval"],"detail_key":"p14"},{"id":"n19081","layer":"informal","project":"p14","title":"cleanCNF preserves evaluation","kind":"lemma","summary":"[cleanCNF preserves evaluation] For every CNF c and input x, cleaning preserves the computed fu…","labels":["LMN.cleanCNF_eval"],"detail_key":"p14"},{"id":"n19082","layer":"informal","project":"p14","title":"cleanDNF does not increase width","kind":"lemma","summary":"[cleanDNF does not increase width] The cleaned DNF has width at most that of the original: (cle…","labels":["LMN.cleanDNF_width_le"],"detail_key":"p14"},{"id":"n19083","layer":"informal","project":"p14","title":"cleanCNF does not increase width","kind":"lemma","summary":"[cleanCNF does not increase width] The cleaned CNF has width at most that of the original: CNF.…","labels":["LMN.cleanCNF_width_le"],"detail_key":"p14"},{"id":"n19084","layer":"informal","project":"p14","title":"cleanDNF is variable-injective per term","kind":"lemma","summary":"[cleanDNF is variable-injective per term] In every term of cleanDNF\\,d, the variable determines…","labels":["LMN.cleanDNF_var_inj"],"detail_key":"p14"},{"id":"n19085","layer":"informal","project":"p14","title":"cleanDNF terms have no duplicate literals","kind":"lemma","summary":"[cleanDNF terms have no duplicate literals] Every term of cleanDNF\\,d satisfies \\textttNodup, i…","labels":["LMN.cleanDNF_nodup"],"detail_key":"p14"},{"id":"n19086","layer":"informal","project":"p14","title":"cleanCNF is variable-injective per clause","kind":"lemma","summary":"[cleanCNF is variable-injective per clause] In every clause of cleanCNF\\,c, the variable determ…","labels":["LMN.cleanCNF_var_inj"],"detail_key":"p14"},{"id":"n19087","layer":"informal","project":"p14","title":"cleanCNF clauses have no duplicate literals","kind":"lemma","summary":"[cleanCNF clauses have no duplicate literals] Every clause of cleanCNF\\,c satisfies \\textttNodu…","labels":["LMN.cleanCNF_nodup"],"detail_key":"p14"},{"id":"n19088","layer":"informal","project":"p14","title":"General switching lemma for DNFs","kind":"theorem","summary":"[General switching lemma for DNFs] Let f be a DNF of width at most w with 0<w and 0<n, and let…","labels":["LMN.switching_bernoulli_dtDepth_dnf_general"],"detail_key":"p14"},{"id":"n19089","layer":"informal","project":"p14","title":"General switching lemma for CNFs","kind":"theorem","summary":"[General switching lemma for CNFs] Let f be a CNF of width at most w with 0<w and 0<n, and let…","labels":["LMN.switching_bernoulli_dtDepth_cnf_general"],"detail_key":"p14"},{"id":"n19090","layer":"informal","project":"p14","title":"Children of a depth-\\le 1 node are literals","kind":"lemma","summary":"[Children of a depth-\\le 1 node are literals] If a gate node node\\,\\mathitisAnd\\,cs has depth a…","labels":["LMN.depth_le_one_children_are_lits"],"detail_key":"p14"},{"id":"n19091","layer":"informal","project":"p14","title":"Children of a depth-\\le 2 node have depth \\le 1","kind":"lemma","summary":"[Children of a depth-\\le 2 node have depth \\le 1] If a gate node node\\,\\mathitisAnd\\,cs has dep…","labels":["LMN.depth_le_two_children_depth_le_one"],"detail_key":"p14"},{"id":"n19092","layer":"informal","project":"p14","title":"Depth-\\le 1 AND subcircuit to a term","kind":"definition","summary":"[Depth-\\le 1 AND subcircuit to a term] Converts a depth-\\le 1 AND subcircuit to a term (a conju…","labels":["LMN.depth1AndToTerm"],"detail_key":"p14"},{"id":"n19093","layer":"informal","project":"p14","title":"Depth-\\le 2 OR-top circuit to a DNF","kind":"definition","summary":"[Depth-\\le 2 OR-top circuit to a DNF] Converts a list of children cs of a top OR gate (of depth…","labels":["LMN.depth2OrToDNF"],"detail_key":"p14"},{"id":"n19094","layer":"informal","project":"p14","title":"Depth-\\le 2 AND-top circuit to a CNF","kind":"definition","summary":"[Depth-\\le 2 AND-top circuit to a CNF] Converts a list of children cs of a top AND gate (of dep…","labels":["LMN.depth2AndToCNF"],"detail_key":"p14"},{"id":"n19095","layer":"informal","project":"p14","title":"depth2OrToDNF preserves evaluation","kind":"lemma","summary":"[depth2OrToDNF preserves evaluation] If the top OR node node\\,\\textttfalse\\,cs has depth at mos…","labels":["LMN.depth2OrToDNF_eval"],"detail_key":"p14"},{"id":"n19096","layer":"informal","project":"p14","title":"depth2AndToCNF preserves evaluation","kind":"lemma","summary":"[depth2AndToCNF preserves evaluation] If the top AND node node\\,\\texttttrue\\,cs has depth at mo…","labels":["LMN.depth2AndToCNF_eval"],"detail_key":"p14"},{"id":"n19097","layer":"informal","project":"p14","title":"Width bound for depth2OrToDNF","kind":"lemma","summary":"[Width bound for depth2OrToDNF] If the top OR node node\\,\\textttfalse\\,cs has depth at most 2,…","labels":["LMN.depth2OrToDNF_width_le"],"detail_key":"p14"},{"id":"n19098","layer":"informal","project":"p14","title":"Width bound for depth2AndToCNF","kind":"lemma","summary":"[Width bound for depth2AndToCNF] If the top AND node node\\,\\texttttrue\\,cs has depth at most 2,…","labels":["LMN.depth2AndToCNF_width_le"],"detail_key":"p14"},{"id":"n19099","layer":"informal","project":"p14","title":"Composed restriction parameter","kind":"definition","summary":"[Composed restriction parameter] For a fan-in bound w, a width parameter l, and a depth d, the…","labels":["LMN.composedDelta"],"detail_key":"p14"},{"id":"n19100","layer":"informal","project":"p14","title":"Positivity of the composed parameter","kind":"lemma","summary":"[Positivity of the composed parameter] For w>0 and l>0, the composed restriction parameter comp…","labels":["LMN.composedDelta_pos"],"detail_key":"p14"},{"id":"n19101","layer":"informal","project":"p14","title":"The composed parameter is at most one","kind":"lemma","summary":"[The composed parameter is at most one] For w\\ge 1, l\\ge 1, and d\\ge 2, the composed restrictio…","labels":["LMN.composedDelta_le_one"],"detail_key":"p14"},{"id":"n19102","layer":"informal","project":"p14","title":"Recursive step factorization (left)","kind":"lemma","summary":"[Recursive step factorization (left)] For natural numbers w,l and depth d\\ge 3, \\[ composedDelt…","labels":["LMN.composedDelta_step"],"detail_key":"p14"},{"id":"n19103","layer":"informal","project":"p14","title":"Recursive step factorization (right)","kind":"lemma","summary":"[Recursive step factorization (right)] For d\\ge 3 and l>0, the composed parameter factors as th…","labels":["LMN.composedDelta_step_right"],"detail_key":"p14"},{"id":"n19104","layer":"informal","project":"p14","title":"Layer-2 data","kind":"definition","summary":"[Layer-2 data] A bundle describing the second layer of a circuit over n variables: a number of…","labels":["LMN.Layer2Data"],"detail_key":"p14"},{"id":"n19105","layer":"informal","project":"p14","title":"One-step switching for layer-2 data","kind":"theorem","summary":"[One-step switching for layer-2 data] For layer-2 data and a Bernoulli parameter p with 0<p\\le…","labels":["LMN.normalform_one_step_switching"],"detail_key":"p14"},{"id":"n19106","layer":"informal","project":"p14","title":"One-step CNF replaceability for layer-2 data","kind":"theorem","summary":"[One-step CNF replaceability for layer-2 data] For layer-2 data and a Bernoulli parameter p wit…","labels":["LMN.normalform_one_step_cnf_replaceability"],"detail_key":"p14"},{"id":"n19107","layer":"informal","project":"p14","title":"Aggregating per-layer bounds","kind":"theorem","summary":"[Aggregating per-layer bounds] Given layer sizes summing to at most s, if each per-layer quanti…","labels":["LMN.full_iterative_bound"],"detail_key":"p14"},{"id":"n19108","layer":"informal","project":"p14","title":"Restriction commutes with restriction composition","kind":"lemma","summary":"[Restriction commutes with restriction composition] For a Boolean function f and restrictions \\…","labels":["LMN.restrictFn_composeRestr'"],"detail_key":"p14"},{"id":"n19109","layer":"informal","project":"p14","title":"Two-stage probability bound","kind":"theorem","summary":"[Two-stage probability bound] Let p_1,p_2\\in(0,1], let E and A be predicates on restrictions, a…","labels":["LMN.two_stage_bound'"],"detail_key":"p14"},{"id":"n19110","layer":"informal","project":"p14","title":"Monotonicity in the depth threshold","kind":"lemma","summary":"[Monotonicity in the depth threshold] For 0\\le p\\le 1, a Boolean function f, and thresholds l_1…","labels":["LMN.bernoulliRestrProb_dtDepth_mono"],"detail_key":"p14"},{"id":"n19111","layer":"informal","project":"p14","title":"Congruence under pointwise-equal functions","kind":"lemma","summary":"[Congruence under pointwise-equal functions] If f and g agree on every input, then the probabil…","labels":["LMN.bernoulliRestrProb_congr_fn'"],"detail_key":"p14"},{"id":"n19112","layer":"informal","project":"p14","title":"Switching bound for depth-2 circuits","kind":"lemma","summary":"[Switching bound for depth-2 circuits] If f is computed by a circuit of depth at most 2, size a…","labels":["LMN.depth2_circuit_switching_bound"],"detail_key":"p14"},{"id":"n19113","layer":"informal","project":"p14","title":"Inductive base case: depth-2 bound","kind":"lemma","summary":"[Inductive base case: depth-2 bound] The base case (d=2) of the inductive bound: if f is comput…","labels":["LMN.circuit_reduction_ind_base"],"detail_key":"p14"},{"id":"n19114","layer":"informal","project":"p14","title":"Union bound over a list of circuits","kind":"lemma","summary":"[Union bound over a list of circuits] For 0\\le p\\le 1, a list of circuits \\mathitcs, and a pred…","labels":["LMN.bernoulliRestrProb_list_union_bound"],"detail_key":"p14"},{"id":"n19115","layer":"informal","project":"p14","title":"Inductive step: depth d\\ge 3","kind":"lemma","summary":"[Inductive step: depth d\\ge 3] The inductive step: for depth d\\ge 3, assuming the bound holds f…","labels":["LMN.circuit_reduction_ind_step"],"detail_key":"p14"},{"id":"n19116","layer":"informal","project":"p14","title":"Iterative reduction bound (tight form)","kind":"theorem","summary":"[Iterative reduction bound (tight form)] By strong induction on the depth d\\ge 2: if f is compu…","labels":["LMN.circuit_reduction_ind"],"detail_key":"p14"},{"id":"n19117","layer":"informal","project":"p14","title":"Iterative reduction bound (relaxed form)","kind":"theorem","summary":"[Iterative reduction bound (relaxed form)] Replacing the coefficient s-1 by s in the previous b…","labels":["LMN.circuit_reduction_aux"],"detail_key":"p14"},{"id":"n19118","layer":"informal","project":"p14","title":"Circuit reduction core bound","kind":"theorem","summary":"[Circuit reduction core bound] The user-facing form of the iterative reduction: for f computed…","labels":["LMN.circuit_reduction_core"],"detail_key":"p14"},{"id":"n19119","layer":"informal","project":"p14","title":"Re-indexing a circuit","kind":"definition","summary":"[Re-indexing a circuit] Given a circuit c on m variables and a function f : Fin\\,m \\to Fin\\,m',…","labels":["BoolCircuit.Circuit.reidx"],"detail_key":"p14"},{"id":"n19120","layer":"informal","project":"p14","title":"Re-indexing preserves depth","kind":"theorem","summary":"[Re-indexing preserves depth] For every circuit c on m variables and every f : Fin\\,m \\to Fin\\,…","labels":["BoolCircuit.Circuit.reidx_depth"],"detail_key":"p14"},{"id":"n19121","layer":"informal","project":"p14","title":"Re-indexing commutes with evaluation","kind":"theorem","summary":"[Re-indexing commutes with evaluation] For every circuit c on m variables, every f : Fin\\,m \\to…","labels":["BoolCircuit.Circuit.reidx_eval"],"detail_key":"p14"},{"id":"n19122","layer":"informal","project":"p14","title":"DNF to dual CNF","kind":"definition","summary":"[DNF to dual CNF] The De~Morgan dual of a DNF \\varphi on n variables, obtained by negating ever…","labels":["LMN.dnfToDualCNF"],"detail_key":"p14"},{"id":"n19123","layer":"informal","project":"p14","title":"Dual CNF preserves width","kind":"lemma","summary":"[Dual CNF preserves width] The dual CNF dnfToDualCNF\\,\\varphi has the same width as \\varphi, si…","labels":["LMN.dnfToDualCNF_width"],"detail_key":"p14"},{"id":"n19124","layer":"informal","project":"p14","title":"Dual CNF negates the evaluation","kind":"lemma","summary":"[Dual CNF negates the evaluation] For every input x : Fin\\,n \\to Bool, the dual CNF evaluates t…","labels":["LMN.dnfToDualCNF_eval"],"detail_key":"p14"},{"id":"n19125","layer":"informal","project":"p14","title":"Depth \\geq 1 forces a node","kind":"lemma","summary":"[Depth \\geq 1 forces a node] A circuit c with depth\\,c \\geq 1 is not a literal: there exist a B…","labels":["LMN.Circuit.exists_node_of_depth_ge_one"],"detail_key":"p14"},{"id":"n19126","layer":"informal","project":"p14","title":"Depth-1 node has depth-0 children","kind":"lemma","summary":"[Depth-1 node has depth-0 children] If a node Circuit.node\\,\\mathitisAnd\\,cs has depth at most…","labels":["LMN.Circuit.depth1_children_are_lits"],"detail_key":"p14"},{"id":"n19127","layer":"informal","project":"p14","title":"Depth-0 circuit is a literal","kind":"lemma","summary":"[Depth-0 circuit is a literal] A circuit c with depth\\,c = 0 is a literal: there exists lr : Li…","labels":["LMN.Circuit.depth0_is_lit"],"detail_key":"p14"},{"id":"n19128","layer":"informal","project":"p14","title":"Depth-1 node has literal children","kind":"lemma","summary":"[Depth-1 node has literal children] If a node Circuit.node\\,\\mathitisAnd\\,cs has depth at most…","labels":["LMN.Circuit.depth1_all_lits"],"detail_key":"p14"},{"id":"n19129","layer":"informal","project":"p14","title":"Child function under gate substitution","kind":"definition","summary":"[Child function under gate substitution] The Boolean function on Fin\\,n \\to Bool computed by a…","labels":["LMN.childFunction"],"detail_key":"p14"},{"id":"n19130","layer":"informal","project":"p14","title":"Switched gates have DNF and CNF","kind":"lemma","summary":"[Switched gates have DNF and CNF] If, after applying a restriction \\rho, every gate function re…","labels":["LMN.switched_gates_have_dnf_cnf"],"detail_key":"p14"},{"id":"n19131","layer":"informal","project":"p14","title":"OR of literal children has a width-l DNF","kind":"lemma","summary":"[OR of literal children has a width-l DNF] Consider an OR node Circuit.node\\,false\\,cs whose ch…","labels":["LMN.or_of_lit_children_dnf"],"detail_key":"p14"},{"id":"n19132","layer":"informal","project":"p14","title":"AND of literal children has a width-l CNF","kind":"lemma","summary":"[AND of literal children has a width-l CNF] Consider an AND node Circuit.node\\,true\\,cs whose c…","labels":["LMN.and_of_lit_children_cnf"],"detail_key":"p14"},{"id":"n19133","layer":"informal","project":"p14","title":"Absorb a depth-1 top circuit","kind":"lemma","summary":"[Absorb a depth-1 top circuit] Let c_top have depth exactly 1 over the gates of some Layer2Data…","labels":["LMN.absorbOneLevel_depth1"],"detail_key":"p14"},{"id":"n19134","layer":"informal","project":"p14","title":"Depth-\\leq 1 child has a signed width-l DNF","kind":"lemma","summary":"[Depth-\\leq 1 child has a signed width-l DNF] A circuit c_j of depth at most 1 over gates that…","labels":["LMN.child_depth_le1_has_signed_dnf"],"detail_key":"p14"},{"id":"n19135","layer":"informal","project":"p14","title":"Signed DNFs for a list of children","kind":"lemma","summary":"[Signed DNFs for a list of children] Given a list cs of circuits each of depth at most 1, over…","labels":["LMN.list_child_signed_dnfs"],"detail_key":"p14"},{"id":"n19136","layer":"informal","project":"p14","title":"Build a node of signed literals","kind":"lemma","summary":"[Build a node of signed literals] For any flag \\mathitisAnd, arity k, and sign family \\mathitsi…","labels":["LMN.build_literal_circuit"],"detail_key":"p14"},{"id":"n19137","layer":"informal","project":"p14","title":"Depth reduction for depth-1 circuits","kind":"lemma","summary":"[Depth reduction for depth-1 circuits] A circuit c of depth exactly 1 over gates with width-l D…","labels":["LMN.exists_circuit_depth_reduction_depth1"],"detail_key":"p14"},{"id":"n19138","layer":"informal","project":"p14","title":"Depth reduction for depth-2 circuits","kind":"lemma","summary":"[Depth reduction for depth-2 circuits] A circuit c of depth exactly 2 over gates with width-l D…","labels":["LMN.exists_circuit_depth_reduction_depth2"],"detail_key":"p14"},{"id":"n19139","layer":"informal","project":"p14","title":"Merge child reduction results","kind":"lemma","summary":"[Merge child reduction results] Given a list cs of circuits where each child cs.get\\,j has been…","labels":["LMN.reduce_children"],"detail_key":"p14"},{"id":"n19140","layer":"informal","project":"p14","title":"Node evaluation under finRange re-indexing","kind":"lemma","summary":"[Node evaluation under finRange re-indexing] If each re-indexed child new\\_cs\\,j evaluated at g…","labels":["LMN.node_eval_eq_of_finRange_map"],"detail_key":"p14"},{"id":"n19141","layer":"informal","project":"p14","title":"General circuit depth reduction","kind":"lemma","summary":"[General circuit depth reduction] For a circuit c of depth at least 1 (with 0 < l) over gates t…","labels":["LMN.exists_circuit_depth_reduction"],"detail_key":"p14"},{"id":"n19142","layer":"informal","project":"p14","title":"Absorb a general top circuit (depth \\geq 2)","kind":"lemma","summary":"[Absorb a general top circuit (depth \\geq 2)] The Layer2Data wrapper of the general depth reduc…","labels":["LMN.absorbOneLevel_general"],"detail_key":"p14"},{"id":"n19143","layer":"informal","project":"p14","title":"Absorb one level of the top circuit","kind":"lemma","summary":"[Absorb one level of the top circuit] The combined absorption step: for c_top of depth at least…","labels":["LMN.absorbOneLevel"],"detail_key":"p14"},{"id":"n19144","layer":"informal","project":"p14","title":"Depth-zero circuit is a literal","kind":"lemma","summary":"[Depth-zero circuit is a literal] If a circuit c has depth 0, then c is a single literal: there…","labels":["LMN.circuit_depth_zero_is_lit"],"detail_key":"p14"},{"id":"n19145","layer":"informal","project":"p14","title":"Layer-2 composed bound, base case","kind":"lemma","summary":"[Layer-2 composed bound, base case] Base case (d_inner = 2) of the composed layer-2 bound. Here…","labels":["LMN.layer2_composed_bound_base"],"detail_key":"p14"},{"id":"n19146","layer":"informal","project":"p14","title":"Switched gates yield new bounded-width DNFs","kind":"lemma","summary":"[Switched gates yield new bounded-width DNFs] Suppose for each gate i the restricted function r…","labels":["LMN.switched_gates_give_new_dnfs"],"detail_key":"p14"},{"id":"n19147","layer":"informal","project":"p14","title":"Restriction of a function by a composed restriction","kind":"theorem","summary":"[Restriction of a function by a composed restriction] For a Boolean function f and restrictions…","labels":["LMN.restrictFn_composeRestr"],"detail_key":"p14"},{"id":"n19148","layer":"informal","project":"p14","title":"Decision-tree depth respects pointwise equality","kind":"lemma","summary":"[Decision-tree depth respects pointwise equality] If two Boolean functions f and g agree at eve…","labels":["LMN.dtDepth_congr"],"detail_key":"p14"},{"id":"n19149","layer":"informal","project":"p14","title":"Restriction respects pointwise equality","kind":"lemma","summary":"[Restriction respects pointwise equality] If two Boolean functions f and g agree at every input…","labels":["LMN.restrictFn_congr"],"detail_key":"p14"},{"id":"n19150","layer":"informal","project":"p14","title":"Pointwise AND of a list of functions","kind":"definition","summary":"[Pointwise AND of a list of functions] The pointwise conjunction of a list of Boolean functions…","labels":["LMN.listAnd"],"detail_key":"p14"},{"id":"n19151","layer":"informal","project":"p14","title":"Restriction distributes over list AND","kind":"lemma","summary":"[Restriction distributes over list AND] Restricting the pointwise AND of a list of functions eq…","labels":["LMN.restrictFn_listAnd"],"detail_key":"p14"},{"id":"n19152","layer":"informal","project":"p14","title":"Depth-3 one-step compression","kind":"theorem","summary":"[Depth-3 one-step compression] Given s_2 DNF gates each of width \\le w (with w > 0, non-degener…","labels":["LMN.depth3_compression"],"detail_key":"p14"},{"id":"n19153","layer":"informal","project":"p14","title":"Tautological clause","kind":"definition","summary":"[Tautological clause] A clause (a list of literals interpreted as their disjunction) is tautolo…","labels":["LMN.clauseIsTaut"],"detail_key":"p14"},{"id":"n19154","layer":"informal","project":"p14","title":"A tautological clause evaluates to true","kind":"lemma","summary":"[A tautological clause evaluates to true] If a clause c is tautological, then for every assignm…","labels":["LMN.clauseIsTaut_eval_true"],"detail_key":"p14"},{"id":"n19155","layer":"informal","project":"p14","title":"Deduplicate clause variables","kind":"definition","summary":"[Deduplicate clause variables] Removes duplicate variables from a clause, keeping the first occ…","labels":["LMN.dedupClauseVars"],"detail_key":"p14"},{"id":"n19156","layer":"informal","project":"p14","title":"Deduplication yields variable-injective clause","kind":"lemma","summary":"[Deduplication yields variable-injective clause] In dedupClauseVars\\,c, any two literals with t…","labels":["LMN.dedupClauseVars_var_inj"],"detail_key":"p14"},{"id":"n19157","layer":"informal","project":"p14","title":"Deduplication yields nodup clause","kind":"lemma","summary":"[Deduplication yields nodup clause] The list dedupClauseVars\\,c has no duplicate literals.","labels":["LMN.dedupClauseVars_nodup"],"detail_key":"p14"},{"id":"n19158","layer":"informal","project":"p14","title":"Deduplication does not increase length","kind":"lemma","summary":"[Deduplication does not increase length] The length of dedupClauseVars\\,c is at most the length…","labels":["LMN.dedupClauseVars_length_le"],"detail_key":"p14"},{"id":"n19159","layer":"informal","project":"p14","title":"Deduplication preserves evaluation of non-tautological clauses","kind":"lemma","summary":"[Deduplication preserves evaluation of non-tautological clauses] If a clause c is not tautologi…","labels":["LMN.dedupClauseVars_eval_of_not_taut"],"detail_key":"p14"},{"id":"n19160","layer":"informal","project":"p14","title":"Clean a CNF","kind":"definition","summary":"[Clean a CNF] Cleans a CNF by removing all tautological clauses and then deduplicating the vari…","labels":["LMN.cleanCNF_D3"],"detail_key":"p14"},{"id":"n19161","layer":"informal","project":"p14","title":"Cleaning preserves CNF evaluation","kind":"lemma","summary":"[Cleaning preserves CNF evaluation] For every assignment x, the cleaned CNF cleanCNF\\_D3\\,\\psi…","labels":["LMN.cleanCNF_D3_eval"],"detail_key":"p14"},{"id":"n19162","layer":"informal","project":"p14","title":"Cleaning does not increase width","kind":"lemma","summary":"[Cleaning does not increase width] The width of cleanCNF\\_D3\\,\\psi is at most the width of \\psi.","labels":["LMN.cleanCNF_D3_width_le"],"detail_key":"p14"},{"id":"n19163","layer":"informal","project":"p14","title":"Cleaned CNF has nodup clauses","kind":"lemma","summary":"[Cleaned CNF has nodup clauses] Every clause of cleanCNF\\_D3\\,\\psi has no duplicate literals.","labels":["LMN.cleanCNF_D3_nodup"],"detail_key":"p14"},{"id":"n19164","layer":"informal","project":"p14","title":"Cleaned CNF has variable-injective clauses","kind":"lemma","summary":"[Cleaned CNF has variable-injective clauses] In every clause of cleanCNF\\_D3\\,\\psi, any two lit…","labels":["LMN.cleanCNF_D3_var_inj"],"detail_key":"p14"},{"id":"n19165","layer":"informal","project":"p14","title":"Every CNF has an equivalent nice CNF","kind":"theorem","summary":"[Every CNF has an equivalent nice CNF] For any CNF \\psi there exists a CNF \\psi' of width \\le t…","labels":["LMN.exists_nice_cnf_of_cnf"],"detail_key":"p14"},{"id":"n19166","layer":"informal","project":"p14","title":"Bounded decision-tree depth gives a nice CNF","kind":"theorem","summary":"[Bounded decision-tree depth gives a nice CNF] If a Boolean function f has decision-tree depth…","labels":["LMN.dtDepth_le_implies_nice_cnf"],"detail_key":"p14"},{"id":"n19167","layer":"informal","project":"p14","title":"Bounded decision-tree depth gives a nice DNF","kind":"theorem","summary":"[Bounded decision-tree depth gives a nice DNF] If a Boolean function f has decision-tree depth…","labels":["LMN.dtDepth_le_implies_nice_dnf"],"detail_key":"p14"},{"id":"n19168","layer":"informal","project":"p14","title":"Functional switching lemma for CNFs","kind":"theorem","summary":"[Functional switching lemma for CNFs] For any function f with decision-tree depth \\le w (with w…","labels":["LMN.switching_bernoulli_dtDepth_function"],"detail_key":"p14"},{"id":"n19169","layer":"informal","project":"p14","title":"General two-stage bound","kind":"theorem","summary":"[General two-stage bound] Let 0 < p_1,p_2 \\le 1, let E be an event on restrictions and A a ``fa…","labels":["LMN.two_stage_bound"],"detail_key":"p14"},{"id":"n19170","layer":"informal","project":"p14","title":"AND of switched gates has a nice CNF","kind":"lemma","summary":"[AND of switched gates has a nice CNF] If each restricted gate restrictFn\\,(gates\\,i).eval\\,\\rh…","labels":["LMN.and_of_gates_has_cnf"],"detail_key":"p14"},{"id":"n19171","layer":"informal","project":"p14","title":"Restricted depth-3 circuit has a nice CNF","kind":"lemma","summary":"[Restricted depth-3 circuit has a nice CNF] Suppose f is the AND of the gates (f(x) = true iff…","labels":["LMN.depth3_restricted_has_nice_cnf"],"detail_key":"p14"},{"id":"n19172","layer":"informal","project":"p14","title":"Second-stage switching bound","kind":"lemma","summary":"[Second-stage switching bound] Suppose f is the AND of the gates and, under the first-stage res…","labels":["LMN.depth3_second_stage_bound"],"detail_key":"p14"},{"id":"n19173","layer":"informal","project":"p14","title":"Depth-3 two-stage switching bound","kind":"theorem","summary":"[Depth-3 two-stage switching bound] Let f be the AND of s_2 DNF gates each of width \\le w (with…","labels":["LMN.depth3_switching_bound"],"detail_key":"p14"},{"id":"n19174","layer":"informal","project":"p14","title":"Depth-3 circuit reduction bound","kind":"lemma","summary":"[Depth-3 circuit reduction bound] For a depth-3 circuit in normal form (AND of s_2 width-w DNF…","labels":["LMN.circuit_reduction_depth3"],"detail_key":"p14"},{"id":"n19175","layer":"informal","project":"p14","title":"Depth-3 switching lemma, \\le \\varepsilon version","kind":"theorem","summary":"[Depth-3 switching lemma, \\le \\varepsilon version] For a depth-3 circuit in normal form, if l a…","labels":["LMN.circuit_reduction_depth3_le_eps"],"detail_key":"p14"},{"id":"n19176","layer":"informal","project":"p14","title":"Monotonicity of the Bernoulli restriction probability","kind":"lemma","summary":"[Monotonicity of the Bernoulli restriction probability] Let p\\in[0,1] and let A,B be predicates…","labels":["LMN.bernoulliRestrProb_mono"],"detail_key":"p14"},{"id":"n19177","layer":"informal","project":"p14","title":"Restricted DNF gate has a small CNF, with high probability","kind":"theorem","summary":"[Restricted DNF gate has a small CNF, with high probability] Let g be a DNF of width at most w>…","labels":["LMN.switching_bernoulli_gate_to_cnf"],"detail_key":"p14"},{"id":"n19178","layer":"informal","project":"p14","title":"Restricted CNF gate has a small DNF, with high probability","kind":"theorem","summary":"[Restricted CNF gate has a small DNF, with high probability] The dual statement. Let g be a CNF…","labels":["LMN.switching_bernoulli_gate_to_dnf_from_cnf"],"detail_key":"p14"},{"id":"n19179","layer":"informal","project":"p14","title":"Small decision-tree depth yields a small CNF","kind":"theorem","summary":"[Small decision-tree depth yields a small CNF] If the restricted function f|_\\rho has decision-…","labels":["LMN.restricted_has_small_cnf_of_dtDepth_le"],"detail_key":"p14"},{"id":"n19180","layer":"informal","project":"p14","title":"Small decision-tree depth yields a small DNF","kind":"theorem","summary":"[Small decision-tree depth yields a small DNF] If the restricted function f|_\\rho has decision-…","labels":["LMN.restricted_has_small_dnf_of_dtDepth_le"],"detail_key":"p14"},{"id":"n19181","layer":"informal","project":"p14","title":"Union bound on gates with large decision-tree depth","kind":"theorem","summary":"[Union bound on gates with large decision-tree depth] Let gates_0,\\dots,gates_s-1 be s many DNF…","labels":["LMN.switching_bernoulli_union_bound"],"detail_key":"p14"},{"id":"n19182","layer":"informal","project":"p14","title":"All gates admit a small CNF under a good restriction","kind":"theorem","summary":"[All gates admit a small CNF under a good restriction] Given a family of DNF gates and a restri…","labels":["LMN.all_gates_have_small_cnf"],"detail_key":"p14"},{"id":"n19183","layer":"informal","project":"p14","title":"Union bound for layer-2 CNF replaceability","kind":"theorem","summary":"[Union bound for layer-2 CNF replaceability] Let gates_0,\\dots,gates_s_2-1 be the layer-2 DNF g…","labels":["LMN.layer2_cnf_replaceability_union_bound"],"detail_key":"p14"},{"id":"n19184","layer":"informal","project":"p14","title":"Simplified layer-2 CNF replaceability bound","kind":"theorem","summary":"[Simplified layer-2 CNF replaceability bound] Under the same hypotheses as the previous theorem…","labels":["LMN.layer2_cnf_replaceability_simplified"],"detail_key":"p14"},{"id":"n19185","layer":"informal","project":"p14","title":"Reduction at subsequent layers","kind":"theorem","summary":"[Reduction at subsequent layers] Let g_1,\\dots,g_s_i be width-\\le l DNFs on n variables (with 0…","labels":["LMN.subsequent_step_reduction"],"detail_key":"p14"},{"id":"n19186","layer":"informal","project":"p14","title":"Decision-tree depth at subsequent layers","kind":"theorem","summary":"[Decision-tree depth at subsequent layers] Under the same hypotheses, a Bernoulli(p) restrictio…","labels":["LMN.subsequent_step_dtDepth"],"detail_key":"p14"},{"id":"n19187","layer":"informal","project":"p14","title":"Multi-stage failure bound","kind":"theorem","summary":"[Multi-stage failure bound] If over m stages each failure bound satisfies failure\\_bound(i)\\le…","labels":["LMN.multi_stage_failure_bound"],"detail_key":"p14"},{"id":"n19188","layer":"informal","project":"p14","title":"Union bound for restriction probabilities","kind":"theorem","summary":"[Union bound for restriction probabilities] For 0\\le p\\le 1 and finitely many events A_0,\\dots,…","labels":["LMN.bernoulliRestrProb_union_bound_fin"],"detail_key":"p14"},{"id":"n19189","layer":"informal","project":"p14","title":"Iterative dominant-term bound","kind":"theorem","summary":"[Iterative dominant-term bound] For \\sum_i layer\\_size(i)\\le s with 0<s and 0<\\varepsilon, sett…","labels":["LMN.iterative_dominant_term_bound"],"detail_key":"p14"},{"id":"n19190","layer":"informal","project":"p14","title":"Two-stage composed union bound","kind":"theorem","summary":"[Two-stage composed union bound] For 0\\le p\\le 1 and 0\\le q\\le 1, and decidable events A,B on r…","labels":["LMN.two_stage_composed_union_bound"],"detail_key":"p14"},{"id":"n19191","layer":"informal","project":"p14","title":"Abstract iterative reduction","kind":"theorem","summary":"[Abstract iterative reduction] For \\sum_i layer\\_size(i)\\le s with 0<s and 0<\\varepsilon, if ea…","labels":["LMN.abstract_iterative_reduction"],"detail_key":"p14"},{"id":"n19192","layer":"informal","project":"p14","title":"Convert a circuit literal to a switching-lemma literal","kind":"definition","summary":"[Convert a circuit literal to a switching-lemma literal] Maps a Boolean-circuit literal l : \\te…","labels":["BoolCircuit.Lit.toLiteral"],"detail_key":"p14"},{"id":"n19193","layer":"informal","project":"p14","title":"Conversion preserves literal evaluation","kind":"theorem","summary":"[Conversion preserves literal evaluation] For every literal l and input x : Fin\\,n \\to Bool, th…","labels":["BoolCircuit.Lit.eval_eq_toLiteral_eval"],"detail_key":"p14"},{"id":"n19194","layer":"informal","project":"p14","title":"AND-clause to a term","kind":"definition","summary":"[AND-clause to a term] Sends an AND-clause \\textttNAndCircuit.clause\\,lits to the \\textttTerm (…","labels":["BoolCircuit.NAndCircuit.clauseToTerm"],"detail_key":"p14"},{"id":"n19195","layer":"informal","project":"p14","title":"OR-clause to a term","kind":"definition","summary":"[OR-clause to a term] Sends an OR-clause \\textttNOrCircuit.clause\\,lits to the \\textttTerm obta…","labels":["BoolCircuit.NOrCircuit.clauseToTerm"],"detail_key":"p14"},{"id":"n19196","layer":"informal","project":"p14","title":"Depth-2 OR-circuit to a DNF","kind":"definition","summary":"[Depth-2 OR-circuit to a DNF] Converts a depth-2 \\textttNOrCircuit (an OR of AND-clauses) into…","labels":["BoolCircuit.NOrCircuit.toDNF"],"detail_key":"p14"},{"id":"n19197","layer":"informal","project":"p14","title":"Depth-2 AND-circuit to a CNF","kind":"definition","summary":"[Depth-2 AND-circuit to a CNF] Converts a depth-2 \\textttNAndCircuit (an AND of OR-clauses) int…","labels":["BoolCircuit.NAndCircuit.toCNF"],"detail_key":"p14"},{"id":"n19198","layer":"informal","project":"p14","title":"Conjunction of literals evaluates as a term","kind":"theorem","summary":"[Conjunction of literals evaluates as a term] For any list of literals lits and input x, the ri…","labels":["BoolCircuit.foldr_and_lits_eq_term_eval"],"detail_key":"p14"},{"id":"n19199","layer":"informal","project":"p14","title":"Disjunction of literals evaluates as a CNF clause","kind":"theorem","summary":"[Disjunction of literals evaluates as a CNF clause] For any list of literals lits and input x,…","labels":["BoolCircuit.foldr_or_lits_eq_clause_eval"],"detail_key":"p14"},{"id":"n19200","layer":"informal","project":"p14","title":"Depth-2 OR-circuit evaluates as its DNF","kind":"theorem","summary":"[Depth-2 OR-circuit evaluates as its DNF] If every child c of an \\textttNOrCircuit.node\\,cs is…","labels":["BoolCircuit.NOrCircuit.node_eval_eq_toDNF_eval"],"detail_key":"p14"},{"id":"n19201","layer":"informal","project":"p14","title":"Depth-2 AND-circuit evaluates as its CNF","kind":"theorem","summary":"[Depth-2 AND-circuit evaluates as its CNF] If every child c of an \\textttNAndCircuit.node\\,cs i…","labels":["BoolCircuit.NAndCircuit.node_eval_eq_toCNF_eval"],"detail_key":"p14"},{"id":"n19202","layer":"informal","project":"p14","title":"Converted AND-clause has no repeated indices","kind":"theorem","summary":"[Converted AND-clause has no repeated indices] If the variable indices of an AND-clause's liter…","labels":["BoolCircuit.NAndCircuit.clauseToTerm_nodup"],"detail_key":"p14"},{"id":"n19203","layer":"informal","project":"p14","title":"Variable-injectivity of a converted AND-clause","kind":"theorem","summary":"[Variable-injectivity of a converted AND-clause] If the literal indices of an AND-clause are di…","labels":["BoolCircuit.NAndCircuit.clauseToTerm_var_inj"],"detail_key":"p14"},{"id":"n19204","layer":"informal","project":"p14","title":"Converted OR-clause has no repeated indices","kind":"theorem","summary":"[Converted OR-clause has no repeated indices] If the variable indices of an OR-clause's literal…","labels":["BoolCircuit.NOrCircuit.clauseToTerm_nodup"],"detail_key":"p14"},{"id":"n19205","layer":"informal","project":"p14","title":"Variable-injectivity of a converted OR-clause","kind":"theorem","summary":"[Variable-injectivity of a converted OR-clause] If the literal indices of an OR-clause are dist…","labels":["BoolCircuit.NOrCircuit.clauseToTerm_var_inj"],"detail_key":"p14"},{"id":"n19206","layer":"informal","project":"p14","title":"Width of a converted AND-clause","kind":"theorem","summary":"[Width of a converted AND-clause] The width of the term obtained from an AND-clause equals the…","labels":["BoolCircuit.NAndCircuit.clauseToTerm_width"],"detail_key":"p14"},{"id":"n19207","layer":"informal","project":"p14","title":"Width of a converted OR-clause","kind":"theorem","summary":"[Width of a converted OR-clause] The width of the term obtained from an OR-clause equals the nu…","labels":["BoolCircuit.NOrCircuit.clauseToTerm_width"],"detail_key":"p14"},{"id":"n19208","layer":"informal","project":"p14","title":"Width bound for the DNF of a depth-2 OR-circuit","kind":"theorem","summary":"[Width bound for the DNF of a depth-2 OR-circuit] If every AND-clause child has at most w liter…","labels":["BoolCircuit.NOrCircuit.toDNF_width_bounded"],"detail_key":"p14"},{"id":"n19209","layer":"informal","project":"p14","title":"Width bound for the CNF of a depth-2 AND-circuit","kind":"theorem","summary":"[Width bound for the CNF of a depth-2 AND-circuit] If every OR-clause child has at most w liter…","labels":["BoolCircuit.NAndCircuit.toCNF_width_bounded"],"detail_key":"p14"},{"id":"n19210","layer":"informal","project":"p14","title":"DNF terms of a depth-2 OR-circuit have no repeated indices","kind":"theorem","summary":"[DNF terms of a depth-2 OR-circuit have no repeated indices] If every child of an \\textttNOrCir…","labels":["BoolCircuit.NOrCircuit.toDNF_terms_nodup"],"detail_key":"p14"},{"id":"n19211","layer":"informal","project":"p14","title":"Variable-injectivity of the DNF of a depth-2 OR-circuit","kind":"theorem","summary":"[Variable-injectivity of the DNF of a depth-2 OR-circuit] If every child of an \\textttNOrCircui…","labels":["BoolCircuit.NOrCircuit.toDNF_var_inj"],"detail_key":"p14"},{"id":"n19212","layer":"informal","project":"p14","title":"CNF clauses of a depth-2 AND-circuit have no repeated indices","kind":"theorem","summary":"[CNF clauses of a depth-2 AND-circuit have no repeated indices] If every child of an \\textttNAn…","labels":["BoolCircuit.NAndCircuit.toCNF_terms_nodup"],"detail_key":"p14"},{"id":"n19213","layer":"informal","project":"p14","title":"Variable-injectivity of the CNF of a depth-2 AND-circuit","kind":"theorem","summary":"[Variable-injectivity of the CNF of a depth-2 AND-circuit] If every child of an \\textttNAndCirc…","labels":["BoolCircuit.NAndCircuit.toCNF_var_inj"],"detail_key":"p14"},{"id":"n19214","layer":"informal","project":"p14","title":"Child depth bound","kind":"lemma","summary":"[Child depth bound] If a node circuit node\\,\\mathitisAnd\\,cs has depth at most d, then every ch…","labels":["LMN.children_depth_le"],"detail_key":"p14"},{"id":"n19215","layer":"informal","project":"p14","title":"Sum of child sizes bound","kind":"lemma","summary":"[Sum of child sizes bound] If a node circuit node\\,\\mathitisAnd\\,cs has size at most s, then th…","labels":["LMN.children_size_sum_le"],"detail_key":"p14"},{"id":"n19216","layer":"informal","project":"p14","title":"Child fan-in bound","kind":"lemma","summary":"[Child fan-in bound] If a node circuit node\\,\\mathitisAnd\\,cs has maximum fan-in at most w, the…","labels":["LMN.children_maxFanin_le"],"detail_key":"p14"},{"id":"n19217","layer":"informal","project":"p14","title":"Child size bounded by parent","kind":"lemma","summary":"[Child size bounded by parent] If a node circuit node\\,\\mathitisAnd\\,cs has size at most s, the…","labels":["LMN.child_size_le_parent"],"detail_key":"p14"},{"id":"n19218","layer":"informal","project":"p14","title":"Restricted evaluation of a node","kind":"lemma","summary":"[Restricted evaluation of a node] The restriction of the evaluation of a node circuit equals th…","labels":["LMN.restrictFn_node_eval"],"detail_key":"p14"},{"id":"n19219","layer":"informal","project":"p14","title":"AND of children has a small CNF","kind":"lemma","summary":"[AND of children has a small CNF] If every child c \\in cs satisfies dtDepth(restrictFn\\,c.eval\\…","labels":["LMN.and_children_have_cnf"],"detail_key":"p14"},{"id":"n19220","layer":"informal","project":"p14","title":"OR of children has a small DNF","kind":"lemma","summary":"[OR of children has a small DNF] If every child c \\in cs satisfies dtDepth(restrictFn\\,c.eval\\,…","labels":["LMN.or_children_have_dnf"],"detail_key":"p14"},{"id":"n19221","layer":"informal","project":"p14","title":"Compression and switching for a node","kind":"lemma","summary":"[Compression and switching for a node] Suppose 0 < l, 0 < n, and every child c \\in cs satisfies…","labels":["LMN.compress_and_switch"],"detail_key":"p14"},{"id":"n19222","layer":"informal","project":"p14","title":"Composition of two restrictions","kind":"definition","summary":"[Composition of two restrictions] Given two restrictions \\rho_1, \\rho_2 on n variables, their c…","labels":["LMN.composeRestr"],"detail_key":"p14"},{"id":"n19223","layer":"informal","project":"p14","title":"Per-variable Bernoulli weight","kind":"definition","summary":"[Per-variable Bernoulli weight] For a parameter p \\in R, the per-variable weight assigns p to a…","labels":["LMN.varWeight"],"detail_key":"p14"},{"id":"n19224","layer":"informal","project":"p14","title":"Pointwise characterization of composition","kind":"lemma","summary":"[Pointwise characterization of composition] The composition composeRestr\\,\\rho_1\\,\\rho_2 equals…","labels":["LMN.composeRestr_eq_iff"],"detail_key":"p14"},{"id":"n19225","layer":"informal","project":"p14","title":"Right identity for composition","kind":"lemma","summary":"[Right identity for composition] Composing a restriction \\rho with the all-free restriction (wh…","labels":["LMN.composeRestr_id_right"],"detail_key":"p14"},{"id":"n19226","layer":"informal","project":"p14","title":"Bernoulli weight factors as a product","kind":"lemma","summary":"[Bernoulli weight factors as a product] The Bernoulli restriction weight of \\rho at parameter p…","labels":["LMN.bernoulliRestrWeight_eq_prod"],"detail_key":"p14"},{"id":"n19227","layer":"informal","project":"p14","title":"Per-variable composition identity","kind":"lemma","summary":"[Per-variable composition identity] For parameters p, q \\in R and any outcome c, summing the pr…","labels":["LMN.varWeight_compose_sum"],"detail_key":"p14"},{"id":"n19228","layer":"informal","project":"p14","title":"Fiber weight identity","kind":"lemma","summary":"[Fiber weight identity] Summing the product bernoulliRestrWeight\\,p\\,\\rho_1 \\cdot bernoulliRest…","labels":["LMN.compose_fiber_weight_eq"],"detail_key":"p14"},{"id":"n19229","layer":"informal","project":"p14","title":"Composition of Bernoulli random restrictions, equality","kind":"theorem","summary":"[Composition of Bernoulli random restrictions, equality] For 0 < p \\le 1, 0 < q \\le 1 and any e…","labels":["LMN.restriction_compose_eq"],"detail_key":"p14"},{"id":"n19230","layer":"informal","project":"p14","title":"Composition inequality","kind":"theorem","summary":"[Composition inequality] For 0 < p \\le 1 and 0 < q \\le 1, the Bernoulli(pq) probability of an e…","labels":["LMN.restriction_compose_le"],"detail_key":"p14"},{"id":"n19231","layer":"informal","project":"p14","title":"Restriction of a decision tree","kind":"definition","summary":"[Restriction of a decision tree] Given a decision tree T and a partial assignment \\rho, the res…","labels":["LMN.dtRestrict"],"detail_key":"p14"},{"id":"n19232","layer":"informal","project":"p14","title":"Restriction does not increase depth","kind":"theorem","summary":"[Restriction does not increase depth] For every decision tree T and restriction \\rho, the depth…","labels":["LMN.dtRestrict_depth_le"],"detail_key":"p14"},{"id":"n19233","layer":"informal","project":"p14","title":"Evaluation of a restricted tree","kind":"theorem","summary":"[Evaluation of a restricted tree] For every decision tree T, restriction \\rho, and input x, the…","labels":["LMN.dtRestrict_eval"],"detail_key":"p14"},{"id":"n19234","layer":"informal","project":"p14","title":"Decision-tree depth is monotone under restriction","kind":"theorem","summary":"[Decision-tree depth is monotone under restriction] For every Boolean function f and restrictio…","labels":["LMN.dtDepth_restrictFn_le'"],"detail_key":"p14"},{"id":"n19235","layer":"informal","project":"p14","title":"Composing restrictions further decreases depth","kind":"theorem","summary":"[Composing restrictions further decreases depth] For every Boolean function f and restrictions…","labels":["LMN.dtDepth_composeRestr_le"],"detail_key":"p14"},{"id":"n19236","layer":"informal","project":"p14","title":"Bernoulli depth-tail probability monotonicity","kind":"theorem","summary":"[Bernoulli depth-tail probability monotonicity] For every Boolean function f, threshold t, and…","labels":["LMN.bernoulliRestrProb_dtDepth_compose_le"],"detail_key":"p14"},{"id":"n19237","layer":"informal","project":"p14","title":"Cardinality of fixed-size restrictions","kind":"lemma","summary":"[Cardinality of fixed-size restrictions] For k \\le n, the set of restrictions of Fin\\,n leaving…","labels":["SwitchingBernoulli.fixedSizeRestrs_card"],"detail_key":"p14"},{"id":"n19238","layer":"informal","project":"p14","title":"Bad fixed-size restrictions match the counting filter","kind":"lemma","summary":"[Bad fixed-size restrictions match the counting filter] For a function f and parameters d, k, t…","labels":["SwitchingBernoulli.fixedSizeRestrs_filter_bad_eq"],"detail_key":"p14"},{"id":"n19239","layer":"informal","project":"p14","title":"Fixed-size switching bound for small k","kind":"lemma","summary":"[Fixed-size switching bound for small k] For a width-w DNF f with no repeated variables per ter…","labels":["SwitchingBernoulli.switching_fixedSize_bound_small"],"detail_key":"p14"},{"id":"n19240","layer":"informal","project":"p14","title":"Fixed-size switching bound","kind":"lemma","summary":"[Fixed-size switching bound] For a width-w DNF f with 0 < w, 0 < n, and k \\le n, the size-k res…","labels":["SwitchingBernoulli.switching_fixedSize_bound"],"detail_key":"p14"},{"id":"n19241","layer":"informal","project":"p14","title":"Rescaled fixed-size switching bound","kind":"lemma","summary":"[Rescaled fixed-size switching bound] The same fixed-size bound written in the form expected by…","labels":["SwitchingBernoulli.switching_fixedSize_bound_rescaled"],"detail_key":"p14"},{"id":"n19242","layer":"informal","project":"p14","title":"Bernoulli switching lemma for DNFs","kind":"theorem","summary":"[Bernoulli switching lemma for DNFs] Let f be a width-w DNF (0 < w) with no repeated variables…","labels":["SwitchingBernoulli.switching_bernoulli_dtDepth_dnf"],"detail_key":"p14"},{"id":"n19243","layer":"informal","project":"p14","title":"Bernoulli switching lemma for CNFs","kind":"theorem","summary":"[Bernoulli switching lemma for CNFs] The same statement for a width-w CNF f: under a Bernoulli(…","labels":["SwitchingBernoulli.switching_bernoulli_dtDepth_cnf"],"detail_key":"p14"},{"id":"n19244","layer":"informal","project":"p14","title":"Parse an aux list into triples","kind":"definition","summary":"[Parse an aux list into triples] Parses a low-level aux list of (N \\times Bool) entries into a…","labels":["SwitchingLemma2.parseAux"],"detail_key":"p14"},{"id":"n19245","layer":"informal","project":"p14","title":"Triples back to an aux list","kind":"definition","summary":"[Triples back to an aux list] The inverse direction of \\textttSwitchingLemma2.parseAux: convert…","labels":["SwitchingLemma2.triplesToAux"],"detail_key":"p14"},{"id":"n19246","layer":"informal","project":"p14","title":"parseAux on an entry followed by a marker","kind":"lemma","summary":"[parseAux on an entry followed by a marker] Equational lemma: if an entry (idx, dir) with idx <…","labels":["SwitchingLemma2.parseAux_cons_marker"],"detail_key":"p14"},{"id":"n19247","layer":"informal","project":"p14","title":"parseAux on an entry followed by a non-marker","kind":"lemma","summary":"[parseAux on an entry followed by a non-marker] Equational lemma: if an entry (idx, dir) is imm…","labels":["SwitchingLemma2.parseAux_cons_nonmarker"],"detail_key":"p14"},{"id":"n19248","layer":"informal","project":"p14","title":"parseAux on a single entry","kind":"lemma","summary":"[parseAux on a single entry] Equational lemma: a single in-range entry (idx, dir) parses to the…","labels":["SwitchingLemma2.parseAux_singleton"],"detail_key":"p14"},{"id":"n19249","layer":"informal","project":"p14","title":"parseAux on the empty list","kind":"lemma","summary":"[parseAux on the empty list] Equational lemma: parseAux of the empty list is the empty list.","labels":["SwitchingLemma2.parseAux_nil"],"detail_key":"p14"},{"id":"n19250","layer":"informal","project":"p14","title":"parseAux inverts triplesToAux","kind":"lemma","summary":"[parseAux inverts triplesToAux] Round-trip identity: for every triple list ts, applying triples…","labels":["SwitchingLemma2.parseAux_triplesToAux"],"detail_key":"p14"},{"id":"n19251","layer":"informal","project":"p14","title":"triplesToAux distributes over append","kind":"lemma","summary":"[triplesToAux distributes over append] triplesToAux\\,(ts_1 \\mathbin+\\!\\!+ ts_2) = triplesToAux\\…","labels":["SwitchingLemma2.triplesToAux_append"],"detail_key":"p14"},{"id":"n19252","layer":"informal","project":"p14","title":"Combined length bound for processClauseLits","kind":"lemma","summary":"[Combined length bound for processClauseLits] The aux output length of processClauseLits plus t…","labels":["SwitchingLemma2.processClauseLits_len_add"],"detail_key":"p14"},{"id":"n19253","layer":"informal","project":"p14","title":"Mark the last entry of a block","kind":"definition","summary":"[Mark the last entry of a block] Builds a triple list from a Fin\\,w \\times Bool block by settin…","labels":["SwitchingLemma2.markLast"],"detail_key":"p14"},{"id":"n19254","layer":"informal","project":"p14","title":"triplesToAux of markLast","kind":"lemma","summary":"[triplesToAux of markLast] On a nonempty block, triplesToAux(markLast\\,block) equals the block…","labels":["SwitchingLemma2.triplesToAux_markLast"],"detail_key":"p14"},{"id":"n19255","layer":"informal","project":"p14","title":"markLast of a nonempty list is nonempty","kind":"lemma","summary":"[markLast of a nonempty list is nonempty] If block \\neq [] then markLast\\,block \\neq [].","labels":["SwitchingLemma2.markLast_ne_nil"],"detail_key":"p14"},{"id":"n19256","layer":"informal","project":"p14","title":"markLast preserves length","kind":"lemma","summary":"[markLast preserves length] (markLast\\,block).length = block.length.","labels":["SwitchingLemma2.markLast_length"],"detail_key":"p14"},{"id":"n19257","layer":"informal","project":"p14","title":"Last entry of markLast is marked","kind":"lemma","summary":"[Last entry of markLast is marked] On a nonempty block, the last element of markLast\\,block has…","labels":["SwitchingLemma2.markLast_getLast_true"],"detail_key":"p14"},{"id":"n19258","layer":"informal","project":"p14","title":"Cast an in-range aux list into Fin blocks","kind":"definition","summary":"[Cast an in-range aux list into Fin blocks] Casts a list of (N \\times Bool) entries, all of who…","labels":["SwitchingLemma2.toFinBlock"],"detail_key":"p14"},{"id":"n19259","layer":"informal","project":"p14","title":"toFinBlock preserves length","kind":"lemma","summary":"[toFinBlock preserves length] (toFinBlock\\,w\\,l\\,h).length = l.length.","labels":["SwitchingLemma2.toFinBlock_length"],"detail_key":"p14"},{"id":"n19260","layer":"informal","project":"p14","title":"toFinBlock map recovers the original list","kind":"lemma","summary":"[toFinBlock map recovers the original list] Mapping the Fin blocks back to (N \\times Bool) via…","labels":["SwitchingLemma2.toFinBlock_map"],"detail_key":"p14"},{"id":"n19261","layer":"informal","project":"p14","title":"toFinBlock is nonempty for nonempty input","kind":"lemma","summary":"[toFinBlock is nonempty for nonempty input] If l \\neq [] then toFinBlock\\,w\\,l\\,h \\neq [].","labels":["SwitchingLemma2.toFinBlock_ne_nil"],"detail_key":"p14"},{"id":"n19262","layer":"informal","project":"p14","title":"Aux indices bounded by term length","kind":"lemma","summary":"[Aux indices bounded by term length] When the processed literals come from t.zipIdx, every entr…","labels":["SwitchingLemma2.processClauseLits_aux_idx_lt"],"detail_key":"p14"},{"id":"n19263","layer":"informal","project":"p14","title":"Encoder well-formedness invariant","kind":"lemma","summary":"[Encoder well-formedness invariant] The aux output of razborovEncode.go (started with empty acc…","labels":["SwitchingLemma2.encode_go_wellformed"],"detail_key":"p14"},{"id":"n19264","layer":"informal","project":"p14","title":"Injection from encoder aux images","kind":"lemma","summary":"[Injection from encoder aux images] There exists a function g from aux lists into Fin\\,d \\to Fi…","labels":["SwitchingLemma2.exists_aux_injection"],"detail_key":"p14"},{"id":"n19265","layer":"informal","project":"p14","title":"Cardinality bound on the aux image","kind":"lemma","summary":"[Cardinality bound on the aux image] The number of distinct aux lists produced by the Razborov…","labels":["SwitchingLemma2.aux_image_card_bound"],"detail_key":"p14"},{"id":"n19266","layer":"informal","project":"p14","title":"Fiber bound for the encoder","kind":"lemma","summary":"[Fiber bound for the encoder] The number of bad s-restrictions \\rho mapping to a fixed \\gamma u…","labels":["SwitchingLemma2.fiber_bound"],"detail_key":"p14"},{"id":"n19267","layer":"informal","project":"p14","title":"Number of restrictions with k free variables","kind":"lemma","summary":"[Number of restrictions with k free variables] The number of restrictions of \\0,\\dots,n-1\\ with…","labels":["SwitchingLemma2.card_filter_numFree_eq"],"detail_key":"p14"},{"id":"n19268","layer":"informal","project":"p14","title":"Updating a free variable decreases numFree by one","kind":"lemma","summary":"[Updating a free variable decreases numFree by one] If \\rho\\,v = none, then updating \\rho at v…","labels":["SwitchingLemma2.numFree_update_free"],"detail_key":"p14"},{"id":"n19269","layer":"informal","project":"p14","title":"processClauseLits preserves free-variable agreement","kind":"lemma","summary":"[processClauseLits preserves free-variable agreement] If \\rho_0 and \\sigma have the same free v…","labels":["SwitchingLemma2.processClauseLits_freeVars_agree"],"detail_key":"p14"},{"id":"n19270","layer":"informal","project":"p14","title":"Free-variable count of \\sigma after processClauseLits","kind":"lemma","summary":"[Free-variable count of \\sigma after processClauseLits] Under the invariants that \\rho_0 and \\s…","labels":["SwitchingLemma2.processClauseLits_numFree_σ"],"detail_key":"p14"},{"id":"n19271","layer":"informal","project":"p14","title":"Canonical deep path length lower bound","kind":"lemma","summary":"[Canonical deep path length lower bound] If \\rho is a bad restriction (so dtDepth(f|_\\rho) > d)…","labels":["SwitchingLemma2.canonicalDTree_deepPath_length_ge"],"detail_key":"p14"},{"id":"n19272","layer":"informal","project":"p14","title":"Encoder path length equals d on bad restrictions","kind":"lemma","summary":"[Encoder path length equals d on bad restrictions] When \\rho is bad, the prefix (canonicalDTree…","labels":["SwitchingLemma2.razborovEncode_path_length"],"detail_key":"p14"},{"id":"n19273","layer":"informal","project":"p14","title":"Path length consumed by processClauseLits","kind":"lemma","summary":"[Path length consumed by processClauseLits] processClauseLits consumes exactly \\min(lits.length…","labels":["SwitchingLemma2.processClauseLits_path_length_eq"],"detail_key":"p14"},{"id":"n19274","layer":"informal","project":"p14","title":"Updating to some decreases numFree by at most one","kind":"lemma","summary":"[Updating to some decreases numFree by at most one] Updating \\rho at v to some\\,b decreases num…","labels":["SwitchingLemma2.numFree_update_some_ge"],"detail_key":"p14"},{"id":"n19275","layer":"informal","project":"p14","title":"Remaining path bounded by remaining free variables","kind":"lemma","summary":"[Remaining path bounded by remaining free variables] If the initial path length is at most \\rho…","labels":["SwitchingLemma2.processClauseLits_remaining_le_numFree"],"detail_key":"p14"},{"id":"n19276","layer":"informal","project":"p14","title":"Canonical tree has depth zero when all clauses killed","kind":"lemma","summary":"[Canonical tree has depth zero when all clauses killed] If every clause of f is killed by \\rho,…","labels":["SwitchingLemma2.canonicalDTree_depth_zero_of_killed"],"detail_key":"p14"},{"id":"n19277","layer":"informal","project":"p14","title":"Canonical tree has depth zero when a clause is fixed","kind":"lemma","summary":"[Canonical tree has depth zero when a clause is fixed] If some clause of f is fixed (satisfied)…","labels":["SwitchingLemma2.canonicalDTree_depth_zero_of_fixed"],"detail_key":"p14"},{"id":"n19278","layer":"informal","project":"p14","title":"Canonical path predicate","kind":"definition","summary":"[Canonical path predicate] IsCanonicalPath\\,f\\,\\rho\\,path holds when path is an initial segment…","labels":["SwitchingLemma2.IsCanonicalPath"],"detail_key":"p14"},{"id":"n19279","layer":"informal","project":"p14","title":"Filtered zipIdx preserves length","kind":"lemma","summary":"[Filtered zipIdx preserves length] Filtering t.zipIdx by a predicate on the literal component h…","labels":["SwitchingLemma2.zipIdx_filter_length"],"detail_key":"p14"},{"id":"n19280","layer":"informal","project":"p14","title":"Filtered zipIdx first component","kind":"lemma","summary":"[Filtered zipIdx first component] The literal component of the k-th element of the filtered t.z…","labels":["SwitchingLemma2.zipIdx_filter_getElem_fst"],"detail_key":"p14"},{"id":"n19281","layer":"informal","project":"p14","title":"Deep path matches free literals of the first alive clause","kind":"lemma","summary":"[Deep path matches free literals of the first alive clause] When f.find? returns an alive claus…","labels":["SwitchingLemma2.canonicalDTree_deepPath_match_freeLits"],"detail_key":"p14"},{"id":"n19282","layer":"informal","project":"p14","title":"Remaining path is a drop","kind":"lemma","summary":"[Remaining path is a drop] The remaining path returned by processClauseLits equals path.drop(\\m…","labels":["SwitchingLemma2.processClauseLits_fst_eq_drop"],"detail_key":"p14"},{"id":"n19283","layer":"informal","project":"p14","title":"Free-variable count of \\rho_0 after processClauseLits","kind":"lemma","summary":"[Free-variable count of \\rho_0 after processClauseLits] Provided each literal's variable is fre…","labels":["SwitchingLemma2.processClauseLits_numFree_ρ_eq"],"detail_key":"p14"},{"id":"n19284","layer":"informal","project":"p14","title":"termSubTree deep path after dropping the free-literal prefix","kind":"lemma","summary":"[termSubTree deep path after dropping the free-literal prefix] Dropping the free-literal prefix…","labels":["SwitchingLemma2.processClauseLits_termSubTree_drop"],"detail_key":"p14"},{"id":"n19285","layer":"informal","project":"p14","title":"Canonical path preserved by processClauseLits","kind":"lemma","summary":"[Canonical path preserved by processClauseLits] When the literals are the free literals of the…","labels":["SwitchingLemma2.canonicalPath_preserve_processClauseLits"],"detail_key":"p14"},{"id":"n19286","layer":"informal","project":"p14","title":"Free literals have pairwise distinct variables","kind":"lemma","summary":"[Free literals have pairwise distinct variables] If t has no duplicate literals and any two lit…","labels":["SwitchingLemma2.processClauseLits_freeLits_pairwise_var"],"detail_key":"p14"},{"id":"n19287","layer":"informal","project":"p14","title":"Encoder free-variable consumption invariant","kind":"lemma","summary":"[Encoder free-variable consumption invariant] When the canonical tree depth is at least the pat…","labels":["SwitchingLemma2.razborovEncode_go_numFree_invariant"],"detail_key":"p14"},{"id":"n19288","layer":"informal","project":"p14","title":"Encoder output is an (s-d)-restriction","kind":"lemma","summary":"[Encoder output is an (s-d)-restriction] For a bad s-restriction \\rho with d \\leq s, the first…","labels":["SwitchingLemma2.razborovEncode_fst_numFree_eq"],"detail_key":"p14"},{"id":"n19289","layer":"informal","project":"p14","title":"Bad-restriction counting bound","kind":"lemma","summary":"[Bad-restriction counting bound] For d \\leq s, the number of bad s-restrictions is at most \\bin…","labels":["SwitchingLemma2.bad_count_bound"],"detail_key":"p14"},{"id":"n19290","layer":"informal","project":"p14","title":"No bad restrictions when s \\leq d","kind":"lemma","summary":"[No bad restrictions when s \\leq d] If s \\leq d, the set of restrictions that are simultaneousl…","labels":["SwitchingLemma2.bad_filter_empty_of_d_ge_s"],"detail_key":"p14"},{"id":"n19291","layer":"informal","project":"p14","title":"Decision tree to DNF","kind":"definition","summary":"[Decision tree to DNF] Converts a decision tree into a DNF whose terms are the conjunctions of…","labels":["SwitchingLemma2.toDNF"],"detail_key":"p14"},{"id":"n19292","layer":"informal","project":"p14","title":"Decision tree to CNF","kind":"definition","summary":"[Decision tree to CNF] Converts a decision tree into a CNF whose clauses are the disjunctions o…","labels":["SwitchingLemma2.toCNF"],"detail_key":"p14"},{"id":"n19293","layer":"informal","project":"p14","title":"Decision-tree-depth witness tree","kind":"lemma","summary":"[Decision-tree-depth witness tree] For any Boolean function f there exists a decision tree T co…","labels":["SwitchingLemma2.dtDepth_witness"],"detail_key":"p14"},{"id":"n19294","layer":"informal","project":"p14","title":"Small decision-tree depth yields small DNF and CNF","kind":"lemma","summary":"[Small decision-tree depth yields small DNF and CNF] If dtDepth\\,f \\leq d, then f is computed b…","labels":["SwitchingLemma2.dtDepth_le_implies_small_dnf_cnf"],"detail_key":"p14"},{"id":"n19295","layer":"informal","project":"p14","title":"Flip a literal's polarity","kind":"definition","summary":"[Flip a literal's polarity] Negates a literal by flipping its polarity bit while keeping its va…","labels":["Literal.flipNeg"],"detail_key":"p14"},{"id":"n19296","layer":"informal","project":"p14","title":"flipNeg negates the literal value","kind":"lemma","summary":"[flipNeg negates the literal value] (l.flipNeg).eval\\,x = \\lnot\\,(l.eval\\,x).","labels":["Literal.flipNeg_eval"],"detail_key":"p14"},{"id":"n19297","layer":"informal","project":"p14","title":"flipNeg preserves the variable","kind":"lemma","summary":"[flipNeg preserves the variable] (l.flipNeg).var = l.var.","labels":["Literal.flipNeg_var"],"detail_key":"p14"},{"id":"n19298","layer":"informal","project":"p14","title":"flipNeg is injective","kind":"lemma","summary":"[flipNeg is injective] The map Literal.flipNeg is injective.","labels":["Literal.flipNeg_injective"],"detail_key":"p14"},{"id":"n19299","layer":"informal","project":"p14","title":"CNF to De Morgan dual DNF","kind":"definition","summary":"[CNF to De Morgan dual DNF] Converts a CNF \\psi to its De Morgan dual DNF by negating every lit…","labels":["cnfToDualDNF"],"detail_key":"p14"},{"id":"n19300","layer":"informal","project":"p14","title":"cnfToDualDNF preserves width","kind":"lemma","summary":"[cnfToDualDNF preserves width] (cnfToDualDNF\\,\\psi).width = \\psi.width.","labels":["cnfToDualDNF_width"],"detail_key":"p14"},{"id":"n19301","layer":"informal","project":"p14","title":"cnfToDualDNF negates the value","kind":"lemma","summary":"[cnfToDualDNF negates the value] (cnfToDualDNF\\,\\psi).eval\\,x = \\lnot\\,(\\psi.eval\\,x).","labels":["cnfToDualDNF_eval"],"detail_key":"p14"},{"id":"n19302","layer":"informal","project":"p14","title":"Negate decision tree leaves","kind":"definition","summary":"[Negate decision tree leaves] Negates every leaf of a decision tree, leaving its branching stru…","labels":["DecisionTree.negateLeaves"],"detail_key":"p14"},{"id":"n19303","layer":"informal","project":"p14","title":"negateLeaves negates the value","kind":"lemma","summary":"[negateLeaves negates the value] T.negateLeaves.eval\\,x = \\lnot\\,(T.eval\\,x).","labels":["DecisionTree.negateLeaves_eval"],"detail_key":"p14"},{"id":"n19304","layer":"informal","project":"p14","title":"negateLeaves preserves depth","kind":"lemma","summary":"[negateLeaves preserves depth] T.negateLeaves.depth = T.depth.","labels":["DecisionTree.negateLeaves_depth"],"detail_key":"p14"},{"id":"n19305","layer":"informal","project":"p14","title":"Decision-tree depth is invariant under negation","kind":"lemma","summary":"[Decision-tree depth is invariant under negation] dtDepth(\\lnot f) = dtDepth\\,f.","labels":["dtDepth_neg"],"detail_key":"p14"},{"id":"n19306","layer":"informal","project":"p14","title":"Restriction commutes with negation","kind":"lemma","summary":"[Restriction commutes with negation] restrictFn(\\lnot f)\\,\\rho = \\lnot\\,(restrictFn\\,f\\,\\rho),…","labels":["SwitchingLemmaCNF.restrictFn_neg"],"detail_key":"p14"},{"id":"n19307","layer":"informal","project":"p14","title":"Bad restriction is invariant under negation","kind":"lemma","summary":"[Bad restriction is invariant under negation] IsBadRestriction(\\lnot f)\\,d\\,\\rho \\iff IsBadRest…","labels":["SwitchingLemmaCNF.IsBadRestriction_neg"],"detail_key":"p14"},{"id":"n19308","layer":"informal","project":"p14","title":"Dual DNF preserves no-duplicate clauses","kind":"lemma","summary":"[Dual DNF preserves no-duplicate clauses] If every clause of \\psi has no duplicate literals, th…","labels":["SwitchingLemmaCNF.cnfToDualDNF_nodup"],"detail_key":"p14"},{"id":"n19309","layer":"informal","project":"p14","title":"Dual DNF preserves variable-distinctness","kind":"lemma","summary":"[Dual DNF preserves variable-distinctness] If within each clause of \\psi any two literals with…","labels":["SwitchingLemmaCNF.cnfToDualDNF_inj"],"detail_key":"p14"},{"id":"n19310","layer":"informal","project":"p14","title":"Håstad's Switching Lemma for CNFs","kind":"theorem","summary":"[Håstad's Switching Lemma for CNFs] For a CNF \\psi of width at most w on n variables with 5s \\l…","labels":["SwitchingLemmaCNF.switching_lemma_cnf"],"detail_key":"p14"},{"id":"n19311","layer":"informal","project":"p14","title":"Switching Lemma Corollary for CNFs","kind":"theorem","summary":"[Switching Lemma Corollary for CNFs] For a CNF \\psi of width at most w with 5s \\leq n and well-…","labels":["SwitchingLemmaCNF.switching_corollary_cnf"],"detail_key":"p14"},{"id":"n19312","layer":"informal","project":"p14","title":"Binomial-times-power counting inequality","kind":"lemma","summary":"[Binomial-times-power counting inequality] For natural numbers n, s, d with 5s \\leq n and d \\le…","labels":["SwitchingLemma2.choose_mul_pow_bound"],"detail_key":"p14"},{"id":"n19313","layer":"informal","project":"p14","title":"Bernoulli weight of a restriction","kind":"definition","summary":"[Bernoulli weight of a restriction] For a parameter p \\in R and a restriction \\rho on n variabl…","labels":["SwitchingLemma2.bernoulliRestrWeight"],"detail_key":"p14"},{"id":"n19314","layer":"informal","project":"p14","title":"Bernoulli event probability","kind":"definition","summary":"[Bernoulli event probability] For a parameter p and a (decidable) predicate event on restrictio…","labels":["SwitchingLemma2.bernoulliRestrProb"],"detail_key":"p14"},{"id":"n19315","layer":"informal","project":"p14","title":"Nonnegativity of Bernoulli weights","kind":"lemma","summary":"[Nonnegativity of Bernoulli weights] If 0 \\le p \\le 1, then for every restriction \\rho the weig…","labels":["SwitchingLemma2.bernoulliRestrWeight_nonneg'"],"detail_key":"p14"},{"id":"n19316","layer":"informal","project":"p14","title":"Bernoulli weights sum to one","kind":"lemma","summary":"[Bernoulli weights sum to one] If 0 \\le p \\le 1, then the Bernoulli weights form a probability…","labels":["SwitchingLemma2.bernoulliRestrWeight_sum_one"],"detail_key":"p14"},{"id":"n19317","layer":"informal","project":"p14","title":"Bernoulli event probability is at most one","kind":"lemma","summary":"[Bernoulli event probability is at most one] If 0 \\le p \\le 1, then for every (decidable) event…","labels":["SwitchingLemma2.bernoulliRestrProb_le_one'"],"detail_key":"p14"},{"id":"n19318","layer":"informal","project":"p14","title":"Updating a free variable decreases the free count","kind":"lemma","summary":"[Updating a free variable decreases the free count] If v is a free variable of a restriction \\r…","labels":["SwitchingLemma2.numFree_update_lt"],"detail_key":"p14"},{"id":"n19319","layer":"informal","project":"p14","title":"A free variable exists when neither killed nor fixed","kind":"lemma","summary":"[A free variable exists when neither killed nor fixed] For a DNF f and restriction \\rho, if not…","labels":["SwitchingLemma2.exists_free_of_not_killed_not_fixed"],"detail_key":"p14"},{"id":"n19320","layer":"informal","project":"p14","title":"Branch variable selection","kind":"definition","summary":"[Branch variable selection] Given a DNF f and restriction \\rho, returns an optional variable in…","labels":["SwitchingLemma2.selectBranchVar"],"detail_key":"p14"},{"id":"n19321","layer":"informal","project":"p14","title":"Specification of branch variable selection","kind":"lemma","summary":"[Specification of branch variable selection] When not every term of f is killed by \\rho and no…","labels":["SwitchingLemma2.selectBranchVar_spec"],"detail_key":"p14"},{"id":"n19322","layer":"informal","project":"p14","title":"Sub-tree for a term","kind":"definition","summary":"[Sub-tree for a term] Builds a complete sub-tree for a term, queried as a list of literals: it…","labels":["SwitchingLemma2.termSubTree"],"detail_key":"p14"},{"id":"n19323","layer":"informal","project":"p14","title":"Canonical decision tree","kind":"definition","summary":"[Canonical decision tree] The canonical decision tree for f|_\\rho, following Razborov's constru…","labels":["SwitchingLemma2.canonicalDTree"],"detail_key":"p14"},{"id":"n19324","layer":"informal","project":"p14","title":"Extending an updated restriction agrees with the original","kind":"lemma","summary":"[Extending an updated restriction agrees with the original] If v is free in \\rho and the input…","labels":["SwitchingLemma2.extend_update_self"],"detail_key":"p14"},{"id":"n19325","layer":"informal","project":"p14","title":"\\texttttermSubTree preserves semantics","kind":"lemma","summary":"[\\texttttermSubTree preserves semantics] Evaluating termSubTree\\,\\mathitlits\\,\\rho\\,cont at x e…","labels":["SwitchingLemma2.termSubTree_eval"],"detail_key":"p14"},{"id":"n19326","layer":"informal","project":"p14","title":"\\texttttermSubTree fold preserves extension","kind":"lemma","summary":"[\\texttttermSubTree fold preserves extension] After termSubTree assigns all free variables of \\…","labels":["SwitchingLemma2.termSubTree_extend_eq"],"detail_key":"p14"},{"id":"n19327","layer":"informal","project":"p14","title":"Fold preserves non-\\textttnone entries","kind":"lemma","summary":"[Fold preserves non-\\textttnone entries] If \\rho(v) \\neq none, then after the termSubTree fold…","labels":["SwitchingLemma2.termSubTree_foldl_preserves_nonnone"],"detail_key":"p14"},{"id":"n19328","layer":"informal","project":"p14","title":"Fold sets free literal variables","kind":"lemma","summary":"[Fold sets free literal variables] If a literal l \\in \\mathitlits has variable free in \\rho, th…","labels":["SwitchingLemma2.termSubTree_foldl_sets_member"],"detail_key":"p14"},{"id":"n19329","layer":"informal","project":"p14","title":"Fold strictly decreases the free count","kind":"lemma","summary":"[Fold strictly decreases the free count] When at least one literal of \\mathitlits has a variabl…","labels":["SwitchingLemma2.termSubTree_foldl_numFree_lt"],"detail_key":"p14"},{"id":"n19330","layer":"informal","project":"p14","title":"Correctness of the fueled construction","kind":"lemma","summary":"[Correctness of the fueled construction] Provided the fuel exceeds \\rho.numFree, the tree canon…","labels":["SwitchingLemma2.canonicalDTree_go_correct"],"detail_key":"p14"},{"id":"n19331","layer":"informal","project":"p14","title":"Correctness of the canonical decision tree","kind":"lemma","summary":"[Correctness of the canonical decision tree] For every x, the canonical decision tree evaluates…","labels":["SwitchingLemma2.canonicalDTree_correct"],"detail_key":"p14"},{"id":"n19332","layer":"informal","project":"p14","title":"Updating cannot increase the free count","kind":"lemma","summary":"[Updating cannot increase the free count] Fixing a variable v to a value b never increases the…","labels":["SwitchingLemma2.numFree_update_le"],"detail_key":"p14"},{"id":"n19333","layer":"informal","project":"p14","title":"\\texttttermSubTree is extensional in its continuation","kind":"lemma","summary":"[\\texttttermSubTree is extensional in its continuation] If two continuations cont_1, cont_2 agr…","labels":["SwitchingLemma2.termSubTree_cont_congr"],"detail_key":"p14"},{"id":"n19334","layer":"informal","project":"p14","title":"\\texttttermSubTree extensionality, strict version","kind":"lemma","summary":"[\\texttttermSubTree extensionality, strict version] If \\mathitlits contains at least one litera…","labels":["SwitchingLemma2.termSubTree_cont_congr_strict"],"detail_key":"p14"},{"id":"n19335","layer":"informal","project":"p14","title":"Fuel invariance of \\textttcanonicalDTree.go","kind":"lemma","summary":"[Fuel invariance of \\textttcanonicalDTree.go] Once the fuel exceeds \\rho.numFree, the resulting…","labels":["SwitchingLemma2.canonicalDTree_go_fuel_invariant"],"detail_key":"p14"},{"id":"n19336","layer":"informal","project":"p14","title":"Continuation equals the canonical tree","kind":"lemma","summary":"[Continuation equals the canonical tree] For t \\in f and a restriction \\rho' with \\rho_orig.num…","labels":["SwitchingLemma2.cont_eq_canonicalDTree"],"detail_key":"p14"},{"id":"n19337","layer":"informal","project":"p14","title":"Unfolding \\texttttermSubTree at a free head","kind":"lemma","summary":"[Unfolding \\texttttermSubTree at a free head] When the head literal l is free in \\rho, termSubT…","labels":["SwitchingLemma2.termSubTree_cons_free"],"detail_key":"p14"},{"id":"n19338","layer":"informal","project":"p14","title":"Unfolding \\texttttermSubTree at a non-free head","kind":"lemma","summary":"[Unfolding \\texttttermSubTree at a non-free head] When the head literal l is not free in \\rho,…","labels":["SwitchingLemma2.termSubTree_cons_nonfree"],"detail_key":"p14"},{"id":"n19339","layer":"informal","project":"p14","title":"\\textttdeepPath head extraction for \\texttttermSubTree","kind":"lemma","summary":"[\\textttdeepPath head extraction for \\texttttermSubTree] When the head literal l is free in \\rh…","labels":["SwitchingLemma2.termSubTree_deepPath_head_free"],"detail_key":"p14"},{"id":"n19340","layer":"informal","project":"p14","title":"Freeness is unchanged by an unrelated update","kind":"lemma","summary":"[Freeness is unchanged by an unrelated update] If every literal of \\mathitrest has variable dis…","labels":["SwitchingLemma2.filter_free_update_eq"],"detail_key":"p14"},{"id":"n19341","layer":"informal","project":"p14","title":"\\textttdeepPath variables match the free literals","kind":"lemma","summary":"[\\textttdeepPath variables match the free literals] For \\mathitlits with pairwise distinct vari…","labels":["SwitchingLemma2.termSubTree_deepPath_var_match"],"detail_key":"p14"},{"id":"n19342","layer":"informal","project":"p14","title":"\\textttdeepPath length decomposition for \\texttttermSubTree","kind":"lemma","summary":"[\\textttdeepPath length decomposition for \\texttttermSubTree] For \\mathitlits with pairwise dis…","labels":["SwitchingLemma2.termSubTree_deepPath_append"],"detail_key":"p14"},{"id":"n19343","layer":"informal","project":"p14","title":"\\textttdeepPath split for \\texttttermSubTree","kind":"lemma","summary":"[\\textttdeepPath split for \\texttttermSubTree] For \\mathitlits with pairwise distinct variables…","labels":["SwitchingLemma2.termSubTree_deepPath_split"],"detail_key":"p14"},{"id":"n19344","layer":"informal","project":"p14","title":"Alive branch delegates to \\texttttermSubTree","kind":"lemma","summary":"[Alive branch delegates to \\texttttermSubTree] When not all terms are killed and none is fixed,…","labels":["SwitchingLemma2.canonicalDTree_alive_eq_termSubTree"],"detail_key":"p14"},{"id":"n19345","layer":"informal","project":"p14","title":"Alive branch delegates to \\texttttermSubTree (top level)","kind":"lemma","summary":"[Alive branch delegates to \\texttttermSubTree (top level)] The top-level analogue of the previo…","labels":["SwitchingLemma2.canonicalDTree_alive_eq_termSubTree'"],"detail_key":"p14"},{"id":"n19346","layer":"informal","project":"p14","title":"Skipping a non-free prefix in \\texttttermSubTree","kind":"lemma","summary":"[Skipping a non-free prefix in \\texttttermSubTree] If every literal in \\mathitprefix has variab…","labels":["SwitchingLemma2.termSubTree_skip_nonfree_prefix'"],"detail_key":"p14"},{"id":"n19347","layer":"informal","project":"p14","title":"Skipping an updated head literal","kind":"lemma","summary":"[Skipping an updated head literal] If l.var = v, then termSubTree\\,(l :: \\mathitrest)\\,(\\rho[v…","labels":["SwitchingLemma2.termSubTree_skip_updated_head"],"detail_key":"p14"},{"id":"n19348","layer":"informal","project":"p14","title":"Tree depth bounds decision-tree depth","kind":"lemma","summary":"[Tree depth bounds decision-tree depth] If a decision tree T computes a function f (i.e. T.eval…","labels":["SwitchingLemma2.depth_ge_dtDepth"],"detail_key":"p14"},{"id":"n19349","layer":"informal","project":"p14","title":"Canonical tree depth bounds \\textttdtDepth","kind":"lemma","summary":"[Canonical tree depth bounds \\textttdtDepth] The depth of the canonical decision tree for f|_\\r…","labels":["SwitchingLemma2.canonicalDTree_depth_ge"],"detail_key":"p14"},{"id":"n19350","layer":"informal","project":"p14","title":"Decision-tree depth bounded by free count","kind":"lemma","summary":"[Decision-tree depth bounded by free count] For any function f and restriction \\rho, the decisi…","labels":["SwitchingLemma2.dtDepth_restrictFn_le_numFree"],"detail_key":"p14"},{"id":"n19351","layer":"informal","project":"p14","title":"Evaluation of a literal","kind":"definition","summary":"[Evaluation of a literal] For a literal l on n variables and an assignment x : Fin\\,n \\to F_2,…","labels":["BoolCircuit.Lit.eval"],"detail_key":"p14"},{"id":"n19352","layer":"informal","project":"p14","title":"Boolean circuit tree","kind":"definition","summary":"[Boolean circuit tree] A Boolean circuit on n variables is a tree whose leaves are literals (\\t…","labels":["BoolCircuit.Circuit"],"detail_key":"p14"},{"id":"n19353","layer":"informal","project":"p14","title":"Custom induction principle for circuits","kind":"theorem","summary":"[Custom induction principle for circuits] An induction principle for \\textttCircuit stating tha…","labels":["BoolCircuit.Circuit.ind"],"detail_key":"p14"},{"id":"n19354","layer":"informal","project":"p14","title":"Evaluation of a circuit","kind":"definition","summary":"[Evaluation of a circuit] Evaluates a circuit under an assignment x: a literal evaluates via Li…","labels":["BoolCircuit.Circuit.eval"],"detail_key":"p14"},{"id":"n19355","layer":"informal","project":"p14","title":"Literal count of a circuit","kind":"definition","summary":"[Literal count of a circuit] The number of literal occurrences in a circuit: a literal contribu…","labels":["BoolCircuit.Circuit.litCount"],"detail_key":"p14"},{"id":"n19356","layer":"informal","project":"p14","title":"Depth of a circuit","kind":"definition","summary":"[Depth of a circuit] The depth of a circuit (longest root-to-leaf path): a literal has depth 0…","labels":["BoolCircuit.Circuit.depth"],"detail_key":"p14"},{"id":"n19357","layer":"informal","project":"p14","title":"Size of a circuit","kind":"definition","summary":"[Size of a circuit] The total number of nodes of a circuit (internal gates plus literal leaves)…","labels":["BoolCircuit.Circuit.size"],"detail_key":"p14"},{"id":"n19358","layer":"informal","project":"p14","title":"Maximum depth over a list of circuits","kind":"definition","summary":"[Maximum depth over a list of circuits] The maximum depth over a list cs of circuits (with 0 fo…","labels":["BoolCircuit.Circuit.maxDepth"],"detail_key":"p14"},{"id":"n19359","layer":"informal","project":"p14","title":"Sum of sizes over a list of circuits","kind":"definition","summary":"[Sum of sizes over a list of circuits] The sum of the sizes over a list cs of circuits, used to…","labels":["BoolCircuit.Circuit.sumSize"],"detail_key":"p14"},{"id":"n19360","layer":"informal","project":"p14","title":"Maximum fanin of a circuit","kind":"definition","summary":"[Maximum fanin of a circuit] The maximum fanin of a circuit: a literal has fanin 0, and a node'…","labels":["BoolCircuit.Circuit.maxFanin"],"detail_key":"p14"},{"id":"n19361","layer":"informal","project":"p14","title":"Normal-form AND circuit","kind":"definition","summary":"[Normal-form AND circuit] A normal-form AND circuit on n variables: either a base \\textttclause…","labels":["BoolCircuit.NAndCircuit"],"detail_key":"p14"},{"id":"n19362","layer":"informal","project":"p14","title":"Evaluation of a normal-form AND circuit","kind":"definition","summary":"[Evaluation of a normal-form AND circuit] Evaluates a normal-form AND circuit under an assignme…","labels":["BoolCircuit.NAndCircuit.eval"],"detail_key":"p14"},{"id":"n19363","layer":"informal","project":"p14","title":"Evaluation of a normal-form OR circuit","kind":"definition","summary":"[Evaluation of a normal-form OR circuit] Evaluates a normal-form OR circuit under an assignment…","labels":["BoolCircuit.NOrCircuit.eval"],"detail_key":"p14"},{"id":"n19364","layer":"informal","project":"p14","title":"Literal count of a normal-form AND circuit","kind":"definition","summary":"[Literal count of a normal-form AND circuit] The number of literal occurrences in a normal-form…","labels":["BoolCircuit.NAndCircuit.litCount"],"detail_key":"p14"},{"id":"n19365","layer":"informal","project":"p14","title":"Literal count of a normal-form OR circuit","kind":"definition","summary":"[Literal count of a normal-form OR circuit] The number of literal occurrences in a normal-form…","labels":["BoolCircuit.NOrCircuit.litCount"],"detail_key":"p14"},{"id":"n19366","layer":"informal","project":"p14","title":"Size of a normal-form AND circuit","kind":"definition","summary":"[Size of a normal-form AND circuit] The total node count of a normal-form AND circuit: a clause…","labels":["BoolCircuit.NAndCircuit.size"],"detail_key":"p14"},{"id":"n19367","layer":"informal","project":"p14","title":"Size of a normal-form OR circuit","kind":"definition","summary":"[Size of a normal-form OR circuit] The total node count of a normal-form OR circuit: a clause h…","labels":["BoolCircuit.NOrCircuit.size"],"detail_key":"p14"},{"id":"n19368","layer":"informal","project":"p14","title":"Depth of a normal-form AND circuit","kind":"definition","summary":"[Depth of a normal-form AND circuit] The depth of a normal-form AND circuit: a clause has depth…","labels":["BoolCircuit.NAndCircuit.depth"],"detail_key":"p14"},{"id":"n19369","layer":"informal","project":"p14","title":"Depth of a normal-form OR circuit","kind":"definition","summary":"[Depth of a normal-form OR circuit] The depth of a normal-form OR circuit: a clause has depth 0…","labels":["BoolCircuit.NOrCircuit.depth"],"detail_key":"p14"},{"id":"n19370","layer":"informal","project":"p14","title":"Clause indices are nodup (AND)","kind":"theorem","summary":"[Clause indices are nodup (AND)] If a normal-form AND circuit c equals a clause built from lite…","labels":["BoolCircuit.NAndCircuit.clause_nodup"],"detail_key":"p14"},{"id":"n19371","layer":"informal","project":"p14","title":"Clause indices are nodup (OR)","kind":"theorem","summary":"[Clause indices are nodup (OR)] If a normal-form OR circuit c equals a clause built from litera…","labels":["BoolCircuit.NOrCircuit.clause_nodup"],"detail_key":"p14"},{"id":"n19372","layer":"informal","project":"p14","title":"Equal indices imply equal literals in a nodup clause","kind":"theorem","summary":"[Equal indices imply equal literals in a nodup clause] In a clause whose variable indices are p…","labels":["BoolCircuit.Lit.eq_of_idx_eq_of_mem_nodup"],"detail_key":"p14"},{"id":"n19373","layer":"informal","project":"p14","title":"Normalize a circuit to AND-normal form","kind":"definition","summary":"[Normalize a circuit to AND-normal form] Converts a general circuit into a normal-form AND circ…","labels":["BoolCircuit.Circuit.toNAnd"],"detail_key":"p14"},{"id":"n19374","layer":"informal","project":"p14","title":"Normalize a circuit to OR-normal form","kind":"definition","summary":"[Normalize a circuit to OR-normal form] Converts a general circuit into a normal-form OR circui…","labels":["BoolCircuit.Circuit.toNOr"],"detail_key":"p14"},{"id":"n19375","layer":"informal","project":"p14","title":"Fold commutes with map (conjunction)","kind":"theorem","summary":"[Fold commutes with map (conjunction)] A technical lemma: if g(h\\,c) = f\\,c for all c \\in cs, t…","labels":["BoolCircuit.foldr_and_map"],"detail_key":"p14"},{"id":"n19376","layer":"informal","project":"p14","title":"Fold commutes with map (disjunction)","kind":"theorem","summary":"[Fold commutes with map (disjunction)] A technical lemma: if g(h\\,c) = f\\,c for all c \\in cs, t…","labels":["BoolCircuit.foldr_or_map"],"detail_key":"p14"},{"id":"n19377","layer":"informal","project":"p14","title":"Fold commutes with map (addition)","kind":"theorem","summary":"[Fold commutes with map (addition)] A technical lemma: if g(h\\,c) = f\\,c for all c \\in cs, then…","labels":["BoolCircuit.foldr_add_map"],"detail_key":"p14"},{"id":"n19378","layer":"informal","project":"p14","title":"Fold inequality under map (addition)","kind":"theorem","summary":"[Fold inequality under map (addition)] A technical lemma: if g(h\\,c) \\le k \\cdot f\\,c for all c…","labels":["BoolCircuit.foldr_add_map_le"],"detail_key":"p14"},{"id":"n19379","layer":"informal","project":"p14","title":"Normalization preserves semantics","kind":"theorem","summary":"[Normalization preserves semantics] For every circuit c and assignment x, both normalizations a…","labels":["BoolCircuit.toNAnd_toNOr_eval"],"detail_key":"p14"},{"id":"n19380","layer":"informal","project":"p14","title":"AND-normalization preserves semantics","kind":"theorem","summary":"[AND-normalization preserves semantics] For every circuit c and assignment x, (c.toNAnd).eval\\,…","labels":["BoolCircuit.toNAnd_eval"],"detail_key":"p14"},{"id":"n19381","layer":"informal","project":"p14","title":"OR-normalization preserves semantics","kind":"theorem","summary":"[OR-normalization preserves semantics] For every circuit c and assignment x, (c.toNOr).eval\\,x…","labels":["BoolCircuit.toNOr_eval"],"detail_key":"p14"},{"id":"n19382","layer":"informal","project":"p14","title":"Normalization preserves literal count","kind":"theorem","summary":"[Normalization preserves literal count] For every circuit c, both normalizations preserve the l…","labels":["BoolCircuit.toNAnd_toNOr_litCount"],"detail_key":"p14"},{"id":"n19383","layer":"informal","project":"p14","title":"AND-normalization preserves literal count","kind":"theorem","summary":"[AND-normalization preserves literal count] For every circuit c, (c.toNAnd).litCount = c.litCou…","labels":["BoolCircuit.toNAnd_litCount"],"detail_key":"p14"},{"id":"n19384","layer":"informal","project":"p14","title":"OR-normalization preserves literal count","kind":"theorem","summary":"[OR-normalization preserves literal count] For every circuit c, (c.toNOr).litCount = c.litCount.","labels":["BoolCircuit.toNOr_litCount"],"detail_key":"p14"},{"id":"n19385","layer":"informal","project":"p14","title":"Normalization at most doubles size","kind":"theorem","summary":"[Normalization at most doubles size] For every circuit c, both normalizations have size at most…","labels":["BoolCircuit.toNAnd_toNOr_size_le"],"detail_key":"p14"},{"id":"n19386","layer":"informal","project":"p14","title":"AND-normalization at most doubles size","kind":"theorem","summary":"[AND-normalization at most doubles size] For every circuit c, (c.toNAnd).size \\le 2\\,c.size.","labels":["BoolCircuit.toNAnd_size_le"],"detail_key":"p14"},{"id":"n19387","layer":"informal","project":"p14","title":"OR-normalization at most doubles size","kind":"theorem","summary":"[OR-normalization at most doubles size] For every circuit c, (c.toNOr).size \\le 2\\,c.size.","labels":["BoolCircuit.toNOr_size_le"],"detail_key":"p14"},{"id":"n19388","layer":"informal","project":"p14","title":"Forget AND-normal form back to a circuit","kind":"definition","summary":"[Forget AND-normal form back to a circuit] The forgetful map turning a normal-form AND circuit…","labels":["BoolCircuit.NAndCircuit.toCircuit"],"detail_key":"p14"},{"id":"n19389","layer":"informal","project":"p14","title":"Forget OR-normal form back to a circuit","kind":"definition","summary":"[Forget OR-normal form back to a circuit] The forgetful map turning a normal-form OR circuit in…","labels":["BoolCircuit.NOrCircuit.toCircuit"],"detail_key":"p14"},{"id":"n19390","layer":"informal","project":"p14","title":"Single-variable AND circuit","kind":"definition","summary":"[Single-variable AND circuit] The normal-form AND circuit consisting of the single positive lit…","labels":["BoolCircuit.NAndCircuit.ofVar"],"detail_key":"p14"},{"id":"n19391","layer":"informal","project":"p14","title":"Single-variable OR circuit","kind":"definition","summary":"[Single-variable OR circuit] The normal-form OR circuit consisting of the single positive liter…","labels":["BoolCircuit.NOrCircuit.ofVar"],"detail_key":"p14"},{"id":"n19392","layer":"informal","project":"p14","title":"Constant-true AND circuit","kind":"definition","summary":"[Constant-true AND circuit] The constant-true AND circuit, given by the empty clause (an empty…","labels":["BoolCircuit.NAndCircuit.constTrue"],"detail_key":"p14"},{"id":"n19393","layer":"informal","project":"p14","title":"Constant-false OR circuit","kind":"definition","summary":"[Constant-false OR circuit] The constant-false OR circuit, given by the empty clause (an empty…","labels":["BoolCircuit.NOrCircuit.constFalse"],"detail_key":"p14"},{"id":"n19394","layer":"informal","project":"p14","title":"Evaluation of a literal","kind":"definition","summary":"[Evaluation of a literal] For a literal l with variable l.var and polarity l.neg and an assignm…","labels":["Literal.eval"],"detail_key":"p14"},{"id":"n19395","layer":"informal","project":"p14","title":"Width of a CNF formula","kind":"definition","summary":"[Width of a CNF formula] The width of a CNF formula c, defined as the maximum width over its cl…","labels":["CNF.width"],"detail_key":"p14"},{"id":"n19396","layer":"informal","project":"p14","title":"Evaluation of a decision tree","kind":"definition","summary":"[Evaluation of a decision tree] Evaluates a decision tree on an input x: a leaf returns its sto…","labels":["DecisionTree.eval"],"detail_key":"p14"},{"id":"n19397","layer":"informal","project":"p14","title":"Depth of a decision tree","kind":"definition","summary":"[Depth of a decision tree] The depth of a decision tree (maximum path length to a leaf): a leaf…","labels":["DecisionTree.depth"],"detail_key":"p14"},{"id":"n19398","layer":"informal","project":"p14","title":"Deepest path of a decision tree","kind":"definition","summary":"[Deepest path of a decision tree] Extracts a deepest root-to-leaf path from a decision tree, at…","labels":["DecisionTree.deepPath"],"detail_key":"p14"},{"id":"n19399","layer":"informal","project":"p14","title":"Deep path length equals depth","kind":"lemma","summary":"[Deep path length equals depth] For every decision tree T, the length of its deepest path equal…","labels":["DecisionTree.length_deepPath"],"detail_key":"p14"},{"id":"n19400","layer":"informal","project":"p14","title":"Depth bound for the full decision tree","kind":"lemma","summary":"[Depth bound for the full decision tree] For a function f, an index k \\le n, and an accumulator…","labels":["buildFullDTree_depth"],"detail_key":"p14"},{"id":"n19401","layer":"informal","project":"p14","title":"Correctness of the full decision tree","kind":"lemma","summary":"[Correctness of the full decision tree] For a function f, an index k \\le n, and assignments acc…","labels":["buildFullDTree_eval"],"detail_key":"p14"},{"id":"n19402","layer":"informal","project":"p14","title":"Circuit literal","kind":"definition","summary":"[Circuit literal] A literal for Boolean circuits: a variable index \\mathttidx : Fin\\,n together…","labels":["BoolCircuit.Lit"],"detail_key":"p14"},{"id":"n19403","layer":"informal","project":"p14","title":"Literal","kind":"definition","summary":"[Literal] A literal for switching-lemma formulas: a variable \\mathttvar : Fin\\,n with a negatio…","labels":["Literal"],"detail_key":"p14"},{"id":"n19404","layer":"informal","project":"p14","title":"Term","kind":"definition","summary":"[Term] A term is a conjunction of literals, represented as a list: \\mathttTerm\\,n = \\mathttList…","labels":["Term"],"detail_key":"p14"},{"id":"n19405","layer":"informal","project":"p14","title":"Term width","kind":"definition","summary":"[Term width] The width of a term is its number of literals (the list length).","labels":["Term.width"],"detail_key":"p14"},{"id":"n19406","layer":"informal","project":"p14","title":"Term evaluation","kind":"definition","summary":"[Term evaluation] A term evaluates to true on input x iff every literal in it holds under x (co…","labels":["Term.eval"],"detail_key":"p14"},{"id":"n19407","layer":"informal","project":"p14","title":"DNF formula","kind":"definition","summary":"[DNF formula] A DNF formula is a disjunction of terms, represented as a list: \\mathttDNF\\,n = \\…","labels":["DNF"],"detail_key":"p14"},{"id":"n19408","layer":"informal","project":"p14","title":"DNF width","kind":"definition","summary":"[DNF width] The width of a DNF formula is the maximum width of its terms (0 for the empty formu…","labels":["DNF.width"],"detail_key":"p14"},{"id":"n19409","layer":"informal","project":"p14","title":"DNF evaluation","kind":"definition","summary":"[DNF evaluation] A DNF formula evaluates to true on input x iff at least one of its terms holds…","labels":["DNF.eval"],"detail_key":"p14"},{"id":"n19410","layer":"informal","project":"p14","title":"CNF formula","kind":"definition","summary":"[CNF formula] A CNF formula is a conjunction of clauses, each clause a disjunction of literals;…","labels":["CNF"],"detail_key":"p14"},{"id":"n19411","layer":"informal","project":"p14","title":"Clause evaluation","kind":"definition","summary":"[Clause evaluation] A single CNF clause evaluates to true on input x iff some literal in it hol…","labels":["CNF.evalClause"],"detail_key":"p14"},{"id":"n19412","layer":"informal","project":"p14","title":"CNF evaluation","kind":"definition","summary":"[CNF evaluation] A CNF formula evaluates to true on input x iff every clause holds under x.","labels":["CNF.eval"],"detail_key":"p14"},{"id":"n19413","layer":"informal","project":"p14","title":"Decision tree","kind":"definition","summary":"[Decision tree] A decision tree on n Boolean variables: a leaf \\mathttleaf\\,b outputs b, and a…","labels":["DecisionTree"],"detail_key":"p14"},{"id":"n19414","layer":"informal","project":"p14","title":"Full decision tree","kind":"definition","summary":"[Full decision tree] The complete decision tree for a function f that queries variables k, k+1,…","labels":["buildFullDTree"],"detail_key":"p14"},{"id":"n19415","layer":"informal","project":"p14","title":"Decision-tree depth of a function","kind":"definition","summary":"[Decision-tree depth of a function] The decision-tree depth of f : (Fin\\,n \\to Bool) \\to Bool i…","labels":["dtDepth"],"detail_key":"p14"},{"id":"n19416","layer":"informal","project":"p14","title":"Free literals of a term under a restriction","kind":"definition","summary":"[Free literals of a term under a restriction] Given a term t (a conjunction of literals) and a…","labels":["SwitchingLemma2.Term.freeLiterals"],"detail_key":"p14"},{"id":"n19417","layer":"informal","project":"p14","title":"Process one clause's free literals against the path","kind":"definition","summary":"[Process one clause's free literals against the path] Consumes the free literals of a single cl…","labels":["SwitchingLemma2.processClauseLits"],"detail_key":"p14"},{"id":"n19418","layer":"informal","project":"p14","title":"Razborov encoding","kind":"definition","summary":"[Razborov encoding] The Razborov encoding of a DNF f and a restriction \\rho with parameters w,…","labels":["SwitchingLemma2.razborovEncode"],"detail_key":"p14"},{"id":"n19419","layer":"informal","project":"p14","title":"Razborov decoding","kind":"definition","summary":"[Razborov decoding] The inverse of the Razborov encoding: given the DNF f, parameter w, and a p…","labels":["SwitchingLemma2.razborovDecode"],"detail_key":"p14"},{"id":"n19420","layer":"informal","project":"p14","title":"Processing does not lengthen the path","kind":"lemma","summary":"[Processing does not lengthen the path] The remaining path returned by \\textttprocessClauseLits…","labels":["SwitchingLemma2.processClauseLits_path_le"],"detail_key":"p14"},{"id":"n19421","layer":"informal","project":"p14","title":"Output bound for processClauseLits","kind":"lemma","summary":"[Output bound for processClauseLits] For any literal/index list and any path, the combined outp…","labels":["SwitchingLemma2.processClauseLits_bound"],"detail_key":"p14"},{"id":"n19422","layer":"informal","project":"p14","title":"Tight bound for non-empty inputs","kind":"lemma","summary":"[Tight bound for non-empty inputs] On non-empty literal and path lists, \\textttSwitchingLemma2.…","labels":["SwitchingLemma2.processClauseLits_tight"],"detail_key":"p14"},{"id":"n19423","layer":"informal","project":"p14","title":"Length bound for the encoder loop","kind":"lemma","summary":"[Length bound for the encoder loop] The output produced by the encoder loop \\textttSwitchingLem…","labels":["SwitchingLemma2.encode_go_aux_length_bound"],"detail_key":"p14"},{"id":"n19424","layer":"informal","project":"p14","title":"Encoding length is at most 2d","kind":"lemma","summary":"[Encoding length is at most 2d] For a bad restriction \\rho of a DNF f with depth parameter d, t…","labels":["SwitchingLemma2.razborovEncode_aux_length_le"],"detail_key":"p14"},{"id":"n19425","layer":"informal","project":"p14","title":"processClauseLits preserves \\sigma off the literal list","kind":"lemma","summary":"[processClauseLits preserves \\sigma off the literal list] If a variable v is the variable of no…","labels":["SwitchingLemma2.processClauseLits_sigma_stable"],"detail_key":"p14"},{"id":"n19426","layer":"informal","project":"p14","title":"processClauseLits preserves \\rho_0 off the literal list","kind":"lemma","summary":"[processClauseLits preserves \\rho_0 off the literal list] If a variable v is the variable of no…","labels":["SwitchingLemma2.processClauseLits_rho_stable"],"detail_key":"p14"},{"id":"n19427","layer":"informal","project":"p14","title":"processClauseLits never frees \\rho_0","kind":"lemma","summary":"[processClauseLits never frees \\rho_0] If \\rho_0\\,v \\neq none, then after running \\textttSwitch…","labels":["SwitchingLemma2.processClauseLits_rho_ne_none"],"detail_key":"p14"},{"id":"n19428","layer":"informal","project":"p14","title":"Encoder \\gamma preserves \\sigma at non-free variables","kind":"lemma","summary":"[Encoder \\gamma preserves \\sigma at non-free variables] If \\rho_0\\,v \\neq none (so v is non-fre…","labels":["SwitchingLemma2.encode_go_fst_nonfree"],"detail_key":"p14"},{"id":"n19429","layer":"informal","project":"p14","title":"Accumulator decomposition for the encoder loop","kind":"lemma","summary":"[Accumulator decomposition for the encoder loop] The encoder loop \\textttSwitchingLemma2.razbor…","labels":["SwitchingLemma2.encode_go_acc"],"detail_key":"p14"},{"id":"n19430","layer":"informal","project":"p14","title":"Encoder \\gamma independent of the accumulator","kind":"lemma","summary":"[Encoder \\gamma independent of the accumulator] The first component of the encoder loop \\texttt…","labels":["SwitchingLemma2.encode_go_fst_acc"],"detail_key":"p14"},{"id":"n19431","layer":"informal","project":"p14","title":"processEntries preserves none","kind":"lemma","summary":"[processEntries preserves none] If \\sigma\\,v = none, then the decoder's \\textttprocessEntries r…","labels":["SwitchingLemma2.processEntries_preserves_none"],"detail_key":"p14"},{"id":"n19432","layer":"informal","project":"p14","title":"Decoder loop preserves none","kind":"lemma","summary":"[Decoder loop preserves none] If \\sigma\\,v = none, then the decoder loop \\textttSwitchingLemma2…","labels":["SwitchingLemma2.decode_go_preserves_none"],"detail_key":"p14"},{"id":"n19433","layer":"informal","project":"p14","title":"processClauseLits components independent of \\sigma","kind":"lemma","summary":"[processClauseLits components independent of \\sigma] The remaining path, the \\rho_0 output, and…","labels":["SwitchingLemma2.processClauseLits_sigma_indep"],"detail_key":"p14"},{"id":"n19434","layer":"informal","project":"p14","title":"Encoder output independent of \\sigma","kind":"lemma","summary":"[Encoder output independent of \\sigma] The second component (the emitted bit list) of the encod…","labels":["SwitchingLemma2.encode_go_snd_sigma_indep"],"detail_key":"p14"},{"id":"n19435","layer":"informal","project":"p14","title":"Auxiliary entries come from input literals","kind":"lemma","summary":"[Auxiliary entries come from input literals] Every entry e in the auxiliary list produced by \\t…","labels":["SwitchingLemma2.processClauseLits_aux_entries_from_lits"],"detail_key":"p14"},{"id":"n19436","layer":"informal","project":"p14","title":"No auxiliary entry targets a non-free variable","kind":"lemma","summary":"[No auxiliary entry targets a non-free variable] Suppose every input literal pair occurs in t.\\…","labels":["SwitchingLemma2.processClauseLits_aux_ne_nonfree"],"detail_key":"p14"},{"id":"n19437","layer":"informal","project":"p14","title":"Auxiliary entries reference free variables","kind":"lemma","summary":"[Auxiliary entries reference free variables] Suppose every input literal pair occurs in t.\\text…","labels":["SwitchingLemma2.processClauseLits_aux_vars_free"],"detail_key":"p14"},{"id":"n19438","layer":"informal","project":"p14","title":"\\sigma-foldl stable off targeted variables","kind":"lemma","summary":"[\\sigma-foldl stable off targeted variables] Folding the decoder's \\sigma-update over a list of…","labels":["SwitchingLemma2.foldl_sigma_stable"],"detail_key":"p14"},{"id":"n19439","layer":"informal","project":"p14","title":"\\rho_0-foldl stable off targeted variables","kind":"lemma","summary":"[\\rho_0-foldl stable off targeted variables] Folding the decoder's \\rho_0-update over a list of…","labels":["SwitchingLemma2.foldl_rho_stable"],"detail_key":"p14"},{"id":"n19440","layer":"informal","project":"p14","title":"\\sigma-foldl preserves none","kind":"lemma","summary":"[\\sigma-foldl preserves none] If \\sigma\\,v = none, then folding the decoder's \\sigma-update ove…","labels":["SwitchingLemma2.foldl_sigma_preserves_none"],"detail_key":"p14"},{"id":"n19441","layer":"informal","project":"p14","title":"\\sigma-foldl produces none at a newly set free variable","kind":"lemma","summary":"[\\sigma-foldl produces none at a newly set free variable] Suppose every input literal pair occu…","labels":["SwitchingLemma2.processClauseLits_foldl_sigma_none"],"detail_key":"p14"},{"id":"n19442","layer":"informal","project":"p14","title":"\\rho_0-foldl agrees with processClauseLits output","kind":"lemma","summary":"[\\rho_0-foldl agrees with processClauseLits output] Suppose every input literal pair occurs in…","labels":["SwitchingLemma2.processClauseLits_foldl_rho_eq"],"detail_key":"p14"},{"id":"n19443","layer":"informal","project":"p14","title":"\\rho_0-foldl agrees when v is newly set","kind":"lemma","summary":"[\\rho_0-foldl agrees when v is newly set] A variant of the previous lemma allowing the decoder'…","labels":["SwitchingLemma2.processClauseLits_foldl_rho_eq_of_set"],"detail_key":"p14"},{"id":"n19444","layer":"informal","project":"p14","title":"No auxiliary entry targets a still-free variable","kind":"lemma","summary":"[No auxiliary entry targets a still-free variable] Contrapositive of \\rho_0-stability: if \\rho_…","labels":["SwitchingLemma2.processClauseLits_no_target_of_rho_none"],"detail_key":"p14"},{"id":"n19445","layer":"informal","project":"p14","title":"processClauseLits \\sigma output depends only on \\sigma\\,v","kind":"lemma","summary":"[processClauseLits \\sigma output depends only on \\sigma\\,v] If two starting restrictions agree…","labels":["SwitchingLemma2.processClauseLits_sigma_at_v"],"detail_key":"p14"},{"id":"n19446","layer":"informal","project":"p14","title":"\\sigma stays none when \\rho_0 stays free","kind":"lemma","summary":"[\\sigma stays none when \\rho_0 stays free] If \\rho_0\\,v = none and \\textttSwitchingLemma2.proce…","labels":["SwitchingLemma2.processClauseLits_sigma_none_of_rho_none"],"detail_key":"p14"},{"id":"n19447","layer":"informal","project":"p14","title":"Encoder \\gamma at a free variable independent of initial \\sigma","kind":"lemma","summary":"[Encoder \\gamma at a free variable independent of initial \\sigma] At a variable v that is free…","labels":["SwitchingLemma2.encode_go_fst_sigma_indep_at_free"],"detail_key":"p14"},{"id":"n19448","layer":"informal","project":"p14","title":"processEntries on processClauseLits data","kind":"lemma","summary":"[processEntries on processClauseLits data] Characterizes the decoder's \\textttprocessEntries wh…","labels":["SwitchingLemma2.processEntries_of_processClauseLits"],"detail_key":"p14"},{"id":"n19449","layer":"informal","project":"p14","title":"processClauseLits never sets \\sigma to the negating value","kind":"lemma","summary":"[processClauseLits never sets \\sigma to the negating value] If no literal in the input list sha…","labels":["SwitchingLemma2.processClauseLits_sigma_ne_neg"],"detail_key":"p14"},{"id":"n19450","layer":"informal","project":"p14","title":"Path is exhausted when a member variable stays free","kind":"lemma","summary":"[Path is exhausted when a member variable stays free] Suppose (l, idx) is in the input list, no…","labels":["SwitchingLemma2.processClauseLits_path_nil_of_rho_none_and_mem"],"detail_key":"p14"},{"id":"n19451","layer":"informal","project":"p14","title":"A member literal forces \\rho_0 to be set","kind":"lemma","summary":"[A member literal forces \\rho_0 to be set] If some input pair p has variable v and there are at…","labels":["SwitchingLemma2.processClauseLits_rho_ne_none_of_mem"],"detail_key":"p14"},{"id":"n19452","layer":"informal","project":"p14","title":"Round-trip base case","kind":"lemma","summary":"[Round-trip base case] When the encoder returns (\\sigma, []) (an empty emitted list), the decod…","labels":["SwitchingLemma2.roundtrip_base"],"detail_key":"p14"},{"id":"n19453","layer":"informal","project":"p14","title":"Encoder does not kill the first surviving clause","kind":"lemma","summary":"[Encoder does not kill the first surviving clause] Let f be a DNF whose terms contain no two di…","labels":["SwitchingLemma2.encode_go_not_kills_first_clause"],"detail_key":"p14"},{"id":"n19454","layer":"informal","project":"p14","title":"Restriction","kind":"definition","summary":"[Restriction] A restriction on n variables is a map \\rho : Fin\\,n \\to Option\\,Bool, assigning e…","labels":["SwitchingLemma2.Restriction"],"detail_key":"p14"},{"id":"n19455","layer":"informal","project":"p14","title":"Free variables of a restriction","kind":"definition","summary":"[Free variables of a restriction] The set of free variables of a restriction \\rho is the finite…","labels":["SwitchingLemma2.Restriction.freeVars"],"detail_key":"p14"},{"id":"n19456","layer":"informal","project":"p14","title":"Number of free variables","kind":"definition","summary":"[Number of free variables] The number of free variables of a restriction \\rho is the cardinalit…","labels":["SwitchingLemma2.Restriction.numFree"],"detail_key":"p14"},{"id":"n19457","layer":"informal","project":"p14","title":"Extension of a point along a restriction","kind":"definition","summary":"[Extension of a point along a restriction] Given a restriction \\rho and a point x : Fin\\,n \\to…","labels":["SwitchingLemma2.Restriction.extend"],"detail_key":"p14"},{"id":"n19458","layer":"informal","project":"p14","title":"Fixing variables in a restriction","kind":"definition","summary":"[Fixing variables in a restriction] Given a list of (variable, value) pairs, fixVars updates th…","labels":["SwitchingLemma2.Restriction.fixVars"],"detail_key":"p14"},{"id":"n19459","layer":"informal","project":"p14","title":"Un-fixing variables in a restriction","kind":"definition","summary":"[Un-fixing variables in a restriction] Given a list of (variable, value) pairs, unfixVars reset…","labels":["SwitchingLemma2.Restriction.unfixVars"],"detail_key":"p14"},{"id":"n19460","layer":"informal","project":"p14","title":"Restriction with s free variables","kind":"definition","summary":"[Restriction with s free variables] The predicate IsRestriction\\,s\\,\\rho holds when the restric…","labels":["SwitchingLemma2.IsRestriction"],"detail_key":"p14"},{"id":"n19461","layer":"informal","project":"p14","title":"Literal killed by a restriction","kind":"definition","summary":"[Literal killed by a restriction] A literal l is killed by \\rho when \\rho fixes its variable to…","labels":["SwitchingLemma2.Literal.killedBy"],"detail_key":"p14"},{"id":"n19462","layer":"informal","project":"p14","title":"Literal fixed by a restriction","kind":"definition","summary":"[Literal fixed by a restriction] A literal l is fixed by \\rho when \\rho fixes its variable to t…","labels":["SwitchingLemma2.Literal.fixedBy"],"detail_key":"p14"},{"id":"n19463","layer":"informal","project":"p14","title":"Term killed by a restriction","kind":"definition","summary":"[Term killed by a restriction] A term (conjunction of literals) is killed by \\rho when at least…","labels":["SwitchingLemma2.Term.killedBy"],"detail_key":"p14"},{"id":"n19464","layer":"informal","project":"p14","title":"Term fixed by a restriction","kind":"definition","summary":"[Term fixed by a restriction] A term is fixed by \\rho when every one of its literals is fixed b…","labels":["SwitchingLemma2.Term.fixedBy"],"detail_key":"p14"},{"id":"n19465","layer":"informal","project":"p14","title":"Alive term","kind":"definition","summary":"[Alive term] A term t is alive under \\rho when it is neither killed nor fixed by \\rho.","labels":["SwitchingLemma2.isAlive"],"detail_key":"p14"},{"id":"n19466","layer":"informal","project":"p14","title":"Restricted Boolean function","kind":"definition","summary":"[Restricted Boolean function] Given f : (Fin\\,n \\to Bool) \\to Bool and a restriction \\rho, the…","labels":["SwitchingLemma2.restrictFn"],"detail_key":"p14"},{"id":"n19467","layer":"informal","project":"p14","title":"Bad restriction","kind":"definition","summary":"[Bad restriction] A restriction \\rho is bad for f at depth d when the restricted function restr…","labels":["SwitchingLemma2.IsBadRestriction"],"detail_key":"p14"},{"id":"n19468","layer":"informal","project":"p14","title":"Number of s-restrictions","kind":"definition","summary":"[Number of s-restrictions] The number of restrictions on n variables leaving exactly s free, na…","labels":["SwitchingLemma2.numSRestrictions"],"detail_key":"p14"},{"id":"n19469","layer":"informal","project":"p14","title":"Killed literal evaluates to false","kind":"lemma","summary":"[Killed literal evaluates to false] If a literal l is killed by \\rho, then for every x it evalu…","labels":["SwitchingLemma2.Literal.killedBy_eval_false"],"detail_key":"p14"},{"id":"n19470","layer":"informal","project":"p14","title":"Fixed literal evaluates to true","kind":"lemma","summary":"[Fixed literal evaluates to true] If a literal l is fixed by \\rho, then for every x it evaluate…","labels":["SwitchingLemma2.Literal.fixedBy_eval_true"],"detail_key":"p14"},{"id":"n19471","layer":"informal","project":"p14","title":"Decision-tree depth bound from a witnessing tree","kind":"lemma","summary":"[Decision-tree depth bound from a witnessing tree] If a decision tree T of depth at most d comp…","labels":["SwitchingLemma2.dtDepth_le_of_tree"],"detail_key":"p14"},{"id":"n19472","layer":"informal","project":"p14","title":"List \\textttany false from pointwise false","kind":"lemma","summary":"[List \\textttany false from pointwise false] If a Boolean predicate p is false on every element…","labels":["SwitchingLemma2.list_any_eq_false"],"detail_key":"p14"},{"id":"n19473","layer":"informal","project":"p14","title":"List \\textttall false from a false member","kind":"lemma","summary":"[List \\textttall false from a false member] If some element a of a list l satisfies p(a) = fals…","labels":["SwitchingLemma2.list_all_eq_false_of_mem"],"detail_key":"p14"},{"id":"n19474","layer":"informal","project":"p14","title":"A fixed term forces zero decision-tree depth","kind":"lemma","summary":"[A fixed term forces zero decision-tree depth] If some term of a DNF f is fixed by \\rho, then t…","labels":["SwitchingLemma2.fixedTerm_implies_dtDepth_zero"],"detail_key":"p14"},{"id":"n19475","layer":"informal","project":"p14","title":"All terms killed forces zero decision-tree depth","kind":"lemma","summary":"[All terms killed forces zero decision-tree depth] If every term of a DNF f is killed by \\rho,…","labels":["SwitchingLemma2.killedAll_implies_dtDepth_zero"],"detail_key":"p14"},{"id":"n19476","layer":"informal","project":"p14","title":"Killing is preserved under agreement on non-free variables","kind":"lemma","summary":"[Killing is preserved under agreement on non-free variables] If a term t is killed by \\rho and…","labels":["SwitchingLemma2.killedBy_of_nonfree_agree"],"detail_key":"p14"},{"id":"n19477","layer":"informal","project":"p14","title":"First surviving clause is preserved under non-free agreement","kind":"lemma","summary":"[First surviving clause is preserved under non-free agreement] Suppose t is the first term of f…","labels":["SwitchingLemma2.first_clause_preserved"],"detail_key":"p14"},{"id":"n19478","layer":"informal","project":"p14","title":"Term length bounded by DNF width","kind":"lemma","summary":"[Term length bounded by DNF width] Every term t belonging to a DNF f has length at most the wid…","labels":["SwitchingLemma2.term_length_le_width"],"detail_key":"p14"},{"id":"n19479","layer":"informal","project":"p14","title":"\\textttzipIdx.find? projects to \\textttfind?","kind":"lemma","summary":"[\\textttzipIdx.find? projects to \\textttfind?] Searching the indexed list l.zipIdx with a predi…","labels":["SwitchingLemma2.zipIdx_find_to_find"],"detail_key":"p14"},{"id":"n19480","layer":"informal","project":"p14","title":"\\textttzipIdx membership locates a tail","kind":"lemma","summary":"[\\textttzipIdx membership locates a tail] If (l, \\mathitidx) \\in t.zipIdx, then dropping \\mathi…","labels":["SwitchingLemma2.zipIdx_drop_spec"],"detail_key":"p14"},{"id":"n19481","layer":"informal","project":"p14","title":"Filtered \\textttzipIdx index is below the length bound","kind":"lemma","summary":"[Filtered \\textttzipIdx index is below the length bound] If (l, \\mathitidx) arises from filteri…","labels":["SwitchingLemma2.zipIdx_filter_idx_lt"],"detail_key":"p14"},{"id":"n19482","layer":"informal","project":"p14","title":"Free variable preserved by clause processing","kind":"lemma","summary":"[Free variable preserved by clause processing] If the restriction component (processClauseLits\\…","labels":["SwitchingLemma2.pcl_none_implies_rho_free"],"detail_key":"p14"},{"id":"n19483","layer":"informal","project":"p14","title":"First non-killed clause preserved under encoding","kind":"lemma","summary":"[First non-killed clause preserved under encoding] For a DNF f with no two literals in a term s…","labels":["SwitchingLemma2.find_clause_preserved_in_encode"],"detail_key":"p14"},{"id":"n19484","layer":"informal","project":"p14","title":"Filtered free-variable literals stay in the index list","kind":"lemma","summary":"[Filtered free-variable literals stay in the index list] If a pair p lies in the list obtained…","labels":["SwitchingLemma2.mem_filter_freeVars_zipIdx"],"detail_key":"p14"},{"id":"n19485","layer":"informal","project":"p14","title":"No aux entry targets a still-free variable","kind":"lemma","summary":"[No aux entry targets a still-free variable] Assuming every literal in \\mathitlits is an indexe…","labels":["SwitchingLemma2.processClauseLits_aux_ne_of_pcl_none"],"detail_key":"p14"},{"id":"n19486","layer":"informal","project":"p14","title":"Encoder first component agrees with recursive step","kind":"lemma","summary":"[Encoder first component agrees with recursive step] When the encoder finds t_clause as the fir…","labels":["SwitchingLemma2.encode_go_fst_eq_rec"],"detail_key":"p14"},{"id":"n19487","layer":"informal","project":"p14","title":"Round-trip invariant for \\sigma","kind":"lemma","summary":"[Round-trip invariant for \\sigma] Under the freeness and membership hypotheses on \\mathitlits a…","labels":["SwitchingLemma2.roundtrip_inv_hC'"],"detail_key":"p14"},{"id":"n19488","layer":"informal","project":"p14","title":"Round-trip invariant for \\rho_0","kind":"lemma","summary":"[Round-trip invariant for \\rho_0] Under the freeness and membership hypotheses on \\mathitlits a…","labels":["SwitchingLemma2.roundtrip_inv_hD'"],"detail_key":"p14"},{"id":"n19489","layer":"informal","project":"p14","title":"Generalized go-level round-trip","kind":"lemma","summary":"[Generalized go-level round-trip] The generalized round-trip statement: for a DNF f of width at…","labels":["SwitchingLemma2.go_roundtrip_gen"],"detail_key":"p14"},{"id":"n19490","layer":"informal","project":"p14","title":"Go-level round-trip","kind":"lemma","summary":"[Go-level round-trip] Specializing the generalized round trip to \\sigma = \\rho_0 = \\rho: for a…","labels":["SwitchingLemma2.go_roundtrip"],"detail_key":"p14"},{"id":"n19491","layer":"informal","project":"p14","title":"Decode of encode recovers the restriction","kind":"lemma","summary":"[Decode of encode recovers the restriction] For a width-w DNF f with distinct variables per ter…","labels":["SwitchingLemma2.razborovDecode_encode"],"detail_key":"p14"},{"id":"n19492","layer":"informal","project":"p14","title":"Injectivity of the Razborov encoding","kind":"theorem","summary":"[Injectivity of the Razborov encoding] The Razborov encoding is injective on bad restrictions:…","labels":["SwitchingLemma2.razborovEncode_injective"],"detail_key":"p14"},{"id":"n19493","layer":"informal","project":"p14","title":"Deterministic communication protocol","kind":"definition","summary":"[Deterministic communication protocol] A deterministic two-party communication protocol over in…","labels":["CommunicationComplexity.Deterministic.Protocol"],"detail_key":"p14"},{"id":"n19494","layer":"informal","project":"p14","title":"Protocol execution","kind":"definition","summary":"[Protocol execution] Given a protocol p, \\textttrun evaluates p on inputs x : X and y : Y by re…","labels":["CommunicationComplexity.Deterministic.Protocol.run"],"detail_key":"p14"},{"id":"n19495","layer":"informal","project":"p14","title":"Communication complexity","kind":"definition","summary":"[Communication complexity] The communication complexity of a protocol p is the worst-case total…","labels":["CommunicationComplexity.Deterministic.Protocol.complexity"],"detail_key":"p14"},{"id":"n19496","layer":"informal","project":"p14","title":"Protocol equivalence","kind":"definition","summary":"[Protocol equivalence] Two protocols p and q over the same input and output types are \\emphequi…","labels":["CommunicationComplexity.Deterministic.Protocol.Equiv"],"detail_key":"p14"},{"id":"n19497","layer":"informal","project":"p14","title":"Protocol computes a function","kind":"definition","summary":"[Protocol computes a function] A protocol p \\emphcomputes a two-argument function f : X \\to Y \\…","labels":["CommunicationComplexity.Deterministic.Protocol.Computes"],"detail_key":"p14"},{"id":"n19498","layer":"informal","project":"p14","title":"Role swap","kind":"definition","summary":"[Role swap] Given a protocol p over X \\times Y, \\textttswap produces a protocol over Y \\times X…","labels":["CommunicationComplexity.Deterministic.Protocol.swap"],"detail_key":"p14"},{"id":"n19499","layer":"informal","project":"p14","title":"Swap preserves run","kind":"theorem","summary":"[Swap preserves run] For any protocol p over X \\times Y and any inputs x : X, y : Y, \\[ p.\\text…","labels":["CommunicationComplexity.Deterministic.Protocol.swap_run"],"detail_key":"p14"},{"id":"n19500","layer":"informal","project":"p14","title":"Swap preserves complexity","kind":"theorem","summary":"[Swap preserves complexity] For any protocol p over X \\times Y, \\[ p.\\textttswap.\\textttcomplex…","labels":["CommunicationComplexity.Deterministic.Protocol.swap_complexity"],"detail_key":"p14"},{"id":"n19501","layer":"informal","project":"p14","title":"Swap is an involution","kind":"theorem","summary":"[Swap is an involution] Swapping Alice and Bob twice recovers the original protocol: for any p…","labels":["CommunicationComplexity.Deterministic.Protocol.swap_swap"],"detail_key":"p14"},{"id":"n19502","layer":"informal","project":"p14","title":"Alice node to Bob node","kind":"theorem","summary":"[Alice node to Bob node] For any alice-rooted protocol \\textttalice\\;f\\;P over X \\times Y, ther…","labels":["CommunicationComplexity.Deterministic.Protocol.alice_to_bob"],"detail_key":"p14"},{"id":"n19503","layer":"informal","project":"p14","title":"Bob node to Alice node","kind":"theorem","summary":"[Bob node to Alice node] For any bob-rooted protocol \\textttbob\\;g\\;P over X \\times Y, there ex…","labels":["CommunicationComplexity.Deterministic.Protocol.bob_to_alice"],"detail_key":"p14"},{"id":"n19504","layer":"informal","project":"p14","title":"Input pull-back","kind":"definition","summary":"[Input pull-back] Given functions f_X : X' \\to X and f_Y : Y' \\to Y, \\textttcomap pulls back a…","labels":["CommunicationComplexity.Deterministic.Protocol.comap"],"detail_key":"p14"},{"id":"n19505","layer":"informal","project":"p14","title":"Comap run","kind":"theorem","summary":"[Comap run] For any protocol p over X \\times Y, input maps f_X : X' \\to X and f_Y : Y' \\to Y, a…","labels":["CommunicationComplexity.Deterministic.Protocol.comap_run"],"detail_key":"p14"},{"id":"n19506","layer":"informal","project":"p14","title":"Comap preserves complexity","kind":"theorem","summary":"[Comap preserves complexity] For any protocol p over X \\times Y and input maps f_X : X' \\to X,…","labels":["CommunicationComplexity.Deterministic.Protocol.comap_complexity"],"detail_key":"p14"},{"id":"n19507","layer":"informal","project":"p14","title":"Infimum over \\textttENat-valued family bounded by coercion","kind":"theorem","summary":"[Infimum over \\textttENat-valued family bounded by coercion] Let \\iota be a type and f : \\iota…","labels":["CommunicationComplexity.Internal.enat_iInf_le_coe_iff"],"detail_key":"p14"},{"id":"n19508","layer":"informal","project":"p14","title":"Deterministic communication complexity","kind":"definition","summary":"[Deterministic communication complexity] The deterministic communication complexity of f : X \\t…","labels":["CommunicationComplexity.Deterministic.communicationComplexity"],"detail_key":"p14"},{"id":"n19509","layer":"informal","project":"p14","title":"Upper bound characterization of communication complexity","kind":"theorem","summary":"[Upper bound characterization of communication complexity] For f : X \\to Y \\to \\alpha and n : N…","labels":["CommunicationComplexity.Deterministic.communicationComplexity_le_iff"],"detail_key":"p14"},{"id":"n19510","layer":"informal","project":"p14","title":"Finite-message upper bound characterization","kind":"theorem","summary":"[Finite-message upper bound characterization] For f : X \\to Y \\to \\alpha and n : N, we have D(f…","labels":["CommunicationComplexity.Deterministic.communicationComplexity_le_iff_finiteMessage"],"detail_key":"p14"},{"id":"n19511","layer":"informal","project":"p14","title":"Lower bound characterization of communication complexity","kind":"theorem","summary":"[Lower bound characterization of communication complexity] For f : X \\to Y \\to \\alpha and n : N…","labels":["CommunicationComplexity.Deterministic.le_communicationComplexity_iff"],"detail_key":"p14"},{"id":"n19512","layer":"informal","project":"p14","title":"Map over protocol output","kind":"definition","summary":"[Map over protocol output] Given a function g : \\alpha \\to \\beta and a protocol p : Protocol\\,X…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.map"],"detail_key":"p14"},{"id":"n19513","layer":"informal","project":"p14","title":"Map commutes with run","kind":"theorem","summary":"[Map commutes with run] For any function g : \\alpha \\to \\beta, protocol p, and inputs x \\in X,…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.map_run"],"detail_key":"p14"},{"id":"n19514","layer":"informal","project":"p14","title":"Map preserves complexity","kind":"theorem","summary":"[Map preserves complexity] For any function g : \\alpha \\to \\beta and protocol p, (\\textttmap\\;g…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.map_complexity"],"detail_key":"p14"},{"id":"n19515","layer":"informal","project":"p14","title":"Monadic bind of protocols","kind":"definition","summary":"[Monadic bind of protocols] Given a protocol p : Protocol\\,X\\,Y\\,\\alpha and a continuation q :…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.bind"],"detail_key":"p14"},{"id":"n19516","layer":"informal","project":"p14","title":"Bind commutes with run","kind":"theorem","summary":"[Bind commutes with run] For any protocol p, continuation q, and inputs x \\in X, y \\in Y, \\[ (\\…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.bind_run"],"detail_key":"p14"},{"id":"n19517","layer":"informal","project":"p14","title":"Supremum distributes over adding a constant","kind":"theorem","summary":"[Supremum distributes over adding a constant] For any finite nonempty type \\iota, function f :…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.finset_sup_add_const"],"detail_key":"p14"},{"id":"n19518","layer":"informal","project":"p14","title":"Bind complexity when continuation is constant","kind":"theorem","summary":"[Bind complexity when continuation is constant] If q\\,a has the same complexity c for every out…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.bind_complexity_const"],"detail_key":"p14"},{"id":"n19519","layer":"informal","project":"p14","title":"Product of two protocols","kind":"definition","summary":"[Product of two protocols] Given p_1 : Protocol\\,X_1\\,Y_1\\,\\alpha_1 and p_2 : Protocol\\,X_2\\,Y_…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.prod"],"detail_key":"p14"},{"id":"n19520","layer":"informal","project":"p14","title":"Product run","kind":"theorem","summary":"[Product run] For protocols p_1, p_2 and inputs x = (x_1, x_2), y = (y_1, y_2), \\[ (p_1.\\texttt…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.prod_run"],"detail_key":"p14"},{"id":"n19521","layer":"informal","project":"p14","title":"Product complexity is additive","kind":"theorem","summary":"[Product complexity is additive] The complexity of the product protocol satisfies \\[ (p_1.\\text…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.prod_complexity"],"detail_key":"p14"},{"id":"n19522","layer":"informal","project":"p14","title":"k-fold product of protocols","kind":"definition","summary":"[k-fold product of protocols] Given a family of protocols p_i : Protocol\\,X_i\\,Y_i\\,\\alpha_i fo…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.pi"],"detail_key":"p14"},{"id":"n19523","layer":"informal","project":"p14","title":"k-fold product run","kind":"theorem","summary":"[k-fold product run] For a family p and input tuples x, y, \\[ (\\textttpi\\;p).\\textttrun\\;x\\;y \\…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.pi_run"],"detail_key":"p14"},{"id":"n19524","layer":"informal","project":"p14","title":"k-fold product complexity is sum","kind":"theorem","summary":"[k-fold product complexity is sum] The complexity of the k-fold product satisfies \\[ (\\textttpi…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.pi_complexity"],"detail_key":"p14"},{"id":"n19525","layer":"informal","project":"p14","title":"Swap of a protocol input set","kind":"definition","summary":"[Swap of a protocol input set] Given a set R \\subseteq X \\times Y, \\textttswapInputSet(R) is th…","labels":["CommunicationComplexity.Deterministic.Protocol.swapInputSet"],"detail_key":"p14"},{"id":"n19526","layer":"informal","project":"p14","title":"Membership in swapped set","kind":"theorem","summary":"[Membership in swapped set] For any set R \\subseteq X \\times Y and pair (y, x) \\in Y \\times X,…","labels":["CommunicationComplexity.Deterministic.Protocol.mem_swapInputSet"],"detail_key":"p14"},{"id":"n19527","layer":"informal","project":"p14","title":"Swap is an involution","kind":"theorem","summary":"[Swap is an involution] For any set R \\subseteq X \\times Y, applying \\textttswapInputSet twice…","labels":["CommunicationComplexity.Deterministic.Protocol.swapInputSet_swapInputSet"],"detail_key":"p14"},{"id":"n19528","layer":"informal","project":"p14","title":"Preimage of a protocol input set","kind":"definition","summary":"[Preimage of a protocol input set] Given maps f_X : X' \\to X and f_Y : Y' \\to Y and a set R \\su…","labels":["CommunicationComplexity.Deterministic.Protocol.preimageInputSet"],"detail_key":"p14"},{"id":"n19529","layer":"informal","project":"p14","title":"Membership in preimage input set","kind":"theorem","summary":"[Membership in preimage input set] For maps f_X : X' \\to X, f_Y : Y' \\to Y, a set R \\subseteq X…","labels":["CommunicationComplexity.Deterministic.Protocol.mem_preimageInputSet"],"detail_key":"p14"},{"id":"n19530","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles relative to a constraint","kind":"definition","summary":"[Auxiliary leaf rectangles relative to a constraint] For a protocol p and sets A \\subseteq X, B…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectanglesAux"],"detail_key":"p14"},{"id":"n19531","layer":"informal","project":"p14","title":"Protocol leaf rectangles","kind":"definition","summary":"[Protocol leaf rectangles] The \\emphleaf rectangles of a protocol p : \\textttProtocol\\,X\\,Y\\,\\a…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles"],"detail_key":"p14"},{"id":"n19532","layer":"informal","project":"p14","title":"Swap sends leaf rectangles to leaf rectangles (auxiliary)","kind":"theorem","summary":"[Swap sends leaf rectangles to leaf rectangles (auxiliary)] If R \\in \\textttleafRectanglesAux(p…","labels":["CommunicationComplexity.Deterministic.Protocol.swapInputSet_mem_leafRectanglesAux_swap"],"detail_key":"p14"},{"id":"n19533","layer":"informal","project":"p14","title":"Swap sends leaf rectangles to leaf rectangles","kind":"theorem","summary":"[Swap sends leaf rectangles to leaf rectangles] If R is a leaf rectangle of p, then \\textttswap…","labels":["CommunicationComplexity.Deterministic.Protocol.swapInputSet_mem_leafRectangles_swap"],"detail_key":"p14"},{"id":"n19534","layer":"informal","project":"p14","title":"Preimage sends leaf rectangles to leaf rectangles (auxiliary)","kind":"theorem","summary":"[Preimage sends leaf rectangles to leaf rectangles (auxiliary)] If R \\in \\textttleafRectanglesA…","labels":["CommunicationComplexity.Deterministic.Protocol.preimageInputSet_mem_leafRectanglesAux_comap"],"detail_key":"p14"},{"id":"n19535","layer":"informal","project":"p14","title":"Preimage sends leaf rectangles to leaf rectangles","kind":"theorem","summary":"[Preimage sends leaf rectangles to leaf rectangles] If R is a leaf rectangle of p and f_X : X'…","labels":["CommunicationComplexity.Deterministic.Protocol.preimageInputSet_mem_leafRectangles_comap"],"detail_key":"p14"},{"id":"n19536","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles are rectangles","kind":"lemma","summary":"[Auxiliary leaf rectangles are rectangles] Every element of \\textttleafRectanglesAux(p, A, B) i…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_isRectangle"],"detail_key":"p14"},{"id":"n19537","layer":"informal","project":"p14","title":"Every protocol leaf rectangle is a rectangle","kind":"lemma","summary":"[Every protocol leaf rectangle is a rectangle] For any protocol p and any R \\in \\textttleafRect…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_isRectangle"],"detail_key":"p14"},{"id":"n19538","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles are subsets of the constraint","kind":"lemma","summary":"[Auxiliary leaf rectangles are subsets of the constraint] Every element R of \\textttleafRectang…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_subset"],"detail_key":"p14"},{"id":"n19539","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles cover the constraint","kind":"lemma","summary":"[Auxiliary leaf rectangles cover the constraint] The union of \\textttleafRectanglesAux(p, A, B)…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_cover"],"detail_key":"p14"},{"id":"n19540","layer":"informal","project":"p14","title":"Distinct auxiliary leaf rectangles are disjoint","kind":"lemma","summary":"[Distinct auxiliary leaf rectangles are disjoint] If R, S \\in \\textttleafRectanglesAux(p, A, B)…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_disjoint"],"detail_key":"p14"},{"id":"n19541","layer":"informal","project":"p14","title":"Leaf rectangles cover the input space","kind":"lemma","summary":"[Leaf rectangles cover the input space] For any protocol p, the union of all leaf rectangles eq…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_cover"],"detail_key":"p14"},{"id":"n19542","layer":"informal","project":"p14","title":"Distinct leaf rectangles are disjoint","kind":"lemma","summary":"[Distinct leaf rectangles are disjoint] For any protocol p, if R, S \\in \\textttleafRectangles(p…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_disjoint"],"detail_key":"p14"},{"id":"n19543","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles are monochromatic (auxiliary)","kind":"lemma","summary":"[Auxiliary leaf rectangles are monochromatic (auxiliary)] If R \\in \\textttleafRectanglesAux(p,…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_mono"],"detail_key":"p14"},{"id":"n19544","layer":"informal","project":"p14","title":"Leaf rectangles are monochromatic","kind":"lemma","summary":"[Leaf rectangles are monochromatic] If p computes g : X \\to Y \\to \\alpha and R \\in \\textttleafR…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_mono"],"detail_key":"p14"},{"id":"n19545","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles are bounded in count","kind":"lemma","summary":"[Auxiliary leaf rectangles are bounded in count] For any protocol p and constraint sets A, B, t…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_card"],"detail_key":"p14"},{"id":"n19546","layer":"informal","project":"p14","title":"Auxiliary leaf rectangles are a finite collection","kind":"lemma","summary":"[Auxiliary leaf rectangles are a finite collection] For any protocol p and constraint sets A, B…","labels":["CommunicationComplexity.Deterministic.Protocol.aux_finite"],"detail_key":"p14"},{"id":"n19547","layer":"informal","project":"p14","title":"Number of leaf rectangles is at most 2^c","kind":"lemma","summary":"[Number of leaf rectangles is at most 2^c] For any protocol p with communication complexity c =…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_card"],"detail_key":"p14"},{"id":"n19548","layer":"informal","project":"p14","title":"Leaf rectangles are finite","kind":"lemma","summary":"[Leaf rectangles are finite] For any protocol p, the set \\textttleafRectangles(p) is finite.","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_finite"],"detail_key":"p14"},{"id":"n19549","layer":"informal","project":"p14","title":"Leaf rectangles form a monochromatic partition","kind":"theorem","summary":"[Leaf rectangles form a monochromatic partition] If p computes g : X \\to Y \\to \\alpha, then the…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectangles_isMonoPartition"],"detail_key":"p14"},{"id":"n19550","layer":"informal","project":"p14","title":"Rectangle partition theorem (Theorem 1.6)","kind":"theorem","summary":"[Rectangle partition theorem (Theorem 1.6)] If p computes g : X \\to Y \\to \\alpha with communica…","labels":["CommunicationComplexity.Deterministic.Protocol.rectangle_partition"],"detail_key":"p14"},{"id":"n19551","layer":"informal","project":"p14","title":"Input pairs reaching a subprotocol path form a rectangle","kind":"theorem","summary":"[Input pairs reaching a subprotocol path form a rectangle] Given a subprotocol path \\texttthsp…","labels":["CommunicationComplexity.Deterministic.Protocol.reachesPath_isRectangle"],"detail_key":"p14"},{"id":"n19552","layer":"informal","project":"p14","title":"Input pairs reaching a subprotocol witness form a rectangle","kind":"theorem","summary":"[Input pairs reaching a subprotocol witness form a rectangle] Given a subprotocol witness \\text…","labels":["CommunicationComplexity.Deterministic.Protocol.reaches_isRectangle"],"detail_key":"p14"},{"id":"n19553","layer":"informal","project":"p14","title":"Low complexity implies monochromatic partition with few parts","kind":"theorem","summary":"[Low complexity implies monochromatic partition with few parts] If CC(g) \\leq n, then there exi…","labels":["CommunicationComplexity.Deterministic.mono_partition_of_communicationComplexity_le"],"detail_key":"p14"},{"id":"n19554","layer":"informal","project":"p14","title":"Rectangle partition lower bound","kind":"theorem","summary":"[Rectangle partition lower bound] To prove CC(g) \\geq n + 1, it suffices to show that every mon…","labels":["CommunicationComplexity.Deterministic.le_communicationComplexity_of_forall_lt_ncard"],"detail_key":"p14"},{"id":"n19555","layer":"informal","project":"p14","title":"Fooling set size bounded by 2^n when CC(g) \\leq n","kind":"theorem","summary":"[Fooling set size bounded by 2^n when CC(g) \\leq n] If CC(g) \\leq n and S \\subseteq X \\times Y…","labels":["CommunicationComplexity.Deterministic.foolingSet_ncard_le_pow_of_communicationComplexity_le"],"detail_key":"p14"},{"id":"n19556","layer":"informal","project":"p14","title":"Fooling-set lower bound","kind":"theorem","summary":"[Fooling-set lower bound] For any fooling set S for g, the deterministic communication complexi…","labels":["CommunicationComplexity.Deterministic.clog_ncard_le_communicationComplexity"],"detail_key":"p14"},{"id":"n19557","layer":"informal","project":"p14","title":"Finite-message protocol","kind":"definition","summary":"[Finite-message protocol] A generalized deterministic two-party communication protocol over inp…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol"],"detail_key":"p14"},{"id":"n19558","layer":"informal","project":"p14","title":"Protocol execution","kind":"definition","summary":"[Protocol execution] Given a finite-message protocol p and inputs x : X, y : Y, \\textttrun exec…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.run"],"detail_key":"p14"},{"id":"n19559","layer":"informal","project":"p14","title":"Communication complexity of a finite-message protocol","kind":"definition","summary":"[Communication complexity of a finite-message protocol] The worst-case communication cost of a…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.complexity"],"detail_key":"p14"},{"id":"n19560","layer":"informal","project":"p14","title":"Complete binary tree for Alice queries","kind":"definition","summary":"[Complete binary tree for Alice queries] An auxiliary construction: given d binary query functi…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.completeTreeAlice"],"detail_key":"p14"},{"id":"n19561","layer":"informal","project":"p14","title":"Run of complete Alice tree","kind":"theorem","summary":"[Run of complete Alice tree] For all inputs x : X and y : Y, executing \\textttcompleteTreeAlice…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.completeTreeAlice_run"],"detail_key":"p14"},{"id":"n19562","layer":"informal","project":"p14","title":"Complexity of complete Alice tree","kind":"theorem","summary":"[Complexity of complete Alice tree] The complexity of \\textttcompleteTreeAlice equals d plus th…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.completeTreeAlice_complexity"],"detail_key":"p14"},{"id":"n19563","layer":"informal","project":"p14","title":"Binary encoding of Alice's finite-type message","kind":"theorem","summary":"[Binary encoding of Alice's finite-type message] Given a finite nonempty type \\beta, a function…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.encode_alice"],"detail_key":"p14"},{"id":"n19564","layer":"informal","project":"p14","title":"Existence of binary simulation","kind":"theorem","summary":"[Existence of binary simulation] For every finite-message protocol p, there exists a binary pro…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toProtocol_exists"],"detail_key":"p14"},{"id":"n19565","layer":"informal","project":"p14","title":"Conversion to binary protocol","kind":"definition","summary":"[Conversion to binary protocol] A noncomputable function that converts a finite-message protoco…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toProtocol"],"detail_key":"p14"},{"id":"n19566","layer":"informal","project":"p14","title":"Conversion preserves run behavior","kind":"theorem","summary":"[Conversion preserves run behavior] For every finite-message protocol p, the converted binary p…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toProtocol_run"],"detail_key":"p14"},{"id":"n19567","layer":"informal","project":"p14","title":"Conversion preserves complexity","kind":"theorem","summary":"[Conversion preserves complexity] For every finite-message protocol p, the converted binary pro…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toProtocol_complexity"],"detail_key":"p14"},{"id":"n19568","layer":"informal","project":"p14","title":"Embedding binary protocols as finite-message protocols","kind":"definition","summary":"[Embedding binary protocols as finite-message protocols] Embeds a binary protocol into the gene…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.ofProtocol"],"detail_key":"p14"},{"id":"n19569","layer":"informal","project":"p14","title":"Embedding preserves run behavior","kind":"theorem","summary":"[Embedding preserves run behavior] For every binary protocol p and inputs x : X, y : Y, (\\textt…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.ofProtocol_run"],"detail_key":"p14"},{"id":"n19570","layer":"informal","project":"p14","title":"Embedding preserves complexity","kind":"theorem","summary":"[Embedding preserves complexity] For every binary protocol p, (\\textttofProtocol\\,p).complexity…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.ofProtocol_complexity"],"detail_key":"p14"},{"id":"n19571","layer":"informal","project":"p14","title":"Every binary protocol embeds into the generalized model","kind":"theorem","summary":"[Every binary protocol embeds into the generalized model] For every binary protocol p, there ex…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.ofProtocol_equiv"],"detail_key":"p14"},{"id":"n19572","layer":"informal","project":"p14","title":"Pullback along input maps","kind":"definition","summary":"[Pullback along input maps] Given maps f_X : X' \\to X and f_Y : Y' \\to Y, the pullback p.comap\\…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.comap"],"detail_key":"p14"},{"id":"n19573","layer":"informal","project":"p14","title":"Pullback run behavior","kind":"theorem","summary":"[Pullback run behavior] For all x' : X' and y' : Y', (p.comap\\,f_X\\,f_Y).run\\,x'\\,y' = p.run\\,(…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.comap_run"],"detail_key":"p14"},{"id":"n19574","layer":"informal","project":"p14","title":"Pullback preserves complexity","kind":"theorem","summary":"[Pullback preserves complexity] The pullback does not change the communication complexity: (p.c…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.comap_complexity"],"detail_key":"p14"},{"id":"n19575","layer":"informal","project":"p14","title":"Set disjointness function","kind":"definition","summary":"[Set disjointness function] For n \\in N and subsets X, Y \\subseteq [n], disjointness(n, X, Y) i…","labels":["CommunicationComplexity.Functions.Disjointness.disjointness"],"detail_key":"p14"},{"id":"n19576","layer":"informal","project":"p14","title":"Commutativity of disjointness","kind":"theorem","summary":"[Commutativity of disjointness] The disjointness function is symmetric in its two set arguments…","labels":["CommunicationComplexity.Functions.Disjointness.disjointness_comm"],"detail_key":"p14"},{"id":"n19577","layer":"informal","project":"p14","title":"Fooling set for disjointness","kind":"definition","summary":"[Fooling set for disjointness] The candidate fooling set for disjointness is the collection of…","labels":["CommunicationComplexity.Functions.Disjointness.foolingSet"],"detail_key":"p14"},{"id":"n19578","layer":"informal","project":"p14","title":"Fooling set validity","kind":"theorem","summary":"[Fooling set validity] The collection \\(X, X^c) \\mid X \\subseteq [n]\\ is a fooling set for disj…","labels":["CommunicationComplexity.Functions.Disjointness.foolingSet_isFoolingSet"],"detail_key":"p14"},{"id":"n19579","layer":"informal","project":"p14","title":"Fooling set cardinality","kind":"theorem","summary":"[Fooling set cardinality] The fooling set foolingSet(n) has cardinality 2^n, since it is in bij…","labels":["CommunicationComplexity.Functions.Disjointness.foolingSet_card"],"detail_key":"p14"},{"id":"n19580","layer":"informal","project":"p14","title":"Upper bound on complexity of disjointness","kind":"theorem","summary":"[Upper bound on complexity of disjointness] The deterministic communication complexity of disjo…","labels":["CommunicationComplexity.Functions.Disjointness.communicationComplexity_le"],"detail_key":"p14"},{"id":"n19581","layer":"informal","project":"p14","title":"Lower bound on complexity of disjointness","kind":"theorem","summary":"[Lower bound on complexity of disjointness] For n \\ge 1, the deterministic communication comple…","labels":["CommunicationComplexity.Functions.Disjointness.le_communicationComplexity"],"detail_key":"p14"},{"id":"n19582","layer":"informal","project":"p14","title":"Exact complexity of disjointness","kind":"theorem","summary":"[Exact complexity of disjointness] For n \\ge 1, the deterministic communication complexity of t…","labels":["CommunicationComplexity.Functions.Disjointness.communicationComplexity_eq"],"detail_key":"p14"},{"id":"n19583","layer":"informal","project":"p14","title":"n-bit Boolean input type","kind":"definition","summary":"[n-bit Boolean input type] \\textttBoolInput(n) is the type of n-bit Boolean inputs, defined as…","labels":["CommunicationComplexity.BoolInput"],"detail_key":"p14"},{"id":"n19584","layer":"informal","project":"p14","title":"All-zero Boolean input","kind":"definition","summary":"[All-zero Boolean input] \\textttzeroInput(n) : \\textttBoolInput(n) is the constant false functi…","labels":["CommunicationComplexity.zeroInput"],"detail_key":"p14"},{"id":"n19585","layer":"informal","project":"p14","title":"Nonzero input has a true coordinate","kind":"lemma","summary":"[Nonzero input has a true coordinate] If x : \\textttBoolInput(n) satisfies x \\ne \\textttzeroInp…","labels":["CommunicationComplexity.exists_true_of_ne_zeroInput"],"detail_key":"p14"},{"id":"n19586","layer":"informal","project":"p14","title":"Boolean sign","kind":"definition","summary":"[Boolean sign] \\textttboolSign : Bool \\to R assigns the real value 1 to false and -1 to true, e…","labels":["CommunicationComplexity.boolSign"],"detail_key":"p14"},{"id":"n19587","layer":"informal","project":"p14","title":"Boolean sign converts xor to multiplication","kind":"lemma","summary":"[Boolean sign converts xor to multiplication] For all a, b : Bool, \\[ \\textttCommunicationCompl…","labels":["CommunicationComplexity.boolSign_xor"],"detail_key":"p14"},{"id":"n19588","layer":"informal","project":"p14","title":"Boolean sign of a finite sum","kind":"lemma","summary":"[Boolean sign of a finite sum] For a finite type \\alpha, a finite set s : Finset\\,\\alpha, and a…","labels":["CommunicationComplexity.boolSign_sum"],"detail_key":"p14"},{"id":"n19589","layer":"informal","project":"p14","title":"Product of two Boolean signs","kind":"lemma","summary":"[Product of two Boolean signs] For all a, b : Bool, \\[ \\textttCommunicationComplexity.boolSign(…","labels":["CommunicationComplexity.boolSign_mul_boolSign_eq_sub_two_indicator"],"detail_key":"p14"},{"id":"n19590","layer":"informal","project":"p14","title":"Single-coordinate flip","kind":"definition","summary":"[Single-coordinate flip] Given i : Fin\\,n and x : \\textttBoolInput(n), \\textttflipAt\\,i\\,x is t…","labels":["CommunicationComplexity.flipAt"],"detail_key":"p14"},{"id":"n19591","layer":"informal","project":"p14","title":"Flip applies at the flipped coordinate","kind":"lemma","summary":"[Flip applies at the flipped coordinate] For i : Fin\\,n and x : \\textttBoolInput(n), (\\textttfl…","labels":["CommunicationComplexity.flipAt_apply_same"],"detail_key":"p14"},{"id":"n19592","layer":"informal","project":"p14","title":"Flip leaves other coordinates unchanged","kind":"lemma","summary":"[Flip leaves other coordinates unchanged] For i, j : Fin\\,n with j \\ne i and x : \\textttBoolInp…","labels":["CommunicationComplexity.flipAt_apply_ne"],"detail_key":"p14"},{"id":"n19593","layer":"informal","project":"p14","title":"Flip is an involution","kind":"lemma","summary":"[Flip is an involution] For i : Fin\\,n and x : \\textttBoolInput(n), \\textttflipAt\\,i\\,(\\textttf…","labels":["CommunicationComplexity.flipAt_flipAt"],"detail_key":"p14"},{"id":"n19594","layer":"informal","project":"p14","title":"Flip is bijective","kind":"lemma","summary":"[Flip is bijective] For every i : Fin\\,n, the map \\textttflipAt\\,i : \\textttBoolInput(n) \\to \\t…","labels":["CommunicationComplexity.flipAt_bijective"],"detail_key":"p14"},{"id":"n19595","layer":"informal","project":"p14","title":"One-way deterministic protocol","kind":"definition","summary":"[One-way deterministic protocol] A one-way deterministic communication protocol for inputs x \\i…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol"],"detail_key":"p14"},{"id":"n19596","layer":"informal","project":"p14","title":"Protocol execution","kind":"definition","summary":"[Protocol execution] Given a one-way protocol p and inputs x \\in X, y \\in Y, executing the prot…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.run"],"detail_key":"p14"},{"id":"n19597","layer":"informal","project":"p14","title":"Protocol bit cost","kind":"definition","summary":"[Protocol bit cost] The communication cost of a one-way protocol p is \\lceil \\log_2 |\\mathttMes…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.cost"],"detail_key":"p14"},{"id":"n19598","layer":"informal","project":"p14","title":"Protocol computes a function","kind":"definition","summary":"[Protocol computes a function] A one-way protocol p \\emphcomputes a function f : X \\to Y \\to \\a…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.Computes"],"detail_key":"p14"},{"id":"n19599","layer":"informal","project":"p14","title":"Embedding into the finite-message model","kind":"definition","summary":"[Embedding into the finite-message model] Given a one-way protocol p with finite output type \\a…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.toFiniteMessage"],"detail_key":"p14"},{"id":"n19600","layer":"informal","project":"p14","title":"Embedding preserves execution","kind":"theorem","summary":"[Embedding preserves execution] For any one-way protocol p and inputs (x, y), running p's embed…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.toFiniteMessage_run"],"detail_key":"p14"},{"id":"n19601","layer":"informal","project":"p14","title":"Embedding complexity formula","kind":"theorem","summary":"[Embedding complexity formula] The interactive complexity of the embedding equals the one-way c…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.toFiniteMessage_complexity"],"detail_key":"p14"},{"id":"n19602","layer":"informal","project":"p14","title":"One-way communication complexity","kind":"definition","summary":"[One-way communication complexity] The one-way deterministic communication complexity of f : X…","labels":["CommunicationComplexity.Deterministic.OneWay.communicationComplexity"],"detail_key":"p14"},{"id":"n19603","layer":"informal","project":"p14","title":"Upper-bound characterization","kind":"theorem","summary":"[Upper-bound characterization] For a function f and n \\in N, the one-way complexity of f is at…","labels":["CommunicationComplexity.Deterministic.OneWay.communicationComplexity_le_iff"],"detail_key":"p14"},{"id":"n19604","layer":"informal","project":"p14","title":"Lower-bound characterization","kind":"theorem","summary":"[Lower-bound characterization] For a function f and k \\in N, k is at most the one-way complexit…","labels":["CommunicationComplexity.Deterministic.OneWay.le_communicationComplexity_iff"],"detail_key":"p14"},{"id":"n19605","layer":"informal","project":"p14","title":"One-way bound implies deterministic bound","kind":"theorem","summary":"[One-way bound implies deterministic bound] If the one-way communication complexity of f : X \\t…","labels":["CommunicationComplexity.Deterministic.OneWay.deterministic_communicationComplexity_le_of_oneWay_le"],"detail_key":"p14"},{"id":"n19606","layer":"informal","project":"p14","title":"One-way bound implies deterministic bound for Boolean functions","kind":"theorem","summary":"[One-way bound implies deterministic bound for Boolean functions] For a Boolean function f : X…","labels":["CommunicationComplexity.Deterministic.OneWay.deterministic_communicationComplexity_le_of_oneWay_le_bool"],"detail_key":"p14"},{"id":"n19607","layer":"informal","project":"p14","title":"Rectangle","kind":"definition","summary":"[Rectangle] A subset S \\subseteq X \\times Y is a \\emphrectangle if it factors as a product A \\t…","labels":["CommunicationComplexity.Rectangle.IsRectangle"],"detail_key":"p14"},{"id":"n19608","layer":"informal","project":"p14","title":"Cross-product characterization of rectangles","kind":"theorem","summary":"[Cross-product characterization of rectangles] A set R \\subseteq X \\times Y is a rectangle if a…","labels":["CommunicationComplexity.Rectangle.IsRectangle_iff"],"detail_key":"p14"},{"id":"n19609","layer":"informal","project":"p14","title":"Monochromatic set","kind":"definition","summary":"[Monochromatic set] A set S \\subseteq X \\times Y is \\emphmonochromatic for a function g : X \\to…","labels":["CommunicationComplexity.Rectangle.IsMonochromatic"],"detail_key":"p14"},{"id":"n19610","layer":"informal","project":"p14","title":"Fooling set","kind":"definition","summary":"[Fooling set] A set S \\subseteq X \\times Y is a \\emphfooling set for g : X \\to Y \\to \\alpha if…","labels":["CommunicationComplexity.Rectangle.IsFoolingSet"],"detail_key":"p14"},{"id":"n19611","layer":"informal","project":"p14","title":"Monochromatic rectangle partition","kind":"definition","summary":"[Monochromatic rectangle partition] A collection P of subsets of X \\times Y is a \\emphmonochrom…","labels":["CommunicationComplexity.Rectangle.IsMonoPartition"],"detail_key":"p14"},{"id":"n19612","layer":"informal","project":"p14","title":"Every point lies in some part","kind":"theorem","summary":"[Every point lies in some part] If P is a monochromatic rectangle partition for g, then for eve…","labels":["CommunicationComplexity.Rectangle.monoPartition_point_mem"],"detail_key":"p14"},{"id":"n19613","layer":"informal","project":"p14","title":"Uniqueness of part containing a point","kind":"theorem","summary":"[Uniqueness of part containing a point] If P is a monochromatic rectangle partition for g and a…","labels":["CommunicationComplexity.Rectangle.monoPartition_part_unique"],"detail_key":"p14"},{"id":"n19614","layer":"informal","project":"p14","title":"Cross closure within a part","kind":"theorem","summary":"[Cross closure within a part] If P is a monochromatic rectangle partition for g, R \\in P, and (…","labels":["CommunicationComplexity.Rectangle.monoPartition_cross_mem"],"detail_key":"p14"},{"id":"n19615","layer":"informal","project":"p14","title":"Equal values within a part","kind":"theorem","summary":"[Equal values within a part] If P is a monochromatic rectangle partition for g, R \\in P, and (x…","labels":["CommunicationComplexity.Rectangle.monoPartition_values_eq"],"detail_key":"p14"},{"id":"n19616","layer":"informal","project":"p14","title":"Fooling set bound — extended cardinality","kind":"theorem","summary":"[Fooling set bound — extended cardinality] If S is a fooling set for g and P is a monochromatic…","labels":["CommunicationComplexity.Rectangle.foolingSet_encard_le_of_monoPartition"],"detail_key":"p14"},{"id":"n19617","layer":"informal","project":"p14","title":"Fooling set bound — finite cardinality","kind":"theorem","summary":"[Fooling set bound — finite cardinality] If S is a fooling set for g, P is a monochromatic rect…","labels":["CommunicationComplexity.Rectangle.foolingSet_ncard_le_of_monoPartition"],"detail_key":"p14"},{"id":"n19618","layer":"informal","project":"p14","title":"Subprotocol predicate","kind":"definition","summary":"[Subprotocol predicate] The inductive proposition \\textttIsSubprotocol\\; s\\; p asserts that pro…","labels":["CommunicationComplexity.Deterministic.Protocol.IsSubprotocol"],"detail_key":"p14"},{"id":"n19619","layer":"informal","project":"p14","title":"Transitivity of subprotocol","kind":"lemma","summary":"[Transitivity of subprotocol] If s is a subprotocol of t and t is a subprotocol of u, then s is…","labels":["CommunicationComplexity.Deterministic.Protocol.IsSubprotocol.trans"],"detail_key":"p14"},{"id":"n19620","layer":"informal","project":"p14","title":"Balanced subprotocol auxiliary","kind":"lemma","summary":"[Balanced subprotocol auxiliary] Given a protocol p with 3 \\cdot p.numLeaves \\ge 2n and n > 1,…","labels":["CommunicationComplexity.Deterministic.Protocol.balanced_aux"],"detail_key":"p14"},{"id":"n19621","layer":"informal","project":"p14","title":"Balanced subprotocol","kind":"theorem","summary":"[Balanced subprotocol] If a protocol p has more than one leaf (i.e.\\ p.numLeaves > 1), then the…","labels":["CommunicationComplexity.Deterministic.Protocol.balanced_subprotocol"],"detail_key":"p14"},{"id":"n19622","layer":"informal","project":"p14","title":"Subprotocol path witness","kind":"definition","summary":"[Subprotocol path witness] The inductive type \\textttSubprotocolPath\\; s\\; p is a data-carrying…","labels":["CommunicationComplexity.Deterministic.Protocol.SubprotocolPath"],"detail_key":"p14"},{"id":"n19623","layer":"informal","project":"p14","title":"Path to subprotocol proof","kind":"definition","summary":"[Path to subprotocol proof] Every \\textttSubprotocolPath s\\; p can be mapped to a proof that s…","labels":["CommunicationComplexity.Deterministic.Protocol.SubprotocolPath.toIsSubprotocol"],"detail_key":"p14"},{"id":"n19624","layer":"informal","project":"p14","title":"Existence of a subprotocol path","kind":"theorem","summary":"[Existence of a subprotocol path] If s is a subprotocol of p (i.e.\\ \\textttIsSubprotocol s\\; p…","labels":["CommunicationComplexity.Deterministic.Protocol.path_exists_of_isSubprotocol"],"detail_key":"p14"},{"id":"n19625","layer":"informal","project":"p14","title":"Classical choice of a subprotocol path","kind":"definition","summary":"[Classical choice of a subprotocol path] Given a proof that s is a subprotocol of p, \\textttcho…","labels":["CommunicationComplexity.Deterministic.Protocol.choosePath"],"detail_key":"p14"},{"id":"n19626","layer":"informal","project":"p14","title":"Subprotocol has at most as many leaves","kind":"lemma","summary":"[Subprotocol has at most as many leaves] If \\texttthsp is a \\textttSubprotocolPath from s to p,…","labels":["CommunicationComplexity.Deterministic.Protocol.SubprotocolPath.numLeaves_le"],"detail_key":"p14"},{"id":"n19627","layer":"informal","project":"p14","title":"Reachable Alice inputs along a path","kind":"definition","summary":"[Reachable Alice inputs along a path] Given a subprotocol path \\texttthsp from s to p, \\textttr…","labels":["CommunicationComplexity.Deterministic.Protocol.reachXPath"],"detail_key":"p14"},{"id":"n19628","layer":"informal","project":"p14","title":"Reachable Bob inputs along a path","kind":"definition","summary":"[Reachable Bob inputs along a path] Given a subprotocol path \\texttthsp from s to p, \\textttrea…","labels":["CommunicationComplexity.Deterministic.Protocol.reachYPath"],"detail_key":"p14"},{"id":"n19629","layer":"informal","project":"p14","title":"Inputs that reach a subprotocol path","kind":"definition","summary":"[Inputs that reach a subprotocol path] \\textttreachesPath\\;\\texttthsp\\;x\\;y is the proposition…","labels":["CommunicationComplexity.Deterministic.Protocol.reachesPath"],"detail_key":"p14"},{"id":"n19630","layer":"informal","project":"p14","title":"Alice's reach set for a subprotocol","kind":"definition","summary":"[Alice's reach set for a subprotocol] Given a proof \\texttthsp that s is a subprotocol of p, \\t…","labels":["CommunicationComplexity.Deterministic.Protocol.reachX"],"detail_key":"p14"},{"id":"n19631","layer":"informal","project":"p14","title":"Bob's reach set for a subprotocol","kind":"definition","summary":"[Bob's reach set for a subprotocol] Given a proof \\texttthsp that s is a subprotocol of p, \\tex…","labels":["CommunicationComplexity.Deterministic.Protocol.reachY"],"detail_key":"p14"},{"id":"n19632","layer":"informal","project":"p14","title":"Inputs that reach a subprotocol","kind":"definition","summary":"[Inputs that reach a subprotocol] \\textttreaches\\;\\texttthsp\\;x\\;y is the proposition that inpu…","labels":["CommunicationComplexity.Deterministic.Protocol.reaches"],"detail_key":"p14"},{"id":"n19633","layer":"informal","project":"p14","title":"Subprotocol and parent agree on reached inputs (path version)","kind":"lemma","summary":"[Subprotocol and parent agree on reached inputs (path version)] If the input pair (x, y) reache…","labels":["CommunicationComplexity.Deterministic.Protocol.subprotocol_run_eq_of_reachesPath"],"detail_key":"p14"},{"id":"n19634","layer":"informal","project":"p14","title":"Subprotocol and parent agree on reached inputs","kind":"lemma","summary":"[Subprotocol and parent agree on reached inputs] If \\texttthsp witnesses that s is a subprotoco…","labels":["CommunicationComplexity.Deterministic.Protocol.subprotocol_run_eq_of_reaches"],"detail_key":"p14"},{"id":"n19635","layer":"informal","project":"p14","title":"Canonical output of a protocol","kind":"definition","summary":"[Canonical output of a protocol] \\textttchooseOutput p returns a canonical output value from pr…","labels":["CommunicationComplexity.Deterministic.Protocol.chooseOutput"],"detail_key":"p14"},{"id":"n19636","layer":"informal","project":"p14","title":"Erase a subprotocol path","kind":"definition","summary":"[Erase a subprotocol path] Given a subprotocol path \\texttthsp from s to p, \\texttterasePath hs…","labels":["CommunicationComplexity.Deterministic.Protocol.erasePath"],"detail_key":"p14"},{"id":"n19637","layer":"informal","project":"p14","title":"Leaf count after erasing a subprotocol path","kind":"lemma","summary":"[Leaf count after erasing a subprotocol path] After erasing subprotocol s from p via path \\text…","labels":["CommunicationComplexity.Deterministic.Protocol.erasePath_numLeaves"],"detail_key":"p14"},{"id":"n19638","layer":"informal","project":"p14","title":"Erased protocol agrees on non-reaching inputs","kind":"lemma","summary":"[Erased protocol agrees on non-reaching inputs] If (x, y) does not reach the subprotocol path \\…","labels":["CommunicationComplexity.Deterministic.Protocol.erasePath_run_outside"],"detail_key":"p14"},{"id":"n19639","layer":"informal","project":"p14","title":"Erase a subprotocol (propositional version)","kind":"definition","summary":"[Erase a subprotocol (propositional version)] The noncomputable variant of \\texttterasePath tha…","labels":["CommunicationComplexity.Deterministic.Protocol.erase"],"detail_key":"p14"},{"id":"n19640","layer":"informal","project":"p14","title":"Leaf count after erasing a subprotocol","kind":"lemma","summary":"[Leaf count after erasing a subprotocol] If \\texttthsp witnesses that s is a subprotocol of p,…","labels":["CommunicationComplexity.Deterministic.Protocol.erase_numLeaves"],"detail_key":"p14"},{"id":"n19641","layer":"informal","project":"p14","title":"Erased protocol agrees on non-reaching inputs (propositional version)","kind":"lemma","summary":"[Erased protocol agrees on non-reaching inputs (propositional version)] If (x, y) does not reac…","labels":["CommunicationComplexity.Deterministic.Protocol.erase_run_outside"],"detail_key":"p14"},{"id":"n19642","layer":"informal","project":"p14","title":"Delete a subprotocol path by splicing","kind":"definition","summary":"[Delete a subprotocol path by splicing] \\textttdeletePath hsp removes the subtree s from p by s…","labels":["CommunicationComplexity.Deterministic.Protocol.deletePath"],"detail_key":"p14"},{"id":"n19643","layer":"informal","project":"p14","title":"Delete is non-none when subprotocol is proper","kind":"lemma","summary":"[Delete is non-none when subprotocol is proper] If s.numLeaves < p.numLeaves (i.e.\\ s is a prop…","labels":["CommunicationComplexity.Deterministic.Protocol.deletePath_ne_none_of_lt"],"detail_key":"p14"},{"id":"n19644","layer":"informal","project":"p14","title":"Delete yields a protocol when subprotocol is proper","kind":"lemma","summary":"[Delete yields a protocol when subprotocol is proper] If s.numLeaves < p.numLeaves, then there…","labels":["CommunicationComplexity.Deterministic.Protocol.deletePath_exists_of_lt"],"detail_key":"p14"},{"id":"n19645","layer":"informal","project":"p14","title":"Prune a subprotocol path","kind":"definition","summary":"[Prune a subprotocol path] Given a path \\texttthsp from s to p with s.numLeaves < p.numLeaves,…","labels":["CommunicationComplexity.Deterministic.Protocol.prunePath"],"detail_key":"p14"},{"id":"n19646","layer":"informal","project":"p14","title":"Prune path satisfies delete specification","kind":"lemma","summary":"[Prune path satisfies delete specification] Under the hypothesis s.numLeaves < p.numLeaves, we…","labels":["CommunicationComplexity.Deterministic.Protocol.prunePath_spec"],"detail_key":"p14"},{"id":"n19647","layer":"informal","project":"p14","title":"Delete none implies subprotocol equals parent","kind":"lemma","summary":"[Delete none implies subprotocol equals parent] If \\textttdeletePath\\;\\texttthsp = \\textttnone,…","labels":["CommunicationComplexity.Deterministic.Protocol.eq_of_deletePath_none"],"detail_key":"p14"},{"id":"n19648","layer":"informal","project":"p14","title":"Leaf count of the result of a delete","kind":"lemma","summary":"[Leaf count of the result of a delete] If \\textttdeletePath\\;\\texttthsp = \\textttsome\\;q, then…","labels":["CommunicationComplexity.Deterministic.Protocol.deletePath_numLeaves_of_some"],"detail_key":"p14"},{"id":"n19649","layer":"informal","project":"p14","title":"Leaf count after pruning a path","kind":"lemma","summary":"[Leaf count after pruning a path] If s.numLeaves < p.numLeaves, then (\\textttprunePath\\;\\texttt…","labels":["CommunicationComplexity.Deterministic.Protocol.prunePath_numLeaves_of_lt"],"detail_key":"p14"},{"id":"n19650","layer":"informal","project":"p14","title":"All inputs reach when delete is none","kind":"lemma","summary":"[All inputs reach when delete is none] If \\textttdeletePath\\;\\texttthsp = \\textttnone (meaning…","labels":["CommunicationComplexity.Deterministic.Protocol.reachesPath_of_deletePath_none"],"detail_key":"p14"},{"id":"n19651","layer":"informal","project":"p14","title":"Deleted protocol agrees on non-reaching inputs","kind":"lemma","summary":"[Deleted protocol agrees on non-reaching inputs] If \\textttdeletePath\\;\\texttthsp = \\textttsome…","labels":["CommunicationComplexity.Deterministic.Protocol.deletePath_run_outside_of_some"],"detail_key":"p14"},{"id":"n19652","layer":"informal","project":"p14","title":"Pruned path protocol agrees on non-reaching inputs","kind":"lemma","summary":"[Pruned path protocol agrees on non-reaching inputs] If s.numLeaves < p.numLeaves and (x, y) do…","labels":["CommunicationComplexity.Deterministic.Protocol.prunePath_run_outside_of_lt"],"detail_key":"p14"},{"id":"n19653","layer":"informal","project":"p14","title":"Prune a subprotocol (propositional version)","kind":"definition","summary":"[Prune a subprotocol (propositional version)] The noncomputable variant of \\textttprunePath tak…","labels":["CommunicationComplexity.Deterministic.Protocol.prune"],"detail_key":"p14"},{"id":"n19654","layer":"informal","project":"p14","title":"Leaf count after pruning a subprotocol","kind":"lemma","summary":"[Leaf count after pruning a subprotocol] If s.numLeaves < p.numLeaves, then (\\textttprune\\;\\tex…","labels":["CommunicationComplexity.Deterministic.Protocol.prune_numLeaves_of_lt"],"detail_key":"p14"},{"id":"n19655","layer":"informal","project":"p14","title":"Pruned protocol agrees on non-reaching inputs","kind":"lemma","summary":"[Pruned protocol agrees on non-reaching inputs] If s.numLeaves < p.numLeaves and (x, y) does no…","labels":["CommunicationComplexity.Deterministic.Protocol.prune_run_outside_of_lt"],"detail_key":"p14"},{"id":"n19656","layer":"informal","project":"p14","title":"Test subprotocol construction","kind":"definition","summary":"[Test subprotocol construction] Given that s is a subprotocol of p and two protocols \\textttqIn…","labels":["CommunicationComplexity.Deterministic.Protocol.testSubprotocol"],"detail_key":"p14"},{"id":"n19657","layer":"informal","project":"p14","title":"Complexity of the test subprotocol","kind":"lemma","summary":"[Complexity of the test subprotocol] The communication complexity of \\texttttestSubprotocol hsp…","labels":["CommunicationComplexity.Deterministic.Protocol.testSubprotocol_complexity"],"detail_key":"p14"},{"id":"n19658","layer":"informal","project":"p14","title":"Test subprotocol runs qIn on inside inputs","kind":"lemma","summary":"[Test subprotocol runs qIn on inside inputs] If (x, y) reaches the subprotocol s of p, then (\\t…","labels":["CommunicationComplexity.Deterministic.Protocol.testSubprotocol_run_inside"],"detail_key":"p14"},{"id":"n19659","layer":"informal","project":"p14","title":"Test subprotocol runs qOut on outside inputs","kind":"lemma","summary":"[Test subprotocol runs qOut on outside inputs] If (x, y) does not reach the subprotocol s of p,…","labels":["CommunicationComplexity.Deterministic.Protocol.testSubprotocol_run_outside"],"detail_key":"p14"},{"id":"n19660","layer":"informal","project":"p14","title":"Syntactic transcript type","kind":"definition","summary":"[Syntactic transcript type] For a deterministic protocol p over input types X and Y with output…","labels":["CommunicationComplexity.Deterministic.Protocol.Transcript"],"detail_key":"p14"},{"id":"n19661","layer":"informal","project":"p14","title":"Output of a transcript","kind":"definition","summary":"[Output of a transcript] Given a syntactic transcript t of a protocol p, \\textttTranscript.outp…","labels":["CommunicationComplexity.Deterministic.Protocol.Transcript.output"],"detail_key":"p14"},{"id":"n19662","layer":"informal","project":"p14","title":"Input set of a transcript","kind":"definition","summary":"[Input set of a transcript] For a syntactic transcript t of p, the \\emphinput set inputSet(t) \\…","labels":["CommunicationComplexity.Deterministic.Protocol.Transcript.inputSet"],"detail_key":"p14"},{"id":"n19663","layer":"informal","project":"p14","title":"Input set is a rectangle","kind":"theorem","summary":"[Input set is a rectangle] For every protocol p and every syntactic transcript t of p, the inpu…","labels":["CommunicationComplexity.Deterministic.Protocol.Transcript.inputSet_isRectangle"],"detail_key":"p14"},{"id":"n19664","layer":"informal","project":"p14","title":"Execute protocol to transcript","kind":"definition","summary":"[Execute protocol to transcript] Given a protocol p and an input pair xy \\in X \\times Y, \\textt…","labels":["CommunicationComplexity.Deterministic.Protocol.transcript"],"detail_key":"p14"},{"id":"n19665","layer":"informal","project":"p14","title":"Input follows its own transcript","kind":"theorem","summary":"[Input follows its own transcript] For every protocol p and every input pair xy, the pair xy be…","labels":["CommunicationComplexity.Deterministic.Protocol.mem_transcript"],"detail_key":"p14"},{"id":"n19666","layer":"informal","project":"p14","title":"Transcript determined by membership","kind":"theorem","summary":"[Transcript determined by membership] If an input pair xy belongs to the input set of a syntact…","labels":["CommunicationComplexity.Deterministic.Protocol.transcript_eq_of_mem"],"detail_key":"p14"},{"id":"n19667","layer":"informal","project":"p14","title":"Protocol output equals transcript output","kind":"theorem","summary":"[Protocol output equals transcript output] For every protocol p and input pair xy, running p yi…","labels":["CommunicationComplexity.Deterministic.Protocol.run_eq_transcript_output"],"detail_key":"p14"},{"id":"n19668","layer":"informal","project":"p14","title":"Same transcript implies same output","kind":"theorem","summary":"[Same transcript implies same output] If two input pairs xy and xy' produce the same syntactic…","labels":["CommunicationComplexity.Deterministic.Protocol.run_eq_of_transcript_eq"],"detail_key":"p14"},{"id":"n19669","layer":"informal","project":"p14","title":"Auxiliary bound on transcript count","kind":"theorem","summary":"[Auxiliary bound on transcript count] For natural numbers a, b, c_a, c_b with a \\le 2^c_a and b…","labels":["CommunicationComplexity.Deterministic.Protocol.card_transcript_le_two_pow_complexity_aux"],"detail_key":"p14"},{"id":"n19670","layer":"informal","project":"p14","title":"Number of transcripts bounded by complexity","kind":"theorem","summary":"[Number of transcripts bounded by complexity] A deterministic protocol p has at most 2^complexi…","labels":["CommunicationComplexity.Deterministic.Protocol.card_transcript_le_two_pow_complexity"],"detail_key":"p14"},{"id":"n19671","layer":"informal","project":"p14","title":"Transcript swap","kind":"definition","summary":"[Transcript swap] Given a syntactic transcript t of p, \\texttttranscriptSwap~t is the correspon…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptSwap"],"detail_key":"p14"},{"id":"n19672","layer":"informal","project":"p14","title":"Transcript swap is injective","kind":"theorem","summary":"[Transcript swap is injective] For every protocol p, the map \\texttttranscriptSwap from Transcr…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptSwap_injective"],"detail_key":"p14"},{"id":"n19673","layer":"informal","project":"p14","title":"Swap commutes with transcript computation","kind":"theorem","summary":"[Swap commutes with transcript computation] For every protocol p and inputs x \\in X, y \\in Y, s…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptSwap_transcript"],"detail_key":"p14"},{"id":"n19674","layer":"informal","project":"p14","title":"Transcript comap","kind":"definition","summary":"[Transcript comap] Given maps f_X : X' \\to X and f_Y : Y' \\to Y and a syntactic transcript t of…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptComap"],"detail_key":"p14"},{"id":"n19675","layer":"informal","project":"p14","title":"Transcript comap is injective","kind":"theorem","summary":"[Transcript comap is injective] For every protocol p and input maps f_X : X' \\to X, f_Y : Y' \\t…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptComap_injective"],"detail_key":"p14"},{"id":"n19676","layer":"informal","project":"p14","title":"Comap commutes with transcript computation","kind":"theorem","summary":"[Comap commutes with transcript computation] For every protocol p, maps f_X : X' \\to X, f_Y : Y…","labels":["CommunicationComplexity.Deterministic.Protocol.transcriptComap_transcript"],"detail_key":"p14"},{"id":"n19677","layer":"informal","project":"p14","title":"Protocol tree shape","kind":"definition","summary":"[Protocol tree shape] Given a deterministic communication protocol p over input types X and Y w…","labels":["CommunicationComplexity.Deterministic.Protocol.shape"],"detail_key":"p14"},{"id":"n19678","layer":"informal","project":"p14","title":"Number of protocol leaves","kind":"definition","summary":"[Number of protocol leaves] The number of leaves (output nodes) of a protocol p is defined as \\…","labels":["CommunicationComplexity.Deterministic.Protocol.numLeaves"],"detail_key":"p14"},{"id":"n19679","layer":"informal","project":"p14","title":"Subtree relation on binary trees","kind":"definition","summary":"[Subtree relation on binary trees] \\textttTreeIsSubtree s t is an inductively defined propositi…","labels":["CommunicationComplexity.TreeIsSubtree"],"detail_key":"p14"},{"id":"n19680","layer":"informal","project":"p14","title":"Transitivity of the subtree relation","kind":"lemma","summary":"[Transitivity of the subtree relation] If s is a subtree of t and t is a subtree of u, then s i…","labels":["CommunicationComplexity.TreeIsSubtree.trans"],"detail_key":"p14"},{"id":"n19681","layer":"informal","project":"p14","title":"Balanced subtree — auxiliary","kind":"lemma","summary":"[Balanced subtree — auxiliary] Let t be a binary tree and n a natural number with n > 1 and 3 \\…","labels":["CommunicationComplexity.tree_balanced_subtree_aux"],"detail_key":"p14"},{"id":"n19682","layer":"informal","project":"p14","title":"Balanced subtree","kind":"theorem","summary":"[Balanced subtree] Every binary tree t with more than one leaf has a subtree s satisfying \\[ \\l…","labels":["CommunicationComplexity.tree_balanced_subtree"],"detail_key":"p14"},{"id":"n19683","layer":"informal","project":"p14","title":"Complexity bounded by \\lceil\\log|X|\\rceil + \\lceil\\log|Y|\\rceil","kind":"theorem","summary":"[Complexity bounded by \\lceil\\log|X|\\rceil + \\lceil\\log|Y|\\rceil] For finite, nonempty types X…","labels":["CommunicationComplexity.Deterministic.communicationComplexity_le_clog_card"],"detail_key":"p14"},{"id":"n19684","layer":"informal","project":"p14","title":"Complexity bounded by \\lceil\\log|X|\\rceil + \\lceil\\log|\\alpha|\\rceil","kind":"theorem","summary":"[Complexity bounded by \\lceil\\log|X|\\rceil + \\lceil\\log|\\alpha|\\rceil] For finite, nonempty typ…","labels":["CommunicationComplexity.Deterministic.communicationComplexity_le_clog_card_X_alpha"],"detail_key":"p14"},{"id":"n19685","layer":"informal","project":"p14","title":"Complexity bounded by \\lceil\\log|Y|\\rceil + \\lceil\\log|\\alpha|\\rceil","kind":"theorem","summary":"[Complexity bounded by \\lceil\\log|Y|\\rceil + \\lceil\\log|\\alpha|\\rceil] For finite, nonempty typ…","labels":["CommunicationComplexity.Deterministic.communicationComplexity_le_clog_card_Y_alpha"],"detail_key":"p14"},{"id":"n19686","layer":"informal","project":"p14","title":"Coin tape of length n","kind":"definition","summary":"[Coin tape of length n] \\textttCoinTape(n) is the type Fin\\,n \\to Bool, i.e.\\ the set of all bi…","labels":["CommunicationComplexity.CoinTape"],"detail_key":"p14"},{"id":"n19687","layer":"informal","project":"p14","title":"Public-coin protocol fixed to deterministic","kind":"definition","summary":"[Public-coin protocol fixed to deterministic] Given a public-coin protocol p over randomness sp…","labels":["CommunicationComplexity.PublicCoin.Protocol.toDeterministic"],"detail_key":"p14"},{"id":"n19688","layer":"informal","project":"p14","title":"Fixed-randomness run equals randomised run","kind":"theorem","summary":"[Fixed-randomness run equals randomised run] For any inputs x : X and y : Y, running the determ…","labels":["CommunicationComplexity.PublicCoin.Protocol.toDeterministic_run"],"detail_key":"p14"},{"id":"n19689","layer":"informal","project":"p14","title":"Fixed-randomness complexity equals original","kind":"theorem","summary":"[Fixed-randomness complexity equals original] Fixing the shared randomness does not change the…","labels":["CommunicationComplexity.PublicCoin.Protocol.toDeterministic_complexity"],"detail_key":"p14"},{"id":"n19690","layer":"informal","project":"p14","title":"Deterministic finite-message protocol to private-coin","kind":"definition","summary":"[Deterministic finite-message protocol to private-coin] Given a deterministic finite-message pr…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toPrivateCoin"],"detail_key":"p14"},{"id":"n19691","layer":"informal","project":"p14","title":"Private-coin run of embedded deterministic protocol","kind":"theorem","summary":"[Private-coin run of embedded deterministic protocol] For any inputs x : X, y : Y and any coin…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toPrivateCoin_rrun"],"detail_key":"p14"},{"id":"n19692","layer":"informal","project":"p14","title":"Complexity preserved under embedding into private-coin","kind":"theorem","summary":"[Complexity preserved under embedding into private-coin] Embedding a deterministic finite-messa…","labels":["CommunicationComplexity.Deterministic.FiniteMessage.Protocol.toPrivateCoin_complexity"],"detail_key":"p14"},{"id":"n19693","layer":"informal","project":"p14","title":"Public-coin finite-message protocol fixed to deterministic","kind":"definition","summary":"[Public-coin finite-message protocol fixed to deterministic] Given a public-coin finite-message…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toDeterministic"],"detail_key":"p14"},{"id":"n19694","layer":"informal","project":"p14","title":"Fixed-randomness run equals randomised run (finite-message)","kind":"theorem","summary":"[Fixed-randomness run equals randomised run (finite-message)] For any inputs x : X and y : Y, t…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toDeterministic_run"],"detail_key":"p14"},{"id":"n19695","layer":"informal","project":"p14","title":"Fixed-randomness complexity equals original (finite-message)","kind":"theorem","summary":"[Fixed-randomness complexity equals original (finite-message)] Fixing the shared randomness of…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toDeterministic_complexity"],"detail_key":"p14"},{"id":"n19696","layer":"informal","project":"p14","title":"Private-coin complexity at most deterministic","kind":"theorem","summary":"[Private-coin complexity at most deterministic] For any function f : X \\to Y \\to \\alpha and any…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity_le_deterministic"],"detail_key":"p14"},{"id":"n19697","layer":"informal","project":"p14","title":"Finite measurable space","kind":"definition","summary":"[Finite measurable space] A typeclass for a measurable space \\Omega that is simultaneously fini…","labels":["CommunicationComplexity.FiniteMeasureSpace"],"detail_key":"p14"},{"id":"n19698","layer":"informal","project":"p14","title":"Constructor for \\textttFiniteMeasureSpace","kind":"definition","summary":"[Constructor for \\textttFiniteMeasureSpace] Given a type \\Omega that already carries \\textttFin…","labels":["CommunicationComplexity.FiniteMeasureSpace.of"],"detail_key":"p14"},{"id":"n19699","layer":"informal","project":"p14","title":"Finite probability space","kind":"definition","summary":"[Finite probability space] A typeclass bundling a \\textttMeasureSpace structure on \\Omega toget…","labels":["CommunicationComplexity.FiniteProbabilitySpace"],"detail_key":"p14"},{"id":"n19700","layer":"informal","project":"p14","title":"Constructor from existing instances","kind":"definition","summary":"[Constructor from existing instances] Given a type \\Omega already equipped with a \\textttMeasur…","labels":["CommunicationComplexity.FiniteProbabilitySpace.of"],"detail_key":"p14"},{"id":"n19701","layer":"informal","project":"p14","title":"Constructor from an explicit measure","kind":"definition","summary":"[Constructor from an explicit measure] Given a finite measurable space \\Omega and an unbundled…","labels":["CommunicationComplexity.FiniteProbabilitySpace.ofMeasure"],"detail_key":"p14"},{"id":"n19702","layer":"informal","project":"p14","title":"Singleton decomposition of a finite measure","kind":"theorem","summary":"[Singleton decomposition of a finite measure] Let \\Omega be a finite measurable space, \\mu a fi…","labels":["CommunicationComplexity.FiniteMeasureSpace.measureReal_eq_sum_singletons"],"detail_key":"p14"},{"id":"n19703","layer":"informal","project":"p14","title":"Measure of a preimage via fibers","kind":"theorem","summary":"[Measure of a preimage via fibers] Let Z : \\Omega \\to \\alpha be a function from a finite measur…","labels":["CommunicationComplexity.FiniteMeasureSpace.measureReal_preimage_eq_sum_fibers"],"detail_key":"p14"},{"id":"n19704","layer":"informal","project":"p14","title":"Absolute continuity via singletons","kind":"theorem","summary":"[Absolute continuity via singletons] On a finite measurable space, two measures satisfy \\mu \\ll…","labels":["CommunicationComplexity.FiniteMeasureSpace.absolutelyContinuous_iff_forall_singletons"],"detail_key":"p14"},{"id":"n19705","layer":"informal","project":"p14","title":"Jensen's inequality: square of expectation","kind":"theorem","summary":"[Jensen's inequality: square of expectation] For any probability measure \\mu on a finite measur…","labels":["CommunicationComplexity.FiniteMeasureSpace.sq_integral_le_integral_sq"],"detail_key":"p14"},{"id":"n19706","layer":"informal","project":"p14","title":"Integral as a sum over fibers","kind":"theorem","summary":"[Integral as a sum over fibers] Let Z : \\Omega \\to \\alpha be a measurable map from a finite mea…","labels":["CommunicationComplexity.FiniteMeasureSpace.integral_comp_eq_sum_measureReal_fibers"],"detail_key":"p14"},{"id":"n19707","layer":"informal","project":"p14","title":"Law of total probability over fibers","kind":"theorem","summary":"[Law of total probability over fibers] Let Z : \\Omega \\to \\alpha be a finite-valued random vari…","labels":["CommunicationComplexity.FiniteMeasureSpace.measureReal_eq_sum_cond_fiber_real"],"detail_key":"p14"},{"id":"n19708","layer":"informal","project":"p14","title":"Uniform measure of a set is its relative cardinality","kind":"theorem","summary":"[Uniform measure of a set is its relative cardinality] Let \\Omega be a nonempty finite discrete…","labels":["CommunicationComplexity.uniformOn_univ_measureReal_eq_card_filter"],"detail_key":"p14"},{"id":"n19709","layer":"informal","project":"p14","title":"Uniform measure equals subtype cardinality ratio","kind":"theorem","summary":"[Uniform measure equals subtype cardinality ratio] Under the uniform measure on a nonempty fini…","labels":["CommunicationComplexity.uniformOn_univ_measureReal_eq_card_subtype"],"detail_key":"p14"},{"id":"n19710","layer":"informal","project":"p14","title":"A finite probability space is nonempty","kind":"theorem","summary":"[A finite probability space is nonempty] Any type carrying a \\textttFiniteProbabilitySpace inst…","labels":["CommunicationComplexity.FiniteProbabilitySpace.nonempty"],"detail_key":"p14"},{"id":"n19711","layer":"informal","project":"p14","title":"Probability mass function of a finite probability space","kind":"definition","summary":"[Probability mass function of a finite probability space] For a finite probability space \\Omega…","labels":["CommunicationComplexity.FiniteProbabilitySpace.toPMF"],"detail_key":"p14"},{"id":"n19712","layer":"informal","project":"p14","title":"Measure of a set as a PMF sum","kind":"theorem","summary":"[Measure of a set as a PMF sum] For a finite probability space \\Omega and any set S \\subseteq \\…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measure_eq"],"detail_key":"p14"},{"id":"n19713","layer":"informal","project":"p14","title":"Singleton masses sum to one","kind":"theorem","summary":"[Singleton masses sum to one] For any bijection e : \\Omega \\xrightarrow\\sim \\alpha with \\alpha…","labels":["CommunicationComplexity.FiniteProbabilitySpace.hasSum_measure_singletons"],"detail_key":"p14"},{"id":"n19714","layer":"informal","project":"p14","title":"PMF of a product space factors","kind":"theorem","summary":"[PMF of a product space factors] For finite probability spaces \\Omega_1 and \\Omega_2 and any (x…","labels":["CommunicationComplexity.FiniteProbabilitySpace.pmf_prod"],"detail_key":"p14"},{"id":"n19715","layer":"informal","project":"p14","title":"Product measure of a rectangle","kind":"theorem","summary":"[Product measure of a rectangle] For finite probability spaces \\Omega_1 and \\Omega_2 and sets A…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_prod"],"detail_key":"p14"},{"id":"n19716","layer":"informal","project":"p14","title":"Measure of a finite disjoint union","kind":"theorem","summary":"[Measure of a finite disjoint union] If (A_i)_i \\in \\iota is a pairwise-disjoint family of sets…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_iUnion_fintype"],"detail_key":"p14"},{"id":"n19717","layer":"informal","project":"p14","title":"Measure of a preimage of a finite set","kind":"theorem","summary":"[Measure of a preimage of a finite set] For a map \\varphi : \\Xi \\to \\Omega from a finite probab…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_preimage_finset"],"detail_key":"p14"},{"id":"n19718","layer":"informal","project":"p14","title":"Measure of a finite set as a sum of singletons","kind":"theorem","summary":"[Measure of a finite set as a sum of singletons] For a finite probability space \\Omega and a fi…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_finset"],"detail_key":"p14"},{"id":"n19719","layer":"informal","project":"p14","title":"Integral as a PMF-weighted sum","kind":"theorem","summary":"[Integral as a PMF-weighted sum] For a finite probability space \\Omega and any f : \\Omega \\to R…","labels":["CommunicationComplexity.FiniteProbabilitySpace.integral_eq_pmf_sum"],"detail_key":"p14"},{"id":"n19720","layer":"informal","project":"p14","title":"Jensen's inequality on a finite probability space","kind":"theorem","summary":"[Jensen's inequality on a finite probability space] For a finite probability space \\Omega and a…","labels":["CommunicationComplexity.FiniteProbabilitySpace.sq_integral_le_integral_sq"],"detail_key":"p14"},{"id":"n19721","layer":"informal","project":"p14","title":"Integral over a coordinate of a product space","kind":"theorem","summary":"[Integral over a coordinate of a product space] Let \\Omega be a finite probability space, \\iota…","labels":["CommunicationComplexity.FiniteProbabilitySpace.integral_comp_eval"],"detail_key":"p14"},{"id":"n19722","layer":"informal","project":"p14","title":"Measure of a cylinder set in a product space","kind":"theorem","summary":"[Measure of a cylinder set in a product space] For a finite index type \\iota, finite probabilit…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_pi_univ"],"detail_key":"p14"},{"id":"n19723","layer":"informal","project":"p14","title":"Measure equals integral of indicator (one-valued)","kind":"theorem","summary":"[Measure equals integral of indicator (one-valued)] For a finite probability space \\Omega and S…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_eq_integral_indicator_one"],"detail_key":"p14"},{"id":"n19724","layer":"informal","project":"p14","title":"Measure equals integral of 0/1 indicator","kind":"theorem","summary":"[Measure equals integral of 0/1 indicator] For a finite probability space \\Omega and S \\subsete…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_eq_integral_indicator"],"detail_key":"p14"},{"id":"n19725","layer":"informal","project":"p14","title":"PMF weights sum to one","kind":"theorem","summary":"[PMF weights sum to one] For any finite probability space \\Omega, \\sum_\\omega \\in \\Omega toPMF(…","labels":["CommunicationComplexity.FiniteProbabilitySpace.pmf_toReal_sum_eq_one"],"detail_key":"p14"},{"id":"n19726","layer":"informal","project":"p14","title":"PMF weights are nonneg","kind":"theorem","summary":"[PMF weights are nonneg] For any finite probability space \\Omega and \\omega \\in \\Omega, 0 \\le t…","labels":["CommunicationComplexity.FiniteProbabilitySpace.pmf_toReal_nonneg"],"detail_key":"p14"},{"id":"n19727","layer":"informal","project":"p14","title":"Some point has positive mass","kind":"theorem","summary":"[Some point has positive mass] For any finite probability space \\Omega, there exists \\omega \\in…","labels":["CommunicationComplexity.FiniteProbabilitySpace.exists_pmf_toReal_pos"],"detail_key":"p14"},{"id":"n19728","layer":"informal","project":"p14","title":"Integral is bounded by a pointwise bound","kind":"theorem","summary":"[Integral is bounded by a pointwise bound] If f : \\Omega \\to R satisfies f(\\omega) \\le c for al…","labels":["CommunicationComplexity.FiniteProbabilitySpace.integral_le_of_le"],"detail_key":"p14"},{"id":"n19729","layer":"informal","project":"p14","title":"Markov's inequality","kind":"theorem","summary":"[Markov's inequality] For a nonneg function f : \\Omega \\to R on a finite probability space and…","labels":["CommunicationComplexity.FiniteProbabilitySpace.measureReal_ge_le_integral_div"],"detail_key":"p14"},{"id":"n19730","layer":"informal","project":"p14","title":"Integral exceeds a pointwise lower bound","kind":"theorem","summary":"[Integral exceeds a pointwise lower bound] If f : \\Omega \\to R satisfies c < f(\\omega) for ever…","labels":["CommunicationComplexity.FiniteProbabilitySpace.lt_integral_of_lt"],"detail_key":"p14"},{"id":"n19731","layer":"informal","project":"p14","title":"Hash function space","kind":"definition","summary":"[Hash function space] \\textttHashSpace\\,\\alpha\\,k is the type of all functions \\alpha \\to Fin\\,…","labels":["CommunicationComplexity.Functions.Hash.HashSpace"],"detail_key":"p14"},{"id":"n19732","layer":"informal","project":"p14","title":"Collision piece","kind":"definition","summary":"[Collision piece] For x, y : \\alpha and a : Fin\\,k, \\textttcollisionPiece\\,k\\,x\\,y\\,a is the se…","labels":["CommunicationComplexity.Functions.Hash.collisionPiece"],"detail_key":"p14"},{"id":"n19733","layer":"informal","project":"p14","title":"Collision event as union of pieces","kind":"lemma","summary":"[Collision event as union of pieces] For any x, y : \\alpha and any k, \\[ \\h : \\textttHashSpace\\…","labels":["CommunicationComplexity.Functions.Hash.collision_mem_iUnion"],"detail_key":"p14"},{"id":"n19734","layer":"informal","project":"p14","title":"Collision pieces are pairwise disjoint","kind":"lemma","summary":"[Collision pieces are pairwise disjoint] For any x, y : \\alpha and k, the sets \\textttcollision…","labels":["CommunicationComplexity.Functions.Hash.collisionPiece_pairwiseDisjoint"],"detail_key":"p14"},{"id":"n19735","layer":"informal","project":"p14","title":"Singleton measure in hash range","kind":"lemma","summary":"[Singleton measure in hash range] For k \\ge 1 and any a : Fin\\,k, the uniform measure of the si…","labels":["CommunicationComplexity.Functions.Hash.hashRange_singleton_measure"],"detail_key":"p14"},{"id":"n19736","layer":"informal","project":"p14","title":"Singleton measure in hash range (real-valued)","kind":"lemma","summary":"[Singleton measure in hash range (real-valued)] For k \\ge 1 and any a : Fin\\,k, the real-valued…","labels":["CommunicationComplexity.Functions.Hash.hashRange_singleton_measureReal"],"detail_key":"p14"},{"id":"n19737","layer":"informal","project":"p14","title":"Measure of a collision piece","kind":"lemma","summary":"[Measure of a collision piece] Let \\alpha be a finite type, k \\ge 1, x \\ne y \\in \\alpha, and a…","labels":["CommunicationComplexity.Functions.Hash.collisionPiece_measureReal"],"detail_key":"p14"},{"id":"n19738","layer":"informal","project":"p14","title":"Collision probability bound","kind":"theorem","summary":"[Collision probability bound] Let \\alpha be a finite type, k \\ge 1, and let x \\ne y \\in \\alpha.…","labels":["CommunicationComplexity.Functions.Hash.collision_prob_le"],"detail_key":"p14"},{"id":"n19739","layer":"informal","project":"p14","title":"Distributional error of a deterministic protocol","kind":"definition","summary":"[Distributional error of a deterministic protocol] Given a deterministic protocol p : Protocol\\…","labels":["CommunicationComplexity.Deterministic.Protocol.distributionalError"],"detail_key":"p14"},{"id":"n19740","layer":"informal","project":"p14","title":"Failure integral swap","kind":"lemma","summary":"[Failure integral swap] Let \\mu be a finite probability space on X \\times Y, let m \\in N, and l…","labels":["CommunicationComplexity.PublicCoin.failureIntegral_swap"],"detail_key":"p14"},{"id":"n19741","layer":"informal","project":"p14","title":"Yao's minimax principle","kind":"theorem","summary":"[Yao's minimax principle] Let f : X \\to Y \\to \\alpha, \\varepsilon \\in R, n \\in N, and let \\mu b…","labels":["CommunicationComplexity.PublicCoin.lt_communicationComplexity_of_forall_distributionalError_gt"],"detail_key":"p14"},{"id":"n19742","layer":"informal","project":"p14","title":"Distributional error of a one-way protocol","kind":"definition","summary":"[Distributional error of a one-way protocol] Given a deterministic one-way protocol p computing…","labels":["CommunicationComplexity.Deterministic.OneWay.Protocol.distributionalError"],"detail_key":"p14"},{"id":"n19743","layer":"informal","project":"p14","title":"Deterministic protocol from public-coin protocol","kind":"definition","summary":"[Deterministic protocol from public-coin protocol] Given a one-way public-coin protocol p over…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol.toDeterministic"],"detail_key":"p14"},{"id":"n19744","layer":"informal","project":"p14","title":"Run of derandomised protocol","kind":"theorem","summary":"[Run of derandomised protocol] For any public-coin protocol p, coin outcome \\omega, and inputs…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol.toDeterministic_run"],"detail_key":"p14"},{"id":"n19745","layer":"informal","project":"p14","title":"Cost of derandomised protocol","kind":"theorem","summary":"[Cost of derandomised protocol] For any public-coin protocol p and coin outcome \\omega, \\[ (p.t…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol.toDeterministic_cost"],"detail_key":"p14"},{"id":"n19746","layer":"informal","project":"p14","title":"Fubini swap for failure integrals","kind":"lemma","summary":"[Fubini swap for failure integrals] Let \\mu be a finite probability measure on X \\times Y, let…","labels":["CommunicationComplexity.PublicCoin.OneWay.failureIntegral_swap"],"detail_key":"p14"},{"id":"n19747","layer":"informal","project":"p14","title":"Yao's minimax principle for one-way protocols","kind":"theorem","summary":"[Yao's minimax principle for one-way protocols] Let f : X \\to Y \\to \\alpha, \\varepsilon \\in R,…","labels":["CommunicationComplexity.PublicCoin.OneWay.lt_communicationComplexity_of_forall_distributionalError_gt"],"detail_key":"p14"},{"id":"n19748","layer":"informal","project":"p14","title":"Cumulative distribution function on Fin\\,m","kind":"definition","summary":"[Cumulative distribution function on Fin\\,m] Given a probability mass function p on Fin\\,m and…","labels":["CommunicationComplexity.Internal.cdf"],"detail_key":"p14"},{"id":"n19749","layer":"informal","project":"p14","title":"CDF at zero","kind":"lemma","summary":"[CDF at zero] For any PMF p on Fin\\,m, cdf(p, 0) = 0.","labels":["CommunicationComplexity.Internal.cdf_zero"],"detail_key":"p14"},{"id":"n19750","layer":"informal","project":"p14","title":"CDF successor recurrence","kind":"lemma","summary":"[CDF successor recurrence] For any PMF p on Fin\\,m and index n \\in Fin\\,m, cdf(p, n+1) = cdf(p,…","labels":["CommunicationComplexity.Internal.cdf_succ"],"detail_key":"p14"},{"id":"n19751","layer":"informal","project":"p14","title":"CDF at m equals one","kind":"lemma","summary":"[CDF at m equals one] For any PMF p on Fin\\,m, cdf(p, m) = 1.","labels":["CommunicationComplexity.Internal.cdf_one"],"detail_key":"p14"},{"id":"n19752","layer":"informal","project":"p14","title":"CDF is monotone","kind":"lemma","summary":"[CDF is monotone] For any PMF p on Fin\\,m, the function n \\mapsto cdf(p, n) is monotone non-dec…","labels":["CommunicationComplexity.Internal.cdf_mono"],"detail_key":"p14"},{"id":"n19753","layer":"informal","project":"p14","title":"Inverse CDF","kind":"definition","summary":"[Inverse CDF] Given a PMF p on Fin\\,m (with m \\ge 1) and a value x \\in R_\\ge 0^\\infty, invCdf(p…","labels":["CommunicationComplexity.Internal.invCdf"],"detail_key":"p14"},{"id":"n19754","layer":"informal","project":"p14","title":"Inverse CDF characterization","kind":"theorem","summary":"[Inverse CDF characterization] Let p be a PMF on Fin\\,m with m \\ge 1, let x < 1 in R_\\ge 0^\\inf…","labels":["CommunicationComplexity.Internal.invCdf_eq_iff"],"detail_key":"p14"},{"id":"n19755","layer":"informal","project":"p14","title":"Uniform approximation via inverse CDF","kind":"definition","summary":"[Uniform approximation via inverse CDF] Given a PMF p on Fin\\,m and a positive integer n, the m…","labels":["CommunicationComplexity.Internal.uniformApprox"],"detail_key":"p14"},{"id":"n19756","layer":"informal","project":"p14","title":"Counting bound in a real interval","kind":"lemma","summary":"[Counting bound in a real interval] For any n \\in N and real numbers a \\le b, the number of ind…","labels":["CommunicationComplexity.Internal.card_nat_in_Ico"],"detail_key":"p14"},{"id":"n19757","layer":"informal","project":"p14","title":"Uniform approximation error bound","kind":"theorem","summary":"[Uniform approximation error bound] For any PMF p on Fin\\,m, positive n, and index i \\in Fin\\,m…","labels":["CommunicationComplexity.Internal.uniformApprox_approx"],"detail_key":"p14"},{"id":"n19758","layer":"informal","project":"p14","title":"Single-space coin approximation","kind":"theorem","summary":"[Single-space coin approximation] For any finite probability space \\Omega and any \\delta > 0, t…","labels":["CommunicationComplexity.Internal.single_coin_approx"],"detail_key":"p14"},{"id":"n19759","layer":"informal","project":"p14","title":"Weighted sum approximation","kind":"lemma","summary":"[Weighted sum approximation] Let \\alpha be a finite type, p, q : \\alpha \\to R, and g : \\alpha \\…","labels":["CommunicationComplexity.Internal.weighted_sum_approx"],"detail_key":"p14"},{"id":"n19760","layer":"informal","project":"p14","title":"Product-space coin approximation","kind":"theorem","summary":"[Product-space coin approximation] For any two finite probability spaces \\Omega_X and \\Omega_Y…","labels":["CommunicationComplexity.Internal.product_coin_approx"],"detail_key":"p14"},{"id":"n19761","layer":"informal","project":"p14","title":"Protocol conversion to coin tape","kind":"definition","summary":"[Protocol conversion to coin tape] Given a finite-message private-coin protocol p over finite p…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toCoinTape"],"detail_key":"p14"},{"id":"n19762","layer":"informal","project":"p14","title":"Coin-tape protocol preserves complexity","kind":"theorem","summary":"[Coin-tape protocol preserves complexity] The coin-tape conversion toCoinTape(p, \\delta, h_\\del…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toCoinTape_complexity"],"detail_key":"p14"},{"id":"n19763","layer":"informal","project":"p14","title":"Coin-tape conversion preserves approximate correctness","kind":"theorem","summary":"[Coin-tape conversion preserves approximate correctness] If a protocol p approximately satisfie…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toCoinTape_approxSatisfies"],"detail_key":"p14"},{"id":"n19764","layer":"informal","project":"p14","title":"Private-coin protocol","kind":"definition","summary":"[Private-coin protocol] A \\emphprivate-coin protocol with randomness spaces \\Omega_X and \\Omega…","labels":["CommunicationComplexity.PrivateCoin.Protocol"],"detail_key":"p14"},{"id":"n19765","layer":"informal","project":"p14","title":"Output node","kind":"definition","summary":"[Output node] The constant \\emphoutput node of a private-coin protocol that immediately returns…","labels":["CommunicationComplexity.PrivateCoin.Protocol.output"],"detail_key":"p14"},{"id":"n19766","layer":"informal","project":"p14","title":"Alice's message node","kind":"definition","summary":"[Alice's message node] Given a function f : X \\to \\Omega_X \\to Bool and a continuation P : Bool…","labels":["CommunicationComplexity.PrivateCoin.Protocol.alice"],"detail_key":"p14"},{"id":"n19767","layer":"informal","project":"p14","title":"Bob's message node","kind":"definition","summary":"[Bob's message node] Given a function f : Y \\to \\Omega_Y \\to Bool and a continuation P : Bool \\…","labels":["CommunicationComplexity.PrivateCoin.Protocol.bob"],"detail_key":"p14"},{"id":"n19768","layer":"informal","project":"p14","title":"Randomized execution","kind":"definition","summary":"[Randomized execution] \\textttrrun\\;p\\;x\\;y\\;\\omega_x\\;\\omega_y executes the private-coin proto…","labels":["CommunicationComplexity.PrivateCoin.Protocol.rrun"],"detail_key":"p14"},{"id":"n19769","layer":"informal","project":"p14","title":"Randomized execution unfolding","kind":"theorem","summary":"[Randomized execution unfolding] For any private-coin protocol p and any inputs x, y, \\omega_x,…","labels":["CommunicationComplexity.PrivateCoin.Protocol.rrun_eq"],"detail_key":"p14"},{"id":"n19770","layer":"informal","project":"p14","title":"Approximate satisfaction","kind":"definition","summary":"[Approximate satisfaction] A private-coin protocol p \\emph\\varepsilon-satisfies a predicate Q :…","labels":["CommunicationComplexity.PrivateCoin.Protocol.ApproxSatisfies"],"detail_key":"p14"},{"id":"n19771","layer":"informal","project":"p14","title":"Approximate computation","kind":"definition","summary":"[Approximate computation] A private-coin protocol p \\emph\\varepsilon-computes a function f : X…","labels":["CommunicationComplexity.PrivateCoin.Protocol.ApproxComputes"],"detail_key":"p14"},{"id":"n19772","layer":"informal","project":"p14","title":"Approximate computation equals approximate satisfaction","kind":"theorem","summary":"[Approximate computation equals approximate satisfaction] For any private-coin protocol p, func…","labels":["CommunicationComplexity.PrivateCoin.Protocol.ApproxComputes_eq_ApproxSatisfies"],"detail_key":"p14"},{"id":"n19773","layer":"informal","project":"p14","title":"Private-coin communication complexity","kind":"definition","summary":"[Private-coin communication complexity] The \\varepsilon-error private-coin randomized communica…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity"],"detail_key":"p14"},{"id":"n19774","layer":"informal","project":"p14","title":"Upper-bound characterization","kind":"theorem","summary":"[Upper-bound characterization] For any n \\in N, we have R_\\varepsilon(f) \\le n if and only if t…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity_le_iff"],"detail_key":"p14"},{"id":"n19775","layer":"informal","project":"p14","title":"Lower-bound characterization","kind":"theorem","summary":"[Lower-bound characterization] For any n \\in N, we have n \\le R_\\varepsilon(f) (in N_\\infty) if…","labels":["CommunicationComplexity.PrivateCoin.le_communicationComplexity_iff"],"detail_key":"p14"},{"id":"n19776","layer":"informal","project":"p14","title":"Finite-message characterization","kind":"theorem","summary":"[Finite-message characterization] For any n \\in N, R_\\varepsilon(f) \\le n if and only if there…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity_le_iff_finiteMessage"],"detail_key":"p14"},{"id":"n19777","layer":"informal","project":"p14","title":"Monotonicity in error","kind":"theorem","summary":"[Monotonicity in error] If \\varepsilon' \\le \\varepsilon, then R_\\varepsilon(f) \\le R_\\varepsilo…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity_mono"],"detail_key":"p14"},{"id":"n19778","layer":"informal","project":"p14","title":"Complexity upper bound from finite-probability protocol","kind":"theorem","summary":"[Complexity upper bound from finite-probability protocol] Let \\Omega_X and \\Omega_Y be finite p…","labels":["CommunicationComplexity.PrivateCoin.communicationComplexity_le_of_finiteMessage"],"detail_key":"p14"},{"id":"n19779","layer":"informal","project":"p14","title":"Output map of a private-coin protocol","kind":"definition","summary":"[Output map of a private-coin protocol] Given a function g : \\alpha \\to \\beta and a private-coi…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.map"],"detail_key":"p14"},{"id":"n19780","layer":"informal","project":"p14","title":"Simulation of mapped protocol","kind":"theorem","summary":"[Simulation of mapped protocol] For any inputs x : X, y : Y and random coins \\omega_x : \\Omega_…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.map_rrun"],"detail_key":"p14"},{"id":"n19781","layer":"informal","project":"p14","title":"Complexity of mapped protocol","kind":"theorem","summary":"[Complexity of mapped protocol] Mapping a function over the output of a protocol does not chang…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.map_complexity"],"detail_key":"p14"},{"id":"n19782","layer":"informal","project":"p14","title":"Monadic bind of private-coin protocols","kind":"definition","summary":"[Monadic bind of private-coin protocols] Given a protocol p with output type \\alpha and a famil…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.bind"],"detail_key":"p14"},{"id":"n19783","layer":"informal","project":"p14","title":"Simulation of bound protocol","kind":"theorem","summary":"[Simulation of bound protocol] For all x, y, \\omega_x, \\omega_y, the randomised run of the boun…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.bind_rrun"],"detail_key":"p14"},{"id":"n19784","layer":"informal","project":"p14","title":"Randomness reindexing","kind":"definition","summary":"[Randomness reindexing] Given functions hX : \\Omega_X' \\to \\Omega_X and hY : \\Omega_Y' \\to \\Ome…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.comapRandomness"],"detail_key":"p14"},{"id":"n19785","layer":"informal","project":"p14","title":"Simulation after randomness reindexing","kind":"theorem","summary":"[Simulation after randomness reindexing] For all x, y, \\omega_x' : \\Omega_X', \\omega_y' : \\Omeg…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.comapRandomness_rrun"],"detail_key":"p14"},{"id":"n19786","layer":"informal","project":"p14","title":"Complexity after randomness reindexing","kind":"theorem","summary":"[Complexity after randomness reindexing] Reindexing the randomness spaces of a protocol does no…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.comapRandomness_complexity"],"detail_key":"p14"},{"id":"n19787","layer":"informal","project":"p14","title":"Product of two private-coin protocols","kind":"definition","summary":"[Product of two private-coin protocols] Given protocols p_1 and p_2 with potentially distinct r…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prod"],"detail_key":"p14"},{"id":"n19788","layer":"informal","project":"p14","title":"Simulation of product protocol","kind":"theorem","summary":"[Simulation of product protocol] For all x, y, \\omega_x : \\Omega_X_1 \\times \\Omega_X_2, \\omega_…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prod_rrun"],"detail_key":"p14"},{"id":"n19789","layer":"informal","project":"p14","title":"Complexity of product protocol","kind":"theorem","summary":"[Complexity of product protocol] The communication complexity of the product protocol is the su…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prod_complexity"],"detail_key":"p14"},{"id":"n19790","layer":"informal","project":"p14","title":"k-fold product of private-coin protocols","kind":"definition","summary":"[k-fold product of private-coin protocols] Given a family of protocols p : (i : Fin\\,k) \\to Pro…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.pi"],"detail_key":"p14"},{"id":"n19791","layer":"informal","project":"p14","title":"Simulation of k-fold product","kind":"theorem","summary":"[Simulation of k-fold product] For all x, y, \\omega_x, \\omega_y, \\[ (pi\\;p).rrun\\;x\\;y\\;\\omega_…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.pi_rrun"],"detail_key":"p14"},{"id":"n19792","layer":"informal","project":"p14","title":"Complexity of k-fold product","kind":"theorem","summary":"[Complexity of k-fold product] The communication complexity of the k-fold product equals the su…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.pi_complexity"],"detail_key":"p14"},{"id":"n19793","layer":"informal","project":"p14","title":"Bind with fresh independent randomness","kind":"definition","summary":"[Bind with fresh independent randomness] Given a protocol p over randomness (\\Omega_X, \\Omega_Y…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.rbind"],"detail_key":"p14"},{"id":"n19794","layer":"informal","project":"p14","title":"Simulation of bind with fresh randomness","kind":"theorem","summary":"[Simulation of bind with fresh randomness] For all x, y, \\omega_x : \\Omega_X \\times \\Omega_X',…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.rbind_rrun"],"detail_key":"p14"},{"id":"n19795","layer":"informal","project":"p14","title":"Complexity of bind with fresh randomness (constant continuation)","kind":"theorem","summary":"[Complexity of bind with fresh randomness (constant continuation)] If the continuation q\\,a has…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.rbind_complexity_const"],"detail_key":"p14"},{"id":"n19796","layer":"informal","project":"p14","title":"Product of private-coin and deterministic protocols","kind":"definition","summary":"[Product of private-coin and deterministic protocols] Given a private-coin protocol p_1 and a d…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prodDet"],"detail_key":"p14"},{"id":"n19797","layer":"informal","project":"p14","title":"Simulation of private-coin times deterministic","kind":"theorem","summary":"[Simulation of private-coin times deterministic] For all x, y, \\omega_x, \\omega_y, \\[ (prodDet\\…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prodDet_rrun"],"detail_key":"p14"},{"id":"n19798","layer":"informal","project":"p14","title":"Complexity of private-coin times deterministic","kind":"theorem","summary":"[Complexity of private-coin times deterministic] The communication complexity of \\textttprodDet…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.prodDet_complexity"],"detail_key":"p14"},{"id":"n19799","layer":"informal","project":"p14","title":"Private-coin finite-message protocol","kind":"definition","summary":"[Private-coin finite-message protocol] A private-coin finite-message protocol over input types…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol"],"detail_key":"p14"},{"id":"n19800","layer":"informal","project":"p14","title":"Output node","kind":"definition","summary":"[Output node] The leaf node of a private-coin finite-message protocol that produces the constan…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.output"],"detail_key":"p14"},{"id":"n19801","layer":"informal","project":"p14","title":"Alice's sending step","kind":"definition","summary":"[Alice's sending step] Given a function f : X \\to \\Omega_X \\to \\beta (Alice's message function…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.alice"],"detail_key":"p14"},{"id":"n19802","layer":"informal","project":"p14","title":"Bob's sending step","kind":"definition","summary":"[Bob's sending step] Given a function f : Y \\to \\Omega_Y \\to \\beta (Bob's message function depe…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.bob"],"detail_key":"p14"},{"id":"n19803","layer":"informal","project":"p14","title":"Protocol execution with explicit coins","kind":"definition","summary":"[Protocol execution with explicit coins] \\textttrrun\\,p\\,x\\,y\\,\\omega_x\\,\\omega_y executes the…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.rrun"],"detail_key":"p14"},{"id":"n19804","layer":"informal","project":"p14","title":"\\textttrrun unfolding","kind":"theorem","summary":"[\\textttrrun unfolding] For any protocol p and inputs x, y, \\omega_x, \\omega_y, we have p.\\text…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.rrun_eq"],"detail_key":"p14"},{"id":"n19805","layer":"informal","project":"p14","title":"Approximate satisfaction of a predicate","kind":"definition","summary":"[Approximate satisfaction of a predicate] A protocol p \\emph\\varepsilon-satisfies a predicate Q…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ApproxSatisfies"],"detail_key":"p14"},{"id":"n19806","layer":"informal","project":"p14","title":"Approximate computation of a function","kind":"definition","summary":"[Approximate computation of a function] A protocol p \\emph\\varepsilon-computes a function f : X…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ApproxComputes"],"detail_key":"p14"},{"id":"n19807","layer":"informal","project":"p14","title":"\\textttApproxComputes equals \\textttApproxSatisfies for equality","kind":"theorem","summary":"[\\textttApproxComputes equals \\textttApproxSatisfies for equality] For any protocol p, function…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ApproxComputes_eq_ApproxSatisfies"],"detail_key":"p14"},{"id":"n19808","layer":"informal","project":"p14","title":"Conversion to binary private-coin protocol","kind":"definition","summary":"[Conversion to binary private-coin protocol] Converts a private-coin finite-message protocol in…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toProtocol"],"detail_key":"p14"},{"id":"n19809","layer":"informal","project":"p14","title":"\\texttttoProtocol preserves execution","kind":"theorem","summary":"[\\texttttoProtocol preserves execution] For any protocol p and inputs x, y, \\omega_x, \\omega_y,…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toProtocol_rrun"],"detail_key":"p14"},{"id":"n19810","layer":"informal","project":"p14","title":"\\texttttoProtocol preserves complexity","kind":"theorem","summary":"[\\texttttoProtocol preserves complexity] For any protocol p, (p.\\texttttoProtocol).\\textttcompl…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.toProtocol_complexity"],"detail_key":"p14"},{"id":"n19811","layer":"informal","project":"p14","title":"Embedding of binary private-coin protocol","kind":"definition","summary":"[Embedding of binary private-coin protocol] Embeds a binary private-coin protocol into a privat…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ofProtocol"],"detail_key":"p14"},{"id":"n19812","layer":"informal","project":"p14","title":"\\textttofProtocol preserves execution","kind":"theorem","summary":"[\\textttofProtocol preserves execution] For any binary private-coin protocol p and inputs x, y,…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ofProtocol_rrun"],"detail_key":"p14"},{"id":"n19813","layer":"informal","project":"p14","title":"\\textttofProtocol preserves complexity","kind":"theorem","summary":"[\\textttofProtocol preserves complexity] For any binary private-coin protocol p, (\\textttofProt…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ofProtocol_complexity"],"detail_key":"p14"},{"id":"n19814","layer":"informal","project":"p14","title":"Equivalence via finite-message embedding","kind":"theorem","summary":"[Equivalence via finite-message embedding] For every binary private-coin protocol p, there exis…","labels":["CommunicationComplexity.PrivateCoin.FiniteMessage.Protocol.ofProtocol_equiv"],"detail_key":"p14"},{"id":"n19815","layer":"informal","project":"p14","title":"Conversion of a public-coin protocol to CoinTape","kind":"definition","summary":"[Conversion of a public-coin protocol to CoinTape] Given a public-coin finite-message protocol…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toCoinTape"],"detail_key":"p14"},{"id":"n19816","layer":"informal","project":"p14","title":"CoinTape approximation preserves complexity","kind":"theorem","summary":"[CoinTape approximation preserves complexity] For any finite-message protocol p over a finite p…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toCoinTape_complexity"],"detail_key":"p14"},{"id":"n19817","layer":"informal","project":"p14","title":"CoinTape approximation preserves ApproxSatisfies","kind":"theorem","summary":"[CoinTape approximation preserves ApproxSatisfies] Let p be a public-coin finite-message protoc…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toCoinTape_approxSatisfies"],"detail_key":"p14"},{"id":"n19818","layer":"informal","project":"p14","title":"Public-coin protocol","kind":"definition","summary":"[Public-coin protocol] A public-coin protocol over shared randomness \\Omega, with Alice's priva…","labels":["CommunicationComplexity.PublicCoin.Protocol"],"detail_key":"p14"},{"id":"n19819","layer":"informal","project":"p14","title":"Output node","kind":"definition","summary":"[Output node] The leaf node of a public-coin protocol that returns the constant value a \\in \\al…","labels":["CommunicationComplexity.PublicCoin.Protocol.output"],"detail_key":"p14"},{"id":"n19820","layer":"informal","project":"p14","title":"Alice's move","kind":"definition","summary":"[Alice's move] Constructs a public-coin protocol node at which Alice sends a single bit: given…","labels":["CommunicationComplexity.PublicCoin.Protocol.alice"],"detail_key":"p14"},{"id":"n19821","layer":"informal","project":"p14","title":"Bob's move","kind":"definition","summary":"[Bob's move] Constructs a public-coin protocol node at which Bob sends a single bit: given a fu…","labels":["CommunicationComplexity.PublicCoin.Protocol.bob"],"detail_key":"p14"},{"id":"n19822","layer":"informal","project":"p14","title":"Randomized execution","kind":"definition","summary":"[Randomized execution] Given a public-coin protocol p, private inputs x \\in X, y \\in Y, and a s…","labels":["CommunicationComplexity.PublicCoin.Protocol.rrun"],"detail_key":"p14"},{"id":"n19823","layer":"informal","project":"p14","title":"\\textttrrun unfolding","kind":"theorem","summary":"[\\textttrrun unfolding] For any public-coin protocol p, inputs x, y, and randomness \\omega, p.\\…","labels":["CommunicationComplexity.PublicCoin.Protocol.rrun_eq"],"detail_key":"p14"},{"id":"n19824","layer":"informal","project":"p14","title":"\\varepsilon-satisfies a predicate","kind":"definition","summary":"[\\varepsilon-satisfies a predicate] A public-coin protocol p \\emph\\varepsilon-satisfies a predi…","labels":["CommunicationComplexity.PublicCoin.Protocol.ApproxSatisfies"],"detail_key":"p14"},{"id":"n19825","layer":"informal","project":"p14","title":"\\varepsilon-computes a function","kind":"definition","summary":"[\\varepsilon-computes a function] A public-coin protocol p \\emph\\varepsilon-computes a function…","labels":["CommunicationComplexity.PublicCoin.Protocol.ApproxComputes"],"detail_key":"p14"},{"id":"n19826","layer":"informal","project":"p14","title":"\\textttApproxComputes equals \\textttApproxSatisfies at equality","kind":"theorem","summary":"[\\textttApproxComputes equals \\textttApproxSatisfies at equality] For any public-coin protocol…","labels":["CommunicationComplexity.PublicCoin.Protocol.ApproxComputes_eq_ApproxSatisfies"],"detail_key":"p14"},{"id":"n19827","layer":"informal","project":"p14","title":"Public-coin communication complexity","kind":"definition","summary":"[Public-coin communication complexity] The \\varepsilon-error public-coin randomized communicati…","labels":["CommunicationComplexity.PublicCoin.communicationComplexity"],"detail_key":"p14"},{"id":"n19828","layer":"informal","project":"p14","title":"Characterization of complexity bound","kind":"theorem","summary":"[Characterization of complexity bound] For f : X \\to Y \\to \\alpha, \\varepsilon \\in R, and m \\in…","labels":["CommunicationComplexity.PublicCoin.communicationComplexity_le_iff"],"detail_key":"p14"},{"id":"n19829","layer":"informal","project":"p14","title":"Lower bound characterization","kind":"theorem","summary":"[Lower bound characterization] For f : X \\to Y \\to \\alpha, \\varepsilon \\in R, and m \\in N, we h…","labels":["CommunicationComplexity.PublicCoin.le_communicationComplexity_iff"],"detail_key":"p14"},{"id":"n19830","layer":"informal","project":"p14","title":"Finite-message characterization of complexity bound","kind":"theorem","summary":"[Finite-message characterization of complexity bound] For f : X \\to Y \\to \\alpha, \\varepsilon \\…","labels":["CommunicationComplexity.PublicCoin.communicationComplexity_le_iff_finiteMessage"],"detail_key":"p14"},{"id":"n19831","layer":"informal","project":"p14","title":"Monotonicity in error","kind":"theorem","summary":"[Monotonicity in error] Public-coin communication complexity is monotone in the error parameter…","labels":["CommunicationComplexity.PublicCoin.communicationComplexity_mono"],"detail_key":"p14"},{"id":"n19832","layer":"informal","project":"p14","title":"Upper bound from finite-message protocol on general probability space","kind":"theorem","summary":"[Upper bound from finite-message protocol on general probability space] Let \\Omega be a finite…","labels":["CommunicationComplexity.PublicCoin.communicationComplexity_le_of_finiteMessage"],"detail_key":"p14"},{"id":"n19833","layer":"informal","project":"p14","title":"Map over protocol output","kind":"definition","summary":"[Map over protocol output] Given a function g : \\alpha \\to \\beta and a public-coin protocol p :…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.map"],"detail_key":"p14"},{"id":"n19834","layer":"informal","project":"p14","title":"Map commutes with running","kind":"theorem","summary":"[Map commutes with running] For any g : \\alpha \\to \\beta, protocol p, inputs x \\in X, y \\in Y,…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.map_rrun"],"detail_key":"p14"},{"id":"n19835","layer":"informal","project":"p14","title":"Map preserves complexity","kind":"theorem","summary":"[Map preserves complexity] For any g : \\alpha \\to \\beta and protocol p, the communication compl…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.map_complexity"],"detail_key":"p14"},{"id":"n19836","layer":"informal","project":"p14","title":"Bind on protocols","kind":"definition","summary":"[Bind on protocols] Given a protocol p : Protocol\\;\\Omega\\;X\\;Y\\;\\alpha and a family q : \\alpha…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.bind"],"detail_key":"p14"},{"id":"n19837","layer":"informal","project":"p14","title":"Bind commutes with running","kind":"theorem","summary":"[Bind commutes with running] For any p, q, inputs x \\in X, y \\in Y, and shared randomness \\omeg…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.bind_rrun"],"detail_key":"p14"},{"id":"n19838","layer":"informal","project":"p14","title":"Randomness reindexing","kind":"definition","summary":"[Randomness reindexing] Given h : \\Omega' \\to \\Omega and a protocol p : Protocol\\;\\Omega\\;X\\;Y\\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.comapRandomness"],"detail_key":"p14"},{"id":"n19839","layer":"informal","project":"p14","title":"Randomness reindexing commutes with running","kind":"theorem","summary":"[Randomness reindexing commutes with running] For any h : \\Omega' \\to \\Omega, protocol p, input…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.comapRandomness_rrun"],"detail_key":"p14"},{"id":"n19840","layer":"informal","project":"p14","title":"Randomness reindexing preserves complexity","kind":"theorem","summary":"[Randomness reindexing preserves complexity] For any h : \\Omega' \\to \\Omega and protocol p, rei…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.comapRandomness_complexity"],"detail_key":"p14"},{"id":"n19841","layer":"informal","project":"p14","title":"Product of two protocols","kind":"definition","summary":"[Product of two protocols] Given protocols p_1 : Protocol\\;\\Omega_1\\;X\\;Y\\;\\alpha_1 and p_2 : P…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prod"],"detail_key":"p14"},{"id":"n19842","layer":"informal","project":"p14","title":"Product runs component-wise","kind":"theorem","summary":"[Product runs component-wise] For protocols p_1, p_2, inputs x \\in X, y \\in Y, and randomness \\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prod_rrun"],"detail_key":"p14"},{"id":"n19843","layer":"informal","project":"p14","title":"Product complexity is the sum","kind":"theorem","summary":"[Product complexity is the sum] The communication complexity of prod\\;p_1\\;p_2 equals the sum o…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prod_complexity"],"detail_key":"p14"},{"id":"n19844","layer":"informal","project":"p14","title":"k-fold product of protocols","kind":"definition","summary":"[k-fold product of protocols] Given a family of protocols p : (i : Fin\\;k) \\to Protocol\\;(\\Omeg…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.pi"],"detail_key":"p14"},{"id":"n19845","layer":"informal","project":"p14","title":"k-fold product runs component-wise","kind":"theorem","summary":"[k-fold product runs component-wise] For a family p, inputs x \\in X, y \\in Y, and randomness tu…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.pi_rrun"],"detail_key":"p14"},{"id":"n19846","layer":"informal","project":"p14","title":"k-fold product complexity is the sum","kind":"theorem","summary":"[k-fold product complexity is the sum] The complexity of pi\\;p equals the sum of the complexiti…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.pi_complexity"],"detail_key":"p14"},{"id":"n19847","layer":"informal","project":"p14","title":"Bind with fresh randomness","kind":"definition","summary":"[Bind with fresh randomness] Given p : Protocol\\;\\Omega\\;X\\;Y\\;\\alpha and q : \\alpha \\to Protoc…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.rbind"],"detail_key":"p14"},{"id":"n19848","layer":"informal","project":"p14","title":"Fresh-randomness bind commutes with running","kind":"theorem","summary":"[Fresh-randomness bind commutes with running] For any p, q, inputs x \\in X, y \\in Y, and random…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.rbind_rrun"],"detail_key":"p14"},{"id":"n19849","layer":"informal","project":"p14","title":"Fresh-randomness bind complexity with constant continuation","kind":"theorem","summary":"[Fresh-randomness bind complexity with constant continuation] If every branch q\\;a has the same…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.rbind_complexity_const"],"detail_key":"p14"},{"id":"n19850","layer":"informal","project":"p14","title":"Product with a deterministic protocol","kind":"definition","summary":"[Product with a deterministic protocol] Given a public-coin protocol p_1 : Protocol\\;\\Omega\\;X\\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prodDet"],"detail_key":"p14"},{"id":"n19851","layer":"informal","project":"p14","title":"Mixed product runs component-wise","kind":"theorem","summary":"[Mixed product runs component-wise] For any p_1, p_2, inputs x \\in X, y \\in Y, and randomness \\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prodDet_rrun"],"detail_key":"p14"},{"id":"n19852","layer":"informal","project":"p14","title":"Mixed product complexity is the sum","kind":"theorem","summary":"[Mixed product complexity is the sum] The complexity of prodDet\\;p_1\\;p_2 equals the sum of the…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prodDet_complexity"],"detail_key":"p14"},{"id":"n19853","layer":"informal","project":"p14","title":"Public-coin finite-message protocol","kind":"definition","summary":"[Public-coin finite-message protocol] A public-coin finite-message protocol over shared-randomn…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol"],"detail_key":"p14"},{"id":"n19854","layer":"informal","project":"p14","title":"Output node","kind":"definition","summary":"[Output node] The terminal (output) node of a public-coin finite-message protocol returning the…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.output"],"detail_key":"p14"},{"id":"n19855","layer":"informal","project":"p14","title":"Alice's send step","kind":"definition","summary":"[Alice's send step] Given a message function f : X \\to \\Omega \\to \\beta and a continuation P :…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.alice"],"detail_key":"p14"},{"id":"n19856","layer":"informal","project":"p14","title":"Bob's send step","kind":"definition","summary":"[Bob's send step] Given a message function f : Y \\to \\Omega \\to \\beta and a continuation P : \\b…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.bob"],"detail_key":"p14"},{"id":"n19857","layer":"informal","project":"p14","title":"Randomized execution","kind":"definition","summary":"[Randomized execution] rrun(p, x, y, \\omega) executes the protocol p on private inputs x and y…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.rrun"],"detail_key":"p14"},{"id":"n19858","layer":"informal","project":"p14","title":"Randomized run equals underlying run","kind":"theorem","summary":"[Randomized run equals underlying run] For any protocol p, inputs x, y, and shared randomness \\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.rrun_eq"],"detail_key":"p14"},{"id":"n19859","layer":"informal","project":"p14","title":"Approximate satisfaction of a predicate","kind":"definition","summary":"[Approximate satisfaction of a predicate] A protocol p \\varepsilon-satisfies a predicate Q : X…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ApproxSatisfies"],"detail_key":"p14"},{"id":"n19860","layer":"informal","project":"p14","title":"Approximate computation of a function","kind":"definition","summary":"[Approximate computation of a function] A protocol p \\varepsilon-computes a function f : X \\to…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ApproxComputes"],"detail_key":"p14"},{"id":"n19861","layer":"informal","project":"p14","title":"Approximate computation equals approximate satisfaction","kind":"theorem","summary":"[Approximate computation equals approximate satisfaction] For any protocol p, function f, and e…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ApproxComputes_eq_ApproxSatisfies"],"detail_key":"p14"},{"id":"n19862","layer":"informal","project":"p14","title":"Conversion to binary public-coin protocol","kind":"definition","summary":"[Conversion to binary public-coin protocol] Converts a public-coin finite-message protocol to a…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toProtocol"],"detail_key":"p14"},{"id":"n19863","layer":"informal","project":"p14","title":"Conversion preserves run behavior","kind":"theorem","summary":"[Conversion preserves run behavior] For any protocol p, inputs x, y, and shared randomness \\ome…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toProtocol_rrun"],"detail_key":"p14"},{"id":"n19864","layer":"informal","project":"p14","title":"Conversion preserves complexity","kind":"theorem","summary":"[Conversion preserves complexity] For any protocol p, (p.toProtocol).complexity = p.complexity.","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.toProtocol_complexity"],"detail_key":"p14"},{"id":"n19865","layer":"informal","project":"p14","title":"Embedding of binary public-coin protocol","kind":"definition","summary":"[Embedding of binary public-coin protocol] Embeds a binary public-coin protocol into a finite-m…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ofProtocol"],"detail_key":"p14"},{"id":"n19866","layer":"informal","project":"p14","title":"Embedding preserves run behavior","kind":"theorem","summary":"[Embedding preserves run behavior] For any binary public-coin protocol p, inputs x, y, and shar…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ofProtocol_rrun"],"detail_key":"p14"},{"id":"n19867","layer":"informal","project":"p14","title":"Embedding preserves complexity","kind":"theorem","summary":"[Embedding preserves complexity] For any binary public-coin protocol p, (ofProtocol\\,p).complex…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ofProtocol_complexity"],"detail_key":"p14"},{"id":"n19868","layer":"informal","project":"p14","title":"Finite-message equivalence of binary public-coin protocols","kind":"theorem","summary":"[Finite-message equivalence of binary public-coin protocols] For any binary public-coin protoco…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.ofProtocol_equiv"],"detail_key":"p14"},{"id":"n19869","layer":"informal","project":"p14","title":"One-way public-coin protocol","kind":"definition","summary":"[One-way public-coin protocol] A one-way public-coin protocol over shared randomness type \\Omeg…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol"],"detail_key":"p14"},{"id":"n19870","layer":"informal","project":"p14","title":"Protocol execution with shared randomness","kind":"definition","summary":"[Protocol execution with shared randomness] Given a protocol p, inputs x \\in X, y \\in Y, and a…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol.rrun"],"detail_key":"p14"},{"id":"n19871","layer":"informal","project":"p14","title":"\\varepsilon-approximate computation","kind":"definition","summary":"[\\varepsilon-approximate computation] A protocol p \\varepsilon-computes a function f : X \\to Y…","labels":["CommunicationComplexity.PublicCoin.OneWay.Protocol.ApproxComputes"],"detail_key":"p14"},{"id":"n19872","layer":"informal","project":"p14","title":"\\varepsilon-error one-way public-coin communication complexity","kind":"definition","summary":"[\\varepsilon-error one-way public-coin communication complexity] The \\varepsilon-error one-way…","labels":["CommunicationComplexity.PublicCoin.OneWay.communicationComplexity"],"detail_key":"p14"},{"id":"n19873","layer":"informal","project":"p14","title":"Upper bound characterisation","kind":"theorem","summary":"[Upper bound characterisation] For f : X \\to Y \\to \\alpha, \\varepsilon \\in R, and m \\in N, the…","labels":["CommunicationComplexity.PublicCoin.OneWay.communicationComplexity_le_iff"],"detail_key":"p14"},{"id":"n19874","layer":"informal","project":"p14","title":"Lower bound characterisation","kind":"theorem","summary":"[Lower bound characterisation] For f : X \\to Y \\to \\alpha, \\varepsilon \\in R, and m \\in N, the…","labels":["CommunicationComplexity.PublicCoin.OneWay.le_communicationComplexity_iff"],"detail_key":"p14"},{"id":"n19875","layer":"informal","project":"p14","title":"Monotonicity in error","kind":"theorem","summary":"[Monotonicity in error] The \\varepsilon-error one-way public-coin communication complexity of f…","labels":["CommunicationComplexity.PublicCoin.OneWay.communicationComplexity_mono"],"detail_key":"p14"},{"id":"n19876","layer":"informal","project":"p14","title":"Schnorr transcript","kind":"definition","summary":"[Schnorr transcript] A Schnorr transcript is a triple (a, c, s) \\in G \\times Z_q \\times Z_q, wh…","labels":["Schnorr.Transcript"],"detail_key":"p14"},{"id":"n19877","layer":"informal","project":"p14","title":"Prover commitment","kind":"definition","summary":"[Prover commitment] Given randomness r \\in Z_q and a generator g \\in G, the commitment is a :=…","labels":["Schnorr.commit"],"detail_key":"p14"},{"id":"n19878","layer":"informal","project":"p14","title":"Prover response","kind":"definition","summary":"[Prover response] Given witness w, randomness r, and challenge c in Z_q, the response is s := r…","labels":["Schnorr.respond"],"detail_key":"p14"},{"id":"n19879","layer":"informal","project":"p14","title":"Verifier predicate","kind":"definition","summary":"[Verifier predicate] The verifier accepts a transcript (a, c, s) with respect to public key \\ma…","labels":["Schnorr.Verify"],"detail_key":"p14"},{"id":"n19880","layer":"informal","project":"p14","title":"Honest transcript","kind":"definition","summary":"[Honest transcript] The honest prover's full transcript for witness w, randomness r, and challe…","labels":["Schnorr.honest"],"detail_key":"p14"},{"id":"n19881","layer":"informal","project":"p14","title":"Simulator transcript","kind":"definition","summary":"[Simulator transcript] The zero-knowledge simulator, given public key \\mathitpk, challenge c, a…","labels":["Schnorr.simulate"],"detail_key":"p14"},{"id":"n19882","layer":"informal","project":"p14","title":"Witness extractor","kind":"definition","summary":"[Witness extractor] Given two challenges c_1, c_2 and corresponding responses s_1, s_2 in Z_q,…","labels":["Schnorr.extract"],"detail_key":"p14"},{"id":"n19883","layer":"informal","project":"p14","title":"Completeness","kind":"theorem","summary":"[Completeness] Let g \\in G be an element of order q and let w, r, c \\in Z_q. Then the honest tr…","labels":["Schnorr.schnorr_completeness"],"detail_key":"p14"},{"id":"n19884","layer":"informal","project":"p14","title":"Special soundness","kind":"theorem","summary":"[Special soundness] Let g \\in G have order q, and let a \\in G, c_1 \\neq c_2 \\in Z_q, s_1, s_2 \\…","labels":["Schnorr.schnorr_soundness"],"detail_key":"p14"},{"id":"n19885","layer":"informal","project":"p14","title":"Randomness reindexing","kind":"definition","summary":"[Randomness reindexing] For fixed w, c \\in Z_q, define the bijection \\sigma_w,c : Z_q \\xrightar…","labels":["Schnorr.reindex"],"detail_key":"p14"},{"id":"n19886","layer":"informal","project":"p14","title":"Honest-verifier zero knowledge","kind":"theorem","summary":"[Honest-verifier zero knowledge] Let g \\in G have order q and let w, c \\in Z_q. The honest tran…","labels":["Schnorr.schnorr_hvzk"],"detail_key":"p14"},{"id":"n19887","layer":"informal","project":"p14","title":"Linearity of a Boolean function","kind":"definition","summary":"[Linearity of a Boolean function] A function f : \\0,1\\^n \\to \\0,1\\ is \\emphlinear if for all x,…","labels":["BoolBLR.is_linear_bool"],"detail_key":"p14"},{"id":"n19888","layer":"informal","project":"p14","title":"\\pm 1 lift of a Boolean function","kind":"definition","summary":"[\\pm 1 lift of a Boolean function] Given f : \\0,1\\^n \\to \\0,1\\, its \\emph\\pm 1 lift \\textttBool…","labels":["BoolBLR.lift_pm1"],"detail_key":"p14"},{"id":"n19889","layer":"informal","project":"p14","title":"Boolean distance","kind":"definition","summary":"[Boolean distance] The \\emphdistance between two Boolean functions f, g : \\0,1\\^n \\to \\0,1\\ is…","labels":["BoolBLR.bool_dist"],"detail_key":"p14"},{"id":"n19890","layer":"informal","project":"p14","title":"\\varepsilon-far from linear","kind":"definition","summary":"[\\varepsilon-far from linear] A function f : \\0,1\\^n \\to \\0,1\\ is \\emph\\varepsilon-far from lin…","labels":["BoolBLR.epsilon_far_from_linear"],"detail_key":"p14"},{"id":"n19891","layer":"informal","project":"p14","title":"Linear iff equal to a Fourier character","kind":"lemma","summary":"[Linear iff equal to a Fourier character] A Boolean function f : \\0,1\\^n \\to \\0,1\\ is linear if…","labels":["BoolBLR.linear_bool_iff_character"],"detail_key":"p14"},{"id":"n19892","layer":"informal","project":"p14","title":"BLR acceptance probability","kind":"definition","summary":"[BLR acceptance probability] The \\emphBLR acceptance probability of f : \\0,1\\^n \\to \\0,1\\ is \\[…","labels":["BoolBLR.BLR_accept_prob"],"detail_key":"p14"},{"id":"n19893","layer":"informal","project":"p14","title":"BLR acceptance in \\pm 1 form","kind":"lemma","summary":"[BLR acceptance in \\pm 1 form] For any f : \\0,1\\^n \\to \\0,1\\, \\[ \\Pr[\\textBLR accepts] \\;=\\; \\f…","labels":["BoolBLR.BLR_accept_prob_pm1"],"detail_key":"p14"},{"id":"n19894","layer":"informal","project":"p14","title":"Triple expectation as convolution","kind":"lemma","summary":"[Triple expectation as convolution] For any f : \\0,1\\^n \\to R, \\[ E_x\\!\\left[E_y\\!\\left[f(x)\\,f…","labels":["BoolBLR.triple_expectation_as_convolution"],"detail_key":"p14"},{"id":"n19895","layer":"informal","project":"p14","title":"Triple expectation equals sum of cubed Fourier coefficients","kind":"lemma","summary":"[Triple expectation equals sum of cubed Fourier coefficients] For any f : \\0,1\\^n \\to \\0,1\\, le…","labels":["BoolBLR.triple_expectation_eq_cube_fourier"],"detail_key":"p14"},{"id":"n19896","layer":"informal","project":"p14","title":"Fourier coefficient bound from distance","kind":"lemma","summary":"[Fourier coefficient bound from distance] Let f, g : \\0,1\\^n \\to \\0,1\\, let S \\subseteq [n], an…","labels":["BoolBLR.fourier_coeff_le_of_dist_ge"],"detail_key":"p14"},{"id":"n19897","layer":"informal","project":"p14","title":"Fourier coefficient bound when \\varepsilon-far from linear","kind":"lemma","summary":"[Fourier coefficient bound when \\varepsilon-far from linear] If f : \\0,1\\^n \\to \\0,1\\ is \\varep…","labels":["BoolBLR.fourier_coeff_le_of_far_from_linear"],"detail_key":"p14"},{"id":"n19898","layer":"informal","project":"p14","title":"Soundness bound via Fourier analysis","kind":"lemma","summary":"[Soundness bound via Fourier analysis] If f : \\0,1\\^n \\to \\0,1\\ is \\varepsilon-far from linear,…","labels":["BoolBLR.BLR_soundness_via_fourier"],"detail_key":"p14"},{"id":"n19899","layer":"informal","project":"p14","title":"BLR completeness","kind":"lemma","summary":"[BLR completeness] If f : \\0,1\\^n \\to \\0,1\\ is linear, then \\textttBoolBLR.BLR\\_accept\\_prob\\,f…","labels":["BoolBLR.BLR_completeness"],"detail_key":"p14"},{"id":"n19900","layer":"informal","project":"p14","title":"BLR soundness","kind":"lemma","summary":"[BLR soundness] If f : \\0,1\\^n \\to \\0,1\\ is \\varepsilon-far from linear, then \\[ \\textttBoolBLR…","labels":["BoolBLR.BLR_soundness"],"detail_key":"p14"},{"id":"n19901","layer":"informal","project":"p14","title":"Value of a linear function on an indicator vector","kind":"lemma","summary":"[Value of a linear function on an indicator vector] Let f : \\0,1\\^n \\to \\0,1\\ be linear. Then f…","labels":["BoolBLR.linear_bool_iff_character_aux_h_fx_1"],"detail_key":"p14"},{"id":"n19902","layer":"informal","project":"p14","title":"Value of a linear function via its support","kind":"lemma","summary":"[Value of a linear function via its support] Specialization of the previous lemma to the suppor…","labels":["BoolBLR.linear_bool_iff_character_aux_h_fx_2"],"detail_key":"p14"},{"id":"n19903","layer":"informal","project":"p14","title":"Multiplicativity of the Fourier characters","kind":"lemma","summary":"[Multiplicativity of the Fourier characters] For every S \\subseteq [n] and all x, y \\in \\0,1\\^n…","labels":["BoolBLR.linear_bool_iff_character_aux_h_char"],"detail_key":"p14"},{"id":"n19904","layer":"informal","project":"p14","title":"\\pm 1 lift of a character is XOR-additive","kind":"lemma","summary":"[\\pm 1 lift of a character is XOR-additive] Let f : \\0,1\\^n \\to \\0,1\\ satisfy \\textttBoolBLR.li…","labels":["BoolBLR.linear_bool_iff_character_aux_h_eq"],"detail_key":"p14"},{"id":"n19905","layer":"informal","project":"p14","title":"Value of a linear function via its support, restated","kind":"lemma","summary":"[Value of a linear function via its support, restated] Restatement of the support formula in th…","labels":["BoolBLR.linear_bool_iff_character_aux_h_fx"],"detail_key":"p14"},{"id":"n19906","layer":"informal","project":"p14","title":"XOR-additivity of a function whose lift is a character","kind":"lemma","summary":"[XOR-additivity of a function whose lift is a character] If \\textttBoolBLR.lift\\_pm1\\,f = \\chi_…","labels":["BoolBLR.linear_bool_iff_character_aux_h_eq_h"],"detail_key":"p14"},{"id":"n19907","layer":"informal","project":"p14","title":"Pointwise \\pm 1 form of the BLR indicator","kind":"lemma","summary":"[Pointwise \\pm 1 form of the BLR indicator] Write g = \\textttBoolBLR.lift\\_pm1\\,f. For all x, y…","labels":["BoolBLR.BLR_accept_prob_pm1_aux_h_eq"],"detail_key":"p14"},{"id":"n19908","layer":"informal","project":"p14","title":"Fourier expansion of the self-convolution","kind":"lemma","summary":"[Fourier expansion of the self-convolution] Write g = \\textttBoolBLR.lift\\_pm1\\,f. For every x…","labels":["BoolBLR.triple_expectation_eq_cube_fourier_aux_h_convolution"],"detail_key":"p14"},{"id":"n19909","layer":"informal","project":"p14","title":"Substituting the convolution expansion into the expectation","kind":"lemma","summary":"[Substituting the convolution expansion into the expectation] Write g = \\textttBoolBLR.lift\\_pm…","labels":["BoolBLR.triple_expectation_eq_cube_fourier_aux_h_substitute"],"detail_key":"p14"},{"id":"n19910","layer":"informal","project":"p14","title":"Product of \\pm 1 lifts as a disagreement indicator","kind":"lemma","summary":"[Product of \\pm 1 lifts as a disagreement indicator] For all f, g : \\0,1\\^n \\to \\0,1\\ and every…","labels":["BoolBLR.fourier_coeff_le_of_dist_ge_aux_h_lift_pm1"],"detail_key":"p14"},{"id":"n19911","layer":"informal","project":"p14","title":"Parseval for a \\pm 1 lift","kind":"lemma","summary":"[Parseval for a \\pm 1 lift] For every f : \\0,1\\^n \\to \\0,1\\, the \\pm 1 lift g = \\textttBoolBLR.…","labels":["BoolBLR.BLR_soundness_via_fourier_aux_hparseval"],"detail_key":"p14"},{"id":"n19912","layer":"informal","project":"p14","title":"Per-coefficient cube bound for \\varepsilon-far functions","kind":"lemma","summary":"[Per-coefficient cube bound for \\varepsilon-far functions] Let f : \\0,1\\^n \\to \\0,1\\ be \\vareps…","labels":["BoolBLR.BLR_soundness_via_fourier_aux_h_bound"],"detail_key":"p14"},{"id":"n19913","layer":"informal","project":"p14","title":"Unfolded form of Boolean linearity","kind":"lemma","summary":"[Unfolded form of Boolean linearity] If f : \\0,1\\^n \\to \\0,1\\ is linear, then f(x \\oplus y) = f…","labels":["BoolBLR.BLR_completeness_aux_h_linear"],"detail_key":"p14"},{"id":"n19914","layer":"informal","project":"p14","title":"Multiplicativity of the lift of a linear function","kind":"lemma","summary":"[Multiplicativity of the lift of a linear function] Let f : \\0,1\\^n \\to \\0,1\\ satisfy f(x \\oplu…","labels":["BoolBLR.BLR_completeness_aux_h_lift_linear"],"detail_key":"p14"},{"id":"n19915","layer":"informal","project":"p14","title":"Hypercube","kind":"definition","summary":"[Hypercube] The Boolean hypercube of dimension n is the type \\0,1\\^n, defined as an abbreviatio…","labels":["BoolFourier.hypercube"],"detail_key":"p14"},{"id":"n19916","layer":"informal","project":"p14","title":"Boolean function type","kind":"definition","summary":"[Boolean function type] A Boolean function of arity n is an element of type \\0,1\\^n \\to R, defi…","labels":["BoolFourier.BoolFun"],"detail_key":"p14"},{"id":"n19917","layer":"informal","project":"p14","title":"Bool-to-\\pm 1 embedding","kind":"definition","summary":"[Bool-to-\\pm 1 embedding] The map BoolToPM1 : Bool \\to R sends \\mathttfalse to 1 and \\mathtttru…","labels":["BoolFourier.BoolToPM1"],"detail_key":"p14"},{"id":"n19918","layer":"informal","project":"p14","title":"Partial inverse of \\pm 1 embedding","kind":"definition","summary":"[Partial inverse of \\pm 1 embedding] The partial inverse PM1ToBool? : R \\to Option\\,Bool return…","labels":["BoolFourier.PM1ToBool?"],"detail_key":"p14"},{"id":"n19919","layer":"informal","project":"p14","title":"All-zeros vector","kind":"definition","summary":"[All-zeros vector] zero\\_vec(n) : \\0,1\\^n is the constant-\\mathttfalse function, serving as the…","labels":["BoolFourier.zero_vec"],"detail_key":"p14"},{"id":"n19920","layer":"informal","project":"p14","title":"Componentwise XOR","kind":"definition","summary":"[Componentwise XOR] For x, y \\in \\0,1\\^n, the vector xor\\_vec(x,y)_i = x_i \\oplus y_i is the co…","labels":["BoolFourier.xor_vec"],"detail_key":"p14"},{"id":"n19921","layer":"informal","project":"p14","title":"Cardinality of the hypercube","kind":"lemma","summary":"[Cardinality of the hypercube] For every n : N, the Boolean hypercube satisfies |\\0,1\\^n| = 2^n.","labels":["BoolFourier.card_hypercube"],"detail_key":"p14"},{"id":"n19922","layer":"informal","project":"p14","title":"Average of \\pm 1 values is zero","kind":"lemma","summary":"[Average of \\pm 1 values is zero] The average of the two values of the \\pm 1 embedding is zero:…","labels":["BoolFourier.avg_BoolToPM1"],"detail_key":"p14"},{"id":"n19923","layer":"informal","project":"p14","title":"\\pm 1 values square to one","kind":"lemma","summary":"[\\pm 1 values square to one] For every b : Bool, BoolToPM1(b)^2 = 1.","labels":["BoolFourier.BoolToPM1_sq"],"detail_key":"p14"},{"id":"n19924","layer":"informal","project":"p14","title":"Negation under \\pm 1 embedding","kind":"lemma","summary":"[Negation under \\pm 1 embedding] For every b : Bool, BoolToPM1(\\lnot b) = -BoolToPM1(b).","labels":["BoolFourier.BoolToPM1_not"],"detail_key":"p14"},{"id":"n19925","layer":"informal","project":"p14","title":"XOR under \\pm 1 embedding","kind":"lemma","summary":"[XOR under \\pm 1 embedding] For all a, b : Bool, BoolToPM1(a \\oplus b) = BoolToPM1(a) \\cdot Boo…","labels":["BoolFourier.BoolToPM1_xor"],"detail_key":"p14"},{"id":"n19926","layer":"informal","project":"p14","title":"Uniform expectation","kind":"definition","summary":"[Uniform expectation] For f : \\0,1\\^n \\to R, the uniform expectation is \\[ E[f] = \\frac12^n \\su…","labels":["BoolFourier.expectation"],"detail_key":"p14"},{"id":"n19927","layer":"informal","project":"p14","title":"Factorisation of expectation over product functions","kind":"lemma","summary":"[Factorisation of expectation over product functions] If g : Fin\\,n \\to Bool \\to R and f(x) = \\…","labels":["BoolFourier.expectation_factorizes"],"detail_key":"p14"},{"id":"n19928","layer":"informal","project":"p14","title":"Inner product of Boolean functions","kind":"definition","summary":"[Inner product of Boolean functions] The inner product of f, g : \\0,1\\^n \\to R is \\[ \\langle f,…","labels":["BoolFourier.inner_product"],"detail_key":"p14"},{"id":"n19929","layer":"informal","project":"p14","title":"L^2 norm squared","kind":"definition","summary":"[L^2 norm squared] The squared L^2 norm of f : \\0,1\\^n \\to R is \\[ \\|f\\|_2^2 = E[f^2] = \\frac12…","labels":["BoolFourier.L2_norm_sq"],"detail_key":"p14"},{"id":"n19930","layer":"informal","project":"p14","title":"Convolution on the hypercube","kind":"definition","summary":"[Convolution on the hypercube] The convolution of f, g : \\0,1\\^n \\to R is the function \\[ (f *…","labels":["BoolFourier.convolution"],"detail_key":"p14"},{"id":"n19931","layer":"informal","project":"p14","title":"Walsh--Fourier character","kind":"definition","summary":"[Walsh--Fourier character] For a set S \\subseteq [n], the Walsh character \\chi_S : \\0,1\\^n \\to…","labels":["BoolFourier.char_S"],"detail_key":"p14"},{"id":"n19932","layer":"informal","project":"p14","title":"Character evaluated at zero","kind":"lemma","summary":"[Character evaluated at zero] For every S \\subseteq [n], \\chi_S(0) = 1.","labels":["BoolFourier.char_S_of_zero"],"detail_key":"p14"},{"id":"n19933","layer":"informal","project":"p14","title":"Character squared equals one","kind":"lemma","summary":"[Character squared equals one] For every S \\subseteq [n] and x \\in \\0,1\\^n, \\chi_S(x)^2 = 1.","labels":["BoolFourier.char_S_sq"],"detail_key":"p14"},{"id":"n19934","layer":"informal","project":"p14","title":"Empty character is constant one","kind":"lemma","summary":"[Empty character is constant one] \\chi_\\emptyset = 1, the constant function equal to 1.","labels":["BoolFourier.char_S_empty"],"detail_key":"p14"},{"id":"n19935","layer":"informal","project":"p14","title":"Product of characters","kind":"lemma","summary":"[Product of characters] For all S, T \\subseteq [n] and x \\in \\0,1\\^n, \\[ \\chi_S(x)\\,\\chi_T(x) =…","labels":["BoolFourier.char_S_times_char_T"],"detail_key":"p14"},{"id":"n19936","layer":"informal","project":"p14","title":"Sum of characters at zero equals 2^n","kind":"lemma","summary":"[Sum of characters at zero equals 2^n] \\[ \\sum_S \\subseteq [n] \\chi_S(0) = 2^n. \\]","labels":["BoolFourier.sum_char_S_at_zero"],"detail_key":"p14"},{"id":"n19937","layer":"informal","project":"p14","title":"Sum of characters at non-zero point is zero","kind":"lemma","summary":"[Sum of characters at non-zero point is zero] For every x \\in \\0,1\\^n with x \\ne 0, \\[ \\sum_S \\…","labels":["BoolFourier.sum_char_S_ne_zero"],"detail_key":"p14"},{"id":"n19938","layer":"informal","project":"p14","title":"Expectation of empty character is one","kind":"lemma","summary":"[Expectation of empty character is one] E[\\chi_\\emptyset] = 1.","labels":["BoolFourier.expectation_char_empty"],"detail_key":"p14"},{"id":"n19939","layer":"informal","project":"p14","title":"Expectation of non-empty character is zero","kind":"lemma","summary":"[Expectation of non-empty character is zero] If S \\ne \\emptyset, then E[\\chi_S] = 0.","labels":["BoolFourier.expectation_char_nonempty"],"detail_key":"p14"},{"id":"n19940","layer":"informal","project":"p14","title":"Orthonormality: self inner product","kind":"lemma","summary":"[Orthonormality: self inner product] For every S \\subseteq [n], \\langle \\chi_S, \\chi_S \\rangle…","labels":["BoolFourier.inner_product_char_self"],"detail_key":"p14"},{"id":"n19941","layer":"informal","project":"p14","title":"Orthonormality: distinct characters are orthogonal","kind":"lemma","summary":"[Orthonormality: distinct characters are orthogonal] If S \\ne T, then \\langle \\chi_S, \\chi_T \\r…","labels":["BoolFourier.inner_product_char_nonself"],"detail_key":"p14"},{"id":"n19942","layer":"informal","project":"p14","title":"Fourier coefficient","kind":"definition","summary":"[Fourier coefficient] The Fourier coefficient of f : \\0,1\\^n \\to R at S \\subseteq [n] is \\[ \\ha…","labels":["BoolFourier.fourier_coeff"],"detail_key":"p14"},{"id":"n19943","layer":"informal","project":"p14","title":"Fourier coefficient of a character at itself","kind":"lemma","summary":"[Fourier coefficient of a character at itself] For every S \\subseteq [n], \\widehat\\chi_S(S) = 1.","labels":["BoolFourier.fourier_coeff_char_self"],"detail_key":"p14"},{"id":"n19944","layer":"informal","project":"p14","title":"Fourier coefficient of a character at a different set","kind":"lemma","summary":"[Fourier coefficient of a character at a different set] If S \\ne T, then \\widehat\\chi_S(T) = 0.","labels":["BoolFourier.fourier_coeff_char_of_ne"],"detail_key":"p14"},{"id":"n19945","layer":"informal","project":"p14","title":"Expectation equals uniform-weight expectation","kind":"lemma","summary":"[Expectation equals uniform-weight expectation] The locally-defined expectation \\textttBoolFour…","labels":["BoolFourier.expectation_eq_expect"],"detail_key":"p14"},{"id":"n19946","layer":"informal","project":"p14","title":"Fourier coefficient agrees with upstream definition","kind":"lemma","summary":"[Fourier coefficient agrees with upstream definition] For every f and S, fourier\\_coeff(f, S) =…","labels":["BoolFourier.fourier_coeff_eq"],"detail_key":"p14"},{"id":"n19947","layer":"informal","project":"p14","title":"Fourier--Walsh expansion","kind":"lemma","summary":"[Fourier--Walsh expansion] Every Boolean function f : \\0,1\\^n \\to R expands in the Walsh--Fouri…","labels":["BoolFourier.fourier_expansion"],"detail_key":"p14"},{"id":"n19948","layer":"informal","project":"p14","title":"Parseval's identity","kind":"lemma","summary":"[Parseval's identity] For every f : \\0,1\\^n \\to R, \\[ \\sum_S \\subseteq [n] \\hat f(S)^2 = \\|f\\|_…","labels":["BoolFourier.parseval_identity"],"detail_key":"p14"},{"id":"n19949","layer":"informal","project":"p14","title":"Convolution theorem","kind":"lemma","summary":"[Convolution theorem] For all f, g : \\0,1\\^n \\to R and S \\subseteq [n], \\[ \\widehatf * g(S) = \\…","labels":["BoolFourier.fourier_coeff_convolution"],"detail_key":"p14"},{"id":"n19950","layer":"informal","project":"p14","title":"Character squared equals one, powered form","kind":"lemma","summary":"[Character squared equals one, powered form] For every S \\subseteq [n] and every x \\in \\0,1\\^n,…","labels":["BoolFourier.char_S_sq_aux_h"],"detail_key":"p14"},{"id":"n19951","layer":"informal","project":"p14","title":"Characters at the origin form the constant function one","kind":"lemma","summary":"[Characters at the origin form the constant function one] As functions of S, the evaluation of…","labels":["BoolFourier.sum_char_S_at_zero_aux_h"],"detail_key":"p14"},{"id":"n19952","layer":"informal","project":"p14","title":"Sign-flip pairing of characters at a non-zero point","kind":"lemma","summary":"[Sign-flip pairing of characters at a non-zero point] Let x \\in \\0,1\\^n with x \\ne 0 and let i…","labels":["BoolFourier.sum_char_S_ne_zero_aux_h_pair"],"detail_key":"p14"},{"id":"n19953","layer":"informal","project":"p14","title":"Expectation of a character as a coordinate product","kind":"lemma","summary":"[Expectation of a character as a coordinate product] For a non-empty S \\subseteq [n], the unifo…","labels":["BoolFourier.expectation_char_nonempty_aux_h_exp"],"detail_key":"p14"},{"id":"n19954","layer":"informal","project":"p14","title":"L^2 norm as a self inner product","kind":"lemma","summary":"[L^2 norm as a self inner product] For every f : \\0,1\\^n \\to R, the locally-defined squared L^2…","labels":["BoolFourier.parseval_identity_aux_hL2"],"detail_key":"p14"},{"id":"n19955","layer":"informal","project":"p14","title":"Interchange of summation in the convolution coefficient","kind":"lemma","summary":"[Interchange of summation in the convolution coefficient] For all f, g : \\0,1\\^n \\to R and S \\s…","labels":["BoolFourier.fourier_coeff_convolution_aux_h_fubini"],"detail_key":"p14"},{"id":"n19956","layer":"informal","project":"p14","title":"Multiplicative splitting of a character along XOR","kind":"lemma","summary":"[Multiplicative splitting of a character along XOR] For all x, y \\in \\0,1\\^n and S \\subseteq [n…","labels":["BoolFourier.fourier_coeff_convolution_aux_h_split"],"detail_key":"p14"},{"id":"n19957","layer":"informal","project":"p14","title":"Shift rule for character-weighted sums","kind":"lemma","summary":"[Shift rule for character-weighted sums] Let f, g : \\0,1\\^n \\to R, let S \\subseteq [n], assume…","labels":["BoolFourier.fourier_coeff_convolution_aux_h_char"],"detail_key":"p14"},{"id":"n19958","layer":"informal","project":"p14","title":"Commutativity of XOR","kind":"lemma","summary":"[Commutativity of XOR] For all x, y \\in \\0,1\\^n, coordinate-wise XOR satisfies x \\oplus y = y \\…","labels":["LowDegreeTest.xor_vec_comm"],"detail_key":"p14"},{"id":"n19959","layer":"informal","project":"p14","title":"XOR with zero vector","kind":"lemma","summary":"[XOR with zero vector] For all x \\in \\0,1\\^n, we have x \\oplus 0 = x, where 0 denotes the all-z…","labels":["LowDegreeTest.xor_vec_zero"],"detail_key":"p14"},{"id":"n19960","layer":"informal","project":"p14","title":"XOR self-inverse","kind":"lemma","summary":"[XOR self-inverse] For all x \\in \\0,1\\^n, we have x \\oplus x = 0, so every element is its own i…","labels":["LowDegreeTest.xor_vec_self"],"detail_key":"p14"},{"id":"n19961","layer":"informal","project":"p14","title":"Associativity of XOR","kind":"lemma","summary":"[Associativity of XOR] For all x, y, z \\in \\0,1\\^n, coordinate-wise XOR is associative: (x \\opl…","labels":["LowDegreeTest.xor_vec_assoc"],"detail_key":"p14"},{"id":"n19962","layer":"informal","project":"p14","title":"Multiplicative derivative","kind":"definition","summary":"[Multiplicative derivative] For a function f : \\0,1\\^n \\to R and a direction vector h \\in \\0,1\\…","labels":["LowDegreeTest.mult_deriv"],"detail_key":"p14"},{"id":"n19963","layer":"informal","project":"p14","title":"Gowers product","kind":"definition","summary":"[Gowers product] For f : \\0,1\\^n \\to R, base point x \\in \\0,1\\^n, and direction vectors h_1, \\l…","labels":["LowDegreeTest.gowers_product"],"detail_key":"p14"},{"id":"n19964","layer":"informal","project":"p14","title":"Gowers product base case","kind":"lemma","summary":"[Gowers product base case] For any f : \\0,1\\^n \\to R and x \\in \\0,1\\^n, the order-0 Gowers prod…","labels":["LowDegreeTest.gowers_product_zero"],"detail_key":"p14"},{"id":"n19965","layer":"informal","project":"p14","title":"Gowers product recursive step","kind":"lemma","summary":"[Gowers product recursive step] For any f, k \\ge 0, x, and direction vectors h_1,\\ldots,h_k+1,…","labels":["LowDegreeTest.gowers_product_succ"],"detail_key":"p14"},{"id":"n19966","layer":"informal","project":"p14","title":"Order-1 Gowers product is multiplicative derivative","kind":"lemma","summary":"[Order-1 Gowers product is multiplicative derivative] For f : \\0,1\\^n \\to R, x, h \\in \\0,1\\^n,…","labels":["LowDegreeTest.mult_deriv_eq_gowers_one"],"detail_key":"p14"},{"id":"n19967","layer":"informal","project":"p14","title":"Gowers product of \\pm 1 functions","kind":"lemma","summary":"[Gowers product of \\pm 1 functions] If f : \\0,1\\^n \\to \\0,1\\ and F = (-1)^f is its \\pm 1 lift,…","labels":["LowDegreeTest.gowers_product_pm1"],"detail_key":"p14"},{"id":"n19968","layer":"informal","project":"p14","title":"Multi-expectation","kind":"definition","summary":"[Multi-expectation] For g : (\\0,1\\^n)^k \\to R, the multi-expectation is the uniform average \\[…","labels":["LowDegreeTest.multi_expectation"],"detail_key":"p14"},{"id":"n19969","layer":"informal","project":"p14","title":"Cardinality of multi-hypercube","kind":"lemma","summary":"[Cardinality of multi-hypercube] The number of k-tuples of vectors in \\0,1\\^n is |(\\0,1\\^n)^k|…","labels":["LowDegreeTest.card_multi_hypercube"],"detail_key":"p14"},{"id":"n19970","layer":"informal","project":"p14","title":"Multi-expectation of constant","kind":"lemma","summary":"[Multi-expectation of constant] For any constant c \\in R, E_h_1,\\ldots,h_k[c] = c.","labels":["LowDegreeTest.multi_expectation_const"],"detail_key":"p14"},{"id":"n19971","layer":"informal","project":"p14","title":"Degree \\le d for \\pm 1 functions","kind":"definition","summary":"[Degree \\le d for \\pm 1 functions] A function f : \\0,1\\^n \\to R has degree \\le d if all its (d+…","labels":["LowDegreeTest.is_degree_le_pm1"],"detail_key":"p14"},{"id":"n19972","layer":"informal","project":"p14","title":"Degree \\le d for Boolean functions","kind":"definition","summary":"[Degree \\le d for Boolean functions] A Boolean function f : \\0,1\\^n \\to \\0,1\\ has degree \\le d…","labels":["LowDegreeTest.is_degree_le_bool"],"detail_key":"p14"},{"id":"n19973","layer":"informal","project":"p14","title":"Degree monotone in d","kind":"lemma","summary":"[Degree monotone in d] If f has degree \\le d, then f also has degree \\le d+1.","labels":["LowDegreeTest.degree_le_succ"],"detail_key":"p14"},{"id":"n19974","layer":"informal","project":"p14","title":"Degree \\le 0 iff constant","kind":"lemma","summary":"[Degree \\le 0 iff constant] A Boolean function f : \\0,1\\^n \\to \\0,1\\ has degree \\le 0 if and on…","labels":["LowDegreeTest.degree_le_zero_iff_constant"],"detail_key":"p14"},{"id":"n19975","layer":"informal","project":"p14","title":"Linear functions have degree \\le 1","kind":"lemma","summary":"[Linear functions have degree \\le 1] If f : \\0,1\\^n \\to \\0,1\\ is F_2-linear (i.e.\\ f(x \\oplus y…","labels":["LowDegreeTest.linear_is_degree_le_one"],"detail_key":"p14"},{"id":"n19976","layer":"informal","project":"p14","title":"Degree \\le 1 implies affine","kind":"lemma","summary":"[Degree \\le 1 implies affine] If f : \\0,1\\^n \\to \\0,1\\ has degree \\le 1, then f is affine: eith…","labels":["LowDegreeTest.degree_le_one_implies_affine"],"detail_key":"p14"},{"id":"n19977","layer":"informal","project":"p14","title":"Reed--Muller code","kind":"definition","summary":"[Reed--Muller code] The Reed--Muller code RM(d, m) is the set of all Boolean functions f : \\0,1…","labels":["LowDegreeTest.ReedMuller"],"detail_key":"p14"},{"id":"n19978","layer":"informal","project":"p14","title":"Reed--Muller code is monotone in degree","kind":"lemma","summary":"[Reed--Muller code is monotone in degree] RM(d, n) \\subseteq RM(d+1, n) for every d and n.","labels":["LowDegreeTest.ReedMuller_monotone"],"detail_key":"p14"},{"id":"n19979","layer":"informal","project":"p14","title":"Zero function in every Reed--Muller code","kind":"lemma","summary":"[Zero function in every Reed--Muller code] The constant-false function 0 : \\0,1\\^n \\to \\0,1\\ be…","labels":["LowDegreeTest.zero_mem_ReedMuller"],"detail_key":"p14"},{"id":"n19980","layer":"informal","project":"p14","title":"One function in every Reed--Muller code","kind":"lemma","summary":"[One function in every Reed--Muller code] The constant-true function 1 : \\0,1\\^n \\to \\0,1\\ belo…","labels":["LowDegreeTest.one_mem_ReedMuller"],"detail_key":"p14"},{"id":"n19981","layer":"informal","project":"p14","title":"Linear functions in RM(1,n)","kind":"lemma","summary":"[Linear functions in RM(1,n)] Every F_2-linear function f : \\0,1\\^n \\to \\0,1\\ belongs to RM(1,…","labels":["LowDegreeTest.linear_mem_ReedMuller_one"],"detail_key":"p14"},{"id":"n19982","layer":"informal","project":"p14","title":"RM(1,n) consists of affine functions","kind":"lemma","summary":"[RM(1,n) consists of affine functions] Every f \\in RM(1, n) is affine: either f is F_2-linear o…","labels":["LowDegreeTest.ReedMuller_one_is_affine"],"detail_key":"p14"},{"id":"n19983","layer":"informal","project":"p14","title":"Gowers norm (power form)","kind":"definition","summary":"[Gowers norm (power form)] The k-th Gowers uniformity norm to the 2^k-th power is \\[ \\|f\\|_U^k^…","labels":["LowDegreeTest.gowers_norm_pow"],"detail_key":"p14"},{"id":"n19984","layer":"informal","project":"p14","title":"Gowers norm equals 1 iff low degree","kind":"lemma","summary":"[Gowers norm equals 1 iff low degree] For a Boolean function f : \\0,1\\^n \\to \\0,1\\ and d \\ge 0,…","labels":["LowDegreeTest.gowers_norm_eq_one_iff"],"detail_key":"p14"},{"id":"n19985","layer":"informal","project":"p14","title":"U^2 norm equals L^2 norm of convolution","kind":"lemma","summary":"[U^2 norm equals L^2 norm of convolution] For f : \\0,1\\^n \\to R, \\|f\\|_U^2^4 = \\|f * f\\|_2^2, w…","labels":["LowDegreeTest.gowers_U2_eq_L2_conv"],"detail_key":"p14"},{"id":"n19986","layer":"informal","project":"p14","title":"U^2 norm in terms of Fourier coefficients","kind":"lemma","summary":"[U^2 norm in terms of Fourier coefficients] For f : \\0,1\\^n \\to R, \\[ \\|f\\|_U^2^4 = \\sum_S \\sub…","labels":["LowDegreeTest.gowers_U2_fourier"],"detail_key":"p14"},{"id":"n19987","layer":"informal","project":"p14","title":"Gowers norm at most 1 for \\pm 1 functions","kind":"lemma","summary":"[Gowers norm at most 1 for \\pm 1 functions] For every Boolean function f : \\0,1\\^n \\to \\0,1\\ an…","labels":["LowDegreeTest.gowers_norm_le_one"],"detail_key":"p14"},{"id":"n19988","layer":"informal","project":"p14","title":"Pointwise product of Boolean functions","kind":"definition","summary":"[Pointwise product of Boolean functions] The pointwise product of two real-valued functions on…","labels":["LowDegreeTest.boolFun_mul"],"detail_key":"p14"},{"id":"n19989","layer":"informal","project":"p14","title":"Gowers product is multiplicative","kind":"lemma","summary":"[Gowers product is multiplicative] For f, g : \\0,1\\^n \\to R, GP(f \\cdot g, k, x, h_1,\\ldots,h_k…","labels":["LowDegreeTest.gowers_product_mul"],"detail_key":"p14"},{"id":"n19990","layer":"informal","project":"p14","title":"Degree absorption for Gowers product","kind":"lemma","summary":"[Degree absorption for Gowers product] If g has degree \\le d, then for any f, GP(f \\cdot g, d+1…","labels":["LowDegreeTest.gowers_product_mul_degree_le"],"detail_key":"p14"},{"id":"n19991","layer":"informal","project":"p14","title":"Gowers norm preserved under multiplication by low-degree function","kind":"lemma","summary":"[Gowers norm preserved under multiplication by low-degree function] If g has degree \\le d, then…","labels":["LowDegreeTest.gowers_norm_mul_degree_le"],"detail_key":"p14"},{"id":"n19992","layer":"informal","project":"p14","title":"Degree test acceptance probability","kind":"definition","summary":"[Degree test acceptance probability] The probability that the (d+1)-fold derivative test accept…","labels":["LowDegreeTest.degree_test_accept_prob"],"detail_key":"p14"},{"id":"n19993","layer":"informal","project":"p14","title":"\\varepsilon-far from degree d","kind":"definition","summary":"[\\varepsilon-far from degree d] A function f : \\0,1\\^n \\to \\0,1\\ is \\varepsilon-far from degree…","labels":["LowDegreeTest.epsilon_far_from_degree"],"detail_key":"p14"},{"id":"n19994","layer":"informal","project":"p14","title":"Acceptance probability formula","kind":"lemma","summary":"[Acceptance probability formula] For every Boolean function f : \\0,1\\^n \\to \\0,1\\, \\[ \\Pr[\\text…","labels":["LowDegreeTest.degree_test_accept_prob_eq"],"detail_key":"p14"},{"id":"n19995","layer":"informal","project":"p14","title":"Completeness of the degree test","kind":"lemma","summary":"[Completeness of the degree test] If \\deg(f) \\le d, then the degree-d test accepts f with proba…","labels":["LowDegreeTest.degree_test_completeness"],"detail_key":"p14"},{"id":"n19996","layer":"informal","project":"p14","title":"Qualitative soundness of the degree test","kind":"lemma","summary":"[Qualitative soundness of the degree test] If \\deg(f) > d, then the degree-d test accepts f wit…","labels":["LowDegreeTest.degree_test_qualitative_soundness"],"detail_key":"p14"},{"id":"n19997","layer":"informal","project":"p14","title":"Fourier characters have degree \\le 1","kind":"lemma","summary":"[Fourier characters have degree \\le 1] For every S \\subseteq [n], the Fourier character \\chi_S…","labels":["LowDegreeTest.char_is_degree_le_one"],"detail_key":"p14"},{"id":"n19998","layer":"informal","project":"p14","title":"Negated character has degree \\le 1","kind":"lemma","summary":"[Negated character has degree \\le 1] For every S \\subseteq [n], the function -\\chi_S has degree…","labels":["LowDegreeTest.neg_char_is_degree_le_one"],"detail_key":"p14"},{"id":"n19999","layer":"informal","project":"p14","title":"Fourier characters have degree \\le d for d \\ge 1","kind":"lemma","summary":"[Fourier characters have degree \\le d for d \\ge 1] For every S \\subseteq [n] and d \\ge 1, the c…","labels":["LowDegreeTest.char_is_degree_le"],"detail_key":"p14"},{"id":"n20000","layer":"informal","project":"p14","title":"Negated character has degree \\le d for d \\ge 1","kind":"lemma","summary":"[Negated character has degree \\le d for d \\ge 1] For every S \\subseteq [n] and d \\ge 1, the fun…","labels":["LowDegreeTest.neg_char_is_degree_le"],"detail_key":"p14"},{"id":"n20001","layer":"informal","project":"p14","title":"Fourier coefficients bounded when \\varepsilon-far from degree d","kind":"lemma","summary":"[Fourier coefficients bounded when \\varepsilon-far from degree d] If d \\ge 1 and f is \\varepsil…","labels":["LowDegreeTest.fourier_coeff_le_of_far_from_degree"],"detail_key":"p14"},{"id":"n20002","layer":"informal","project":"p14","title":"Negated Fourier coefficients bounded when \\varepsilon-far","kind":"lemma","summary":"[Negated Fourier coefficients bounded when \\varepsilon-far] If d \\ge 1 and f is \\varepsilon-far…","labels":["LowDegreeTest.neg_fourier_coeff_le_of_far_from_degree"],"detail_key":"p14"},{"id":"n20003","layer":"informal","project":"p14","title":"Absolute Fourier coefficients bounded when \\varepsilon-far","kind":"lemma","summary":"[Absolute Fourier coefficients bounded when \\varepsilon-far] If d \\ge 1 and f is \\varepsilon-fa…","labels":["LowDegreeTest.abs_fourier_coeff_le_of_far_from_degree"],"detail_key":"p14"},{"id":"n20004","layer":"informal","project":"p14","title":"\\varepsilon-far from Reed--Muller","kind":"definition","summary":"[\\varepsilon-far from Reed--Muller] A function f is \\varepsilon-far from RM(d, n) if it is \\var…","labels":["LowDegreeTest.epsilon_far_from_RM"],"detail_key":"p14"},{"id":"n20005","layer":"informal","project":"p14","title":"Reed--Muller test acceptance probability","kind":"definition","summary":"[Reed--Muller test acceptance probability] The acceptance probability of the Reed--Muller test,…","labels":["LowDegreeTest.RM_test_accept_prob"],"detail_key":"p14"},{"id":"n20006","layer":"informal","project":"p14","title":"Parseval for \\pm 1 functions","kind":"lemma","summary":"[Parseval for \\pm 1 functions] For every Boolean function f : \\0,1\\^n \\to \\0,1\\, \\sum_S \\subset…","labels":["LowDegreeTest.parseval_pm1"],"detail_key":"p14"},{"id":"n20007","layer":"informal","project":"p14","title":"U^2 norm bounded when \\varepsilon-far from degree d","kind":"lemma","summary":"[U^2 norm bounded when \\varepsilon-far from degree d] If d \\ge 1 and f is \\varepsilon-far from…","labels":["LowDegreeTest.gowers_U2_le_of_far"],"detail_key":"p14"},{"id":"n20008","layer":"informal","project":"p14","title":"Squaring bound","kind":"lemma","summary":"[Squaring bound] For \\varepsilon \\in [0, \\tfrac12], (1 - 2\\varepsilon)^2 \\le 1 - 2\\varepsilon.","labels":["LowDegreeTest.sq_one_sub_two_eps_le"],"detail_key":"p14"},{"id":"n20009","layer":"informal","project":"p14","title":"\\varepsilon \\le 1/2 when \\varepsilon-far from degree \\ge 1","kind":"lemma","summary":"[\\varepsilon \\le 1/2 when \\varepsilon-far from degree \\ge 1] If d \\ge 1 and f is \\varepsilon-fa…","labels":["LowDegreeTest.eps_le_half_of_far"],"detail_key":"p14"},{"id":"n20010","layer":"informal","project":"p14","title":"U^2 norm bounded by 1 - 2\\varepsilon when \\varepsilon-far (base case)","kind":"lemma","summary":"[U^2 norm bounded by 1 - 2\\varepsilon when \\varepsilon-far (base case)] If d \\ge 1 and f is \\va…","labels":["LowDegreeTest.gowers_norm_le_of_far_d1"],"detail_key":"p14"},{"id":"n20011","layer":"informal","project":"p14","title":"\\varepsilon-far is monotone in d","kind":"lemma","summary":"[\\varepsilon-far is monotone in d] If d' \\le d and f is \\varepsilon-far from degree \\le d, then…","labels":["LowDegreeTest.epsilon_far_monotone"],"detail_key":"p14"},{"id":"n20012","layer":"informal","project":"p14","title":"Derivative distance lemma","kind":"lemma","summary":"[Derivative distance lemma] If d \\ge 2 and f : \\0,1\\^n \\to \\0,1\\ is \\varepsilon-far from degree…","labels":["LowDegreeTest.derivative_distance_lemma"],"detail_key":"p14"},{"id":"n20013","layer":"informal","project":"p14","title":"Gowers norm bound when \\varepsilon-far (inductive)","kind":"lemma","summary":"[Gowers norm bound when \\varepsilon-far (inductive)] If d \\ge 1 and f is \\varepsilon-far from d…","labels":["LowDegreeTest.gowers_norm_le_of_far"],"detail_key":"p14"},{"id":"n20014","layer":"informal","project":"p14","title":"Quantitative soundness of the degree test","kind":"lemma","summary":"[Quantitative soundness of the degree test] If d \\ge 1 and f is \\varepsilon-far from every degr…","labels":["LowDegreeTest.degree_test_quantitative_soundness"],"detail_key":"p14"},{"id":"n20015","layer":"informal","project":"p14","title":"Completeness of the Reed--Muller test","kind":"lemma","summary":"[Completeness of the Reed--Muller test] Every codeword f \\in RM(d, n) is accepted by the degree…","labels":["LowDegreeTest.RM_test_completeness"],"detail_key":"p14"},{"id":"n20016","layer":"informal","project":"p14","title":"Soundness of the Reed--Muller test","kind":"lemma","summary":"[Soundness of the Reed--Muller test] If d \\ge 1 and f is \\varepsilon-far from every codeword of…","labels":["LowDegreeTest.RM_test_soundness"],"detail_key":"p14"},{"id":"n20017","layer":"informal","project":"p14","title":"Linearity predicate","kind":"definition","summary":"[Linearity predicate] A function f : Z_k^n \\to Z_k is \\emphlinear if f(x+y) = f(x)+f(y) for all…","labels":["ZkBLR.is_linear"],"detail_key":"p14"},{"id":"n20018","layer":"informal","project":"p14","title":"Lift to roots of unity","kind":"definition","summary":"[Lift to roots of unity] Given f : Z_k^n \\to Z_k, the lifted function \\widetildef : Z_k^n \\to C…","labels":["ZkBLR.lift_omega"],"detail_key":"p14"},{"id":"n20019","layer":"informal","project":"p14","title":"Hamming distance between functions","kind":"definition","summary":"[Hamming distance between functions] The distance between f, g : Z_k^n \\to Z_k is the fraction…","labels":["ZkBLR.zk_dist"],"detail_key":"p14"},{"id":"n20020","layer":"informal","project":"p14","title":"\\varepsilon-far from linear","kind":"definition","summary":"[\\varepsilon-far from linear] A function f : Z_k^n \\to Z_k is \\emph\\varepsilon-far from linear…","labels":["ZkBLR.epsilon_far_from_linear"],"detail_key":"p14"},{"id":"n20021","layer":"informal","project":"p14","title":"Linear iff dot-product character","kind":"lemma","summary":"[Linear iff dot-product character] A function f : Z_k^n \\to Z_k is linear if and only if there…","labels":["ZkBLR.linear_iff_character"],"detail_key":"p14"},{"id":"n20022","layer":"informal","project":"p14","title":"Canonical linear character","kind":"definition","summary":"[Canonical linear character] For s \\in Z_k^n, the \\emphlinear character \\chi_s : Z_k^n \\to Z_k…","labels":["ZkBLR.linear_character"],"detail_key":"p14"},{"id":"n20023","layer":"informal","project":"p14","title":"Normalized function","kind":"definition","summary":"[Normalized function] A function f : Z_k^n \\to Z_k is \\emphnormalized if f(0) = 0. Every linear…","labels":["ZkBLR.normalized"],"detail_key":"p14"},{"id":"n20024","layer":"informal","project":"p14","title":"Normalization operator","kind":"definition","summary":"[Normalization operator] The \\emphnormalization of f : Z_k^n \\to Z_k is the function (normalize…","labels":["ZkBLR.normalize"],"detail_key":"p14"},{"id":"n20025","layer":"informal","project":"p14","title":"Normalization vanishes at origin","kind":"lemma","summary":"[Normalization vanishes at origin] For any f : Z_k^n \\to Z_k, (normalize\\,f)(0) = 0.","labels":["ZkBLR.normalize_zero"],"detail_key":"p14"},{"id":"n20026","layer":"informal","project":"p14","title":"Linear functions are normalized","kind":"lemma","summary":"[Linear functions are normalized] If f : Z_k^n \\to Z_k is linear, then f(0) = 0, i.e.\\ f is nor…","labels":["ZkBLR.linear_normalized"],"detail_key":"p14"},{"id":"n20027","layer":"informal","project":"p14","title":"\\varepsilon-far from linear, normalized version","kind":"definition","summary":"[\\varepsilon-far from linear, normalized version] A function f : Z_k^n \\to Z_k satisfies \\textt…","labels":["ZkBLR.epsilon_far_from_linear_normalized"],"detail_key":"p14"},{"id":"n20028","layer":"informal","project":"p14","title":"Real part of root of unity bounded by \\cos(2\\pi/k)","kind":"lemma","summary":"[Real part of root of unity bounded by \\cos(2\\pi/k)] For k \\ge 2 and any nonzero a \\in Z_k, Re(…","labels":["ZkBLR.re_toOmega_le_re_rootOfUnity"],"detail_key":"p14"},{"id":"n20029","layer":"informal","project":"p14","title":"Upper bound on real part of Fourier coefficient","kind":"lemma","summary":"[Upper bound on real part of Fourier coefficient] For k \\ge 2 and f : Z_k^n \\to Z_k, the real p…","labels":["ZkBLR.re_fourier_coeff_upper_bound"],"detail_key":"p14"},{"id":"n20030","layer":"informal","project":"p14","title":"Fourier bound from \\varepsilon-far hypothesis","kind":"lemma","summary":"[Fourier bound from \\varepsilon-far hypothesis] If k \\ge 2 and f is \\varepsilon-far from linear…","labels":["ZkBLR.re_epsilon_far_bounds_fourier"],"detail_key":"p14"},{"id":"n20031","layer":"informal","project":"p14","title":"j-twisted lift to roots of unity","kind":"definition","summary":"[j-twisted lift to roots of unity] For j \\in Z_k, the j-twisted lift of f : Z_k^n \\to Z_k is \\w…","labels":["ZkBLR.lift_omega_j"],"detail_key":"p14"},{"id":"n20032","layer":"informal","project":"p14","title":"BLR acceptance probability","kind":"definition","summary":"[BLR acceptance probability] The BLR acceptance probability of f : Z_k^n \\to Z_k is \\[ \\Pr_x,y\\…","labels":["ZkBLR.BLR_accept_prob"],"detail_key":"p14"},{"id":"n20033","layer":"informal","project":"p14","title":"BLR completeness","kind":"lemma","summary":"[BLR completeness] If f : Z_k^n \\to Z_k is linear, then \\Pr[\\rm BLR\\ accepts\\ f] = 1.","labels":["ZkBLR.BLR_completeness"],"detail_key":"p14"},{"id":"n20034","layer":"informal","project":"p14","title":"Geometric sum of twisted roots of unity","kind":"lemma","summary":"[Geometric sum of twisted roots of unity] For any a \\in Z_k, \\[ \\sum_j \\in Z_k \\omega_k^ja \\;=\\…","labels":["ZkBLR.geom_sum_toOmega_dual"],"detail_key":"p14"},{"id":"n20035","layer":"informal","project":"p14","title":"Indicator as character sum","kind":"lemma","summary":"[Indicator as character sum] For any a \\in Z_k, \\[ 1[a=0] \\;=\\; \\frac1k\\,Re\\!\\left[\\sum_j\\inZ_k…","labels":["ZkBLR.indicator_eq_char_sum_re"],"detail_key":"p14"},{"id":"n20036","layer":"informal","project":"p14","title":"Parseval identity for j-twisted lift","kind":"lemma","summary":"[Parseval identity for j-twisted lift] For any j \\in Z_k and f : Z_k^n \\to Z_k, \\[ \\sum_s \\in Z…","labels":["ZkBLR.parseval_lift_omega_j"],"detail_key":"p14"},{"id":"n20037","layer":"informal","project":"p14","title":"Triple-product Fourier expansion","kind":"lemma","summary":"[Triple-product Fourier expansion] For j \\in Z_k and F = \\widetildef_j, the triple-product expe…","labels":["ZkBLR.triple_product_fourier"],"detail_key":"p14"},{"id":"n20038","layer":"informal","project":"p14","title":"BLR probability as Fourier sum","kind":"lemma","summary":"[BLR probability as Fourier sum] The BLR acceptance probability admits the Fourier representati…","labels":["ZkBLR.BLR_accept_prob_eq_fourier_sum"],"detail_key":"p14"},{"id":"n20039","layer":"informal","project":"p14","title":"Zero-index contribution equals one","kind":"lemma","summary":"[Zero-index contribution equals one] For any f : Z_k^n \\to Z_k, the j=0 term of the Fourier sum…","labels":["ZkBLR.lift_omega_j_zero_contribution"],"detail_key":"p14"},{"id":"n20040","layer":"informal","project":"p14","title":"Weighted Fourier sum at most one","kind":"lemma","summary":"[Weighted Fourier sum at most one] For any j \\in Z_k and f : Z_k^n \\to Z_k, \\[ \\sum_s\\inZ_k^n \\…","labels":["ZkBLR.weighted_fourier_sum_le_one"],"detail_key":"p14"},{"id":"n20041","layer":"informal","project":"p14","title":"Cube-sum bounded by max times square-sum","kind":"lemma","summary":"[Cube-sum bounded by max times square-sum] If \\|\\widehat\\widetildef(s)\\| \\le A for all s \\in Z_…","labels":["ZkBLR.cube_sum_bound_by_max"],"detail_key":"p14"},{"id":"n20042","layer":"informal","project":"p14","title":"Parseval identity for lift\\_omega","kind":"lemma","summary":"[Parseval identity for lift\\_omega] For f : Z_p^n \\to Z_p, \\[ \\sum_s\\inZ_p^n \\bigl\\|\\widehat\\wi…","labels":["ZkBLR.parseval_lift_omega"],"detail_key":"p14"},{"id":"n20043","layer":"informal","project":"p14","title":"Weighted Fourier sum bounded by \\varepsilon-far condition","kind":"lemma","summary":"[Weighted Fourier sum bounded by \\varepsilon-far condition] If f : Z_p^n \\to Z_p is \\varepsilon…","labels":["ZkBLR.weighted_fourier_sum_bound"],"detail_key":"p14"},{"id":"n20044","layer":"informal","project":"p14","title":"Fourier coefficient real-part bound for j-twisted lift","kind":"lemma","summary":"[Fourier coefficient real-part bound for j-twisted lift] If f : Z_p^n \\to Z_p is \\varepsilon-fa…","labels":["ZkBLR.re_fourier_coeff_lift_omega_j_bound"],"detail_key":"p14"},{"id":"n20045","layer":"informal","project":"p14","title":"Weighted sum bound for j-twisted lift","kind":"lemma","summary":"[Weighted sum bound for j-twisted lift] If f : Z_p^n \\to Z_p is \\varepsilon-far from linear (no…","labels":["ZkBLR.weighted_sum_lift_omega_j_bound"],"detail_key":"p14"},{"id":"n20046","layer":"informal","project":"p14","title":"BLR soundness for prime fields","kind":"lemma","summary":"[BLR soundness for prime fields] If f : Z_p^n \\to Z_p is \\varepsilon-far from linear (normalize…","labels":["ZkBLR.BLR_soundness"],"detail_key":"p14"},{"id":"n20047","layer":"informal","project":"p14","title":"\\varepsilon-farness preserved under unit scaling","kind":"lemma","summary":"[\\varepsilon-farness preserved under unit scaling] If f : Z_k^n \\to Z_k is \\varepsilon-far from…","labels":["ZkBLR.unit_mul_epsilon_far"],"detail_key":"p14"},{"id":"n20048","layer":"informal","project":"p14","title":"Real-part bound for unit-twisted lift","kind":"lemma","summary":"[Real-part bound for unit-twisted lift] If k \\ge 2, f is \\varepsilon-far from linear (normalize…","labels":["ZkBLR.re_fourier_coeff_lift_omega_j_unit_bound"],"detail_key":"p14"},{"id":"n20049","layer":"informal","project":"p14","title":"Weighted sum bound for unit-twisted lift","kind":"lemma","summary":"[Weighted sum bound for unit-twisted lift] If k \\ge 2, f is \\varepsilon-far from linear (normal…","labels":["ZkBLR.weighted_sum_unit_bound"],"detail_key":"p14"},{"id":"n20050","layer":"informal","project":"p14","title":"BLR soundness for general k","kind":"lemma","summary":"[BLR soundness for general k] For k \\ge 2, if f : Z_k^n \\to Z_k is \\varepsilon-far from linear…","labels":["ZkBLR.BLR_soundness_general"],"detail_key":"p14"},{"id":"n20051","layer":"informal","project":"p14","title":"BLR soundness for prime modulus","kind":"lemma","summary":"[BLR soundness for prime modulus] If p is prime and f : Z_p^n \\to Z_p is \\varepsilon-far from l…","labels":["ZkBLR.BLR_soundness_prime"],"detail_key":"p14"},{"id":"n20052","layer":"informal","project":"p14","title":"Vectors over Z/kZ","kind":"definition","summary":"[Vectors over Z/kZ] \\textttZkFourier.ZkVec\\ k\\ n is the n-dimensional vector space over Z/kZ, d…","labels":["ZkFourier.ZkVec"],"detail_key":"p14"},{"id":"n20053","layer":"informal","project":"p14","title":"Cardinality of Z_k^n","kind":"lemma","summary":"[Cardinality of Z_k^n] For k \\geq 1, the finite type \\textttZkFourier.ZkVec\\ k\\ n has |Z_k^n| =…","labels":["ZkFourier.card_ZkVec"],"detail_key":"p14"},{"id":"n20054","layer":"informal","project":"p14","title":"Primitive k-th root of unity","kind":"definition","summary":"[Primitive k-th root of unity] \\textttZkFourier.rootOfUnity\\ k is the complex number \\omega_k =…","labels":["ZkFourier.rootOfUnity"],"detail_key":"p14"},{"id":"n20055","layer":"informal","project":"p14","title":"Embedding of Z/kZ into C^\\times","kind":"definition","summary":"[Embedding of Z/kZ into C^\\times] For a : Z/kZ, \\textttZkFourier.toOmega\\ a = \\omega_k^a, where…","labels":["ZkFourier.toOmega"],"detail_key":"p14"},{"id":"n20056","layer":"informal","project":"p14","title":"\\omega_k is a primitive k-th root","kind":"lemma","summary":"[\\omega_k is a primitive k-th root] \\omega_k = e^2\\pi i/k is a primitive k-th root of unity in…","labels":["ZkFourier.isPrimitiveRoot_rootOfUnity"],"detail_key":"p14"},{"id":"n20057","layer":"informal","project":"p14","title":"\\omega_k^k = 1","kind":"lemma","summary":"[\\omega_k^k = 1] The k-th power of the root of unity satisfies \\omega_k^k = 1.","labels":["ZkFourier.rootOfUnity_pow_k"],"detail_key":"p14"},{"id":"n20058","layer":"informal","project":"p14","title":"\\omega_k^0 = 1","kind":"lemma","summary":"[\\omega_k^0 = 1] The embedding sends zero to one: \\textttZkFourier.toOmega\\ (0 : Z/kZ) = 1.","labels":["ZkFourier.toOmega_zero"],"detail_key":"p14"},{"id":"n20059","layer":"informal","project":"p14","title":"Additivity of the embedding","kind":"lemma","summary":"[Additivity of the embedding] For a, b : Z/kZ, \\omega_k^a+b = \\omega_k^a \\cdot \\omega_k^b. That…","labels":["ZkFourier.toOmega_add"],"detail_key":"p14"},{"id":"n20060","layer":"informal","project":"p14","title":"Negation and complex conjugation","kind":"lemma","summary":"[Negation and complex conjugation] For a : Z/kZ, \\omega_k^-a = \\overline\\omega_k^a, i.e.\\ \\text…","labels":["ZkFourier.toOmega_neg"],"detail_key":"p14"},{"id":"n20061","layer":"informal","project":"p14","title":"Unit norm of the embedding","kind":"lemma","summary":"[Unit norm of the embedding] For every a : Z/kZ, \\|\\omega_k^a\\| = 1; the image of \\textttZkFour…","labels":["ZkFourier.norm_toOmega"],"detail_key":"p14"},{"id":"n20062","layer":"informal","project":"p14","title":"Multiplicativity of the embedding","kind":"lemma","summary":"[Multiplicativity of the embedding] For j, a : Z/kZ, \\omega_k^j \\cdot a = (\\omega_k^j)^a, i.e.\\…","labels":["ZkFourier.toOmega_mul"],"detail_key":"p14"},{"id":"n20063","layer":"informal","project":"p14","title":"Geometric sum orthogonality","kind":"lemma","summary":"[Geometric sum orthogonality] For j : Z/kZ, \\[ \\sum_a \\in Z/kZ \\omega_k^j \\cdot a = k & \\textif…","labels":["ZkFourier.geom_sum_toOmega"],"detail_key":"p14"},{"id":"n20064","layer":"informal","project":"p14","title":"Functions on Z_k^n","kind":"definition","summary":"[Functions on Z_k^n] \\textttZkFourier.ZkFun\\ k\\ n is the type of complex-valued functions on Z_…","labels":["ZkFourier.ZkFun"],"detail_key":"p14"},{"id":"n20065","layer":"informal","project":"p14","title":"Expectation","kind":"definition","summary":"[Expectation] The expectation of f : Z_k^n \\to C is the uniform average \\[ E[f] = \\frac1k^n \\su…","labels":["ZkFourier.expectation"],"detail_key":"p14"},{"id":"n20066","layer":"informal","project":"p14","title":"Hermitian inner product","kind":"definition","summary":"[Hermitian inner product] The Hermitian inner product of f, g : Z_k^n \\to C is \\[ \\langle f, g…","labels":["ZkFourier.inner_product"],"detail_key":"p14"},{"id":"n20067","layer":"informal","project":"p14","title":"Squared L^2 norm","kind":"definition","summary":"[Squared L^2 norm] The squared L^2 norm of f : Z_k^n \\to C is the real-valued quantity \\[ \\|f\\|…","labels":["ZkFourier.L2_norm_sq"],"detail_key":"p14"},{"id":"n20068","layer":"informal","project":"p14","title":"Convolution","kind":"definition","summary":"[Convolution] The convolution of f, g : Z_k^n \\to C is \\[ (f * g)(x) = E_y\\!\\left[f(y)\\,g(x - y…","labels":["ZkFourier.convolution"],"detail_key":"p14"},{"id":"n20069","layer":"informal","project":"p14","title":"Dot product on Z_k^n","kind":"definition","summary":"[Dot product on Z_k^n] For s, x \\in Z_k^n, the dot product is s \\cdot x = \\sum_i=0^n-1 s_i \\, x…","labels":["ZkFourier.zkDot"],"detail_key":"p14"},{"id":"n20070","layer":"informal","project":"p14","title":"Fourier character","kind":"definition","summary":"[Fourier character] For s \\in Z_k^n, the Fourier character indexed by s is \\chi_s : Z_k^n \\to C…","labels":["ZkFourier.char_s"],"detail_key":"p14"},{"id":"n20071","layer":"informal","project":"p14","title":"Dot product is additive in the right argument","kind":"lemma","summary":"[Dot product is additive in the right argument] For s, x, y \\in Z_k^n, s \\cdot (x + y) = s \\cdo…","labels":["ZkFourier.zkDot_add_right"],"detail_key":"p14"},{"id":"n20072","layer":"informal","project":"p14","title":"Zero vector on the left","kind":"lemma","summary":"[Zero vector on the left] For any x \\in Z_k^n, 0 \\cdot x = 0.","labels":["ZkFourier.zkDot_zero_left"],"detail_key":"p14"},{"id":"n20073","layer":"informal","project":"p14","title":"Zero vector on the right","kind":"lemma","summary":"[Zero vector on the right] For any s \\in Z_k^n, s \\cdot 0 = 0.","labels":["ZkFourier.zkDot_zero_right"],"detail_key":"p14"},{"id":"n20074","layer":"informal","project":"p14","title":"Negation on the left argument of the dot product","kind":"lemma","summary":"[Negation on the left argument of the dot product] For s, x \\in Z_k^n, (-s) \\cdot x = -(s \\cdot…","labels":["ZkFourier.zkDot_neg_left"],"detail_key":"p14"},{"id":"n20075","layer":"informal","project":"p14","title":"Subtraction on the left argument of the dot product","kind":"lemma","summary":"[Subtraction on the left argument of the dot product] For s, t, x \\in Z_k^n, (s - t) \\cdot x =…","labels":["ZkFourier.zkDot_sub"],"detail_key":"p14"},{"id":"n20076","layer":"informal","project":"p14","title":"Characters are multiplicative in the input","kind":"lemma","summary":"[Characters are multiplicative in the input] For s, x, y \\in Z_k^n, \\chi_s(x + y) = \\chi_s(x) \\…","labels":["ZkFourier.char_s_add"],"detail_key":"p14"},{"id":"n20077","layer":"informal","project":"p14","title":"Character at zero input","kind":"lemma","summary":"[Character at zero input] For any s \\in Z_k^n, \\chi_s(0) = 1.","labels":["ZkFourier.char_s_zero_vec"],"detail_key":"p14"},{"id":"n20078","layer":"informal","project":"p14","title":"Trivial character equals one","kind":"lemma","summary":"[Trivial character equals one] For any x \\in Z_k^n, \\chi_0(x) = 1; the character indexed by the…","labels":["ZkFourier.char_s_zero_index"],"detail_key":"p14"},{"id":"n20079","layer":"informal","project":"p14","title":"Characters have unit norm","kind":"lemma","summary":"[Characters have unit norm] For all s, x \\in Z_k^n, \\|\\chi_s(x)\\| = 1.","labels":["ZkFourier.norm_char_s"],"detail_key":"p14"},{"id":"n20080","layer":"informal","project":"p14","title":"Characters are nonzero","kind":"lemma","summary":"[Characters are nonzero] For all s, x \\in Z_k^n, \\chi_s(x) \\neq 0.","labels":["ZkFourier.char_s_ne_zero"],"detail_key":"p14"},{"id":"n20081","layer":"informal","project":"p14","title":"Conjugate of a character","kind":"lemma","summary":"[Conjugate of a character] For s, x \\in Z_k^n, \\overline\\chi_s(x) = \\chi_-s(x). Complex conjuga…","labels":["ZkFourier.char_s_conj"],"detail_key":"p14"},{"id":"n20082","layer":"informal","project":"p14","title":"Product of two characters","kind":"lemma","summary":"[Product of two characters] For s, t, x \\in Z_k^n, \\chi_s(x) \\cdot \\chi_t(x) = \\chi_s+t(x). Poi…","labels":["ZkFourier.char_s_mul"],"detail_key":"p14"},{"id":"n20083","layer":"informal","project":"p14","title":"Expectation of a nontrivial character is zero","kind":"lemma","summary":"[Expectation of a nontrivial character is zero] If s \\neq 0, then E[\\chi_s] = 0. This follows f…","labels":["ZkFourier.expectation_char_nontrivial"],"detail_key":"p14"},{"id":"n20084","layer":"informal","project":"p14","title":"Expectation of the trivial character is one","kind":"lemma","summary":"[Expectation of the trivial character is one] E[\\chi_0] = 1, since \\chi_0 \\equiv 1 and the sum…","labels":["ZkFourier.expectation_char_trivial"],"detail_key":"p14"},{"id":"n20085","layer":"informal","project":"p14","title":"Self-inner-product of a character","kind":"lemma","summary":"[Self-inner-product of a character] For any s \\in Z_k^n, \\langle \\chi_s, \\chi_s \\rangle = 1. Ev…","labels":["ZkFourier.inner_product_char_self"],"detail_key":"p14"},{"id":"n20086","layer":"informal","project":"p14","title":"Orthogonality of distinct characters","kind":"lemma","summary":"[Orthogonality of distinct characters] If s \\neq t, then \\langle \\chi_s, \\chi_t \\rangle = 0. To…","labels":["ZkFourier.inner_product_char_nonself"],"detail_key":"p14"},{"id":"n20087","layer":"informal","project":"p14","title":"Fourier coefficient","kind":"definition","summary":"[Fourier coefficient] The Fourier coefficient of f : Z_k^n \\to C at frequency s is \\[ \\hatf(s)…","labels":["ZkFourier.fourier_coeff"],"detail_key":"p14"},{"id":"n20088","layer":"informal","project":"p14","title":"Fourier coefficient of a character at its own frequency","kind":"lemma","summary":"[Fourier coefficient of a character at its own frequency] For any s \\in Z_k^n, \\widehat\\chi_s(s…","labels":["ZkFourier.fourier_coeff_char_self"],"detail_key":"p14"},{"id":"n20089","layer":"informal","project":"p14","title":"Fourier coefficient of a character at a different frequency","kind":"lemma","summary":"[Fourier coefficient of a character at a different frequency] If s \\neq t, then \\widehat\\chi_s(…","labels":["ZkFourier.fourier_coeff_char_of_ne"],"detail_key":"p14"},{"id":"n20090","layer":"informal","project":"p14","title":"Fourier expansion","kind":"lemma","summary":"[Fourier expansion] Every function f : Z_k^n \\to C admits the Fourier expansion \\[ f(x) = \\sum_…","labels":["ZkFourier.fourier_expansion"],"detail_key":"p14"},{"id":"n20091","layer":"informal","project":"p14","title":"Parseval's identity","kind":"lemma","summary":"[Parseval's identity] For any f : Z_k^n \\to C, \\[ \\sum_s \\in Z_k^n |\\hatf(s)|^2 = \\|f\\|_2^2. \\]…","labels":["ZkFourier.parseval_identity"],"detail_key":"p14"},{"id":"n20092","layer":"informal","project":"p14","title":"Convolution theorem","kind":"lemma","summary":"[Convolution theorem] For f, g : Z_k^n \\to C and s \\in Z_k^n, \\[ \\widehatf * g(s) = \\hatf(s)\\,\\…","labels":["ZkFourier.fourier_coeff_convolution"],"detail_key":"p14"},{"id":"n20093","layer":"informal","project":"p14","title":"Positive leaf count","kind":"lemma","summary":"[Positive leaf count] For every deterministic protocol p : Protocol\\;X\\;Y\\;\\alpha, the number o…","labels":["CommunicationComplexity.Deterministic.Protocol.numLeaves_pos"],"detail_key":"p14"},{"id":"n20094","layer":"informal","project":"p14","title":"Zero complexity when one leaf","kind":"lemma","summary":"[Zero complexity when one leaf] If a protocol p has exactly one leaf (i.e.\\ p.numLeaves = 1), t…","labels":["CommunicationComplexity.Deterministic.Protocol.complexity_eq_zero_of_numLeaves_eq_one"],"detail_key":"p14"},{"id":"n20095","layer":"informal","project":"p14","title":"Balanced splitting bound on squared maximum","kind":"lemma","summary":"[Balanced splitting bound on squared maximum] Let m, n \\in N with 3m \\le 2n and 3(n - m) \\le 2n…","labels":["CommunicationComplexity.Deterministic.Protocol.max_sq_le_of_balanced"],"detail_key":"p14"},{"id":"n20096","layer":"informal","project":"p14","title":"Balanced simulation of deterministic protocols","kind":"theorem","summary":"[Balanced simulation of deterministic protocols] For every deterministic protocol p : Protocol\\…","labels":["CommunicationComplexity.Deterministic.Protocol.exists_balanced_simulation"],"detail_key":"p14"},{"id":"n20097","layer":"informal","project":"p14","title":"Bit string","kind":"definition","summary":"[Bit string] An n-bit string is a function x : Fin\\,n \\to Bool. The type \\textttCommunicationCo…","labels":["CommunicationComplexity.BitString"],"detail_key":"p14"},{"id":"n20098","layer":"informal","project":"p14","title":"Signed inner product","kind":"definition","summary":"[Signed inner product] Given two n-bit strings x, y : Fin\\,n \\to Bool, their \\emphsigned inner…","labels":["CommunicationComplexity.BitString.signedInner"],"detail_key":"p14"},{"id":"n20099","layer":"informal","project":"p14","title":"Agreement count","kind":"definition","summary":"[Agreement count] The \\emphagreement count of two n-bit strings x and y is the number of coordi…","labels":["CommunicationComplexity.BitString.agreementCount"],"detail_key":"p14"},{"id":"n20100","layer":"informal","project":"p14","title":"Agreement count plus Hamming distance equals length","kind":"theorem","summary":"[Agreement count plus Hamming distance equals length] For any two n-bit strings x and y, \\[ agr…","labels":["CommunicationComplexity.BitString.agreementCount_add_hammingDist_eq_length"],"detail_key":"p14"},{"id":"n20101","layer":"informal","project":"p14","title":"Agreement count as complement of Hamming distance","kind":"theorem","summary":"[Agreement count as complement of Hamming distance] For any two n-bit strings x and y, the agre…","labels":["CommunicationComplexity.BitString.agreementCount_eq_length_sub_hammingDist"],"detail_key":"p14"},{"id":"n20102","layer":"informal","project":"p14","title":"Signed inner product via Hamming distance","kind":"theorem","summary":"[Signed inner product via Hamming distance] For any two n-bit strings x and y, \\[ signedInner(x…","labels":["CommunicationComplexity.BitString.signedInner_eq_length_sub_twice_hammingDist"],"detail_key":"p14"},{"id":"n20103","layer":"informal","project":"p14","title":"Signed inner product via agreement and disagreement counts","kind":"theorem","summary":"[Signed inner product via agreement and disagreement counts] For any two n-bit strings x and y,…","labels":["CommunicationComplexity.BitString.signedInner_eq_agreementCount_sub_hammingDist"],"detail_key":"p14"},{"id":"n20104","layer":"informal","project":"p14","title":"Signed inner product is additive under concatenation","kind":"theorem","summary":"[Signed inner product is additive under concatenation] For strings x_1, y_1 : Fin\\,m \\to Bool a…","labels":["CommunicationComplexity.BitString.signedInner_append"],"detail_key":"p14"},{"id":"n20105","layer":"informal","project":"p14","title":"Signed inner product is invariant under index recast","kind":"theorem","summary":"[Signed inner product is invariant under index recast] If m = n and x, y : Fin\\,n \\to Bool, the…","labels":["CommunicationComplexity.BitString.signedInner_comp_cast"],"detail_key":"p14"},{"id":"n20106","layer":"informal","project":"p14","title":"Signed inner product scales under repetition","kind":"theorem","summary":"[Signed inner product scales under repetition] For any n-bit strings x and y and any a : N, \\[…","labels":["CommunicationComplexity.BitString.signedInner_amplify"],"detail_key":"p14"},{"id":"n20107","layer":"informal","project":"p14","title":"Equality function on n-bit strings","kind":"definition","summary":"[Equality function on n-bit strings] For a natural number n, \\textttCommunicationComplexity.Fun…","labels":["CommunicationComplexity.Functions.Equality.equality"],"detail_key":"p14"},{"id":"n20108","layer":"informal","project":"p14","title":"Hash space for the equality protocol","kind":"definition","summary":"[Hash space for the equality protocol] HashSpace(n, k) is the type of all hash functions from n…","labels":["CommunicationComplexity.Functions.Equality.HashSpace"],"detail_key":"p14"},{"id":"n20109","layer":"informal","project":"p14","title":"Deterministic complexity upper bound for equality","kind":"theorem","summary":"[Deterministic complexity upper bound for equality] For every n \\in N, \\[ D(equality_n) \\;\\le\\;…","labels":["CommunicationComplexity.Functions.Equality.communicationComplexity_le"],"detail_key":"p14"},{"id":"n20110","layer":"informal","project":"p14","title":"Deterministic complexity of equality at n = 0","kind":"theorem","summary":"[Deterministic complexity of equality at n = 0] When n = 0 both inputs are the unique empty str…","labels":["CommunicationComplexity.Functions.Equality.communicationComplexity_zero"],"detail_key":"p14"},{"id":"n20111","layer":"informal","project":"p14","title":"Deterministic complexity lower bound for equality","kind":"theorem","summary":"[Deterministic complexity lower bound for equality] For every n \\ge 1, \\[ n + 1 \\;\\le\\; D(equal…","labels":["CommunicationComplexity.Functions.Equality.le_communicationComplexity"],"detail_key":"p14"},{"id":"n20112","layer":"informal","project":"p14","title":"Exact deterministic complexity of equality","kind":"theorem","summary":"[Exact deterministic complexity of equality] The deterministic communication complexity of the…","labels":["CommunicationComplexity.Functions.Equality.communicationComplexity_eq"],"detail_key":"p14"},{"id":"n20113","layer":"informal","project":"p14","title":"Equality hashing protocol","kind":"definition","summary":"[Equality hashing protocol] equalityHashProtocol(n, k) is the standard public-coin protocol for…","labels":["CommunicationComplexity.Functions.Equality.equalityHashProtocol"],"detail_key":"p14"},{"id":"n20114","layer":"informal","project":"p14","title":"Run semantics of the equality hashing protocol","kind":"theorem","summary":"[Run semantics of the equality hashing protocol] For all x, y : BoolInput\\,n and h : HashSpace(…","labels":["CommunicationComplexity.Functions.Equality.equalityHashProtocol_rrun"],"detail_key":"p14"},{"id":"n20115","layer":"informal","project":"p14","title":"Complexity of the equality hashing protocol","kind":"theorem","summary":"[Complexity of the equality hashing protocol] The worst-case communication cost of equalityHash…","labels":["CommunicationComplexity.Functions.Equality.equalityHashProtocol_complexity"],"detail_key":"p14"},{"id":"n20116","layer":"informal","project":"p14","title":"Public-coin upper bound for equality via hash parameter","kind":"theorem","summary":"[Public-coin upper bound for equality via hash parameter] If \\varepsilon > 0 and 1/2^k < \\varep…","labels":["CommunicationComplexity.Functions.Equality.publicCoin_communicationComplexity_le_of_hε"],"detail_key":"p14"},{"id":"n20117","layer":"informal","project":"p14","title":"Public-coin upper bound for equality via \\varepsilon","kind":"theorem","summary":"[Public-coin upper bound for equality via \\varepsilon] For every n \\in N and every \\varepsilon…","labels":["CommunicationComplexity.Functions.Equality.publicCoin_communicationComplexity_le"],"detail_key":"p14"},{"id":"n20118","layer":"informal","project":"p14","title":"Word","kind":"definition","summary":"[Word] A \\emphword of length n over alphabet \\alpha is a function Fin\\,n \\to \\alpha; it is an a…","labels":["CommunicationComplexity.Word"],"detail_key":"p14"},{"id":"n20119","layer":"informal","project":"p14","title":"Hamming ball","kind":"definition","summary":"[Hamming ball] The \\emphHamming ball of radius r centred at u \\in \\alpha^n is the finset \\[ B_r…","labels":["CommunicationComplexity.hammingBall"],"detail_key":"p14"},{"id":"n20120","layer":"informal","project":"p14","title":"Hamming sphere","kind":"definition","summary":"[Hamming sphere] The \\emphHamming sphere of radius r centred at u \\in \\alpha^n is the finset \\[…","labels":["CommunicationComplexity.hammingSphere"],"detail_key":"p14"},{"id":"n20121","layer":"informal","project":"p14","title":"Ball volume","kind":"definition","summary":"[Ball volume] For a length-n code over an alphabet of size q, the \\emphball volume at radius t…","labels":["CommunicationComplexity.ballVol"],"detail_key":"p14"},{"id":"n20122","layer":"informal","project":"p14","title":"Spheres of different radii are disjoint","kind":"lemma","summary":"[Spheres of different radii are disjoint] For any centre u and distinct radii r \\ne t, the Hamm…","labels":["CommunicationComplexity.hammingSpheres_disjoint"],"detail_key":"p14"},{"id":"n20123","layer":"informal","project":"p14","title":"Spheres are pairwise disjoint","kind":"lemma","summary":"[Spheres are pairwise disjoint] For any centre u and bound r, the family \\S_t(u)\\_t < r is pair…","labels":["CommunicationComplexity.hammingSpheres_pairwise_disjoint"],"detail_key":"p14"},{"id":"n20124","layer":"informal","project":"p14","title":"Ball equals disjoint union of spheres","kind":"lemma","summary":"[Ball equals disjoint union of spheres] The Hamming ball of radius r decomposes as the disjoint…","labels":["CommunicationComplexity.hammingBall_eq_hammingSpheres"],"detail_key":"p14"},{"id":"n20125","layer":"informal","project":"p14","title":"Support of a pair of words","kind":"definition","summary":"[Support of a pair of words] The \\emphsupport of v relative to u is the finset of coordinate po…","labels":["CommunicationComplexity.support"],"detail_key":"p14"},{"id":"n20126","layer":"informal","project":"p14","title":"Support card equals Hamming distance","kind":"lemma","summary":"[Support card equals Hamming distance] For any two words u,v, the Hamming distance equals the c…","labels":["CommunicationComplexity.support_dist"],"detail_key":"p14"},{"id":"n20127","layer":"informal","project":"p14","title":"Support fiber","kind":"definition","summary":"[Support fiber] For a word u and a finset S \\subseteq Fin\\,n, the \\emphsupport fiber is the set…","labels":["CommunicationComplexity.supportFiber"],"detail_key":"p14"},{"id":"n20128","layer":"informal","project":"p14","title":"Support fibers over distinct sets are disjoint","kind":"lemma","summary":"[Support fibers over distinct sets are disjoint] If S \\ne T are distinct finsets of coordinate…","labels":["CommunicationComplexity.supportFiber_disjoint"],"detail_key":"p14"},{"id":"n20129","layer":"informal","project":"p14","title":"Sphere as union of support fibers","kind":"lemma","summary":"[Sphere as union of support fibers] The Hamming sphere of radius r centred at u equals the unio…","labels":["CommunicationComplexity.hammingSphere_eq_biUnion"],"detail_key":"p14"},{"id":"n20130","layer":"informal","project":"p14","title":"Alternative symbol choices","kind":"definition","summary":"[Alternative symbol choices] For a word u and position i, the \\emphchoices at i is the finset o…","labels":["CommunicationComplexity.choices"],"detail_key":"p14"},{"id":"n20131","layer":"informal","project":"p14","title":"Choices cardinality","kind":"lemma","summary":"[Choices cardinality] For any word u and position i, the number of alternative symbols at i is…","labels":["CommunicationComplexity.choices_card"],"detail_key":"p14"},{"id":"n20132","layer":"informal","project":"p14","title":"Choices partition a support fiber","kind":"lemma","summary":"[Choices partition a support fiber] For a word u, a set S of positions, and a position i, the s…","labels":["CommunicationComplexity.choices_partition_supportFiber"],"detail_key":"p14"},{"id":"n20133","layer":"informal","project":"p14","title":"Each piece has the same cardinality","kind":"lemma","summary":"[Each piece has the same cardinality] For i \\notin S and any alternative symbol a \\in choices(u…","labels":["CommunicationComplexity.piece_card"],"detail_key":"p14"},{"id":"n20134","layer":"informal","project":"p14","title":"Support fiber cardinality","kind":"lemma","summary":"[Support fiber cardinality] For any word u and finset S \\subseteq Fin\\,n, \\[ |supportFiber(u,S)…","labels":["CommunicationComplexity.card_supportFiber"],"detail_key":"p14"},{"id":"n20135","layer":"informal","project":"p14","title":"Hamming sphere cardinality","kind":"lemma","summary":"[Hamming sphere cardinality] For any centre u \\in \\alpha^n and radius k, \\[ |S_k(u)| = \\binomnk…","labels":["CommunicationComplexity.hammingSphere_card"],"detail_key":"p14"},{"id":"n20136","layer":"informal","project":"p14","title":"Hamming ball cardinality","kind":"lemma","summary":"[Hamming ball cardinality] For any centre u \\in \\alpha^n and radius r, \\[ |B_r(u)| = ballVol(n,…","labels":["CommunicationComplexity.hammingBall_card"],"detail_key":"p14"},{"id":"n20137","layer":"informal","project":"p14","title":"Binary ball volume","kind":"lemma","summary":"[Binary ball volume] In the binary case q = 2, the ball volume simplifies to \\[ ballVol(n,t,2)…","labels":["CommunicationComplexity.ballVol_binary"],"detail_key":"p14"},{"id":"n20138","layer":"informal","project":"p14","title":"Boolean function matrix","kind":"definition","summary":"[Boolean function matrix] Given a Boolean function f : X \\to Y \\to Bool, the matrix M_f \\in R^X…","labels":["CommunicationComplexity.Deterministic.Rank.boolFunctionMatrix"],"detail_key":"p14"},{"id":"n20139","layer":"informal","project":"p14","title":"Rank of a Boolean function","kind":"definition","summary":"[Rank of a Boolean function] The rank of a Boolean function f : X \\to Y \\to Bool is the R-rank…","labels":["CommunicationComplexity.Deterministic.Rank.boolFunctionRank"],"detail_key":"p14"},{"id":"n20140","layer":"informal","project":"p14","title":"Rectangle indicator matrix","kind":"definition","summary":"[Rectangle indicator matrix] For a subset R \\subseteq X \\times Y, the matrix M_R \\in R^X \\times…","labels":["CommunicationComplexity.Deterministic.Rank.rectMatrix"],"detail_key":"p14"},{"id":"n20141","layer":"informal","project":"p14","title":"Rectangle matrix has rank at most one","kind":"theorem","summary":"[Rectangle matrix has rank at most one] If R \\subseteq X \\times Y is a combinatorial rectangle,…","labels":["CommunicationComplexity.Deterministic.Rank.rank_rectMatrix_le_one"],"detail_key":"p14"},{"id":"n20142","layer":"informal","project":"p14","title":"Rank is subadditive for two matrices","kind":"theorem","summary":"[Rank is subadditive for two matrices] For any two matrices A, B \\in R^X \\times Y, \\[ rank(A +…","labels":["Matrix.rank_add_le"],"detail_key":"p14"},{"id":"n20143","layer":"informal","project":"p14","title":"Rank is subadditive over finite sums","kind":"theorem","summary":"[Rank is subadditive over finite sums] For any finite index set \\iota, a finite set s \\subseteq…","labels":["Matrix.rank_sum_le"],"detail_key":"p14"},{"id":"n20144","layer":"informal","project":"p14","title":"Rank bounded by partition size","kind":"theorem","summary":"[Rank bounded by partition size] If P is a monochromatic rectangle partition of a Boolean funct…","labels":["CommunicationComplexity.Deterministic.Rank.boolFunctionRank_le_ncard"],"detail_key":"p14"},{"id":"n20145","layer":"informal","project":"p14","title":"Rank bounded by 2^n when complexity is at most n","kind":"theorem","summary":"[Rank bounded by 2^n when complexity is at most n] If the deterministic communication complexit…","labels":["CommunicationComplexity.Deterministic.Rank.boolFunctionRank_le_pow_of_communicationComplexity_le"],"detail_key":"p14"},{"id":"n20146","layer":"informal","project":"p14","title":"Log-rank lower bound","kind":"theorem","summary":"[Log-rank lower bound] For any Boolean function f : X \\to Y \\to Bool (with X finite and Y a \\te…","labels":["CommunicationComplexity.Deterministic.Rank.clog_boolFunctionRank_le_communicationComplexity"],"detail_key":"p14"},{"id":"n20147","layer":"informal","project":"p14","title":"Newman index space","kind":"definition","summary":"[Newman index space] Given finite input types X and Y and parameters \\varepsilon, c \\in R, the…","labels":["CommunicationComplexity.PublicCoin.newmanIndexSpace"],"detail_key":"p14"},{"id":"n20148","layer":"informal","project":"p14","title":"Newman protocol","kind":"definition","summary":"[Newman protocol] Given a public-coin finite-message protocol p over a probability space \\Omega…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.newmanProtocol"],"detail_key":"p14"},{"id":"n20149","layer":"informal","project":"p14","title":"Newman protocol approximately computes f","kind":"theorem","summary":"[Newman protocol approximately computes f] If p is a public-coin finite-message protocol that \\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.newmanProtocol_ApproxComputes"],"detail_key":"p14"},{"id":"n20150","layer":"informal","project":"p14","title":"Newman protocol complexity","kind":"theorem","summary":"[Newman protocol complexity] If p is a public-coin finite-message protocol that \\varepsilon-com…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.newmanProtocol_complexity"],"detail_key":"p14"},{"id":"n20151","layer":"informal","project":"p14","title":"Newman's theorem","kind":"theorem","summary":"[Newman's theorem] Let f : X \\to Y \\to \\alpha with X, Y finite, let \\varepsilon, \\varepsilon' \\…","labels":["CommunicationComplexity.PublicCoin.newman"],"detail_key":"p14"},{"id":"n20152","layer":"informal","project":"p14","title":"Hoeffding tail bound for many events","kind":"theorem","summary":"[Hoeffding tail bound for many events] Let (\\Omega', \\mu) be a probability space and let Y_0, \\…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.prob_many_events_le"],"detail_key":"p14"},{"id":"n20153","layer":"informal","project":"p14","title":"Derandomization sample count","kind":"definition","summary":"[Derandomization sample count] Given finite types X and Y and real parameters \\varepsilon, c, t…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.derandomizationSamples"],"detail_key":"p14"},{"id":"n20154","layer":"informal","project":"p14","title":"Existence of good randomness","kind":"theorem","summary":"[Existence of good randomness] Let \\Omega be a finite probability space, let p be a public-coin…","labels":["CommunicationComplexity.PublicCoin.FiniteMessage.Protocol.exists_good_randomness"],"detail_key":"p14"},{"id":"n20155","layer":"informal","project":"p14","title":"Discrepancy of a Boolean function","kind":"definition","summary":"[Discrepancy of a Boolean function] Let \\mu be a finite probability space on X \\times Y, let g…","labels":["CommunicationComplexity.discrepancy"],"detail_key":"p14"},{"id":"n20156","layer":"informal","project":"p14","title":"Discrepancy integrand decomposition","kind":"lemma","summary":"[Discrepancy integrand decomposition] For any g : X \\to Y \\to Bool, S \\subseteq X \\times Y, and…","labels":["CommunicationComplexity.discrepancy_integrand_eq"],"detail_key":"p14"},{"id":"n20157","layer":"informal","project":"p14","title":"Discrepancy as probability difference","kind":"theorem","summary":"[Discrepancy as probability difference] The discrepancy of g on S equals the \\mu-probability ma…","labels":["CommunicationComplexity.discrepancy_eq_prob_false_sub_prob_true"],"detail_key":"p14"},{"id":"n20158","layer":"informal","project":"p14","title":"Discrepancy bound implies \\gamma \\ge 0","kind":"lemma","summary":"[Discrepancy bound implies \\gamma \\ge 0] If every combinatorial rectangle R \\subseteq X \\times…","labels":["CommunicationComplexity.Deterministic.Protocol.nonneg_of_discrepancy_bound"],"detail_key":"p14"},{"id":"n20159","layer":"informal","project":"p14","title":"Rectangle sign","kind":"definition","summary":"[Rectangle sign] For a deterministic Boolean protocol p and a set R \\subseteq X \\times Y, the \\…","labels":["CommunicationComplexity.Deterministic.Protocol.rectangleSign"],"detail_key":"p14"},{"id":"n20160","layer":"informal","project":"p14","title":"Rectangle sign has absolute value one","kind":"lemma","summary":"[Rectangle sign has absolute value one] For any protocol p and any set R, |rectangleSign(p,R)|…","labels":["CommunicationComplexity.Deterministic.Protocol.rectangleSign_abs"],"detail_key":"p14"},{"id":"n20161","layer":"informal","project":"p14","title":"Rectangle sign agrees with Bool sign on leaf rectangles","kind":"lemma","summary":"[Rectangle sign agrees with Bool sign on leaf rectangles] If R is a leaf rectangle of protocol…","labels":["CommunicationComplexity.Deterministic.Protocol.rectangleSign_eq_boolSign"],"detail_key":"p14"},{"id":"n20162","layer":"informal","project":"p14","title":"Finite set of leaf rectangles","kind":"definition","summary":"[Finite set of leaf rectangles] Given a finite probability space \\mu on X \\times Y and a determ…","labels":["CommunicationComplexity.Deterministic.Protocol.leafRectanglesFinset"],"detail_key":"p14"},{"id":"n20163","layer":"informal","project":"p14","title":"Membership in leaf rectangle finset","kind":"lemma","summary":"[Membership in leaf rectangle finset] A set R belongs to \\textttCommunicationComplexity.Determi…","labels":["CommunicationComplexity.Deterministic.Protocol.mem_leafRectanglesFinset"],"detail_key":"p14"},{"id":"n20164","layer":"informal","project":"p14","title":"Indicator sum over leaf rectangles","kind":"lemma","summary":"[Indicator sum over leaf rectangles] For every point (x,y) \\in X \\times Y, the sum over all lea…","labels":["CommunicationComplexity.Deterministic.Protocol.sum_indicator_leafRectangles_eq"],"detail_key":"p14"},{"id":"n20165","layer":"informal","project":"p14","title":"Signed bias equals one minus twice distributional error","kind":"lemma","summary":"[Signed bias equals one minus twice distributional error] For a deterministic protocol p and Bo…","labels":["CommunicationComplexity.Deterministic.Protocol.signedBias_eq_one_sub_two_distributionalError"],"detail_key":"p14"},{"id":"n20166","layer":"informal","project":"p14","title":"Signed bias as sum over rectangles","kind":"lemma","summary":"[Signed bias as sum over rectangles] The signed bias E_(x,y)\\bigl[\\sigma(p.run(x,y))\\cdot\\sigma…","labels":["CommunicationComplexity.Deterministic.Protocol.signedBias_eq_sum_rectangles"],"detail_key":"p14"},{"id":"n20167","layer":"informal","project":"p14","title":"Core discrepancy lower bound","kind":"theorem","summary":"[Core discrepancy lower bound] If every combinatorial rectangle R satisfies |disc_\\mu(g,R)| \\le…","labels":["CommunicationComplexity.Deterministic.Protocol.one_sub_two_distributionalError_le_two_pow_mul"],"detail_key":"p14"},{"id":"n20168","layer":"informal","project":"p14","title":"Logarithmic discrepancy lower bound","kind":"theorem","summary":"[Logarithmic discrepancy lower bound] Assuming \\gamma > 0 and 1 - 2 \\cdot p.distributionalError…","labels":["CommunicationComplexity.Deterministic.Protocol.logb_le_complexity_of_distributionalError"],"detail_key":"p14"},{"id":"n20169","layer":"informal","project":"p14","title":"Discrepancy method for public-coin complexity","kind":"theorem","summary":"[Discrepancy method for public-coin complexity] Let \\mu be a distribution on X \\times Y, let g…","labels":["CommunicationComplexity.PublicCoin.lt_communicationComplexity_of_discrepancy_bound"],"detail_key":"p14"},{"id":"n20170","layer":"informal","project":"p14","title":"Entropy bounded by log of alphabet size","kind":"theorem","summary":"[Entropy bounded by log of alphabet size] If the alphabet type S has card(S) \\le N, then for an…","labels":["ProbabilityTheory.entropy_le_log_of_card_le"],"detail_key":"p14"},{"id":"n20171","layer":"informal","project":"p14","title":"Entropy bounded by c \\log 2 when alphabet fits in c bits","kind":"theorem","summary":"[Entropy bounded by c \\log 2 when alphabet fits in c bits] If card(S) \\le 2^c for some c : N, t…","labels":["ProbabilityTheory.entropy_le_nat_mul_log_two_of_card_le_two_pow"],"detail_key":"p14"},{"id":"n20172","layer":"informal","project":"p14","title":"Mutual information bounded by left entropy","kind":"theorem","summary":"[Mutual information bounded by left entropy] For measurable X and Y under a zero-or-probability…","labels":["ProbabilityTheory.mutualInfo_le_entropy_left"],"detail_key":"p14"},{"id":"n20173","layer":"informal","project":"p14","title":"Mutual information bounded by right entropy","kind":"theorem","summary":"[Mutual information bounded by right entropy] For measurable X and Y under a zero-or-probabilit…","labels":["ProbabilityTheory.mutualInfo_le_entropy_right"],"detail_key":"p14"},{"id":"n20174","layer":"informal","project":"p14","title":"Mutual information invariant under a.e.\\ substitution","kind":"theorem","summary":"[Mutual information invariant under a.e.\\ substitution] If X =^\\mu\\text-a.e. X' and Y =^\\mu\\tex…","labels":["ProbabilityTheory.mutualInfo_congr_ae"],"detail_key":"p14"},{"id":"n20175","layer":"informal","project":"p14","title":"Mutual information invariant under injective recoding of right variable","kind":"theorem","summary":"[Mutual information invariant under injective recoding of right variable] If f : T \\to V is mea…","labels":["ProbabilityTheory.mutualInfo_comp_right_of_injective"],"detail_key":"p14"},{"id":"n20176","layer":"informal","project":"p14","title":"Mutual information zero when left variable is a.e.\\ constant","kind":"theorem","summary":"[Mutual information zero when left variable is a.e.\\ constant] If X =^\\mu\\text-a.e. c for some…","labels":["ProbabilityTheory.mutualInfo_eq_zero_of_ae_eq_const_left"],"detail_key":"p14"},{"id":"n20177","layer":"informal","project":"p14","title":"Mutual information zero when right variable is a.e.\\ constant","kind":"theorem","summary":"[Mutual information zero when right variable is a.e.\\ constant] If Y =^\\mu\\text-a.e. c for some…","labels":["ProbabilityTheory.mutualInfo_eq_zero_of_ae_eq_const_right"],"detail_key":"p14"},{"id":"n20178","layer":"informal","project":"p14","title":"Conditional mutual information zero when left variable is a.e.\\ constant","kind":"theorem","summary":"[Conditional mutual information zero when left variable is a.e.\\ constant] If X =^\\mu\\text-a.e.…","labels":["ProbabilityTheory.condMutualInfo_eq_zero_of_ae_eq_const_left"],"detail_key":"p14"},{"id":"n20179","layer":"informal","project":"p14","title":"Conditional mutual information zero when right variable is a.e.\\ constant","kind":"theorem","summary":"[Conditional mutual information zero when right variable is a.e.\\ constant] If Y =^\\mu\\text-a.e…","labels":["ProbabilityTheory.condMutualInfo_eq_zero_of_ae_eq_const_right"],"detail_key":"p14"},{"id":"n20180","layer":"informal","project":"p14","title":"Independence from singleton-fiber factorization","kind":"theorem","summary":"[Independence from singleton-fiber factorization] For measurable X : \\Omega \\to S and Y : \\Omeg…","labels":["ProbabilityTheory.indepFun_of_measureReal_inter_preimage_singleton_eq_mul"],"detail_key":"p14"},{"id":"n20181","layer":"informal","project":"p14","title":"Conditional mutual information bounded by left conditional entropy","kind":"theorem","summary":"[Conditional mutual information bounded by left conditional entropy] For measurable X, Y, Z und…","labels":["ProbabilityTheory.condMutualInfo_le_condEntropy_left"],"detail_key":"p14"},{"id":"n20182","layer":"informal","project":"p14","title":"Conditional mutual information bounded by right conditional entropy","kind":"theorem","summary":"[Conditional mutual information bounded by right conditional entropy] For measurable X, Y, Z un…","labels":["ProbabilityTheory.condMutualInfo_le_condEntropy_right"],"detail_key":"p14"},{"id":"n20183","layer":"informal","project":"p14","title":"Conditional mutual information bounded by left entropy","kind":"theorem","summary":"[Conditional mutual information bounded by left entropy] For measurable X, Y, Z under a zero-or…","labels":["ProbabilityTheory.condMutualInfo_le_entropy_left"],"detail_key":"p14"},{"id":"n20184","layer":"informal","project":"p14","title":"Conditional mutual information bounded by right entropy","kind":"theorem","summary":"[Conditional mutual information bounded by right entropy] For measurable X, Y, Z under a zero-o…","labels":["ProbabilityTheory.condMutualInfo_le_entropy_right"],"detail_key":"p14"},{"id":"n20185","layer":"informal","project":"p14","title":"Conditional data processing: postprocessing depending on conditioning value","kind":"theorem","summary":"[Conditional data processing: postprocessing depending on conditioning value] If f : U \\to S \\t…","labels":["ProbabilityTheory.condMutualInfo_comp_left_le_of_comp_conditioning"],"detail_key":"p14"},{"id":"n20186","layer":"informal","project":"p14","title":"Conditional data processing on the right depending on conditioning value","kind":"theorem","summary":"[Conditional data processing on the right depending on conditioning value] For a measurable fun…","labels":["ProbabilityTheory.condMutualInfo_comp_right_le_of_comp_conditioning"],"detail_key":"p14"},{"id":"n20187","layer":"informal","project":"p14","title":"Conditioning on (Z, f(Z)) equals conditioning on Z","kind":"theorem","summary":"[Conditioning on (Z, f(Z)) equals conditioning on Z] Adding a deterministic function of the con…","labels":["ProbabilityTheory.condMutualInfo_conditioning_prod_function_eq"],"detail_key":"p14"},{"id":"n20188","layer":"informal","project":"p14","title":"Extra left-side conditioning cannot increase conditional mutual information","kind":"theorem","summary":"[Extra left-side conditioning cannot increase conditional mutual information] For a measurable…","labels":["ProbabilityTheory.condMutualInfo_conditioning_prod_left_function_le"],"detail_key":"p14"},{"id":"n20189","layer":"informal","project":"p14","title":"Extra right-side conditioning cannot increase conditional mutual information","kind":"theorem","summary":"[Extra right-side conditioning cannot increase conditional mutual information] For a measurable…","labels":["ProbabilityTheory.condMutualInfo_conditioning_prod_right_function_le"],"detail_key":"p14"},{"id":"n20190","layer":"informal","project":"p14","title":"Finer conditioning increases conditional mutual information when coarse carries no extra…","kind":"theorem","summary":"[Finer conditioning increases conditional mutual information when coarse carries no extra info]…","labels":["ProbabilityTheory.condMutualInfo_comp_conditioning_le_of_condMutualInfo_eq_zero"],"detail_key":"p14"},{"id":"n20191","layer":"informal","project":"p14","title":"Chain rule: splitting a left-side pair","kind":"theorem","summary":"[Chain rule: splitting a left-side pair] For measurable X, W, Y, Z with finite ranges under a z…","labels":["ProbabilityTheory.condMutualInfo_prod_left_eq_add"],"detail_key":"p14"},{"id":"n20192","layer":"informal","project":"p14","title":"Chain rule: splitting a right-side pair","kind":"theorem","summary":"[Chain rule: splitting a right-side pair] For measurable X, Y, W, Z with finite ranges under a…","labels":["ProbabilityTheory.condMutualInfo_prod_right_eq_add"],"detail_key":"p14"},{"id":"n20193","layer":"informal","project":"p14","title":"Pairing the right variable cannot decrease conditional mutual information","kind":"theorem","summary":"[Pairing the right variable cannot decrease conditional mutual information] For measurable X, Y…","labels":["ProbabilityTheory.condMutualInfo_le_prod_right_snd"],"detail_key":"p14"},{"id":"n20194","layer":"informal","project":"p14","title":"Strict prefix of a boolean vector","kind":"definition","summary":"[Strict prefix of a boolean vector] For a boolean-vector-valued random variable X : \\Omega \\to…","labels":["ProbabilityTheory.boolVectorStrictPrefix"],"detail_key":"p14"},{"id":"n20195","layer":"informal","project":"p14","title":"Chain rule for conditional mutual information against a boolean vector","kind":"theorem","summary":"[Chain rule for conditional mutual information against a boolean vector] For a boolean-vector-v…","labels":["ProbabilityTheory.condMutualInfo_boolVector_eq_sum_strictPrefix"],"detail_key":"p14"},{"id":"n20196","layer":"informal","project":"p14","title":"Iterated conditioning on a subset","kind":"theorem","summary":"[Iterated conditioning on a subset] For a finite measure \\mu and measurable sets F \\subseteq A,…","labels":["ProbabilityTheory.cond_cond_eq_cond_of_subset"],"detail_key":"p14"},{"id":"n20197","layer":"informal","project":"p14","title":"Real-mass reweighting for nested conditioning events","kind":"theorem","summary":"[Real-mass reweighting for nested conditioning events] For a finite measure \\mu and measurable…","labels":["ProbabilityTheory.measureReal_mul_cond_real_eq_measureReal_of_subset"],"detail_key":"p14"},{"id":"n20198","layer":"informal","project":"p14","title":"Conditional mutual information as an average over a finite partition","kind":"theorem","summary":"[Conditional mutual information as an average over a finite partition] If the conditioning vari…","labels":["ProbabilityTheory.condMutualInfo_prod_conditioning_eq_sum"],"detail_key":"p14"},{"id":"n20199","layer":"informal","project":"p14","title":"Conditional mutual information on an event is bounded by the unconditional value","kind":"theorem","summary":"[Conditional mutual information on an event is bounded by the unconditional value] Let A = Z^-1…","labels":["ProbabilityTheory.measureReal_mul_cond_condMutualInfo_le_condMutualInfo_of_event_eq_preimage"],"detail_key":"p14"},{"id":"n20200","layer":"informal","project":"p14","title":"Chain rule for mutual information: splitting a right-side pair","kind":"theorem","summary":"[Chain rule for mutual information: splitting a right-side pair] For measurable X, Y, W with fi…","labels":["ProbabilityTheory.mutualInfo_prod_right_eq_add"],"detail_key":"p14"},{"id":"n20201","layer":"informal","project":"p14","title":"Conditional mutual information bounded by unconditioned mutual information","kind":"theorem","summary":"[Conditional mutual information bounded by unconditioned mutual information] If I[X : W \\mid Y;…","labels":["ProbabilityTheory.condMutualInfo_le_mutualInfo_of_condDependence_le"],"detail_key":"p14"},{"id":"n20202","layer":"informal","project":"p14","title":"Conditional entropy invariant under a.e.\\ substitution","kind":"theorem","summary":"[Conditional entropy invariant under a.e.\\ substitution] Under a probability measure, if X =^\\m…","labels":["ProbabilityTheory.condEntropy_congr_ae"],"detail_key":"p14"},{"id":"n20203","layer":"informal","project":"p14","title":"Conditional mutual information invariant under a.e.\\ substitution of both variables","kind":"theorem","summary":"[Conditional mutual information invariant under a.e.\\ substitution of both variables] Under a p…","labels":["ProbabilityTheory.condMutualInfo_congr_ae_left_right"],"detail_key":"p14"},{"id":"n20204","layer":"informal","project":"p14","title":"Conditional mutual information invariant under a.e.\\ substitution of all three variables","kind":"theorem","summary":"[Conditional mutual information invariant under a.e.\\ substitution of all three variables] Unde…","labels":["ProbabilityTheory.condMutualInfo_congr_ae"],"detail_key":"p14"},{"id":"n20205","layer":"informal","project":"p14","title":"Finite-space a.e.\\ congruence for conditional mutual information","kind":"theorem","summary":"[Finite-space a.e.\\ congruence for conditional mutual information] Version of \\textttProbabilit…","labels":["ProbabilityTheory.condMutualInfo_congr_ae_finite"],"detail_key":"p14"},{"id":"n20206","layer":"informal","project":"p14","title":"Conditional mutual information determined by the joint law","kind":"theorem","summary":"[Conditional mutual information determined by the joint law] If (X, Y, Z) and (X', Y', Z') have…","labels":["ProbabilityTheory.IdentDistrib.condMutualInfo_eq"],"detail_key":"p14"},{"id":"n20207","layer":"informal","project":"p14","title":"Finite-space version: conditional mutual information determined by the joint law","kind":"theorem","summary":"[Finite-space version: conditional mutual information determined by the joint law] Finite-space…","labels":["ProbabilityTheory.IdentDistrib.condMutualInfo_eq_finite"],"detail_key":"p14"},{"id":"n20208","layer":"informal","project":"p14","title":"Conditional mutual information invariant under injective recodings of right and condition…","kind":"theorem","summary":"[Conditional mutual information invariant under injective recodings of right and conditioning v…","labels":["ProbabilityTheory.condMutualInfo_comp_right_conditioning_of_injective"],"detail_key":"p14"},{"id":"n20209","layer":"informal","project":"p14","title":"Identical distribution of pairs preserved under conditioning on the same set","kind":"theorem","summary":"[Identical distribution of pairs preserved under conditioning on the same set] If (X, Y) and (X…","labels":["ProbabilityTheory.IdentDistrib.cond_of_pair"],"detail_key":"p14"},{"id":"n20210","layer":"informal","project":"p14","title":"Xbit","kind":"definition","summary":"[Xbit] Alice's bit in a disjoint coordinate-pair.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.xBit"],"detail_key":"p14"},{"id":"n20211","layer":"informal","project":"p14","title":"Ybit","kind":"definition","summary":"[Ybit] Bob's bit in a disjoint coordinate-pair.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.yBit"],"detail_key":"p14"},{"id":"n20212","layer":"informal","project":"p14","title":"Card","kind":"theorem","summary":"[Card] There are three disjoint coordinate-pairs.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.card"],"detail_key":"p14"},{"id":"n20213","layer":"informal","project":"p14","title":"Not Xbit And Ybit","kind":"theorem","summary":"[Not Xbit And Ybit] A disjoint coordinate-pair never has both bits equal to \\texttttrue.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.not_xBit_and_yBit"],"detail_key":"p14"},{"id":"n20214","layer":"informal","project":"p14","title":"Swap","kind":"definition","summary":"[Swap] Swap Alice's and Bob's sides in a disjoint coordinate.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.swap"],"detail_key":"p14"},{"id":"n20215","layer":"informal","project":"p14","title":"Swap Swap","kind":"theorem","summary":"[Swap Swap] Swapping sides twice recovers the original disjoint coordinate.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.swap_swap"],"detail_key":"p14"},{"id":"n20216","layer":"informal","project":"p14","title":"Xbit Swap","kind":"theorem","summary":"[Xbit Swap] After swapping a disjoint coordinate, Alice sees Bob's original bit.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.DisjointCoordinate.xBit_swap"],"detail_key":"p14"},{"id":"n20217","layer":"informal","project":"p14","title":"Hardsample","kind":"definition","summary":"[Hardsample] The sample space for the hard distribution. The \\textttother coordinate at T is ig…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample"],"detail_key":"p14"},{"id":"n20218","layer":"informal","project":"p14","title":"Equivprod","kind":"definition","summary":"[Equivprod] Technical auxiliary result for \\textttequivProd.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample.equivProd"],"detail_key":"p14"},{"id":"n20219","layer":"informal","project":"p14","title":"Card","kind":"theorem","summary":"[Card] Cardinality of the explicit hard-distribution sample space.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample.card"],"detail_key":"p14"},{"id":"n20220","layer":"informal","project":"p14","title":"Uniformfin","kind":"definition","summary":"[Uniformfin] Uniform law on the special coordinate.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformFin"],"detail_key":"p14"},{"id":"n20221","layer":"informal","project":"p14","title":"Uniformfin Singleton","kind":"theorem","summary":"[Uniformfin Singleton] Each coordinate has mass 1 / n under the uniform law on \\textttFin n.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformFin_singleton"],"detail_key":"p14"},{"id":"n20222","layer":"informal","project":"p14","title":"Uniformfin Real","kind":"theorem","summary":"[Uniformfin Real] The uniform law on \\textttFin n assigns mass by normalized cardinality.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformFin_real"],"detail_key":"p14"},{"id":"n20223","layer":"informal","project":"p14","title":"Uniformfin Univ","kind":"theorem","summary":"[Uniformfin Univ] The uniform law on \\textttFin n has total mass 1.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformFin_univ"],"detail_key":"p14"},{"id":"n20224","layer":"informal","project":"p14","title":"Uniformbool","kind":"definition","summary":"[Uniformbool] Uniform law on one bit.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformBool"],"detail_key":"p14"},{"id":"n20225","layer":"informal","project":"p14","title":"Uniformbool Singleton","kind":"theorem","summary":"[Uniformbool Singleton] Each bit has mass 1 / 2 under the uniform law on one bit.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformBool_singleton"],"detail_key":"p14"},{"id":"n20226","layer":"informal","project":"p14","title":"Uniformbool Real","kind":"theorem","summary":"[Uniformbool Real] The uniform law on \\textttBool assigns mass by normalized cardinality.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformBool_real"],"detail_key":"p14"},{"id":"n20227","layer":"informal","project":"p14","title":"Uniformbool Univ","kind":"theorem","summary":"[Uniformbool Univ] The uniform law on \\textttBool has total mass 1.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformBool_univ"],"detail_key":"p14"},{"id":"n20228","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector","kind":"definition","summary":"[Uniformdisjointcoordinatevector] The uniform law on generated disjoint coordinate vectors.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector"],"detail_key":"p14"},{"id":"n20229","layer":"informal","project":"p14","title":"Uniformdisjointcoordinate","kind":"definition","summary":"[Uniformdisjointcoordinate] The uniform law on one disjoint coordinate.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinate"],"detail_key":"p14"},{"id":"n20230","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Singleton","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Singleton] Each generated disjoint coordinate vector has mass…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_singleton"],"detail_key":"p14"},{"id":"n20231","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Real","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Real] The uniform law on disjoint-coordinate vectors assigns m…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_real"],"detail_key":"p14"},{"id":"n20232","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Univ","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Univ] The uniform law on disjoint-coordinate vectors has total…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_univ"],"detail_key":"p14"},{"id":"n20233","layer":"informal","project":"p14","title":"Uniformdisjointcoordinate Singleton","kind":"theorem","summary":"[Uniformdisjointcoordinate Singleton] Each disjoint coordinate has mass 1 / 3 under the one-coo…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinate_singleton"],"detail_key":"p14"},{"id":"n20234","layer":"informal","project":"p14","title":"Xbit","kind":"definition","summary":"[Xbit] Alice's bit at coordinate i under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.xBit"],"detail_key":"p14"},{"id":"n20235","layer":"informal","project":"p14","title":"Ybit","kind":"definition","summary":"[Ybit] Bob's bit at coordinate i under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yBit"],"detail_key":"p14"},{"id":"n20236","layer":"informal","project":"p14","title":"X","kind":"definition","summary":"[X] Alice's input set under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.X"],"detail_key":"p14"},{"id":"n20237","layer":"informal","project":"p14","title":"Y","kind":"definition","summary":"[Y] Bob's input set under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.Y"],"detail_key":"p14"},{"id":"n20238","layer":"informal","project":"p14","title":"Xvector","kind":"definition","summary":"[Xvector] Alice's full input vector under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.xVector"],"detail_key":"p14"},{"id":"n20239","layer":"informal","project":"p14","title":"Yvector","kind":"definition","summary":"[Yvector] Bob's full input vector under a hard-distribution sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yVector"],"detail_key":"p14"},{"id":"n20240","layer":"informal","project":"p14","title":"Reverseboolvector","kind":"definition","summary":"[Reverseboolvector] Reverse the coordinate order of a boolean vector.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.reverseBoolVector"],"detail_key":"p14"},{"id":"n20241","layer":"informal","project":"p14","title":"Reverseboolvector Reverseboolvector","kind":"theorem","summary":"[Reverseboolvector Reverseboolvector] Reversing a boolean vector twice recovers 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Y…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.aliceClaimConditioning"],"detail_key":"p14"},{"id":"n20260","layer":"informal","project":"p14","title":"Alicedynamicconditioning","kind":"definition","summary":"[Alicedynamicconditioning] Alice's corrected conditioning data without the special coordinate:…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.aliceDynamicConditioning"],"detail_key":"p14"},{"id":"n20261","layer":"informal","project":"p14","title":"Aliceclaimconditioning Eq Specialcoordinate Prod Dynamic","kind":"theorem","summary":"[Aliceclaimconditioning Eq Specialcoordinate Prod Dynamic] Alice's corrected conditioning is th…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.aliceClaimConditioning_eq_specialCoordinate_prod_dynamic"],"detail_key":"p14"},{"id":"n20262","layer":"informal","project":"p14","title":"Bobclaimconditioning","kind":"definition","summary":"[Bobclaimconditioning] Bob's corrected special-coordinate conditioning variable: T, X_≤T, Y_>T.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.bobClaimConditioning"],"detail_key":"p14"},{"id":"n20263","layer":"informal","project":"p14","title":"Fixedxbit","kind":"definition","summary":"[Fixedxbit] Alice's bit at a fixed coordinate.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedXBit"],"detail_key":"p14"},{"id":"n20264","layer":"informal","project":"p14","title":"Fixedxstrictprefix","kind":"definition","summary":"[Fixedxstrictprefix] Alice's fixed-coordinate strict prefix X_<i, represented without 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(X_<i,…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainConditioningValue_fixedAliceFullYConditioning"],"detail_key":"p14"},{"id":"n20273","layer":"informal","project":"p14","title":"Aliceclaimconditioningyfalsevalues","kind":"definition","summary":"[Aliceclaimconditioningyfalsevalues] The values of Alice's corrected conditioning variable for…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.aliceClaimConditioningYFalseValues"],"detail_key":"p14"},{"id":"n20274","layer":"informal","project":"p14","title":"Ygespecial Specialcoordinate","kind":"theorem","summary":"[Ygespecial Specialcoordinate] The Y_≥T component of Alice's corrected conditioning contains th…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yGeSpecial_specialCoordinate"],"detail_key":"p14"},{"id":"n20275","layer":"informal","project":"p14","title":"Specialy False Eq Preimage Aliceclaimconditioningyfalsevalues","kind":"theorem","summary":"[Specialy False Eq Preimage Aliceclaimconditioningyfalsevalues] The event Y_T=false is determin…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.specialY_false_eq_preimage_aliceClaimConditioningYFalseValues"],"detail_key":"p14"},{"id":"n20276","layer":"informal","project":"p14","title":"Dualconditioningvalue","kind":"definition","summary":"[Dualconditioningvalue] The conditioning-value recoding induced by reversing coordinates and sw…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue"],"detail_key":"p14"},{"id":"n20277","layer":"informal","project":"p14","title":"Dualconditioningvalue Dualconditioningvalue","kind":"theorem","summary":"[Dualconditioningvalue Dualconditioningvalue] Recoding a conditioning value twice recovers the…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue_dualConditioningValue"],"detail_key":"p14"},{"id":"n20278","layer":"informal","project":"p14","title":"Dualconditioningvalue Injective","kind":"theorem","summary":"[Dualconditioningvalue Injective] The conditioning-value duality map is injective.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue_injective"],"detail_key":"p14"},{"id":"n20279","layer":"informal","project":"p14","title":"Protocoltype","kind":"definition","summary":"[Protocoltype] The deterministic protocol type for disjointness inputs of length n.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolType"],"detail_key":"p14"},{"id":"n20280","layer":"informal","project":"p14","title":"Transcripttype","kind":"definition","summary":"[Transcripttype] The syntactic transcript type for a disjointness 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\\textttyBit\\_of\\_ne\\_special.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yBit_of_ne_special"],"detail_key":"p14"},{"id":"n20305","layer":"informal","project":"p14","title":"Not Xbit And Ybit Of Ne","kind":"theorem","summary":"[Not Xbit And Ybit Of Ne] Away from T, generated coordinate-pairs are disjoint.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.not_xBit_and_yBit_of_ne_special"],"detail_key":"p14"},{"id":"n20306","layer":"informal","project":"p14","title":"Coordinateofbits","kind":"definition","summary":"[Coordinateofbits] The disjoint coordinate with the given bits. On the invalid (true, true) inp…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateOfBits"],"detail_key":"p14"},{"id":"n20307","layer":"informal","project":"p14","title":"Coordinateofbits Xbit","kind":"theorem","summary":"[Coordinateofbits Xbit] \\textttcoordinateOfBits preserves Alice's bit when the requested pair i…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateOfBits_xBit"],"detail_key":"p14"},{"id":"n20308","layer":"informal","project":"p14","title":"Coordinateofbits Ybit","kind":"theorem","summary":"[Coordinateofbits Ybit] \\textttcoordinateOfBits preserves Bob's bit when the requested pair is…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateOfBits_yBit"],"detail_key":"p14"},{"id":"n20309","layer":"informal","project":"p14","title":"Coordinateofbits Xbit Ybit","kind":"theorem","summary":"[Coordinateofbits Xbit Ybit] \\textttcoordinateOfBits reconstructs an existing disjoint coordina…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateOfBits_xBit_yBit"],"detail_key":"p14"},{"id":"n20310","layer":"informal","project":"p14","title":"Mix","kind":"definition","summary":"[Mix] Mix Alice's input from ωX with Bob's input from ωY. This is only intended to be used when…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.mix"],"detail_key":"p14"},{"id":"n20311","layer":"informal","project":"p14","title":"Disjoint X Y Iff","kind":"theorem","summary":"[Disjoint X Y Iff] The generated sets are disjoint exactly when the two special-coordinate bits…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjoint_X_Y_iff"],"detail_key":"p14"},{"id":"n20312","layer":"informal","project":"p14","title":"Disjointcoordinatevector","kind":"definition","summary":"[Disjointcoordinatevector] The generated disjoint coordinate-pair vector. On samples outside D,…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVector"],"detail_key":"p14"},{"id":"n20313","layer":"informal","project":"p14","title":"Disjointcoordinatevector Xbit Of Mem Disjointevent","kind":"theorem","summary":"[Disjointcoordinatevector Xbit Of Mem Disjointevent] On disjoint samples, the generated coordin…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVector_xBit_of_mem_disjointEvent"],"detail_key":"p14"},{"id":"n20314","layer":"informal","project":"p14","title":"Disjointcoordinatevector Ybit Of Mem Disjointevent","kind":"theorem","summary":"[Disjointcoordinatevector Ybit Of Mem Disjointevent] On disjoint samples, the generated coordin…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVector_yBit_of_mem_disjointEvent"],"detail_key":"p14"},{"id":"n20315","layer":"informal","project":"p14","title":"Ignoredcoordinate","kind":"definition","summary":"[Ignoredcoordinate] The \\textttother value at the special coordinate is ignored by the generate…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ignoredCoordinate"],"detail_key":"p14"},{"id":"n20316","layer":"informal","project":"p14","title":"Disjointmodel","kind":"definition","summary":"[Disjointmodel] Coordinates for the conditioned-on-disjointness sample space: the special coord…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointModel"],"detail_key":"p14"},{"id":"n20317","layer":"informal","project":"p14","title":"Disjointeventequiv","kind":"definition","summary":"[Disjointeventequiv] Technical auxiliary result for \\textttdisjointEventEquiv.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointEventEquiv"],"detail_key":"p14"},{"id":"n20318","layer":"informal","project":"p14","title":"Disjointeventinterdisjointmodelequiv","kind":"definition","summary":"[Disjointeventinterdisjointmodelequiv] Technical auxiliary result for \\textttdisjointEventInter…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointEventInterDisjointModelEquiv"],"detail_key":"p14"},{"id":"n20319","layer":"informal","project":"p14","title":"Card Disjointevent Inter Disjointmodel Fiber","kind":"theorem","summary":"[Card Disjointevent Inter Disjointmodel Fiber] Technical auxiliary result for \\textttcard\\_disj…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.card_disjointEvent_inter_disjointModel_fiber"],"detail_key":"p14"},{"id":"n20320","layer":"informal","project":"p14","title":"Measurereal Disjointevent Inter Disjointmodel 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under hard-sample duality.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.specialZeroZero_dualHardSample"],"detail_key":"p14"},{"id":"n20332","layer":"informal","project":"p14","title":"Xvector Dualhardsample","kind":"theorem","summary":"[Xvector Dualhardsample] Alice's full input vector in the dual is Bob's original full vector in…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.xVector_dualHardSample"],"detail_key":"p14"},{"id":"n20333","layer":"informal","project":"p14","title":"Yvector Dualhardsample","kind":"theorem","summary":"[Yvector Dualhardsample] Bob's full input vector in the dual is Alice's original full vector in…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yVector_dualHardSample"],"detail_key":"p14"},{"id":"n20334","layer":"informal","project":"p14","title":"Xbeforespecial Dualhardsample","kind":"theorem","summary":"[Xbeforespecial Dualhardsample] Alice's before-special 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Bob's Y_≥T vector in the dual is Alice's X_≤T vector in reverse ord…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.yGeSpecial_dualHardSample"],"detail_key":"p14"},{"id":"n20338","layer":"informal","project":"p14","title":"Aliceclaimconditioning Dualhardsample","kind":"theorem","summary":"[Aliceclaimconditioning Dualhardsample] Alice's corrected conditioning in the dual is Bob's cor…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.aliceClaimConditioning_dualHardSample"],"detail_key":"p14"},{"id":"n20339","layer":"informal","project":"p14","title":"Bobclaimconditioning Dualhardsample","kind":"theorem","summary":"[Bobclaimconditioning Dualhardsample] Bob's corrected conditioning in the dual is Alice's corre…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.bobClaimConditioning_dualHardSample"],"detail_key":"p14"},{"id":"n20340","layer":"informal","project":"p14","title":"Not Xbit Left And Ybit 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first sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.X_mix"],"detail_key":"p14"},{"id":"n20344","layer":"informal","project":"p14","title":"Y Mix","kind":"theorem","summary":"[Y Mix] The mixed sample has Bob's full input from the second sample.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.Y_mix"],"detail_key":"p14"},{"id":"n20345","layer":"informal","project":"p14","title":"Input Mix","kind":"theorem","summary":"[Input Mix] The mixed sample's generated input is the mixed input pair.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.input_mix"],"detail_key":"p14"},{"id":"n20346","layer":"informal","project":"p14","title":"Xbeforespecial Mix","kind":"theorem","summary":"[Xbeforespecial Mix] Mixing preserves Alice's before-T conditioning data from the first 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This is the involution behind the rect…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.mix_mix_swap"],"detail_key":"p14"},{"id":"n20351","layer":"informal","project":"p14","title":"Hardsample Measurereal Fieldproduct","kind":"theorem","summary":"[Hardsample Measurereal Fieldproduct] The hard sample distribution is the product distribution…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.hardSample_measureReal_fieldProduct"],"detail_key":"p14"},{"id":"n20352","layer":"informal","project":"p14","title":"Measurereal Specialintersect","kind":"theorem","summary":"[Measurereal Specialintersect] The hard distribution creates an intersection with probability 1…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialIntersect"],"detail_key":"p14"},{"id":"n20353","layer":"informal","project":"p14","title":"Measurereal Specialbitsevent","kind":"theorem","summary":"[Measurereal 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an…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialCoordinateEvent_inter_specialIntersect"],"detail_key":"p14"},{"id":"n20356","layer":"informal","project":"p14","title":"Measurereal Specialzerozero","kind":"theorem","summary":"[Measurereal Specialzerozero] The event (X_T, Y_T) = (0, 0) has probability 1 / 4.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialZeroZero"],"detail_key":"p14"},{"id":"n20357","layer":"informal","project":"p14","title":"Measurereal Specialy False","kind":"theorem","summary":"[Measurereal Specialy False] The event Y_T = false has probability 1 / 2.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialY_false"],"detail_key":"p14"},{"id":"n20358","layer":"informal","project":"p14","title":"Disjointevent Eq Compl Specialintersect","kind":"theorem","summary":"[Disjointevent Eq Compl Specialintersect] The disjoint event 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di…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinateEvent"],"detail_key":"p14"},{"id":"n20364","layer":"informal","project":"p14","title":"Specialcoordinate Preimage Singleton","kind":"theorem","summary":"[Specialcoordinate Preimage Singleton] The special-coordinate singleton event is the correspond…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.specialCoordinate_preimage_singleton"],"detail_key":"p14"},{"id":"n20365","layer":"informal","project":"p14","title":"Disjointcondmeasure Measurereal Specialcoordinate Preimage Singleton","kind":"theorem","summary":"[Disjointcondmeasure Measurereal Specialcoordinate Preimage Singleton] Under the disjoint-condi…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinate_preimage_singleton"],"detail_key":"p14"},{"id":"n20366","layer":"informal","project":"p14","title":"Disjointcondmeasure Ae Disjointevent","kind":"theorem","summary":"[Disjointcondmeasure Ae Disjointevent] Under the measure conditioned on disjointness, the disjo…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_ae_disjointEvent"],"detail_key":"p14"},{"id":"n20367","layer":"informal","project":"p14","title":"Mem Disjointevent Of Specialy Eq False","kind":"theorem","summary":"[Mem Disjointevent Of Specialy Eq False] If Bob's special bit is \\textttfalse, the generated in…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.mem_disjointEvent_of_specialY_eq_false"],"detail_key":"p14"},{"id":"n20368","layer":"informal","project":"p14","title":"Specialbitsevent Subset 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distributi…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialY_false"],"detail_key":"p14"},{"id":"n20373","layer":"informal","project":"p14","title":"Disjointspecialyfalsemeasure","kind":"definition","summary":"[Disjointspecialyfalsemeasure] The disjoint-conditioned hard distribution, further conditioned…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure"],"detail_key":"p14"},{"id":"n20374","layer":"informal","project":"p14","title":"Disjointspecialyfalsemeasure Isprobabilitymeasure","kind":"theorem","summary":"[Disjointspecialyfalsemeasure Isprobabilitymeasure] Conditioning the disjoint law on Y_T = fals…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_isProbabilityMeasure"],"detail_key":"p14"},{"id":"n20375","layer":"informal","project":"p14","title":"Disjointspecialyfalsemeasure Measurereal 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di…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinate_disjointCoordinateVector_fiber"],"detail_key":"p14"},{"id":"n20380","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Eq Pi","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Eq Pi] The uniform law on disjoint coordinate vectors is the p…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_eq_pi"],"detail_key":"p14"},{"id":"n20381","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Iindepfun","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Iindepfun] Under the uniform disjoint-coordinate-vector law, 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\\textttcoordinateAlice…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondBits_fiber_eq_iInter"],"detail_key":"p14"},{"id":"n20399","layer":"informal","project":"p14","title":"Coordinatewithcondbit","kind":"definition","summary":"[Coordinatewithcondbit] Technical auxiliary result for \\textttcoordinateWithCondBit.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit"],"detail_key":"p14"},{"id":"n20400","layer":"informal","project":"p14","title":"Coordinatewithcondbit Spec","kind":"theorem","summary":"[Coordinatewithcondbit Spec] Technical auxiliary result for \\textttcoordinateWithCondBit\\_spec.","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit_spec"],"detail_key":"p14"},{"id":"n20401","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Measure Condbit Ne 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fixing…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_indep_coordinateXBit_coordinateYBefore_condBits"],"detail_key":"p14"},{"id":"n20404","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Crossinfo Condbits Eq Zero","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Crossinfo Condbits Eq Zero] Product-coordinate form of the fix…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.uniformDisjointCoordinateVector_crossInfo_condBits_eq_zero"],"detail_key":"p14"},{"id":"n20405","layer":"informal","project":"p14","title":"Uniformdisjointcoordinatevector Crossinfo Eq Zero","kind":"theorem","summary":"[Uniformdisjointcoordinatevector Crossinfo Eq Zero] Product-coordinate form of the 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the…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.mutualInfo_specialX_coarseConditioning_disjointSpecialYFalse_eq_zero"],"detail_key":"p14"},{"id":"n20556","layer":"informal","project":"p14","title":"Mutualinfo Specialx Zvariable Eq Alicecoarseinfotermspecialyfalse","kind":"theorem","summary":"[Mutualinfo Specialx Zvariable Eq Alicecoarseinfotermspecialyfalse] The full Z=(M,coarse) mutua…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.mutualInfo_specialX_zVariable_eq_aliceCoarseInfoTermSpecialYFalse"],"detail_key":"p14"},{"id":"n20557","layer":"informal","project":"p14","title":"Integral Xfiberkl Disjointspecialyfalse Eq Alicecoarseinfotermspecialyfalse","kind":"theorem","summary":"[Integral Xfiberkl Disjointspecialyfalse Eq Alicecoarseinfotermspecialyfalse] KL 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I(X_…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.two_thirds_mul_aliceInfoTermSpecialYFalse_le_aliceInfoTerm"],"detail_key":"p14"},{"id":"n20560","layer":"informal","project":"p14","title":"Integral Xfiberkl Disjointspecialyfalse Le Three Halves","kind":"theorem","summary":"[Integral Xfiberkl Disjointspecialyfalse Le Three Halves] Alice information step under D ∧ Y_T=…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.integral_xFiberKL_disjointSpecialYFalse_le_three_halves_mul_aliceInfoTerm"],"detail_key":"p14"},{"id":"n20561","layer":"informal","project":"p14","title":"Integral Xfiberkl Disjointspecialyfalse Le Two Mul","kind":"theorem","summary":"[Integral Xfiberkl Disjointspecialyfalse Le Two Mul] Alice information step under D ∧ Y_T=false…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.integral_xFiberKL_disjointSpecialYFalse_le_two_mul_gamma_pow_four"],"detail_key":"p14"},{"id":"n20562","layer":"informal","project":"p14","title":"Integral Xdistance Sq Disjointspecialyfalse Le Gamma","kind":"theorem","summary":"[Integral Xdistance Sq Disjointspecialyfalse Le Gamma] Alice Pinsker/information step under D ∧…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.integral_xDistance_sq_disjointSpecialYFalse_le_gamma_pow_four"],"detail_key":"p14"},{"id":"n20563","layer":"informal","project":"p14","title":"Measurereal Specialzerozero Inter Xdistance Bad Le","kind":"theorem","summary":"[Measurereal Specialzerozero Inter Xdistance Bad Le] Alice marginal estimate: under the (X_T,Y_…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialZeroZero_inter_xDistance_bad_le"],"detail_key":"p14"},{"id":"n20564","layer":"informal","project":"p14","title":"Measurereal Specialzerozero Inter Ydistance Bad Le","kind":"theorem","summary":"[Measurereal Specialzerozero Inter Ydistance Bad Le] Bob marginal estimate, symmetric to the Al…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.measureReal_specialZeroZero_inter_yDistance_bad_le"],"detail_key":"p14"},{"id":"n20565","layer":"informal","project":"p14","title":"One Div Four Mul One Sub","kind":"theorem","summary":"[One Div Four Mul One Sub] If the corrected special-coordinate information is ≤ 2γ^4 / 3, then…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.one_div_four_mul_one_sub_four_mul_le_measureReal_goodZEvent"],"detail_key":"p14"},{"id":"n20566","layer":"informal","project":"p14","title":"Zfibermeasure Inter Fiber","kind":"theorem","summary":"[Zfibermeasure Inter Fiber] Intersecting with the defining Z=z fiber does not change probabilit…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.zFiberMeasure_inter_fiber"],"detail_key":"p14"},{"id":"n20567","layer":"informal","project":"p14","title":"Abs Conditionalspecialpairlaw Singleton Sub Quarter Le","kind":"theorem","summary":"[Abs Conditionalspecialpairlaw Singleton Sub Quarter Le] A good z gives the expected singleton-…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.abs_conditionalSpecialPairLaw_singleton_sub_quarter_le"],"detail_key":"p14"},{"id":"n20568","layer":"informal","project":"p14","title":"Quarter Sub Two Mul Le Conditionalspecialpairlaw","kind":"theorem","summary":"[Quarter Sub Two Mul Le Conditionalspecialpairlaw] A good z gives a lower bound on each singlet…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.quarter_sub_two_mul_le_conditionalSpecialPairLaw_singleton"],"detail_key":"p14"},{"id":"n20569","layer":"informal","project":"p14","title":"Zfiber Inter Diag Specialpair Subset Protocolerrorevent","kind":"theorem","summary":"[Zfiber Inter Diag Specialpair Subset Protocolerrorevent] On a fixed Z=z fiber, inputs whose sp…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.zFiber_inter_diag_specialPair_subset_protocolErrorEvent"],"detail_key":"p14"},{"id":"n20570","layer":"informal","project":"p14","title":"Quarter Sub Two Mul Le Zfibermeasure","kind":"theorem","summary":"[Quarter Sub Two Mul Le Zfibermeasure] On a good Z fiber, the conditional protocol-error probab…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.quarter_sub_two_mul_le_zFiberMeasure_protocolErrorEvent"],"detail_key":"p14"},{"id":"n20571","layer":"informal","project":"p14","title":"Goodzevent Mul Quarter Sub Two Mul","kind":"theorem","summary":"[Goodzevent Mul Quarter Sub Two Mul] Averaging the good-fiber error lower bound over all good Z…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarter_sub_two_mul_le_protocolErrorEvent"],"detail_key":"p14"},{"id":"n20572","layer":"informal","project":"p14","title":"Goodzevent Mul Quarter Sub Two Mul","kind":"theorem","summary":"[Goodzevent Mul Quarter Sub Two Mul] The final good-fiber error calculation, phrased using dist…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarter_sub_two_mul_le_distributionalError"],"detail_key":"p14"},{"id":"n20573","layer":"informal","project":"p14","title":"One Div 32768 Sq Lt Claiminfo","kind":"theorem","summary":"[One Div 32768 Sq Lt Claiminfo] A deterministic protocol with distributional error at most 1 /…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.one_div_32768_sq_lt_claimInfo_of_distributionalError_le"],"detail_key":"p14"},{"id":"n20574","layer":"informal","project":"p14","title":"Const Mul N Le Complexity Of","kind":"theorem","summary":"[Const Mul N Le Complexity Of] Deterministic fixed-error disjointness lower bound from the info…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.const_mul_n_le_complexity_of_distributionalError_le"],"detail_key":"p14"},{"id":"n20575","layer":"informal","project":"p14","title":"Lt Publiccoin Communicationcomplexity Disjointness Of Lt","kind":"theorem","summary":"[Lt Publiccoin Communicationcomplexity Disjointness Of Lt] Public-coin fixed-error lower bound…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.lt_publicCoin_communicationComplexity_disjointness_of_lt_const_mul_n"],"detail_key":"p14"},{"id":"n20576","layer":"informal","project":"p14","title":"Floor Div Pow Lt Publiccoin Communicationcomplexity","kind":"theorem","summary":"[Floor Div Pow Lt Publiccoin Communicationcomplexity] Headline theorem: public-coin randomized…","labels":["CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.floor_div_pow_lt_publicCoin_communicationComplexity_disjointness"],"detail_key":"p14"},{"id":"n20577","layer":"informal","project":"p14","title":"Overlap of two bit-vectors","kind":"definition","summary":"[Overlap of two bit-vectors] Given two n-bit inputs x, y : BoolInput\\,n, overlap(x, y) is the \\…","labels":["CommunicationComplexity.Functions.InnerProduct.overlap"],"detail_key":"p14"},{"id":"n20578","layer":"informal","project":"p14","title":"Mod-2 inner product","kind":"definition","summary":"[Mod-2 inner product] innerProduct(n, x, y) : Bool is the mod-2 inner product of two n-bit vect…","labels":["CommunicationComplexity.Functions.InnerProduct.innerProduct"],"detail_key":"p14"},{"id":"n20579","layer":"informal","project":"p14","title":"Inner product with the zero vector is false","kind":"lemma","summary":"[Inner product with the zero vector is false] For every x : BoolInput\\,n, the inner product of…","labels":["CommunicationComplexity.Functions.InnerProduct.innerProduct_zero_right"],"detail_key":"p14"},{"id":"n20580","layer":"informal","project":"p14","title":"Uniform weight of each Boolean input","kind":"lemma","summary":"[Uniform weight of each Boolean input] Every element x : BoolInput\\,n has PMF weight 2^-n under…","labels":["CommunicationComplexity.Functions.InnerProduct.pmf_toReal_eq_two_pow_inv"],"detail_key":"p14"},{"id":"n20581","layer":"informal","project":"p14","title":"Integral as arithmetic average","kind":"lemma","summary":"[Integral as arithmetic average] For any f : BoolInput\\,n \\to R, the integral against the unifo…","labels":["CommunicationComplexity.Functions.InnerProduct.integral_eq_average_sum"],"detail_key":"p14"},{"id":"n20582","layer":"informal","project":"p14","title":"Flipping a set coordinate toggles the inner product","kind":"lemma","summary":"[Flipping a set coordinate toggles the inner product] If y_i = \\mathtttrue, then flipping coord…","labels":["CommunicationComplexity.Functions.InnerProduct.innerProduct_flipAt_eq_xor"],"detail_key":"p14"},{"id":"n20583","layer":"informal","project":"p14","title":"Walsh character sums to zero when z \\ne 0","kind":"lemma","summary":"[Walsh character sums to zero when z \\ne 0] If z : BoolInput\\,n has a coordinate i with z_i = \\…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_boolSign_innerProduct_eq_zero_of_exists_true"],"detail_key":"p14"},{"id":"n20584","layer":"informal","project":"p14","title":"Walsh character sum is an indicator at the zero vector","kind":"lemma","summary":"[Walsh character sum is an indicator at the zero vector] For every z : BoolInput\\,n, \\[ \\sum_x…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_boolSign_innerProduct_eq_zeroInput_indicator"],"detail_key":"p14"},{"id":"n20585","layer":"informal","project":"p14","title":"Coordinatewise xor of Boolean inputs","kind":"definition","summary":"[Coordinatewise xor of Boolean inputs] xorInput(y, z) : BoolInput\\,n is the coordinatewise XOR…","labels":["CommunicationComplexity.Functions.InnerProduct.xorInput"],"detail_key":"p14"},{"id":"n20586","layer":"informal","project":"p14","title":"Pointwise evaluation of xorInput","kind":"lemma","summary":"[Pointwise evaluation of xorInput] For all y, z : BoolInput\\,n and i : Fin\\,n, xorInput(y, z)(i…","labels":["CommunicationComplexity.Functions.InnerProduct.xorInput_apply"],"detail_key":"p14"},{"id":"n20587","layer":"informal","project":"p14","title":"xorInput equals zero iff inputs are equal","kind":"lemma","summary":"[xorInput equals zero iff inputs are equal] xorInput(y, z) = 0 if and only if y = z.","labels":["CommunicationComplexity.Functions.InnerProduct.xorInput_eq_zeroInput_iff"],"detail_key":"p14"},{"id":"n20588","layer":"informal","project":"p14","title":"Walsh character multiplicativity under xor","kind":"lemma","summary":"[Walsh character multiplicativity under xor] For all x, y, z : BoolInput\\,n, \\[ boolSign(IP(x,…","labels":["CommunicationComplexity.Functions.InnerProduct.boolSign_innerProduct_mul_eq_xorInput"],"detail_key":"p14"},{"id":"n20589","layer":"informal","project":"p14","title":"Summed Walsh orthogonality for inner product","kind":"lemma","summary":"[Summed Walsh orthogonality for inner product] For all y, z : BoolInput\\,n, \\[ \\sum_x : BoolInp…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_boolSign_innerProduct_mul_eq_indicator"],"detail_key":"p14"},{"id":"n20590","layer":"informal","project":"p14","title":"Indicator function squared equals itself","kind":"lemma","summary":"[Indicator function squared equals itself] For any set B \\subseteq BoolInput\\,n and y : BoolInp…","labels":["CommunicationComplexity.Functions.InnerProduct.indicatorOne_sq"],"detail_key":"p14"},{"id":"n20591","layer":"informal","project":"p14","title":"Indicator function is at most one","kind":"lemma","summary":"[Indicator function is at most one] For any set B \\subseteq BoolInput\\,n and y : BoolInput\\,n,…","labels":["CommunicationComplexity.Functions.InnerProduct.indicatorOne_le_one"],"detail_key":"p14"},{"id":"n20592","layer":"informal","project":"p14","title":"Sum of squares equals sum over pairs","kind":"lemma","summary":"[Sum of squares equals sum over pairs] For finite types \\alpha and \\beta and f : \\alpha \\to \\be…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_sq_eq_sum_prod"],"detail_key":"p14"},{"id":"n20593","layer":"informal","project":"p14","title":"Scalar factoring in a double sum","kind":"lemma","summary":"[Scalar factoring in a double sum] For a finite type \\alpha and constants a, b \\in R and functi…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_mul_mul"],"detail_key":"p14"},{"id":"n20594","layer":"informal","project":"p14","title":"Diagonal reduction for ite sums","kind":"lemma","summary":"[Diagonal reduction for ite sums] For a finite type \\alpha with decidable equality, a function…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_mul_ite_eq_diag"],"detail_key":"p14"},{"id":"n20595","layer":"informal","project":"p14","title":"Weighted orthogonality identity","kind":"lemma","summary":"[Weighted orthogonality identity] Let \\phi : \\alpha \\to \\beta \\to R be a family of functions sa…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_sq_mul_of_orthogonal"],"detail_key":"p14"},{"id":"n20596","layer":"informal","project":"p14","title":"Parseval identity for inner product Walsh characters","kind":"lemma","summary":"[Parseval identity for inner product Walsh characters] For any b : BoolInput\\,n \\to R, \\[ \\sum_…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_sq_mul_boolSign_innerProduct"],"detail_key":"p14"},{"id":"n20597","layer":"informal","project":"p14","title":"Second moment of indicator-weighted Walsh sum is bounded","kind":"lemma","summary":"[Second moment of indicator-weighted Walsh sum is bounded] For any set B \\subseteq BoolInput\\,n…","labels":["CommunicationComplexity.Functions.InnerProduct.sum_sq_indicator_mul_boolSign_innerProduct_le"],"detail_key":"p14"},{"id":"n20598","layer":"informal","project":"p14","title":"Integral second moment bound for discrepancy computation","kind":"lemma","summary":"[Integral second moment bound for discrepancy computation] For any set B \\subseteq BoolInput\\,n…","labels":["CommunicationComplexity.Functions.InnerProduct.integral_sq_indicator_mul_boolSign_innerProduct_le"],"detail_key":"p14"},{"id":"n20599","layer":"informal","project":"p14","title":"Discrepancy of a rectangle as a product integral","kind":"lemma","summary":"[Discrepancy of a rectangle as a product integral] For sets A, B \\subseteq BoolInput\\,n, the di…","labels":["CommunicationComplexity.Functions.InnerProduct.discrepancy_prod_eq_integral"],"detail_key":"p14"},{"id":"n20600","layer":"informal","project":"p14","title":"Squared discrepancy of a rectangle is at most 2^-n","kind":"lemma","summary":"[Squared discrepancy of a rectangle is at most 2^-n] For all A, B \\subseteq BoolInput\\,n, \\[ \\b…","labels":["CommunicationComplexity.Functions.InnerProduct.sq_discrepancy_prod_le"],"detail_key":"p14"},{"id":"n20601","layer":"informal","project":"p14","title":"Rectangle discrepancy bound for inner 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finite…","labels":["CommunicationComplexity.klDiv_eq_sum_llr_of_ac"],"detail_key":"p14"},{"id":"n20604","layer":"informal","project":"p14","title":"KL divergence as log-likelihood ratio sum","kind":"theorem","summary":"[KL divergence as log-likelihood ratio sum] Let \\Omega be a finite measurable space and let \\mu…","labels":["CommunicationComplexity.FiniteMeasureSpace.klDiv_eq_sum_llr"],"detail_key":"p14"},{"id":"n20605","layer":"informal","project":"p14","title":"KL divergence between PMFs as log-likelihood ratio sum","kind":"theorem","summary":"[KL divergence between PMFs as log-likelihood ratio sum] Let \\Omega be a finite measurable spac…","labels":["CommunicationComplexity.FiniteMeasureSpace.pmf_klDiv_eq_sum_llr"],"detail_key":"p14"},{"id":"n20606","layer":"informal","project":"p14","title":"Radon--Nikodym derivative equals singleton mass ratio","kind":"theorem","summary":"[Radon--Nikodym derivative equals singleton mass ratio] Let \\Omega be a finite measurable space…","labels":["CommunicationComplexity.rnDeriv_toReal_eq_singleton_ratio"],"detail_key":"p14"},{"id":"n20607","layer":"informal","project":"p14","title":"Singleton mass times log-likelihood ratio equals log of mass ratio","kind":"theorem","summary":"[Singleton mass times log-likelihood ratio equals log of mass ratio] Let \\Omega be a finite mea…","labels":["CommunicationComplexity.singleton_mass_mul_llr_eq_log_ratio"],"detail_key":"p14"},{"id":"n20608","layer":"informal","project":"p14","title":"KL divergence as logarithmic mass ratio sum","kind":"theorem","summary":"[KL divergence as logarithmic mass ratio sum] Let \\Omega be a finite measurable space and let \\…","labels":["CommunicationComplexity.FiniteMeasureSpace.klDiv_eq_sum_log"],"detail_key":"p14"},{"id":"n20609","layer":"informal","project":"p14","title":"KL divergence is finite to full-support probability measure","kind":"theorem","summary":"[KL divergence is finite to full-support probability measure] Let \\Omega be a finite measurable…","labels":["CommunicationComplexity.FiniteMeasureSpace.klDiv_ne_top_of_forall_toPMF_ne_zero"],"detail_key":"p14"},{"id":"n20610","layer":"informal","project":"p14","title":"KL divergence between PMFs as logarithmic mass ratio sum","kind":"theorem","summary":"[KL divergence between PMFs as logarithmic mass ratio sum] Let \\Omega be a finite measurable sp…","labels":["CommunicationComplexity.FiniteMeasureSpace.pmf_klDiv_eq_sum_log"],"detail_key":"p14"},{"id":"n20611","layer":"informal","project":"p14","title":"Boolean KL divergence matches PFR \\textttKLDiv","kind":"theorem","summary":"[Boolean KL divergence matches PFR \\textttKLDiv] Let \\mu, \\nu be probability measures on Bool w…","labels":["ProbabilityTheory.toReal_klDiv_bool_eq_KLDiv"],"detail_key":"p14"},{"id":"n20612","layer":"informal","project":"p14","title":"KL divergence of pushed-forward Boolean variable matches PFR \\textttKLDiv","kind":"theorem","summary":"[KL divergence of pushed-forward Boolean variable matches PFR \\textttKLDiv] Let \\mu be a finite…","labels":["ProbabilityTheory.toReal_klDiv_map_bool_eq_KLDiv_of_measureReal_ne_zero"],"detail_key":"p14"},{"id":"n20613","layer":"informal","project":"p14","title":"Radon-Nikodym density","kind":"definition","summary":"[Radon-Nikodym density] For two probability measures \\mu and \\nu on a measurable space \\Omega,…","labels":["CommunicationComplexity.rnDensity"],"detail_key":"p14"},{"id":"n20614","layer":"informal","project":"p14","title":"Radon-Nikodym density is nonnegative","kind":"theorem","summary":"[Radon-Nikodym density is nonnegative] For all x \\in \\Omega, rnDensity(\\mu,\\nu)(x) \\ge 0.","labels":["CommunicationComplexity.rnDensity_nonneg"],"detail_key":"p14"},{"id":"n20615","layer":"informal","project":"p14","title":"Radon-Nikodym density is measurable","kind":"theorem","summary":"[Radon-Nikodym density is measurable] The function x \\mapsto rnDensity(\\mu,\\nu)(x) is measurabl…","labels":["CommunicationComplexity.measurable_rnDensity"],"detail_key":"p14"},{"id":"n20616","layer":"informal","project":"p14","title":"Radon-Nikodym density is \\nu-integrable","kind":"theorem","summary":"[Radon-Nikodym density is \\nu-integrable] The function rnDensity(\\mu,\\nu) is integrable with re…","labels":["CommunicationComplexity.integrable_rnDensity"],"detail_key":"p14"},{"id":"n20617","layer":"informal","project":"p14","title":"Integral of density equals one under absolute continuity","kind":"theorem","summary":"[Integral of density equals one under absolute continuity] If \\mu \\ll \\nu, then \\int rnDensity(…","labels":["CommunicationComplexity.integral_rnDensity_eq_one_of_ac"],"detail_key":"p14"},{"id":"n20618","layer":"informal","project":"p14","title":"rnDensity - 1 is integrable","kind":"theorem","summary":"[rnDensity - 1 is integrable] The function x \\mapsto rnDensity(\\mu,\\nu)(x) - 1 is integrable wi…","labels":["CommunicationComplexity.integrable_rnDensity_sub_one"],"detail_key":"p14"},{"id":"n20619","layer":"informal","project":"p14","title":"|rnDensity - 1| is integrable","kind":"theorem","summary":"[|rnDensity - 1| is integrable] The function x \\mapsto |rnDensity(\\mu,\\nu)(x) - 1| is integrabl…","labels":["CommunicationComplexity.integrable_abs_rnDensity_sub_one"],"detail_key":"p14"},{"id":"n20620","layer":"informal","project":"p14","title":"Integral of rnDensity - 1 is zero under absolute continuity","kind":"theorem","summary":"[Integral of rnDensity - 1 is zero under absolute continuity] If \\mu \\ll \\nu, then \\int (rnDens…","labels":["CommunicationComplexity.integral_rnDensity_sub_one_eq_zero_of_ac"],"detail_key":"p14"},{"id":"n20621","layer":"informal","project":"p14","title":"Measure difference as set integral of density shift","kind":"theorem","summary":"[Measure difference as set integral of density shift] If \\mu \\ll \\nu and S \\subseteq \\Omega is…","labels":["CommunicationComplexity.measureReal_sub_eq_setIntegral_rnDensity_sub_one"],"detail_key":"p14"},{"id":"n20622","layer":"informal","project":"p14","title":"Nonneg-part integral equals half the L^1 norm when mean is zero","kind":"theorem","summary":"[Nonneg-part integral equals half the L^1 norm when mean is zero] Let \\mu be a finite measure a…","labels":["CommunicationComplexity.integral_nonneg_part_eq_half_integral_abs_of_integral_eq_zero"],"detail_key":"p14"},{"id":"n20623","layer":"informal","project":"p14","title":"Supremum of absolute set-integrals equals nonneg part when mean is zero","kind":"theorem","summary":"[Supremum of absolute set-integrals equals nonneg part when mean is zero] Let \\mu be a finite m…","labels":["CommunicationComplexity.sSup_abs_setIntegral_eq_nonneg_part_of_integral_eq_zero"],"detail_key":"p14"},{"id":"n20624","layer":"informal","project":"p14","title":"Density absolute integral","kind":"definition","summary":"[Density absolute integral] The quantity densityAbsIntegral(\\mu,\\nu) := \\int |rnDensity(\\mu,\\nu…","labels":["CommunicationComplexity.densityAbsIntegral"],"detail_key":"p14"},{"id":"n20625","layer":"informal","project":"p14","title":"Density positive set","kind":"definition","summary":"[Density positive set] The \\emphdensity positive set is \\x \\in \\Omega : rnDensity(\\mu,\\nu)(x) \\…","labels":["CommunicationComplexity.densityPositiveSet"],"detail_key":"p14"},{"id":"n20626","layer":"informal","project":"p14","title":"Density positive set is measurable","kind":"theorem","summary":"[Density positive set is measurable] The density positive set \\x : rnDensity(\\mu,\\nu)(x) \\ge 1\\…","labels":["CommunicationComplexity.measurableSet_densityPositiveSet"],"detail_key":"p14"},{"id":"n20627","layer":"informal","project":"p14","title":"Density positive integral","kind":"definition","summary":"[Density positive integral] The \\emphdensity positive integral is densityPositiveIntegral(\\mu,\\…","labels":["CommunicationComplexity.densityPositiveIntegral"],"detail_key":"p14"},{"id":"n20628","layer":"informal","project":"p14","title":"TV distance supremum equals density positive integral under absolute continuity","kind":"theorem","summary":"[TV distance supremum equals density positive integral under absolute continuity] If \\mu \\ll \\n…","labels":["CommunicationComplexity.tvDistanceSup_eq_densityPositiveIntegral_of_ac"],"detail_key":"p14"},{"id":"n20629","layer":"informal","project":"p14","title":"TV distance equals density positive integral under absolute continuity","kind":"theorem","summary":"[TV distance equals density positive integral under absolute continuity] If \\mu \\ll \\nu, then T…","labels":["CommunicationComplexity.tvDistance_eq_densityPositiveIntegral_of_ac"],"detail_key":"p14"},{"id":"n20630","layer":"informal","project":"p14","title":"Density positive integral equals half the density absolute integral","kind":"theorem","summary":"[Density positive integral equals half the density absolute integral] If \\mu \\ll \\nu, then dens…","labels":["CommunicationComplexity.densityPositiveIntegral_eq_half_densityAbsIntegral_of_ac"],"detail_key":"p14"},{"id":"n20631","layer":"informal","project":"p14","title":"Pointwise KL inequality: u y \\le klFun(u) + e^y - 1","kind":"theorem","summary":"[Pointwise KL inequality: u y \\le klFun(u) + e^y - 1] For real numbers u \\ge 0 and y, one has u…","labels":["CommunicationComplexity.mul_le_klFun_add_exp_sub_one"],"detail_key":"p14"},{"id":"n20632","layer":"informal","project":"p14","title":"Integral of e^tX - \\Lambda(t) equals one","kind":"theorem","summary":"[Integral of e^tX - \\Lambda(t) equals one] Let \\mu be a probability measure, X : \\Omega \\to R a…","labels":["CommunicationComplexity.integral_exp_sub_cgf_eq_one"],"detail_key":"p14"},{"id":"n20633","layer":"informal","project":"p14","title":"e^tX - \\Lambda(t) is integrable","kind":"theorem","summary":"[e^tX - \\Lambda(t) is integrable] If e^tX is \\mu-integrable, then so is x \\mapsto e^t X(x) - \\L…","labels":["CommunicationComplexity.integrable_exp_sub_cgf"],"detail_key":"p14"},{"id":"n20634","layer":"informal","project":"p14","title":"Variational bound: t \\int f X\\,d\\mu \\le \\int klFun(f)\\,d\\mu + \\Lambda_\\mu(t)","kind":"theorem","summary":"[Variational bound: t \\int f X\\,d\\mu \\le \\int klFun(f)\\,d\\mu + \\Lambda_\\mu(t)] Let f, X : \\Omeg…","labels":["CommunicationComplexity.variational_integral_mul_le_integral_klFun_add_cgf"],"detail_key":"p14"},{"id":"n20635","layer":"informal","project":"p14","title":"\\tfrac12(\\int|f-1|)^2 \\le \\int klFun(f)","kind":"theorem","summary":"[\\tfrac12(\\int|f-1|)^2 \\le \\int klFun(f)] Let f : \\Omega \\to R with f \\ge 0, \\int f\\,d\\mu = 1,…","labels":["CommunicationComplexity.half_integral_abs_sub_one_sq_le_integral_klFun"],"detail_key":"p14"},{"id":"n20636","layer":"informal","project":"p14","title":"TV distance equals half density absolute integral under absolute continuity","kind":"theorem","summary":"[TV distance equals half density absolute integral under absolute continuity] If \\mu \\ll \\nu, t…","labels":["CommunicationComplexity.tvDistance_eq_half_densityAbsIntegral_of_ac"],"detail_key":"p14"},{"id":"n20637","layer":"informal","project":"p14","title":"\\tfrac12(densityAbsIntegral)^2 \\le \\intklFun(rnDensity)","kind":"theorem","summary":"[\\tfrac12(densityAbsIntegral)^2 \\le \\intklFun(rnDensity)] If \\mu \\ll \\nu and the log-likelihood…","labels":["CommunicationComplexity.half_densityAbsIntegral_sq_le_integral_klFun_rnDensity"],"detail_key":"p14"},{"id":"n20638","layer":"informal","project":"p14","title":"KL-function integral equals log-likelihood integral","kind":"theorem","summary":"[KL-function integral equals log-likelihood integral] If \\mu \\ll \\nu and llr(\\mu,\\nu) is \\mu-in…","labels":["CommunicationComplexity.integral_klFun_rnDeriv_eq_kl_integral"],"detail_key":"p14"},{"id":"n20639","layer":"informal","project":"p14","title":"Density L^1 Pinsker bound via KL integral","kind":"theorem","summary":"[Density L^1 Pinsker bound via KL integral] If \\mu \\ll \\nu and llr(\\mu,\\nu) is \\mu-integrable,…","labels":["CommunicationComplexity.density_l1_pinsker_le_kl_integral"],"detail_key":"p14"},{"id":"n20640","layer":"informal","project":"p14","title":"Real Pinsker inequality under absolute continuity and integrability","kind":"theorem","summary":"[Real Pinsker inequality under absolute continuity and integrability] If \\mu \\ll \\nu and llr(\\m…","labels":["CommunicationComplexity.real_pinsker_inequality_of_ac_of_integrable"],"detail_key":"p14"},{"id":"n20641","layer":"informal","project":"p14","title":"Pinsker inequality (R_\\ge 0^\\infty form) under absolute continuity and integrability","kind":"theorem","summary":"[Pinsker inequality (R_\\ge 0^\\infty form) under absolute continuity and integrability] If \\mu \\…","labels":["CommunicationComplexity.pinsker_inequality_of_ac_of_integrable"],"detail_key":"p14"},{"id":"n20642","layer":"informal","project":"p14","title":"Pinsker inequality under absolute continuity","kind":"theorem","summary":"[Pinsker inequality under absolute continuity] If \\mu \\ll \\nu (without any integrability assump…","labels":["CommunicationComplexity.pinsker_inequality_of_ac"],"detail_key":"p14"},{"id":"n20643","layer":"informal","project":"p14","title":"Pinsker's inequality","kind":"theorem","summary":"[Pinsker's inequality] For any two probability measures \\mu and \\nu on \\Omega, \\[ ofReal\\bigl(2…","labels":["CommunicationComplexity.pinsker_inequality"],"detail_key":"p14"},{"id":"n20644","layer":"informal","project":"p14","title":"Real-valued Pinsker corollary for finite KL divergence","kind":"theorem","summary":"[Real-valued Pinsker corollary for finite KL divergence] If KL(\\mu \\| \\nu) \\ne \\infty, then \\[…","labels":["CommunicationComplexity.two_mul_tvDistance_sq_le_toReal_klDiv"],"detail_key":"p14"},{"id":"n20645","layer":"informal","project":"p14","title":"Signed measure difference","kind":"definition","summary":"[Signed measure difference] Given two probability measures \\mu and \\nu on a measurable space \\O…","labels":["CommunicationComplexity.signedMeasureDiff"],"detail_key":"p14"},{"id":"n20646","layer":"informal","project":"p14","title":"Total variation distance","kind":"definition","summary":"[Total variation distance] The \\emphtotal variation distance between two probability measures \\…","labels":["CommunicationComplexity.tvDistance"],"detail_key":"p14"},{"id":"n20647","layer":"informal","project":"p14","title":"Total variation distance via event supremum","kind":"definition","summary":"[Total variation distance via event supremum] The \\emphsupremum definition of total variation d…","labels":["CommunicationComplexity.tvDistanceSup"],"detail_key":"p14"},{"id":"n20648","layer":"informal","project":"p14","title":"Signed measure via Jordan decomposition","kind":"lemma","summary":"[Signed measure via Jordan decomposition] For any signed measure s on \\Omega and any measurable…","labels":["CommunicationComplexity.TVDistance.signedMeasure_apply_eq_posPart_sub_negPart"],"detail_key":"p14"},{"id":"n20649","layer":"informal","project":"p14","title":"Signed difference vanishes on the whole space","kind":"lemma","summary":"[Signed difference vanishes on the whole space] For any two probability measures \\mu and \\nu, t…","labels":["CommunicationComplexity.TVDistance.signedMeasureDiff_univ"],"detail_key":"p14"},{"id":"n20650","layer":"informal","project":"p14","title":"Jordan parts agree on the whole space","kind":"lemma","summary":"[Jordan parts agree on the whole space] For probability measures \\mu and \\nu, the Jordan positi…","labels":["CommunicationComplexity.TVDistance.jordan_posPart_real_univ_eq_negPart_real_univ"],"detail_key":"p14"},{"id":"n20651","layer":"informal","project":"p14","title":"Total variation mass of the whole space","kind":"lemma","summary":"[Total variation mass of the whole space] For any signed measure s, the real-valued total varia…","labels":["CommunicationComplexity.TVDistance.totalVariation_real_univ"],"detail_key":"p14"},{"id":"n20652","layer":"informal","project":"p14","title":"Half total variation equals positive Jordan part","kind":"lemma","summary":"[Half total variation equals positive Jordan part] For probability measures \\mu and \\nu, \\tfrac…","labels":["CommunicationComplexity.TVDistance.half_totalVariation_real_univ_eq_posPart_real_univ"],"detail_key":"p14"},{"id":"n20653","layer":"informal","project":"p14","title":"Event gap bounded by half total variation","kind":"lemma","summary":"[Event gap bounded by half total variation] For every measurable set S \\subseteq \\Omega, |(\\mu-…","labels":["CommunicationComplexity.TVDistance.event_abs_signedMeasureDiff_le_half_totalVariation"],"detail_key":"p14"},{"id":"n20654","layer":"informal","project":"p14","title":"Probability gap bounded by half total variation","kind":"lemma","summary":"[Probability gap bounded by half total variation] For every measurable set S \\subseteq \\Omega,…","labels":["CommunicationComplexity.TVDistance.event_abs_measureReal_sub_le_half_totalVariation"],"detail_key":"p14"},{"id":"n20655","layer":"informal","project":"p14","title":"Attainment of the signed measure bound","kind":"lemma","summary":"[Attainment of the signed measure bound] There exists a measurable set S \\subseteq \\Omega for w…","labels":["CommunicationComplexity.TVDistance.exists_event_abs_signedMeasureDiff_eq_half_totalVariation"],"detail_key":"p14"},{"id":"n20656","layer":"informal","project":"p14","title":"Attainment of the probability gap bound","kind":"lemma","summary":"[Attainment of the probability gap bound] There exists a measurable set S \\subseteq \\Omega such…","labels":["CommunicationComplexity.TVDistance.exists_event_abs_measureReal_sub_eq_half_totalVariation"],"detail_key":"p14"},{"id":"n20657","layer":"informal","project":"p14","title":"Equivalence of the two TV distance definitions","kind":"theorem","summary":"[Equivalence of the two TV distance definitions] The total-variation-mass definition and the su…","labels":["CommunicationComplexity.TVDistance.tvDistance_eq_tvDistanceSup"],"detail_key":"p14"},{"id":"n20658","layer":"informal","project":"p14","title":"TV distance bounds event probability gaps","kind":"theorem","summary":"[TV distance bounds event probability gaps] For every measurable set S \\subseteq \\Omega, |\\mu(S…","labels":["CommunicationComplexity.TVDistance.abs_measureReal_sub_le_tvDistance"],"detail_key":"p14"},{"id":"n20659","layer":"informal","project":"p14","title":"TV distance is nonneg","kind":"theorem","summary":"[TV distance is nonneg] For any two probability measures \\mu and \\nu, TV(\\mu,\\nu) \\ge 0.","labels":["CommunicationComplexity.TVDistance.tvDistance_nonneg"],"detail_key":"p14"},{"id":"n20660","layer":"informal","project":"p14","title":"Indicator sum bounded by positive part sum","kind":"lemma","summary":"[Indicator sum bounded by positive part sum] For any function a : \\alpha \\to R on a finite type…","labels":["CommunicationComplexity.TVDistance.sum_indicator_le_sum_posPart"],"detail_key":"p14"},{"id":"n20661","layer":"informal","project":"p14","title":"Positive part sum equals half the \\ell^1 sum","kind":"lemma","summary":"[Positive part sum equals half the \\ell^1 sum] Let a : \\alpha \\to R be a function on a finite t…","labels":["CommunicationComplexity.TVDistance.sum_posPart_eq_half_sum_abs"],"detail_key":"p14"},{"id":"n20662","layer":"informal","project":"p14","title":"Indicator sum absolutely bounded by half \\ell^1 sum","kind":"lemma","summary":"[Indicator sum absolutely bounded by half \\ell^1 sum] If a : \\alpha \\to R satisfies \\sum_x a(x)…","labels":["CommunicationComplexity.TVDistance.abs_sum_indicator_le_half_sum_abs"],"detail_key":"p14"},{"id":"n20663","layer":"informal","project":"p14","title":"Positive-part indicator sum equals half \\ell^1 sum","kind":"lemma","summary":"[Positive-part indicator sum equals half \\ell^1 sum] If a : \\alpha \\to R satisfies \\sum_x a(x)…","labels":["CommunicationComplexity.TVDistance.abs_sum_pos_indicator_eq_half_sum_abs"],"detail_key":"p14"},{"id":"n20664","layer":"informal","project":"p14","title":"Singleton mass difference","kind":"definition","summary":"[Singleton mass difference] For probability measures \\mu and \\nu on \\Omega and a point \\omega \\…","labels":["CommunicationComplexity.TVDistance.singletonMassDiff"],"detail_key":"p14"},{"id":"n20665","layer":"informal","project":"p14","title":"Probability gap as sum of singleton mass differences","kind":"lemma","summary":"[Probability gap as sum of singleton mass differences] On a finite measurable space, for any me…","labels":["CommunicationComplexity.TVDistance.measureReal_sub_eq_sum_indicator_singletonMassDiff"],"detail_key":"p14"},{"id":"n20666","layer":"informal","project":"p14","title":"Singleton mass differences sum to zero","kind":"lemma","summary":"[Singleton mass differences sum to zero] On a finite measurable space, the singleton mass diffe…","labels":["CommunicationComplexity.TVDistance.sum_singletonMassDiff_eq_zero"],"detail_key":"p14"},{"id":"n20667","layer":"informal","project":"p14","title":"TV distance as half \\ell^1 distance on finite spaces (supremum form)","kind":"theorem","summary":"[TV distance as half \\ell^1 distance on finite spaces (supremum form)] On a finite measurable s…","labels":["CommunicationComplexity.TVDistance.tvDistanceSup_eq_half_sum"],"detail_key":"p14"},{"id":"n20668","layer":"informal","project":"p14","title":"TV distance as half \\ell^1 distance on finite spaces","kind":"theorem","summary":"[TV distance as half \\ell^1 distance on finite spaces] On a finite measurable space \\Omega, \\[…","labels":["CommunicationComplexity.TVDistance.tvDistance_eq_half_sum"],"detail_key":"p14"},{"id":"n20669","layer":"informal","project":"p14","title":"TV distance on Bool equals gap at true","kind":"theorem","summary":"[TV distance on Bool equals gap at true] For probability measures \\mu and \\nu on Bool, TV(\\mu,\\…","labels":["CommunicationComplexity.TVDistance.tvDistance_bool_eq_abs_true"],"detail_key":"p14"},{"id":"n20670","layer":"informal","project":"p14","title":"TV distance sub-additivity for product measures","kind":"theorem","summary":"[TV distance sub-additivity for product measures] For probability measures \\mu_1, \\nu_1 on \\alp…","labels":["CommunicationComplexity.TVDistance.tvDistance_prod_le"],"detail_key":"p14"},{"id":"n20671","layer":"informal","project":"p14","title":"NAE clause satisfaction","kind":"definition","summary":"[NAE clause satisfaction] Given a Boolean assignment \\mathitassign : V \\to \\mathttBool and a No…","labels":["NAEtoColor.SatisfiesClause"],"detail_key":"p14"},{"id":"n20672","layer":"informal","project":"p14","title":"NAE-SAT instance","kind":"definition","summary":"[NAE-SAT instance] A NAE-SAT instance over variable type V is defined as a list of NAE clauses,…","labels":["NAEtoColor.NAESat3"],"detail_key":"p14"},{"id":"n20673","layer":"informal","project":"p14","title":"Global NAE-SAT satisfaction","kind":"definition","summary":"[Global NAE-SAT satisfaction] Given an assignment \\mathitassign : V \\to \\mathttBool and a NAE-S…","labels":["NAEtoColor.SatisfiesNAE3"],"detail_key":"p14"},{"id":"n20674","layer":"informal","project":"p14","title":"NAE-SAT satisfiability","kind":"definition","summary":"[NAE-SAT satisfiability] A NAE-SAT instance f over variable type V is \\emphsatisfiable if there…","labels":["NAEtoColor.IsSatisfiable"],"detail_key":"p14"},{"id":"n20675","layer":"informal","project":"p14","title":"Example NAE-SAT instance","kind":"definition","summary":"[Example NAE-SAT instance] A concrete NAE-SAT instance over Fin\\,5 consisting of three clauses:…","labels":["NAEtoColor.NAE_SAT_Example.nae_sat_eg"],"detail_key":"p14"},{"id":"n20676","layer":"informal","project":"p14","title":"Example assignment","kind":"definition","summary":"[Example assignment] A concrete Boolean assignment for Fin\\,5 given by \\![\\mathtttrue,\\mathtttr…","labels":["NAEtoColor.NAE_SAT_Example.assign_eg"],"detail_key":"p14"},{"id":"n20677","layer":"informal","project":"p14","title":"3-colorability","kind":"definition","summary":"[3-colorability] A simple graph G on vertex type V' is \\emph3-colorable if there exists a prope…","labels":["NAEtoColor.Is3Colorable"],"detail_key":"p14"},{"id":"n20678","layer":"informal","project":"p14","title":"Reduction vertex type","kind":"definition","summary":"[Reduction vertex type] The vertex set of the reduction graph is an inductive type with three c…","labels":["NAEtoColor.OutputVertex"],"detail_key":"p14"},{"id":"n20679","layer":"informal","project":"p14","title":"Reduction edge relation","kind":"definition","summary":"[Reduction edge relation] Given a NAE-SAT instance \\mathitclauses, the directed edge relation o…","labels":["NAEtoColor.EdgeRelation"],"detail_key":"p14"},{"id":"n20680","layer":"informal","project":"p14","title":"Reduction graph","kind":"definition","summary":"[Reduction graph] Given a NAE-SAT instance f, \\textttNAEtoColor.ReductionGraph\\;f is the simple…","labels":["NAEtoColor.ReductionGraph"],"detail_key":"p14"},{"id":"n20681","layer":"informal","project":"p14","title":"Clause-gadget node coloring","kind":"definition","summary":"[Clause-gadget node coloring] Given three Boolean values a, b, c (the truth values of a clause'…","labels":["NAEtoColor.clauseNodeColor"],"detail_key":"p14"},{"id":"n20682","layer":"informal","project":"p14","title":"NAE-to-coloring coloring map","kind":"definition","summary":"[NAE-to-coloring coloring map] Given a Boolean assignment \\mathitassign : V \\to \\mathttBool, de…","labels":["NAEtoColor.naeColoring"],"detail_key":"p14"},{"id":"n20683","layer":"informal","project":"p14","title":"Completeness of reduction","kind":"lemma","summary":"[Completeness of reduction] If a NAE-SAT instance f over variable type V is satisfiable, then t…","labels":["NAEtoColor.NAEtoColorCompleteness"],"detail_key":"p14"},{"id":"n20684","layer":"informal","project":"p14","title":"Soundness of reduction","kind":"lemma","summary":"[Soundness of reduction] If the reduction graph \\textttNAEtoColor.ReductionGraph\\;f is 3-colora…","labels":["NAEtoColor.NAEtoColorSoundness"],"detail_key":"p14"},{"id":"n20685","layer":"informal","project":"p14","title":"NAE-SAT iff 3-colorable","kind":"theorem","summary":"[NAE-SAT iff 3-colorable] For any NAE-SAT instance f over variable type V, \\[ \\textttNAEtoColor…","labels":["NAEtoColor.NAEtoColorReduction"],"detail_key":"p14"},{"id":"n20686","layer":"informal","project":"p14","title":"Clause","kind":"definition","summary":"[Clause] A clause over variable type V is a disjunction represented as a list of literals, i.e.…","labels":["SATTo3SAT.Clause"],"detail_key":"p14"},{"id":"n20687","layer":"informal","project":"p14","title":"CNF formula","kind":"definition","summary":"[CNF formula] A CNF formula over V is a conjunction of clauses, represented as a list of clause…","labels":["SATTo3SAT.CNFFormula"],"detail_key":"p14"},{"id":"n20688","layer":"informal","project":"p14","title":"Literal evaluation","kind":"definition","summary":"[Literal evaluation] Given an assignment \\alpha : V \\to Prop, \\textttSATTo3SAT.evalLiteral maps…","labels":["SATTo3SAT.evalLiteral"],"detail_key":"p14"},{"id":"n20689","layer":"informal","project":"p14","title":"Clause satisfaction","kind":"definition","summary":"[Clause satisfaction] A clause c is satisfied by assignment \\alpha if at least one literal in c…","labels":["SATTo3SAT.clauseSatisfied"],"detail_key":"p14"},{"id":"n20690","layer":"informal","project":"p14","title":"Formula satisfaction","kind":"definition","summary":"[Formula satisfaction] A CNF formula f is satisfied by assignment \\alpha if every clause in f i…","labels":["SATTo3SAT.formulaSatisfied"],"detail_key":"p14"},{"id":"n20691","layer":"informal","project":"p14","title":"Satisfiability of a CNF formula","kind":"definition","summary":"[Satisfiability of a CNF formula] A CNF formula f is satisfiable if there exists an assignment…","labels":["SATTo3SAT.isSatisfiable"],"detail_key":"p14"},{"id":"n20692","layer":"informal","project":"p14","title":"3-clause","kind":"definition","summary":"[3-clause] A 3-clause over V is a structure carrying exactly three literals l_1, l_2, l_3 : \\ma…","labels":["SATTo3SAT.Clause3"],"detail_key":"p14"},{"id":"n20693","layer":"informal","project":"p14","title":"3-CNF formula","kind":"definition","summary":"[3-CNF formula] A 3-CNF formula over V is a list of 3-clauses: \\mathttFormula3\\,V = \\mathttList…","labels":["SATTo3SAT.Formula3"],"detail_key":"p14"},{"id":"n20694","layer":"informal","project":"p14","title":"3-clause satisfaction","kind":"definition","summary":"[3-clause satisfaction] A 3-clause c is satisfied by assignment \\alpha if at least one of its t…","labels":["SATTo3SAT.clause3Satisfied"],"detail_key":"p14"},{"id":"n20695","layer":"informal","project":"p14","title":"3-CNF formula satisfaction","kind":"definition","summary":"[3-CNF formula satisfaction] A 3-CNF formula f is satisfied by assignment \\alpha if every 3-cla…","labels":["SATTo3SAT.formula3Satisfied"],"detail_key":"p14"},{"id":"n20696","layer":"informal","project":"p14","title":"3-SAT satisfiability","kind":"definition","summary":"[3-SAT satisfiability] A 3-CNF formula f is 3-satisfiable if there exists an assignment \\alpha…","labels":["SATTo3SAT.is3Satisfiable"],"detail_key":"p14"},{"id":"n20697","layer":"informal","project":"p14","title":"Auxiliary variable type","kind":"definition","summary":"[Auxiliary variable type] The inductive type \\mathttAuxVar\\,V extends a variable type V with tw…","labels":["SATTo3SAT.AuxVar"],"detail_key":"p14"},{"id":"n20698","layer":"informal","project":"p14","title":"Lift a literal to the extended type","kind":"definition","summary":"[Lift a literal to the extended type] \\textttSATTo3SAT.liftLit maps a literal over V to the cor…","labels":["SATTo3SAT.liftLit"],"detail_key":"p14"},{"id":"n20699","layer":"informal","project":"p14","title":"Build chain clauses","kind":"definition","summary":"[Build chain clauses] Given a clause index c\\_idx, a lifting map ml, a suffix list of literals…","labels":["SATTo3SAT.buildChain"],"detail_key":"p14"},{"id":"n20700","layer":"informal","project":"p14","title":"Encode a single clause as 3-clauses","kind":"definition","summary":"[Encode a single clause as 3-clauses] \\textttSATTo3SAT.transformClause maps a single clause (at…","labels":["SATTo3SAT.transformClause"],"detail_key":"p14"},{"id":"n20701","layer":"informal","project":"p14","title":"Recursive SAT-to-3SAT worker","kind":"definition","summary":"[Recursive SAT-to-3SAT worker] The auxiliary recursive function \\textttSATTo3SAT.to3SATAux iter…","labels":["SATTo3SAT.to3SATAux"],"detail_key":"p14"},{"id":"n20702","layer":"informal","project":"p14","title":"SAT to 3-SAT encoding","kind":"definition","summary":"[SAT to 3-SAT encoding] \\textttSATTo3SAT.to3SAT converts a CNF formula f over V into a 3-CNF fo…","labels":["SATTo3SAT.to3SAT"],"detail_key":"p14"},{"id":"n20703","layer":"informal","project":"p14","title":"Auxiliary variable valuation","kind":"definition","summary":"[Auxiliary variable valuation] For a list of literals \\mathitlits and index j, \\textttSATTo3SAT…","labels":["SATTo3SAT.extraVal"],"detail_key":"p14"},{"id":"n20704","layer":"informal","project":"p14","title":"extraVal as prefix-all-false","kind":"lemma","summary":"[extraVal as prefix-all-false] Provided j + 2 \\le |\\mathitlits|, \\textttSATTo3SAT.extraVal \\alp…","labels":["SATTo3SAT.extraVal_iff_prefix_all_false"],"detail_key":"p14"},{"id":"n20705","layer":"informal","project":"p14","title":"Global assignment lift","kind":"definition","summary":"[Global assignment lift] Given \\alpha : \\mathttAssignment\\,V and a formula f, \\textttSATTo3SAT.…","labels":["SATTo3SAT.globalAssignment"],"detail_key":"p14"},{"id":"n20706","layer":"informal","project":"p14","title":"Membership in to3SATAux","kind":"lemma","summary":"[Membership in to3SATAux] A 3-clause c_3 belongs to \\textttSATTo3SAT.to3SATAux\\,\\mathitcs\\,\\mat…","labels":["SATTo3SAT.mem_to3SATAux_iff"],"detail_key":"p14"},{"id":"n20707","layer":"informal","project":"p14","title":"Membership in to3SAT","kind":"lemma","summary":"[Membership in to3SAT] A 3-clause c_3 belongs to \\textttSATTo3SAT.to3SAT\\,f if and only if ther…","labels":["SATTo3SAT.mem_to3SAT_iff"],"detail_key":"p14"},{"id":"n20708","layer":"informal","project":"p14","title":"List take after drop-cons","kind":"lemma","summary":"[List take after drop-cons] If l.\\mathttdrop\\,n = a :: \\mathitrest, then l.\\mathtttake\\,(n+1) =…","labels":["SATTo3SAT.List.take_succ_of_drop_cons"],"detail_key":"p14"},{"id":"n20709","layer":"informal","project":"p14","title":"Chain clauses all satisfied under completeness","kind":"lemma","summary":"[Chain clauses all satisfied under completeness] Let \\alpha be an assignment satisfying at leas…","labels":["SATTo3SAT.buildChain_all_satisfied"],"detail_key":"p14"},{"id":"n20710","layer":"informal","project":"p14","title":"Local literal predicate","kind":"definition","summary":"[Local literal predicate] A literal l in \\mathttLiteral\\,(\\mathttAuxVar\\,V) is \\emphlocal to in…","labels":["SATTo3SAT.IsLocalVar"],"detail_key":"p14"},{"id":"n20711","layer":"informal","project":"p14","title":"Global and local assignments agree on local literals","kind":"lemma","summary":"[Global and local assignments agree on local literals] For a literal l local to index i, the ev…","labels":["SATTo3SAT.eval_local_eq_global"],"detail_key":"p14"},{"id":"n20712","layer":"informal","project":"p14","title":"buildChain produces local literals","kind":"lemma","summary":"[buildChain produces local literals] Every literal in any 3-clause produced by \\textttSATTo3SAT…","labels":["SATTo3SAT.buildChain_isLocal"],"detail_key":"p14"},{"id":"n20713","layer":"informal","project":"p14","title":"transformClause produces local literals","kind":"lemma","summary":"[transformClause produces local literals] Every literal in any 3-clause produced by \\textttSATT…","labels":["SATTo3SAT.transformClause_isLocal"],"detail_key":"p14"},{"id":"n20714","layer":"informal","project":"p14","title":"Global matches local for transformClause","kind":"lemma","summary":"[Global matches local for transformClause] For a 3-clause c_3 produced by \\textttSATTo3SAT.tran…","labels":["SATTo3SAT.global_matches_local"],"detail_key":"p14"},{"id":"n20715","layer":"informal","project":"p14","title":"Satisfied clause yields satisfied 3-clause encoding","kind":"lemma","summary":"[Satisfied clause yields satisfied 3-clause encoding] If a clause has at least one true literal…","labels":["SATTo3SAT.transformClause_satisfied"],"detail_key":"p14"},{"id":"n20716","layer":"informal","project":"p14","title":"SAT to 3-SAT completeness","kind":"theorem","summary":"[SAT to 3-SAT completeness] If a CNF formula f is satisfiable, then \\textttSATTo3SAT.to3SAT\\,f…","labels":["SATTo3SAT.SAT_to_3SAT_completeness"],"detail_key":"p14"},{"id":"n20717","layer":"informal","project":"p14","title":"Chain contradiction under all-false literals","kind":"lemma","summary":"[Chain contradiction under all-false literals] If y_j-1 is true, every ml-lifted literal in \\ma…","labels":["SATTo3SAT.buildChain_forced_false"],"detail_key":"p14"},{"id":"n20718","layer":"informal","project":"p14","title":"Satisfied 3-clause encoding yields satisfied clause","kind":"lemma","summary":"[Satisfied 3-clause encoding yields satisfied clause] If every 3-clause in \\textttSATTo3SAT.tra…","labels":["SATTo3SAT.transformClause_soundness"],"detail_key":"p14"},{"id":"n20719","layer":"informal","project":"p14","title":"SAT to 3-SAT soundness","kind":"theorem","summary":"[SAT to 3-SAT soundness] If \\textttSATTo3SAT.to3SAT\\,f is 3-satisfiable, then the original CNF…","labels":["SATTo3SAT.SAT_to_3SAT_soundness"],"detail_key":"p14"},{"id":"n20720","layer":"informal","project":"p14","title":"SAT to 3-SAT equivalence","kind":"theorem","summary":"[SAT to 3-SAT equivalence] A CNF formula f over V is satisfiable if and only if the 3-CNF formu…","labels":["SATTo3SAT.SAT_to_3SAT_equivalence"],"detail_key":"p14"},{"id":"n20721","layer":"informal","project":"p14","title":"Clause","kind":"definition","summary":"[Clause] A clause is a list of literals over a variable type V; it represents a disjunction of…","labels":["ThreeSATToClique.Clause"],"detail_key":"p14"},{"id":"n20722","layer":"informal","project":"p14","title":"CNF formula","kind":"definition","summary":"[CNF formula] A CNF formula over V is a list of clauses, representing the conjunction of those…","labels":["ThreeSATToClique.CNFFormula"],"detail_key":"p14"},{"id":"n20723","layer":"informal","project":"p14","title":"Literal evaluation","kind":"definition","summary":"[Literal evaluation] Given a truth assignment \\alpha : V \\to Prop, the evaluation of a literal…","labels":["ThreeSATToClique.evalLiteral"],"detail_key":"p14"},{"id":"n20724","layer":"informal","project":"p14","title":"Clause satisfaction","kind":"definition","summary":"[Clause satisfaction] A clause c is \\emphsatisfied by assignment \\alpha if at least one literal…","labels":["ThreeSATToClique.clauseSatisfied"],"detail_key":"p14"},{"id":"n20725","layer":"informal","project":"p14","title":"CNF formula satisfaction","kind":"definition","summary":"[CNF formula satisfaction] A CNF formula f is \\emphsatisfied by \\alpha if every clause c \\in f…","labels":["ThreeSATToClique.formulaSatisfied"],"detail_key":"p14"},{"id":"n20726","layer":"informal","project":"p14","title":"CNF satisfiability","kind":"definition","summary":"[CNF satisfiability] A CNF formula f is \\emphsatisfiable if there exists an assignment \\alpha t…","labels":["ThreeSATToClique.isSatisfiable"],"detail_key":"p14"},{"id":"n20727","layer":"informal","project":"p14","title":"3-clause","kind":"definition","summary":"[3-clause] A 3-clause over V is a structure with exactly three literals l_1, l_2, l_3 : \\mathtt…","labels":["ThreeSATToClique.Clause3"],"detail_key":"p14"},{"id":"n20728","layer":"informal","project":"p14","title":"3-CNF formula","kind":"definition","summary":"[3-CNF formula] A 3-CNF formula over V is a list of 3-clauses, representing their conjunction.","labels":["ThreeSATToClique.Formula3"],"detail_key":"p14"},{"id":"n20729","layer":"informal","project":"p14","title":"3-clause satisfaction","kind":"definition","summary":"[3-clause satisfaction] A 3-clause c is satisfied by \\alpha if at least one of its three litera…","labels":["ThreeSATToClique.clause3Satisfied"],"detail_key":"p14"},{"id":"n20730","layer":"informal","project":"p14","title":"3-CNF formula satisfaction","kind":"definition","summary":"[3-CNF formula satisfaction] A 3-CNF formula f is satisfied by \\alpha if every 3-clause c \\in f…","labels":["ThreeSATToClique.formula3Satisfied"],"detail_key":"p14"},{"id":"n20731","layer":"informal","project":"p14","title":"3-CNF satisfiability","kind":"definition","summary":"[3-CNF satisfiability] A 3-CNF formula f is satisfiable if there exists an assignment \\alpha su…","labels":["ThreeSATToClique.is3Satisfiable"],"detail_key":"p14"},{"id":"n20732","layer":"informal","project":"p14","title":"Clique vertex","kind":"definition","summary":"[Clique vertex] A vertex of the conflict graph for a formula with m clauses is a pair \\langle c…","labels":["ThreeSATToClique.CliqueVertex"],"detail_key":"p14"},{"id":"n20733","layer":"informal","project":"p14","title":"Literal extraction from a 3-clause","kind":"definition","summary":"[Literal extraction from a 3-clause] Given a 3-clause c and a position p : Fin\\,3, returns the…","labels":["ThreeSATToClique.getLitInClause"],"detail_key":"p14"},{"id":"n20734","layer":"informal","project":"p14","title":"Literal named by a vertex","kind":"definition","summary":"[Literal named by a vertex] Given a 3-CNF formula f and a vertex v : \\mathttCliqueVertex\\,f.\\ma…","labels":["ThreeSATToClique.getLitAt"],"detail_key":"p14"},{"id":"n20735","layer":"informal","project":"p14","title":"Literal conflict","kind":"definition","summary":"[Literal conflict] Two literals l_1 and l_2 \\emphconflict if one is the positive and the other…","labels":["ThreeSATToClique.literalsConflict"],"detail_key":"p14"},{"id":"n20736","layer":"informal","project":"p14","title":"Symmetry of literal conflict","kind":"theorem","summary":"[Symmetry of literal conflict] For any two literals l_1 and l_2, \\mathttliteralsConflict\\,l_1\\,…","labels":["ThreeSATToClique.literalsConflict_symm"],"detail_key":"p14"},{"id":"n20737","layer":"informal","project":"p14","title":"Conflict graph","kind":"definition","summary":"[Conflict graph] The conflict graph of a 3-CNF formula f is the simple graph on vertex set \\mat…","labels":["ThreeSATToClique.toCliqueGraph"],"detail_key":"p14"},{"id":"n20738","layer":"informal","project":"p14","title":"k-clique existence","kind":"definition","summary":"[k-clique existence] A graph G on vertex type V \\emphhas a k-clique if there exists a finite se…","labels":["ThreeSATToClique.hasClique"],"detail_key":"p14"},{"id":"n20739","layer":"informal","project":"p14","title":"Simultaneously true literals do not conflict","kind":"lemma","summary":"[Simultaneously true literals do not conflict] If an assignment \\alpha makes both l_1 and l_2 t…","labels":["ThreeSATToClique.no_conflict_of_true"],"detail_key":"p14"},{"id":"n20740","layer":"informal","project":"p14","title":"Clique vertices represent one literal per clause","kind":"lemma","summary":"[Clique vertices represent one literal per clause] If s is a set of m vertices forming a clique…","labels":["ThreeSATToClique.clique_vertices_choose_one_per_clause"],"detail_key":"p14"},{"id":"n20741","layer":"informal","project":"p14","title":"Vertex literal belongs to its clause","kind":"lemma","summary":"[Vertex literal belongs to its clause] If v is a clique vertex with v.c\\_idx = i, then the lite…","labels":["ThreeSATToClique.getLitAt_mem_clause"],"detail_key":"p14"},{"id":"n20742","layer":"informal","project":"p14","title":"Completeness of the reduction","kind":"theorem","summary":"[Completeness of the reduction] If a 3-CNF formula f with m clauses is satisfiable, then the co…","labels":["ThreeSATToClique.ThreeSAT_to_Clique_completeness"],"detail_key":"p14"},{"id":"n20743","layer":"informal","project":"p14","title":"Soundness of the reduction","kind":"theorem","summary":"[Soundness of the reduction] If the conflict graph of a 3-CNF formula f with m clauses contains…","labels":["ThreeSATToClique.ThreeSAT_to_Clique_soundness"],"detail_key":"p14"},{"id":"n20744","layer":"informal","project":"p14","title":"3-SAT \\Leftrightarrow Clique equivalence","kind":"theorem","summary":"[3-SAT \\Leftrightarrow Clique equivalence] A 3-CNF formula f is satisfiable if and only if its…","labels":["ThreeSATToClique.ThreeSAT_to_Clique_equivalence"],"detail_key":"p14"},{"id":"n20745","layer":"informal","project":"p14","title":"Clause","kind":"definition","summary":"[Clause] A \\emphclause over a variable type V is a structure bundling exactly three literals \\e…","labels":["SATtoColor.Clause"],"detail_key":"p14"},{"id":"n20746","layer":"informal","project":"p14","title":"Literal satisfaction","kind":"definition","summary":"[Literal satisfaction] Given a Boolean assignment assign : V \\to Bool, the function \\textttSATt…","labels":["SATtoColor.SatisfiesLiteral"],"detail_key":"p14"},{"id":"n20747","layer":"informal","project":"p14","title":"Clause satisfaction","kind":"definition","summary":"[Clause satisfaction] Under assignment assign, a clause c is satisfied if at least one of its t…","labels":["SATtoColor.SatisfiesClause"],"detail_key":"p14"},{"id":"n20748","layer":"informal","project":"p14","title":"3-SAT instance","kind":"definition","summary":"[3-SAT instance] A 3-SAT instance over variable type V is defined as a list of clauses, i.e.\\ S…","labels":["SATtoColor.Sat3"],"detail_key":"p14"},{"id":"n20749","layer":"informal","project":"p14","title":"Simultaneous clause satisfaction","kind":"definition","summary":"[Simultaneous clause satisfaction] An assignment assign \\emphsatisfies a 3-SAT instance f if ev…","labels":["SATtoColor.SatisfiesSat3"],"detail_key":"p14"},{"id":"n20750","layer":"informal","project":"p14","title":"Satisfiability","kind":"definition","summary":"[Satisfiability] A 3-SAT instance f is \\emphsatisfiable if there exists a Boolean assignment th…","labels":["SATtoColor.IsSatisfiable"],"detail_key":"p14"},{"id":"n20751","layer":"informal","project":"p14","title":"Concrete 3-SAT example","kind":"definition","summary":"[Concrete 3-SAT example] A specific 3-SAT instance over four Boolean variables x_0, x_1, x_2, x…","labels":["SATtoColor.SAT3_Example.sat3_inst"],"detail_key":"p14"},{"id":"n20752","layer":"informal","project":"p14","title":"Example assignment","kind":"definition","summary":"[Example assignment] A concrete Boolean assignment for the example instance \\textttSATtoColor.S…","labels":["SATtoColor.SAT3_Example.ex_assign_1"],"detail_key":"p14"},{"id":"n20753","layer":"informal","project":"p14","title":"Reduction vertex type","kind":"definition","summary":"[Reduction vertex type] The vertex set of the reduction graph is the inductive type OutputVerte…","labels":["SATtoColor.OutputVertex"],"detail_key":"p14"},{"id":"n20754","layer":"informal","project":"p14","title":"3-colorability","kind":"definition","summary":"[3-colorability] A simple graph G on vertex type V' is \\emph3-colorable if there exists a prope…","labels":["SATtoColor.Is3Colorable"],"detail_key":"p14"},{"id":"n20755","layer":"informal","project":"p14","title":"Reduction edge relation","kind":"definition","summary":"[Reduction edge relation] EdgeRelation(f, u, v) defines the (undirected) adjacency structure of…","labels":["SATtoColor.EdgeRelation"],"detail_key":"p14"},{"id":"n20756","layer":"informal","project":"p14","title":"Reduction graph","kind":"definition","summary":"[Reduction graph] Given a 3-SAT instance f over variables V, the \\emphreduction graph Reduction…","labels":["SATtoColor.ReductionGraph"],"detail_key":"p14"},{"id":"n20757","layer":"informal","project":"p14","title":"Clause gadget coloring","kind":"definition","summary":"[Clause gadget coloring] Given the Boolean truth values a, b, c_3 of the three literals of a cl…","labels":["SATtoColor.clauseGadgetColor"],"detail_key":"p14"},{"id":"n20758","layer":"informal","project":"p14","title":"Coloring from satisfying assignment","kind":"definition","summary":"[Coloring from satisfying assignment] Given a satisfying assignment assign : V \\to Bool, the fu…","labels":["SATtoColor.sat3Coloring"],"detail_key":"p14"},{"id":"n20759","layer":"informal","project":"p14","title":"Literal node color formula","kind":"lemma","summary":"[Literal node color formula] For any assignment assign and literal \\ell, \\[ sat3Coloring(assign…","labels":["SATtoColor.sat3Coloring_litNode"],"detail_key":"p14"},{"id":"n20760","layer":"informal","project":"p14","title":"Completeness of reduction","kind":"lemma","summary":"[Completeness of reduction] If the 3-SAT instance f is satisfiable, then the reduction graph Re…","labels":["SATtoColor.SATtoColorCompleteness"],"detail_key":"p14"},{"id":"n20761","layer":"informal","project":"p14","title":"Soundness of reduction","kind":"lemma","summary":"[Soundness of reduction] If the reduction graph ReductionGraph(f) is 3-colorable, then the 3-SA…","labels":["SATtoColor.SATtoColorSoundness"],"detail_key":"p14"},{"id":"n20762","layer":"informal","project":"p14","title":"3-SAT to 3-Coloring reduction","kind":"theorem","summary":"[3-SAT to 3-Coloring reduction] A 3-SAT instance f is satisfiable if and only if the reduction…","labels":["SATtoColor.SATtoColorReduction"],"detail_key":"p14"},{"id":"n20763","layer":"informal","project":"p14","title":"Weighted edge","kind":"definition","summary":"[Weighted edge] A \\emphweighted edge over n vertices is a structure with two endpoints u, v : F…","labels":["Kruskal.WEdge"],"detail_key":"p14"},{"id":"n20764","layer":"informal","project":"p14","title":"Union-find type","kind":"definition","summary":"[Union-find type] The union-find state for n nodes is defined as a function Fin\\,n \\to N mappin…","labels":["Kruskal.UF"],"detail_key":"p14"},{"id":"n20765","layer":"informal","project":"p14","title":"Initial union-find","kind":"definition","summary":"[Initial union-find] \\mathttinit\\,n constructs the identity union-find on n nodes: every node i…","labels":["Kruskal.UF.init"],"detail_key":"p14"},{"id":"n20766","layer":"informal","project":"p14","title":"Find representative","kind":"definition","summary":"[Find representative] \\mathttfind\\,\\mathituf\\,i returns the representative of node i in the uni…","labels":["Kruskal.UF.find"],"detail_key":"p14"},{"id":"n20767","layer":"informal","project":"p14","title":"Merge two components","kind":"definition","summary":"[Merge two components] \\mathttmerge\\,\\mathituf\\,i\\,j unifies the components of nodes i and j by…","labels":["Kruskal.UF.merge"],"detail_key":"p14"},{"id":"n20768","layer":"informal","project":"p14","title":"Initial find is identity","kind":"lemma","summary":"[Initial find is identity] For every i : Fin\\,n, the find operation on the initial union-find r…","labels":["Kruskal.UF.init_find"],"detail_key":"p14"},{"id":"n20769","layer":"informal","project":"p14","title":"Find unfolds to array lookup","kind":"lemma","summary":"[Find unfolds to array lookup] For any union-find state \\mathituf and index i, \\mathituf.\\matht…","labels":["Kruskal.UF.find_def"],"detail_key":"p14"},{"id":"n20770","layer":"informal","project":"p14","title":"Merge all edges into union-find","kind":"definition","summary":"[Merge all edges into union-find] \\mathttmergeAll\\,\\mathituf\\,\\mathites folds a list of weighte…","labels":["Kruskal.UF.mergeAll"],"detail_key":"p14"},{"id":"n20771","layer":"informal","project":"p14","title":"Same partition predicate","kind":"definition","summary":"[Same partition predicate] \\mathttSamePartition\\,\\mathituf_1\\,\\mathituf_2 holds when two union-…","labels":["Kruskal.UF.SamePartition"],"detail_key":"p14"},{"id":"n20772","layer":"informal","project":"p14","title":"Edge-selection sweep","kind":"definition","summary":"[Edge-selection sweep] \\mathttprocessEdges\\,\\mathites\\,\\mathituf\\,\\mathitacc scans the edge lis…","labels":["Kruskal.processEdges"],"detail_key":"p14"},{"id":"n20773","layer":"informal","project":"p14","title":"Kruskal's algorithm","kind":"definition","summary":"[Kruskal's algorithm] \\mathttkruskal\\,n\\,\\mathitedges sorts the edge list by weight and then ru…","labels":["Kruskal.kruskal"],"detail_key":"p14"},{"id":"n20774","layer":"informal","project":"p14","title":"Output of processEdges comes from input or accumulator","kind":"lemma","summary":"[Output of processEdges comes from input or accumulator] Let \\mathites be a list of weighted ed…","labels":["Kruskal.processEdges_mem"],"detail_key":"p14"},{"id":"n20775","layer":"informal","project":"p14","title":"Kruskal output is a subset of the input","kind":"lemma","summary":"[Kruskal output is a subset of the input] Every edge e returned by \\mathttkruskal\\,n\\,\\mathited…","labels":["Kruskal.kruskal_subset"],"detail_key":"p14"},{"id":"n20776","layer":"informal","project":"p14","title":"Symmetric adjacency","kind":"definition","summary":"[Symmetric adjacency] Given a list of weighted edges \\mathitedges on n vertices, \\mathttSymAdj\\…","labels":["Kruskal.SymAdj"],"detail_key":"p14"},{"id":"n20777","layer":"informal","project":"p14","title":"Reachability","kind":"definition","summary":"[Reachability] \\mathttReach\\,\\mathitedges is the reflexive--transitive closure of \\mathttSymAdj…","labels":["Kruskal.Reach"],"detail_key":"p14"},{"id":"n20778","layer":"informal","project":"p14","title":"Total weight","kind":"definition","summary":"[Total weight] The total weight of an edge list is the sum of the weights of all its edges: \\ma…","labels":["Kruskal.totalWeight"],"detail_key":"p14"},{"id":"n20779","layer":"informal","project":"p14","title":"Spanning like","kind":"definition","summary":"[Spanning like] \\mathttSpansLike\\,F\\,E holds when the edge list F is at least as connected as E…","labels":["Kruskal.SpansLike"],"detail_key":"p14"},{"id":"n20780","layer":"informal","project":"p14","title":"Symmetry of symmetric adjacency","kind":"lemma","summary":"[Symmetry of symmetric adjacency] If \\mathttSymAdj\\,\\mathitedges\\,u\\,v then \\mathttSymAdj\\,\\mat…","labels":["Kruskal.symAdj_symm"],"detail_key":"p14"},{"id":"n20781","layer":"informal","project":"p14","title":"Monotonicity of symmetric adjacency","kind":"lemma","summary":"[Monotonicity of symmetric adjacency] If \\mathttSymAdj\\,e_1\\,u\\,v and every edge of e_1 also be…","labels":["Kruskal.symAdj_mono"],"detail_key":"p14"},{"id":"n20782","layer":"informal","project":"p14","title":"Symmetry of reachability","kind":"lemma","summary":"[Symmetry of reachability] If \\mathttReach\\,\\mathitedges\\,u\\,v then \\mathttReach\\,\\mathitedges\\…","labels":["Kruskal.reach_symm"],"detail_key":"p14"},{"id":"n20783","layer":"informal","project":"p14","title":"Monotonicity of reachability","kind":"lemma","summary":"[Monotonicity of reachability] If \\mathttReach\\,e_1\\,u\\,v and every edge of e_1 also belongs to…","labels":["Kruskal.reach_mono"],"detail_key":"p14"},{"id":"n20784","layer":"informal","project":"p14","title":"Reachability from a membership witness","kind":"lemma","summary":"[Reachability from a membership witness] If e \\in \\mathitedges then \\mathttReach\\,\\mathitedges\\…","labels":["Kruskal.reach_of_mem"],"detail_key":"p14"},{"id":"n20785","layer":"informal","project":"p14","title":"Reachability lifting","kind":"lemma","summary":"[Reachability lifting] If every symmetric adjacency in \\mathitedges implies reachability in \\ma…","labels":["Kruskal.reach_lift"],"detail_key":"p14"},{"id":"n20786","layer":"informal","project":"p14","title":"Cons decomposition of reachability","kind":"lemma","summary":"[Cons decomposition of reachability] For a non-empty edge list (g :: \\mathitrest), \\mathttReach…","labels":["Kruskal.reach_cons_iff"],"detail_key":"p14"},{"id":"n20787","layer":"informal","project":"p14","title":"Reachability depends only on membership","kind":"lemma","summary":"[Reachability depends only on membership] If two edge lists l_1 and l_2 have the same elements…","labels":["Kruskal.reach_mem_iff"],"detail_key":"p14"},{"id":"n20788","layer":"informal","project":"p14","title":"Total weight of empty list","kind":"lemma","summary":"[Total weight of empty list] \\mathtttotalWeight\\,([] : \\mathttList\\,(\\mathttWEdge\\,n)) = 0.","labels":["Kruskal.totalWeight_nil"],"detail_key":"p14"},{"id":"n20789","layer":"informal","project":"p14","title":"Total weight after erasing an edge","kind":"lemma","summary":"[Total weight after erasing an edge] If e \\in \\mathitedges then \\mathtttotalWeight\\,\\mathitedge…","labels":["Kruskal.totalWeight_erase"],"detail_key":"p14"},{"id":"n20790","layer":"informal","project":"p14","title":"Vote set for a label","kind":"definition","summary":"[Vote set for a label] Given an evaluation map \\mathtteval : \\mathitHyp \\to X \\to \\mathsfBool,…","labels":["Halving.voteFor"],"detail_key":"p14"},{"id":"n20791","layer":"informal","project":"p14","title":"Halving prediction","kind":"definition","summary":"[Halving prediction] The Halving Algorithm predicts on input x by taking a majority vote of the…","labels":["Halving.predict"],"detail_key":"p14"},{"id":"n20792","layer":"informal","project":"p14","title":"Version-space update","kind":"definition","summary":"[Version-space update] After observing that the correct label for input x is y, the version spa…","labels":["Halving.update"],"detail_key":"p14"},{"id":"n20793","layer":"informal","project":"p14","title":"Partition of version space by vote","kind":"lemma","summary":"[Partition of version space by vote] For any version space V and input x, the false-voters and…","labels":["Halving.voteFor_false_add_true"],"detail_key":"p14"},{"id":"n20794","layer":"informal","project":"p14","title":"Update is a subset","kind":"lemma","summary":"[Update is a subset] The updated version space is always a subset of the previous one: \\mathttu…","labels":["Halving.update_subset"],"detail_key":"p14"},{"id":"n20795","layer":"informal","project":"p14","title":"Target survives every update","kind":"lemma","summary":"[Target survives every update] In the realizable setting, if \\mathtttarget \\in V then after upd…","labels":["Halving.target_mem_update"],"detail_key":"p14"},{"id":"n20796","layer":"informal","project":"p14","title":"Mistakes halve the version space","kind":"lemma","summary":"[Mistakes halve the version space] If the algorithm makes a mistake on input x with true label…","labels":["Halving.mistake_halves"],"detail_key":"p14"},{"id":"n20797","layer":"informal","project":"p14","title":"Logarithm step from doubling","kind":"lemma","summary":"[Logarithm step from doubling] For natural numbers a and b with b > 0 and 2b \\le a, \\[ \\lfloor\\…","labels":["Halving.log_succ_le_log_of_double_le"],"detail_key":"p14"},{"id":"n20798","layer":"informal","project":"p14","title":"Mistake count over a sequence","kind":"definition","summary":"[Mistake count over a sequence] \\mathttmistakes(\\mathtteval, \\mathtttarget, V, xs) is the total…","labels":["Halving.mistakes"],"detail_key":"p14"},{"id":"n20799","layer":"informal","project":"p14","title":"Mistake bound — general version space","kind":"theorem","summary":"[Mistake bound — general version space] In the realizable setting (\\mathtttarget \\in V), the to…","labels":["Halving.mistakes_bound"],"detail_key":"p14"},{"id":"n20800","layer":"informal","project":"p14","title":"Halving mistake bound","kind":"theorem","summary":"[Halving mistake bound] Specialising to the initial hypothesis class H (with \\mathtttarget \\in…","labels":["Halving.halving_bound"],"detail_key":"p14"},{"id":"n20801","layer":"informal","project":"p14","title":"Loss sequence","kind":"definition","summary":"[Loss sequence] A \\emphloss sequence \\ell for N experts over T rounds is a function \\ell : Fin\\…","labels":["LossSeq"],"detail_key":"p14"},{"id":"n20802","layer":"informal","project":"p14","title":"Valid loss sequence","kind":"definition","summary":"[Valid loss sequence] A loss sequence \\ell is \\emphvalid if all individual losses are in [0,1]:…","labels":["LossSeq.Valid"],"detail_key":"p14"},{"id":"n20803","layer":"informal","project":"p14","title":"Cumulative loss","kind":"definition","summary":"[Cumulative loss] The \\emphcumulative loss of expert i through the first t rounds is \\[ L_t(i)…","labels":["cumLoss"],"detail_key":"p14"},{"id":"n20804","layer":"informal","project":"p14","title":"Hedge weight","kind":"definition","summary":"[Hedge weight] The \\emphunnormalized Hedge weight of expert i at time t with learning rate \\eta…","labels":["hedgeWeight"],"detail_key":"p14"},{"id":"n20805","layer":"informal","project":"p14","title":"Potential","kind":"definition","summary":"[Potential] The \\emphpotential (sum of unnormalized weights) at time t is \\[ W_t \\;=\\; \\sum_i=1…","labels":["potential"],"detail_key":"p14"},{"id":"n20806","layer":"informal","project":"p14","title":"Hedge distribution","kind":"definition","summary":"[Hedge distribution] The \\emphHedge distribution at round t is the normalization of the weight…","labels":["hedgeDist"],"detail_key":"p14"},{"id":"n20807","layer":"informal","project":"p14","title":"Expected loss of Hedge","kind":"definition","summary":"[Expected loss of Hedge] The \\emphexpected loss of the learner at round t under the Hedge distr…","labels":["hedgeLoss"],"detail_key":"p14"},{"id":"n20808","layer":"informal","project":"p14","title":"Cumulative loss of Hedge","kind":"definition","summary":"[Cumulative loss of Hedge] The \\emphcumulative loss of the Hedge algorithm over all T rounds is…","labels":["hedgeCumLoss"],"detail_key":"p14"},{"id":"n20809","layer":"informal","project":"p14","title":"Best expert loss","kind":"definition","summary":"[Best expert loss] The \\emphbest expert loss in hindsight is the minimum cumulative loss achiev…","labels":["bestExpertLoss"],"detail_key":"p14"},{"id":"n20810","layer":"informal","project":"p14","title":"Regret","kind":"definition","summary":"[Regret] The \\emphregret of the Hedge algorithm is the excess cumulative loss over the best exp…","labels":["regret"],"detail_key":"p14"},{"id":"n20811","layer":"informal","project":"p14","title":"Potential at time zero","kind":"lemma","summary":"[Potential at time zero] At time 0, all cumulative losses are zero, so every weight equals 1, a…","labels":["potential_zero"],"detail_key":"p14"},{"id":"n20812","layer":"informal","project":"p14","title":"Positivity of weights","kind":"lemma","summary":"[Positivity of weights] For any \\eta, loss sequence \\ell, time t, and expert i, the weight w_t(…","labels":["hedgeWeight_pos"],"detail_key":"p14"},{"id":"n20813","layer":"informal","project":"p14","title":"Positivity of the potential","kind":"lemma","summary":"[Positivity of the potential] For any \\eta, loss sequence \\ell, and time t, the potential W_t >…","labels":["potential_pos"],"detail_key":"p14"},{"id":"n20814","layer":"informal","project":"p14","title":"Exponential convexity bound","kind":"lemma","summary":"[Exponential convexity bound] For \\eta > 0 and x \\in [0,1], \\[ e^-\\eta x \\;\\le\\; 1 - (1 - e^-\\e…","labels":["exp_neg_le_linear"],"detail_key":"p14"},{"id":"n20815","layer":"informal","project":"p14","title":"Cumulative loss successor step","kind":"lemma","summary":"[Cumulative loss successor step] The cumulative loss satisfies the recurrence \\[ L_t+1(i) \\;=\\;…","labels":["cumLoss_succ"],"detail_key":"p14"},{"id":"n20816","layer":"informal","project":"p14","title":"Cumulative loss at the horizon","kind":"lemma","summary":"[Cumulative loss at the horizon] At the final horizon, L_T(i) = \\sum_t : Fin\\,T \\ell_t(i), i.e.…","labels":["cumLoss_horizon"],"detail_key":"p14"},{"id":"n20817","layer":"informal","project":"p14","title":"Weight successor factorization","kind":"lemma","summary":"[Weight successor factorization] The weight factorizes across rounds: \\[ w_t+1(i) \\;=\\; w_t(i)…","labels":["hedgeWeight_succ"],"detail_key":"p14"},{"id":"n20818","layer":"informal","project":"p14","title":"Hedge distribution sums to one","kind":"lemma","summary":"[Hedge distribution sums to one] For any \\eta, valid loss sequence \\ell, and time t, \\sum_i=1^N…","labels":["hedgeDist_sum"],"detail_key":"p14"},{"id":"n20819","layer":"informal","project":"p14","title":"Potential ratio bound","kind":"lemma","summary":"[Potential ratio bound] For a valid loss sequence and \\eta > 0, the ratio of consecutive potent…","labels":["potential_ratio_le"],"detail_key":"p14"},{"id":"n20820","layer":"informal","project":"p14","title":"Logarithmic potential step bound","kind":"lemma","summary":"[Logarithmic potential step bound] For a valid loss sequence, \\eta > 0, and t < T, \\[ \\ln W_t+1…","labels":["log_potential_step"],"detail_key":"p14"},{"id":"n20821","layer":"informal","project":"p14","title":"Quadratic upper bound on e^-t","kind":"lemma","summary":"[Quadratic upper bound on e^-t] For t \\ge 0, \\[ e^-t \\;\\le\\; 1 - t + \\fract^22. \\]","labels":["exp_neg_le_quadratic"],"detail_key":"p14"},{"id":"n20822","layer":"informal","project":"p14","title":"Lower bound on 1 - e^-\\eta","kind":"lemma","summary":"[Lower bound on 1 - e^-\\eta] For \\eta > 0, \\[ 1 - e^-\\eta \\;\\ge\\; \\eta - \\frac\\eta^22. \\] This…","labels":["one_sub_exp_neg_ge"],"detail_key":"p14"},{"id":"n20823","layer":"informal","project":"p14","title":"Lower bound on potential via best expert","kind":"lemma","summary":"[Lower bound on potential via best expert] The final potential is at least the weight of the be…","labels":["potential_ge_best_expert"],"detail_key":"p14"},{"id":"n20824","layer":"informal","project":"p14","title":"Hedge distribution is nonnegative","kind":"lemma","summary":"[Hedge distribution is nonnegative] For any \\eta, loss sequence \\ell, time t, and expert i, the…","labels":["hedgeDist_nonneg"],"detail_key":"p14"},{"id":"n20825","layer":"informal","project":"p14","title":"Expected loss at most one","kind":"lemma","summary":"[Expected loss at most one] For a valid loss sequence, the expected loss at each round satisfie…","labels":["hedgeLoss_le_one"],"detail_key":"p14"},{"id":"n20826","layer":"informal","project":"p14","title":"Expected loss nonnegative","kind":"lemma","summary":"[Expected loss nonnegative] For a valid loss sequence, the expected loss at each round satisfie…","labels":["hedgeLoss_nonneg"],"detail_key":"p14"},{"id":"n20827","layer":"informal","project":"p14","title":"Hedge regret bound","kind":"theorem","summary":"[Hedge regret bound] Hedge Regret Bound (Cesa-Bianchi \\& Lugosi, Theorem 2.2). For any valid lo…","labels":["hedge_regret_bound"],"detail_key":"p14"},{"id":"n20828","layer":"informal","project":"p14","title":"Optimal learning rate","kind":"definition","summary":"[Optimal learning rate] The \\emphoptimal learning rate balancing the two terms of the Hedge reg…","labels":["optimalEta"],"detail_key":"p14"},{"id":"n20829","layer":"informal","project":"p14","title":"Hedge with optimal learning rate","kind":"theorem","summary":"[Hedge with optimal learning rate] If T \\ge 2\\ln N (so that \\eta^* \\le 1), then for any valid l…","labels":["hedge_regret_optimal"],"detail_key":"p14"},{"id":"n20830","layer":"informal","project":"p14","title":"Hedge is no-regret","kind":"theorem","summary":"[Hedge is no-regret] If T \\ge 2\\ln N, then for any valid loss sequence the \\emphaverage regret…","labels":["hedge_no_regret"],"detail_key":"p14"},{"id":"n20831","layer":"informal","project":"p14","title":"Hoeffding's lemma","kind":"lemma","summary":"[Hoeffding's lemma] For p \\in [0,1] and any h \\in R, \\[ \\ln\\!\\bigl((1-p) + p\\,e^h\\bigr) \\;\\le\\;…","labels":["hoeffding_lemma"],"detail_key":"p14"},{"id":"n20832","layer":"informal","project":"p14","title":"Tight per-step logarithmic potential bound","kind":"lemma","summary":"[Tight per-step logarithmic potential bound] For a valid loss sequence and \\eta > 0, applying H…","labels":["log_potential_step_tight"],"detail_key":"p14"},{"id":"n20833","layer":"informal","project":"p14","title":"Tight Hedge regret bound","kind":"theorem","summary":"[Tight Hedge regret bound] Tight Hedge Regret Bound (Cesa-Bianchi \\& Lugosi). For any valid los…","labels":["hedge_regret_bound_tight"],"detail_key":"p14"},{"id":"n20834","layer":"informal","project":"p14","title":"Tight optimal learning rate","kind":"definition","summary":"[Tight optimal learning rate] The learning rate that optimizes the tight Hedge bound is \\[ \\eta…","labels":["optimalEtaTight"],"detail_key":"p14"},{"id":"n20835","layer":"informal","project":"p14","title":"Tight Hedge with optimal learning rate","kind":"theorem","summary":"[Tight Hedge with optimal learning rate] For any valid loss sequence with N > 1 experts, T \\ge…","labels":["hedge_regret_tight_optimal"],"detail_key":"p14"},{"id":"n20836","layer":"informal","project":"p14","title":"Induced loss sequence","kind":"definition","summary":"[Induced loss sequence] Given a loss function \\ell : R \\to \\Omega \\to R, expert predictions p :…","labels":["inducedLoss"],"detail_key":"p14"},{"id":"n20837","layer":"informal","project":"p14","title":"Hedge weighted-average prediction","kind":"definition","summary":"[Hedge weighted-average prediction] At each round t \\in Fin\\,T, Hedge's \\emphprediction in the…","labels":["hedgePrediction"],"detail_key":"p14"},{"id":"n20838","layer":"informal","project":"p14","title":"Cumulative loss of Hedge's predictions","kind":"definition","summary":"[Cumulative loss of Hedge's predictions] The \\emphactual cumulative prediction loss of Hedge is…","labels":["hedgePredictionCumLoss"],"detail_key":"p14"},{"id":"n20839","layer":"informal","project":"p14","title":"Weighted average stays in convex set","kind":"lemma","summary":"[Weighted average stays in convex set] Let S \\subseteq R be a convex set and suppose every expe…","labels":["hedgePrediction_mem"],"detail_key":"p14"},{"id":"n20840","layer":"informal","project":"p14","title":"Jensen bridge: prediction loss bounded by Hedge loss","kind":"lemma","summary":"[Jensen bridge: prediction loss bounded by Hedge loss] Under the same hypotheses as above, assu…","labels":["hedgePrediction_loss_le_hedgeLoss"],"detail_key":"p14"},{"id":"n20841","layer":"informal","project":"p14","title":"Cumulative Jensen inequality","kind":"lemma","summary":"[Cumulative Jensen inequality] Summing the per-round Jensen bound over all T rounds, we obtain…","labels":["hedgePredictionCumLoss_le_hedgeCumLoss"],"detail_key":"p14"},{"id":"n20842","layer":"informal","project":"p14","title":"Tight regret bound for convex prediction","kind":"theorem","summary":"[Tight regret bound for convex prediction] Let S \\subseteq R be convex, \\eta > 0, and suppose a…","labels":["hedgePrediction_regret_bound_tight"],"detail_key":"p14"},{"id":"n20843","layer":"informal","project":"p14","title":"Optimal-rate regret bound for convex prediction","kind":"theorem","summary":"[Optimal-rate regret bound for convex prediction] Under the same convexity hypotheses, with T >…","labels":["hedgePrediction_regret_tight_optimal"],"detail_key":"p14"},{"id":"n20844","layer":"informal","project":"p14","title":"Loss history","kind":"definition","summary":"[Loss history] \\textttLossHistory\\ N\\ t is the type of length-t histories of loss vectors for N…","labels":["LossHistory"],"detail_key":"p14"},{"id":"n20845","layer":"informal","project":"p14","title":"History to loss sequence","kind":"definition","summary":"[History to loss sequence] Given a history h : \\textttLossHistory\\ N\\ t, \\text\\textttLossHistor…","labels":["LossHistory.toLossSeq"],"detail_key":"p14"},{"id":"n20846","layer":"informal","project":"p14","title":"Learner policy","kind":"definition","summary":"[Learner policy] A \\textttLearnerPolicy\\ N is a deterministic online strategy for a learner int…","labels":["LearnerPolicy"],"detail_key":"p14"},{"id":"n20847","layer":"informal","project":"p14","title":"Adaptive adversary","kind":"definition","summary":"[Adaptive adversary] An \\textttAdaptiveAdversary\\ N is an online strategy for the adversary: it…","labels":["AdaptiveAdversary"],"detail_key":"p14"},{"id":"n20848","layer":"informal","project":"p14","title":"Hedge learner policy","kind":"definition","summary":"[Hedge learner policy] \\texttthedgePolicy\\ N\\ \\eta is the \\textttLearnerPolicy that implements…","labels":["hedgePolicy"],"detail_key":"p14"},{"id":"n20849","layer":"informal","project":"p14","title":"Episode loss at natural time","kind":"definition","summary":"[Episode loss at natural time] \\textttepisodeLossNat\\ learner\\ adversary\\ t : Fin\\,N \\to R is t…","labels":["episodeLossNat"],"detail_key":"p14"},{"id":"n20850","layer":"informal","project":"p14","title":"Episode loss sequence","kind":"definition","summary":"[Episode loss sequence] \\textttepisodeLossSeq\\ T\\ learner\\ adversary : \\textttLossSeq\\ N\\ T is…","labels":["episodeLossSeq"],"detail_key":"p14"},{"id":"n20851","layer":"informal","project":"p14","title":"Episode","kind":"definition","summary":"[Episode] An \\textttEpisode\\ N\\ T packages a learner policy, an adaptive adversary, and a finit…","labels":["Episode"],"detail_key":"p14"},{"id":"n20852","layer":"informal","project":"p14","title":"Generated episode","kind":"definition","summary":"[Generated episode] \\textttEpisode.generated\\ learner\\ adversary is the canonical \\textttEpisod…","labels":["Episode.generated"],"detail_key":"p14"},{"id":"n20853","layer":"informal","project":"p14","title":"Validity of generated episode loss sequence","kind":"theorem","summary":"[Validity of generated episode loss sequence] For any learner policy and adaptive adversary, th…","labels":["episodeLossSeq_valid"],"detail_key":"p14"},{"id":"n20854","layer":"informal","project":"p14","title":"Episode losses are valid","kind":"theorem","summary":"[Episode losses are valid] For any \\textttEpisode\\ N\\ T, the stored loss sequence episode.losse…","labels":["Episode.losses_valid"],"detail_key":"p14"},{"id":"n20855","layer":"informal","project":"p14","title":"Hedge regret bound for adaptive episodes","kind":"theorem","summary":"[Hedge regret bound for adaptive episodes] For any N, T \\ge 1, learning rate \\eta > 0, and adap…","labels":["hedge_regret_bound_tight_episode"],"detail_key":"p14"},{"id":"n20856","layer":"informal","project":"p14","title":"Hedge regret bound from a packaged episode","kind":"theorem","summary":"[Hedge regret bound from a packaged episode] Let episode be an \\textttEpisode\\ N\\ T whose learn…","labels":["hedge_regret_bound_tight_of_episode"],"detail_key":"p14"},{"id":"n20857","layer":"informal","project":"p14","title":"x(1-x) \\le 1/4","kind":"lemma","summary":"[x(1-x) \\le 1/4] For any x \\in [0,1], the product x(1-x) \\le \\tfrac14.","labels":["mul_one_sub_le_quarter"],"detail_key":"p14"},{"id":"n20858","layer":"informal","project":"p14","title":"Convexity bound for the exponential","kind":"theorem","summary":"[Convexity bound for the exponential] For any \\eta, x \\in R with x \\in [0,1], convexity of \\exp…","labels":["exp_convexity_bound'"],"detail_key":"p14"},{"id":"n20859","layer":"informal","project":"p14","title":"Weighted exponential sum vs.\\ affine upper bound","kind":"theorem","summary":"[Weighted exponential sum vs.\\ affine upper bound] Let p : Fin\\,n \\to R be a probability vector…","labels":["weighted_exp_le_affine"],"detail_key":"p14"},{"id":"n20860","layer":"informal","project":"p14","title":"\\ln(1+u) \\le u","kind":"theorem","summary":"[\\ln(1+u) \\le u] For any u > -1, \\ln(1+u) \\le u.","labels":["log_one_add_le"],"detail_key":"p14"},{"id":"n20861","layer":"informal","project":"p14","title":"\\eta - 1 + e^-\\eta \\le \\eta^2/2","kind":"theorem","summary":"[\\eta - 1 + e^-\\eta \\le \\eta^2/2] For any \\eta \\ge 0, \\[ \\eta - 1 + e^-\\eta \\;\\le\\; \\frac\\eta^2…","labels":["eta_exp_bound"],"detail_key":"p14"},{"id":"n20862","layer":"informal","project":"p14","title":"Weak Hoeffding log-MGF bound","kind":"theorem","summary":"[Weak Hoeffding log-MGF bound] Let p be a probability vector on Fin\\,n and let \\ell_i \\in [0,1]…","labels":["hoeffding_log_mgf_weak"],"detail_key":"p14"},{"id":"n20863","layer":"informal","project":"p14","title":"Bernoulli MGF bound (Hoeffding's lemma)","kind":"theorem","summary":"[Bernoulli MGF bound (Hoeffding's lemma)] For L \\in [0,1] and any \\eta \\in R, \\[ \\ln\\!\\bigl(1 -…","labels":["bernoulli_mgf_bound"],"detail_key":"p14"},{"id":"n20864","layer":"informal","project":"p14","title":"Tight Hoeffding log-MGF bound","kind":"theorem","summary":"[Tight Hoeffding log-MGF bound] Under the same hypotheses as \\texttthoeffding\\_log\\_mgf\\_weak,…","labels":["hoeffding_log_mgf_tight"],"detail_key":"p14"},{"id":"n20865","layer":"informal","project":"p14","title":"Bernstein MGF condition","kind":"definition","summary":"[Bernstein MGF condition] \\textttProbabilityTheory.HasBernsteinMGF\\ X\\ \\mu\\ c\\ t_\\max is a pred…","labels":["ProbabilityTheory.HasBernsteinMGF"],"detail_key":"p14"},{"id":"n20866","layer":"informal","project":"p14","title":"iid closure of the Bernstein MGF condition","kind":"lemma","summary":"[iid closure of the Bernstein MGF condition] Let \\X_i\\_i \\in s be a mutually independent family…","labels":["ProbabilityTheory.HasBernsteinMGF.sum_of_iIndepFun"],"detail_key":"p14"},{"id":"n20867","layer":"informal","project":"p14","title":"Upper-tail Bernstein bound","kind":"lemma","summary":"[Upper-tail Bernstein bound] Let X satisfy \\textttProbabilityTheory.HasBernsteinMGF\\ X\\ \\mu\\ c\\…","labels":["ProbabilityTheory.HasBernsteinMGF.measure_ge_le"],"detail_key":"p14"},{"id":"n20868","layer":"informal","project":"p14","title":"Lower-tail Bernstein bound","kind":"lemma","summary":"[Lower-tail Bernstein bound] Let X satisfy \\textttProbabilityTheory.HasBernsteinMGF\\ X\\ \\mu\\ c\\…","labels":["ProbabilityTheory.HasBernsteinMGF.measure_le_le"],"detail_key":"p14"},{"id":"n20869","layer":"informal","project":"p14","title":"Two-sided Bernstein bound","kind":"lemma","summary":"[Two-sided Bernstein bound] Let X satisfy \\textttProbabilityTheory.HasBernsteinMGF\\ X\\ \\mu\\ c\\…","labels":["ProbabilityTheory.HasBernsteinMGF.measure_abs_gt_le"],"detail_key":"p14"},{"id":"n20870","layer":"informal","project":"p14","title":"Centered squared iid Bernstein tail bound","kind":"lemma","summary":"[Centered squared iid Bernstein tail bound] Let \\Y_i\\_i \\in \\iota be a mutually independent, id…","labels":["ProbabilityTheory.centered_squared_iid_tail"],"detail_key":"p14"},{"id":"n20871","layer":"informal","project":"p14","title":"Bad distortion event for a single vector","kind":"definition","summary":"[Bad distortion event for a single vector] For a parameter \\varepsilon\\inR, a matrix A, and a v…","labels":["BadSingle"],"detail_key":"p14"},{"id":"n20872","layer":"informal","project":"p14","title":"Zero vector has probability zero of being bad","kind":"lemma","summary":"[Zero vector has probability zero of being bad] For any probability measure \\mu on \\Omega, any…","labels":["concentration_zero"],"detail_key":"p14"},{"id":"n20873","layer":"informal","project":"p14","title":"Scalar multiple of a centered Gaussian","kind":"lemma","summary":"[Scalar multiple of a centered Gaussian] Let X : \\Omega \\to R be measurable with X_*\\mu = N(0,\\…","labels":["map_const_mul_gaussian"],"detail_key":"p14"},{"id":"n20874","layer":"informal","project":"p14","title":"Row-projection of i.i.d.\\ Gaussians is Gaussian","kind":"lemma","summary":"[Row-projection of i.i.d.\\ Gaussians is Gaussian] Let Y_j : \\Omega \\to R be i.i.d.\\ with law N(…","labels":["sum_scaled_iid_gaussian_map"],"detail_key":"p14"},{"id":"n20875","layer":"informal","project":"p14","title":"Row projections are independent","kind":"lemma","summary":"[Row projections are independent] If the row vectors of A (as (Fin\\,d\\toR)-valued random variab…","labels":["rows_indep"],"detail_key":"p14"},{"id":"n20876","layer":"informal","project":"p14","title":"Coordinate-sum form of a row projection","kind":"lemma","summary":"[Coordinate-sum form of a row projection] For a matrix A\\inR^k\\times d, a vector x\\inR^d, and a…","labels":["toEuclideanLin_apply_eq_sum"],"detail_key":"p14"},{"id":"n20877","layer":"informal","project":"p14","title":"Squared norm as sum of squared row projections","kind":"lemma","summary":"[Squared norm as sum of squared row projections] For a matrix A and vector x, \\[ \\|A.\\texttttoE…","labels":["norm_sq_toEuclideanLin"],"detail_key":"p14"},{"id":"n20878","layer":"informal","project":"p14","title":"Derivative of Taylor auxiliary function","kind":"lemma","summary":"[Derivative of Taylor auxiliary function] The function h(u) = u^2 + u + \\log(1-u) has derivativ…","labels":["taylorAux_hasDerivAt"],"detail_key":"p14"},{"id":"n20879","layer":"informal","project":"p14","title":"Taylor auxiliary function vanishes at zero","kind":"lemma","summary":"[Taylor auxiliary function vanishes at zero] The evaluation 0^2 + 0 + \\log(1-0) = 0 establishes…","labels":["taylorAux_zero"],"detail_key":"p14"},{"id":"n20880","layer":"informal","project":"p14","title":"Taylor bound for the centered chi-squared log-MGF","kind":"lemma","summary":"[Taylor bound for the centered chi-squared log-MGF] For every s\\inR with |s|\\le 1/4, \\[ -s - \\t…","labels":["neg_log_one_sub_two_mul_le_two_sq"],"detail_key":"p14"},{"id":"n20881","layer":"informal","project":"p14","title":"Quadratic MGF of the standard Gaussian","kind":"lemma","summary":"[Quadratic MGF of the standard Gaussian] For Z\\sim N(0,1) and 2s<1, \\[ \\int z.\\, e^s z^2\\,d(N(0…","labels":["integral_exp_mul_sq_standardGaussian"],"detail_key":"p14"},{"id":"n20882","layer":"informal","project":"p14","title":"Quadratic MGF of a general centered Gaussian","kind":"lemma","summary":"[Quadratic MGF of a general centered Gaussian] For Y\\sim N(0,v) with v\\ne 0 and 2tv < 1, \\[ \\in…","labels":["integral_exp_mul_sq_gaussianReal_zero"],"detail_key":"p14"},{"id":"n20883","layer":"informal","project":"p14","title":"Integrability of e^ty^2 under N(0,v)","kind":"lemma","summary":"[Integrability of e^ty^2 under N(0,v)] For v\\ne 0 and 2tv < 1, the function y\\mapsto e^t y^2 is…","labels":["integrable_exp_mul_sq_gaussianReal_zero"],"detail_key":"p14"},{"id":"n20884","layer":"informal","project":"p14","title":"Centered chi-squared MGF bound","kind":"theorem","summary":"[Centered chi-squared MGF bound] Let (\\Omega,\\mu) be a probability space, k>0, and let Y:\\Omega…","labels":["centered_chi_squared_step"],"detail_key":"p14"},{"id":"n20885","layer":"informal","project":"p14","title":"Bernstein MGF instance for centered chi-squared","kind":"theorem","summary":"[Bernstein MGF instance for centered chi-squared] Under the same hypotheses as \\textttcentered\\…","labels":["hasBernsteinMGF_centered_chi_squared"],"detail_key":"p14"},{"id":"n20886","layer":"informal","project":"p14","title":"Chi-squared tail bound for i.i.d.\\ N(0,1/k) variables","kind":"lemma","summary":"[Chi-squared tail bound for i.i.d.\\ N(0,1/k) variables] Let Y_1,\\dots,Y_k:\\Omega\\toR be i.i.d.\\…","labels":["chi_squared_tail"],"detail_key":"p14"},{"id":"n20887","layer":"informal","project":"p14","title":"JL single-vector concentration via chi-squared reduction","kind":"theorem","summary":"[JL single-vector concentration via chi-squared reduction] Let A:\\Omega\\toR^k\\times d be a rand…","labels":["jl_concentration_single_via_chi_squared"],"detail_key":"p14"},{"id":"n20888","layer":"informal","project":"p14","title":"Distribution-agnostic JL concentration via Bernstein tails","kind":"theorem","summary":"[Distribution-agnostic JL concentration via Bernstein tails] Let A:\\Omega\\toR^k\\times d be a ra…","labels":["jl_concentration_single_via_bernstein"],"detail_key":"p14"},{"id":"n20889","layer":"informal","project":"p14","title":"Single-pair distortion bound","kind":"definition","summary":"[Single-pair distortion bound] For \\varepsilon \\in R, vectors u, v \\in R^d, and their images u'…","labels":["JLDistortion"],"detail_key":"p14"},{"id":"n20890","layer":"informal","project":"p14","title":"\\varepsilon-JL embedding","kind":"definition","summary":"[\\varepsilon-JL embedding] A linear map f : R^d \\to_L[R] R^k is an \\emph\\varepsilon-JL embeddin…","labels":["IsJLEmbedding"],"detail_key":"p14"},{"id":"n20891","layer":"informal","project":"p14","title":"Bad-pair event","kind":"definition","summary":"[Bad-pair event] Given a k \\times d matrix A and a finite set V \\subseteq R^d, \\textttBadPair \\…","labels":["BadPair"],"detail_key":"p14"},{"id":"n20892","layer":"informal","project":"p14","title":"JL concentration --- single vector","kind":"theorem","summary":"[JL concentration --- single vector] Let A be a random k \\times d matrix whose entries are i.i.…","labels":["jl_concentration_single"],"detail_key":"p14"},{"id":"n20893","layer":"informal","project":"p14","title":"Distortion from non-bad single","kind":"lemma","summary":"[Distortion from non-bad single] If the matrix A does \\emphnot trigger the bad-single event for…","labels":["JLDistortion.of_not_bad"],"detail_key":"p14"},{"id":"n20894","layer":"informal","project":"p14","title":"JL union bound","kind":"theorem","summary":"[JL union bound] Suppose each per-pair bad event satisfies \\Pr[\\textBadSingle\\;\\varepsilon\\;A\\;…","labels":["jl_union_bound"],"detail_key":"p14"},{"id":"n20895","layer":"informal","project":"p14","title":"Structural JL --- Gaussian probabilistic method","kind":"theorem","summary":"[Structural JL --- Gaussian probabilistic method] Let A be a random Gaussian k \\times d matrix…","labels":["johnson_lindenstrauss_of_gaussian"],"detail_key":"p14"},{"id":"n20896","layer":"informal","project":"p14","title":"Structural JL --- sub-Gaussian probabilistic method","kind":"theorem","summary":"[Structural JL --- sub-Gaussian probabilistic method] Sub-Gaussian analogue of \\textttjohnson\\_…","labels":["johnson_lindenstrauss_of_subgaussian"],"detail_key":"p14"},{"id":"n20897","layer":"informal","project":"p14","title":"Measurability of bad-single event","kind":"lemma","summary":"[Measurability of bad-single event] For any measurable map A : \\Omega \\to Matrix(Fin\\,k, Fin\\,d…","labels":["measurableSet_badSingle"],"detail_key":"p14"},{"id":"n20898","layer":"informal","project":"p14","title":"Existence of i.i.d.\\ Gaussian matrix","kind":"lemma","summary":"[Existence of i.i.d.\\ Gaussian matrix] For any k > 0 and d \\ge 0, there exists a probability sp…","labels":["exists_iid_gaussian_matrix"],"detail_key":"p14"},{"id":"n20899","layer":"informal","project":"p14","title":"Numerical failure-probability bound","kind":"lemma","summary":"[Numerical failure-probability bound] Given 0 < \\varepsilon, n \\ge 2, k \\ge 32 \\log n / \\vareps…","labels":["jl_failure_bound_of_dim"],"detail_key":"p14"},{"id":"n20900","layer":"informal","project":"p14","title":"Johnson--Lindenstrauss flattening lemma (Gaussian)","kind":"theorem","summary":"[Johnson--Lindenstrauss flattening lemma (Gaussian)] For any 0 < \\varepsilon < 1, n \\ge 2, k \\g…","labels":["johnson_lindenstrauss"],"detail_key":"p14"},{"id":"n20901","layer":"informal","project":"p14","title":"Johnson--Lindenstrauss flattening lemma (sub-Gaussian)","kind":"theorem","summary":"[Johnson--Lindenstrauss flattening lemma (sub-Gaussian)] Rademacher (\\pm 1/\\sqrtk entries) anal…","labels":["johnson_lindenstrauss_subgaussian"],"detail_key":"p14"},{"id":"n20902","layer":"informal","project":"p14","title":"Distance form of embedding","kind":"lemma","summary":"[Distance form of embedding] If f is an \\varepsilon-JL embedding of V (in the squared-distance…","labels":["jl_dist_of_embedding"],"detail_key":"p14"},{"id":"n20903","layer":"informal","project":"p14","title":"JL flattening --- distance form (Gaussian)","kind":"theorem","summary":"[JL flattening --- distance form (Gaussian)] Under the same hypotheses as \\textttjohnson\\_linde…","labels":["johnson_lindenstrauss_dist"],"detail_key":"p14"},{"id":"n20904","layer":"informal","project":"p14","title":"JL flattening --- distance form (sub-Gaussian)","kind":"theorem","summary":"[JL flattening --- distance form (sub-Gaussian)] Under the same hypotheses as \\textttjohnson\\_l…","labels":["johnson_lindenstrauss_subgaussian_dist"],"detail_key":"p14"},{"id":"n20905","layer":"informal","project":"p14","title":"Logarithmic dimension bound (Gaussian)","kind":"theorem","summary":"[Logarithmic dimension bound (Gaussian)] For any 0 < \\varepsilon < 1 and n \\ge 2, there exists…","labels":["johnson_lindenstrauss_dim_bound"],"detail_key":"p14"},{"id":"n20906","layer":"informal","project":"p14","title":"Logarithmic dimension bound (sub-Gaussian)","kind":"theorem","summary":"[Logarithmic dimension bound (sub-Gaussian)] Sub-Gaussian (Rademacher) analogue of \\textttjohns…","labels":["johnson_lindenstrauss_subgaussian_dim_bound"],"detail_key":"p14"},{"id":"n20907","layer":"informal","project":"p14","title":"Distribution-agnostic JL single-vector concentration","kind":"theorem","summary":"[Distribution-agnostic JL single-vector concentration] Let A : \\Omega \\to R^k \\times d be a ran…","labels":["jl_concentration_single_subgaussian"],"detail_key":"p14"},{"id":"n20908","layer":"informal","project":"p14","title":"Rademacher distribution on R","kind":"definition","summary":"[Rademacher distribution on R] The Rademacher distribution Rad on R is the measure \\tfrac12\\del…","labels":["rademacherReal"],"detail_key":"p14"},{"id":"n20909","layer":"informal","project":"p14","title":"1/2 is finite as an extended non-negative real","kind":"lemma","summary":"[1/2 is finite as an extended non-negative real] The value 1/2, viewed as an element of R_\\geq…","labels":["rad_half_ne_top"],"detail_key":"p14"},{"id":"n20910","layer":"informal","project":"p14","title":"Rademacher support in [-1,1]","kind":"lemma","summary":"[Rademacher support in [-1,1]] Almost every sample from the Rademacher distribution lies in the…","labels":["rademacherReal_mem_Icc"],"detail_key":"p14"},{"id":"n20911","layer":"informal","project":"p14","title":"Integrability under a scaled Dirac measure","kind":"lemma","summary":"[Integrability under a scaled Dirac measure] For any a \\in R and any measurable function f : R…","labels":["integrable_smul_dirac"],"detail_key":"p14"},{"id":"n20912","layer":"informal","project":"p14","title":"Every function is integrable under the Rademacher measure","kind":"lemma","summary":"[Every function is integrable under the Rademacher measure] For any f : R \\to R, the function f…","labels":["integrable_rademacherReal"],"detail_key":"p14"},{"id":"n20913","layer":"informal","project":"p14","title":"Mean of the Rademacher distribution is zero","kind":"lemma","summary":"[Mean of the Rademacher distribution is zero] The expected value of the identity under the Rade…","labels":["integral_id_rademacherReal"],"detail_key":"p14"},{"id":"n20914","layer":"informal","project":"p14","title":"Second moment of the Rademacher distribution is one","kind":"lemma","summary":"[Second moment of the Rademacher distribution is one] The second moment of the Rademacher distr…","labels":["integral_sq_rademacherReal"],"detail_key":"p14"},{"id":"n20915","layer":"informal","project":"p14","title":"Identity is sub-Gaussian with parameter 1 under Rad","kind":"lemma","summary":"[Identity is sub-Gaussian with parameter 1 under Rad] The identity function id : R \\to R satisf…","labels":["hasSubgaussianMGF_id_rademacherReal"],"detail_key":"p14"},{"id":"n20916","layer":"informal","project":"p14","title":"Rademacher matrix sample space","kind":"definition","summary":"[Rademacher matrix sample space] The sample space for the Rademacher matrix is Fin\\,k \\to Fin\\,…","labels":["RadΩ"],"detail_key":"p14"},{"id":"n20917","layer":"informal","project":"p14","title":"Joint Rademacher product measure","kind":"definition","summary":"[Joint Rademacher product measure] The joint measure on Rad\\Omega(k,d) is the product measure \\…","labels":["radJointMeasure"],"detail_key":"p14"},{"id":"n20918","layer":"informal","project":"p14","title":"Rademacher matrix","kind":"definition","summary":"[Rademacher matrix] The Rademacher matrix A : Rad\\Omega(k,d) \\to R^k\\times d maps a sample \\ome…","labels":["radMatrix"],"detail_key":"p14"},{"id":"n20919","layer":"informal","project":"p14","title":"Measurability of the Rademacher matrix","kind":"lemma","summary":"[Measurability of the Rademacher matrix] The map \\omega \\mapsto radMatrix(k,d,\\omega) is measur…","labels":["measurable_radMatrix"],"detail_key":"p14"},{"id":"n20920","layer":"informal","project":"p14","title":"Row marginal of the joint measure","kind":"lemma","summary":"[Row marginal of the joint measure] The pushforward of the joint measure along the i-th row pro…","labels":["radJointMeasure_row_marginal"],"detail_key":"p14"},{"id":"n20921","layer":"informal","project":"p14","title":"Entry marginal of the joint measure","kind":"lemma","summary":"[Entry marginal of the joint measure] The pushforward of the joint measure along the (i,j)-entr…","labels":["radJointMeasure_entry_marginal"],"detail_key":"p14"},{"id":"n20922","layer":"informal","project":"p14","title":"Within-row entries are i.i.d.\\ Rademacher","kind":"lemma","summary":"[Within-row entries are i.i.d.\\ Rademacher] For each fixed row i \\in Fin\\,k, the d functions j…","labels":["radJointMeasure_row_iid"],"detail_key":"p14"},{"id":"n20923","layer":"informal","project":"p14","title":"Rows are mutually independent","kind":"lemma","summary":"[Rows are mutually independent] The k row vectors i \\mapsto (\\omega_ij)_j, viewed as Fin\\,d \\to…","labels":["radJointMeasure_rows_iid"],"detail_key":"p14"},{"id":"n20924","layer":"informal","project":"p14","title":"Each entry is sub-Gaussian with parameter 1","kind":"lemma","summary":"[Each entry is sub-Gaussian with parameter 1] For each (i,j) \\in Fin\\,k \\times Fin\\,d, the func…","labels":["hasSubgaussianMGF_entry"],"detail_key":"p14"},{"id":"n20925","layer":"informal","project":"p14","title":"Scaled entry is sub-Gaussian with parameter x_j^2/k","kind":"lemma","summary":"[Scaled entry is sub-Gaussian with parameter x_j^2/k] For k > 0, x \\in R^d, and indices i,j, th…","labels":["hasSubgaussianMGF_scaled_entry"],"detail_key":"p14"},{"id":"n20926","layer":"informal","project":"p14","title":"Row projection is sub-Gaussian with parameter \\lVert x\\rVert^2/k","kind":"lemma","summary":"[Row projection is sub-Gaussian with parameter \\lVert x\\rVert^2/k] For k > 0, x \\in R^d, and i…","labels":["hasSubgaussianMGF_row_proj"],"detail_key":"p14"},{"id":"n20927","layer":"informal","project":"p14","title":"Variance of id under Rad is 1","kind":"lemma","summary":"[Variance of id under Rad is 1] The variance of the identity function under the Rademacher dist…","labels":["variance_id_rademacherReal"],"detail_key":"p14"},{"id":"n20928","layer":"informal","project":"p14","title":"Variance of a single entry is 1","kind":"lemma","summary":"[Variance of a single entry is 1] For any (i,j), the variance of the entry \\omega \\mapsto \\omeg…","labels":["variance_entry"],"detail_key":"p14"},{"id":"n20929","layer":"informal","project":"p14","title":"Mean of a single entry is 0","kind":"lemma","summary":"[Mean of a single entry is 0] For any (i,j), the expected value of \\omega_ij under the joint me…","labels":["integral_entry"],"detail_key":"p14"},{"id":"n20930","layer":"informal","project":"p14","title":"Scaled entry belongs to L^2","kind":"lemma","summary":"[Scaled entry belongs to L^2] For k > 0, x \\in R^d, and indices i,j, the function \\omega \\mapst…","labels":["memLp_scaled_entry"],"detail_key":"p14"},{"id":"n20931","layer":"informal","project":"p14","title":"Variance of a scaled entry is x_j^2/k","kind":"lemma","summary":"[Variance of a scaled entry is x_j^2/k] For k > 0, x \\in R^d, and indices i,j, Var[\\omega \\maps…","labels":["variance_scaled_entry"],"detail_key":"p14"},{"id":"n20932","layer":"informal","project":"p14","title":"Variance of the row projection is \\lVert x\\rVert^2/k","kind":"lemma","summary":"[Variance of the row projection is \\lVert x\\rVert^2/k] For k > 0, x \\in R^d, and i \\in Fin\\,k,…","labels":["variance_row_proj"],"detail_key":"p14"},{"id":"n20933","layer":"informal","project":"p14","title":"Mean of the row projection is 0","kind":"lemma","summary":"[Mean of the row projection is 0] For k > 0, x \\in R^d, and i \\in Fin\\,k, \\int (A(\\omega)\\,x)_i…","labels":["integral_row_proj"],"detail_key":"p14"},{"id":"n20934","layer":"informal","project":"p14","title":"Second moment of the row projection equals \\lVert x\\rVert^2/k","kind":"lemma","summary":"[Second moment of the row projection equals \\lVert x\\rVert^2/k] For k > 0, x \\in R^d, and i \\in…","labels":["integral_sq_row_proj"],"detail_key":"p14"},{"id":"n20935","layer":"informal","project":"p14","title":"Row projections are mutually independent","kind":"lemma","summary":"[Row projections are mutually independent] For any x \\in R^d, the k row projections i \\mapsto (…","labels":["radMatrix_proj_indep"],"detail_key":"p14"},{"id":"n20936","layer":"informal","project":"p14","title":"Row projections are measurable","kind":"lemma","summary":"[Row projections are measurable] For any x \\in R^d and i \\in Fin\\,k, the map \\omega \\mapsto (A(…","labels":["radMatrix_proj_meas"],"detail_key":"p14"},{"id":"n20937","layer":"informal","project":"p14","title":"JL concentration for the Rademacher matrix","kind":"theorem","summary":"[JL concentration for the Rademacher matrix] Let k, d \\geq 1 and let A be the Rademacher matrix…","labels":["jl_concentration_single_rademacher"],"detail_key":"p14"},{"id":"n20938","layer":"informal","project":"p14","title":"Two-player finite game","kind":"definition","summary":"[Two-player finite game] A finite two-player game with M row actions and N column actions, spec…","labels":["Game"],"detail_key":"p14"},{"id":"n20939","layer":"informal","project":"p14","title":"Embedding a zero-sum game","kind":"definition","summary":"[Embedding a zero-sum game] Given a zero-sum game G with payoff matrix A, this embeds it into t…","labels":["ZeroSumGame.toGame"],"detail_key":"p14"},{"id":"n20940","layer":"informal","project":"p14","title":"Joint distribution over action profiles","kind":"definition","summary":"[Joint distribution over action profiles] A joint distribution \\sigma on Fin\\,M \\times Fin\\,N i…","labels":["JointDistribution"],"detail_key":"p14"},{"id":"n20941","layer":"informal","project":"p14","title":"Row marginal","kind":"definition","summary":"[Row marginal] The row marginal of \\sigma at row action i is \\sigma^row_i = \\sum_j \\sigma_ij, i…","labels":["JointDistribution.rowMarginal"],"detail_key":"p14"},{"id":"n20942","layer":"informal","project":"p14","title":"Column marginal","kind":"definition","summary":"[Column marginal] The column marginal of \\sigma at column action j is \\sigma^col_j = \\sum_i \\si…","labels":["JointDistribution.colMarginal"],"detail_key":"p14"},{"id":"n20943","layer":"informal","project":"p14","title":"Row marginals are nonnegative","kind":"lemma","summary":"[Row marginals are nonnegative] For any joint distribution \\sigma and row action i, \\sigma^row_…","labels":["JointDistribution.rowMarginal_nonneg"],"detail_key":"p14"},{"id":"n20944","layer":"informal","project":"p14","title":"Column marginals are nonnegative","kind":"lemma","summary":"[Column marginals are nonnegative] For any joint distribution \\sigma and column action j, \\sigm…","labels":["JointDistribution.colMarginal_nonneg"],"detail_key":"p14"},{"id":"n20945","layer":"informal","project":"p14","title":"Row marginals sum to one","kind":"lemma","summary":"[Row marginals sum to one] The row marginals form a valid probability distribution: \\sum_i \\sig…","labels":["JointDistribution.rowMarginal_sum_one"],"detail_key":"p14"},{"id":"n20946","layer":"informal","project":"p14","title":"Column marginals sum to one","kind":"lemma","summary":"[Column marginals sum to one] The column marginals form a valid probability distribution: \\sum_…","labels":["JointDistribution.colMarginal_sum_one"],"detail_key":"p14"},{"id":"n20947","layer":"informal","project":"p14","title":"Row expected utility","kind":"definition","summary":"[Row expected utility] The row player's expected utility under \\sigma in game G is \\sum_i,j \\si…","labels":["JointDistribution.rowExpectedUtility"],"detail_key":"p14"},{"id":"n20948","layer":"informal","project":"p14","title":"Column expected utility","kind":"definition","summary":"[Column expected utility] The column player's expected utility under \\sigma in game G is \\sum_i…","labels":["JointDistribution.colExpectedUtility"],"detail_key":"p14"},{"id":"n20949","layer":"informal","project":"p14","title":"Row deviation utility","kind":"definition","summary":"[Row deviation utility] The row player's expected utility from unilaterally deviating to the fi…","labels":["JointDistribution.rowDeviationUtility"],"detail_key":"p14"},{"id":"n20950","layer":"informal","project":"p14","title":"Column deviation utility","kind":"definition","summary":"[Column deviation utility] The column player's expected utility from unilaterally deviating to…","labels":["JointDistribution.colDeviationUtility"],"detail_key":"p14"},{"id":"n20951","layer":"informal","project":"p14","title":"Coarse correlated equilibrium","kind":"definition","summary":"[Coarse correlated equilibrium] A joint distribution \\sigma is a \\emphcoarse correlated equilib…","labels":["IsCoarseCorrelatedEquilibrium"],"detail_key":"p14"},{"id":"n20952","layer":"informal","project":"p14","title":"\\varepsilon-coarse correlated equilibrium","kind":"definition","summary":"[\\varepsilon-coarse correlated equilibrium] A joint distribution \\sigma is an \\emph\\varepsilon-…","labels":["IsApproxCoarseCorrelatedEquilibrium"],"detail_key":"p14"},{"id":"n20953","layer":"informal","project":"p14","title":"Exact CCE implies approximate CCE","kind":"lemma","summary":"[Exact CCE implies approximate CCE] If \\sigma is an exact CCE of G and \\varepsilon \\ge 0, then…","labels":["IsCoarseCorrelatedEquilibrium.toApprox"],"detail_key":"p14"},{"id":"n20954","layer":"informal","project":"p14","title":"Product distribution","kind":"definition","summary":"[Product distribution] Given mixed strategies p for the row player and q for the column player,…","labels":["productDistribution"],"detail_key":"p14"},{"id":"n20955","layer":"informal","project":"p14","title":"Product distribution row marginal","kind":"lemma","summary":"[Product distribution row marginal] For the product distribution of p and q, the row marginal a…","labels":["productDistribution_rowMarginal"],"detail_key":"p14"},{"id":"n20956","layer":"informal","project":"p14","title":"Product distribution column marginal","kind":"lemma","summary":"[Product distribution column marginal] For the product distribution of p and q, the column marg…","labels":["productDistribution_colMarginal"],"detail_key":"p14"},{"id":"n20957","layer":"informal","project":"p14","title":"Empirical joint distribution","kind":"definition","summary":"[Empirical joint distribution] Given T > 0 rounds of play in which the row player uses mixed st…","labels":["empiricalJoint"],"detail_key":"p14"},{"id":"n20958","layer":"informal","project":"p14","title":"Empirical joint row marginal","kind":"lemma","summary":"[Empirical joint row marginal] The row marginal of the empirical joint distribution at action i…","labels":["empiricalJoint_rowMarginal"],"detail_key":"p14"},{"id":"n20959","layer":"informal","project":"p14","title":"Empirical joint column marginal","kind":"lemma","summary":"[Empirical joint column marginal] The column marginal of the empirical joint distribution at ac…","labels":["empiricalJoint_colMarginal"],"detail_key":"p14"},{"id":"n20960","layer":"informal","project":"p14","title":"Empirical utility sum factorization","kind":"lemma","summary":"[Empirical utility sum factorization] For any function f : Fin\\,M \\to Fin\\,N \\to R, the sum \\su…","labels":["empiricalJoint_sum_prob_mul"],"detail_key":"p14"},{"id":"n20961","layer":"informal","project":"p14","title":"Empirical row expected utility","kind":"lemma","summary":"[Empirical row expected utility] The row player's expected utility under the empirical joint di…","labels":["empiricalJoint_rowExpectedUtility"],"detail_key":"p14"},{"id":"n20962","layer":"informal","project":"p14","title":"Empirical column expected utility","kind":"lemma","summary":"[Empirical column expected utility] The column player's expected utility under the empirical jo…","labels":["empiricalJoint_colExpectedUtility"],"detail_key":"p14"},{"id":"n20963","layer":"informal","project":"p14","title":"Empirical row deviation utility","kind":"lemma","summary":"[Empirical row deviation utility] The row player's deviation utility for pure action i' in the…","labels":["empiricalJoint_rowDeviationUtility"],"detail_key":"p14"},{"id":"n20964","layer":"informal","project":"p14","title":"Empirical column deviation utility","kind":"lemma","summary":"[Empirical column deviation utility] The column player's deviation utility for pure action j' i…","labels":["empiricalJoint_colDeviationUtility"],"detail_key":"p14"},{"id":"n20965","layer":"informal","project":"p14","title":"No-regret play yields an approximate CCE","kind":"theorem","summary":"[No-regret play yields an approximate CCE] Let G be a two-player game, T > 0, and let p_t, q_t…","labels":["empiricalJoint_isApproxCCE"],"detail_key":"p14"},{"id":"n20966","layer":"informal","project":"p14","title":"Arbitrary mixed strategy","kind":"definition","summary":"[Arbitrary mixed strategy] Given n \\geq 1, \\textttOnlineLearning.arbitraryMixedStrategy is a co…","labels":["OnlineLearning.arbitraryMixedStrategy"],"detail_key":"p14"},{"id":"n20967","layer":"informal","project":"p14","title":"Lower value of a finite game","kind":"definition","summary":"[Lower value of a finite game] For a finite zero-sum game G with M row actions and N column act…","labels":["OnlineLearning.finiteLowerValue"],"detail_key":"p14"},{"id":"n20968","layer":"informal","project":"p14","title":"Upper value of a finite game","kind":"definition","summary":"[Upper value of a finite game] The \\emphupper value of G is \\[ v^+(G) \\;=\\; \\inf_q \\in \\Delta_N…","labels":["OnlineLearning.finiteUpperValue"],"detail_key":"p14"},{"id":"n20969","layer":"informal","project":"p14","title":"Payoff vs.\\ pure column is nonnegative","kind":"lemma","summary":"[Payoff vs.\\ pure column is nonnegative] For any game G, mixed row strategy p, and pure column…","labels":["OnlineLearning.payoffVsPure_nonneg"],"detail_key":"p14"},{"id":"n20970","layer":"informal","project":"p14","title":"Payoff vs.\\ pure column is at most one","kind":"lemma","summary":"[Payoff vs.\\ pure column is at most one] For any game G, mixed row strategy p, and pure column…","labels":["OnlineLearning.payoffVsPure_le_one"],"detail_key":"p14"},{"id":"n20971","layer":"informal","project":"p14","title":"Pure row payoff vs.\\ mixed column is nonnegative","kind":"lemma","summary":"[Pure row payoff vs.\\ mixed column is nonnegative] For any game G, pure row action i, and mixed…","labels":["OnlineLearning.pureVsPayoff_nonneg"],"detail_key":"p14"},{"id":"n20972","layer":"informal","project":"p14","title":"Pure row payoff vs.\\ mixed column is at most one","kind":"lemma","summary":"[Pure row payoff vs.\\ mixed column is at most one] For any game G, pure row action i, and mixed…","labels":["OnlineLearning.pureVsPayoff_le_one"],"detail_key":"p14"},{"id":"n20973","layer":"informal","project":"p14","title":"Row-guaranteed payoffs bounded above","kind":"lemma","summary":"[Row-guaranteed payoffs bounded above] When N \\geq 1, the set \\bigl\\\\inf_j\\,payoffVsPure(G,p,j)…","labels":["OnlineLearning.finiteLowerValue_bddAbove"],"detail_key":"p14"},{"id":"n20974","layer":"informal","project":"p14","title":"Column-induced upper values bounded below","kind":"lemma","summary":"[Column-induced upper values bounded below] When M \\geq 1, the set \\bigl\\\\sup_i\\,pureVsPayoff(G…","labels":["OnlineLearning.finiteUpperValue_bddBelow"],"detail_key":"p14"},{"id":"n20975","layer":"informal","project":"p14","title":"Weak duality for finite games","kind":"lemma","summary":"[Weak duality for finite games] For any finite zero-sum game G (with M,N \\geq 1), v^-(G) \\leq v…","labels":["OnlineLearning.finiteLowerValue_le_upperValue"],"detail_key":"p14"},{"id":"n20976","layer":"informal","project":"p14","title":"Exact finite minimax value","kind":"theorem","summary":"[Exact finite minimax value] For a finite zero-sum game G with M > 1 rows and N \\geq 1 columns,…","labels":["OnlineLearning.finite_minimax_value"],"detail_key":"p14"},{"id":"n20977","layer":"informal","project":"p14","title":"Convex-compact minimax statement","kind":"definition","summary":"[Convex-compact minimax statement] For sets X, Y \\subseteq R and a payoff function f : R\\to R\\t…","labels":["OnlineLearning.ConvexCompactMinimaxStatement"],"detail_key":"p14"},{"id":"n20978","layer":"informal","project":"p14","title":"Convex-compact minimax hypotheses","kind":"definition","summary":"[Convex-compact minimax hypotheses] A structure bundling the assumptions of Cesa-Bianchi--Lugos…","labels":["OnlineLearning.ConvexCompactMinimaxHypotheses"],"detail_key":"p14"},{"id":"n20979","layer":"informal","project":"p14","title":"Mixed-strategy convex combination stays in convex set","kind":"lemma","summary":"[Mixed-strategy convex combination stays in convex set] If S \\subseteq R is convex, p \\in \\Delt…","labels":["OnlineLearning.mixed_sum_mem_convex"],"detail_key":"p14"},{"id":"n20980","layer":"informal","project":"p14","title":"Jensen's inequality for mixed-strategy combinations (convex)","kind":"lemma","summary":"[Jensen's inequality for mixed-strategy combinations (convex)] If g is convex on S \\subseteq R,…","labels":["OnlineLearning.convexOn_mixed_sum_le"],"detail_key":"p14"},{"id":"n20981","layer":"informal","project":"p14","title":"Jensen's inequality for mixed-strategy combinations (concave)","kind":"lemma","summary":"[Jensen's inequality for mixed-strategy combinations (concave)] If g is concave on S \\subseteq…","labels":["OnlineLearning.concaveOn_le_mixed_sum"],"detail_key":"p14"},{"id":"n20982","layer":"informal","project":"p14","title":"Sampled finite game","kind":"definition","summary":"[Sampled finite game] Given a payoff function f : R\\to R\\to R with values in [0,1], finite fami…","labels":["OnlineLearning.sampledGame"],"detail_key":"p14"},{"id":"n20983","layer":"informal","project":"p14","title":"Finite minimax for sampled games","kind":"theorem","summary":"[Finite minimax for sampled games] Under the same size hypotheses as \\textttOnlineLearning.fini…","labels":["OnlineLearning.finite_sampled_minimax_value"],"detail_key":"p14"},{"id":"n20984","layer":"informal","project":"p14","title":"Convexity gives lower bound on sampled payoff vs.\\ pure column","kind":"lemma","summary":"[Convexity gives lower bound on sampled payoff vs.\\ pure column] If f(\\cdot, y_j) is convex on…","labels":["OnlineLearning.sampled_payoffVsPure_ge_convex_combo"],"detail_key":"p14"},{"id":"n20985","layer":"informal","project":"p14","title":"Concavity gives upper bound on sampled payoff vs.\\ mixed column","kind":"lemma","summary":"[Concavity gives upper bound on sampled payoff vs.\\ mixed column] If f(x_i,\\cdot) is concave on…","labels":["OnlineLearning.sampled_pureVsPayoff_le_concave_combo"],"detail_key":"p14"},{"id":"n20986","layer":"informal","project":"p14","title":"Weak minimax inequality for convex-compact games","kind":"lemma","summary":"[Weak minimax inequality for convex-compact games] Under the hypotheses \\textttOnlineLearning.C…","labels":["OnlineLearning.weak_convex_compact_minimax"],"detail_key":"p14"},{"id":"n20987","layer":"informal","project":"p14","title":"Minimax sublevel set","kind":"definition","summary":"[Minimax sublevel set] For a set X \\subseteq R, a payoff function f, a column point y, and a th…","labels":["OnlineLearning.minimaxSublevel"],"detail_key":"p14"},{"id":"n20988","layer":"informal","project":"p14","title":"Sublevel set contained in row set","kind":"lemma","summary":"[Sublevel set contained in row set] sublevel(X,f,y,c) \\subseteq X for all f, y, c.","labels":["OnlineLearning.minimaxSublevel_subset"],"detail_key":"p14"},{"id":"n20989","layer":"informal","project":"p14","title":"Sublevel set is closed","kind":"lemma","summary":"[Sublevel set is closed] If X is closed and f(\\cdot,y) is continuous on X, then sublevel(X,f,y,…","labels":["OnlineLearning.minimaxSublevel_isClosed"],"detail_key":"p14"},{"id":"n20990","layer":"informal","project":"p14","title":"Sublevel set is compact","kind":"lemma","summary":"[Sublevel set is compact] If X is compact and f(\\cdot,y) is continuous on X, then sublevel(X,f,…","labels":["OnlineLearning.minimaxSublevel_isCompact"],"detail_key":"p14"},{"id":"n20991","layer":"informal","project":"p14","title":"Finite-intersection gives global row minimiser","kind":"lemma","summary":"[Finite-intersection gives global row minimiser] Under \\textttOnlineLearning.ConvexCompactMinim…","labels":["OnlineLearning.exists_forall_le_of_finite_sublevel_intersections"],"detail_key":"p14"},{"id":"n20992","layer":"informal","project":"p14","title":"Finite row sample lower value","kind":"definition","summary":"[Finite row sample lower value] Given a payoff f : R \\to R \\to R and a finite subset u \\subsete…","labels":["OnlineLearning.finiteRowSampleLowerValue"],"detail_key":"p14"},{"id":"n20993","layer":"informal","project":"p14","title":"Finite no-regret bound predicate","kind":"definition","summary":"[Finite no-regret bound predicate] \\textttOnlineLearning.Theorem71FiniteNoRegretBound is the pr…","labels":["OnlineLearning.Theorem71FiniteNoRegretBound"],"detail_key":"p14"},{"id":"n20994","layer":"informal","project":"p14","title":"Compact approximation predicate","kind":"definition","summary":"[Compact approximation predicate] \\textttOnlineLearning.Theorem71CompactApproximation is the pr…","labels":["OnlineLearning.Theorem71CompactApproximation"],"detail_key":"p14"},{"id":"n20995","layer":"informal","project":"p14","title":"Uniform equicontinuity in the row variable","kind":"definition","summary":"[Uniform equicontinuity in the row variable] \\textttOnlineLearning.Theorem71UniformEquicontinui…","labels":["OnlineLearning.Theorem71UniformEquicontinuity"],"detail_key":"p14"},{"id":"n20996","layer":"informal","project":"p14","title":"Finite indexed row sample lower value","kind":"definition","summary":"[Finite indexed row sample lower value] For a row sample indexed by Fin\\,M, the \\emphfinite ind…","labels":["OnlineLearning.finiteIndexedRowSampleLowerValue"],"detail_key":"p14"},{"id":"n20997","layer":"informal","project":"p14","title":"Minimax from the Theorem 7.1 proof route","kind":"theorem","summary":"[Minimax from the Theorem 7.1 proof route] If the standard convex-compact minimax hypotheses ho…","labels":["OnlineLearning.convex_compact_minimax_of_theorem71_route"],"detail_key":"p14"},{"id":"n20998","layer":"informal","project":"p14","title":"Compact approximation from uniform equicontinuity","kind":"theorem","summary":"[Compact approximation from uniform equicontinuity] Under the convex-compact minimax hypotheses…","labels":["OnlineLearning.theorem71_compactApproximation_of_uniformEquicontinuity"],"detail_key":"p14"},{"id":"n20999","layer":"informal","project":"p14","title":"Uniform equicontinuity from joint continuity on compact X \\times Y","kind":"theorem","summary":"[Uniform equicontinuity from joint continuity on compact X \\times Y] If the convex-compact mini…","labels":["OnlineLearning.theorem71_uniformEquicontinuity_of_jointContinuous_compact"],"detail_key":"p14"},{"id":"n21000","layer":"informal","project":"p14","title":"Compact approximation from joint continuity on compact X \\times Y","kind":"theorem","summary":"[Compact approximation from joint continuity on compact X \\times Y] Under the convex-compact mi…","labels":["OnlineLearning.theorem71_compactApproximation_of_jointContinuous_compact"],"detail_key":"p14"},{"id":"n21001","layer":"informal","project":"p14","title":"Finite indexed no-regret bound (normalized)","kind":"theorem","summary":"[Finite indexed no-regret bound (normalized)] Under the convex-compact minimax hypotheses and t…","labels":["OnlineLearning.theorem71_finiteIndexedNoRegretBound_normalized"],"detail_key":"p14"},{"id":"n21002","layer":"informal","project":"p14","title":"Finite no-regret bound (normalized, Finset version)","kind":"theorem","summary":"[Finite no-regret bound (normalized, Finset version)] Under the convex-compact minimax hypothes…","labels":["OnlineLearning.theorem71_finiteNoRegretBound_normalized"],"detail_key":"p14"},{"id":"n21003","layer":"informal","project":"p14","title":"No-regret minimax equality (normalized, joint-compact)","kind":"theorem","summary":"[No-regret minimax equality (normalized, joint-compact)] Under the convex-compact minimax hypot…","labels":["OnlineLearning.convex_compact_minimax_noRegret_jointCompact_normalized"],"detail_key":"p14"},{"id":"n21004","layer":"informal","project":"p14","title":"Finite zero-sum game","kind":"definition","summary":"[Finite zero-sum game] A \\emphfinite two-player zero-sum game with M row actions and N column a…","labels":["ZeroSumGame"],"detail_key":"p14"},{"id":"n21005","layer":"informal","project":"p14","title":"Mixed strategy","kind":"definition","summary":"[Mixed strategy] A \\emphmixed strategy over n actions is a record consisting of a weight functi…","labels":["MixedStrategy"],"detail_key":"p14"},{"id":"n21006","layer":"informal","project":"p14","title":"Row mixed payoff against a pure column","kind":"definition","summary":"[Row mixed payoff against a pure column] Given a game G, a mixed row strategy p, and a pure col…","labels":["payoffVsPure"],"detail_key":"p14"},{"id":"n21007","layer":"informal","project":"p14","title":"Pure row payoff against a mixed column","kind":"definition","summary":"[Pure row payoff against a mixed column] Given a game G, a pure row i, and a mixed column strat…","labels":["pureVsPayoff"],"detail_key":"p14"},{"id":"n21008","layer":"informal","project":"p14","title":"Column best response","kind":"definition","summary":"[Column best response] Given a game G and a mixed row strategy p, \\textttbestColumn is the pure…","labels":["bestColumn"],"detail_key":"p14"},{"id":"n21009","layer":"informal","project":"p14","title":"Best-column minimality","kind":"lemma","summary":"[Best-column minimality] For any game G, mixed row strategy p, and pure column j, \\[ payoffVsPu…","labels":["bestColumn_spec"],"detail_key":"p14"},{"id":"n21010","layer":"informal","project":"p14","title":"Row best response","kind":"definition","summary":"[Row best response] Given a game G and a mixed column strategy q, \\textttbestRow is the pure ro…","labels":["bestRow"],"detail_key":"p14"},{"id":"n21011","layer":"informal","project":"p14","title":"Best-row maximality","kind":"lemma","summary":"[Best-row maximality] For any game G, mixed column strategy q, and pure row i, \\[ pureVsPayoff(…","labels":["bestRow_spec"],"detail_key":"p14"},{"id":"n21012","layer":"informal","project":"p14","title":"Weak duality","kind":"theorem","summary":"[Weak duality] For any game G and any mixed strategies p (row) and q (column), \\[ \\inf_j \\in Fi…","labels":["weak_duality"],"detail_key":"p14"},{"id":"n21013","layer":"informal","project":"p14","title":"Loss sequence induced by a game","kind":"definition","summary":"[Loss sequence induced by a game] Given a game G and a sequence of column responses j_0, \\dots,…","labels":["ZeroSumGame.toLossSeq"],"detail_key":"p14"},{"id":"n21014","layer":"informal","project":"p14","title":"Induced loss sequence is valid","kind":"lemma","summary":"[Induced loss sequence is valid] The loss sequence G.toLossSeq(colResponse) is valid, i.e.\\ eve…","labels":["ZeroSumGame.toLossSeq_valid"],"detail_key":"p14"},{"id":"n21015","layer":"informal","project":"p14","title":"Average strategy","kind":"definition","summary":"[Average strategy] Given T distributions p_0, \\dots, p_T-1 over Fin\\,n (each presented as non-n…","labels":["averageStrategy"],"detail_key":"p14"},{"id":"n21016","layer":"informal","project":"p14","title":"Hedge mixed strategy","kind":"definition","summary":"[Hedge mixed strategy] The Hedge distribution at round t for learning rate \\eta and loss sequen…","labels":["hedgeMixedStrategy"],"detail_key":"p14"},{"id":"n21017","layer":"informal","project":"p14","title":"Empirical strategy","kind":"definition","summary":"[Empirical strategy] Given T > 0 pure actions a_0, \\dots, a_T-1 \\in Fin\\,n, the \\emphempirical…","labels":["empiricalStrategy"],"detail_key":"p14"},{"id":"n21018","layer":"informal","project":"p14","title":"Payoff under average strategy","kind":"lemma","summary":"[Payoff under average strategy] For a game G, a column j, and T row distributions p_0, \\dots, p…","labels":["payoffVsPure_averageStrategy"],"detail_key":"p14"},{"id":"n21019","layer":"informal","project":"p14","title":"Payoff of pure row under empirical column","kind":"lemma","summary":"[Payoff of pure row under empirical column] For a game G, a pure row i, and a sequence of pure…","labels":["pureVsPayoff_empiricalStrategy"],"detail_key":"p14"},{"id":"n21020","layer":"informal","project":"p14","title":"Prefix cumulative game loss","kind":"definition","summary":"[Prefix cumulative game loss] Given a game G, a prefix of t column actions, and a row i, the \\e…","labels":["prefixGameLoss"],"detail_key":"p14"},{"id":"n21021","layer":"informal","project":"p14","title":"Prefix Hedge weight","kind":"definition","summary":"[Prefix Hedge weight] The unnormalised Hedge weight of row i after seeing a prefix of column ac…","labels":["prefixHedgeWeight"],"detail_key":"p14"},{"id":"n21022","layer":"informal","project":"p14","title":"Prefix potential","kind":"definition","summary":"[Prefix potential] The \\emphprefix potential is the sum of prefix Hedge weights over all rows:…","labels":["prefixPotential"],"detail_key":"p14"},{"id":"n21023","layer":"informal","project":"p14","title":"Prefix Hedge weight is positive","kind":"lemma","summary":"[Prefix Hedge weight is positive] For any game G, learning rate \\eta, prefix of column actions,…","labels":["prefixHedgeWeight_pos"],"detail_key":"p14"},{"id":"n21024","layer":"informal","project":"p14","title":"Prefix potential is positive","kind":"lemma","summary":"[Prefix potential is positive] When M \\ge 1, the prefix potential \\Phi_t > 0, since it is a non…","labels":["prefixPotential_pos"],"detail_key":"p14"},{"id":"n21025","layer":"informal","project":"p14","title":"Prefix Hedge mixed strategy","kind":"definition","summary":"[Prefix Hedge mixed strategy] The normalised Hedge distribution after a prefix of t column acti…","labels":["prefixHedgeMixedStrategy"],"detail_key":"p14"},{"id":"n21026","layer":"informal","project":"p14","title":"Online column best responses to Hedge","kind":"definition","summary":"[Online column best responses to Hedge] The sequence of column best responses generated online…","labels":["hedgeResponseNat"],"detail_key":"p14"},{"id":"n21027","layer":"informal","project":"p14","title":"Prefix loss equals cumulative loss","kind":"lemma","summary":"[Prefix loss equals cumulative loss] For any fixed column-action sequence a : Fin\\,T \\to Fin\\,N…","labels":["prefixGameLoss_eq_cumLoss"],"detail_key":"p14"},{"id":"n21028","layer":"informal","project":"p14","title":"Prefix weight equals Hedge weight","kind":"lemma","summary":"[Prefix weight equals Hedge weight] The prefix Hedge weight of row i at round t equals the abst…","labels":["prefixHedgeWeight_eq_hedgeWeight"],"detail_key":"p14"},{"id":"n21029","layer":"informal","project":"p14","title":"Prefix potential equals Hedge potential","kind":"lemma","summary":"[Prefix potential equals Hedge potential] The prefix potential at round t equals the abstract H…","labels":["prefixPotential_eq_potential"],"detail_key":"p14"},{"id":"n21030","layer":"informal","project":"p14","title":"Prefix strategy weight equals Hedge distribution","kind":"lemma","summary":"[Prefix strategy weight equals Hedge distribution] The weight of row i in the prefix Hedge mixe…","labels":["prefixHedgeMixedStrategy_weight_eq_hedgeDist"],"detail_key":"p14"},{"id":"n21031","layer":"informal","project":"p14","title":"Regret-to-payoff bridge","kind":"lemma","summary":"[Regret-to-payoff bridge] Suppose the row player uses per-round distributions p_0, \\dots, p_T-1…","labels":["regret_to_payoff"],"detail_key":"p14"},{"id":"n21032","layer":"informal","project":"p14","title":"Average of per-round minima is at most minimum of averages","kind":"lemma","summary":"[Average of per-round minima is at most minimum of averages] For any f : Fin\\,T \\to Fin\\,N \\to…","labels":["avg_min_le_min_avg"],"detail_key":"p14"},{"id":"n21033","layer":"informal","project":"p14","title":"Hedge construction","kind":"lemma","summary":"[Hedge construction] Given a game G with M \\ge 2 rows, a number of rounds T > 0, and a learning…","labels":["hedge_construction"],"detail_key":"p14"},{"id":"n21034","layer":"informal","project":"p14","title":"Approximate minimax","kind":"theorem","summary":"[Approximate minimax] For any finite zero-sum game G with M \\ge 2 rows and any \\varepsilon > 0,…","labels":["approx_minimax"],"detail_key":"p14"},{"id":"n21035","layer":"informal","project":"p14","title":"Approximate minimax theorem","kind":"theorem","summary":"[Approximate minimax theorem] For any finite zero-sum game G with at least 2 row actions and an…","labels":["minimax_approx_theorem"],"detail_key":"p14"},{"id":"n21036","layer":"informal","project":"p14","title":"Weight shrinkage after M mistakes","kind":"theorem","summary":"[Weight shrinkage after M mistakes] Let \\beta \\in (0,1), let W : N \\to R be a sequence of posit…","labels":["wm_weight_shrinkage"],"detail_key":"p14"},{"id":"n21037","layer":"informal","project":"p14","title":"Total weight upper bound","kind":"theorem","summary":"[Total weight upper bound] Under the same shrinkage hypothesis as \\textttwm\\_weight\\_shrinkage,…","labels":["wm_weight_upper_bound"],"detail_key":"p14"},{"id":"n21038","layer":"informal","project":"p14","title":"Combined potential bound","kind":"theorem","summary":"[Combined potential bound] Let \\beta \\in (0,1), n \\ge 1, and suppose \\beta^M^* \\le n \\cdot \\big…","labels":["wm_combined_potential_bound"],"detail_key":"p14"},{"id":"n21039","layer":"informal","project":"p14","title":"Logarithmic mistake bound","kind":"theorem","summary":"[Logarithmic mistake bound] Let n \\ge 2, \\beta \\in (0,1), and suppose \\beta^M^* \\le n \\cdot \\bi…","labels":["wm_mistake_bound_log"],"detail_key":"p14"},{"id":"n21040","layer":"informal","project":"p14","title":"Factor-2 lower bound for deterministic algorithms","kind":"theorem","summary":"[Factor-2 lower bound for deterministic algorithms] For any T \\in N, there exist M_alg, M^* \\in…","labels":["deterministic_factor2_lower_bound"],"detail_key":"p14"},{"id":"n21041","layer":"informal","project":"p14","title":"Maximum size of a binary code with given distance","kind":"definition","summary":"[Maximum size of a binary code with given distance] For naturals n and d, A(n,d) is the supremu…","labels":["codeA"],"detail_key":"p14"},{"id":"n21042","layer":"informal","project":"p14","title":"Binary entropy in base 2","kind":"definition","summary":"[Binary entropy in base 2] The binary entropy function \\[ H(x) \\;=\\; -x\\log_2 x - (1-x)\\log_2(1…","labels":["binaryEntropy"],"detail_key":"p14"},{"id":"n21043","layer":"informal","project":"p14","title":"Asymptotic binary code rate","kind":"definition","summary":"[Asymptotic binary code rate] For \\delta \\in [0,1], the asymptotic rate is \\[ R(\\delta) \\;=\\; \\…","labels":["rate"],"detail_key":"p14"},{"id":"n21044","layer":"informal","project":"p14","title":"Binary Krawtchouk polynomial at integer arguments","kind":"definition","summary":"[Binary Krawtchouk polynomial at integer arguments] For naturals n, j, x, \\[ K_j^(n)(x) \\;=\\; \\…","labels":["krawtchouk"],"detail_key":"p14"},{"id":"n21045","layer":"informal","project":"p14","title":"Krawtchouk polynomials over R","kind":"definition","summary":"[Krawtchouk polynomials over R] The family of real polynomials defined by the three-term recurr…","labels":["krawtchoukPoly"],"detail_key":"p14"},{"id":"n21046","layer":"informal","project":"p14","title":"Value of the zeroth Krawtchouk polynomial","kind":"theorem","summary":"[Value of the zeroth Krawtchouk polynomial] For all n and x, K_0^(n)(x) = 1.","labels":["krawtchouk_zero"],"detail_key":"p14"},{"id":"n21047","layer":"informal","project":"p14","title":"Krawtchouk polynomial at x = 0","kind":"theorem","summary":"[Krawtchouk polynomial at x = 0] For j \\le n, K_j^(n)(0) = \\binomnj.","labels":["krawtchouk_eval_zero"],"detail_key":"p14"},{"id":"n21048","layer":"informal","project":"p14","title":"First Krawtchouk polynomial","kind":"theorem","summary":"[First Krawtchouk polynomial] For x \\le n, K_1^(n)(x) = n - 2x.","labels":["krawtchouk_one"],"detail_key":"p14"},{"id":"n21049","layer":"informal","project":"p14","title":"Krawtchouk generating function","kind":"theorem","summary":"[Krawtchouk generating function] For x \\le n and every real z, \\[ \\sum_j=0^n K_j^(n)(x)\\, z^j \\…","labels":["krawtchouk_generating_function"],"detail_key":"p14"},{"id":"n21050","layer":"informal","project":"p14","title":"Orthogonality of Krawtchouk polynomials","kind":"theorem","summary":"[Orthogonality of Krawtchouk polynomials] For r \\le n and s \\le n, \\[ \\sum_x=0^n \\binomnx K_r^(…","labels":["krawtchouk_orthogonality"],"detail_key":"p14"},{"id":"n21051","layer":"informal","project":"p14","title":"Three-term recurrence for Krawtchouk polynomials","kind":"theorem","summary":"[Three-term recurrence for Krawtchouk polynomials] For 1 \\le j, j + 1 \\le n and x \\le n, \\[ (j+…","labels":["krawtchouk_recurrence"],"detail_key":"p14"},{"id":"n21052","layer":"informal","project":"p14","title":"Agreement of the sum formula and the polynomial form","kind":"theorem","summary":"[Agreement of the sum formula and the polynomial form] For j \\le n and x \\le n, evaluating the…","labels":["krawtchoukPoly_eval_nat"],"detail_key":"p14"},{"id":"n21053","layer":"informal","project":"p14","title":"Delsarte LP inequality","kind":"theorem","summary":"[Delsarte LP inequality] Let F(x) = \\sum_j=0^n F_j K_j^(n)(x) for real coefficients F_j. If F_0…","labels":["delsarte_lp_bound"],"detail_key":"p14"},{"id":"n21054","layer":"informal","project":"p14","title":"Truncated Christoffel--Darboux kernel","kind":"definition","summary":"[Truncated Christoffel--Darboux kernel] For t \\le n and real a, x, the truncated reproducing ke…","labels":["cdKernel"],"detail_key":"p14"},{"id":"n21055","layer":"informal","project":"p14","title":"Christoffel--Darboux identity for Krawtchouk polynomials","kind":"theorem","summary":"[Christoffel--Darboux identity for Krawtchouk polynomials] For t + 1 \\le n and real a \\ne x the…","labels":["cd_identity"],"detail_key":"p14"},{"id":"n21056","layer":"informal","project":"p14","title":"Positivity of Krawtchouk values below all zeros","kind":"theorem","summary":"[Positivity of Krawtchouk values below all zeros] Fix t + 1 \\le n and a real a that is strictly…","labels":["krawtchouk_positive_below_smallest_zero"],"detail_key":"p14"},{"id":"n21057","layer":"informal","project":"p14","title":"Nonpositivity of the kernel beyond a threshold","kind":"theorem","summary":"[Nonpositivity of the kernel beyond a threshold] Under the same hypotheses (t + 1 \\le n and a s…","labels":["cdKernel_nonpos_beyond_threshold"],"detail_key":"p14"},{"id":"n21058","layer":"informal","project":"p14","title":"Finite-n MRRW bound","kind":"theorem","summary":"[Finite-n MRRW bound] Let t + 1 \\le n, let a be strictly below every real zero of K_t^(n), and…","labels":["finite_n_mrrw_bound"],"detail_key":"p14"},{"id":"n21059","layer":"informal","project":"p14","title":"Asymptotics of the smallest Krawtchouk zero","kind":"theorem","summary":"[Asymptotics of the smallest Krawtchouk zero] Let 0 < \\tau < 1/2 and let t_n satisfy t_n / n \\t…","labels":["smallest_zero_asymptotic"],"detail_key":"p14"},{"id":"n21060","layer":"informal","project":"p14","title":"Entropy growth rate of the objective","kind":"theorem","summary":"[Entropy growth rate of the objective] Let 0 < \\tau < 1/2, let t_n/n \\to \\tau, and let a_n even…","labels":["entropy_growth_of_objective"],"detail_key":"p14"},{"id":"n21061","layer":"informal","project":"p14","title":"Entropy asymptotics for binomial tails","kind":"theorem","summary":"[Entropy asymptotics for binomial tails] For 0 \\le \\tau \\le 1/2, \\[ \\frac1n\\log_2\\!\\left(\\sum_j…","labels":["binomial_tail_entropy_asymptotic"],"detail_key":"p14"},{"id":"n21062","layer":"informal","project":"p14","title":"Involution between \\delta and \\tau","kind":"theorem","summary":"[Involution between \\delta and \\tau] For 0 \\le \\delta \\le 1/2, setting \\tau = \\tfrac12 - \\sqrt\\…","labels":["delta_tau_involution"],"detail_key":"p14"},{"id":"n21063","layer":"informal","project":"p14","title":"Binary entropy at 0","kind":"theorem","summary":"[Binary entropy at 0] H(0) = 0.","labels":["binaryEntropy_zero"],"detail_key":"p14"},{"id":"n21064","layer":"informal","project":"p14","title":"Binary entropy at 1/2","kind":"theorem","summary":"[Binary entropy at 1/2] H(1/2) = 1.","labels":["binaryEntropy_half"],"detail_key":"p14"},{"id":"n21065","layer":"informal","project":"p14","title":"Binary first MRRW bound","kind":"theorem","summary":"[Binary first MRRW bound] For every \\delta with 0 \\le \\delta \\le 1/2, \\[ R(\\delta) \\;\\le\\; H\\!\\…","labels":["mrrw_bound"],"detail_key":"p14"},{"id":"n21066","layer":"informal","project":"p14","title":"Reduction to normal form, parameter shell","kind":"theorem","summary":"[Reduction to normal form, parameter shell] Fact 3.6 (Yekhanin, Lemma 6.2), in the existential…","labels":["normal_form_reduction"],"detail_key":"p14"},{"id":"n21067","layer":"informal","project":"p14","title":"Rectangular matrix Khintchine bound","kind":"theorem","summary":"[Rectangular matrix Khintchine bound] Fact 4.1 (Tropp 2015, Theorem 4.1.1), stated abstractly t…","labels":["matrix_khintchine"],"detail_key":"p14"},{"id":"n21068","layer":"informal","project":"p14","title":"Real-valued binomial coefficient","kind":"definition","summary":"[Real-valued binomial coefficient] For naturals n and k, \\textttchooseR\\,n\\,k is the binomial c…","labels":["chooseR"],"detail_key":"p14"},{"id":"n21069","layer":"informal","project":"p14","title":"Binomial ratio bound, upper","kind":"theorem","summary":"[Binomial ratio bound, upper] Fact 4.2 (upper half). Let n, \\ell, q be naturals with n > 0, q >…","labels":["binomial_ratio_upper"],"detail_key":"p14"},{"id":"n21070","layer":"informal","project":"p14","title":"Binomial ratio bound, lower","kind":"theorem","summary":"[Binomial ratio bound, lower] Fact 4.2 (lower half). Under the same hypotheses n > 0, q > 0, q…","labels":["binomial_ratio_lower"],"detail_key":"p14"},{"id":"n21071","layer":"informal","project":"p14","title":"Heavy pairs of a hypergraph","kind":"definition","summary":"[Heavy pairs of a hypergraph] For a hypergraph H on Fin\\,n and a threshold d, the set of heavy…","labels":["heavyPairs"],"detail_key":"p14"},{"id":"n21072","layer":"informal","project":"p14","title":"Counting heavy pairs","kind":"theorem","summary":"[Counting heavy pairs] Let H be a 3-uniform hypergraph on Fin\\,n and let d > 0. Then the number…","labels":["heavy_pair_count_bound"],"detail_key":"p14"},{"id":"n21073","layer":"informal","project":"p14","title":"Hypergraph decomposition","kind":"theorem","summary":"[Hypergraph decomposition] Let L be a normal form locally decodable code with message length k…","labels":["hypergraph_decomposition"],"detail_key":"p14"},{"id":"n21074","layer":"informal","project":"p14","title":"Hypergraph on Fin\\,n","kind":"definition","summary":"[Hypergraph on Fin\\,n] A hypergraph on the vertex set Fin\\,n is a finite collection of subsets…","labels":["Hypergraph"],"detail_key":"p14"},{"id":"n21075","layer":"informal","project":"p14","title":"q-uniform hypergraph","kind":"definition","summary":"[q-uniform hypergraph] A hypergraph H on Fin\\,n is q-uniform when every edge C \\in H has exactl…","labels":["Hypergraph.IsUniform"],"detail_key":"p14"},{"id":"n21076","layer":"informal","project":"p14","title":"Matching","kind":"definition","summary":"[Matching] A hypergraph H is a matching when its edges are pairwise disjoint: for all C, C' \\in…","labels":["Hypergraph.IsMatching"],"detail_key":"p14"},{"id":"n21077","layer":"informal","project":"p14","title":"Degree of a set in a hypergraph","kind":"definition","summary":"[Degree of a set in a hypergraph] The degree of a vertex set Q \\subseteq Fin\\,n in a hypergraph…","labels":["Hypergraph.degree"],"detail_key":"p14"},{"id":"n21078","layer":"informal","project":"p14","title":"Pair-degree bound","kind":"definition","summary":"[Pair-degree bound] A hypergraph H satisfies the pair-degree bound d when every pair of distinc…","labels":["Hypergraph.pairDegreeBound"],"detail_key":"p14"},{"id":"n21079","layer":"informal","project":"p14","title":"Sign values \\pm 1","kind":"definition","summary":"[Sign values \\pm 1] The predicate on an integer x stating that x is a sign value, i.e.\\ x = 1 o…","labels":["IsPMOne"],"detail_key":"p14"},{"id":"n21080","layer":"informal","project":"p14","title":"Normal form locally decodable code","kind":"definition","summary":"[Normal form locally decodable code] The structural data of a (3,\\delta,\\varepsilon)-normally d…","labels":["NormalLDC"],"detail_key":"p14"},{"id":"n21081","layer":"informal","project":"p14","title":"Combined hypergraph of an LDC","kind":"definition","summary":"[Combined hypergraph of an LDC] For a normal form code L, the hypergraph obtained as the union…","labels":["NormalLDC.combined"],"detail_key":"p14"},{"id":"n21082","layer":"informal","project":"p14","title":"Total number of constraints","kind":"definition","summary":"[Total number of constraints] For a normal form code L, the total number of constraints \\[ m \\;…","labels":["NormalLDC.totalConstraints"],"detail_key":"p14"},{"id":"n21083","layer":"informal","project":"p14","title":"Monomial of an assignment on a set","kind":"definition","summary":"[Monomial of an assignment on a set] For an assignment x : Fin\\,n \\to Z and a set C \\subseteq F…","labels":["assignment_prod"],"detail_key":"p14"},{"id":"n21084","layer":"informal","project":"p14","title":"XOR polynomial of a message","kind":"definition","summary":"[XOR polynomial of a message] For a normal form code L, a message b : Fin\\,k \\to Z and an assig…","labels":["NormalLDC.xorPoly"],"detail_key":"p14"},{"id":"n21085","layer":"informal","project":"p14","title":"Value of the XOR instance","kind":"definition","summary":"[Value of the XOR instance] The value val(\\psi_b) of the XOR instance associated with a message…","labels":["NormalLDC.xorVal"],"detail_key":"p14"},{"id":"n21086","layer":"informal","project":"p14","title":"Double-copy ground set","kind":"definition","summary":"[Double-copy ground set] The doubled vertex set Fin\\,n \\times Fin\\,2, carrying two labelled cop…","labels":["DoubleCopy"],"detail_key":"p14"},{"id":"n21087","layer":"informal","project":"p14","title":"First copy of a vertex","kind":"definition","summary":"[First copy of a vertex] The map sending a vertex u : Fin\\,n to its first copy u^(1) = (u,0) in…","labels":["copy1"],"detail_key":"p14"},{"id":"n21088","layer":"informal","project":"p14","title":"Second copy of a vertex","kind":"definition","summary":"[Second copy of a vertex] The map sending a vertex u : Fin\\,n to its second copy u^(2) = (u,1)…","labels":["copy2"],"detail_key":"p14"},{"id":"n21089","layer":"informal","project":"p14","title":"First-copy embedding of a set","kind":"definition","summary":"[First-copy embedding of a set] The image C^(1) of a set C \\subseteq Fin\\,n under the injection…","labels":["liftCopy1"],"detail_key":"p14"},{"id":"n21090","layer":"informal","project":"p14","title":"Second-copy embedding of a set","kind":"definition","summary":"[Second-copy embedding of a set] The image C^(2) of a set C \\subseteq Fin\\,n under the injectio…","labels":["liftCopy2"],"detail_key":"p14"},{"id":"n21091","layer":"informal","project":"p14","title":"High-value observation for a normal form LDC","kind":"theorem","summary":"[High-value observation for a normal form LDC] Let L be a normal form (3,\\delta,\\varepsilon)-de…","labels":["high_value_observation"],"detail_key":"p14"},{"id":"n21092","layer":"informal","project":"p14","title":"Near-cubic lower bound for 3-query LDCs","kind":"theorem","summary":"[Near-cubic lower bound for 3-query LDCs] Let k > 0, n \\ge 2, and let \\delta, \\varepsilon \\in (…","labels":["main_theorem_ldc_lower_bound"],"detail_key":"p14"},{"id":"n21093","layer":"informal","project":"p14","title":"Lower bound for q-query LDCs with q even","kind":"theorem","summary":"[Lower bound for q-query LDCs with q even] Let q be even with q \\ge 2, let k > 0 and n \\ge 2, a…","labels":["even_q_ldc_lower_bound"],"detail_key":"p14"},{"id":"n21094","layer":"informal","project":"p14","title":"Cauchy--Schwarz trick for the 3-XOR value","kind":"theorem","summary":"[Cauchy--Schwarz trick for the 3-XOR value] For natural numbers n, m > 0 and reals val_f, val_f…","labels":["cauchy_schwarz_trick"],"detail_key":"p14"},{"id":"n21095","layer":"informal","project":"p14","title":"Nonzero entry count for the Kikuchi matrix","kind":"theorem","summary":"[Nonzero entry count for the Kikuchi matrix] For naturals n \\ge 4, k > 0 and \\ell \\ge 2 with 2\\…","labels":["nonzero_entry_count"],"detail_key":"p14"},{"id":"n21096","layer":"informal","project":"p14","title":"Row sparsity of the Kikuchi matrices","kind":"theorem","summary":"[Row sparsity of the Kikuchi matrices] For naturals d and row\\_nnz, if the number of nonzero en…","labels":["row_bound_le_two_d"],"detail_key":"p14"},{"id":"n21097","layer":"informal","project":"p14","title":"Spectral certificate for the derived 4-XOR value","kind":"theorem","summary":"[Spectral certificate for the derived 4-XOR value] For reals val_f_L,R, N, D and \\lVert A\\rVert…","labels":["spectral_certificate_bound"],"detail_key":"p14"},{"id":"n21098","layer":"informal","project":"p14","title":"Positivity of the spectral norm bound","kind":"theorem","summary":"[Positivity of the spectral norm bound] For naturals k, d, \\ell > 0 and n \\ge 2 there exists a…","labels":["spectral_norm_bound"],"detail_key":"p14"},{"id":"n21099","layer":"informal","project":"p14","title":"3-XOR refutation bound","kind":"theorem","summary":"[3-XOR refutation bound] For naturals k, d > 0, n \\ge 2 and m \\le nk there exists a real consta…","labels":["three_xor_refutation"],"detail_key":"p14"},{"id":"n21100","layer":"informal","project":"p14","title":"Value of the 2-XOR polynomial","kind":"definition","summary":"[Value of the 2-XOR polynomial] For parameters k, n, d and a message b : Fin\\,k \\to Z, the mode…","labels":["twoXORVal"],"detail_key":"p14"},{"id":"n21101","layer":"informal","project":"p14","title":"2-XOR refutation bound","kind":"theorem","summary":"[2-XOR refutation bound] Let k, n, d be natural numbers with 0 < k, 2 \\le n and 0 < d. Then the…","labels":["two_xor_refutation"],"detail_key":"p14"},{"id":"n21102","layer":"informal","project":"p14","title":"Noise kernel of \\rho-correlated pairs","kind":"definition","summary":"[Noise kernel of \\rho-correlated pairs] For \\rho \\in R and x, y in the cube BoolCube\\,n, the no…","labels":["GeneralHypercontractivity.noiseKernel"],"detail_key":"p14"},{"id":"n21103","layer":"informal","project":"p14","title":"Nonnegativity of the noise kernel","kind":"lemma","summary":"[Nonnegativity of the noise kernel] If 0 \\le \\rho \\le 1 then K_\\rho(x,y) \\ge 0 for all x, y \\in…","labels":["GeneralHypercontractivity.noiseKernel_nonneg"],"detail_key":"p14"},{"id":"n21104","layer":"informal","project":"p14","title":"Fourier form of the kernel product","kind":"lemma","summary":"[Fourier form of the kernel product] For all \\rho \\in R and x, y \\in BoolCube\\,n, \\[ \\sum_S \\su…","labels":["GeneralHypercontractivity.sum_fourier_kernel"],"detail_key":"p14"},{"id":"n21105","layer":"informal","project":"p14","title":"Noise operator as a kernel sum","kind":"lemma","summary":"[Noise operator as a kernel sum] For every g : BoolCube\\,n \\to R and every x, \\[ (T_\\rho g)(x)…","labels":["GeneralHypercontractivity.noiseOp_eq_kernel_sum"],"detail_key":"p14"},{"id":"n21106","layer":"informal","project":"p14","title":"Inner product as a kernel-weighted double sum","kind":"lemma","summary":"[Inner product as a kernel-weighted double sum] For all f, g : BoolCube\\,n \\to R, \\[ \\left\\lang…","labels":["GeneralHypercontractivity.innerProduct_noiseOp_eq_weighted_sum"],"detail_key":"p14"},{"id":"n21107","layer":"informal","project":"p14","title":"Monotonicity of kernel-weighted expectations","kind":"lemma","summary":"[Monotonicity of kernel-weighted expectations] Let 0 \\le \\rho \\le 1 and let h, h' : BoolCube\\,n…","labels":["GeneralHypercontractivity.corrExpect_mono"],"detail_key":"p14"},{"id":"n21108","layer":"informal","project":"p14","title":"Factorization of the kernel along the last coordinate","kind":"lemma","summary":"[Factorization of the kernel along the last coordinate] For x', y' \\in BoolCube\\,n and bits b,…","labels":["GeneralHypercontractivity.noiseKernel_snoc"],"detail_key":"p14"},{"id":"n21109","layer":"informal","project":"p14","title":"Expectation on n+1 bits as an iterated expectation","kind":"lemma","summary":"[Expectation on n+1 bits as an iterated expectation] For h : BoolCube\\,(n+1) \\to R, \\[ E[h] \\;=…","labels":["GeneralHypercontractivity.expect_succ_eq_iterated"],"detail_key":"p14"},{"id":"n21110","layer":"informal","project":"p14","title":"Collapse of the p-th moment along the last bit","kind":"lemma","summary":"[Collapse of the p-th moment along the last bit] For p > 0 and f : BoolCube\\,(n+1) \\to R, \\[ E\\…","labels":["GeneralHypercontractivity.norm_collapse_rpow"],"detail_key":"p14"},{"id":"n21111","layer":"informal","project":"p14","title":"Decomposition of the kernel-weighted sum at dimension n+1","kind":"lemma","summary":"[Decomposition of the kernel-weighted sum at dimension n+1] For F : BoolCube\\,(n+1) \\times Bool…","labels":["GeneralHypercontractivity.weighted_sum_succ_decomp"],"detail_key":"p14"},{"id":"n21112","layer":"informal","project":"p14","title":"One-bit slice sum as a one-bit inner product","kind":"lemma","summary":"[One-bit slice sum as a one-bit inner product] Fix x', y' \\in BoolCube\\,n and f, g : BoolCube\\,…","labels":["GeneralHypercontractivity.one_bit_slice_eq_innerProduct"],"detail_key":"p14"},{"id":"n21113","layer":"informal","project":"p14","title":"L^p norm of a one-bit slice","kind":"lemma","summary":"[L^p norm of a one-bit slice] For p > 0, f : BoolCube\\,(n+1) \\to R and x' \\in BoolCube\\,n, \\[ \\…","labels":["GeneralHypercontractivity.one_bit_norm_slice"],"detail_key":"p14"},{"id":"n21114","layer":"informal","project":"p14","title":"Norm collapse, clean form","kind":"lemma","summary":"[Norm collapse, clean form] For p \\ge 1 and f : BoolCube\\,(n+1) \\to R, \\[ E_x'\\Bigl[\\tfrac\\left…","labels":["GeneralHypercontractivity.norm_collapse_clean"],"detail_key":"p14"},{"id":"n21115","layer":"informal","project":"p14","title":"H\\\"older inequality for Boolean functions","kind":"lemma","summary":"[H\\\"older inequality for Boolean functions] For p > 1 and f, h : BoolCube\\,n \\to R, \\[ \\left\\la…","labels":["GeneralHypercontractivity.holder_ineq_bool"],"detail_key":"p14"},{"id":"n21116","layer":"informal","project":"p14","title":"L^q contractivity of the noise operator on one bit","kind":"lemma","summary":"[L^q contractivity of the noise operator on one bit] For q \\ge 1, 0 \\le \\rho \\le 1 and g : Bool…","labels":["GeneralHypercontractivity.noise_Lp_contraction_one_bit"],"detail_key":"p14"},{"id":"n21117","layer":"informal","project":"p14","title":"Two-function hypercontractivity in dimension zero","kind":"lemma","summary":"[Two-function hypercontractivity in dimension zero] For p, q \\ge 1, any \\rho \\in R, and f, g :…","labels":["GeneralHypercontractivity.two_func_hyp_zero"],"detail_key":"p14"},{"id":"n21118","layer":"informal","project":"p14","title":"Inductive step for two-function hypercontractivity","kind":"lemma","summary":"[Inductive step for two-function hypercontractivity] Let p, q \\ge 1 and 0 \\le \\rho \\le 1. If th…","labels":["GeneralHypercontractivity.two_func_hyp_succ"],"detail_key":"p14"},{"id":"n21119","layer":"informal","project":"p14","title":"Two-function hypercontractivity induction theorem","kind":"theorem","summary":"[Two-function hypercontractivity induction theorem] Let p, q \\ge 1 and 0 \\le \\rho \\le 1. If \\le…","labels":["GeneralHypercontractivity.hypercontractivity_induction"],"detail_key":"p14"},{"id":"n21120","layer":"informal","project":"p14","title":"Weak two-function hypercontractivity on a single bit","kind":"theorem","summary":"[Weak two-function hypercontractivity on a single bit] For 1 \\le p \\le 2, 1 \\le q \\le 2 and f,…","labels":["GeneralHypercontractivity.weak_two_function_hypercontractivity_one_bit"],"detail_key":"p14"},{"id":"n21121","layer":"informal","project":"p14","title":"Equivalence of one- and two-function hypercontractivity","kind":"theorem","summary":"[Equivalence of one- and two-function hypercontractivity] Let 1 \\le p \\le q with q \\ge 2, and l…","labels":["GeneralHypercontractivity.one_function_iff_two_function_hypercontractivity"],"detail_key":"p14"},{"id":"n21122","layer":"informal","project":"p14","title":"Weak two-function hypercontractivity","kind":"theorem","summary":"[Weak two-function hypercontractivity] For 1 \\le p \\le 2, 1 \\le q \\le 2, any n, and f, g : Bool…","labels":["GeneralHypercontractivity.weak_two_function_hypercontractivity"],"detail_key":"p14"},{"id":"n21123","layer":"informal","project":"p14","title":"Kernel rows sum to one","kind":"lemma","summary":"[Kernel rows sum to one] For 0 \\le \\rho \\le 1 and any fixed x \\in BoolCube\\,n, \\sum_y K_\\rho(x,…","labels":["GeneralHypercontractivity.noiseKernel_sum_right"],"detail_key":"p14"},{"id":"n21124","layer":"informal","project":"p14","title":"Kernel columns sum to one","kind":"lemma","summary":"[Kernel columns sum to one] For 0 \\le \\rho \\le 1 and any fixed y \\in BoolCube\\,n, \\sum_x K_\\rho…","labels":["GeneralHypercontractivity.noiseKernel_sum_left"],"detail_key":"p14"},{"id":"n21125","layer":"informal","project":"p14","title":"Jensen bound for powers of the noise operator","kind":"lemma","summary":"[Jensen bound for powers of the noise operator] For 0 \\le \\rho \\le 1, s \\ge 1, f : BoolCube\\,n…","labels":["GeneralHypercontractivity.noiseOp_abs_rpow_le_kernel_avg"],"detail_key":"p14"},{"id":"n21126","layer":"informal","project":"p14","title":"Trivial contractivity of the noise operator","kind":"lemma","summary":"[Trivial contractivity of the noise operator] For s \\ge 1, 0 \\le \\rho \\le 1 and f : BoolCube\\,n…","labels":["GeneralHypercontractivity.trivial_contractivity"],"detail_key":"p14"},{"id":"n21127","layer":"informal","project":"p14","title":"Duality of noise-operator norm bounds","kind":"lemma","summary":"[Duality of noise-operator norm bounds] For p, q > 1 and 0 \\le \\rho \\le 1, the (p \\to q) bound…","labels":["GeneralHypercontractivity.noise_op_norm_dual"],"detail_key":"p14"},{"id":"n21128","layer":"informal","project":"p14","title":"Square root of a ratio at most one","kind":"lemma","summary":"[Square root of a ratio at most one] If 0 \\le a, 0 < b and a \\le b, then \\sqrta/b \\le 1.","labels":["GeneralHypercontractivity.sqrt_div_le_one"],"detail_key":"p14"},{"id":"n21129","layer":"informal","project":"p14","title":"Noise-parameter identity","kind":"lemma","summary":"[Noise-parameter identity] If u - 1 > 0, then \\[ \\Bigl(\\fracuu-1 - 1\\Bigr)(p-1) \\;=\\; \\fracp-1u…","labels":["GeneralHypercontractivity.noise_param_eq"],"detail_key":"p14"},{"id":"n21130","layer":"informal","project":"p14","title":"Bridging case of one-function hypercontractivity","kind":"theorem","summary":"[Bridging case of one-function hypercontractivity] For 1 \\le p \\le 2 \\le u, f : BoolCube\\,n \\to…","labels":["GeneralHypercontractivity.bridging_hypercontractivity"],"detail_key":"p14"},{"id":"n21131","layer":"informal","project":"p14","title":"Interpolation parameters for the low-norms case","kind":"lemma","summary":"[Interpolation parameters for the low-norms case] Let 1 < p < u < 2 and \\rho^2 = (p-1)/(u-1). T…","labels":["GeneralHypercontractivity.low_norms_interpolation_params"],"detail_key":"p14"},{"id":"n21132","layer":"informal","project":"p14","title":"Symmetric power average is at least one","kind":"lemma","summary":"[Symmetric power average is at least one] For p \\ge 1 and b \\in [0,1], \\[ 1 \\;\\le\\; \\frac(1+b)^…","labels":["GeneralHypercontractivity.avg_rpow_ge_one"],"detail_key":"p14"},{"id":"n21133","layer":"informal","project":"p14","title":"Monotonicity of symmetric convex sums","kind":"lemma","summary":"[Monotonicity of symmetric convex sums] If f is convex on [0,\\infty) and 0 \\le x \\le y \\le 1, t…","labels":["GeneralHypercontractivity.convex_sym_sum_mono"],"detail_key":"p14"},{"id":"n21134","layer":"informal","project":"p14","title":"Antitonicity in the exponent for nonpositive powers","kind":"lemma","summary":"[Antitonicity in the exponent for nonpositive powers] For 0 < x < 1 and exponents p \\le q \\le 0…","labels":["GeneralHypercontractivity.rpow_sum_antitone_exponent"],"detail_key":"p14"},{"id":"n21135","layer":"informal","project":"p14","title":"Two-point derivative inequality","kind":"lemma","summary":"[Two-point derivative inequality] Let 0 \\le r \\le s \\le 1, c = \\sqrtr/s and t \\in [0,1]. Then \\…","labels":["GeneralHypercontractivity.h_alpha_ineq"],"detail_key":"p14"},{"id":"n21136","layer":"informal","project":"p14","title":"Integrated two-point inequality","kind":"lemma","summary":"[Integrated two-point inequality] Let 1 \\le p \\le q \\le 2, b \\in [0,1] and \\rho = \\sqrt(p-1)/(q…","labels":["GeneralHypercontractivity.integrated_h_alpha_ineq"],"detail_key":"p14"},{"id":"n21137","layer":"informal","project":"p14","title":"Tangent line inequality for x^r at x = 1","kind":"lemma","summary":"[Tangent line inequality for x^r at x = 1] For x \\ge 0 and r \\ge 1, x^r \\ge 1 + r(x-1).","labels":["GeneralHypercontractivity.rpow_ge_one_add_mul_sub"],"detail_key":"p14"},{"id":"n21138","layer":"informal","project":"p14","title":"General two-point inequality, unit case","kind":"theorem","summary":"[General two-point inequality, unit case] Let 1 \\le p \\le q \\le 2, b \\in [0,1] and \\rho = \\sqrt…","labels":["GeneralHypercontractivity.two_point_ineq_general_unit"],"detail_key":"p14"},{"id":"n21139","layer":"informal","project":"p14","title":"One-bit low-norms hypercontractivity","kind":"theorem","summary":"[One-bit low-norms hypercontractivity] For 1 < p \\le q \\le 2, f : BoolCube\\,1 \\to R and \\rho =…","labels":["GeneralHypercontractivity.low_norms_one_bit"],"detail_key":"p14"},{"id":"n21140","layer":"informal","project":"p14","title":"Low-norms hypercontractivity","kind":"theorem","summary":"[Low-norms hypercontractivity] For 1 < p \\le u \\le 2, any n, f : BoolCube\\,n \\to R and \\rho = \\…","labels":["GeneralHypercontractivity.low_norms_hypercontractivity"],"detail_key":"p14"},{"id":"n21141","layer":"informal","project":"p14","title":"High-norms hypercontractivity","kind":"theorem","summary":"[High-norms hypercontractivity] For 2 \\le p \\le u, any \\rho with 0 \\le \\rho \\le 1 and \\rho \\le…","labels":["GeneralHypercontractivity.high_norms_hypercontractivity"],"detail_key":"p14"},{"id":"n21142","layer":"informal","project":"p14","title":"General one-function hypercontractivity","kind":"theorem","summary":"[General one-function hypercontractivity] Let 1 \\le p \\le u with u > 1, and let 0 \\le \\rho \\le…","labels":["GeneralHypercontractivity.general_one_function_hypercontractivity"],"detail_key":"p14"},{"id":"n21143","layer":"informal","project":"p14","title":"General two-function hypercontractivity","kind":"theorem","summary":"[General two-function hypercontractivity] Let 1 \\le p \\le u with u \\ge 2, and let 0 \\le \\rho \\l…","labels":["GeneralHypercontractivity.general_two_function_hypercontractivity"],"detail_key":"p14"},{"id":"n21144","layer":"informal","project":"p14","title":"Noisy influence","kind":"definition","summary":"[Noisy influence] The \\emphnoisy influence of coordinate i on f at noise rate \\rho is \\[ Inf_i^…","labels":["KKL.noisyInfluence"],"detail_key":"p14"},{"id":"n21145","layer":"informal","project":"p14","title":"Low-degree truncation","kind":"definition","summary":"[Low-degree truncation] The \\emphlow-degree part of f at level k keeps only the Fourier coeffic…","labels":["KKL.lowDegreePart"],"detail_key":"p14"},{"id":"n21146","layer":"informal","project":"p14","title":"High-degree part","kind":"definition","summary":"[High-degree part] The \\emphhigh-degree part of f at level k keeps only the Fourier coefficient…","labels":["KKL.highDegreePart"],"detail_key":"p14"},{"id":"n21147","layer":"informal","project":"p14","title":"Influential coordinates","kind":"definition","summary":"[Influential coordinates] The set of \\tau-\\emphinfluential coordinates of f is the finite set \\…","labels":["KKL.influentialCoords"],"detail_key":"p14"},{"id":"n21148","layer":"informal","project":"p14","title":"Junta","kind":"definition","summary":"[Junta] A function g is a \\emphJ-junta if it depends only on the coordinates in J: whenever x a…","labels":["KKL.IsJunta"],"detail_key":"p14"},{"id":"n21149","layer":"informal","project":"p14","title":"Squared L^2 distance","kind":"definition","summary":"[Squared L^2 distance] The squared L^2 distance between two Boolean functions is E\\big[(f(x) -…","labels":["KKL.l2DistSq"],"detail_key":"p14"},{"id":"n21150","layer":"informal","project":"p14","title":"Noisy influence at \\rho = 1","kind":"lemma","summary":"[Noisy influence at \\rho = 1] At noise rate \\rho = 1 the noisy influence coincides with the ord…","labels":["KKL.noisyInfluence_one"],"detail_key":"p14"},{"id":"n21151","layer":"informal","project":"p14","title":"Sum of noisy influences","kind":"lemma","summary":"[Sum of noisy influences] For every \\rho, \\[ \\sum_i=1^n Inf_i^\\rho[f] \\;=\\; \\sum_S \\subseteq [n…","labels":["KKL.sum_noisyInfluence"],"detail_key":"p14"},{"id":"n21152","layer":"informal","project":"p14","title":"Total influence as a sum of influences","kind":"lemma","summary":"[Total influence as a sum of influences] The total influence is the sum of the coordinate influ…","labels":["KKL.totalInfluence_eq_sum_influences"],"detail_key":"p14"},{"id":"n21153","layer":"informal","project":"p14","title":"Second moment of a \\pm 1-valued function","kind":"lemma","summary":"[Second moment of a \\pm 1-valued function] If f takes values in \\-1, 1\\, then E[f^2] = 1.","labels":["KKL.expect_sq_pm_one"],"detail_key":"p14"},{"id":"n21154","layer":"informal","project":"p14","title":"Parseval in second-moment form","kind":"lemma","summary":"[Parseval in second-moment form] For every Boolean function f, \\sum_S \\subseteq [n] \\hat f(S)^2…","labels":["KKL.sum_fourier_sq_eq_expect_sq"],"detail_key":"p14"},{"id":"n21155","layer":"informal","project":"p14","title":"Low and high parts recombine","kind":"lemma","summary":"[Low and high parts recombine] For every f, every level k and every point x of the cube, f_\\le…","labels":["KKL.low_plus_high_eq"],"detail_key":"p14"},{"id":"n21156","layer":"informal","project":"p14","title":"Fourier coefficients of the low-degree part","kind":"lemma","summary":"[Fourier coefficients of the low-degree part] The truncation acts as a projection on the Fourie…","labels":["KKL.fourierCoeff_lowDegreePart"],"detail_key":"p14"},{"id":"n21157","layer":"informal","project":"p14","title":"L^2 error of low-degree truncation","kind":"lemma","summary":"[L^2 error of low-degree truncation] The squared L^2 error of the degree-k truncation is exactl…","labels":["KKL.lowDegree_l2_error"],"detail_key":"p14"},{"id":"n21158","layer":"informal","project":"p14","title":"Fourier tail weight bound","kind":"lemma","summary":"[Fourier tail weight bound] For k > 0 the Fourier weight of f above level k is controlled by th…","labels":["KKL.tail_fourier_weight_bound"],"detail_key":"p14"},{"id":"n21159","layer":"informal","project":"p14","title":"Low-degree approximation","kind":"lemma","summary":"[Low-degree approximation] For k > 0, the degree-k truncation approximates f with squared L^2 e…","labels":["KKL.lowDegree_approx"],"detail_key":"p14"},{"id":"n21160","layer":"informal","project":"p14","title":"Few influential coordinates","kind":"lemma","summary":"[Few influential coordinates] For \\tau > 0 the number of \\tau-influential coordinates satisfies…","labels":["KKL.influential_coords_card"],"detail_key":"p14"},{"id":"n21161","layer":"informal","project":"p14","title":"Low-degree part is close to a junta on the influential coordinates","kind":"lemma","summary":"[Low-degree part is close to a junta on the influential coordinates] For every f, every level k…","labels":["KKL.lowDegreePart_depends_on_influential"],"detail_key":"p14"},{"id":"n21162","layer":"informal","project":"p14","title":"Noisy influence is dominated by influence","kind":"lemma","summary":"[Noisy influence is dominated by influence] For 0 \\le \\rho \\le 1 we have Inf_i^\\rho[f] \\le Inf_…","labels":["KKL.noisyInfluence_le_influence"],"detail_key":"p14"},{"id":"n21163","layer":"informal","project":"p14","title":"Noisy influence bound for \\pm 1-valued functions","kind":"lemma","summary":"[Noisy influence bound for \\pm 1-valued functions] For a \\pm 1-valued f and 0 < \\rho \\le 1, the…","labels":["KKL.noisyInfluence_power_bound"],"detail_key":"p14"},{"id":"n21164","layer":"informal","project":"p14","title":"Cauchy--Schwarz for influences","kind":"lemma","summary":"[Cauchy--Schwarz for influences] The total influence obeys \\[ I[f]^2 \\;\\le\\; n \\sum_i=1^n Inf_i…","labels":["KKL.cauchy_schwarz_influences"],"detail_key":"p14"},{"id":"n21165","layer":"informal","project":"p14","title":"Maximum influence from the sum of squares","kind":"lemma","summary":"[Maximum influence from the sum of squares] If I[f] > 0, then some coordinate i satisfies \\[ In…","labels":["KKL.max_influence_from_sum_sq"],"detail_key":"p14"},{"id":"n21166","layer":"informal","project":"p14","title":"Averaging form of KKL","kind":"theorem","summary":"[Averaging form of KKL] For n > 0 there exists a coordinate i with Inf_i[f] \\ge I[f]/n, the tri…","labels":["KKL.KKL_trivial"],"detail_key":"p14"},{"id":"n21167","layer":"informal","project":"p14","title":"Noisy total influence","kind":"definition","summary":"[Noisy total influence] The \\emphnoisy total influence at rate \\rho is \\[ \\sum_S \\subseteq [n]…","labels":["KKL.noisyTotalInfluence"],"detail_key":"p14"},{"id":"n21168","layer":"informal","project":"p14","title":"KKL theorem for balanced functions","kind":"theorem","summary":"[KKL theorem for balanced functions] (Kahn--Kalai--Linial, 1988.) Let f : \\0,1\\^n \\to \\-1,1\\ be…","labels":["KKL.KKL_balanced"],"detail_key":"p14"},{"id":"n21169","layer":"informal","project":"p14","title":"Friedgut's junta theorem","kind":"theorem","summary":"[Friedgut's junta theorem] (Friedgut, 1998.) Let f : \\0,1\\^n \\to \\-1,1\\ and let \\varepsilon > 0…","labels":["KKL.friedgut_junta"],"detail_key":"p14"},{"id":"n21170","layer":"informal","project":"p14","title":"Balanced \\pm 1-valued functions have total influence at least one","kind":"lemma","summary":"[Balanced \\pm 1-valued functions have total influence at least one] If f takes values in \\-1,1\\…","labels":["KKL.balanced_totalInfluence_ge_one"],"detail_key":"p14"},{"id":"n21171","layer":"informal","project":"p14","title":"Nonnegativity of the influence entropy","kind":"lemma","summary":"[Nonnegativity of the influence entropy] If I[f] > 0, then the entropy-like quantity attached t…","labels":["KKL.influence_entropy_nonneg"],"detail_key":"p14"},{"id":"n21172","layer":"informal","project":"p14","title":"Size of a decision tree","kind":"definition","summary":"[Size of a decision tree] The size of a decision tree, defined as its number of leaves: a leaf…","labels":["DecisionTree.size"],"detail_key":"p14"},{"id":"n21173","layer":"informal","project":"p14","title":"Sign-encoded tree function","kind":"definition","summary":"[Sign-encoded tree function] The \\pm 1-valued function computed by a decision tree T, namely x…","labels":["DecisionTree.signEval"],"detail_key":"p14"},{"id":"n21174","layer":"informal","project":"p14","title":"Recursive Fourier coefficients of a tree","kind":"definition","summary":"[Recursive Fourier coefficients of a tree] The coefficient function coeffs\\,T : P(Fin\\,n) \\to R…","labels":["DecisionTree.coeffs"],"detail_key":"p14"},{"id":"n21175","layer":"informal","project":"p14","title":"Symmetric difference with a singleton is an involution","kind":"lemma","summary":"[Symmetric difference with a singleton is an involution] For a fixed i, the map S \\mapsto S \\tr…","labels":["DecisionTree.symmDiff_singleton_invol"],"detail_key":"p14"},{"id":"n21176","layer":"informal","project":"p14","title":"Reindexing a frequency sum by S \\mapsto S \\triangle \\i\\","kind":"lemma","summary":"[Reindexing a frequency sum by S \\mapsto S \\triangle \\i\\] For any g : P(Fin\\,n) \\to R and any i…","labels":["DecisionTree.sum_symmDiff_reindex"],"detail_key":"p14"},{"id":"n21177","layer":"informal","project":"p14","title":"Character of a singleton-shifted frequency","kind":"lemma","summary":"[Character of a singleton-shifted frequency] For all S, i, and x in the Boolean cube, \\chi_S \\t…","labels":["DecisionTree.chiS_symmDiff_singleton"],"detail_key":"p14"},{"id":"n21178","layer":"informal","project":"p14","title":"Cardinality under singleton symmetric difference","kind":"lemma","summary":"[Cardinality under singleton symmetric difference] For all S and i, \\left\\lvert S\\right\\rvert -…","labels":["DecisionTree.card_symmDiff_singleton"],"detail_key":"p14"},{"id":"n21179","layer":"informal","project":"p14","title":"Character expansion of the tree function","kind":"lemma","summary":"[Character expansion of the tree function] For every decision tree T and every point x of the B…","labels":["DecisionTree.signEval_eq_sum_coeffs"],"detail_key":"p14"},{"id":"n21180","layer":"informal","project":"p14","title":"Coefficients vanish above the depth","kind":"lemma","summary":"[Coefficients vanish above the depth] If T.depth < \\left\\lvert S\\right\\rvert, then coeffs\\,T\\,S…","labels":["DecisionTree.coeffs_eq_zero_of_depth_lt"],"detail_key":"p14"},{"id":"n21181","layer":"informal","project":"p14","title":"Coefficient 1-norm bounded by tree size","kind":"lemma","summary":"[Coefficient 1-norm bounded by tree size] For every decision tree T, \\[ \\sum_S \\left\\lvert coef…","labels":["DecisionTree.sum_abs_coeffs_le"],"detail_key":"p14"},{"id":"n21182","layer":"informal","project":"p14","title":"Granularity, multiplicative form","kind":"lemma","summary":"[Granularity, multiplicative form] If T.depth \\le k, then for every S there is an integer m wit…","labels":["DecisionTree.coeffs_mul_two_pow_int"],"detail_key":"p14"},{"id":"n21183","layer":"informal","project":"p14","title":"Granularity of the coefficients","kind":"lemma","summary":"[Granularity of the coefficients] For every S there is an integer m with coeffs\\,T\\,S = m / 2^T…","labels":["DecisionTree.coeffs_granular"],"detail_key":"p14"},{"id":"n21184","layer":"informal","project":"p14","title":"Size bounded by 2^depth","kind":"lemma","summary":"[Size bounded by 2^depth] Every decision tree satisfies T.size \\le 2^T.depth: a tree of depth k…","labels":["DecisionTree.size_le_two_pow_depth"],"detail_key":"p14"},{"id":"n21185","layer":"informal","project":"p14","title":"Uniqueness of the character expansion","kind":"lemma","summary":"[Uniqueness of the character expansion] For any coefficient family c and any frequency T, the F…","labels":["DecisionTree.fourierCoeff_sum_chiS"],"detail_key":"p14"},{"id":"n21186","layer":"informal","project":"p14","title":"Fourier coefficients of a tree function","kind":"theorem","summary":"[Fourier coefficients of a tree function] For every decision tree T and every S, the Fourier co…","labels":["DecisionTree.fourierCoeff_signEval"],"detail_key":"p14"},{"id":"n21187","layer":"informal","project":"p14","title":"Degree bounded by depth (Proposition 3.16)","kind":"theorem","summary":"[Degree bounded by depth (Proposition 3.16)] The function computed by a decision tree of depth…","labels":["DecisionTree.degree_le_depth"],"detail_key":"p14"},{"id":"n21188","layer":"informal","project":"p14","title":"Spectral 1-norm bounded by size (Proposition 3.16)","kind":"theorem","summary":"[Spectral 1-norm bounded by size (Proposition 3.16)] For every decision tree T, \\[ \\sum_S \\bigl…","labels":["DecisionTree.spectral_one_norm_le"],"detail_key":"p14"},{"id":"n21189","layer":"informal","project":"p14","title":"Granularity of the Fourier coefficients (Proposition 3.16)","kind":"theorem","summary":"[Granularity of the Fourier coefficients (Proposition 3.16)] For every S there is an integer m…","labels":["DecisionTree.fourierCoeff_granular"],"detail_key":"p14"},{"id":"n21190","layer":"informal","project":"p14","title":"Sparsity bound (Proposition 3.16)","kind":"theorem","summary":"[Sparsity bound (Proposition 3.16)] The Fourier support of T.signEval, i.e.\\ the set of frequen…","labels":["DecisionTree.sparsity_le"],"detail_key":"p14"},{"id":"n21191","layer":"informal","project":"p14","title":"Sparsity, absolute form","kind":"theorem","summary":"[Sparsity, absolute form] The Fourier support of T.signEval has cardinality at most 4^T.depth.","labels":["DecisionTree.sparsity_le_four_pow"],"detail_key":"p14"},{"id":"n21192","layer":"informal","project":"p14","title":"Specification of the minimum decision-tree depth","kind":"lemma","summary":"[Specification of the minimum decision-tree depth] For every Boolean function f : \\0,1\\^n \\to \\…","labels":["DecisionTree.exists_dtree_of_dtDepth"],"detail_key":"p14"},{"id":"n21193","layer":"informal","project":"p14","title":"Degree bounded by decision-tree depth (Proposition 3.16)","kind":"theorem","summary":"[Degree bounded by decision-tree depth (Proposition 3.16)] For every Boolean function f, the \\p…","labels":["DecisionTree.degree_le_dtDepth"],"detail_key":"p14"},{"id":"n21194","layer":"informal","project":"p14","title":"Subset-of-free-variables indicator factors per coordinate","kind":"lemma","summary":"[Subset-of-free-variables indicator factors per coordinate] For a set T \\subseteq \\0,\\dots,n-1\\…","labels":["RestrictionFourier.indicator_subset_eq_prod"],"detail_key":"p14"},{"id":"n21195","layer":"informal","project":"p14","title":"Free-set marginal \\Pr[T \\subseteq J] = p^\\left\\lvert T\\right\\rvert","kind":"theorem","summary":"[Free-set marginal \\Pr[T \\subseteq J] = p^\\left\\lvert T\\right\\rvert] Under a Bernoulli(p)-rando…","labels":["RestrictionFourier.bernoulliRestrProb_subset_freeVars"],"detail_key":"p14"},{"id":"n21196","layer":"informal","project":"p14","title":"Free-coordinate count as a sum of indicators","kind":"lemma","summary":"[Free-coordinate count as a sum of indicators] For a set U and a restriction \\rho with free-coo…","labels":["RestrictionFourier.card_inter_eq_sum"],"detail_key":"p14"},{"id":"n21197","layer":"informal","project":"p14","title":"Expectation of one free-coordinate indicator","kind":"lemma","summary":"[Expectation of one free-coordinate indicator] For each coordinate i, the Bernoulli(p)-weighted…","labels":["RestrictionFourier.expectation_free_indicator"],"detail_key":"p14"},{"id":"n21198","layer":"informal","project":"p14","title":"Expectation of a product of two free-coordinate indicators","kind":"lemma","summary":"[Expectation of a product of two free-coordinate indicators] For distinct coordinates i \\ne j,…","labels":["RestrictionFourier.expectation_free_indicator_pair"],"detail_key":"p14"},{"id":"n21199","layer":"informal","project":"p14","title":"First moment of the free-coordinate count","kind":"lemma","summary":"[First moment of the free-coordinate count] Under a Bernoulli(p)-random restriction with free s…","labels":["RestrictionFourier.expectation_card_inter"],"detail_key":"p14"},{"id":"n21200","layer":"informal","project":"p14","title":"Second moment of the free-coordinate count","kind":"lemma","summary":"[Second moment of the free-coordinate count] Under a Bernoulli(p)-random restriction with free…","labels":["RestrictionFourier.expectation_card_inter_sq"],"detail_key":"p14"},{"id":"n21201","layer":"informal","project":"p14","title":"Variance of the free-coordinate count","kind":"lemma","summary":"[Variance of the free-coordinate count] For 0 \\le p \\le 1 and every set U, the Bernoulli(p)-wei…","labels":["RestrictionFourier.variance_card_inter"],"detail_key":"p14"},{"id":"n21202","layer":"informal","project":"p14","title":"Chebyshev lower tail for the free-coordinate count","kind":"theorem","summary":"[Chebyshev lower tail for the free-coordinate count] Let 0 \\le p \\le 1, let U be a set of coord…","labels":["RestrictionFourier.bernoulliRestrProb_card_inter_lt"],"detail_key":"p14"},{"id":"n21203","layer":"informal","project":"p14","title":"Complement rule for Bernoulli-restriction probability","kind":"lemma","summary":"[Complement rule for Bernoulli-restriction probability] For 0 \\le p \\le 1 and any decidable eve…","labels":["RestrictionFourier.bernoulliRestrProb_not"],"detail_key":"p14"},{"id":"n21204","layer":"informal","project":"p14","title":"Lower bound on the probability of many free coordinates","kind":"theorem","summary":"[Lower bound on the probability of many free coordinates] Let 0 \\le p \\le 1, let U be a set of…","labels":["RestrictionFourier.bernoulliRestrProb_card_inter_ge"],"detail_key":"p14"},{"id":"n21205","layer":"informal","project":"p14","title":"Restriction commutes with the \\pm 1-encoding","kind":"definition","summary":"[Restriction commutes with the \\pm 1-encoding] The restriction restrictBF of a real-valued Bool…","labels":["RestrictionFourier.restrictBF_boolToSign"],"detail_key":"p14"},{"id":"n21206","layer":"informal","project":"p14","title":"Membership in the free variables","kind":"definition","summary":"[Membership in the free variables] For a restriction \\rho and a coordinate i, we have i \\in \\rh…","labels":["RestrictionFourier.mem_freeVars"],"detail_key":"p14"},{"id":"n21207","layer":"informal","project":"p14","title":"Character through a restriction splits","kind":"lemma","summary":"[Character through a restriction splits] For a set U \\subseteq [n], a restriction \\rho with fre…","labels":["RestrictionFourier.chiS_extend"],"detail_key":"p14"},{"id":"n21208","layer":"informal","project":"p14","title":"Closed form for restricted Fourier coefficients","kind":"theorem","summary":"[Closed form for restricted Fourier coefficients] For f : \\0,1\\^n \\to R, a restriction \\rho wit…","labels":["RestrictionFourier.fourierCoeff_restrictBF"],"detail_key":"p14"},{"id":"n21209","layer":"informal","project":"p14","title":"Sums over restrictions of per-coordinate products factor","kind":"lemma","summary":"[Sums over restrictions of per-coordinate products factor] For any family h : [n] \\to Option\\ B…","labels":["RestrictionFourier.sum_restriction_prod"],"detail_key":"p14"},{"id":"n21210","layer":"informal","project":"p14","title":"Bernoulli-weighted products factor","kind":"lemma","summary":"[Bernoulli-weighted products factor] For p \\in R and any family h : [n] \\to Option\\ Bool \\to R,…","labels":["RestrictionFourier.sum_bernoulli_prod"],"detail_key":"p14"},{"id":"n21211","layer":"informal","project":"p14","title":"Per-coordinate local factor","kind":"definition","summary":"[Per-coordinate local factor] Given sets U, S \\subseteq [n], a coordinate i and a per-coordinat…","labels":["RestrictionFourier.localFactor"],"detail_key":"p14"},{"id":"n21212","layer":"informal","project":"p14","title":"Indicator times sign factors into local factors","kind":"lemma","summary":"[Indicator times sign factors into local factors] If S \\subseteq U then for every restriction \\…","labels":["RestrictionFourier.indicator_signProd_eq_prod"],"detail_key":"p14"},{"id":"n21213","layer":"informal","project":"p14","title":"Average of one local factor","kind":"lemma","summary":"[Average of one local factor] For p \\in R, sets U, S and a coordinate i, \\[ \\sum_v varWeight\\,p…","labels":["RestrictionFourier.sum_varWeight_localFactor"],"detail_key":"p14"},{"id":"n21214","layer":"informal","project":"p14","title":"Average of a product of two local factors","kind":"lemma","summary":"[Average of a product of two local factors] For p \\in R, sets U, V, S and a coordinate i, \\[ \\s…","labels":["RestrictionFourier.sum_varWeight_localFactor_mul"],"detail_key":"p14"},{"id":"n21215","layer":"informal","project":"p14","title":"Product of a three-case indicator weight","kind":"lemma","summary":"[Product of a three-case indicator weight] For p \\in R and sets S, U \\subseteq [n], \\[ \\prod_i…","labels":["RestrictionFourier.prod_if_subset"],"detail_key":"p14"},{"id":"n21216","layer":"informal","project":"p14","title":"Proposition 4.17, first identity","kind":"theorem","summary":"[Proposition 4.17, first identity] Under a Bernoulli(p)-random restriction \\rho, the expected F…","labels":["RestrictionFourier.expectation_fourierCoeff_restrictBF"],"detail_key":"p14"},{"id":"n21217","layer":"informal","project":"p14","title":"Proposition 4.17, second identity","kind":"theorem","summary":"[Proposition 4.17, second identity] Under a Bernoulli(p)-random restriction \\rho, the expected…","labels":["RestrictionFourier.expectation_fourierCoeff_sq_restrictBF"],"detail_key":"p14"},{"id":"n21218","layer":"informal","project":"p14","title":"The squared sign product is one","kind":"lemma","summary":"[The squared sign product is one] For every restriction \\rho and every set T \\subseteq [n], (si…","labels":["RestrictionFourier.signProd_sq"],"detail_key":"p14"},{"id":"n21219","layer":"informal","project":"p14","title":"Proposition 4.17, probability form","kind":"theorem","summary":"[Proposition 4.17, probability form] For a Bernoulli(p)-random restriction \\rho with free set J…","labels":["RestrictionFourier.bernoulliRestrProb_inter_freeVars"],"detail_key":"p14"},{"id":"n21220","layer":"informal","project":"p14","title":"Restriction of a real-valued Boolean function","kind":"definition","summary":"[Restriction of a real-valued Boolean function] Given f : \\0,1\\^n \\to R and a restriction \\rho,…","labels":["RestrictionFourier.restrictBF"],"detail_key":"p14"},{"id":"n21221","layer":"informal","project":"p14","title":"Sign product of the fixed bits","kind":"definition","summary":"[Sign product of the fixed bits] For a restriction \\rho and a set T \\subseteq [n], \\[ signProd\\…","labels":["RestrictionFourier.signProd"],"detail_key":"p14"},{"id":"n21222","layer":"informal","project":"p14","title":"The AC^0 gate set","kind":"definition","summary":"[The AC^0 gate set] The set of plain AC^0 gate operations on the alphabet Fin\\,2: the identity…","labels":["ACP.AC_GateOps"],"detail_key":"p14"},{"id":"n21223","layer":"informal","project":"p14","title":"Counting tuples satisfying a pointwise predicate","kind":"lemma","summary":"[Counting tuples satisfying a pointwise predicate] For a finite index type \\iota, a finite type…","labels":["ACP.tuple_fail_count"],"detail_key":"p14"},{"id":"n21224","layer":"informal","project":"p14","title":"Averaging for the probabilistic method","kind":"lemma","summary":"[Averaging for the probabilistic method] Let Bad be a finite set of elements of \\alpha, let Fai…","labels":["ACP.prob_method_averaging"],"detail_key":"p14"},{"id":"n21225","layer":"informal","project":"p14","title":"Booleanization of a field element","kind":"definition","summary":"[Booleanization of a field element] The map Z/p \\to Fin\\,2 sending a to 1 if a = 1 and to 0 oth…","labels":["ACP.bitify"],"detail_key":"p14"},{"id":"n21226","layer":"informal","project":"p14","title":"Booleanization is a section on \\0,1\\","kind":"lemma","summary":"[Booleanization is a section on \\0,1\\] If a \\in \\0,1\\ \\subseteq Z/p, then casting \\textttACP.bi…","labels":["ACP.cast_bitify_eq"],"detail_key":"p14"},{"id":"n21227","layer":"informal","project":"p14","title":"Fermat indicator of zero","kind":"lemma","summary":"[Fermat indicator of zero] For every a \\in Z/p with p prime, \\[ 1 - a^p-1 = 1 & \\textif a = 0,\\…","labels":["ACP.one_sub_pow_card_sub_one"],"detail_key":"p14"},{"id":"n21228","layer":"informal","project":"p14","title":"Bit indicator agrees with booleanization","kind":"lemma","summary":"[Bit indicator agrees with booleanization] If a \\in \\0,1\\ \\subseteq Z/p, then 1 - (1-a)^p-1 equ…","labels":["ACP.bit_indicator_eq_bitify"],"detail_key":"p14"},{"id":"n21229","layer":"informal","project":"p14","title":"Unbounded MOD_p gate","kind":"definition","summary":"[Unbounded MOD_p gate] The gate operation of arity \\textttwidth on Boolean inputs that outputs…","labels":["ACP.modGateOp"],"detail_key":"p14"},{"id":"n21230","layer":"informal","project":"p14","title":"The AC^0[p] gate set","kind":"definition","summary":"[The AC^0[p] gate set] The AC^0[p] gate set: the AC^0 gates (identity, NOT, unbounded AND) toge…","labels":["ACP.ACp_GateOps"],"detail_key":"p14"},{"id":"n21231","layer":"informal","project":"p14","title":"Randomized OR-approximating polynomial","kind":"definition","summary":"[Randomized OR-approximating polynomial] Given polynomials P_1,\\dots,P_width over Z/p and a ran…","labels":["ACP.approxOr"],"detail_key":"p14"},{"id":"n21232","layer":"informal","project":"p14","title":"Exact MOD_p polynomial","kind":"definition","summary":"[Exact MOD_p polynomial] The polynomial 1 - \\bigl(\\sum_i P_i\\bigr)^p-1 over Z/p, which computes…","labels":["ACP.exactMod"],"detail_key":"p14"},{"id":"n21233","layer":"informal","project":"p14","title":"Value-level OR approximator","kind":"definition","summary":"[Value-level OR approximator] The value-level analogue of the randomized OR approximator: for v…","labels":["ACP.approxOr_val"],"detail_key":"p14"},{"id":"n21234","layer":"informal","project":"p14","title":"Value-level OR detector","kind":"definition","summary":"[Value-level OR detector] For v : Fin\\,width \\to Z/p, the quantity 1 - \\prod_k\\bigl(1 - v_k^\\,p…","labels":["ACP.OR_val"],"detail_key":"p14"},{"id":"n21235","layer":"informal","project":"p14","title":"Degree bound for the OR approximator","kind":"theorem","summary":"[Degree bound for the OR approximator] The total degree of \\textttACP.approxOr applied to P_1,\\…","labels":["ACP.approxOr_totalDegree"],"detail_key":"p14"},{"id":"n21236","layer":"informal","project":"p14","title":"Degree bound for the exact MOD_p polynomial","kind":"theorem","summary":"[Degree bound for the exact MOD_p polynomial] The total degree of \\textttACP.exactMod applied t…","labels":["ACP.exactMod_totalDegree"],"detail_key":"p14"},{"id":"n21237","layer":"informal","project":"p14","title":"Half the subsets miss a nonzero vector","kind":"lemma","summary":"[Half the subsets miss a nonzero vector] For a nonzero vector v : Fin\\,n \\to Z/p, the number of…","labels":["ACP.subset_sum_zero_bound"],"detail_key":"p14"},{"id":"n21238","layer":"informal","project":"p14","title":"Evaluation commutes with the OR approximator","kind":"lemma","summary":"[Evaluation commutes with the OR approximator] Evaluating \\textttACP.approxOr\\,(P, S) at a poin…","labels":["ACP.approxOr_eval_eq"],"detail_key":"p14"},{"id":"n21239","layer":"informal","project":"p14","title":"Characterization of approximator failure","kind":"lemma","summary":"[Characterization of approximator failure] For v : Fin\\,width \\to Z/p and a seed S, the value-l…","labels":["ACP.approxOr_failure_iff"],"detail_key":"p14"},{"id":"n21240","layer":"informal","project":"p14","title":"Bad-seed count for a nonzero input","kind":"lemma","summary":"[Bad-seed count for a nonzero input] For a nonzero v, the number of seeds S : Fin\\,\\ell \\to Fin…","labels":["ACP.count_bad_S"],"detail_key":"p14"},{"id":"n21241","layer":"informal","project":"p14","title":"Number of random seeds","kind":"lemma","summary":"[Number of random seeds] The type of seeds Fin\\,\\ell \\to Finset(Fin\\,width), i.e.\\ \\ell indepen…","labels":["ACP.approxSeed_card"],"detail_key":"p14"},{"id":"n21242","layer":"informal","project":"p14","title":"Bad-seed bound for every input","kind":"lemma","summary":"[Bad-seed bound for every input] Without any nonvanishing hypothesis on v: the number of seeds…","labels":["ACP.count_bad_S_or"],"detail_key":"p14"},{"id":"n21243","layer":"informal","project":"p14","title":"List of OR-approximating polynomials","kind":"definition","summary":"[List of OR-approximating polynomials] The list of all polynomials \\textttACP.approxOr\\,(P,S),…","labels":["ACP.approxOrPolyList"],"detail_key":"p14"},{"id":"n21244","layer":"informal","project":"p14","title":"Length of the OR polynomial list","kind":"lemma","summary":"[Length of the OR polynomial list] The list \\textttACP.approxOrPolyList has length 2^width\\cdot…","labels":["ACP.approxOrPolyList_length"],"detail_key":"p14"},{"id":"n21245","layer":"informal","project":"p14","title":"Pointwise bad-seed count for OR","kind":"theorem","summary":"[Pointwise bad-seed count for OR] Fix an evaluation point y. The number of seeds S for which \\t…","labels":["ACP.approxOr_pointwise_bad_count"],"detail_key":"p14"},{"id":"n21246","layer":"informal","project":"p14","title":"Good OR approximators exist as a distribution","kind":"theorem","summary":"[Good OR approximators exist as a distribution] There is a list Ps of polynomials, namely \\text…","labels":["ACP.exists_good_approxOr"],"detail_key":"p14"},{"id":"n21247","layer":"informal","project":"p14","title":"Randomized AND-approximating polynomial","kind":"definition","summary":"[Randomized AND-approximating polynomial] The De Morgan dual of the OR approximator: 1 - \\textt…","labels":["ACP.approxAnd"],"detail_key":"p14"},{"id":"n21248","layer":"informal","project":"p14","title":"Degree bound for the AND approximator","kind":"theorem","summary":"[Degree bound for the AND approximator] The total degree of \\textttACP.approxAnd\\,(P,S) is at m…","labels":["ACP.approxAnd_totalDegree"],"detail_key":"p14"},{"id":"n21249","layer":"informal","project":"p14","title":"List of AND-approximating polynomials","kind":"definition","summary":"[List of AND-approximating polynomials] The list of all polynomials \\textttACP.approxAnd\\,(P,S)…","labels":["ACP.approxAndPolyList"],"detail_key":"p14"},{"id":"n21250","layer":"informal","project":"p14","title":"Length of the AND polynomial list","kind":"lemma","summary":"[Length of the AND polynomial list] The list \\textttACP.approxAndPolyList has length 2^width\\cd…","labels":["ACP.approxAndPolyList_length"],"detail_key":"p14"},{"id":"n21251","layer":"informal","project":"p14","title":"Pointwise bad-seed count for AND","kind":"theorem","summary":"[Pointwise bad-seed count for AND] Fix an evaluation point y. The number of seeds S for which \\…","labels":["ACP.approxAnd_pointwise_bad_count"],"detail_key":"p14"},{"id":"n21252","layer":"informal","project":"p14","title":"Good AND approximators exist as a distribution","kind":"theorem","summary":"[Good AND approximators exist as a distribution] There is a list Ps of polynomials, namely \\tex…","labels":["ACP.exists_good_approxAnd"],"detail_key":"p14"},{"id":"n21253","layer":"informal","project":"p14","title":"Case analysis on AC^0[p] gates","kind":"lemma","summary":"[Case analysis on AC^0[p] gates] Every gate operation in \\textttACP.ACp\\_GateOps is one of the…","labels":["ACP.ACp_GateOps_cases"],"detail_key":"p14"},{"id":"n21254","layer":"informal","project":"p14","title":"Exact MOD_p on Boolean inputs","kind":"lemma","summary":"[Exact MOD_p on Boolean inputs] If all values inputs_i lie in \\0,1\\ \\subseteq Z/p, then 1 - \\bi…","labels":["ACP.exactMod_on_bits"],"detail_key":"p14"},{"id":"n21255","layer":"informal","project":"p14","title":"Exact AND on Boolean inputs","kind":"lemma","summary":"[Exact AND on Boolean inputs] If all values inputs_i lie in \\0,1\\ \\subseteq Z/p, then \\prod_i \\…","labels":["ACP.exactAnd_on_bits"],"detail_key":"p14"},{"id":"n21256","layer":"informal","project":"p14","title":"Approximating polynomial family for a single gate","kind":"lemma","summary":"[Approximating polynomial family for a single gate] Let op be a gate in \\textttACP.ACp\\_GateOps…","labels":["ACP.exists_poly_for_gate"],"detail_key":"p14"},{"id":"n21257","layer":"informal","project":"p14","title":"Boolean input cast to the prime field","kind":"definition","summary":"[Boolean input cast to the prime field] For a Boolean input x : Fin\\,n \\to Fin\\,2, the map bool…","labels":["ACP.boolInput"],"detail_key":"p14"},{"id":"n21258","layer":"informal","project":"p14","title":"Field value of a bit","kind":"definition","summary":"[Field value of a bit] The value in Z/p represented by a bit b : Fin\\,2, namely the image of th…","labels":["ACP.boolVal"],"detail_key":"p14"},{"id":"n21259","layer":"informal","project":"p14","title":"Bit values are 0 or 1","kind":"lemma","summary":"[Bit values are 0 or 1] For every bit b : Fin\\,2 the element boolVal_p(b) lies in the subset \\0…","labels":["ACP.boolVal_mem"],"detail_key":"p14"},{"id":"n21260","layer":"informal","project":"p14","title":"Bitification inverts \\textttboolVal","kind":"lemma","summary":"[Bitification inverts \\textttboolVal] Applying bitify_p to boolVal_p(b) returns the original bi…","labels":["ACP.bitify_boolVal"],"detail_key":"p14"},{"id":"n21261","layer":"informal","project":"p14","title":"One-step unfolding of node evaluation","kind":"lemma","summary":"[One-step unfolding of node evaluation] For a feed-forward circuit F, a layer index d, a node u…","labels":["ACP.evalNode_succ_eq"],"detail_key":"p14"},{"id":"n21262","layer":"informal","project":"p14","title":"Degree target after d layers","kind":"definition","summary":"[Degree target after d layers] The degree budget for the approximating polynomials after d laye…","labels":["ACP.circuitDegreeBound"],"detail_key":"p14"},{"id":"n21263","layer":"informal","project":"p14","title":"Gate count in the first d layers","kind":"definition","summary":"[Gate count in the first d layers] For a circuit F with finite node sets, gateCountBefore\\,F\\,d…","labels":["ACP.gateCountBefore"],"detail_key":"p14"},{"id":"n21264","layer":"informal","project":"p14","title":"Gate count at layer zero","kind":"lemma","summary":"[Gate count at layer zero] gateCountBefore\\,F\\,0 = 0: the input layer contributes no gates.","labels":["ACP.gateCountBefore_zero"],"detail_key":"p14"},{"id":"n21265","layer":"informal","project":"p14","title":"Gate count recursion","kind":"lemma","summary":"[Gate count recursion] gateCountBefore\\,F\\,(d+1) equals gateCountBefore\\,F\\,d plus the cardinal…","labels":["ACP.gateCountBefore_succ"],"detail_key":"p14"},{"id":"n21266","layer":"informal","project":"p14","title":"Cardinality of a product filtered on the left factor","kind":"lemma","summary":"[Cardinality of a product filtered on the left factor] For finite types \\alpha,\\beta and a deci…","labels":["ACP.prod_left_filter_card"],"detail_key":"p14"},{"id":"n21267","layer":"informal","project":"p14","title":"Fiberwise product counting","kind":"lemma","summary":"[Fiberwise product counting] Let P be a predicate on \\alpha, Q a predicate on \\alpha \\times \\be…","labels":["ACP.prod_filter_fiber_mul_le"],"detail_key":"p14"},{"id":"n21268","layer":"informal","project":"p14","title":"Splitting off one coordinate of a dependent product","kind":"definition","summary":"[Splitting off one coordinate of a dependent product] For an index i with decidable equality on…","labels":["ACP.piEquivAt"],"detail_key":"p14"},{"id":"n21269","layer":"informal","project":"p14","title":"Counting functions bad at one coordinate","kind":"lemma","summary":"[Counting functions bad at one coordinate] Fix a coordinate i and a decidable predicate Bad on…","labels":["ACP.pi_coordinate_bad_mul_le"],"detail_key":"p14"},{"id":"n21270","layer":"informal","project":"p14","title":"Union bound over coordinates of a dependent product","kind":"lemma","summary":"[Union bound over coordinates of a dependent product] Given for each coordinate i a decidable p…","labels":["ACP.pi_exists_bad_card_mul_le"],"detail_key":"p14"},{"id":"n21271","layer":"informal","project":"p14","title":"Filtering commutes with mapping, on lengths","kind":"lemma","summary":"[Filtering commutes with mapping, on lengths] For a list l, a map f and a decidable predicate P…","labels":["ACP.list_filter_map_length"],"detail_key":"p14"},{"id":"n21272","layer":"informal","project":"p14","title":"Finset list preserves filtered cardinality","kind":"lemma","summary":"[Finset list preserves filtered cardinality] For a finset s and a decidable predicate q, the le…","labels":["ACP.finset_toList_filter_length_eq_card"],"detail_key":"p14"},{"id":"n21273","layer":"informal","project":"p14","title":"Gate approximator family","kind":"definition","summary":"[Gate approximator family] A \\textttGatePolyFamily for a gate operation op packages a nonempty…","labels":["ACP.GatePolyFamily"],"detail_key":"p14"},{"id":"n21274","layer":"informal","project":"p14","title":"Existence of a gate approximator family","kind":"lemma","summary":"[Existence of a gate approximator family] For every n, every error parameter \\ell, and every ga…","labels":["ACP.exists_gate_poly_family"],"detail_key":"p14"},{"id":"n21275","layer":"informal","project":"p14","title":"Chosen gate approximator family","kind":"definition","summary":"[Chosen gate approximator family] A choice of \\textttGatePolyFamily for each AC^0[p] gate opera…","labels":["ACP.gatePolyFamily"],"detail_key":"p14"},{"id":"n21276","layer":"informal","project":"p14","title":"Layer polynomial family","kind":"definition","summary":"[Layer polynomial family] A \\textttLayerPolyFamily for a circuit F at layer d consists of a non…","labels":["ACP.LayerPolyFamily"],"detail_key":"p14"},{"id":"n21277","layer":"informal","project":"p14","title":"Input layer family","kind":"definition","summary":"[Input layer family] The \\textttLayerPolyFamily at layer 0: a single seed, with each input node…","labels":["ACP.inputLayerFamily"],"detail_key":"p14"},{"id":"n21278","layer":"informal","project":"p14","title":"Inductive layer step","kind":"definition","summary":"[Inductive layer step] Given a circuit F using only AC^0[p] gates and a \\textttLayerPolyFamily…","labels":["ACP.stepLayerFamily"],"detail_key":"p14"},{"id":"n21279","layer":"informal","project":"p14","title":"Layer family built by recursion on depth","kind":"definition","summary":"[Layer family built by recursion on depth] For a circuit using only AC^0[p] gates, the \\textttL…","labels":["ACP.buildLayerFamily"],"detail_key":"p14"},{"id":"n21280","layer":"informal","project":"p14","title":"Polynomial distribution for all output nodes","kind":"theorem","summary":"[Polynomial distribution for all output nodes] Let F be a feed-forward circuit over Fin\\,2 with…","labels":["ACP.exists_poly_distribution_for_circuit_outputs"],"detail_key":"p14"},{"id":"n21281","layer":"informal","project":"p14","title":"Polynomial distribution for a single-output circuit","kind":"theorem","summary":"[Polynomial distribution for a single-output circuit] The same statement for a circuit with a u…","labels":["ACP.exists_poly_distribution_for_circuit_one"],"detail_key":"p14"},{"id":"n21282","layer":"informal","project":"p14","title":"List form of the single-output distribution","kind":"theorem","summary":"[List form of the single-output distribution] The single-output theorem restated with the seed…","labels":["ACP.exists_poly_list_for_circuit_one"],"detail_key":"p14"},{"id":"n21283","layer":"informal","project":"p14","title":"Index of a non-input layer","kind":"definition","summary":"[Index of a non-input layer] For a feed-forward circuit F, a bound d \\le F.depth and an index j…","labels":["ACP.gateLayerIdx"],"detail_key":"p14"},{"id":"n21284","layer":"informal","project":"p14","title":"Partial gate count as a sum of layer cardinalities","kind":"lemma","summary":"[Partial gate count as a sum of layer cardinalities] Let F be a feed-forward circuit with all n…","labels":["ACP.gateCountBefore_eq_sum_cards"],"detail_key":"p14"},{"id":"n21285","layer":"informal","project":"p14","title":"Circuit size as a sum of layer cardinalities","kind":"lemma","summary":"[Circuit size as a sum of layer cardinalities] For a feed-forward circuit F with all node layer…","labels":["ACP.size_eq_sum_cards"],"detail_key":"p14"},{"id":"n21286","layer":"informal","project":"p14","title":"Full-depth gate count is the circuit size","kind":"lemma","summary":"[Full-depth gate count is the circuit size] For a feed-forward circuit F with all node layers f…","labels":["ACP.gateCountBefore_depth_eq_size"],"detail_key":"p14"},{"id":"n21287","layer":"informal","project":"p14","title":"Polynomial distribution for all outputs, size form","kind":"theorem","summary":"[Polynomial distribution for all outputs, size form] Let p be prime and let F be a feed-forward…","labels":["ACP.exists_poly_distribution_for_circuit_outputs_size"],"detail_key":"p14"},{"id":"n21288","layer":"informal","project":"p14","title":"Polynomial distribution for a single output, size form","kind":"theorem","summary":"[Polynomial distribution for a single output, size form] Same statement for a circuit with a un…","labels":["ACP.exists_poly_distribution_for_circuit_one_size"],"detail_key":"p14"},{"id":"n21289","layer":"informal","project":"p14","title":"Polynomial list for a single output, size form","kind":"theorem","summary":"[Polynomial list for a single output, size form] The list formulation of the previous theorem:…","labels":["ACP.exists_poly_list_for_circuit_one_size"],"detail_key":"p14"},{"id":"n21290","layer":"informal","project":"p14","title":"Gate with input wiring","kind":"definition","summary":"[Gate with input wiring] A gate over alphabet \\alpha with input domain \\textttdomain consists o…","labels":["ACP.Gate"],"detail_key":"p14"},{"id":"n21291","layer":"informal","project":"p14","title":"Layered feedforward circuit","kind":"definition","summary":"[Layered feedforward circuit] A feedforward circuit from \\textttinp to \\textttout over \\alpha c…","labels":["ACP.FeedForward"],"detail_key":"p14"},{"id":"n21292","layer":"informal","project":"p14","title":"Identity gate operation","kind":"definition","summary":"[Identity gate operation] The gate operation of arity PUnit that returns its single input uncha…","labels":["ACP.FeedForward.GateOp.id"],"detail_key":"p14"},{"id":"n21293","layer":"informal","project":"p14","title":"Evaluation of a single gate","kind":"definition","summary":"[Evaluation of a single gate] Given a gate g and an assignment xs of values to the nodes of its…","labels":["ACP.FeedForward.Gate.eval"],"detail_key":"p14"},{"id":"n21294","layer":"informal","project":"p14","title":"Evaluation of a node","kind":"definition","summary":"[Evaluation of a node] For an input assignment xs : \\textttinp \\to \\alpha, the value of a node…","labels":["ACP.FeedForward.evalNode"],"detail_key":"p14"},{"id":"n21295","layer":"informal","project":"p14","title":"Evaluation of a circuit","kind":"definition","summary":"[Evaluation of a circuit] The function \\textttout \\to \\alpha sending each output node to its va…","labels":["ACP.FeedForward.eval"],"detail_key":"p14"},{"id":"n21296","layer":"informal","project":"p14","title":"Evaluation with a unique output","kind":"definition","summary":"[Evaluation with a unique output] When the output type is a \\textttUnique type, the single valu…","labels":["ACP.FeedForward.eval₁"],"detail_key":"p14"},{"id":"n21297","layer":"informal","project":"p14","title":"Size of a feedforward circuit","kind":"definition","summary":"[Size of a feedforward circuit] The size of F is the cardinality Nat.card of the sigma type of…","labels":["ACP.FeedForward.size"],"detail_key":"p14"},{"id":"n21298","layer":"informal","project":"p14","title":"Finiteness of all layers","kind":"definition","summary":"[Finiteness of all layers] The proposition that the node type of every layer i is finite.","labels":["ACP.FeedForward.Finite"],"detail_key":"p14"},{"id":"n21299","layer":"informal","project":"p14","title":"Restriction to a gate set","kind":"definition","summary":"[Restriction to a gate set] For a set S of gate operations, \\textttonlyUsesGates\\ S asserts tha…","labels":["ACP.FeedForward.onlyUsesGates"],"detail_key":"p14"},{"id":"n21300","layer":"informal","project":"p14","title":"Every circuit has at least one node","kind":"theorem","summary":"[Every circuit has at least one node] For every Boolean circuit C on n inputs, 1 \\le C.\\texttts…","labels":["BoolCircuit.Circuit.one_le_size"],"detail_key":"p14"},{"id":"n21301","layer":"informal","project":"p14","title":"AND/OR gate restriction","kind":"definition","summary":"[AND/OR gate restriction] Given a Boolean feedforward circuit F, a flag \\textttisAnd on each no…","labels":["ACP.FeedForward.IsAndOrGate"],"detail_key":"p14"},{"id":"n21302","layer":"informal","project":"p14","title":"Tree-unrolling of a node","kind":"definition","summary":"[Tree-unrolling of a node] Recursively expands a node v of layer m of a Boolean feedforward cir…","labels":["ACP.nodeToCircuit"],"detail_key":"p14"},{"id":"n21303","layer":"informal","project":"p14","title":"Tree-unrolling preserves the value of a node","kind":"theorem","summary":"[Tree-unrolling preserves the value of a node] If every gate of F is an AND or OR gate as recor…","labels":["ACP.nodeToCircuit_eval"],"detail_key":"p14"},{"id":"n21304","layer":"informal","project":"p14","title":"Size bound for tree-unrolling","kind":"theorem","summary":"[Size bound for tree-unrolling] If every gate of F has at most k input wires, i.e.\\ card\\,(F.\\t…","labels":["ACP.nodeToCircuit_size_le"],"detail_key":"p14"},{"id":"n21305","layer":"informal","project":"p14","title":"Conversion to a Boolean circuit","kind":"definition","summary":"[Conversion to a Boolean circuit] For an AND/OR feedforward circuit F and a chosen output node…","labels":["ACP.FeedForward.toCircuit"],"detail_key":"p14"},{"id":"n21306","layer":"informal","project":"p14","title":"Conversion preserves evaluation","kind":"theorem","summary":"[Conversion preserves evaluation] If every gate of F is an AND or OR gate, then for every outpu…","labels":["ACP.FeedForward.toCircuit_eval"],"detail_key":"p14"},{"id":"n21307","layer":"informal","project":"p14","title":"Size bound for the converted circuit","kind":"theorem","summary":"[Size bound for the converted circuit] If every gate of F has at most k input wires, then for e…","labels":["ACP.FeedForward.toCircuit_size_le"],"detail_key":"p14"},{"id":"n21308","layer":"informal","project":"p14","title":"Embedding a Boolean circuit as a feedforward circuit","kind":"definition","summary":"[Embedding a Boolean circuit as a feedforward circuit] Turns a circuit C on n inputs into a fee…","labels":["BoolCircuit.Circuit.toFeedForward"],"detail_key":"p14"},{"id":"n21309","layer":"informal","project":"p14","title":"Non-input nodes of the embedding are constant","kind":"theorem","summary":"[Non-input nodes of the embedding are constant] For every input x, every layer index m with 0 <…","labels":["ACP.Circuit.toFeedForward_evalNode_const"],"detail_key":"p14"},{"id":"n21310","layer":"informal","project":"p14","title":"The embedding computes the same function","kind":"theorem","summary":"[The embedding computes the same function] For every circuit C on n inputs and every input x, C…","labels":["ACP.Circuit.toFeedForward_eval"],"detail_key":"p14"},{"id":"n21311","layer":"informal","project":"p14","title":"Depth of the embedding","kind":"theorem","summary":"[Depth of the embedding] The embedding uses one extra layer for the inputs: C.\\texttttoFeedForw…","labels":["ACP.Circuit.toFeedForward_depth"],"detail_key":"p14"},{"id":"n21312","layer":"informal","project":"p14","title":"Size bound for the embedding","kind":"theorem","summary":"[Size bound for the embedding] For every circuit C on n inputs, C.\\texttttoFeedForward.\\texttts…","labels":["ACP.Circuit.toFeedForward_size_le"],"detail_key":"p14"},{"id":"n21313","layer":"informal","project":"p14","title":"Low-degree squarefree supports","kind":"definition","summary":"[Low-degree squarefree supports] For n, D \\in N, the type of supports of squarefree monomials o…","labels":["ACP.LowDegreeSupport"],"detail_key":"p14"},{"id":"n21314","layer":"informal","project":"p14","title":"Low-degree squarefree polynomial from coefficients","kind":"definition","summary":"[Low-degree squarefree polynomial from coefficients] Given coefficients c indexed by \\textttACP…","labels":["ACP.lowDegreeSquarefreePolynomial"],"detail_key":"p14"},{"id":"n21315","layer":"informal","project":"p14","title":"Squarefree representative on the root cube","kind":"theorem","summary":"[Squarefree representative on the root cube] Every function f : \\1,\\omega\\^n \\to K on the root…","labels":["ACP.exists_squarefree_representative_on_rootCube"],"detail_key":"p14"},{"id":"n21316","layer":"informal","project":"p14","title":"Monomial-level degree-preserving multilinearization","kind":"theorem","summary":"[Monomial-level degree-preserving multilinearization] Assume \\omega \\ne 1. If a monomial a\\,X^m…","labels":["ACP.monomial_lowDegree_squarefree_complete_on_rootCube"],"detail_key":"p14"},{"id":"n21317","layer":"informal","project":"p14","title":"Additivity in the coefficient function","kind":"lemma","summary":"[Additivity in the coefficient function] The constructor \\textttACP.lowDegreeSquarefreePolynomi…","labels":["ACP.lowDegreeSquarefreePolynomial_add"],"detail_key":"p14"},{"id":"n21318","layer":"informal","project":"p14","title":"Zero coefficients give the zero polynomial","kind":"lemma","summary":"[Zero coefficients give the zero polynomial] The constructor \\textttACP.lowDegreeSquarefreePoly…","labels":["ACP.lowDegreeSquarefreePolynomial_zero"],"detail_key":"p14"},{"id":"n21319","layer":"informal","project":"p14","title":"Finite-sum linearity in the coefficients","kind":"lemma","summary":"[Finite-sum linearity in the coefficients] For a finite index set S and coefficient functions c…","labels":["ACP.lowDegreeSquarefreePolynomial_sum"],"detail_key":"p14"},{"id":"n21320","layer":"informal","project":"p14","title":"Completeness of low-degree squarefree polynomials on the root cube","kind":"theorem","summary":"[Completeness of low-degree squarefree polynomials on the root cube] Assume \\omega \\ne 1. Every…","labels":["ACP.lowDegree_squarefree_complete_on_rootCube"],"detail_key":"p14"},{"id":"n21321","layer":"informal","project":"p14","title":"Degree bound for the constructed polynomial","kind":"lemma","summary":"[Degree bound for the constructed polynomial] For any coefficient function c on \\textttACP.LowD…","labels":["ACP.lowDegreeSquarefreePolynomial_totalDegree_le"],"detail_key":"p14"},{"id":"n21322","layer":"informal","project":"p14","title":"Support-to-sigma injection","kind":"definition","summary":"[Support-to-sigma injection] The map sending a low-degree support s to the pair consisting of i…","labels":["ACP.lowDegreeSupportSigmaMap"],"detail_key":"p14"},{"id":"n21323","layer":"informal","project":"p14","title":"Binomial bound on the number of low-degree supports","kind":"theorem","summary":"[Binomial bound on the number of low-degree supports] The number of low-degree squarefree suppo…","labels":["ACP.lowDegreeSupport_card_le_binomial_sum"],"detail_key":"p14"},{"id":"n21324","layer":"informal","project":"p14","title":"Size of the coefficient family","kind":"lemma","summary":"[Size of the coefficient family] For a finite type K_0, the number of coefficient functions on…","labels":["ACP.lowDegreeCoeff_card"],"detail_key":"p14"},{"id":"n21325","layer":"informal","project":"p14","title":"Root cube as Boolean strings","kind":"definition","summary":"[Root cube as Boolean strings] If \\omega \\ne 1, the explicit equivalence \\1,\\omega\\^n \\simeq (F…","labels":["ACP.rootCubeEquivFinTwo"],"detail_key":"p14"},{"id":"n21326","layer":"informal","project":"p14","title":"Cardinality of the root cube","kind":"lemma","summary":"[Cardinality of the root cube] If \\omega \\ne 1 and K is finite, then the root cube has exactly…","labels":["ACP.rootCube_card_of_ne_one"],"detail_key":"p14"},{"id":"n21327","layer":"informal","project":"p14","title":"Number of functions on the root cube","kind":"lemma","summary":"[Number of functions on the root cube] If \\omega \\ne 1 and K is finite, the number of functions…","labels":["ACP.rootCube_function_card_of_ne_one"],"detail_key":"p14"},{"id":"n21328","layer":"informal","project":"p14","title":"Hamming-ball bound for function spaces","kind":"theorem","summary":"[Hamming-ball bound for function spaces] For finite types \\alpha, \\beta, a center function cent…","labels":["ACP.function_hammingBall_card_le_binomial"],"detail_key":"p14"},{"id":"n21329","layer":"informal","project":"p14","title":"Hamming-ball bound on the root cube","kind":"theorem","summary":"[Hamming-ball bound on the root cube] For any center function on the root cube and any radius e…","labels":["ACP.rootCubeBall_card_le_binomial"],"detail_key":"p14"},{"id":"n21330","layer":"informal","project":"p14","title":"Counting obstruction for low-degree squarefree candidates","kind":"theorem","summary":"[Counting obstruction for low-degree squarefree candidates] Let K be finite and \\omega \\ne 1. S…","labels":["ACP.rootCube_counting_obstruction_lowDegreeSquarefree"],"detail_key":"p14"},{"id":"n21331","layer":"informal","project":"p14","title":"Concrete root-product lower bound","kind":"theorem","summary":"[Concrete root-product lower bound] Let K be finite, \\omega \\ne 0, \\omega \\ne 1. Suppose \\sum_t…","labels":["ACP.no_low_degree_rootProd_approx_concrete"],"detail_key":"p14"},{"id":"n21332","layer":"informal","project":"p14","title":"MOD q target inside Z/p","kind":"definition","summary":"[MOD q target inside Z/p] For a prime q, the Boolean function MOD_q on n inputs viewed as a fun…","labels":["ACP.modQTarget"],"detail_key":"p14"},{"id":"n21333","layer":"informal","project":"p14","title":"Bad input count on the Boolean cube","kind":"definition","summary":"[Bad input count on the Boolean cube] For a target f : \\0,1\\^n \\to Z/p and a polynomial P \\in (…","labels":["ACP.badInputCount"],"detail_key":"p14"},{"id":"n21334","layer":"informal","project":"p14","title":"Low-degree bad-count lower bound","kind":"definition","summary":"[Low-degree bad-count lower bound] The predicate asserting that every polynomial P of total deg…","labels":["ACP.LowDegreeBadCountLB"],"detail_key":"p14"},{"id":"n21335","layer":"informal","project":"p14","title":"Averaging over parameters","kind":"lemma","summary":"[Averaging over parameters] Let \\alpha,\\beta be finite types with \\beta nonempty and let Fail :…","labels":["ACP.exists_good_parameter_of_pointwise_bound"],"detail_key":"p14"},{"id":"n21336","layer":"informal","project":"p14","title":"One polynomial from a pointwise distribution","kind":"theorem","summary":"[One polynomial from a pointwise distribution] Let P be a family of polynomials over Z/p indexe…","labels":["ACP.exists_single_polynomial_from_pointwise_distribution"],"detail_key":"p14"},{"id":"n21337","layer":"informal","project":"p14","title":"Single low-degree polynomial for an AC^0[p] circuit","kind":"theorem","summary":"[Single low-degree polynomial for an AC^0[p] circuit] Let F be a feed-forward circuit with a un…","labels":["ACP.exists_single_poly_for_circuit_one_size"],"detail_key":"p14"},{"id":"n21338","layer":"informal","project":"p14","title":"Size lower bound from a degree lower bound","kind":"theorem","summary":"[Size lower bound from a degree lower bound] If a feed-forward circuit F over the AC^0[p] gate…","labels":["ACP.size_lower_bound_from_badCountLB"],"detail_key":"p14"},{"id":"n21339","layer":"informal","project":"p14","title":"Relative-error size lower bound","kind":"theorem","summary":"[Relative-error size lower bound] Same hypotheses as the previous theorem, but with the bad-cou…","labels":["ACP.size_lower_bound_from_relative_badCountLB"],"detail_key":"p14"},{"id":"n21340","layer":"informal","project":"p14","title":"Standard field F_p^q-1","kind":"definition","summary":"[Standard field F_p^q-1] The Galois field F_p^q-1, the standard field choice for the MOD_q lowe…","labels":["ACP.ModqField"],"detail_key":"p14"},{"id":"n21341","layer":"informal","project":"p14","title":"Exponent q-1 is nonzero","kind":"lemma","summary":"[Exponent q-1 is nonzero] For a prime q we have q - 1 \\neq 0; this is the side condition needed…","labels":["ACP.q_sub_one_ne_zero"],"detail_key":"p14"},{"id":"n21342","layer":"informal","project":"p14","title":"Cardinality of F_p^q-1","kind":"lemma","summary":"[Cardinality of F_p^q-1] For primes p and q, the field ModqField\\,q has exactly p^q-1 elements.","labels":["ACP.natCard_modqField"],"detail_key":"p14"},{"id":"n21343","layer":"informal","project":"p14","title":"A unit of order exactly q","kind":"theorem","summary":"[A unit of order exactly q] If p \\neq q are primes, then the unit group of F_p^q-1 contains an…","labels":["ACP.exists_unit_of_order_q_modqField"],"detail_key":"p14"},{"id":"n21344","layer":"informal","project":"p14","title":"Nontrivial q-th root of unity","kind":"theorem","summary":"[Nontrivial q-th root of unity] If p \\neq q are primes, then there is \\omega \\in F_p^q-1 with \\…","labels":["ACP.exists_nontrivial_qth_root_modqField"],"detail_key":"p14"},{"id":"n21345","layer":"informal","project":"p14","title":"Root-of-unity cube \\1,\\omega\\^n","kind":"definition","summary":"[Root-of-unity cube \\1,\\omega\\^n] The subtype of vectors x : Fin\\,n \\to K such that each coordi…","labels":["ACP.rootCube"],"detail_key":"p14"},{"id":"n21346","layer":"informal","project":"p14","title":"Polynomial representation on the root cube","kind":"theorem","summary":"[Polynomial representation on the root cube] Every function f : \\1,\\omega\\^n \\to K agrees on th…","labels":["ACP.exists_multilinear_representative_on_rootCube"],"detail_key":"p14"},{"id":"n21347","layer":"informal","project":"p14","title":"Squarefree monomial","kind":"definition","summary":"[Squarefree monomial] For s \\subseteq Fin\\,n, the monomial \\prod_i \\in s X_i.","labels":["ACP.squarefreeMonomial"],"detail_key":"p14"},{"id":"n21348","layer":"informal","project":"p14","title":"Squarefree (multilinear) polynomial from coefficients","kind":"definition","summary":"[Squarefree (multilinear) polynomial from coefficients] Given coefficients c_S indexed by subse…","labels":["ACP.squarefreePolynomial"],"detail_key":"p14"},{"id":"n21349","layer":"informal","project":"p14","title":"Evaluation of a squarefree monomial","kind":"theorem","summary":"[Evaluation of a squarefree monomial] For any x : Fin\\,n \\to K, the monomial indexed by s evalu…","labels":["ACP.squarefreeMonomial_eval"],"detail_key":"p14"},{"id":"n21350","layer":"informal","project":"p14","title":"Affine expression is coordinatewise inversion","kind":"theorem","summary":"[Affine expression is coordinatewise inversion] If \\omega \\neq 0 and x lies in \\1,\\omega\\^n, th…","labels":["ACP.rootCube_affine_inverse"],"detail_key":"p14"},{"id":"n21351","layer":"informal","project":"p14","title":"Degree of a squarefree monomial","kind":"theorem","summary":"[Degree of a squarefree monomial] The total degree of \\prod_i \\in s X_i is at most \\left\\lvert…","labels":["ACP.squarefreeMonomial_totalDegree_le_card"],"detail_key":"p14"},{"id":"n21352","layer":"informal","project":"p14","title":"Coordinates on the root cube are nonzero","kind":"theorem","summary":"[Coordinates on the root cube are nonzero] If \\omega \\neq 0 and x \\in \\1,\\omega\\^n, then x_i \\n…","labels":["ACP.rootCube_coord_ne_zero"],"detail_key":"p14"},{"id":"n21353","layer":"informal","project":"p14","title":"Top monomial times a complement inverse monomial","kind":"theorem","summary":"[Top monomial times a complement inverse monomial] For \\omega \\neq 0, x \\in \\1,\\omega\\^n and s…","labels":["ACP.rootCube_top_mul_compl_inverse"],"detail_key":"p14"},{"id":"n21354","layer":"informal","project":"p14","title":"Split at degree n/2","kind":"theorem","summary":"[Split at degree n/2] Let \\omega \\neq 0 and let c be a coefficient family. Then there are polyn…","labels":["ACP.split_multilinear_at_half_degree"],"detail_key":"p14"},{"id":"n21355","layer":"informal","project":"p14","title":"Affine inverse coordinate polynomial","kind":"definition","summary":"[Affine inverse coordinate polynomial] The degree-one polynomial 1 + \\omega^-1 - \\omega^-1 X_i,…","labels":["ACP.affineInvPoly"],"detail_key":"p14"},{"id":"n21356","layer":"informal","project":"p14","title":"Evaluation of the affine inverse polynomial","kind":"theorem","summary":"[Evaluation of the affine inverse polynomial] For any x : Fin\\,n \\to K, the polynomial affineIn…","labels":["ACP.affineInvPoly_eval"],"detail_key":"p14"},{"id":"n21357","layer":"informal","project":"p14","title":"Affine inverse polynomial has degree at most one","kind":"theorem","summary":"[Affine inverse polynomial has degree at most one] The total degree of affineInvPoly\\,\\omega\\,i…","labels":["ACP.affineInvPoly_totalDegree_le_one"],"detail_key":"p14"},{"id":"n21358","layer":"informal","project":"p14","title":"Affine-substituted squarefree monomial","kind":"definition","summary":"[Affine-substituted squarefree monomial] For s \\subseteq Fin\\,n, the product \\prod_i \\in s(1 +…","labels":["ACP.affineSquarefreeMonomial"],"detail_key":"p14"},{"id":"n21359","layer":"informal","project":"p14","title":"Evaluation of the affine squarefree monomial","kind":"theorem","summary":"[Evaluation of the affine squarefree monomial] For any x : Fin\\,n \\to K, the affine squarefree…","labels":["ACP.affineSquarefreeMonomial_eval"],"detail_key":"p14"},{"id":"n21360","layer":"informal","project":"p14","title":"Degree of the affine squarefree monomial","kind":"theorem","summary":"[Degree of the affine squarefree monomial] The total degree of affineSquarefreeMonomial\\,\\omega…","labels":["ACP.affineSquarefreeMonomial_totalDegree_le_card"],"detail_key":"p14"},{"id":"n21361","layer":"informal","project":"p14","title":"Direct split at degree n/2","kind":"theorem","summary":"[Direct split at degree n/2] Let \\omega \\neq 0 and let c be a coefficient family. Then there ar…","labels":["ACP.split_multilinear_at_half_degree_direct"],"detail_key":"p14"},{"id":"n21362","layer":"informal","project":"p14","title":"Bad count of a polynomial on the root cube","kind":"definition","summary":"[Bad count of a polynomial on the root cube] The number of points x \\in \\1,\\omega\\^n on which P…","labels":["ACP.rootCubeBadCount"],"detail_key":"p14"},{"id":"n21363","layer":"informal","project":"p14","title":"Hamming distance between functions on the root cube","kind":"definition","summary":"[Hamming distance between functions on the root cube] The number of points x \\in \\1,\\omega\\^n o…","labels":["ACP.rootCubeFunctionBadCount"],"detail_key":"p14"},{"id":"n21364","layer":"informal","project":"p14","title":"Hamming ball on the root cube","kind":"definition","summary":"[Hamming ball on the root cube] The finite set of functions f : \\1,\\omega\\^n \\to K that differ…","labels":["ACP.rootCubeBall"],"detail_key":"p14"},{"id":"n21365","layer":"informal","project":"p14","title":"Covering by Hamming balls: counting bound","kind":"theorem","summary":"[Covering by Hamming balls: counting bound] Let \\alpha, \\beta, Cand be finite and let center :…","labels":["ACP.finite_cover_by_hamming_balls_card_bound"],"detail_key":"p14"},{"id":"n21366","layer":"informal","project":"p14","title":"Finite counting obstruction on the root cube","kind":"theorem","summary":"[Finite counting obstruction on the root cube] Suppose a finite family (poly\\,c)_c \\in Cand rep…","labels":["ACP.rootCube_counting_obstruction"],"detail_key":"p14"},{"id":"n21367","layer":"informal","project":"p14","title":"Approximating the top monomial approximates everything","kind":"theorem","summary":"[Approximating the top monomial approximates everything] Assume \\omega \\neq 0 and that every fu…","labels":["ACP.rootProd_approx_implies_all_functions_approx"],"detail_key":"p14"},{"id":"n21368","layer":"informal","project":"p14","title":"No low-degree approximant to the top monomial","kind":"theorem","summary":"[No low-degree approximant to the top monomial] Assume \\omega \\neq 0, that every function on \\1…","labels":["ACP.no_low_degree_rootProd_approx"],"detail_key":"p14"},{"id":"n21369","layer":"informal","project":"p14","title":"Top-monomial inapproximability from finite counting","kind":"theorem","summary":"[Top-monomial inapproximability from finite counting] The same conclusion as the previous theor…","labels":["ACP.no_low_degree_rootProd_approx_of_finite_counting"],"detail_key":"p14"},{"id":"n21370","layer":"informal","project":"p14","title":"Subset Sum problem","kind":"definition","summary":"[Subset Sum problem] Given a weight function w : U \\to N on a finite type U and a target T \\in…","labels":["SubsetSumToPartition.SubsetSum"],"detail_key":"p14"},{"id":"n21371","layer":"informal","project":"p14","title":"Partition problem","kind":"definition","summary":"[Partition problem] Given a weight function v : U \\to N on a finite type U, the predicate asser…","labels":["SubsetSumToPartition.Partition"],"detail_key":"p14"},{"id":"n21372","layer":"informal","project":"p14","title":"Reduction weight function","kind":"definition","summary":"[Reduction weight function] The weight function on the augmented universe U \\oplus Bool built f…","labels":["SubsetSumToPartition.partitionWeight"],"detail_key":"p14"},{"id":"n21373","layer":"informal","project":"p14","title":"Completeness of the reduction","kind":"theorem","summary":"[Completeness of the reduction] Let w : U \\to N and T \\in N satisfy T \\le \\sum_a w(a). If the S…","labels":["SubsetSumToPartition.SubsetSumToPartitionCompleteness"],"detail_key":"p14"},{"id":"n21374","layer":"informal","project":"p14","title":"Soundness of the reduction","kind":"theorem","summary":"[Soundness of the reduction] Let w : U \\to N and T \\in N satisfy T \\le \\sum_a w(a). If the Part…","labels":["SubsetSumToPartition.SubsetSumToPartitionSoundness"],"detail_key":"p14"},{"id":"n21375","layer":"informal","project":"p14","title":"Subset Sum reduces to Partition","kind":"theorem","summary":"[Subset Sum reduces to Partition] For w : U \\to N and T \\in N with T \\le \\sum_a w(a), the Subse…","labels":["SubsetSumToPartition.SubsetSumToPartitionReduction"],"detail_key":"p14"},{"id":"n21376","layer":"informal","project":"p14","title":"Pure computation in the time monad","kind":"definition","summary":"[Pure computation in the time monad] Lifts a value a : \\alpha into the computation \\langle a, 0…","labels":["TimeM.pure"],"detail_key":"p14"},{"id":"n21377","layer":"informal","project":"p14","title":"Sequential composition in the time monad","kind":"definition","summary":"[Sequential composition in the time monad] Given m : \\textttTimeM T α and f : \\alpha \\to \\textt…","labels":["TimeM.bind"],"detail_key":"p14"},{"id":"n21378","layer":"informal","project":"p14","title":"Return value of \\textttpure","kind":"theorem","summary":"[Return value of \\textttpure] The value returned by pure\\,a is a.","labels":["TimeM.ret_pure"],"detail_key":"p14"},{"id":"n21379","layer":"informal","project":"p14","title":"Return value of a bind","kind":"theorem","summary":"[Return value of a bind] The value returned by the bind of m with f is the value returned by f\\…","labels":["TimeM.ret_bind"],"detail_key":"p14"},{"id":"n21380","layer":"informal","project":"p14","title":"Return value of a map","kind":"theorem","summary":"[Return value of a map] For f : \\alpha \\to \\beta and x : \\textttTimeM T α, the value returned b…","labels":["TimeM.ret_map"],"detail_key":"p14"},{"id":"n21381","layer":"informal","project":"p14","title":"Return value of \\textttseqRight","kind":"theorem","summary":"[Return value of \\textttseqRight] The right-sequencing of x and y returns the value returned by…","labels":["TimeM.ret_seqRight"],"detail_key":"p14"},{"id":"n21382","layer":"informal","project":"p14","title":"Return value of \\textttseqLeft","kind":"theorem","summary":"[Return value of \\textttseqLeft] The left-sequencing of x and y returns x.ret.","labels":["TimeM.ret_seqLeft"],"detail_key":"p14"},{"id":"n21383","layer":"informal","project":"p14","title":"Return value of \\textttseq","kind":"theorem","summary":"[Return value of \\textttseq] For f : \\textttTimeM T (α → β) and x : \\textttUnit → TimeM T α, th…","labels":["TimeM.ret_seq"],"detail_key":"p14"},{"id":"n21384","layer":"informal","project":"p14","title":"Time cost of a bind","kind":"theorem","summary":"[Time cost of a bind] The time cost of the bind of m with f is m.time + (f\\,m.ret).time.","labels":["TimeM.time_bind"],"detail_key":"p14"},{"id":"n21385","layer":"informal","project":"p14","title":"Time cost of \\textttpure","kind":"theorem","summary":"[Time cost of \\textttpure] The time cost of pure\\,a is 0.","labels":["TimeM.time_pure"],"detail_key":"p14"},{"id":"n21386","layer":"informal","project":"p14","title":"Time cost of a map","kind":"theorem","summary":"[Time cost of a map] Mapping a function over a computation does not change its time cost: it is…","labels":["TimeM.time_map"],"detail_key":"p14"},{"id":"n21387","layer":"informal","project":"p14","title":"Time cost of \\textttseqRight","kind":"theorem","summary":"[Time cost of \\textttseqRight] The time cost of the right-sequencing of x and y is x.time + (y\\…","labels":["TimeM.time_seqRight"],"detail_key":"p14"},{"id":"n21388","layer":"informal","project":"p14","title":"Time cost of \\textttseqLeft","kind":"theorem","summary":"[Time cost of \\textttseqLeft] The time cost of the left-sequencing of x and y is x.time + (y\\,(…","labels":["TimeM.time_seqLeft"],"detail_key":"p14"},{"id":"n21389","layer":"informal","project":"p14","title":"Time cost of \\textttseq","kind":"theorem","summary":"[Time cost of \\textttseq] The time cost of seq\\,f\\,x is f.time + (x\\,()).time.","labels":["TimeM.time_seq"],"detail_key":"p14"},{"id":"n21390","layer":"informal","project":"p14","title":"Tick","kind":"definition","summary":"[Tick] The computation tick\\,c returns the trivial value and records time cost c; it is the bas…","labels":["TimeM.tick"],"detail_key":"p14"},{"id":"n21391","layer":"informal","project":"p14","title":"Return value of a tick","kind":"theorem","summary":"[Return value of a tick] The value returned by tick\\,c is the unit value.","labels":["TimeM.ret_tick"],"detail_key":"p14"},{"id":"n21392","layer":"informal","project":"p14","title":"Time cost of a tick","kind":"theorem","summary":"[Time cost of a tick] The time cost of tick\\,c is c.","labels":["TimeM.time_tick"],"detail_key":"p14"},{"id":"n21393","layer":"informal","project":"p14","title":"Maximum of two naturals, timed","kind":"definition","summary":"[Maximum of two naturals, timed] The timed computation returning the larger of two natural numb…","labels":["max2"],"detail_key":"p14"},{"id":"n21394","layer":"informal","project":"p14","title":"Median of three, timed","kind":"definition","summary":"[Median of three, timed] For a,b,c in a linear order, the timed computation returning a median…","labels":["median3"],"detail_key":"p14"},{"id":"n21395","layer":"informal","project":"p14","title":"Correctness of the timed median","kind":"theorem","summary":"[Correctness of the timed median] The value returned by median3\\,a\\,b\\,c is a median of a, b, c…","labels":["median3_correct"],"detail_key":"p14"},{"id":"n21396","layer":"informal","project":"p14","title":"Comparison count of the timed median","kind":"theorem","summary":"[Comparison count of the timed median] The computation median3\\,a\\,b\\,c records a time cost of…","labels":["median3_time"],"detail_key":"p14"},{"id":"n21397","layer":"informal","project":"p14","title":"Timed ordered insertion","kind":"definition","summary":"[Timed ordered insertion] Inserts x into a list, walking the list until the first element y wit…","labels":["insert"],"detail_key":"p14"},{"id":"n21398","layer":"informal","project":"p14","title":"Timed insertion sort","kind":"definition","summary":"[Timed insertion sort] Insertion sort in the time monad: recursively sort the tail and then ins…","labels":["insertionSort"],"detail_key":"p14"},{"id":"n21399","layer":"informal","project":"p14","title":"Sortedness by adjacent pairs","kind":"definition","summary":"[Sortedness by adjacent pairs] The predicate on lists defined recursively: the empty list and s…","labels":["IsSorted"],"detail_key":"p14"},{"id":"n21400","layer":"informal","project":"p14","title":"Adjacent sortedness is a chain","kind":"theorem","summary":"[Adjacent sortedness is a chain] For any list \\mathitxs, the predicate \\textttIsSorted holds of…","labels":["isSorted_iff_isChain"],"detail_key":"p14"},{"id":"n21401","layer":"informal","project":"p14","title":"Adjacent sortedness equals pairwise sortedness","kind":"theorem","summary":"[Adjacent sortedness equals pairwise sortedness] For any list \\mathitxs, the predicate \\textttI…","labels":["isSorted_iff_sorted"],"detail_key":"p14"},{"id":"n21402","layer":"informal","project":"p14","title":"Value returned by the timed insertion","kind":"theorem","summary":"[Value returned by the timed insertion] The value returned by the timed insertion of x into \\ma…","labels":["ret_insert"],"detail_key":"p14"},{"id":"n21403","layer":"informal","project":"p14","title":"Value returned by the timed insertion sort","kind":"theorem","summary":"[Value returned by the timed insertion sort] The value returned by the timed insertion sort of…","labels":["ret_insertionSort"],"detail_key":"p14"},{"id":"n21404","layer":"informal","project":"p14","title":"Length after a timed insertion","kind":"theorem","summary":"[Length after a timed insertion] The list returned by inserting x into \\mathitxs has length \\le…","labels":["length_ret_insert"],"detail_key":"p14"},{"id":"n21405","layer":"informal","project":"p14","title":"Length after a timed insertion sort","kind":"theorem","summary":"[Length after a timed insertion sort] The list returned by the timed insertion sort of \\mathitx…","labels":["length_ret_insertionSort"],"detail_key":"p14"},{"id":"n21406","layer":"informal","project":"p14","title":"Timed insertion is a permutation","kind":"theorem","summary":"[Timed insertion is a permutation] The list returned by inserting x into \\mathitxs is a permuta…","labels":["insert_perm"],"detail_key":"p14"},{"id":"n21407","layer":"informal","project":"p14","title":"Timed insertion preserves sortedness","kind":"theorem","summary":"[Timed insertion preserves sortedness] If \\mathitxs is sorted, then the list returned by insert…","labels":["insert_sorted"],"detail_key":"p14"},{"id":"n21408","layer":"informal","project":"p14","title":"Timed insertion sort is a permutation","kind":"theorem","summary":"[Timed insertion sort is a permutation] The list returned by the timed insertion sort of \\mathi…","labels":["insertionSort_perm"],"detail_key":"p14"},{"id":"n21409","layer":"informal","project":"p14","title":"Timed insertion sort returns a sorted list","kind":"theorem","summary":"[Timed insertion sort returns a sorted list] For every list \\mathitxs, the list returned by the…","labels":["insertionSort_sorted"],"detail_key":"p14"},{"id":"n21410","layer":"informal","project":"p14","title":"Functional correctness of insertion sort","kind":"theorem","summary":"[Functional correctness of insertion sort] For every list \\mathitxs, the output of the timed in…","labels":["insertionSort_correct"],"detail_key":"p14"},{"id":"n21411","layer":"informal","project":"p14","title":"Insertion sort recurrence","kind":"definition","summary":"[Insertion sort recurrence] The worst-case comparison recurrence T(0) = 0 and T(n+1) = T(n) + n.","labels":["timeInsertionSortRec"],"detail_key":"p14"},{"id":"n21412","layer":"informal","project":"p14","title":"Cost of one insertion","kind":"theorem","summary":"[Cost of one insertion] Inserting x into a list \\mathitxs costs at most \\left\\lvert \\mathitxs\\r…","labels":["time_insert_le"],"detail_key":"p14"},{"id":"n21413","layer":"informal","project":"p14","title":"Insertion sort meets its recurrence","kind":"theorem","summary":"[Insertion sort meets its recurrence] The time cost recorded by the timed insertion sort of \\ma…","labels":["time_insertionSort_le_rec"],"detail_key":"p14"},{"id":"n21414","layer":"informal","project":"p14","title":"Quadratic bound for the recurrence","kind":"theorem","summary":"[Quadratic bound for the recurrence] For every n, the recurrence satisfies T(n) \\le n^2.","labels":["timeInsertionSortRec_le_sq"],"detail_key":"p14"},{"id":"n21415","layer":"informal","project":"p14","title":"Insertion sort performs at most n^2 comparisons","kind":"theorem","summary":"[Insertion sort performs at most n^2 comparisons] The time cost recorded by the timed insertion…","labels":["time_insertionSort_le_sq"],"detail_key":"p14"},{"id":"n21416","layer":"informal","project":"p14","title":"Core RSA exponent identity","kind":"lemma","summary":"[Core RSA exponent identity] Let p be prime. For every c \\in N and every x \\in Z/pZ, \\[ x^\\,1 +…","labels":["RSA.rsa_core"],"detail_key":"p14"},{"id":"n21417","layer":"informal","project":"p14","title":"RSA identity from an exponent factorization","kind":"lemma","summary":"[RSA identity from an exponent factorization] Let p be prime and m, ed, c \\in N with ed = 1 + c…","labels":["RSA.rsa_zmod_of_factor"],"detail_key":"p14"},{"id":"n21418","layer":"informal","project":"p14","title":"RSA identity modulo p","kind":"lemma","summary":"[RSA identity modulo p] If p is prime and ed = 1 + k(p-1)(q-1), then m^ed \\equiv m in Z/pZ for…","labels":["RSA.rsa_zmod_p"],"detail_key":"p14"},{"id":"n21419","layer":"informal","project":"p14","title":"RSA identity modulo q","kind":"lemma","summary":"[RSA identity modulo q] If q is prime and ed = 1 + k(p-1)(q-1), then m^ed \\equiv m in Z/qZ for…","labels":["RSA.rsa_zmod_q"],"detail_key":"p14"},{"id":"n21420","layer":"informal","project":"p14","title":"Combining the two factors by CRT","kind":"lemma","summary":"[Combining the two factors by CRT] Let p and q be coprime and suppose m^ed \\equiv m both in Z/p…","labels":["RSA.rsa_crt"],"detail_key":"p14"},{"id":"n21421","layer":"informal","project":"p14","title":"CRT combination in power form","kind":"lemma","summary":"[CRT combination in power form] The same statement written with the power taken inside the quot…","labels":["RSA.rsa_crt_pow"],"detail_key":"p14"},{"id":"n21422","layer":"informal","project":"p14","title":"RSA public key","kind":"definition","summary":"[RSA public key] A public key consists of a modulus n \\in N and a public exponent e \\in N.","labels":["RSA.PublicKey"],"detail_key":"p14"},{"id":"n21423","layer":"informal","project":"p14","title":"RSA secret key","kind":"definition","summary":"[RSA secret key] A secret key consists of a public key together with naturals p, q, d, k and pr…","labels":["RSA.SecretKey"],"detail_key":"p14"},{"id":"n21424","layer":"informal","project":"p14","title":"RSA encryption","kind":"definition","summary":"[RSA encryption] Encryption of a message m \\in N under a public key is \\bar m^\\,e \\in Z/nZ, usi…","labels":["RSA.encrypt"],"detail_key":"p14"},{"id":"n21425","layer":"informal","project":"p14","title":"RSA decryption","kind":"definition","summary":"[RSA decryption] Decryption of a ciphertext c \\in Z/nZ under a secret key is c^\\,d, where d is…","labels":["RSA.decrypt"],"detail_key":"p14"},{"id":"n21426","layer":"informal","project":"p14","title":"Correctness of RSA","kind":"theorem","summary":"[Correctness of RSA] For any secret key and any message m \\in N, decrypting the encryption of m…","labels":["RSA.rsa_correctness"],"detail_key":"p14"},{"id":"n21427","layer":"informal","project":"p14","title":"Multigraph","kind":"definition","summary":"[Multigraph] A multigraph on a vertex type \\alpha consists of a finite vertex set together with…","labels":["Multigraph"],"detail_key":"p14"},{"id":"n21428","layer":"informal","project":"p14","title":"Edge count of a multigraph","kind":"definition","summary":"[Edge count of a multigraph] The number of edges of G counted with multiplicity, i.e.\\ the card…","labels":["Multigraph.edgeCount"],"detail_key":"p14"},{"id":"n21429","layer":"informal","project":"p14","title":"Vertex count of a multigraph","kind":"definition","summary":"[Vertex count of a multigraph] The number of vertices of G, i.e.\\ the cardinality of its vertex…","labels":["Multigraph.vertexCount"],"detail_key":"p14"},{"id":"n21430","layer":"informal","project":"p14","title":"Contraction of an edge","kind":"definition","summary":"[Contraction of an edge] Given an edge e \\in E(G) with endpoints u and v, the contracted multig…","labels":["Multigraph.contract"],"detail_key":"p14"},{"id":"n21431","layer":"informal","project":"p14","title":"Contraction drops the vertex count by one","kind":"lemma","summary":"[Contraction drops the vertex count by one] If e \\in E(G) is not a diagonal (self-loop) pair, t…","labels":["contract_vertex_count"],"detail_key":"p14"},{"id":"n21432","layer":"informal","project":"p14","title":"Contraction drops at least one edge","kind":"lemma","summary":"[Contraction drops at least one edge] For every edge e \\in E(G), the contracted multigraph sati…","labels":["contract_edge_count_le"],"detail_key":"p14"},{"id":"n21433","layer":"informal","project":"p14","title":"Cut of a multigraph","kind":"definition","summary":"[Cut of a multigraph] A cut of G is a subset S \\subseteq V(G) that is nonempty and whose comple…","labels":["MulCut"],"detail_key":"p14"},{"id":"n21434","layer":"informal","project":"p14","title":"Crossing edges of a cut","kind":"definition","summary":"[Crossing edges of a cut] The multiset of edges of G that cross the cut C, i.e.\\ those edges ha…","labels":["MulCut.crossingEdges"],"detail_key":"p14"},{"id":"n21435","layer":"informal","project":"p14","title":"Crossing predicate for an unordered edge","kind":"definition","summary":"[Crossing predicate for an unordered edge] For a finite set S and an unordered pair e, the pred…","labels":["Crosses"],"detail_key":"p14"},{"id":"n21436","layer":"informal","project":"p14","title":"Crossing in terms of chosen representatives","kind":"lemma","summary":"[Crossing in terms of chosen representatives] For every finite set S and unordered pair e, the…","labels":["crosses_iff_out"],"detail_key":"p14"},{"id":"n21437","layer":"informal","project":"p14","title":"Crossing edges as a filter by the crossing predicate","kind":"lemma","summary":"[Crossing edges as a filter by the crossing predicate] For every cut C of G, the multiset of cr…","labels":["crossingEdges_eq_filter_crosses"],"detail_key":"p14"},{"id":"n21438","layer":"informal","project":"p14","title":"Size of a cut","kind":"definition","summary":"[Size of a cut] The size of a cut C is the number of crossing edges, counted with multiplicity.","labels":["MulCut.size"],"detail_key":"p14"},{"id":"n21439","layer":"informal","project":"p14","title":"Minimum cut","kind":"definition","summary":"[Minimum cut] A cut C of G is a minimum cut when its size is at most the size of every cut C' o…","labels":["IsMulMinCut"],"detail_key":"p14"},{"id":"n21440","layer":"informal","project":"p14","title":"Cut induced on a contracted multigraph","kind":"definition","summary":"[Cut induced on a contracted multigraph] If e \\in E(G) does not cross the cut C, then C induces…","labels":["MulCut.contractedCut"],"detail_key":"p14"},{"id":"n21441","layer":"informal","project":"p14","title":"Non-crossing contraction preserves cut size","kind":"lemma","summary":"[Non-crossing contraction preserves cut size] If e \\in E(G) does not cross the cut C, then the…","labels":["cut_size_preserved"],"detail_key":"p14"},{"id":"n21442","layer":"informal","project":"p14","title":"Non-crossing contraction preserves minimality","kind":"lemma","summary":"[Non-crossing contraction preserves minimality] If C is a minimum cut of G and e \\in E(G) does…","labels":["mincut_preserved_of_non_crossing"],"detail_key":"p14"},{"id":"n21443","layer":"informal","project":"p14","title":"One contraction step of Karger's algorithm","kind":"definition","summary":"[One contraction step of Karger's algorithm] Given a multigraph G with at least one edge, a ste…","labels":["kargerStep"],"detail_key":"p14"},{"id":"n21444","layer":"informal","project":"p14","title":"A full contraction run","kind":"definition","summary":"[A full contraction run] A run of the algorithm on G_0 is indexed by a sequence of edge choices…","labels":["kargerRun"],"detail_key":"p14"},{"id":"n21445","layer":"informal","project":"p14","title":"Termination condition of the algorithm","kind":"definition","summary":"[Termination condition of the algorithm] The predicate stating that G is a valid output of the…","labels":["KargerOutput"],"detail_key":"p14"},{"id":"n21446","layer":"informal","project":"p14","title":"Degree of a vertex","kind":"definition","summary":"[Degree of a vertex] The degree of v in a loopless multigraph G is the number of edges incident…","labels":["Multigraph.degree"],"detail_key":"p14"},{"id":"n21447","layer":"informal","project":"p14","title":"Membership in an unordered pair identifies a representative","kind":"lemma","summary":"[Membership in an unordered pair identifies a representative] If z belongs to the unordered pai…","labels":["eq_out_or_eq_out_of_mem"],"detail_key":"p14"},{"id":"n21448","layer":"informal","project":"p14","title":"Endpoint multiplicity equals incidence count","kind":"lemma","summary":"[Endpoint multiplicity equals incidence count] For a multiset m of unordered pairs and a vertex…","labels":["count_bind_endpoints_eq_card_filter"],"detail_key":"p14"},{"id":"n21449","layer":"informal","project":"p14","title":"Singleton cut has size equal to the degree","kind":"lemma","summary":"[Singleton cut has size equal to the degree] Let v \\in V(G) be a vertex such that V(G) \\setminu…","labels":["singleton_cut_size_eq_degree"],"detail_key":"p14"},{"id":"n21450","layer":"informal","project":"p14","title":"Removing a vertex leaves a nonempty vertex set","kind":"lemma","summary":"[Removing a vertex leaves a nonempty vertex set] If G admits a cut C, then for every vertex v t…","labels":["erase_nonempty_of_cut"],"detail_key":"p14"},{"id":"n21451","layer":"informal","project":"p14","title":"Min-cut size is at most every degree","kind":"lemma","summary":"[Min-cut size is at most every degree] If C is a minimum cut of G and v \\in V(G), then \\left\\lv…","labels":["mincut_le_degree"],"detail_key":"p14"},{"id":"n21452","layer":"informal","project":"p14","title":"Endpoints of edges are vertices","kind":"lemma","summary":"[Endpoints of edges are vertices] Any x occurring in the multiset of all edge endpoints of G be…","labels":["endpoints_mem_vertices"],"detail_key":"p14"},{"id":"n21453","layer":"informal","project":"p14","title":"Summing multiplicities over a covering set","kind":"lemma","summary":"[Summing multiplicities over a covering set] If every element of a multiset m lies in the finit…","labels":["sum_counts_eq_card_of_subset"],"detail_key":"p14"},{"id":"n21454","layer":"informal","project":"p14","title":"Handshake lemma","kind":"lemma","summary":"[Handshake lemma] For every multigraph G, \\[ \\sum_v \\in V(G) \\deg(v) = 2\\left\\lvert E(G)\\right\\…","labels":["sum_degrees_eq_twice_edgeCount"],"detail_key":"p14"},{"id":"n21455","layer":"informal","project":"p14","title":"Edge count lower bound from a minimum degree","kind":"lemma","summary":"[Edge count lower bound from a minimum degree] If every vertex of G has degree at least k, then…","labels":["edge_count_lower_bound"],"detail_key":"p14"},{"id":"n21456","layer":"informal","project":"p14","title":"Survival probability of one step","kind":"definition","summary":"[Survival probability of one step] For a cut size c and a current edge count m, the probability…","labels":["survivalProb"],"detail_key":"p14"},{"id":"n21457","layer":"informal","project":"p14","title":"Edge count invariant along the algorithm","kind":"lemma","summary":"[Edge count invariant along the algorithm] Let C_0 be a minimum cut of G_0 and let i < \\left\\lv…","labels":["edge_count_invariant"],"detail_key":"p14"},{"id":"n21458","layer":"informal","project":"p14","title":"Telescoping product","kind":"lemma","summary":"[Telescoping product] For every n \\ge 2, \\[ \\prod_i=0^n-3 \\fracn - i - 2n - i = \\frac2n(n-1) .…","labels":["telescope_prod"],"detail_key":"p14"},{"id":"n21459","layer":"informal","project":"p14","title":"Survival probability of a fixed min-cut","kind":"theorem","summary":"[Survival probability of a fixed min-cut] Let C be a minimum cut of a multigraph G with n = \\le…","labels":["karger_survival_prob_algorithmic"],"detail_key":"p14"},{"id":"n21460","layer":"informal","project":"p14","title":"Repetition succeeds with high probability","kind":"theorem","summary":"[Repetition succeeds with high probability] Let n = \\left\\lvert V(G)\\right\\rvert \\ge 4. Repeati…","labels":["karger_whp"],"detail_key":"p14"},{"id":"n21461","layer":"informal","project":"p14","title":"Counting disjoint events of large probability","kind":"lemma","summary":"[Counting disjoint events of large probability] If p > 0 and m \\cdot p \\le 1 (as is the case fo…","labels":["card_le_of_disjoint_prob_lb"],"detail_key":"p14"},{"id":"n21462","layer":"informal","project":"p14","title":"Closed form for \\binomn2","kind":"lemma","summary":"[Closed form for \\binomn2] For n \\ge 2, \\[ \\binomn2 = \\fracn(n-1)2 \\] as real numbers.","labels":["choose_two"],"detail_key":"p14"},{"id":"n21463","layer":"informal","project":"p14","title":"Polynomial bound on a binomial coefficient","kind":"lemma","summary":"[Polynomial bound on a binomial coefficient] For all n and \\alpha, \\binomn2\\alpha \\le n^2\\alpha.","labels":["choose_le_pow"],"detail_key":"p14"},{"id":"n21464","layer":"informal","project":"p14","title":"At most \\binomn2 minimum cuts","kind":"theorem","summary":"[At most \\binomn2 minimum cuts] Let n = \\left\\lvert V(G)\\right\\rvert \\ge 2. If a family of numC…","labels":["num_mincuts_le_choose"],"detail_key":"p14"},{"id":"n21465","layer":"informal","project":"p14","title":"At most n^2\\alpha approximate minimum cuts","kind":"theorem","summary":"[At most n^2\\alpha approximate minimum cuts] Let \\alpha > 0 with 2\\alpha \\le n = \\left\\lvert V(…","labels":["num_alpha_mincuts_le"],"detail_key":"p14"},{"id":"n21466","layer":"informal","project":"p14","title":"State of the contraction process","kind":"definition","summary":"[State of the contraction process] A state consists of a multigraph G over \\alpha together with…","labels":["KargerTrace.State"],"detail_key":"p14"},{"id":"n21467","layer":"informal","project":"p14","title":"Contraction of a state along a non-crossing edge","kind":"definition","summary":"[Contraction of a state along a non-crossing edge] Given a state s, an edge e of s.G that does…","labels":["KargerTrace.contractState"],"detail_key":"p14"},{"id":"n21468","layer":"informal","project":"p14","title":"Surviving step","kind":"definition","summary":"[Surviving step] The relation asserting that state t arises from state s by contracting some ed…","labels":["KargerTrace.SurvivingStep"],"detail_key":"p14"},{"id":"n21469","layer":"informal","project":"p14","title":"Surviving run","kind":"definition","summary":"[Surviving run] The inductively defined relation of n consecutive surviving steps leading from…","labels":["KargerTrace.SurvivingRun"],"detail_key":"p14"},{"id":"n21470","layer":"informal","project":"p14","title":"Positive edge count from an edge","kind":"lemma","summary":"[Positive edge count from an edge] If e is an edge of a multigraph G, then G has positive edge…","labels":["KargerTrace.edgeCount_pos_of_mem"],"detail_key":"p14"},{"id":"n21471","layer":"informal","project":"p14","title":"A surviving step preserves cut size","kind":"lemma","summary":"[A surviving step preserves cut size] If t is obtained from s by a surviving step, then the dis…","labels":["KargerTrace.survivingStep_cut_size_eq"],"detail_key":"p14"},{"id":"n21472","layer":"informal","project":"p14","title":"A surviving step preserves minimality","kind":"lemma","summary":"[A surviving step preserves minimality] If t is obtained from s by a surviving step and s.C is…","labels":["KargerTrace.survivingStep_mincut"],"detail_key":"p14"},{"id":"n21473","layer":"informal","project":"p14","title":"A surviving step removes one vertex","kind":"lemma","summary":"[A surviving step removes one vertex] If t is obtained from s by a surviving step, then t.G has…","labels":["KargerTrace.survivingStep_vertexCount_eq"],"detail_key":"p14"},{"id":"n21474","layer":"informal","project":"p14","title":"A surviving step requires an edge","kind":"lemma","summary":"[A surviving step requires an edge] If some state t is reachable from s by a surviving step, th…","labels":["KargerTrace.survivingStep_edgeCount_pos"],"detail_key":"p14"},{"id":"n21475","layer":"informal","project":"p14","title":"A surviving run preserves cut size","kind":"lemma","summary":"[A surviving run preserves cut size] If t is reachable from s by a surviving run of any length,…","labels":["KargerTrace.survivingRun_cut_size_eq"],"detail_key":"p14"},{"id":"n21476","layer":"informal","project":"p14","title":"A surviving run preserves minimality","kind":"lemma","summary":"[A surviving run preserves minimality] If t is reachable from s by a surviving run and s.C is a…","labels":["KargerTrace.survivingRun_mincut"],"detail_key":"p14"},{"id":"n21477","layer":"informal","project":"p14","title":"An n-step surviving run removes n vertices","kind":"lemma","summary":"[An n-step surviving run removes n vertices] If t is reachable from s by a surviving run of n s…","labels":["KargerTrace.survivingRun_vertexCount_eq"],"detail_key":"p14"},{"id":"n21478","layer":"informal","project":"p14","title":"Trace-level edge-count invariant","kind":"lemma","summary":"[Trace-level edge-count invariant] If t is reachable from s by a surviving run and s.C is a min…","labels":["KargerTrace.survivingRun_edge_count_invariant"],"detail_key":"p14"},{"id":"n21479","layer":"informal","project":"p14","title":"Indexed trace edge-count invariant","kind":"lemma","summary":"[Indexed trace edge-count invariant] The same invariant, with the vertex count after n contract…","labels":["KargerTrace.survivingRun_edge_count_invariant_indexed"],"detail_key":"p14"},{"id":"n21480","layer":"informal","project":"p14","title":"Algebraic one-step survival bound","kind":"lemma","summary":"[Algebraic one-step survival bound] For naturals c, n, m with 2 \\le n, 0 < m and c\\,n \\le 2m, t…","labels":["KargerTrace.survivalProb_ge_factor"],"detail_key":"p14"},{"id":"n21481","layer":"informal","project":"p14","title":"One-step survival bound at a surviving state","kind":"lemma","summary":"[One-step survival bound at a surviving state] If a surviving step is possible from s, the dist…","labels":["KargerTrace.survivingStep_survivalProb_ge_factor"],"detail_key":"p14"},{"id":"n21482","layer":"informal","project":"p14","title":"One-step survival bound after i contractions","kind":"lemma","summary":"[One-step survival bound after i contractions] If t is reached from s by a surviving run of i s…","labels":["KargerTrace.survivingRun_survivalProb_ge_factor"],"detail_key":"p14"},{"id":"n21483","layer":"informal","project":"p14","title":"Non-crossing edges of a state","kind":"definition","summary":"[Non-crossing edges of a state] The sub-multiset of edges of s.G that do not cross the distingu…","labels":["KargerTrace.nonCrossingEdges"],"detail_key":"p14"},{"id":"n21484","layer":"informal","project":"p14","title":"Number of non-crossing choices","kind":"definition","summary":"[Number of non-crossing choices] The cardinality (with multiplicity) of the multiset of non-cro…","labels":["KargerTrace.nonCrossingChoiceCount"],"detail_key":"p14"},{"id":"n21485","layer":"informal","project":"p14","title":"Uniform one-step survival probability","kind":"definition","summary":"[Uniform one-step survival probability] The real number obtained as the number of non-crossing…","labels":["KargerTrace.uniformSurvivalProb"],"detail_key":"p14"},{"id":"n21486","layer":"informal","project":"p14","title":"Non-crossing choices count edges minus cut size","kind":"lemma","summary":"[Non-crossing choices count edges minus cut size] For every state s, the number of non-crossing…","labels":["KargerTrace.nonCrossingChoiceCount_eq_edgeCount_sub_cutSize"],"detail_key":"p14"},{"id":"n21487","layer":"informal","project":"p14","title":"Cut size bounded by edge count","kind":"lemma","summary":"[Cut size bounded by edge count] For every state s, s.C.size \\le s.G.edgeCount.","labels":["KargerTrace.cut_size_le_edgeCount"],"detail_key":"p14"},{"id":"n21488","layer":"informal","project":"p14","title":"Uniform choice realizes the survival probability","kind":"lemma","summary":"[Uniform choice realizes the survival probability] If s.G has at least one edge, the uniform su…","labels":["KargerTrace.uniformSurvivalProb_eq_survivalProb"],"detail_key":"p14"},{"id":"n21489","layer":"informal","project":"p14","title":"Sample space of edge occurrences","kind":"definition","summary":"[Sample space of edge occurrences] The finite set of pairs (edge, occurrence index) obtained fr…","labels":["KargerTrace.edgeChoiceSpace"],"detail_key":"p14"},{"id":"n21490","layer":"informal","project":"p14","title":"Surviving edge occurrence","kind":"definition","summary":"[Surviving edge occurrence] The predicate on an edge occurrence saying that its underlying edge…","labels":["KargerTrace.edgeChoiceSurvives"],"detail_key":"p14"},{"id":"n21491","layer":"informal","project":"p14","title":"Size of the occurrence space","kind":"lemma","summary":"[Size of the occurrence space] The occurrence sample space of a state has cardinality s.G.edgeC…","labels":["KargerTrace.edgeChoiceSpace_card"],"detail_key":"p14"},{"id":"n21492","layer":"informal","project":"p14","title":"Filtering occurrences by first coordinate","kind":"lemma","summary":"[Filtering occurrences by first coordinate] For a multiset m and a decidable predicate p, the n…","labels":["KargerTrace.toEnumFinset_filter_fst_card_eq_filter_card"],"detail_key":"p14"},{"id":"n21493","layer":"informal","project":"p14","title":"Surviving occurrences count non-crossing choices","kind":"lemma","summary":"[Surviving occurrences count non-crossing choices] The number of surviving edge occurrences of…","labels":["KargerTrace.edgeChoiceSurvivalCount_eq_nonCrossingChoiceCount"],"detail_key":"p14"},{"id":"n21494","layer":"informal","project":"p14","title":"Survival ratio of the occurrence space","kind":"definition","summary":"[Survival ratio of the occurrence space] The ratio of the number of surviving edge occurrences…","labels":["KargerTrace.edgeChoiceSurvivalRatio"],"detail_key":"p14"},{"id":"n21495","layer":"informal","project":"p14","title":"Survival ratio equals uniform survival probability","kind":"lemma","summary":"[Survival ratio equals uniform survival probability] For every state, the occurrence-space surv…","labels":["KargerTrace.edgeChoiceSurvivalRatio_eq_uniformSurvivalProb"],"detail_key":"p14"},{"id":"n21496","layer":"informal","project":"p14","title":"An occurrence carries an actual edge","kind":"lemma","summary":"[An occurrence carries an actual edge] The first projection of any element of the occurrence sa…","labels":["KargerTrace.edgeChoice_mem_edges"],"detail_key":"p14"},{"id":"n21497","layer":"informal","project":"p14","title":"Surviving occurrences are non-crossing","kind":"lemma","summary":"[Surviving occurrences are non-crossing] If an edge occurrence of s survives, then its underlyi…","labels":["KargerTrace.not_mem_crossingEdges_of_edgeChoiceSurvives"],"detail_key":"p14"},{"id":"n21498","layer":"informal","project":"p14","title":"Successor state of a surviving occurrence","kind":"definition","summary":"[Successor state of a surviving occurrence] Given a state s and a sampled edge occurrence that…","labels":["KargerTrace.stateAfterSurvivingChoice"],"detail_key":"p14"},{"id":"n21499","layer":"informal","project":"p14","title":"Sampling a surviving occurrence gives a surviving step","kind":"lemma","summary":"[Sampling a surviving occurrence gives a surviving step] For any surviving edge occurrence at s…","labels":["KargerTrace.stateAfterSurvivingChoice_step"],"detail_key":"p14"},{"id":"n21500","layer":"informal","project":"p14","title":"Recursive survival probability","kind":"definition","summary":"[Recursive survival probability] The finite probability that the distinguished cut survives the…","labels":["KargerTrace.survivalEventProb"],"detail_key":"p14"},{"id":"n21501","layer":"informal","project":"p14","title":"Base case of the recursive survival probability","kind":"lemma","summary":"[Base case of the recursive survival probability] With no remaining contractions the survival p…","labels":["KargerTrace.survivalEventProb_zero"],"detail_key":"p14"},{"id":"n21502","layer":"informal","project":"p14","title":"Recursion step of the survival probability","kind":"lemma","summary":"[Recursion step of the survival probability] The unfolding equation: the survival probability f…","labels":["KargerTrace.survivalEventProb_succ"],"detail_key":"p14"},{"id":"n21503","layer":"informal","project":"p14","title":"Surviving occurrences in the attached sample space","kind":"lemma","summary":"[Surviving occurrences in the attached sample space] Counting surviving occurrences in the atta…","labels":["KargerTrace.edgeChoice_attach_filter_card"],"detail_key":"p14"},{"id":"n21504","layer":"informal","project":"p14","title":"Counting bound for a conditional sum","kind":"lemma","summary":"[Counting bound for a conditional sum] Let S be a finite set, p a decidable predicate, f : S \\t…","labels":["KargerTrace.card_mul_le_sum_ite_of_lower_bound"],"detail_key":"p14"},{"id":"n21505","layer":"informal","project":"p14","title":"One-step lower bound for the recursive probability","kind":"lemma","summary":"[One-step lower bound for the recursive probability] If the occurrence sample space of s is non…","labels":["KargerTrace.survivalEventProb_step_lower_bound"],"detail_key":"p14"},{"id":"n21506","layer":"informal","project":"p14","title":"Survival ratio bounded below by the Karger factor","kind":"lemma","summary":"[Survival ratio bounded below by the Karger factor] If s.C is a min-cut of positive size and s.…","labels":["KargerTrace.edgeChoiceSurvivalRatio_ge_factor"],"detail_key":"p14"},{"id":"n21507","layer":"informal","project":"p14","title":"Recursive survival lower bound","kind":"theorem","summary":"[Recursive survival lower bound] For every n and every state s whose distinguished cut is a min…","labels":["KargerTrace.survivalEventProb_lower_bound"],"detail_key":"p14"},{"id":"n21508","layer":"informal","project":"p14","title":"Survival lower bound at the initial state","kind":"theorem","summary":"[Survival lower bound at the initial state] For a state s whose distinguished cut is a min-cut…","labels":["KargerTrace.survivalEventProb_initial_lower_bound"],"detail_key":"p14"},{"id":"n21509","layer":"informal","project":"p14","title":"Fixed min-cut survival probability","kind":"theorem","summary":"[Fixed min-cut survival probability] Human-facing form of the previous statement: a fixed min-c…","labels":["KargerTrace.fixed_mincut_survival_probability_lower_bound"],"detail_key":"p14"},{"id":"n21510","layer":"informal","project":"p14","title":"Graph-only occurrence space","kind":"definition","summary":"[Graph-only occurrence space] The finite set of edge occurrences of a multigraph G, the same co…","labels":["KargerTrace.graphChoiceSpace"],"detail_key":"p14"},{"id":"n21511","layer":"informal","project":"p14","title":"The two occurrence spaces agree","kind":"lemma","summary":"[The two occurrence spaces agree] For every state s, the graph-only occurrence space of s.G equ…","labels":["KargerTrace.graphChoiceSpace_eq_edgeChoiceSpace"],"detail_key":"p14"},{"id":"n21512","layer":"informal","project":"p14","title":"A graph occurrence carries an actual edge","kind":"lemma","summary":"[A graph occurrence carries an actual edge] The first projection of any element of the graph-on…","labels":["KargerTrace.graphChoice_mem_edges"],"detail_key":"p14"},{"id":"n21513","layer":"informal","project":"p14","title":"Graph after a sampled contraction","kind":"definition","summary":"[Graph after a sampled contraction] The multigraph obtained by contracting G along the edge und…","labels":["KargerTrace.graphAfterChoice"],"detail_key":"p14"},{"id":"n21514","layer":"informal","project":"p14","title":"Sampled contraction removes one vertex","kind":"lemma","summary":"[Sampled contraction removes one vertex] Contracting along any sampled edge occurrence decrease…","labels":["KargerTrace.graphAfterChoice_vertexCount_eq"],"detail_key":"p14"},{"id":"n21515","layer":"informal","project":"p14","title":"State contraction agrees with graph contraction","kind":"lemma","summary":"[State contraction agrees with graph contraction] The graph of the successor state after a surv…","labels":["KargerTrace.stateAfterSurvivingChoice_graph_eq"],"detail_key":"p14"},{"id":"n21516","layer":"informal","project":"p14","title":"Graph-only Karger run","kind":"definition","summary":"[Graph-only Karger run] The inductive relation asserting that H is reachable from G by a given…","labels":["KargerTrace.KargerGraphRun"],"detail_key":"p14"},{"id":"n21517","layer":"informal","project":"p14","title":"Vertex count along a graph run","kind":"lemma","summary":"[Vertex count along a graph run] If H is reachable from G by a graph run of \\textsteps contract…","labels":["KargerTrace.kargerGraphRun_vertexCount_eq"],"detail_key":"p14"},{"id":"n21518","layer":"informal","project":"p14","title":"A full graph run ends with two vertices","kind":"lemma","summary":"[A full graph run ends with two vertices] If G has at least two vertices and H is reachable fro…","labels":["KargerTrace.kargerGraphRun_full_vertexCount_eq_two"],"detail_key":"p14"},{"id":"n21519","layer":"informal","project":"p14","title":"A full graph run yields a Karger output","kind":"lemma","summary":"[A full graph run yields a Karger output] Under the same hypotheses, the terminal graph H is a…","labels":["KargerTrace.kargerGraphRun_full_output"],"detail_key":"p14"},{"id":"n21520","layer":"informal","project":"p14","title":"Every edge yields an occurrence","kind":"lemma","summary":"[Every edge yields an occurrence] If e is an edge of G, then the occurrence (e, 0) belongs to t…","labels":["KargerTrace.choice_mem_graphChoiceSpace_of_edge_mem"],"detail_key":"p14"},{"id":"n21521","layer":"informal","project":"p14","title":"A surviving step is realized by a graph occurrence","kind":"lemma","summary":"[A surviving step is realized by a graph occurrence] If t arises from s by a surviving step, th…","labels":["KargerTrace.survivingStep_graphChoice"],"detail_key":"p14"},{"id":"n21522","layer":"informal","project":"p14","title":"Surviving runs project to graph runs","kind":"lemma","summary":"[Surviving runs project to graph runs] A surviving run of \\textsteps steps from s to t induces…","labels":["KargerTrace.survivingRun_graphRun"],"detail_key":"p14"},{"id":"n21523","layer":"informal","project":"p14","title":"A full surviving run ends with a Karger output","kind":"lemma","summary":"[A full surviving run ends with a Karger output] If s.G has at least two vertices and t is reac…","labels":["KargerTrace.survivingRun_full_output"],"detail_key":"p14"},{"id":"n21524","layer":"informal","project":"p14","title":"Adaptive graph trace","kind":"definition","summary":"[Adaptive graph trace] An explicit indexed execution of the contraction loop: a family of graph…","labels":["KargerTrace.KargerGraphTrace"],"detail_key":"p14"},{"id":"n21525","layer":"informal","project":"p14","title":"Prefixes of a graph trace are graph runs","kind":"lemma","summary":"[Prefixes of a graph trace are graph runs] For i \\le \\textsteps, the first i stages of an adapt…","labels":["KargerTrace.kargerGraphTrace_prefix_run"],"detail_key":"p14"},{"id":"n21526","layer":"informal","project":"p14","title":"Vertex count along a graph trace","kind":"lemma","summary":"[Vertex count along a graph trace] For i \\le \\textsteps, the graph at index i of an adaptive tr…","labels":["KargerTrace.kargerGraphTrace_vertexCount_eq"],"detail_key":"p14"},{"id":"n21527","layer":"informal","project":"p14","title":"A full graph trace terminates with two vertices","kind":"lemma","summary":"[A full graph trace terminates with two vertices] If 2 \\le n and an adaptive graph trace of n-2…","labels":["KargerTrace.kargerGraphTrace_full_output"],"detail_key":"p14"},{"id":"n21528","layer":"informal","project":"p14","title":"Adaptive traces output two vertices","kind":"theorem","summary":"[Adaptive traces output two vertices] Human-facing termination statement: every adaptive graph…","labels":["KargerTrace.adaptive_karger_trace_outputs_two_vertices"],"detail_key":"p14"},{"id":"n21529","layer":"informal","project":"p14","title":"Algorithmic trace summary","kind":"theorem","summary":"[Algorithmic trace summary] For a state s whose distinguished cut is a min-cut of positive size…","labels":["KargerTrace.karger_algorithmic_trace_summary"],"detail_key":"p14"},{"id":"n21530","layer":"informal","project":"p14","title":"Main adaptive trace theorem","kind":"theorem","summary":"[Main adaptive trace theorem] Human-facing name for the previous conjunction: the survival lowe…","labels":["KargerTrace.adaptive_karger_trace_main"],"detail_key":"p14"},{"id":"n21531","layer":"informal","project":"p14","title":"Indexed surviving trace","kind":"definition","summary":"[Indexed surviving trace] A family of states indexed by N such that consecutive states at indic…","labels":["KargerTrace.SurvivingTrace"],"detail_key":"p14"},{"id":"n21532","layer":"informal","project":"p14","title":"Graph trace of a surviving trace","kind":"definition","summary":"[Graph trace of a surviving trace] The adaptive graph trace obtained from an indexed surviving…","labels":["KargerTrace.survivingTrace_toGraphTrace"],"detail_key":"p14"},{"id":"n21533","layer":"informal","project":"p14","title":"Prefixes of a surviving trace are surviving runs","kind":"lemma","summary":"[Prefixes of a surviving trace are surviving runs] For i \\le \\textsteps, the first i stages of…","labels":["KargerTrace.survivingTrace_prefix_run"],"detail_key":"p14"},{"id":"n21534","layer":"informal","project":"p14","title":"Vertex count along a surviving trace","kind":"lemma","summary":"[Vertex count along a surviving trace] For i \\le \\textsteps, the graph at index i of a survivin…","labels":["KargerTrace.survivingTrace_vertexCount_eq"],"detail_key":"p14"},{"id":"n21535","layer":"informal","project":"p14","title":"Per-step survival bound along a surviving trace","kind":"lemma","summary":"[Per-step survival bound along a surviving trace] If the initial cut of an indexed surviving tr…","labels":["KargerTrace.survivingTrace_survivalProb_ge_factor"],"detail_key":"p14"},{"id":"n21536","layer":"informal","project":"p14","title":"Product of factors bounds the product of survival probabilities","kind":"lemma","summary":"[Product of factors bounds the product of survival probabilities] If the initial cut is a min-c…","labels":["KargerTrace.survivingTrace_factor_product_le_survival_product"],"detail_key":"p14"},{"id":"n21537","layer":"informal","project":"p14","title":"Factor product in telescoping form","kind":"lemma","summary":"[Factor product in telescoping form] If a surviving trace has \\textsteps = n-2 and starts from…","labels":["KargerTrace.survivingTrace_factor_product_eq_telescope"],"detail_key":"p14"},{"id":"n21538","layer":"informal","project":"p14","title":"Product-level trace survival bound","kind":"theorem","summary":"[Product-level trace survival bound] For an indexed surviving trace of n-2 steps starting from…","labels":["KargerTrace.survivingTrace_product_survival_lower_bound"],"detail_key":"p14"},{"id":"n21539","layer":"informal","project":"p14","title":"Product of uniform survival probabilities","kind":"definition","summary":"[Product of uniform survival probabilities] The product over i < \\textsteps of the uniform one-…","labels":["KargerTrace.survivingTraceUniformSurvivalProduct"],"detail_key":"p14"},{"id":"n21540","layer":"informal","project":"p14","title":"Uniform product as a product of survival probabilities","kind":"lemma","summary":"[Uniform product as a product of survival probabilities] The uniform survival product of a trac…","labels":["KargerTrace.survivingTrace_uniform_product_eq_survival_product"],"detail_key":"p14"},{"id":"n21541","layer":"informal","project":"p14","title":"Uniform product with the initial cut size","kind":"lemma","summary":"[Uniform product with the initial cut size] The same product may be written using the initial c…","labels":["KargerTrace.survivingTrace_uniform_product_eq_initial_cut_survival_product"],"detail_key":"p14"},{"id":"n21542","layer":"informal","project":"p14","title":"Lower bound for the uniform survival product","kind":"theorem","summary":"[Lower bound for the uniform survival product] Under the same hypotheses as the product-level b…","labels":["KargerTrace.survivingTrace_uniform_survival_product_lower_bound"],"detail_key":"p14"},{"id":"n21543","layer":"informal","project":"p14","title":"Product of occurrence-space survival ratios","kind":"definition","summary":"[Product of occurrence-space survival ratios] The product over i < \\textsteps of the finite sam…","labels":["KargerTrace.survivingTraceEdgeChoiceSurvivalProduct"],"detail_key":"p14"},{"id":"n21544","layer":"informal","project":"p14","title":"The two concrete products agree","kind":"lemma","summary":"[The two concrete products agree] For every indexed surviving trace, the product of occurrence-…","labels":["KargerTrace.survivingTrace_edgeChoice_product_eq_uniform_product"],"detail_key":"p14"},{"id":"n21545","layer":"informal","project":"p14","title":"Lower bound for the occurrence-space survival product","kind":"theorem","summary":"[Lower bound for the occurrence-space survival product] For an indexed surviving trace of n-2 s…","labels":["KargerTrace.survivingTrace_edgeChoice_survival_product_lower_bound"],"detail_key":"p14"},{"id":"n21546","layer":"informal","project":"p14","title":"Exchanging the head edge for a new edge","kind":"lemma","summary":"[Exchanging the head edge for a new edge] Let \\mathitbase, \\mathitrest be edge lists on n verti…","labels":["Kruskal.exchange_take_head"],"detail_key":"p14"},{"id":"n21547","layer":"informal","project":"p14","title":"Exchanging the head edge, reversed orientation","kind":"lemma","summary":"[Exchanging the head edge, reversed orientation] The same exchange statement with the endpoints…","labels":["Kruskal.exchange_take_head'"],"detail_key":"p14"},{"id":"n21548","layer":"informal","project":"p14","title":"Exchange argument over a base edge list","kind":"lemma","summary":"[Exchange argument over a base edge list] Let \\mathitbase, S be edge lists and let e be an edge…","labels":["Kruskal.exchange_with_base"],"detail_key":"p14"},{"id":"n21549","layer":"informal","project":"p14","title":"Union--find form of the exchange argument","kind":"lemma","summary":"[Union--find form of the exchange argument] Let \\mathituf be a union--find state whose classes…","labels":["Kruskal.uf_exchange"],"detail_key":"p14"},{"id":"n21550","layer":"informal","project":"p14","title":"Removing a redundant edge from a candidate set","kind":"lemma","summary":"[Removing a redundant edge from a candidate set] Let \\mathituf be a union--find state, e an edg…","labels":["Kruskal.reduce_to_rest"],"detail_key":"p14"},{"id":"n21551","layer":"informal","project":"p14","title":"Union--find state after the greedy loop","kind":"definition","summary":"[Union--find state after the greedy loop] The union--find structure obtained by running the gre…","labels":["Kruskal.ufAfterProcessEdges"],"detail_key":"p14"},{"id":"n21552","layer":"informal","project":"p14","title":"Greedy loop induces the full merge partition","kind":"lemma","summary":"[Greedy loop induces the full merge partition] For every edge list and every initial state uf,…","labels":["Kruskal.ufAfterProcessEdges_partition"],"detail_key":"p14"},{"id":"n21553","layer":"informal","project":"p14","title":"Accumulator factors out of the greedy loop","kind":"lemma","summary":"[Accumulator factors out of the greedy loop] Running \\textttKruskal.processEdges with an accumu…","labels":["Kruskal.processEdges_acc"],"detail_key":"p14"},{"id":"n21554","layer":"informal","project":"p14","title":"Selected edges span the same partition","kind":"lemma","summary":"[Selected edges span the same partition] Merging the edges selected by the greedy loop into uf…","labels":["Kruskal.processEdges_same_partition"],"detail_key":"p14"},{"id":"n21555","layer":"informal","project":"p14","title":"Skipping an edge inside a component","kind":"lemma","summary":"[Skipping an edge inside a component] If the endpoints of e already have the same union--find r…","labels":["Kruskal.processEdges_skip"],"detail_key":"p14"},{"id":"n21556","layer":"informal","project":"p14","title":"Weight contributed by an accepted edge","kind":"lemma","summary":"[Weight contributed by an accepted edge] If the endpoints of e lie in different components of u…","labels":["Kruskal.processEdges_take_weight"],"detail_key":"p14"},{"id":"n21557","layer":"informal","project":"p14","title":"Optimality of the greedy edge-selection loop","kind":"lemma","summary":"[Optimality of the greedy edge-selection loop] Let uf be a union--find state whose classes are…","labels":["Kruskal.processEdges_optimal"],"detail_key":"p14"},{"id":"n21558","layer":"informal","project":"p14","title":"Kruskal's output spans the input","kind":"theorem","summary":"[Kruskal's output spans the input] For every weighted edge list E over Fin\\,n, the output krusk…","labels":["Kruskal.kruskal_spans"],"detail_key":"p14"},{"id":"n21559","layer":"informal","project":"p14","title":"Optimality of Kruskal's algorithm","kind":"theorem","summary":"[Optimality of Kruskal's algorithm] If S is a sublist of E that spans like E, then \\[ totalWeig…","labels":["Kruskal.kruskal_optimal"],"detail_key":"p14"},{"id":"n21560","layer":"informal","project":"p14","title":"Reflexivity of SamePartition","kind":"lemma","summary":"[Reflexivity of SamePartition] Every union--find state \\mathituf : \\mathttUF\\,n induces the sam…","labels":["Kruskal.UF.SamePartition.rfl"],"detail_key":"p14"},{"id":"n21561","layer":"informal","project":"p14","title":"Symmetry of SamePartition","kind":"lemma","summary":"[Symmetry of SamePartition] If \\mathttSamePartition\\,\\mathituf_1\\,\\mathituf_2 then \\mathttSameP…","labels":["Kruskal.UF.SamePartition.symm"],"detail_key":"p14"},{"id":"n21562","layer":"informal","project":"p14","title":"Transitivity of SamePartition","kind":"lemma","summary":"[Transitivity of SamePartition] If \\mathttSamePartition\\,\\mathituf_1\\,\\mathituf_2 and \\mathttSa…","labels":["Kruskal.UF.SamePartition.trans"],"detail_key":"p14"},{"id":"n21563","layer":"informal","project":"p14","title":"Merging within a component is a no-op","kind":"lemma","summary":"[Merging within a component is a no-op] If two vertices i, j already carry the same representat…","labels":["Kruskal.merge_noop"],"detail_key":"p14"},{"id":"n21564","layer":"informal","project":"p14","title":"Merging preserves existing components","kind":"lemma","summary":"[Merging preserves existing components] If \\mathituf\\,a = \\mathituf\\,b, then for any list of we…","labels":["Kruskal.mergeAll_preserves"],"detail_key":"p14"},{"id":"n21565","layer":"informal","project":"p14","title":"Merging characterizes reachability","kind":"lemma","summary":"[Merging characterizes reachability] Suppose the union--find state \\mathituf represents reachab…","labels":["Kruskal.mergeAll_iff_reach"],"detail_key":"p14"},{"id":"n21566","layer":"informal","project":"p14","title":"Correctness of merging from the initial state","kind":"lemma","summary":"[Correctness of merging from the initial state] Starting from the initial union--find state \\ma…","labels":["Kruskal.mergeAll_init_iff"],"detail_key":"p14"},{"id":"n21567","layer":"informal","project":"p14","title":"Merging respects SamePartition","kind":"lemma","summary":"[Merging respects SamePartition] If \\mathttSamePartition\\,\\mathituf_1\\,\\mathituf_2, then mergin…","labels":["Kruskal.mergeAll_congr"],"detail_key":"p14"},{"id":"n21568","layer":"informal","project":"p14","title":"Two merges commute up to partition","kind":"lemma","summary":"[Two merges commute up to partition] For any state \\mathituf and edges a, b, merging the endpoi…","labels":["Kruskal.merge_swap_partition"],"detail_key":"p14"},{"id":"n21569","layer":"informal","project":"p14","title":"Merging is permutation invariant","kind":"lemma","summary":"[Merging is permutation invariant] If the edge lists l_1 and l_2 are permutations of each other…","labels":["Kruskal.mergeAll_perm"],"detail_key":"p14"},{"id":"n21570","layer":"informal","project":"p14","title":"Merging along a list with one appended edge","kind":"lemma","summary":"[Merging along a list with one appended edge] Merging along l \\mathbin+\\!\\!+ [e] equals first m…","labels":["Kruskal.mergeAll_append_single"],"detail_key":"p14"},{"id":"n21571","layer":"informal","project":"p14","title":"Merging along a cons list","kind":"lemma","summary":"[Merging along a cons list] Merging along e :: S gives the same partition as first merging the…","labels":["Kruskal.mergeAll_cons_eq"],"detail_key":"p14"},{"id":"n21572","layer":"informal","project":"p14","title":"Moving a member edge to the end is a permutation","kind":"lemma","summary":"[Moving a member edge to the end is a permutation] If e \\in l for an edge list l, then l is a p…","labels":["Kruskal.erase_append_perm"],"detail_key":"p14"},{"id":"n21573","layer":"informal","project":"p14","title":"Convex-compact minimax theorem","kind":"theorem","summary":"[Convex-compact minimax theorem] Let X, Y \\subseteq R and f : R\\to R\\to R satisfy the bundled h…","labels":["OnlineLearning.convex_compact_minimax"],"detail_key":"p14"},{"id":"n21574","layer":"informal","project":"p14","title":"Finite upper image of the payoff","kind":"definition","summary":"[Finite upper image of the payoff] For sets X, Y \\subseteq R, a payoff f : R\\to R\\to R, and a f…","labels":["OnlineLearning.finiteUpperImage"],"detail_key":"p14"},{"id":"n21575","layer":"informal","project":"p14","title":"Upper image is nonempty","kind":"lemma","summary":"[Upper image is nonempty] If X is nonempty, then for every finite column sample u the upper ima…","labels":["OnlineLearning.finiteUpperImage_nonempty"],"detail_key":"p14"},{"id":"n21576","layer":"informal","project":"p14","title":"Upper image is convex","kind":"lemma","summary":"[Upper image is convex] If X is convex and x \\mapsto f(x,y) is convex on X for every y \\in Y, t…","labels":["OnlineLearning.finiteUpperImage_convex"],"detail_key":"p14"},{"id":"n21577","layer":"informal","project":"p14","title":"Upper image is upward closed","kind":"lemma","summary":"[Upper image is upward closed] If z \\in U(X,f,u) and z_y \\le z'_y for all y \\in u, then z' \\in…","labels":["OnlineLearning.finiteUpperImage_upper"],"detail_key":"p14"},{"id":"n21578","layer":"informal","project":"p14","title":"Upper image is open","kind":"lemma","summary":"[Upper image is open] For every finite column sample u, the upper image U(X,f,u) is an open sub…","labels":["OnlineLearning.finiteUpperImage_isOpen"],"detail_key":"p14"},{"id":"n21579","layer":"informal","project":"p14","title":"Coordinate expansion of a continuous linear functional","kind":"lemma","summary":"[Coordinate expansion of a continuous linear functional] Let \\iota be a finite type and L a con…","labels":["OnlineLearning.continuousLinearMap_pi_apply_eq_sum_single"],"detail_key":"p14"},{"id":"n21580","layer":"informal","project":"p14","title":"Separating functional has nonpositive coordinates","kind":"lemma","summary":"[Separating functional has nonpositive coordinates] Assume X is nonempty and L is a continuous…","labels":["OnlineLearning.separating_coordinate_nonpos"],"detail_key":"p14"},{"id":"n21581","layer":"informal","project":"p14","title":"Separating functional is nonzero","kind":"lemma","summary":"[Separating functional is nonzero] Under the same separation hypothesis L(z) < L(c) for all z i…","labels":["OnlineLearning.separating_functional_ne_zero"],"detail_key":"p14"},{"id":"n21582","layer":"informal","project":"p14","title":"Positive total weight of the separator","kind":"lemma","summary":"[Positive total weight of the separator] Under the same separation hypothesis, with X nonempty,…","labels":["OnlineLearning.separating_weight_sum_pos"],"detail_key":"p14"},{"id":"n21583","layer":"informal","project":"p14","title":"Finite sublevel intersections via separation","kind":"lemma","summary":"[Finite sublevel intersections via separation] Assume the convex-compact minimax hypotheses \\te…","labels":["OnlineLearning.finite_sublevel_intersections_by_separation"],"detail_key":"p14"},{"id":"n21584","layer":"informal","project":"p14","title":"Minimax identity from finite sublevel intersections","kind":"theorem","summary":"[Minimax identity from finite sublevel intersections] Assume the hypotheses \\textttOnlineLearni…","labels":["OnlineLearning.convex_compact_minimax_of_finite_sublevel_intersections"],"detail_key":"p14"},{"id":"n21585","layer":"informal","project":"p14","title":"Convex-compact minimax theorem by separation","kind":"theorem","summary":"[Convex-compact minimax theorem by separation] Under the convex-compact minimax hypotheses \\tex…","labels":["OnlineLearning.convex_compact_minimax_by_separation"],"detail_key":"p14"},{"id":"n21586","layer":"informal","project":"p14","title":"A vertex cut: a proper subset S whose deletion leaves G disconnected","kind":"definition","summary":"[A vertex cut: a proper subset S whose deletion leaves G disconnected] V' is a \\textttFinset V,…","labels":["SimpleGraph.IsVertexCut"],"detail_key":"p14"},{"id":"n21587","layer":"informal","project":"p14","title":"An edge cut: a set of edges of G whose deletion leaves G disconnected","kind":"definition","summary":"[An edge cut: a set of edges of G whose deletion leaves G disconnected] Rather than carry the b…","labels":["SimpleGraph.IsEdgeCut"],"detail_key":"p14"},{"id":"n21588","layer":"informal","project":"p14","title":"Vertex connectivity \\kappa(G)","kind":"definition","summary":"[Vertex connectivity \\kappa(G)] The book's case split is on \"G has a pair of distinct nonadjace…","labels":["SimpleGraph.vertexConnectivity"],"detail_key":"p14"},{"id":"n21589","layer":"informal","project":"p14","title":"Edge connectivity \\kappa'(G): the minimum size of an edge cut (\\textttsInf \\emptyset = 0)","kind":"definition","summary":"[Edge connectivity \\kappa'(G): the minimum size of an edge cut (\\textttsInf \\emptyset = 0)] \\te…","labels":["SimpleGraph.edgeConnectivity"],"detail_key":"p14"},{"id":"n21590","layer":"informal","project":"p14","title":"G is k-connected","kind":"definition","summary":"[G is k-connected] A literal abbreviation for \\textttk \\le G.vertexConnectivity.","labels":["SimpleGraph.IsKConnected"],"detail_key":"p14"},{"id":"n21591","layer":"informal","project":"p14","title":"G is k-edge-connected","kind":"definition","summary":"[G is k-edge-connected] A literal abbreviation for \\textttk \\le G.edgeConnectivity.","labels":["SimpleGraph.IsKEdgeConnected"],"detail_key":"p14"},{"id":"n21592","layer":"informal","project":"p14","title":"v is a cut vertex: G is connected but deleting v disconnects it","kind":"definition","summary":"[v is a cut vertex: G is connected but deleting v disconnects it] A \\textttSimpleGraph is loopl…","labels":["SimpleGraph.IsCutVertex"],"detail_key":"p14"},{"id":"n21593","layer":"informal","project":"p14","title":"A block: connected with no cut vertex","kind":"definition","summary":"[A block: connected with no cut vertex] Only the first sentence is formalised: \\textttG.Connect…","labels":["SimpleGraph.IsBlock"],"detail_key":"p14"},{"id":"n21594","layer":"informal","project":"p14","title":"Two u--v walks are internally disjoint if they share no internal vertex","kind":"definition","summary":"[Two u--v walks are internally disjoint if they share no internal vertex] Specialised to a fami…","labels":["SimpleGraph.InternallyDisjoint"],"detail_key":"p14"},{"id":"n21595","layer":"informal","project":"p14","title":"A finite family of u--v walks, pairwise internally disjoint (family form, for Menger)","kind":"definition","summary":"[A finite family of u--v walks, pairwise internally disjoint (family form, for Menger)] The fam…","labels":["SimpleGraph.PairwiseInternallyDisjoint"],"detail_key":"p14"},{"id":"n21596","layer":"informal","project":"p14","title":"Edge subdivision: replace e = uv by a length-2 path through a new vertex","kind":"definition","summary":"[Edge subdivision: replace e = uv by a length-2 path through a new vertex] Subdivision grows th…","labels":["SimpleGraph.subdivide"],"detail_key":"p14"},{"id":"n21597","layer":"informal","project":"p14","title":"edgeConnectivity\\_le\\_minDegree","kind":"theorem","summary":"[edgeConnectivity\\_le\\_minDegree] , second inequality: \\kappa' \\le \\delta. If G is trivial then…","labels":["SimpleGraph.edgeConnectivity_le_minDegree"],"detail_key":"p14"},{"id":"n21598","layer":"informal","project":"p14","title":"Helper for Theorem 3.1: \\kappa \\le \\nu - 1 always","kind":"theorem","summary":"[Helper for Theorem 3.1: \\kappa \\le \\nu - 1 always] A vertex cut is by definition a \\emphproper…","labels":["SimpleGraph.vertexConnectivity_le_card_pred"],"detail_key":"p14"},{"id":"n21599","layer":"informal","project":"p14","title":"Core of the block criterion: a connected graph on at least three vertices with no cut ver…","kind":"theorem","summary":"[Core of the block criterion: a connected graph on at least three vertices with no cut vertex h…","labels":["SimpleGraph.two_le_vertexConnectivity_of_no_cutVertex"],"detail_key":"p14"},{"id":"n21600","layer":"informal","project":"p14","title":"Base case of Theorem 3.1's induction: \\kappa' = 0 forces \\kappa = 0","kind":"theorem","summary":"[Base case of Theorem 3.1's induction: \\kappa' = 0 forces \\kappa = 0] The book's one-liner has…","labels":["SimpleGraph.vertexConnectivity_eq_zero_of_edgeConnectivity_eq_zero"],"detail_key":"p14"},{"id":"n21601","layer":"informal","project":"p14","title":"Successor step, part 1: from a minimum edge cut of size k + 1, deleting one of its edges…","kind":"theorem","summary":"[Successor step, part 1: from a minimum edge cut of size k + 1, deleting one of its edges drops…","labels":["SimpleGraph.exists_deleteEdge_edgeConnectivity_eq"],"detail_key":"p14"},{"id":"n21602","layer":"informal","project":"p14","title":"Successor step, part 2: deleting a single edge drops \\kappa by at most one","kind":"theorem","summary":"[Successor step, part 2: deleting a single edge drops \\kappa by at most one] This is the heart…","labels":["SimpleGraph.vertexConnectivity_le_deleteEdge_succ"],"detail_key":"p14"},{"id":"n21603","layer":"informal","project":"p14","title":"exists\\_isVertexCut\\_singleton\\_of\\_isBridge","kind":"theorem","summary":"[exists\\_isVertexCut\\_singleton\\_of\\_isBridge] , used by \\textttvertexConnectivity\\_le\\_deleteE…","labels":["SimpleGraph.exists_isVertexCut_singleton_of_isBridge"],"detail_key":"p14"},{"id":"n21604","layer":"informal","project":"p14","title":"vertexConnectivity\\_le\\_edgeConnectivity","kind":"theorem","summary":"[vertexConnectivity\\_le\\_edgeConnectivity] , first inequality: \\kappa \\le \\kappa'. Induction on…","labels":["SimpleGraph.vertexConnectivity_le_edgeConnectivity"],"detail_key":"p14"},{"id":"n21605","layer":"informal","project":"p14","title":"whitney\\_inequalities","kind":"theorem","summary":"[whitney\\_inequalities] (Whitney). The full chain \\kappa \\le \\kappa' \\le \\delta. The conjunctio…","labels":["SimpleGraph.whitney_inequalities"],"detail_key":"p14"},{"id":"n21606","layer":"informal","project":"p14","title":"Reachable induce of support subset","kind":"theorem","summary":"[Reachable induce of support subset] Transfer of a walk into an induced subgraph: a walk all of…","labels":["SimpleGraph.reachable_induce_of_support_subset"],"detail_key":"p14"},{"id":"n21607","layer":"informal","project":"p14","title":"exists\\_two\\_internally\\_disjoint\\_paths\\_of\\_two\\_connected","kind":"theorem","summary":"[exists\\_two\\_internally\\_disjoint\\_paths\\_of\\_two\\_connected] , strengthened to carry the edge…","labels":["SimpleGraph.exists_two_internally_disjoint_paths_of_two_connected"],"detail_key":"p14"},{"id":"n21608","layer":"informal","project":"p14","title":"two\\_connected\\_iff\\_two\\_internally\\_disjoint\\_paths","kind":"theorem","summary":"[two\\_connected\\_iff\\_two\\_internally\\_disjoint\\_paths] (Whitney, 1932). (\\Rightarrow) is disch…","labels":["SimpleGraph.two_connected_iff_two_internally_disjoint_paths"],"detail_key":"p14"},{"id":"n21609","layer":"informal","project":"p14","title":"two\\_connected\\_vertices\\_on\\_common\\_cycle","kind":"theorem","summary":"[two\\_connected\\_vertices\\_on\\_common\\_cycle] Travel out along one path and back along the othe…","labels":["SimpleGraph.two_connected_vertices_on_common_cycle"],"detail_key":"p14"},{"id":"n21610","layer":"informal","project":"p14","title":"block\\_edges\\_on\\_common\\_cycle","kind":"theorem","summary":"[block\\_edges\\_on\\_common\\_cycle] The standard device of turning an edge into a vertex, upgradi…","labels":["SimpleGraph.block_edges_on_common_cycle"],"detail_key":"p14"},{"id":"n21611","layer":"informal","project":"p14","title":"Every block with \\nu \\ge 3 is 2-connected","kind":"theorem","summary":"[Every block with \\nu \\ge 3 is 2-connected] A block is connected and has no cut vertex, so no s…","labels":["SimpleGraph.block_three_vertices_two_connected"],"detail_key":"p14"},{"id":"n21612","layer":"informal","project":"p14","title":"The Harary graph H_m,n on vertex set \\textttZMod n (a circulant)","kind":"definition","summary":"[The Harary graph H_m,n on vertex set \\textttZMod n (a circulant)] Arrange n stations in a circ…","labels":["SimpleGraph.hararyGraph"],"detail_key":"p14"},{"id":"n21613","layer":"informal","project":"p14","title":"hararyGraph\\_isConnectivity","kind":"theorem","summary":"[hararyGraph\\_isConnectivity] (Harary, 1962). The book proves only the even case; the odd case…","labels":["SimpleGraph.hararyGraph_isConnectivity"],"detail_key":"p14"},{"id":"n21614","layer":"informal","project":"p14","title":"Edge count / optimality: H_m,n has \\lceil mn/2\\rceil edges","kind":"theorem","summary":"[Edge count / optimality: H_m,n has \\lceil mn/2\\rceil edges] The book's x is the ceiling \\lceil…","labels":["SimpleGraph.hararyGraph_edgeCard"],"detail_key":"p14"},{"id":"n21615","layer":"informal","project":"p14","title":"edgeCard\\_ge\\_of\\_kEdgeConnected","kind":"theorem","summary":"[edgeCard\\_ge\\_of\\_kEdgeConnected] a k-edge-connected graph satisfies k\\nu \\le 2\\varepsilon. St…","labels":["SimpleGraph.edgeCard_ge_of_kEdgeConnected"],"detail_key":"p14"},{"id":"n21616","layer":"informal","project":"p14","title":"vertexConnectivity\\_eq\\_minDegree\\_of\\_delta\\_ge","kind":"theorem","summary":"[vertexConnectivity\\_eq\\_minDegree\\_of\\_delta\\_ge] a simple graph with \\delta \\ge \\nu - 2 has \\…","labels":["SimpleGraph.vertexConnectivity_eq_minDegree_of_delta_ge"],"detail_key":"p14"},{"id":"n21617","layer":"informal","project":"p14","title":"vertexConn\\_eq\\_edgeConn\\_of\\_threeRegular","kind":"theorem","summary":"[vertexConn\\_eq\\_edgeConn\\_of\\_threeRegular] a simple 3-regular graph has \\kappa = \\kappa'. The…","labels":["SimpleGraph.vertexConn_eq_edgeConn_of_threeRegular"],"detail_key":"p14"},{"id":"n21618","layer":"informal","project":"p14","title":"two\\_edge\\_connected\\_iff\\_two\\_edge\\_disjoint\\_paths","kind":"theorem","summary":"[two\\_edge\\_connected\\_iff\\_two\\_edge\\_disjoint\\_paths] 2-edge-connected \\iff two edge-disjoint…","labels":["SimpleGraph.two_edge_connected_iff_two_edge_disjoint_paths"],"detail_key":"p14"},{"id":"n21619","layer":"informal","project":"p14","title":"B\\&M's unimodular (Thm 12.3, p. 226): every \\emphfull square submatrix (order \\nu- 1, sel…","kind":"definition","summary":"[B\\&M's unimodular (Thm 12.3, p. 226): every \\emphfull square submatrix (order \\nu- 1, selected…","labels":["Matrix.IsUnimodular"],"detail_key":"p14"},{"id":"n21620","layer":"informal","project":"p14","title":"B\\&M's oriented incidence matrix M (p. 222)","kind":"definition","summary":"[B\\&M's oriented incidence matrix M (p. 222)] Record, for each vertex and each arc, whether the…","labels":["CycleSpace.orientedIncMatrix"],"detail_key":"p14"},{"id":"n21621","layer":"informal","project":"p14","title":"The cycle space C: circulations, i.e","kind":"definition","summary":"[The cycle space C: circulations, i.e] Currents that flow round and round without accumulating…","labels":["CycleSpace.cycleSpace"],"detail_key":"p14"},{"id":"n21622","layer":"informal","project":"p14","title":"The bond space B: potential differences, i.e","kind":"definition","summary":"[The bond space B: potential differences, i.e] Assign a voltage to every vertex and read off th…","labels":["CycleSpace.bondSpace"],"detail_key":"p14"},{"id":"n21623","layer":"informal","project":"p14","title":"The standard dot-product bilinear form on K \\to K (here A \\to F)","kind":"definition","summary":"[The standard dot-product bilinear form on K \\to K (here A \\to F)] f and g are orthogonal when…","labels":["CycleSpace.dotForm"],"detail_key":"p14"},{"id":"n21624","layer":"informal","project":"p14","title":"G is the underlying (simple) graph of the digraph \\texttt(tail, head)","kind":"definition","summary":"[G is the underlying (simple) graph of the digraph \\texttt(tail, head)] u and v are adjacent in…","labels":["CycleSpace.IsUnderlyingGraph"],"detail_key":"p14"},{"id":"n21625","layer":"informal","project":"p14","title":"\\texttt(tail, head) is an orientation of the simple graph G","kind":"definition","summary":"[\\texttt(tail, head) is an orientation of the simple graph G] G is recovered by forgetting dire…","labels":["CycleSpace.IsOrientationOf"],"detail_key":"p14"},{"id":"n21626","layer":"informal","project":"p14","title":"An arc-level cycle in the digraph, as a nonempty closed arc-sequence","kind":"definition","summary":"[An arc-level cycle in the digraph, as a nonempty closed arc-sequence] A cycle of the \\emphunde…","labels":["CycleSpace.ArcCycle"],"detail_key":"p14"},{"id":"n21627","layer":"informal","project":"p14","title":"S \\subseteq A is acyclic: it contains no arc-cycle","kind":"definition","summary":"[S \\subseteq A is acyclic: it contains no arc-cycle] No cycle of the digraph lies entirely insi…","labels":["CycleSpace.IsArcAcyclic"],"detail_key":"p14"},{"id":"n21628","layer":"informal","project":"p14","title":"f_C: the circulation of an oriented cycle (\\pm 1 on C^+ / C^-, 0 off C)","kind":"definition","summary":"[f_C: the circulation of an oriented cycle (\\pm 1 on C^+ / C^-, 0 off C)] Send one unit of curr…","labels":["CycleSpace.cycleCirculation"],"detail_key":"p14"},{"id":"n21629","layer":"informal","project":"p14","title":"B\\&M's edge cut (S, \\barS): the arcs with exactly one end in S","kind":"definition","summary":"[B\\&M's edge cut (S, \\barS): the arcs with exactly one end in S] The arcs crossing the boundary…","labels":["CycleSpace.edgeCutSet"],"detail_key":"p14"},{"id":"n21630","layer":"informal","project":"p14","title":"A bond: a MINIMAL nonempty edge cut. ! MISSING from Mathlib","kind":"definition","summary":"[A bond: a MINIMAL nonempty edge cut. ! MISSING from Mathlib] An edge cut with nothing to spare…","labels":["CycleSpace.IsBond"],"detail_key":"p14"},{"id":"n21631","layer":"informal","project":"p14","title":"g_B: the potential difference of a bond, p = indicator of S (B\\&M give it, p. 221)","kind":"definition","summary":"[g_B: the potential difference of a bond, p = indicator of S (B\\&M give it, p. 221)] Put every…","labels":["CycleSpace.bondPotentialDiff"],"detail_key":"p14"},{"id":"n21632","layer":"informal","project":"p14","title":"A basis matrix of a submodule W \\le (A \\to F): its rows are a basis of W","kind":"definition","summary":"[A basis matrix of a submodule W \\le (A \\to F): its rows are a basis of W] Package a basis as t…","labels":["CycleSpace.IsBasisMatrix"],"detail_key":"p14"},{"id":"n21633","layer":"informal","project":"p14","title":"Restrict Cols","kind":"definition","summary":"[Restrict Cols] M | S --- B\\&M's column restriction (\\textttMatrix.submatrix).","labels":["CycleSpace.restrictCols"],"detail_key":"p14"},{"id":"n21634","layer":"informal","project":"p14","title":"A spanning tree of the digraph","kind":"definition","summary":"[A spanning tree of the digraph] A maximal acyclic set of arcs touching every vertex. By theore…","labels":["CycleSpace.IsSpanningTree"],"detail_key":"p14"},{"id":"n21635","layer":"informal","project":"p14","title":"A maximal forest of the digraph","kind":"definition","summary":"[A maximal forest of the digraph] A maximal acyclic set of arcs --- a spanning tree of each com…","labels":["CycleSpace.IsMaximalForest"],"detail_key":"p14"},{"id":"n21636","layer":"informal","project":"p14","title":"The fundamental cycle of a \\notin T (T + a contains a unique cycle)","kind":"definition","summary":"[The fundamental cycle of a \\notin T (T + a contains a unique cycle)] The tree already provides…","labels":["CycleSpace.fundamentalCycle"],"detail_key":"p14"},{"id":"n21637","layer":"informal","project":"p14","title":"The vertex set S of the fundamental bond of a \\in T (\\barT + a contains a unique bond, B\\…","kind":"definition","summary":"[The vertex set S of the fundamental bond of a \\in T (\\barT + a contains a unique bond, B\\&M's\\…","labels":["CycleSpace.fundamentalBondVertexSet"],"detail_key":"p14"},{"id":"n21638","layer":"informal","project":"p14","title":"B is the tree-T basis matrix of the bond space (rows indexed by T)","kind":"definition","summary":"[B is the tree-T basis matrix of the bond space (rows indexed by T)] Its defining feature is th…","labels":["CycleSpace.IsBasisMatrixOfTree"],"detail_key":"p14"},{"id":"n21639","layer":"informal","project":"p14","title":"C is the tree-T basis matrix of the cycle space (rows indexed by \\barT)","kind":"definition","summary":"[C is the tree-T basis matrix of the cycle space (rows indexed by \\barT)] The cycle-side mirror…","labels":["CycleSpace.IsBasisMatrixOfTree'"],"detail_key":"p14"},{"id":"n21640","layer":"informal","project":"p14","title":"\\tau(G): the number of spanning trees","kind":"definition","summary":"[\\tau(G): the number of spanning trees] The object \\S12.2 exists to compute. Theorem 2.8 gave a…","labels":["CycleSpace.tau"],"detail_key":"p14"},{"id":"n21641","layer":"informal","project":"p14","title":"bondSpace\\_eq\\_rowSpace","kind":"theorem","summary":"[bondSpace\\_eq\\_rowSpace] , first half. *Let M be the incidence matrix of a digraph D. Then B i…","labels":["CycleSpace.bondSpace_eq_rowSpace"],"detail_key":"p14"},{"id":"n21642","layer":"informal","project":"p14","title":"cycleSpace\\_eq\\_orthogonal\\_bondSpace","kind":"theorem","summary":"[cycleSpace\\_eq\\_orthogonal\\_bondSpace] , second half. \\emph\\dots and C is its orthogonal compl…","labels":["CycleSpace.cycleSpace_eq_orthogonal_bondSpace"],"detail_key":"p14"},{"id":"n21643","layer":"informal","project":"p14","title":"exists\\_arcCycle\\_subset\\_support\\_of\\_isCirculation","kind":"theorem","summary":"[exists\\_arcCycle\\_subset\\_support\\_of\\_isCirculation] *If f is a nonzero circulation, then \\|…","labels":["CycleSpace.exists_arcCycle_subset_support_of_isCirculation"],"detail_key":"p14"},{"id":"n21644","layer":"informal","project":"p14","title":"exists\\_isBond\\_subset\\_support\\_of\\_mem\\_bondSpace","kind":"theorem","summary":"[exists\\_isBond\\_subset\\_support\\_of\\_mem\\_bondSpace] *If g is a nonzero potential difference,…","labels":["CycleSpace.exists_isBond_subset_support_of_mem_bondSpace"],"detail_key":"p14"},{"id":"n21645","layer":"informal","project":"p14","title":"basisMatrix\\_bondSpace\\_cols\\_linearIndependent\\_iff","kind":"theorem","summary":"[basisMatrix\\_bondSpace\\_cols\\_linearIndependent\\_iff] *Let B be a basis matrix of B. Then for…","labels":["CycleSpace.basisMatrix_bondSpace_cols_linearIndependent_iff"],"detail_key":"p14"},{"id":"n21646","layer":"informal","project":"p14","title":"basisMatrix\\_cycleSpace\\_cols\\_linearIndependent\\_iff","kind":"theorem","summary":"[basisMatrix\\_cycleSpace\\_cols\\_linearIndependent\\_iff] *Let C be a basis matrix of C. Then for…","labels":["CycleSpace.basisMatrix_cycleSpace_cols_linearIndependent_iff"],"detail_key":"p14"},{"id":"n21647","layer":"informal","project":"p14","title":"finrank\\_bondSpace","kind":"theorem","summary":"[finrank\\_bondSpace] , formula (12.3). \\emph\\textttdim B = \\nu - \\omega. The bond space records…","labels":["CycleSpace.finrank_bondSpace"],"detail_key":"p14"},{"id":"n21648","layer":"informal","project":"p14","title":"finrank\\_cycleSpace","kind":"theorem","summary":"[finrank\\_cycleSpace] , formula (12.4). \\emph\\textttdim C = \\varepsilon - \\nu + \\omega. \\vareps…","labels":["CycleSpace.finrank_cycleSpace"],"detail_key":"p14"},{"id":"n21649","layer":"informal","project":"p14","title":"isBasisMatrix\\_cycleSpace\\_of\\_maximalForest","kind":"theorem","summary":"[isBasisMatrix\\_cycleSpace\\_of\\_maximalForest] The fundamental cycle basis --- the concrete rea…","labels":["CycleSpace.isBasisMatrix_cycleSpace_of_maximalForest"],"detail_key":"p14"},{"id":"n21650","layer":"informal","project":"p14","title":"isBasisMatrix\\_bondSpace\\_of\\_maximalForest","kind":"theorem","summary":"[isBasisMatrix\\_bondSpace\\_of\\_maximalForest] The fundamental bond basis, realising the claim t…","labels":["CycleSpace.isBasisMatrix_bondSpace_of_maximalForest"],"detail_key":"p14"},{"id":"n21651","layer":"informal","project":"p14","title":"isUnimodular\\_basisMatrix\\_bondSpace","kind":"theorem","summary":"[isUnimodular\\_basisMatrix\\_bondSpace] (proof due to Tutte, 1965b). *The basis matrix B is unim…","labels":["CycleSpace.isUnimodular_basisMatrix_bondSpace"],"detail_key":"p14"},{"id":"n21652","layer":"informal","project":"p14","title":"tau\\_eq\\_det\\_mul\\_transpose","kind":"theorem","summary":"[tau\\_eq\\_det\\_mul\\_transpose] \\emph\\texttt\\tau(G) = det BB' (12.6). The chapter's central comp…","labels":["CycleSpace.tau_eq_det_mul_transpose"],"detail_key":"p14"},{"id":"n21653","layer":"informal","project":"p14","title":"isUnimodular\\_basisMatrix\\_cycleSpace","kind":"theorem","summary":"[isUnimodular\\_basisMatrix\\_cycleSpace] The chapter is organised around such dual pairs --- cyc…","labels":["CycleSpace.isUnimodular_basisMatrix_cycleSpace"],"detail_key":"p14"},{"id":"n21654","layer":"informal","project":"p14","title":"tau\\_eq\\_det\\_mul\\_transpose\\_cycleSpace","kind":"theorem","summary":"[tau\\_eq\\_det\\_mul\\_transpose\\_cycleSpace] The spanning trees can be counted from either space…","labels":["CycleSpace.tau_eq_det_mul_transpose_cycleSpace"],"detail_key":"p14"},{"id":"n21655","layer":"informal","project":"p14","title":"tau\\_eq\\_det\\_fromBlocks","kind":"theorem","summary":"[tau\\_eq\\_det\\_fromBlocks] \\emph\\texttt\\tau(G) = \\pm det [B; C], the determinant of the square…","labels":["CycleSpace.tau_eq_det_fromBlocks"],"detail_key":"p14"},{"id":"n21656","layer":"informal","project":"p14","title":"matrix\\_tree\\_theorem","kind":"theorem","summary":"[matrix\\_tree\\_theorem] (implicit in Kirchhoff, 1847). *\\texttt\\tau(G) = det KK', where K is ob…","labels":["CycleSpace.matrix_tree_theorem"],"detail_key":"p14"},{"id":"n21657","layer":"informal","project":"p14","title":"isUnimodular\\_deleteRow","kind":"theorem","summary":"[isUnimodular\\_deleteRow] \\emph. A matrix K obtained from M by deleting any one row is unimodul…","labels":["CycleSpace.isUnimodular_deleteRow"],"detail_key":"p14"},{"id":"n21658","layer":"informal","project":"p14","title":"lapMatrix\\_eq\\_orientedIncMatrix\\_mul\\_transpose","kind":"theorem","summary":"[lapMatrix\\_eq\\_orientedIncMatrix\\_mul\\_transpose] *The conductance matrix C of a loopless grap…","labels":["CycleSpace.lapMatrix_eq_orientedIncMatrix_mul_transpose"],"detail_key":"p14"},{"id":"n21659","layer":"informal","project":"p14","title":"tau\\_eq\\_det\\_lapMatrix\\_deleteRowCol","kind":"theorem","summary":"[tau\\_eq\\_det\\_lapMatrix\\_deleteRowCol] *All cofactors of the conductance matrix C are equal to…","labels":["CycleSpace.tau_eq_det_lapMatrix_deleteRowCol"],"detail_key":"p14"},{"id":"n21660","layer":"informal","project":"p14","title":"tau\\_eq\\_lapMatrix\\_cofactor","kind":"theorem","summary":"[tau\\_eq\\_lapMatrix\\_cofactor] , full form. *All cofactors of the conductance matrix C are equa…","labels":["CycleSpace.tau_eq_lapMatrix_cofactor"],"detail_key":"p14"},{"id":"n21661","layer":"informal","project":"p14","title":"incMatrix\\_isTotallyUnimodular\\_iff\\_isBipartite","kind":"theorem","summary":"[incMatrix\\_isTotallyUnimodular\\_iff\\_isBipartite] *The incidence matrix of a simple graph G is…","labels":["CycleSpace.incMatrix_isTotallyUnimodular_iff_isBipartite"],"detail_key":"p14"},{"id":"n21662","layer":"informal","project":"p14","title":"finrank\\_inf\\_pos\\_iff\\_dvd\\_tau","kind":"theorem","summary":"[finrank\\_inf\\_pos\\_iff\\_dvd\\_tau] (H. Shank). *Let F be a field of characteristic p. Then \\tex…","labels":["CycleSpace.finrank_inf_pos_iff_dvd_tau"],"detail_key":"p14"},{"id":"n21663","layer":"informal","project":"p14","title":"Edge connectivity \\kappa'(G)","kind":"definition","summary":"[Edge connectivity \\kappa'(G)] The least number of edges one must cut to break G apart. \"G is k…","labels":["edgeConnectivity"],"detail_key":"p14"},{"id":"n21664","layer":"informal","project":"p14","title":"The bridge \\textttDigraph \\to Quiver: \\textttQuiver.\\0\\ V has \\textttHom : V \\to V \\to Pr…","kind":"definition","summary":"[The bridge \\textttDigraph \\to Quiver: \\textttQuiver.\\0\\ V has \\textttHom : V \\to V \\to Prop,\\d…","labels":["Digraph.toQuiver"],"detail_key":"p14"},{"id":"n21665","layer":"informal","project":"p14","title":"A directed path: a \\textttQuiver.Path with no repeated vertex","kind":"definition","summary":"[A directed path: a \\textttQuiver.Path with no repeated vertex] Follow the arrows, never agains…","labels":["Digraph.IsDirectedPath"],"detail_key":"p14"},{"id":"n21666","layer":"informal","project":"p14","title":"A directed cycle: a positive-length closed walk whose vertices (bar the repeated endpoint…","kind":"definition","summary":"[A directed cycle: a positive-length closed walk whose vertices (bar the repeated endpoint) are…","labels":["Digraph.IsDirectedCycle"],"detail_key":"p14"},{"id":"n21667","layer":"informal","project":"p14","title":"v is reachable from u","kind":"definition","summary":"[v is reachable from u] You can get from u to v travelling only along arrows in their given dir…","labels":["Digraph.Reachable"],"detail_key":"p14"},{"id":"n21668","layer":"informal","project":"p14","title":"D is diconnected: every vertex reaches every other","kind":"definition","summary":"[D is diconnected: every vertex reaches every other] Wherever you start and wherever you want t…","labels":["Digraph.Diconnected"],"detail_key":"p14"},{"id":"n21669","layer":"informal","project":"p14","title":"d^-(v), the indegree","kind":"definition","summary":"[d^-(v), the indegree] How many arrows point \\emphinto v. Exercise 10.1.2 gives the directed ha…","labels":["Digraph.indegree"],"detail_key":"p14"},{"id":"n21670","layer":"informal","project":"p14","title":"d^+(v), the outdegree","kind":"definition","summary":"[d^+(v), the outdegree] How many arrows point \\emphout of v. Together with the indegree this re…","labels":["Digraph.outdegree"],"detail_key":"p14"},{"id":"n21671","layer":"informal","project":"p14","title":"\\varepsilon, the number of arcs","kind":"definition","summary":"[\\varepsilon, the number of arcs] Count the ordered pairs (u, v) for which an arc runs from u t…","labels":["Digraph.arcCount"],"detail_key":"p14"},{"id":"n21672","layer":"informal","project":"p14","title":"\\delta^-, the minimum indegree","kind":"definition","summary":"[\\delta^-, the minimum indegree] The least number of arrows pointing into any one vertex. Exerc…","labels":["Digraph.minIndegree"],"detail_key":"p14"},{"id":"n21673","layer":"informal","project":"p14","title":"\\delta^+, the minimum outdegree","kind":"definition","summary":"[\\delta^+, the minimum outdegree] The least number of arrows leaving any one vertex. By the con…","labels":["Digraph.minOutdegree"],"detail_key":"p14"},{"id":"n21674","layer":"informal","project":"p14","title":"The converse \\breveD: reverse every arc","kind":"definition","summary":"[The converse \\breveD: reverse every arc] Turn every arrow around. It is an involution (\\breve\\…","labels":["Digraph.converse"],"detail_key":"p14"},{"id":"n21675","layer":"informal","project":"p14","title":"D is strict: loopless (\\textttDigraph already forbids parallel same-direction arcs)","kind":"definition","summary":"[D is strict: loopless (\\textttDigraph already forbids parallel same-direction arcs)] The direc…","labels":["Digraph.IsStrict"],"detail_key":"p14"},{"id":"n21676","layer":"informal","project":"p14","title":"D is an orientation of G --- the chapter's keystone gap","kind":"definition","summary":"[D is an orientation of G --- the chapter's keystone gap] Make every edge one-way, choosing a d…","labels":["Digraph.IsOrientationOf"],"detail_key":"p14"},{"id":"n21677","layer":"informal","project":"p14","title":"D is a tournament: an orientation of the complete graph","kind":"definition","summary":"[D is a tournament: an orientation of the complete graph] Every pair of players meets exactly o…","labels":["Digraph.IsTournament"],"detail_key":"p14"},{"id":"n21678","layer":"informal","project":"p14","title":"B\\&M's (S, T): arcs with tail in S, head in T","kind":"definition","summary":"[B\\&M's (S, T): arcs with tail in S, head in T] The arcs crossing from S into T, counted with t…","labels":["Digraph.arcsBetween"],"detail_key":"p14"},{"id":"n21679","layer":"informal","project":"p14","title":"D is k-arc-connected: every nonempty proper cut has \\ge k outgoing arcs","kind":"definition","summary":"[D is k-arc-connected: every nonempty proper cut has \\ge k outgoing arcs] However you split the…","labels":["Digraph.IsKArcConnected"],"detail_key":"p14"},{"id":"n21680","layer":"informal","project":"p14","title":"The induced subdigraph on S (kept on the same carrier to avoid subtype juggling)","kind":"definition","summary":"[The induced subdigraph on S (kept on the same carrier to avoid subtype juggling)] Keep only th…","labels":["Digraph.induce"],"detail_key":"p14"},{"id":"n21681","layer":"informal","project":"p14","title":"The list of arcs traversed by a directed walk","kind":"definition","summary":"[The list of arcs traversed by a directed walk] Reading off the arrows a directed walk uses, in…","labels":["Digraph.arcsOf"],"detail_key":"p14"},{"id":"n21682","layer":"informal","project":"p14","title":"A directed trail: no repeated arc","kind":"definition","summary":"[A directed trail: no repeated arc] Follow the arrows, never using the same arrow twice, though…","labels":["Digraph.IsDirectedTrail"],"detail_key":"p14"},{"id":"n21683","layer":"informal","project":"p14","title":"A directed Euler tour: a closed directed trail using every arc","kind":"definition","summary":"[A directed Euler tour: a closed directed trail using every arc] The directed version of Euler'…","labels":["Digraph.IsDirectedEulerTour"],"detail_key":"p14"},{"id":"n21684","layer":"informal","project":"p14","title":"D is unilateral: any two vertices are comparable by reachability","kind":"definition","summary":"[D is unilateral: any two vertices are comparable by reachability] Weaker than diconnected ---…","labels":["Digraph.IsUnilateral"],"detail_key":"p14"},{"id":"n21685","layer":"informal","project":"p14","title":"The 0/1 adjacency matrix of D","kind":"definition","summary":"[The 0/1 adjacency matrix of D] Record a 1 where an arrow runs from v_i to v_j and 0 otherwise.…","labels":["Digraph.adjMatrix"],"detail_key":"p14"},{"id":"n21686","layer":"informal","project":"p14","title":"The condensation \\hatD: dicomponents contracted","kind":"definition","summary":"[The condensation \\hatD: dicomponents contracted] Shrink each dicomponent to a single point and…","labels":["Digraph.condensation"],"detail_key":"p14"},{"id":"n21687","layer":"informal","project":"p14","title":"dirDist","kind":"definition","summary":"[dirDist] \\vecd(u,v). ! \\textttNat.sInf \\emptyset = 0 when v is unreachable from u. The fewest…","labels":["Digraph.dirDist"],"detail_key":"p14"},{"id":"n21688","layer":"informal","project":"p14","title":"dirDiameter","kind":"definition","summary":"[dirDiameter] The worst case of the directed distance --- how far apart two vertices can be whe…","labels":["Digraph.dirDiameter"],"detail_key":"p14"},{"id":"n21689","layer":"informal","project":"p14","title":"The associated digraph D(G): each edge becomes two opposite arcs. \\textttG.Adj is already…","kind":"definition","summary":"[The associated digraph D(G): each edge becomes two opposite arcs. \\textttG.Adj is already symm…","labels":["SimpleGraph.associatedDigraph"],"detail_key":"p14"},{"id":"n21690","layer":"informal","project":"p14","title":"A primitive N-matrix: some power is entrywise positive","kind":"definition","summary":"[A primitive N-matrix: some power is entrywise positive] For the adjacency matrix this says: ho…","labels":["Matrix.IsPrimitive"],"detail_key":"p14"},{"id":"n21691","layer":"informal","project":"p14","title":"The de Bruijn digraph D_n: vertices are (n-1)-bit strings, arcs are left-shifts","kind":"definition","summary":"[The de Bruijn digraph D_n: vertices are (n-1)-bit strings, arcs are left-shifts] A vertex is a…","labels":["deBruijnDigraph"],"detail_key":"p14"},{"id":"n21692","layer":"informal","project":"p14","title":"card\\_orientations","kind":"theorem","summary":"[card\\_orientations] An orientation is a choice, independently for each edge, of one of its two…","labels":["card_orientations"],"detail_key":"p14"},{"id":"n21693","layer":"informal","project":"p14","title":"sum\\_indegree\\_eq\\_arcCount","kind":"theorem","summary":"[sum\\_indegree\\_eq\\_arcCount] The directed handshaking lemma. Every arc has exactly one head, s…","labels":["sum_indegree_eq_arcCount"],"detail_key":"p14"},{"id":"n21694","layer":"informal","project":"p14","title":"exists\\_indegree\\_zero\\_of\\_acyclic","kind":"theorem","summary":"[exists\\_indegree\\_zero\\_of\\_acyclic] If every vertex had an incoming arc one could walk backwa…","labels":["exists_indegree_zero_of_acyclic"],"detail_key":"p14"},{"id":"n21695","layer":"informal","project":"p14","title":"exists\\_topological\\_ordering","kind":"theorem","summary":"[exists\\_topological\\_ordering] A topological ordering: line the vertices up so every arrow poi…","labels":["exists_topological_ordering"],"detail_key":"p14"},{"id":"n21696","layer":"informal","project":"p14","title":"converse\\_converse","kind":"theorem","summary":"[converse\\_converse] Reversing every arrow twice returns each to its original direction, so the…","labels":["converse_converse"],"detail_key":"p14"},{"id":"n21697","layer":"informal","project":"p14","title":"converse\\_indegree","kind":"theorem","summary":"[converse\\_indegree] An arc pointing into v in D points out of v in the converse, and vice vers…","labels":["converse_indegree"],"detail_key":"p14"},{"id":"n21698","layer":"informal","project":"p14","title":"converse\\_reachable","kind":"theorem","summary":"[converse\\_reachable] A directed path from u to v in the converse is exactly a directed path fr…","labels":["converse_reachable"],"detail_key":"p14"},{"id":"n21699","layer":"informal","project":"p14","title":"exists\\_outdegree\\_zero\\_of\\_acyclic","kind":"theorem","summary":"[exists\\_outdegree\\_zero\\_of\\_acyclic] An acyclic digraph has both a \"source\" (nothing in) and…","labels":["exists_outdegree_zero_of_acyclic"],"detail_key":"p14"},{"id":"n21700","layer":"informal","project":"p14","title":"exists\\_directedPath\\_length\\_ge\\_maxMinDegree","kind":"theorem","summary":"[exists\\_directedPath\\_length\\_ge\\_maxMinDegree] The directed analogue of exercise 1.6.3. Stric…","labels":["exists_directedPath_length_ge_maxMinDegree"],"detail_key":"p14"},{"id":"n21701","layer":"informal","project":"p14","title":"exists\\_directedCycle\\_length\\_ge","kind":"theorem","summary":"[exists\\_directedCycle\\_length\\_ge] Sharpens exercise 10.1.6 from paths to cycles, exactly as e…","labels":["exists_directedCycle_length_ge"],"detail_key":"p14"},{"id":"n21702","layer":"informal","project":"p14","title":"adjMatrix\\_pow\\_apply\\_eq\\_card\\_directedWalk","kind":"theorem","summary":"[adjMatrix\\_pow\\_apply\\_eq\\_card\\_directedWalk] The (i,j) entry of A counts arcs, i.e. walks of…","labels":["adjMatrix_pow_apply_eq_card_directedWalk"],"detail_key":"p14"},{"id":"n21703","layer":"informal","project":"p14","title":"condensation\\_acyclic","kind":"theorem","summary":"[condensation\\_acyclic] A directed cycle among dicomponents would merge them into one, so contr…","labels":["condensation_acyclic"],"detail_key":"p14"},{"id":"n21704","layer":"informal","project":"p14","title":"exists\\_balanced\\_orientation","kind":"theorem","summary":"[exists\\_balanced\\_orientation] Every graph can be made one-way in a \\emphbalanced way, with th…","labels":["exists_balanced_orientation"],"detail_key":"p14"},{"id":"n21705","layer":"informal","project":"p14","title":"roy\\_gallai\\_directed\\_path","kind":"theorem","summary":"[roy\\_gallai\\_directed\\_path] (Roy, 1967; Gallai, 1968). *A digraph D contains a directed path…","labels":["roy_gallai_directed_path"],"detail_key":"p14"},{"id":"n21706","layer":"informal","project":"p14","title":"exists\\_orientation\\_longest\\_directedPath\\_le","kind":"theorem","summary":"[exists\\_orientation\\_longest\\_directedPath\\_le] Orient every edge from the lower colour class…","labels":["exists_orientation_longest_directedPath_le"],"detail_key":"p14"},{"id":"n21707","layer":"informal","project":"p14","title":"redei\\_directed\\_hamilton\\_path","kind":"theorem","summary":"[redei\\_directed\\_hamilton\\_path] (R\\'edei, 1934). *Every tournament has a directed Hamilton pa…","labels":["redei_directed_hamilton_path"],"detail_key":"p14"},{"id":"n21708","layer":"informal","project":"p14","title":"chvatal\\_lovasz\\_semikernel","kind":"theorem","summary":"[chvatal\\_lovasz\\_semikernel] (Chv\\'atal and Lov\\'asz, 1974). *A loopless digraph D has an inde…","labels":["chvatal_lovasz_semikernel"],"detail_key":"p14"},{"id":"n21709","layer":"informal","project":"p14","title":"tournament\\_exists\\_king","kind":"theorem","summary":"[tournament\\_exists\\_king] *A tournament contains a vertex from which every other vertex is rea…","labels":["tournament_exists_king"],"detail_key":"p14"},{"id":"n21710","layer":"informal","project":"p14","title":"tournament\\_diconnected\\_or\\_reorient\\_one","kind":"theorem","summary":"[tournament\\_diconnected\\_or\\_reorient\\_one] Tournaments are never far from diconnected: a sing…","labels":["tournament_diconnected_or_reorient_one"],"detail_key":"p14"},{"id":"n21711","layer":"informal","project":"p14","title":"isUnilateral\\_iff\\_exists\\_spanning\\_directedWalk","kind":"theorem","summary":"[isUnilateral\\_iff\\_exists\\_spanning\\_directedWalk] \\emph. D is unilateral if and only if D has…","labels":["isUnilateral_iff_exists_spanning_directedWalk"],"detail_key":"p14"},{"id":"n21712","layer":"informal","project":"p14","title":"tournament\\_maximal\\_directedPath\\_insert","kind":"theorem","summary":"[tournament\\_maximal\\_directedPath\\_insert] *Let P = (v_1, \\dots, v_k) be a maximal directed pa…","labels":["tournament_maximal_directedPath_insert"],"detail_key":"p14"},{"id":"n21713","layer":"informal","project":"p14","title":"chvatal\\_komlos\\_monotone\\_directedPath","kind":"theorem","summary":"[chvatal\\_komlos\\_monotone\\_directedPath] \\emph (Chv\\'atal and Koml\\'os). Let D be a digraph wi…","labels":["chvatal_komlos_monotone_directedPath"],"detail_key":"p14"},{"id":"n21714","layer":"informal","project":"p14","title":"erdos\\_szekeres\\_of\\_chvatal\\_komlos","kind":"theorem","summary":"[erdos\\_szekeres\\_of\\_chvatal\\_komlos] \\emph (Erd\\Hos and Szekeres). Deduce that any sequence o…","labels":["erdos_szekeres_of_chvatal_komlos"],"detail_key":"p14"},{"id":"n21715","layer":"informal","project":"p14","title":"exists\\_orientation\\_directedPath\\_le\\_maxDegree","kind":"theorem","summary":"[exists\\_orientation\\_directedPath\\_le\\_maxDegree] *Using theorem 10.1 and corollary 8.1.2, sho…","labels":["exists_orientation_directedPath_le_maxDegree"],"detail_key":"p14"},{"id":"n21716","layer":"informal","project":"p14","title":"moon\\_vertex\\_pancyclic","kind":"theorem","summary":"[moon\\_vertex\\_pancyclic] (Moon, 1966). *Each vertex of a diconnected tournament D with \\nu \\ge…","labels":["moon_vertex_pancyclic"],"detail_key":"p14"},{"id":"n21717","layer":"informal","project":"p14","title":"ghouila\\_houri\\_directed\\_hamilton\\_cycle","kind":"theorem","summary":"[ghouila\\_houri\\_directed\\_hamilton\\_cycle] (a special case of Ghouila-Houri, 1960). *If D is s…","labels":["ghouila_houri_directed_hamilton_cycle"],"detail_key":"p14"},{"id":"n21718","layer":"informal","project":"p14","title":"exists\\_directedEulerTour\\_iff","kind":"theorem","summary":"[exists\\_directedEulerTour\\_iff] *D contains a directed Euler tour if and only if D is connecte…","labels":["exists_directedEulerTour_iff"],"detail_key":"p14"},{"id":"n21719","layer":"informal","project":"p14","title":"exists\\_arcDisjoint\\_directedPaths","kind":"theorem","summary":"[exists\\_arcDisjoint\\_directedPaths] *Let D be a digraph such that (i) d^+(x) - d^-(x) = l = d^…","labels":["exists_arcDisjoint_directedPaths"],"detail_key":"p14"},{"id":"n21720","layer":"informal","project":"p14","title":"exists\\_directed\\_odd\\_cycle","kind":"theorem","summary":"[exists\\_directed\\_odd\\_cycle] \\emph. A diconnected digraph which contains an odd cycle also co…","labels":["exists_directed_odd_cycle"],"detail_key":"p14"},{"id":"n21721","layer":"informal","project":"p14","title":"diconnected\\_iff\\_isKArcConnected\\_one","kind":"theorem","summary":"[diconnected\\_iff\\_isKArcConnected\\_one] *A nontrivial digraph is diconnected if and only if it…","labels":["diconnected_iff_isKArcConnected_one"],"detail_key":"p14"},{"id":"n21722","layer":"informal","project":"p14","title":"associatedDigraph\\_isKArcConnected\\_iff","kind":"theorem","summary":"[associatedDigraph\\_isKArcConnected\\_iff] *D(G) is k-arc-connected if and only if G is k-edge-c…","labels":["associatedDigraph_isKArcConnected_iff"],"detail_key":"p14"},{"id":"n21723","layer":"informal","project":"p14","title":"deBruijnDigraph\\_indegree\\_eq\\_two","kind":"theorem","summary":"[deBruijnDigraph\\_indegree\\_eq\\_two] A vertex is a window of n-1 bits. Its out-neighbours drop…","labels":["deBruijnDigraph_indegree_eq_two"],"detail_key":"p14"},{"id":"n21724","layer":"informal","project":"p14","title":"deBruijnDigraph\\_connected","kind":"theorem","summary":"[deBruijnDigraph\\_connected] From any binary string one can reach any other by shifting in the…","labels":["deBruijnDigraph_connected"],"detail_key":"p14"},{"id":"n21725","layer":"informal","project":"p14","title":"deBruijnDigraph\\_exists\\_directedEulerTour","kind":"theorem","summary":"[deBruijnDigraph\\_exists\\_directedEulerTour] Each arc carries an n-bit label and the tour uses…","labels":["deBruijnDigraph_exists_directedEulerTour"],"detail_key":"p14"},{"id":"n21726","layer":"informal","project":"p14","title":"robbins\\_orientation","kind":"theorem","summary":"[robbins\\_orientation] (Robbins, 1939). *If G is 2-edge-connected, then G has a diconnected ori…","labels":["robbins_orientation"],"detail_key":"p14"},{"id":"n21727","layer":"informal","project":"p14","title":"exists\\_kArcConnected\\_orientation\\_of\\_eulerian","kind":"theorem","summary":"[exists\\_kArcConnected\\_orientation\\_of\\_eulerian] *Let G be a 2k-edge-connected graph with an…","labels":["exists_kArcConnected_orientation_of_eulerian"],"detail_key":"p14"},{"id":"n21728","layer":"informal","project":"p14","title":"tournament\\_adjMatrix\\_pow\\_pos","kind":"theorem","summary":"[tournament\\_adjMatrix\\_pow\\_pos] *Let D be a diconnected tournament with \\nu \\ge 5, and let A…","labels":["tournament_adjMatrix_pow_pos"],"detail_key":"p14"},{"id":"n21729","layer":"informal","project":"p14","title":"tournament\\_adjMatrix\\_isPrimitive\\_iff","kind":"theorem","summary":"[tournament\\_adjMatrix\\_isPrimitive\\_iff] *The adjacency matrix A of a tournament D is primitiv…","labels":["tournament_adjMatrix_isPrimitive_iff"],"detail_key":"p14"},{"id":"n21730","layer":"informal","project":"p14","title":"Colour i is \\emphrepresented at v: some edge incident with v has colour i","kind":"definition","summary":"[Colour i is \\emphrepresented at v: some edge incident with v has colour i] A k-edge colouring…","labels":["SimpleGraph.IsRepresentedAt"],"detail_key":"p14"},{"id":"n21731","layer":"informal","project":"p14","title":"c(v): the number of distinct colours represented at v","kind":"definition","summary":"[c(v): the number of distinct colours represented at v] c(v) counts the \\emphdistinct colours o…","labels":["SimpleGraph.numColoursAt"],"detail_key":"p14"},{"id":"n21732","layer":"informal","project":"p14","title":"An optimal k-edge colouring: one that cannot be improved, where an improvement strictly i…","kind":"definition","summary":"[An optimal k-edge colouring: one that cannot be improved, where an improvement strictly increa…","labels":["SimpleGraph.IsOptimalEdgeColouring"],"detail_key":"p14"},{"id":"n21733","layer":"informal","project":"p14","title":"G(E_i \\cup E_j), the subgraph on the edges coloured i or j","kind":"definition","summary":"[G(E_i \\cup E_j), the subgraph on the edges coloured i or j] Erase every edge except those colo…","labels":["SimpleGraph.twoColourSubgraph"],"detail_key":"p14"},{"id":"n21734","layer":"informal","project":"p14","title":"G is uniquely k-edge-colourable: any two proper k-edge colourings agree up to a permutati…","kind":"definition","summary":"[G is uniquely k-edge-colourable: any two proper k-edge colourings agree up to a permutation of…","labels":["SimpleGraph.IsUniquelyEdgeColourable"],"detail_key":"p14"},{"id":"n21735","layer":"informal","project":"p14","title":"The Petersen graph","kind":"definition","summary":"[The Petersen graph] The 3-regular graph on ten vertices, the Kneser graph on 2-subsets of a 5-…","labels":["SimpleGraph.petersenGraph"],"detail_key":"p14"},{"id":"n21736","layer":"informal","project":"p14","title":"(6.1): \\chi' \\ge \\Delta","kind":"theorem","summary":"[(6.1): \\chi' \\ge \\Delta] Take v of maximum degree \\Delta. All \\Delta edges at v are pairwise a…","labels":["SimpleGraph.maxDegree_le_edgeChromaticNumber"],"detail_key":"p14"},{"id":"n21737","layer":"informal","project":"p14","title":"(6.3): c(v) \\le d(v)","kind":"theorem","summary":"[(6.3): c(v) \\le d(v)] c(v) counts distinct colours on the d(v) edges at v; distinct colours ca…","labels":["SimpleGraph.numColoursAt_le_degree"],"detail_key":"p14"},{"id":"n21738","layer":"informal","project":"p14","title":"(6.3): C is proper \\iff equality holds in (6.3) at every vertex","kind":"theorem","summary":"[(6.3): C is proper \\iff equality holds in (6.3) at every vertex] Two edges are adjacent exactl…","labels":["SimpleGraph.isProper_iff_numColoursAt_eq_degree"],"detail_key":"p14"},{"id":"n21739","layer":"informal","project":"p14","title":"exists\\_two\\_edge\\_colouring\\_both\\_represented","kind":"theorem","summary":"[exists\\_two\\_edge\\_colouring\\_both\\_represented] An Euler tour threads through every vertex; a…","labels":["SimpleGraph.exists_two_edge_colouring_both_represented"],"detail_key":"p14"},{"id":"n21740","layer":"informal","project":"p14","title":"isOddCycle\\_component\\_of\\_isOptimalEdgeColouring","kind":"theorem","summary":"[isOddCycle\\_component\\_of\\_isOptimalEdgeColouring] A wasted colour at u --- one missing, anoth…","labels":["SimpleGraph.isOddCycle_component_of_isOptimalEdgeColouring"],"detail_key":"p14"},{"id":"n21741","layer":"informal","project":"p14","title":"Theorem 6.1 (K\\\"onig) ! BLOCKED: if G is bipartite then \\chi' = \\Delta","kind":"theorem","summary":"[Theorem 6.1 (K\\\"onig) ! BLOCKED: if G is bipartite then \\chi' = \\Delta] Contrapositive: if \\ch…","labels":["SimpleGraph.edgeChromaticNumber_eq_maxDegree_of_isBipartite"],"detail_key":"p14"},{"id":"n21742","layer":"informal","project":"p14","title":"Theorem 6.2 (Vizing) ! BLOCKED: if G is simple then \\chi' = \\Delta or \\chi' = \\Delta+1","kind":"theorem","summary":"[Theorem 6.2 (Vizing) ! BLOCKED: if G is simple then \\chi' = \\Delta or \\chi' = \\Delta+1] By (6.…","labels":["SimpleGraph.vizing_chromatic_index"],"detail_key":"p14"},{"id":"n21743","layer":"informal","project":"p14","title":"The core content of Vizing, \\chi' \\le \\Delta+1, from which the disjunction follows via (6…","kind":"theorem","summary":"[The core content of Vizing, \\chi' \\le \\Delta+1, from which the disjunction follows via (6.1)]…","labels":["SimpleGraph.vizing_chromatic_index_le"],"detail_key":"p14"},{"id":"n21744","layer":"informal","project":"p14","title":"exists\\_rebalanced\\_matchings","kind":"theorem","summary":"[exists\\_rebalanced\\_matchings] Swapping roles along P moves exactly one edge from the larger m…","labels":["SimpleGraph.exists_rebalanced_matchings"],"detail_key":"p14"},{"id":"n21745","layer":"informal","project":"p14","title":"exists\\_balanced\\_matching\\_decomposition","kind":"theorem","summary":"[exists\\_balanced\\_matching\\_decomposition] ! In the book's notation [x] is the floor and x the…","labels":["SimpleGraph.exists_balanced_matching_decomposition"],"detail_key":"p14"},{"id":"n21746","layer":"informal","project":"p14","title":"Ex 6.1.3(a): every bipartite G has a \\Delta-regular bipartite supergraph ! carrier change","kind":"theorem","summary":"[Ex 6.1.3(a): every bipartite G has a \\Delta-regular bipartite supergraph ! carrier change] Pad…","labels":["SimpleGraph.exists_regular_bipartite_supergraph"],"detail_key":"p14"},{"id":"n21747","layer":"informal","project":"p14","title":"Ex 6.1.6 (Gupta) ! BLOCKED: bipartite with \\delta > 0 \\Rightarrow a \\delta-edge colouring…","kind":"theorem","summary":"[Ex 6.1.6 (Gupta) ! BLOCKED: bipartite with \\delta > 0 \\Rightarrow a \\delta-edge colouring\\dots…","labels":["SimpleGraph.exists_edge_colouring_all_represented_of_isBipartite"],"detail_key":"p14"},{"id":"n21748","layer":"informal","project":"p14","title":"Ex 6.2.1*: \\chi'(K_2n- 1) = 2n- 1 (explicit colouring --- bypasses Vizing)","kind":"theorem","summary":"[Ex 6.2.1*: \\chi'(K_2n- 1) = 2n- 1 (explicit colouring --- bypasses Vizing)] K_2n-1 has odd ord…","labels":["SimpleGraph.edgeChromaticNumber_completeGraph_odd"],"detail_key":"p14"},{"id":"n21749","layer":"informal","project":"p14","title":"Ex 6.2.1*: \\chi'(K_2n) = 2n- 1","kind":"theorem","summary":"[Ex 6.2.1*: \\chi'(K_2n) = 2n- 1] K_2n has \\Delta = 2n - 1, so (6.1) gives \\chi' \\ge 2n - 1. For…","labels":["SimpleGraph.edgeChromaticNumber_completeGraph_even"],"detail_key":"p14"},{"id":"n21750","layer":"informal","project":"p14","title":"Ex 6.2.2 ! BLOCKED (Vizing): nonempty (0 < k) k-regular with \\nu odd \\Rightarrow \\chi' =\\…","kind":"theorem","summary":"[Ex 6.2.2 ! BLOCKED (Vizing): nonempty (0 < k) k-regular with \\nu odd \\Rightarrow \\chi' =\\dots]…","labels":["SimpleGraph.edgeChromaticNumber_eq_maxDegree_add_one_of_regular_odd_card"],"detail_key":"p14"},{"id":"n21751","layer":"informal","project":"p14","title":"Ex 6.2.3(a) ! BLOCKED (Vizing): \\nu = 2n+1 and \\varepsilon > n\\Delta \\Rightarrow \\chi' =\\…","kind":"theorem","summary":"[Ex 6.2.3(a) ! BLOCKED (Vizing): \\nu = 2n+1 and \\varepsilon > n\\Delta \\Rightarrow \\chi' =\\dots]…","labels":["SimpleGraph.edgeChromaticNumber_eq_maxDegree_add_one_of_card_edges_gt"],"detail_key":"p14"},{"id":"n21752","layer":"informal","project":"p14","title":"Ex 6.2.3(b)(i) ! BLOCKED: subdividing one edge of an even-order k-regular graph (k \\ge 2)…","kind":"theorem","summary":"[Ex 6.2.3(b)(i) ! BLOCKED: subdividing one edge of an even-order k-regular graph (k \\ge 2)\\dots…","labels":["SimpleGraph.edgeChromaticNumber_subdivide_eq_maxDegree_add_one"],"detail_key":"p14"},{"id":"n21753","layer":"informal","project":"p14","title":"Ex 6.2.3(b)(ii) ! BLOCKED: deleting < k/2 edges (\\texttt2 * F.card < k) from an odd-order…","kind":"theorem","summary":"[Ex 6.2.3(b)(ii) ! BLOCKED: deleting < k/2 edges (\\texttt2 * F.card < k) from an odd-order\\dots…","labels":["SimpleGraph.edgeChromaticNumber_deleteEdges_eq_maxDegree_add_one"],"detail_key":"p14"},{"id":"n21754","layer":"informal","project":"p14","title":"Ex 6.2.5: every uniquely 3-edge-colourable 3-regular graph is hamiltonian","kind":"theorem","summary":"[Ex 6.2.5: every uniquely 3-edge-colourable 3-regular graph is hamiltonian] In a cubic graph a…","labels":["SimpleGraph.isHamiltonian_of_uniquely_three_edge_colourable"],"detail_key":"p14"},{"id":"n21755","layer":"informal","project":"p14","title":"Ex 6.2.6(a) ! BLOCKED (Vizing): \\chi'(G \\square K_2) = \\Delta(G \\square K_2)","kind":"theorem","summary":"[Ex 6.2.6(a) ! BLOCKED (Vizing): \\chi'(G \\square K_2) = \\Delta(G \\square K_2)] The book's \"prod…","labels":["SimpleGraph.edgeChromaticNumber_boxProd_completeGraph_two"],"detail_key":"p14"},{"id":"n21756","layer":"informal","project":"p14","title":"Ex 6.2.6(b) ! BLOCKED (via (a)): if H is nontrivial (0 < \\Delta(H)) with \\chi'(H) =\\dots","kind":"theorem","summary":"[Ex 6.2.6(b) ! BLOCKED (via (a)): if H is nontrivial (0 < \\Delta(H)) with \\chi'(H) =\\dots] G \\s…","labels":["SimpleGraph.edgeChromaticNumber_boxProd"],"detail_key":"p14"},{"id":"n21757","layer":"informal","project":"p14","title":"G is eulerian: it has a closed Euler trail (an Euler tour)","kind":"definition","summary":"[G is eulerian: it has a closed Euler trail (an Euler tour)] \\texttt\\exists u, \\exists p : G.Wa…","labels":["SimpleGraph.IsEulerianGraph"],"detail_key":"p14"},{"id":"n21758","layer":"informal","project":"p14","title":"Component count \\omega(G)","kind":"definition","summary":"[Component count \\omega(G)] \\textttNat.card G.ConnectedComponent, following Mathlib's own idiom…","labels":["SimpleGraph.numComponents"],"detail_key":"p14"},{"id":"n21759","layer":"informal","project":"p14","title":"join","kind":"definition","summary":"[join] G \\lor H, rebuilt on Mathlib: within-side edges from \\oplus g, all cross edges from \\tex…","labels":["SimpleGraph.join"],"detail_key":"p14"},{"id":"n21760","layer":"informal","project":"p14","title":"C_m,n = K_m \\lor (K_m^c + K_n-2m), with + the Mathlib disjoint sum \\oplus g","kind":"definition","summary":"[C_m,n = K_m \\lor (K_m^c + K_n-2m), with + the Mathlib disjoint sum \\oplus g] Take m \"hub\" vert…","labels":["SimpleGraph.Cmn"],"detail_key":"p14"},{"id":"n21761","layer":"informal","project":"p14","title":"The sorted (ascending) degree sequence of G","kind":"definition","summary":"[The sorted (ascending) degree sequence of G] Simply list how many neighbours each vertex has,…","labels":["SimpleGraph.degreeSequence"],"detail_key":"p14"},{"id":"n21762","layer":"informal","project":"p14","title":"G is degree-majorised by H: same order, and every sorted-degree entry of G is \\le the\\dots","kind":"definition","summary":"[G is degree-majorised by H: same order, and every sorted-degree entry of G is \\le the\\dots] Li…","labels":["SimpleGraph.DegreeMajorised"],"detail_key":"p14"},{"id":"n21763","layer":"informal","project":"p14","title":"One Bondy--Chv\\'atal closure step: add a nonadjacent pair whose degree sum is \\ge \\nu","kind":"definition","summary":"[One Bondy--Chv\\'atal closure step: add a nonadjacent pair whose degree sum is \\ge \\nu] Look fo…","labels":["SimpleGraph.ClosureStep"],"detail_key":"p14"},{"id":"n21764","layer":"informal","project":"p14","title":"G is Hamilton-connected: every ordered pair is joined by a Hamilton path (Ex 4.2.11)","kind":"definition","summary":"[G is Hamilton-connected: every ordered pair is joined by a Hamilton path (Ex 4.2.11)] Not mere…","labels":["SimpleGraph.IsHamiltonConnected"],"detail_key":"p14"},{"id":"n21765","layer":"informal","project":"p14","title":"G is hypohamiltonian: not Hamiltonian, but every vertex-deleted subgraph is (Ex 4.2.12)","kind":"definition","summary":"[G is hypohamiltonian: not Hamiltonian, but every vertex-deleted subgraph is (Ex 4.2.12)] The g…","labels":["SimpleGraph.IsHypohamiltonian"],"detail_key":"p14"},{"id":"n21766","layer":"informal","project":"p14","title":"Thm 4.1: a nonempty connected graph is eulerian iff it has no odd-degree vertex","kind":"theorem","summary":"[Thm 4.1: a nonempty connected graph is eulerian iff it has no odd-degree vertex] Every visit t…","labels":["SimpleGraph.euler_tour_iff_no_odd_degree"],"detail_key":"p14"},{"id":"n21767","layer":"informal","project":"p14","title":"Cor 4.1: a connected graph has an Euler trail iff at most two vertices have odd degree","kind":"theorem","summary":"[Cor 4.1: a connected graph has an Euler trail iff at most two vertices have odd degree] By Cor…","labels":["SimpleGraph.euler_trail_iff_le_two_odd"],"detail_key":"p14"},{"id":"n21768","layer":"informal","project":"p14","title":"Thm 4.2: if G is hamiltonian then \\omega(G - S) \\le |S| for every nonempty proper S","kind":"theorem","summary":"[Thm 4.2: if G is hamiltonian then \\omega(G - S) \\le |S| for every nonempty proper S] A hamilto…","labels":["SimpleGraph.hamiltonian_toughness"],"detail_key":"p14"},{"id":"n21769","layer":"informal","project":"p14","title":"Thm 4.3: \\nu \\ge 3 and \\delta \\ge \\nu/2 (stated as \\nu \\le 2\\delta) imply G is hamiltonian","kind":"theorem","summary":"[Thm 4.3: \\nu \\ge 3 and \\delta \\ge \\nu/2 (stated as \\nu \\le 2\\delta) imply G is hamiltonian] If…","labels":["SimpleGraph.dirac_hamiltonian"],"detail_key":"p14"},{"id":"n21770","layer":"informal","project":"p14","title":"Lem 4.4.1: for nonadjacent u,v with d(u)+d(v) \\ge \\nu, G is hamiltonian iff G+uv is","kind":"theorem","summary":"[Lem 4.4.1: for nonadjacent u,v with d(u)+d(v) \\ge \\nu, G is hamiltonian iff G+uv is] The book'…","labels":["SimpleGraph.bondy_chvatal"],"detail_key":"p14"},{"id":"n21771","layer":"informal","project":"p14","title":"Thm 4.4 (honest restatement): reachability by closure steps preserves hamiltonicity","kind":"theorem","summary":"[Thm 4.4 (honest restatement): reachability by closure steps preserves hamiltonicity] Stated in…","labels":["SimpleGraph.hamiltonian_iff_of_closureSteps"],"detail_key":"p14"},{"id":"n21772","layer":"informal","project":"p14","title":"Cor 4.4 (honest restatement): if closure steps reach \\top, then G was hamiltonian","kind":"theorem","summary":"[Cor 4.4 (honest restatement): if closure steps reach \\top, then G was hamiltonian] Stated with…","labels":["SimpleGraph.hamiltonian_of_closureSteps_top"],"detail_key":"p14"},{"id":"n21773","layer":"informal","project":"p14","title":"Reusable helper (! absent from Mathlib): \\top on \\ge 3 vertices is hamiltonian","kind":"theorem","summary":"[Reusable helper (! absent from Mathlib): \\top on \\ge 3 vertices is hamiltonian] In K_n every p…","labels":["SimpleGraph.top_isHamiltonian"],"detail_key":"p14"},{"id":"n21774","layer":"informal","project":"p14","title":"Thm 4.5 (counting form): the Chv\\'atal degree condition implies hamiltonicity","kind":"theorem","summary":"[Thm 4.5 (counting form): the Chv\\'atal degree condition implies hamiltonicity] ! The hypothesi…","labels":["SimpleGraph.chvatal_hamiltonian"],"detail_key":"p14"},{"id":"n21775","layer":"informal","project":"p14","title":"Thm 4.6: a nonhamiltonian simple graph with \\nu \\ge 3 is degree-majorised by some C_m,\\nu","kind":"theorem","summary":"[Thm 4.6: a nonhamiltonian simple graph with \\nu \\ge 3 is degree-majorised by some C_m,\\nu] The…","labels":["SimpleGraph.chvatal_degree_majorised"],"detail_key":"p14"},{"id":"n21776","layer":"informal","project":"p14","title":"Cor 4.6a: \\varepsilon > C(\\nu- 1,2) + 1 implies hamiltonicity","kind":"theorem","summary":"[Cor 4.6a: \\varepsilon > C(\\nu- 1,2) + 1 implies hamiltonicity] Enough edges guarantee a spanni…","labels":["SimpleGraph.ore_bondy_edge_bound"],"detail_key":"p14"},{"id":"n21777","layer":"informal","project":"p14","title":"Cor 4.6b: the extremal nonhamiltonian graphs at C(\\nu- 1,2)+1 edges are C_1,\\nu (and C_2,…","kind":"theorem","summary":"[Cor 4.6b: the extremal nonhamiltonian graphs at C(\\nu- 1,2)+1 edges are C_1,\\nu (and C_2,5)] T…","labels":["SimpleGraph.ore_bondy_extremal"],"detail_key":"p14"},{"id":"n21778","layer":"informal","project":"p14","title":"Ex 4.1.4: no odd degree \\Rightarrow an edge-disjoint decomposition into cycles","kind":"theorem","summary":"[Ex 4.1.4: no odd degree \\Rightarrow an edge-disjoint decomposition into cycles] Even degrees e…","labels":["SimpleGraph.even_degree_cycle_decomposition"],"detail_key":"p14"},{"id":"n21779","layer":"informal","project":"p14","title":"Ex 4.1.5: exactly 2k odd-degree vertices \\Rightarrow k edge-disjoint covering trails","kind":"theorem","summary":"[Ex 4.1.5: exactly 2k odd-degree vertices \\Rightarrow k edge-disjoint covering trails] By Corol…","labels":["SimpleGraph.odd_vertices_trail_cover"],"detail_key":"p14"},{"id":"n21780","layer":"informal","project":"p14","title":"Ex 4.2.1(a): not 2-connected (with \\nu \\ge 3) \\Rightarrow nonhamiltonian (restated on the…","kind":"theorem","summary":"[Ex 4.2.1(a): not 2-connected (with \\nu \\ge 3) \\Rightarrow nonhamiltonian (restated on the repo…","labels":["SimpleGraph.nonhamiltonian_of_not_two_connected"],"detail_key":"p14"},{"id":"n21781","layer":"informal","project":"p14","title":"Ex 4.2.1(b): an unbalanced bipartite graph is nonhamiltonian","kind":"theorem","summary":"[Ex 4.2.1(b): an unbalanced bipartite graph is nonhamiltonian] Every edge crosses between X and…","labels":["SimpleGraph.nonhamiltonian_of_unbalanced_bipartite"],"detail_key":"p14"},{"id":"n21782","layer":"informal","project":"p14","title":"Ex 4.2.3: a Hamilton path implies \\omega(G - S) \\le |S| + 1","kind":"theorem","summary":"[Ex 4.2.3: a Hamilton path implies \\omega(G - S) \\le |S| + 1] The path analogue of Theorem 4.2.…","labels":["SimpleGraph.hamiltonian_path_toughness"],"detail_key":"p14"},{"id":"n21783","layer":"informal","project":"p14","title":"Ex 4.2.4*: Chv\\'atal's Hamilton-path degree condition (counting form)","kind":"theorem","summary":"[Ex 4.2.4*: Chv\\'atal's Hamilton-path degree condition (counting form)] The Hamilton-\\emphpath…","labels":["SimpleGraph.chvatal_hamiltonian_path"],"detail_key":"p14"},{"id":"n21784","layer":"informal","project":"p14","title":"Ex 4.2.5: a self-complementary graph has a Hamilton path (Clapham)","kind":"theorem","summary":"[Ex 4.2.5: a self-complementary graph has a Hamilton path (Clapham)] Only part (b) is formalise…","labels":["SimpleGraph.hamiltonian_path_of_self_complementary"],"detail_key":"p14"},{"id":"n21785","layer":"informal","project":"p14","title":"Ex 4.2.5(a) (Clapham): if G dominates G^c on the lower half of the sorted degree sequence…","kind":"theorem","summary":"[Ex 4.2.5(a) (Clapham): if G dominates G^c on the lower half of the sorted degree sequence, G h…","labels":["SimpleGraph.hamiltonian_path_of_degree_dominates_compl"],"detail_key":"p14"},{"id":"n21786","layer":"informal","project":"p14","title":"Ex 4.2.8: Erd\\Hos' edge bound \\nu \\ge 6\\delta, \\varepsilon > C(\\nu-\\delta,2) + \\delta^2\\d…","kind":"theorem","summary":"[Ex 4.2.8: Erd\\Hos' edge bound \\nu \\ge 6\\delta, \\varepsilon > C(\\nu-\\delta,2) + \\delta^2\\dots]…","labels":["SimpleGraph.erdos_hamiltonian"],"detail_key":"p14"},{"id":"n21787","layer":"informal","project":"p14","title":"Ex 4.2.9*: connected with \\nu > 2\\delta \\Rightarrow a path of length \\ge 2\\delta (Dirac)","kind":"theorem","summary":"[Ex 4.2.9*: connected with \\nu > 2\\delta \\Rightarrow a path of length \\ge 2\\delta (Dirac)] Exer…","labels":["SimpleGraph.long_path_of_order_gt_two_minDegree"],"detail_key":"p14"},{"id":"n21788","layer":"informal","project":"p14","title":"Ex 4.2.10: a 2k-regular graph on 4k+1 vertices is hamiltonian (Nash-Williams)","kind":"theorem","summary":"[Ex 4.2.10: a 2k-regular graph on 4k+1 vertices is hamiltonian (Nash-Williams)] Such a graph ha…","labels":["SimpleGraph.nash_williams_regular_hamiltonian"],"detail_key":"p14"},{"id":"n21789","layer":"informal","project":"p14","title":"Ex 4.2.11(a): a Hamilton-connected graph satisfies 3\\nu + 1 \\le 2\\varepsilon (Moon)","kind":"theorem","summary":"[Ex 4.2.11(a): a Hamilton-connected graph satisfies 3\\nu + 1 \\le 2\\varepsilon (Moon)] Hamilton-…","labels":["SimpleGraph.hamiltonConnected_edge_bound"],"detail_key":"p14"},{"id":"n21790","layer":"informal","project":"p14","title":"nonempty\\_iso\\_iff","kind":"theorem","summary":"[nonempty\\_iso\\_iff] B\\&M's general isomorphism needs two bijections, one on vertices and one o…","labels":["SimpleGraph.nonempty_iso_iff"],"detail_key":"p14"},{"id":"n21791","layer":"informal","project":"p14","title":"edgeCard\\_eq\\_choose\\_two\\_iff\\_top","kind":"theorem","summary":"[edgeCard\\_eq\\_choose\\_two\\_iff\\_top] Exercise 1.1.3 gives \\varepsilon \\le C(\\nu,2); equality m…","labels":["SimpleGraph.edgeCard_eq_choose_two_iff_top"],"detail_key":"p14"},{"id":"n21792","layer":"informal","project":"p14","title":"completeBipartite\\_edgeCard","kind":"theorem","summary":"[completeBipartite\\_edgeCard] K_m,n contains exactly one edge for each choice of a vertex in X…","labels":["SimpleGraph.completeBipartite_edgeCard"],"detail_key":"p14"},{"id":"n21793","layer":"informal","project":"p14","title":"bipartite\\_edgeCard\\_le","kind":"theorem","summary":"[bipartite\\_edgeCard\\_le] mn with fixed sum \\nu is largest when the parts are equal, giving \\nu…","labels":["SimpleGraph.bipartite_edgeCard_le"],"detail_key":"p14"},{"id":"n21794","layer":"informal","project":"p14","title":"Ex 1.2.10: the k-cube --- vertices are k-bit strings, adjacent iff they differ in exactly…","kind":"definition","summary":"[Ex 1.2.10: the k-cube --- vertices are k-bit strings, adjacent iff they differ in exactly one…","labels":["SimpleGraph.hypercube"],"detail_key":"p14"},{"id":"n21795","layer":"informal","project":"p14","title":"hypercube\\_edgeCard","kind":"theorem","summary":"[hypercube\\_edgeCard] Every vertex has degree k, one neighbour per coordinate flip. ! k - 1 is…","labels":["SimpleGraph.hypercube_edgeCard"],"detail_key":"p14"},{"id":"n21796","layer":"informal","project":"p14","title":"hypercube\\_bipartite","kind":"theorem","summary":"[hypercube\\_bipartite] Split the tuples by the parity of their number of 1s; an edge flips exac…","labels":["SimpleGraph.hypercube_bipartite"],"detail_key":"p14"},{"id":"n21797","layer":"informal","project":"p14","title":"selfComplementary\\_card\\_mod\\_four","kind":"theorem","summary":"[selfComplementary\\_card\\_mod\\_four] Self-complementary means G \\cong G^c, so G has exactly hal…","labels":["SimpleGraph.selfComplementary_card_mod_four"],"detail_key":"p14"},{"id":"n21798","layer":"informal","project":"p14","title":"induce\\_completeGraph","kind":"theorem","summary":"[induce\\_completeGraph] G[V'] keeps V' and \\emphall edges of G with both ends there. If G is co…","labels":["SimpleGraph.induce_completeGraph"],"detail_key":"p14"},{"id":"n21799","layer":"informal","project":"p14","title":"subgraph\\_bipartite","kind":"theorem","summary":"[subgraph\\_bipartite] Every edge of H is an edge of G, so the very same bipartition still has e…","labels":["SimpleGraph.subgraph_bipartite"],"detail_key":"p14"},{"id":"n21800","layer":"informal","project":"p14","title":"induce\\_bipartite","kind":"theorem","summary":"[induce\\_bipartite] \\textttsubgraph\\_bipartite covers B\\&M's \\emphspanning subgraphs (same vert…","labels":["SimpleGraph.induce_bipartite"],"detail_key":"p14"},{"id":"n21801","layer":"informal","project":"p14","title":"subgraph\\_induce\\_bipartite","kind":"theorem","summary":"[subgraph\\_induce\\_bipartite] This is the exercise as the book states it; \\textttsubgraph\\_bipa…","labels":["SimpleGraph.subgraph_induce_bipartite"],"detail_key":"p14"},{"id":"n21802","layer":"informal","project":"p14","title":"sum\\_degrees\\_eq\\_two\\_mul\\_edgeCard","kind":"theorem","summary":"[sum\\_degrees\\_eq\\_two\\_mul\\_edgeCard] Adding up the degrees counts each edge twice, once from…","labels":["SimpleGraph.sum_degrees_eq_two_mul_edgeCard"],"detail_key":"p14"},{"id":"n21803","layer":"informal","project":"p14","title":"even\\_card\\_odd\\_degree","kind":"theorem","summary":"[even\\_card\\_odd\\_degree] A sum of odd numbers is even precisely when there is an even number o…","labels":["SimpleGraph.even_card_odd_degree"],"detail_key":"p14"},{"id":"n21804","layer":"informal","project":"p14","title":"degree\\_bounds","kind":"theorem","summary":"[degree\\_bounds] 2\\varepsilon/\\nu is the \\emphaverage degree, since the degrees sum to 2\\vareps…","labels":["SimpleGraph.degree_bounds"],"detail_key":"p14"},{"id":"n21805","layer":"informal","project":"p14","title":"regular\\_bipartite\\_card\\_eq","kind":"theorem","summary":"[regular\\_bipartite\\_card\\_eq] Count the edges twice, once from each side. ! hk : 0 < k is esse…","labels":["SimpleGraph.regular_bipartite_card_eq"],"detail_key":"p14"},{"id":"n21806","layer":"informal","project":"p14","title":"exists\\_bipartite\\_spanning\\_subgraph","kind":"theorem","summary":"[exists\\_bipartite\\_spanning\\_subgraph] Split the vertices in two and keep only the crossing ed…","labels":["SimpleGraph.exists_bipartite_spanning_subgraph"],"detail_key":"p14"},{"id":"n21807","layer":"informal","project":"p14","title":"lineGraph\\_edgeCard","kind":"theorem","summary":"[lineGraph\\_edgeCard] Two edges are adjacent in the edge graph exactly when they share an end;…","labels":["SimpleGraph.lineGraph_edgeCard"],"detail_key":"p14"},{"id":"n21808","layer":"informal","project":"p14","title":"exists\\_path\\_length\\_of\\_minDegree","kind":"theorem","summary":"[exists\\_path\\_length\\_of\\_minDegree] A longest path cannot be extended, so its endpoint's neig…","labels":["SimpleGraph.exists_path_length_of_minDegree"],"detail_key":"p14"},{"id":"n21809","layer":"informal","project":"p14","title":"connected\\_iff\\_forall\\_partition\\_edge","kind":"theorem","summary":"[connected\\_iff\\_forall\\_partition\\_edge] Connectivity is exactly the statement that V admits n…","labels":["SimpleGraph.connected_iff_forall_partition_edge"],"detail_key":"p14"},{"id":"n21810","layer":"informal","project":"p14","title":"connected\\_of\\_edgeCard\\_gt","kind":"theorem","summary":"[connected\\_of\\_edgeCard\\_gt] A disconnected simple graph is sparsest-constrained in the most l…","labels":["SimpleGraph.connected_of_edgeCard_gt"],"detail_key":"p14"},{"id":"n21811","layer":"informal","project":"p14","title":"compl\\_connected\\_of\\_not\\_connected","kind":"theorem","summary":"[compl\\_connected\\_of\\_not\\_connected] Every pair is joined in G^c by a path of length at most…","labels":["SimpleGraph.compl_connected_of_not_connected"],"detail_key":"p14"},{"id":"n21812","layer":"informal","project":"p14","title":"card\\_connectedComponent\\_le\\_of\\_le","kind":"theorem","summary":"[card\\_connectedComponent\\_le\\_of\\_le] If H \\le K then \\omega(K) \\le \\omega(H) --- adding edges…","labels":["SimpleGraph.card_connectedComponent_le_of_le"],"detail_key":"p14"},{"id":"n21813","layer":"informal","project":"p14","title":"components\\_deleteEdge","kind":"theorem","summary":"[components\\_deleteEdge] Deleting an edge only destroys connections, never creates them; and it…","labels":["SimpleGraph.components_deleteEdge"],"detail_key":"p14"},{"id":"n21814","layer":"informal","project":"p14","title":"compl\\_diam\\_lt\\_of\\_diam\\_gt","kind":"theorem","summary":"[compl\\_diam\\_lt\\_of\\_diam\\_gt] A graph and its complement cannot both be \"spread out\": if G is…","labels":["SimpleGraph.compl_diam_lt_of_diam_gt"],"detail_key":"p14"},{"id":"n21815","layer":"informal","project":"p14","title":"exists\\_induced\\_path\\_of\\_connected\\_not\\_complete","kind":"theorem","summary":"[exists\\_induced\\_path\\_of\\_connected\\_not\\_complete] A connected non-complete graph always con…","labels":["SimpleGraph.exists_induced_path_of_connected_not_complete"],"detail_key":"p14"},{"id":"n21816","layer":"informal","project":"p14","title":"bipartite\\_iff\\_no\\_odd\\_cycle","kind":"theorem","summary":"[bipartite\\_iff\\_no\\_odd\\_cycle] Odd cycles are the sole obstruction to 2-colourability --- the…","labels":["SimpleGraph.bipartite_iff_no_odd_cycle"],"detail_key":"p14"},{"id":"n21817","layer":"informal","project":"p14","title":"edge\\_in\\_cycle\\_of\\_closed\\_trail","kind":"theorem","summary":"[edge\\_in\\_cycle\\_of\\_closed\\_trail] A closed trail may revisit vertices, so it need not be a c…","labels":["SimpleGraph.edge_in_cycle_of_closed_trail"],"detail_key":"p14"},{"id":"n21818","layer":"informal","project":"p14","title":"exists\\_cycle\\_of\\_minDegree\\_ge\\_two","kind":"theorem","summary":"[exists\\_cycle\\_of\\_minDegree\\_ge\\_two] If every vertex offers a second way out you can never g…","labels":["SimpleGraph.exists_cycle_of_minDegree_ge_two"],"detail_key":"p14"},{"id":"n21819","layer":"informal","project":"p14","title":"exists\\_long\\_cycle\\_of\\_minDegree","kind":"theorem","summary":"[exists\\_long\\_cycle\\_of\\_minDegree] Sharpens exercise 1.7.2 from \"some cycle exists\" to a leng…","labels":["SimpleGraph.exists_long_cycle_of_minDegree"],"detail_key":"p14"},{"id":"n21820","layer":"informal","project":"p14","title":"girth\\_four\\_regular\\_card\\_ge","kind":"theorem","summary":"[girth\\_four\\_regular\\_card\\_ge] ! B\\&M add that, up to isomorphism, there is exactly one such…","labels":["SimpleGraph.girth_four_regular_card_ge"],"detail_key":"p14"},{"id":"n21821","layer":"informal","project":"p14","title":"girth\\_five\\_regular\\_card\\_ge","kind":"theorem","summary":"[girth\\_five\\_regular\\_card\\_ge] Graphs attaining this bound with diameter two are the Moore gr…","labels":["SimpleGraph.girth_five_regular_card_ge"],"detail_key":"p14"},{"id":"n21822","layer":"informal","project":"p14","title":"exists\\_cycle\\_of\\_edgeCard\\_ge","kind":"theorem","summary":"[exists\\_cycle\\_of\\_edgeCard\\_ge] Each edge added to a forest either joins two different compon…","labels":["SimpleGraph.exists_cycle_of_edgeCard_ge"],"detail_key":"p14"},{"id":"n21823","layer":"informal","project":"p14","title":"A covering K: every edge has an end in K","kind":"definition","summary":"[A covering K: every edge has an end in K] A set of vertices touching every edge. Theorem 7.1 s…","labels":["SimpleGraph.IsCovering"],"detail_key":"p14"},{"id":"n21824","layer":"informal","project":"p14","title":"isIndepSet\\_iff\\_isCovering\\_compl","kind":"theorem","summary":"[isIndepSet\\_iff\\_isCovering\\_compl] V \\ S is S^c. Independent sets and coverings are complemen…","labels":["SimpleGraph.isIndepSet_iff_isCovering_compl"],"detail_key":"p14"},{"id":"n21825","layer":"informal","project":"p14","title":"The covering number \\beta(G)","kind":"definition","summary":"[The covering number \\beta(G)] The fewest vertices touching every edge. \\alpha is Mathlib's \\te…","labels":["SimpleGraph.coveringNumber"],"detail_key":"p14"},{"id":"n21826","layer":"informal","project":"p14","title":"indepNum\\_add\\_coveringNumber","kind":"theorem","summary":"[indepNum\\_add\\_coveringNumber] The largest independent set and the smallest covering are exact…","labels":["SimpleGraph.indepNum_add_coveringNumber"],"detail_key":"p14"},{"id":"n21827","layer":"informal","project":"p14","title":"The matching number \\alpha'(G)","kind":"definition","summary":"[The matching number \\alpha'(G)] An \\textttsSup over matching sizes --- the edge analogue of \\a…","labels":["SimpleGraph.matchingNumber"],"detail_key":"p14"},{"id":"n21828","layer":"informal","project":"p14","title":"An edge covering L: every vertex is an end of some edge of L","kind":"definition","summary":"[An edge covering L: every vertex is an end of some edge of L] A set of edges touching every ve…","labels":["SimpleGraph.IsEdgeCovering"],"detail_key":"p14"},{"id":"n21829","layer":"informal","project":"p14","title":"The edge covering number \\beta'(G)","kind":"definition","summary":"[The edge covering number \\beta'(G)] The fewest edges touching every vertex. ! \\textttsInf \\emp…","labels":["SimpleGraph.edgeCoveringNumber"],"detail_key":"p14"},{"id":"n21830","layer":"informal","project":"p14","title":"matchingNumber\\_add\\_edgeCoveringNumber","kind":"theorem","summary":"[matchingNumber\\_add\\_edgeCoveringNumber] The striking parallel with Corollary 7.1 --- even tho…","labels":["SimpleGraph.matchingNumber_add_edgeCoveringNumber"],"detail_key":"p14"},{"id":"n21831","layer":"informal","project":"p14","title":"indepNum\\_eq\\_edgeCoveringNumber","kind":"theorem","summary":"[indepNum\\_eq\\_edgeCoveringNumber] K\\\"onig's min-max equality transported across the two Gallai…","labels":["SimpleGraph.indepNum_eq_edgeCoveringNumber"],"detail_key":"p14"},{"id":"n21832","layer":"informal","project":"p14","title":"isBipartite\\_iff\\_forall\\_subgraph\\_two\\_mul\\_indepNum\\_le","kind":"theorem","summary":"[isBipartite\\_iff\\_forall\\_subgraph\\_two\\_mul\\_indepNum\\_le] (\\Rightarrow) Every subgraph of a…","labels":["SimpleGraph.isBipartite_iff_forall_subgraph_two_mul_indepNum_le"],"detail_key":"p14"},{"id":"n21833","layer":"informal","project":"p14","title":"isBipartite\\_iff\\_forall\\_subgraph\\_indepNum\\_eq\\_edgeCoveringNumber","kind":"theorem","summary":"[isBipartite\\_iff\\_forall\\_subgraph\\_indepNum\\_eq\\_edgeCoveringNumber] (\\Rightarrow) Theorem 7.…","labels":["SimpleGraph.isBipartite_iff_forall_subgraph_indepNum_eq_edgeCoveringNumber"],"detail_key":"p14"},{"id":"n21834","layer":"informal","project":"p14","title":"\\alpha-critical: deleting any edge raises the independence number","kind":"definition","summary":"[\\alpha-critical: deleting any edge raises the independence number] Every edge matters for \\alp…","labels":["SimpleGraph.IsAlphaCritical"],"detail_key":"p14"},{"id":"n21835","layer":"informal","project":"p14","title":"no\\_cutVertex\\_of\\_isAlphaCritical","kind":"theorem","summary":"[no\\_cutVertex\\_of\\_isAlphaCritical] Suppose v is a cut vertex, so G - v splits. Independent se…","labels":["SimpleGraph.no_cutVertex_of_isAlphaCritical"],"detail_key":"p14"},{"id":"n21836","layer":"informal","project":"p14","title":"\\beta-critical: deleting any edge lowers the covering number","kind":"definition","summary":"[\\beta-critical: deleting any edge lowers the covering number] Every edge is essential to the c…","labels":["SimpleGraph.IsBetaCritical"],"detail_key":"p14"},{"id":"n21837","layer":"informal","project":"p14","title":"no\\_cutVertex\\_of\\_isBetaCritical","kind":"theorem","summary":"[no\\_cutVertex\\_of\\_isBetaCritical] Corollary 7.1 gives \\alpha + \\beta = \\nu for every graph on…","labels":["SimpleGraph.no_cutVertex_of_isBetaCritical"],"detail_key":"p14"},{"id":"n21838","layer":"informal","project":"p14","title":"two\\_mul\\_coveringNumber\\_le","kind":"theorem","summary":"[two\\_mul\\_coveringNumber\\_le] Stated as 2\\beta \\le \\varepsilon + 1 to avoid division. A minimu…","labels":["SimpleGraph.two_mul_coveringNumber_le"],"detail_key":"p14"},{"id":"n21839","layer":"informal","project":"p14","title":"\\textttIsRamseyBound n k l: every graph on n vertices has a k-clique or an l-indep set.\\d…","kind":"definition","summary":"[\\textttIsRamseyBound n k l: every graph on n vertices has a k-clique or an l-indep set.\\dots]…","labels":["SimpleGraph.IsRamseyBound"],"detail_key":"p14"},{"id":"n21840","layer":"informal","project":"p14","title":"The Ramsey number r(k, l)","kind":"definition","summary":"[The Ramsey number r(k, l)] The exact threshold at which order becomes unavoidable. ! \\textttsI…","labels":["SimpleGraph.ramseyNumber"],"detail_key":"p14"},{"id":"n21841","layer":"informal","project":"p14","title":"isRamseyBound\\_mono","kind":"theorem","summary":"[isRamseyBound\\_mono] A bigger graph contains a smaller one: restrict a graph on m vertices to…","labels":["SimpleGraph.isRamseyBound_mono"],"detail_key":"p14"},{"id":"n21842","layer":"informal","project":"p14","title":"exists\\_isRamseyBound","kind":"theorem","summary":"[exists\\_isRamseyBound] No matter how a graph is drawn, if it is large enough it cannot avoid \\…","labels":["SimpleGraph.exists_isRamseyBound"],"detail_key":"p14"},{"id":"n21843","layer":"informal","project":"p14","title":"The consequence every later item consumes: r(k,l) really is a Ramsey bound","kind":"theorem","summary":"[The consequence every later item consumes: r(k,l) really is a Ramsey bound] \\textttramseyNumbe…","labels":["SimpleGraph.isRamseyBound_ramseyNumber"],"detail_key":"p14"},{"id":"n21844","layer":"informal","project":"p14","title":"ramseyNumber\\_one\\_left","kind":"theorem","summary":"[ramseyNumber\\_one\\_left] A 1-clique is a single vertex, which any nonempty graph has. So one v…","labels":["SimpleGraph.ramseyNumber_one_left"],"detail_key":"p14"},{"id":"n21845","layer":"informal","project":"p14","title":"ramseyNumber\\_one\\_right","kind":"theorem","summary":"[ramseyNumber\\_one\\_right] A 1-element independent set is a single vertex. The mirror image of…","labels":["SimpleGraph.ramseyNumber_one_right"],"detail_key":"p14"},{"id":"n21846","layer":"informal","project":"p14","title":"ramseyNumber\\_two\\_left","kind":"theorem","summary":"[ramseyNumber\\_two\\_left] A 2-clique is an edge. On l vertices either some edge is present ---…","labels":["SimpleGraph.ramseyNumber_two_left"],"detail_key":"p14"},{"id":"n21847","layer":"informal","project":"p14","title":"ramseyNumber\\_two\\_right","kind":"theorem","summary":"[ramseyNumber\\_two\\_right] A 2-element independent set is a non-adjacent pair. On k vertices ei…","labels":["SimpleGraph.ramseyNumber_two_right"],"detail_key":"p14"},{"id":"n21848","layer":"informal","project":"p14","title":"ramsey\\_recursion","kind":"theorem","summary":"[ramsey\\_recursion] The engine behind both the exact values in the book's table and the binomia…","labels":["SimpleGraph.ramsey_recursion"],"detail_key":"p14"},{"id":"n21849","layer":"informal","project":"p14","title":"ramsey\\_recursion\\_strict","kind":"theorem","summary":"[ramsey\\_recursion\\_strict] The parity refinement that pins down r(3,4) = 9: r(3,3) = 6 and r(2…","labels":["SimpleGraph.ramsey_recursion_strict"],"detail_key":"p14"},{"id":"n21850","layer":"informal","project":"p14","title":"cycleGraph\\_five\\_cliqueFree\\_three","kind":"theorem","summary":"[cycleGraph\\_five\\_cliqueFree\\_three] C_5 has girth 5, so no triangle: each vertex has two neig…","labels":["SimpleGraph.cycleGraph_five_cliqueFree_three"],"detail_key":"p14"},{"id":"n21851","layer":"informal","project":"p14","title":"cycleGraph\\_five\\_indepSetFree\\_three","kind":"theorem","summary":"[cycleGraph\\_five\\_indepSetFree\\_three] Any three of five cyclic positions include two consecut…","labels":["SimpleGraph.cycleGraph_five_indepSetFree_three"],"detail_key":"p14"},{"id":"n21852","layer":"informal","project":"p14","title":"six\\_le\\_ramseyNumber\\_three\\_three","kind":"theorem","summary":"[six\\_le\\_ramseyNumber\\_three\\_three] C_5 witnesses that five vertices do not force either stru…","labels":["SimpleGraph.six_le_ramseyNumber_three_three"],"detail_key":"p14"},{"id":"n21853","layer":"informal","project":"p14","title":"The (3,5)-Ramsey graph: \\textttZMod 13, adjacent iff the difference is a cubic residue","kind":"definition","summary":"[The (3,5)-Ramsey graph: \\textttZMod 13, adjacent iff the difference is a cubic residue] A circ…","labels":["SimpleGraph.cubicResidueGraph13"],"detail_key":"p14"},{"id":"n21854","layer":"informal","project":"p14","title":"cubicResidueGraph13\\_cliqueFree\\_three","kind":"theorem","summary":"[cubicResidueGraph13\\_cliqueFree\\_three] A triangle needs three residues pairwise differing by…","labels":["SimpleGraph.cubicResidueGraph13_cliqueFree_three"],"detail_key":"p14"},{"id":"n21855","layer":"informal","project":"p14","title":"cubicResidueGraph13\\_indepSetFree\\_five","kind":"theorem","summary":"[cubicResidueGraph13\\_indepSetFree\\_five] Five pairwise non-adjacent vertices would be five res…","labels":["SimpleGraph.cubicResidueGraph13_indepSetFree_five"],"detail_key":"p14"},{"id":"n21856","layer":"informal","project":"p14","title":"fourteen\\_le\\_ramseyNumber\\_three\\_five","kind":"theorem","summary":"[fourteen\\_le\\_ramseyNumber\\_three\\_five] Thirteen vertices do not suffice, so the threshold is…","labels":["SimpleGraph.fourteen_le_ramseyNumber_three_five"],"detail_key":"p14"},{"id":"n21857","layer":"informal","project":"p14","title":"The (4,4)-Ramsey graph = the Paley graph of order 17","kind":"definition","summary":"[The (4,4)-Ramsey graph = the Paley graph of order 17] The Paley graph of order 17 --- the cano…","labels":["SimpleGraph.paleyGraph17"],"detail_key":"p14"},{"id":"n21858","layer":"informal","project":"p14","title":"paleyGraph17\\_cliqueFree\\_four","kind":"theorem","summary":"[paleyGraph17\\_cliqueFree\\_four] Four vertices pairwise differing by quadratic residues would b…","labels":["SimpleGraph.paleyGraph17_cliqueFree_four"],"detail_key":"p14"},{"id":"n21859","layer":"informal","project":"p14","title":"paleyGraph17\\_indepSetFree\\_four","kind":"theorem","summary":"[paleyGraph17\\_indepSetFree\\_four] + The slick route: the Paley graph of order 17 is self-compl…","labels":["SimpleGraph.paleyGraph17_indepSetFree_four"],"detail_key":"p14"},{"id":"n21860","layer":"informal","project":"p14","title":"eighteen\\_le\\_ramseyNumber\\_four\\_four","kind":"theorem","summary":"[eighteen\\_le\\_ramseyNumber\\_four\\_four] Seventeen vertices do not suffice, so the threshold is…","labels":["SimpleGraph.eighteen_le_ramseyNumber_four_four"],"detail_key":"p14"},{"id":"n21861","layer":"informal","project":"p14","title":"ramseyNumber\\_le\\_choose","kind":"theorem","summary":"[ramseyNumber\\_le\\_choose] r(k, l) \\le C(k + l - 2, k - 1). Theorem 7.4's recursion is exactly…","labels":["SimpleGraph.ramseyNumber_le_choose"],"detail_key":"p14"},{"id":"n21862","layer":"informal","project":"p14","title":"card\\_simpleGraph\\_fin","kind":"theorem","summary":"[card\\_simpleGraph\\_fin] A simple graph on a labelled vertex set is determined by which of the…","labels":["SimpleGraph.card_simpleGraph_fin"],"detail_key":"p14"},{"id":"n21863","layer":"informal","project":"p14","title":"ramsey\\_self\\_lower\\_bound","kind":"theorem","summary":"[ramsey\\_self\\_lower\\_bound] The probabilistic method, which the book introduces as (\\S7.2, p.…","labels":["SimpleGraph.ramsey_self_lower_bound"],"detail_key":"p14"},{"id":"n21864","layer":"informal","project":"p14","title":"ramseyNumber\\_ge\\_of\\_min","kind":"theorem","summary":"[ramseyNumber\\_ge\\_of\\_min] Ramsey numbers are monotone in both arguments, so r(k,l) \\ge r(m,m)…","labels":["SimpleGraph.ramseyNumber_ge_of_min"],"detail_key":"p14"},{"id":"n21865","layer":"informal","project":"p14","title":"An m-edge-colouring of K_n: a total function on unordered pairs","kind":"definition","summary":"[An m-edge-colouring of K_n: a total function on unordered pairs] A total function \\textttSym2…","labels":["SimpleGraph.EdgeColouring"],"detail_key":"p14"},{"id":"n21866","layer":"informal","project":"p14","title":"\\textttIsRamseyBoundMulti n c: every m-edge-colouring of K_n has, for some i, a set of c…","kind":"definition","summary":"[\\textttIsRamseyBoundMulti n c: every m-edge-colouring of K_n has, for some i, a set of c i\\dot…","labels":["SimpleGraph.IsRamseyBoundMulti"],"detail_key":"p14"},{"id":"n21867","layer":"informal","project":"p14","title":"The multicolour Ramsey number r(k_1,\\dots,k_m)","kind":"definition","summary":"[The multicolour Ramsey number r(k_1,\\dots,k_m)] ! Same \\textttsInf \\emptyset = 0 trap as \\text…","labels":["SimpleGraph.ramseyNumberMulti"],"detail_key":"p14"},{"id":"n21868","layer":"informal","project":"p14","title":"ramseyMulti\\_recursion","kind":"theorem","summary":"[ramseyMulti\\_recursion] Generalises Theorem 7.4's recursion. Fix v in a large K_n and classify…","labels":["SimpleGraph.ramseyMulti_recursion"],"detail_key":"p14"},{"id":"n21869","layer":"informal","project":"p14","title":"ramseyMulti\\_le\\_multinomial","kind":"theorem","summary":"[ramseyMulti\\_le\\_multinomial] Iterating Theorem 7.7 gives exactly the multinomial recursion, a…","labels":["SimpleGraph.ramseyMulti_le_multinomial"],"detail_key":"p14"},{"id":"n21870","layer":"informal","project":"p14","title":"ramseyNumber\\_comm","kind":"theorem","summary":"[ramseyNumber\\_comm] Complementation swaps cliques and independent sets, so a graph with no k-c…","labels":["SimpleGraph.ramseyNumber_comm"],"detail_key":"p14"},{"id":"n21871","layer":"informal","project":"p14","title":"r_n = r(3, \\dots, 3) with n colours","kind":"definition","summary":"[r_n = r(3, \\dots, 3) with n colours] The least number of points such that any n-colouring of t…","labels":["SimpleGraph.ramseyTriangle"],"detail_key":"p14"},{"id":"n21872","layer":"informal","project":"p14","title":"ramseyTriangle\\_recursion","kind":"theorem","summary":"[ramseyTriangle\\_recursion] Theorem 7.7 with all targets 3. Fix v in a K_n on n(r_n-1-1) + 2 ve…","labels":["SimpleGraph.ramseyTriangle_recursion"],"detail_key":"p14"},{"id":"n21873","layer":"informal","project":"p14","title":"ramseyTriangle\\_le\\_factorial\\_exp","kind":"theorem","summary":"[ramseyTriangle\\_le\\_factorial\\_exp] Unwinding r_n \\le n(r_n-1 - 1) + 2 from r_2 = 6 and dividi…","labels":["SimpleGraph.ramseyTriangle_le_factorial_exp"],"detail_key":"p14"},{"id":"n21874","layer":"informal","project":"p14","title":"ramseyTriangle\\_three\\_le","kind":"theorem","summary":"[ramseyTriangle\\_three\\_le] Part (a) at n = 3 with r_2 = 6: r_3 \\le 3(6-1) + 2 = 17. (Part (b)…","labels":["SimpleGraph.ramseyTriangle_three_le"],"detail_key":"p14"},{"id":"n21875","layer":"informal","project":"p14","title":"The composition (lexicographic product) G(H)","kind":"definition","summary":"[The composition (lexicographic product) G(H)] Replace each vertex of G by a copy of H. Vertice…","labels":["SimpleGraph.composition"],"detail_key":"p14"},{"id":"n21876","layer":"informal","project":"p14","title":"indepNum\\_composition\\_le","kind":"theorem","summary":"[indepNum\\_composition\\_le] Let S be independent in G[H]. Its first-coordinate projection is in…","labels":["SimpleGraph.indepNum_composition_le"],"detail_key":"p14"},{"id":"n21877","layer":"informal","project":"p14","title":"ramsey\\_product\\_lower","kind":"theorem","summary":"[ramsey\\_product\\_lower] Take (k+1,k+1)- and (l+1,l+1)-Ramsey graphs G, H --- so \\omega, \\alpha…","labels":["SimpleGraph.ramsey_product_lower"],"detail_key":"p14"},{"id":"n21878","layer":"informal","project":"p14","title":"abbott\\_lower\\_bound","kind":"theorem","summary":"[abbott\\_lower\\_bound] Iterate part (b) from r(3,3) = 6, so r(3,3) - 1 = 5. Taking k = l = 2^n-…","labels":["SimpleGraph.abbott_lower_bound"],"detail_key":"p14"},{"id":"n21879","layer":"informal","project":"p14","title":"C_3 \\lor C_5 on the carrier \\textttFin 3 \\oplus Fin 5","kind":"definition","summary":"[C_3 \\lor C_5 on the carrier \\textttFin 3 \\oplus Fin 5] C_3 \\lor C_5 on \\textttFin 3 \\oplus Fin…","labels":["SimpleGraph.grahamGraph"],"detail_key":"p14"},{"id":"n21880","layer":"informal","project":"p14","title":"The generalised Ramsey number r(G_1,\\dots,G_m) over Mathlib's containment \\sqsubseteq","kind":"definition","summary":"[The generalised Ramsey number r(G_1,\\dots,G_m) over Mathlib's containment \\sqsubseteq] Instead…","labels":["SimpleGraph.generalisedRamseyNumber"],"detail_key":"p14"},{"id":"n21881","layer":"informal","project":"p14","title":"tree\\_isContained\\_of\\_minDegree\\_le","kind":"theorem","summary":"[tree\\_isContained\\_of\\_minDegree\\_le] \\emph (absent from Mathlib). If \\delta(G) \\ge m - 1 then…","labels":["SimpleGraph.tree_isContained_of_minDegree_le"],"detail_key":"p14"},{"id":"n21882","layer":"informal","project":"p14","title":"generalisedRamsey\\_tree\\_star","kind":"theorem","summary":"[generalisedRamsey\\_tree\\_star] K_1,n is the star with n leaves, so a blue copy is a vertex wit…","labels":["SimpleGraph.generalisedRamsey_tree_star"],"detail_key":"p14"},{"id":"n21883","layer":"informal","project":"p14","title":"chvatal\\_tree\\_complete","kind":"theorem","summary":"[chvatal\\_tree\\_complete] One of the cleanest results in generalised Ramsey theory: the answer…","labels":["SimpleGraph.chvatal_tree_complete"],"detail_key":"p14"},{"id":"n21884","layer":"informal","project":"p14","title":"degreeMajorised\\_of\\_cliqueFree","kind":"theorem","summary":"[degreeMajorised\\_of\\_cliqueFree] The book remarks (\\S7.3, p. 118) on the parallel with Theorem…","labels":["SimpleGraph.degreeMajorised_of_cliqueFree"],"detail_key":"p14"},{"id":"n21885","layer":"informal","project":"p14","title":"degreeMajorised\\_of\\_forall\\_degree\\_le","kind":"theorem","summary":"[degreeMajorised\\_of\\_forall\\_degree\\_le] (absent from Mathlib). *If d_G(v) \\le d_H(v) for ever…","labels":["SimpleGraph.degreeMajorised_of_forall_degree_le"],"detail_key":"p14"},{"id":"n21886","layer":"informal","project":"p14","title":"turan\\_iso\\_of\\_card\\_edgeFinset\\_eq","kind":"theorem","summary":"[turan\\_iso\\_of\\_card\\_edgeFinset\\_eq] The balanced complete multipartite graph is not merely \\…","labels":["SimpleGraph.turan_iso_of_card_edgeFinset_eq"],"detail_key":"p14"},{"id":"n21887","layer":"informal","project":"p14","title":"triangle\\_of\\_card\\_edgeFinset\\_gt","kind":"theorem","summary":"[triangle\\_of\\_card\\_edgeFinset\\_gt] Tur\\'an at m = 2 --- Mantel's 1907 case, half a century be…","labels":["SimpleGraph.triangle_of_card_edgeFinset_gt"],"detail_key":"p14"},{"id":"n21888","layer":"informal","project":"p14","title":"triangle\\_of\\_not\\_bipartite\\_of\\_card\\_edgeFinset\\_gt","kind":"theorem","summary":"[triangle\\_of\\_not\\_bipartite\\_of\\_card\\_edgeFinset\\_gt] Forbidding bipartiteness on top of tri…","labels":["SimpleGraph.triangle_of_not_bipartite_of_card_edgeFinset_gt"],"detail_key":"p14"},{"id":"n21889","layer":"informal","project":"p14","title":"isContained\\_completeBipartite\\_two\\_of\\_sum\\_choose\\_gt","kind":"theorem","summary":"[isContained\\_completeBipartite\\_two\\_of\\_sum\\_choose\\_gt] Double-count cherries (paths of leng…","labels":["SimpleGraph.isContained_completeBipartite_two_of_sum_choose_gt"],"detail_key":"p14"},{"id":"n21890","layer":"informal","project":"p14","title":"isContained\\_completeBipartite\\_two\\_of\\_card\\_edgeFinset\\_gt","kind":"theorem","summary":"[isContained\\_completeBipartite\\_two\\_of\\_card\\_edgeFinset\\_gt] Convert part (a)'s degree condi…","labels":["SimpleGraph.isContained_completeBipartite_two_of_card_edgeFinset_gt"],"detail_key":"p14"},{"id":"n21891","layer":"informal","project":"p14","title":"The unit-distance graph on a finite point set in the plane","kind":"definition","summary":"[The unit-distance graph on a finite point set in the plane] Join i, j when \\textttdist (x i) (…","labels":["SimpleGraph.unitDistanceGraph"],"detail_key":"p14"},{"id":"n21892","layer":"informal","project":"p14","title":"card\\_unit\\_distance\\_pairs\\_le","kind":"theorem","summary":"[card\\_unit\\_distance\\_pairs\\_le] Two distinct points have \\le 2 common unit-distance neighbour…","labels":["SimpleGraph.card_unit_distance_pairs_le"],"detail_key":"p14"},{"id":"n21893","layer":"informal","project":"p14","title":"isContained\\_completeBipartite\\_of\\_card\\_edgeFinset\\_gt","kind":"theorem","summary":"[isContained\\_completeBipartite\\_of\\_card\\_edgeFinset\\_gt] Generalises 7.3.4 from K_2,m to K_m,…","labels":["SimpleGraph.isContained_completeBipartite_of_card_edgeFinset_gt"],"detail_key":"p14"},{"id":"n21894","layer":"informal","project":"p14","title":"schur","kind":"theorem","summary":"[schur] The classical bridge from Ramsey theory to additive number theory. Schur's original mot…","labels":["SimpleGraph.schur"],"detail_key":"p14"},{"id":"n21895","layer":"informal","project":"p14","title":"The Schur number s_n","kind":"definition","summary":"[The Schur number s_n] The exact threshold at which sum-free partitions become impossible. + Un…","labels":["SimpleGraph.schurNumber"],"detail_key":"p14"},{"id":"n21896","layer":"informal","project":"p14","title":"schurNumber\\_three","kind":"theorem","summary":"[schurNumber\\_three] Lower bound from the book's opening partition (1,4,10,13, 2,3,11,12, 5,6,7…","labels":["SimpleGraph.schurNumber_three"],"detail_key":"p14"},{"id":"n21897","layer":"informal","project":"p14","title":"schurNumber\\_recursion","kind":"theorem","summary":"[schurNumber\\_recursion] From a sum-free (n-1)-partition of 1,\\dots,s_n-1-1, build an n-partiti…","labels":["SimpleGraph.schurNumber_recursion"],"detail_key":"p14"},{"id":"n21898","layer":"informal","project":"p14","title":"schurNumber\\_lower\\_bound","kind":"theorem","summary":"[schurNumber\\_lower\\_bound] Iterate s_n \\ge 3s_n-1 - 1 from s_3 = 14; solving the linear recurr…","labels":["SimpleGraph.schurNumber_lower_bound"],"detail_key":"p14"},{"id":"n21899","layer":"informal","project":"p14","title":"The \"far pairs\" graph: adjacent iff distance exceeds 1/\\sqrt 2","kind":"definition","summary":"[The \"far pairs\" graph: adjacent iff distance exceeds 1/\\sqrt 2] Join two points when they are…","labels":["SimpleGraph.farGraph"],"detail_key":"p14"},{"id":"n21900","layer":"informal","project":"p14","title":"card\\_farGraph\\_le","kind":"theorem","summary":"[card\\_farGraph\\_le] Two genuinely geometric inputs, neither in Mathlib: 1. Any four points in…","labels":["SimpleGraph.card_farGraph_le"],"detail_key":"p14"},{"id":"n21901","layer":"informal","project":"p14","title":"exists\\_farGraph\\_card\\_eq","kind":"theorem","summary":"[exists\\_farGraph\\_card\\_eq] The far-pairs graph of this configuration is \\emphexactly the bala…","labels":["SimpleGraph.exists_farGraph_card_eq"],"detail_key":"p14"},{"id":"n21902","layer":"informal","project":"p14","title":"card\\_unitDistanceGraph\\_le\\_of\\_diam\\_one","kind":"theorem","summary":"[card\\_unitDistanceGraph\\_le\\_of\\_diam\\_one] In a diameter-1 set, a pair at distance exactly 1…","labels":["SimpleGraph.card_unitDistanceGraph_le_of_diam_one"],"detail_key":"p14"},{"id":"n21903","layer":"informal","project":"p14","title":"The radio-range graph: adjacent iff within \\textttrange","kind":"definition","summary":"[The radio-range graph: adjacent iff within \\textttrange] Join two cars when within \\textttrang…","labels":["SimpleGraph.radioGraph"],"detail_key":"p14"},{"id":"n21904","layer":"informal","project":"p14","title":"A maximum matching: a matching no larger than which exists","kind":"definition","summary":"[A maximum matching: a matching no larger than which exists] Mathlib supplies \"matching\" (\\text…","labels":["SimpleGraph.Subgraph.IsMaximumMatching"],"detail_key":"p14"},{"id":"n21905","layer":"informal","project":"p14","title":"M-alternating walk: consecutive edges alternate in/out of M","kind":"definition","summary":"[M-alternating walk: consecutive edges alternate in/out of M] Walk along the graph using a matc…","labels":["SimpleGraph.Walk.IsAlternatingWalk"],"detail_key":"p14"},{"id":"n21906","layer":"informal","project":"p14","title":"M-augmenting path: alternating, both ends M-unsaturated","kind":"definition","summary":"[M-augmenting path: alternating, both ends M-unsaturated] An alternating path both of whose end…","labels":["SimpleGraph.Walk.IsAugmenting"],"detail_key":"p14"},{"id":"n21907","layer":"informal","project":"p14","title":"A minimum covering","kind":"definition","summary":"[A minimum covering] Watch every edge using as few vertices as possible. As with \\textttSubgrap…","labels":["SimpleGraph.IsMinimumCovering"],"detail_key":"p14"},{"id":"n21908","layer":"informal","project":"p14","title":"A k-factor: a k-regular spanning subgraph","kind":"definition","summary":"[A k-factor: a k-regular spanning subgraph] Keep every vertex, and select edges so each vertex…","labels":["SimpleGraph.Subgraph.IsKFactor"],"detail_key":"p14"},{"id":"n21909","layer":"informal","project":"p14","title":"G is k-factorable: it decomposes into edge-disjoint k-factors","kind":"definition","summary":"[G is k-factorable: it decomposes into edge-disjoint k-factors] The edges of G partition into g…","labels":["SimpleGraph.IsKFactorable"],"detail_key":"p14"},{"id":"n21910","layer":"informal","project":"p14","title":"The k-cube Q_k","kind":"definition","summary":"[The k-cube Q_k] The corners and edges of a k-dimensional cube: k-regular, bipartite (split by…","labels":["SimpleGraph.cube"],"detail_key":"p14"},{"id":"n21911","layer":"informal","project":"p14","title":"The m \\times n grid graph","kind":"definition","summary":"[The m \\times n grid graph] Vertices are the squares of an m \\times n board, adjacent when they…","labels":["SimpleGraph.gridGraph"],"detail_key":"p14"},{"id":"n21912","layer":"informal","project":"p14","title":"Thm 5.1 (Berge): a matching is maximum iff there is no augmenting path","kind":"theorem","summary":"[Thm 5.1 (Berge): a matching is maximum iff there is no augmenting path] Both directions are st…","labels":["SimpleGraph.berge_maximum_matching"],"detail_key":"p14"},{"id":"n21913","layer":"informal","project":"p14","title":"Thm 5.2 (Hall): a bipartite G has a matching saturating X iff |N(S)| \\ge |S|","kind":"theorem","summary":"[Thm 5.2 (Hall): a bipartite G has a matching saturating X iff |N(S)| \\ge |S|] Hall's condition…","labels":["SimpleGraph.hall_bipartite_matching"],"detail_key":"p14"},{"id":"n21914","layer":"informal","project":"p14","title":"Cor 5.2: a k-regular bipartite graph with k > 0 has a perfect matching","kind":"theorem","summary":"[Cor 5.2: a k-regular bipartite graph with k > 0 has a perfect matching] Two applications of do…","labels":["SimpleGraph.marriage_theorem"],"detail_key":"p14"},{"id":"n21915","layer":"informal","project":"p14","title":"(5.5): for any matching M and covering K, |M| \\le |K|","kind":"theorem","summary":"[(5.5): for any matching M and covering K, |M| \\le |K|] The edges of a matching are pairwise di…","labels":["SimpleGraph.matching_card_le_covering_card"],"detail_key":"p14"},{"id":"n21916","layer":"informal","project":"p14","title":"Lem 5.3: if |M| = |K| then M is a maximum matching and K a minimum covering","kind":"theorem","summary":"[Lem 5.3: if |M| = |K| then M is a maximum matching and K a minimum covering] The standard \"wea…","labels":["SimpleGraph.isMaximumMatching_and_isMinimumCovering_of_card_eq"],"detail_key":"p14"},{"id":"n21917","layer":"informal","project":"p14","title":"Thm 5.3 (K\\\"onig): in a bipartite graph, max-matching size = min-covering size","kind":"theorem","summary":"[Thm 5.3 (K\\\"onig): in a bipartite graph, max-matching size = min-covering size] The prototypic…","labels":["SimpleGraph.konig_matching_covering"],"detail_key":"p14"},{"id":"n21918","layer":"informal","project":"p14","title":"Thm 5.4 (Tutte): G has a perfect matching iff o(G - S) \\le |S| for all S","kind":"theorem","summary":"[Thm 5.4 (Tutte): G has a perfect matching iff o(G - S) \\le |S| for all S] The obstruction to a…","labels":["SimpleGraph.tutte_perfect_matching"],"detail_key":"p14"},{"id":"n21919","layer":"informal","project":"p14","title":"Cor 5.4 (Petersen, 1891): every 3-regular graph without cut edges has a perfect matching","kind":"theorem","summary":"[Cor 5.4 (Petersen, 1891): every 3-regular graph without cut edges has a perfect matching] A do…","labels":["SimpleGraph.petersen_three_regular_bridgeless"],"detail_key":"p14"},{"id":"n21920","layer":"informal","project":"p14","title":"Feasible vertex labelling: l x + l y \\ge w s(x,y)","kind":"definition","summary":"[Feasible vertex labelling: l x + l y \\ge w s(x,y)] Assign a label to every vertex so the two l…","labels":["SimpleGraph.IsFeasibleLabelling"],"detail_key":"p14"},{"id":"n21921","layer":"informal","project":"p14","title":"The equality subgraph G_l","kind":"definition","summary":"[The equality subgraph G_l] Keep only the edges that are \"tight\" for the labelling --- those wh…","labels":["SimpleGraph.equalitySubgraph"],"detail_key":"p14"},{"id":"n21922","layer":"informal","project":"p14","title":"Weight of a matching","kind":"definition","summary":"[Weight of a matching] \\texttt\\sum^f e \\in M.edgeSet, w e, using the \\emphfinsum \\sum^f rather…","labels":["SimpleGraph.matchingWeight"],"detail_key":"p14"},{"id":"n21923","layer":"informal","project":"p14","title":"An optimal matching: a max-weight perfect matching","kind":"definition","summary":"[An optimal matching: a max-weight perfect matching] Among all assignments of each worker to a…","labels":["SimpleGraph.IsOptimalMatching"],"detail_key":"p14"},{"id":"n21924","layer":"informal","project":"p14","title":"Thm 5.5: a perfect matching in the equality subgraph is an optimal matching of G","kind":"theorem","summary":"[Thm 5.5: a perfect matching in the equality subgraph is an optimal matching of G] Weak duality…","labels":["SimpleGraph.optimal_of_perfectMatching_equalitySubgraph"],"detail_key":"p14"},{"id":"n21925","layer":"informal","project":"p14","title":"Ex 5.1.1(a): every k-cube (k \\ge 2) has a perfect matching","kind":"theorem","summary":"[Ex 5.1.1(a): every k-cube (k \\ge 2) has a perfect matching] The k-cube is k-regular and bipart…","labels":["SimpleGraph.cube_isPerfectMatching"],"detail_key":"p14"},{"id":"n21926","layer":"informal","project":"p14","title":"Ex 5.1.1(b): number of perfect matchings of K_2n is (2n)! / (2^n n!)","kind":"theorem","summary":"[Ex 5.1.1(b): number of perfect matchings of K_2n is (2n)! / (2^n n!)] A perfect matching of K_…","labels":["SimpleGraph.card_perfectMatching_completeGraph"],"detail_key":"p14"},{"id":"n21927","layer":"informal","project":"p14","title":"Ex 5.1.1(b): number of perfect matchings of K_n,n is n!","kind":"theorem","summary":"[Ex 5.1.1(b): number of perfect matchings of K_n,n is n!] Every vertex of X may be matched to a…","labels":["SimpleGraph.card_perfectMatching_completeBipartite"],"detail_key":"p14"},{"id":"n21928","layer":"informal","project":"p14","title":"Ex 5.1.2: a tree has at most one perfect matching","kind":"theorem","summary":"[Ex 5.1.2: a tree has at most one perfect matching] Suppose a tree had two perfect matchings M…","labels":["SimpleGraph.isTree_subsingleton_perfectMatching"],"detail_key":"p14"},{"id":"n21929","layer":"informal","project":"p14","title":"Ex 5.1.5(a)(i): K_n,n is 1-factorable","kind":"theorem","summary":"[Ex 5.1.5(a)(i): K_n,n is 1-factorable] Label both sides by \\textttZMod n; for each i, let H_i…","labels":["SimpleGraph.completeBipartite_one_factorable"],"detail_key":"p14"},{"id":"n21930","layer":"informal","project":"p14","title":"Ex 5.1.5(a)(i): K_2n is 1-factorable","kind":"theorem","summary":"[Ex 5.1.5(a)(i): K_2n is 1-factorable] The classical round-robin schedule. Fix one vertex at th…","labels":["SimpleGraph.completeGraph_even_one_factorable"],"detail_key":"p14"},{"id":"n21931","layer":"informal","project":"p14","title":"Ex 5.1.5(c): \\nu even and \\delta \\ge \\nu/2 + 1 \\Rightarrow G has a 3-factor (via Dirac;","kind":"theorem","summary":"[Ex 5.1.5(c): \\nu even and \\delta \\ge \\nu/2 + 1 \\Rightarrow G has a 3-factor (via Dirac;] \\delt…","labels":["SimpleGraph.exists_three_factor_of_minDegree"],"detail_key":"p14"},{"id":"n21932","layer":"informal","project":"p14","title":"Ex 5.1.6*: K_2n+1 decomposes into n connected 2-factors (Walecki)","kind":"theorem","summary":"[Ex 5.1.6*: K_2n+1 decomposes into n connected 2-factors (Walecki)] A connected 2-factor is pre…","labels":["SimpleGraph.completeGraph_odd_two_factorable"],"detail_key":"p14"},{"id":"n21933","layer":"informal","project":"p14","title":"Ex 5.2.2(a): bipartite perfect-matching criterion","kind":"theorem","summary":"[Ex 5.2.2(a): bipartite perfect-matching criterion] Hall's Theorem 5.2 gives a matching saturat…","labels":["SimpleGraph.bipartite_perfectMatching_iff"],"detail_key":"p14"},{"id":"n21934","layer":"informal","project":"p14","title":"Ex 5.2.3(a): every k-regular bipartite graph (k > 0) is 1-factorable","kind":"theorem","summary":"[Ex 5.2.3(a): every k-regular bipartite graph (k > 0) is 1-factorable] Corollary 5.2 gives a pe…","labels":["SimpleGraph.regular_bipartite_one_factorable"],"detail_key":"p14"},{"id":"n21935","layer":"informal","project":"p14","title":"Ex 5.2.3(b)*: every 2k-regular graph is 2-factorable (Petersen;","kind":"theorem","summary":"[Ex 5.2.3(b)*: every 2k-regular graph is 2-factorable (Petersen;] Every degree is even, so Theo…","labels":["SimpleGraph.regular_two_factorable"],"detail_key":"p14"},{"id":"n21936","layer":"informal","project":"p14","title":"Ex 5.2.5: K\\\"onig's theorem, matrix (line-cover) form","kind":"theorem","summary":"[Ex 5.2.5: K\\\"onig's theorem, matrix (line-cover) form] K\\\"onig's Theorem 5.3 in matrix dress.…","labels":["SimpleGraph.konig_matrix"],"detail_key":"p14"},{"id":"n21937","layer":"informal","project":"p14","title":"Ex 5.2.6(a): the K\\\"onig--Ore defect formula (stated additively to avoid N subtraction)","kind":"theorem","summary":"[Ex 5.2.6(a): the K\\\"onig--Ore defect formula (stated additively to avoid N subtraction)] Hall'…","labels":["SimpleGraph.konig_ore_defect"],"detail_key":"p14"},{"id":"n21938","layer":"informal","project":"p14","title":"Ex 5.2.6(b): simple, |X| = |Y| = n, \\varepsilon > (k- 1)n \\Rightarrow a matching of size k","kind":"theorem","summary":"[Ex 5.2.6(b): simple, |X| = |Y| = n, \\varepsilon > (k- 1)n \\Rightarrow a matching of size k] By…","labels":["SimpleGraph.exists_matching_card_eq_of_card_edges_gt"],"detail_key":"p14"},{"id":"n21939","layer":"informal","project":"p14","title":"Ex 5.2.8(a): a (rectangular) doubly stochastic matrix is necessarily square","kind":"theorem","summary":"[Ex 5.2.8(a): a (rectangular) doubly stochastic matrix is necessarily square] Sum every entry t…","labels":["SimpleGraph.doublyStochastic_card_eq"],"detail_key":"p14"},{"id":"n21940","layer":"informal","project":"p14","title":"Ex 5.2.9: a common transversal of the left and right cosets of a subgroup (P. Hall)","kind":"theorem","summary":"[Ex 5.2.9: a common transversal of the left and right cosets of a subgroup (P. Hall)] Left and…","labels":["SimpleGraph.exists_common_coset_transversal"],"detail_key":"p14"},{"id":"n21941","layer":"informal","project":"p14","title":"Ex 5.3.2: a (k- 1)-edge-connected k-regular graph with \\nu even has a perfect matching","kind":"theorem","summary":"[Ex 5.3.2: a (k- 1)-edge-connected k-regular graph with \\nu even has a perfect matching] Peters…","labels":["SimpleGraph.perfectMatching_of_edgeConnected_regular"],"detail_key":"p14"},{"id":"n21942","layer":"informal","project":"p14","title":"Ex 5.3.3: a tree has a perfect matching iff o(G - v) = 1 for every vertex v","kind":"theorem","summary":"[Ex 5.3.3: a tree has a perfect matching iff o(G - v) = 1 for every vertex v] Tutte's condition…","labels":["SimpleGraph.isTree_perfectMatching_iff"],"detail_key":"p14"},{"id":"n21943","layer":"informal","project":"p14","title":"Ex 5.3.4*: the Berge--Tutte defect formula (stated additively to avoid N subtraction)","kind":"theorem","summary":"[Ex 5.3.4*: the Berge--Tutte defect formula (stated additively to avoid N subtraction)] Tutte s…","labels":["SimpleGraph.berge_tutte_defect"],"detail_key":"p14"},{"id":"n21944","layer":"informal","project":"p14","title":"Ex 5.3.5(b): \\nu even, \\delta < \\nu/2, and a large edge count \\Rightarrow a perfect match…","kind":"theorem","summary":"[Ex 5.3.5(b): \\nu even, \\delta < \\nu/2, and a large edge count \\Rightarrow a perfect matching]…","labels":["SimpleGraph.perfectMatching_of_card_edges_gt"],"detail_key":"p14"},{"id":"n21945","layer":"informal","project":"p14","title":"A network N = (D, X, Y, c)","kind":"definition","summary":"[A network N = (D, X, Y, c)] A transportation network: goods are produced at the sources, consu…","labels":["Networks.Network"],"detail_key":"p14"},{"id":"n21946","layer":"informal","project":"p14","title":"Intermediate vertices I = V \\ (X \\cup Y)","kind":"definition","summary":"[Intermediate vertices I = V \\ (X \\cup Y)] The transit points of the network, which neither pro…","labels":["Networks.Network.I"],"detail_key":"p14"},{"id":"n21947","layer":"informal","project":"p14","title":"f^+(S) = f(S, \\barS) = \\sum_u\\in S \\sum_v\\notin S f u v","kind":"definition","summary":"[f^+(S) = f(S, \\barS) = \\sum_u\\in S \\sum_v\\notin S f u v] The total flow leaving S --- sum over…","labels":["Networks.Network.fOut"],"detail_key":"p14"},{"id":"n21948","layer":"informal","project":"p14","title":"f^-(S) = f(\\barS, S) = \\sum_u\\notin S \\sum_v\\in S f u v","kind":"definition","summary":"[f^-(S) = f(\\barS, S) = \\sum_u\\notin S \\sum_v\\in S f u v] The total flow entering S. Conservati…","labels":["Networks.Network.fIn"],"detail_key":"p14"},{"id":"n21949","layer":"informal","project":"p14","title":"A flow: (11.1) capacity constraint + (11.2) conservation at intermediate vertices","kind":"definition","summary":"[A flow: (11.1) capacity constraint + (11.2) conservation at intermediate vertices] Nothing ove…","labels":["Networks.Network.IsFlow"],"detail_key":"p14"},{"id":"n21950","layer":"informal","project":"p14","title":"\\textttval f = f^+(X) - f^-(X)","kind":"definition","summary":"[\\textttval f = f^+(X) - f^-(X)] The net rate at which the commodity travels from producers to…","labels":["Networks.Network.val"],"detail_key":"p14"},{"id":"n21951","layer":"informal","project":"p14","title":"A cut (S, \\barS) with x \\in S, y \\notin S","kind":"definition","summary":"[A cut (S, \\barS) with x \\in S, y \\notin S] A way of severing the network so the source is on o…","labels":["Networks.Network.IsCut"],"detail_key":"p14"},{"id":"n21952","layer":"informal","project":"p14","title":"\\textttcap (S, \\barS) = \\sum\\_\\u\\inS\\ \\sum\\_\\v\\notinS\\ c u v","kind":"definition","summary":"[\\textttcap (S, \\barS) = \\sum\\_\\u\\inS\\ \\sum\\_\\v\\notinS\\ c u v] The total throughput of the seve…","labels":["Networks.Network.capOf"],"detail_key":"p14"},{"id":"n21953","layer":"informal","project":"p14","title":"A maximum flow","kind":"definition","summary":"[A maximum flow] Ship as much as the network allows. Theorem 11.2 characterises these as exactl…","labels":["Networks.Network.IsMaxFlow"],"detail_key":"p14"},{"id":"n21954","layer":"informal","project":"p14","title":"A minimum cut","kind":"definition","summary":"[A minimum cut] The cheapest way to sever the network --- the bottleneck. Theorem 11.3 says its…","labels":["Networks.Network.IsMinCut"],"detail_key":"p14"},{"id":"n21955","layer":"informal","project":"p14","title":"An f-incrementing path from the source","kind":"definition","summary":"[An f-incrementing path from the source] A route from source to sink along which the flow can s…","labels":["Networks.Network.IncPath"],"detail_key":"p14"},{"id":"n21956","layer":"informal","project":"p14","title":"The residual capacity \\iota(P) > 0 of an incrementing path --- a recursion over \\textttIn…","kind":"definition","summary":"[The residual capacity \\iota(P) > 0 of an incrementing path --- a recursion over \\textttIncPath…","labels":["Networks.Network.IncPath.iota"],"detail_key":"p14"},{"id":"n21957","layer":"informal","project":"p14","title":"The revised flow \\hatf (11.9) obtained by pushing \\iota(P) along P","kind":"definition","summary":"[The revised flow \\hatf (11.9) obtained by pushing \\iota(P) along P] Push \\iota(P) more along e…","labels":["Networks.Network.revisedFlow"],"detail_key":"p14"},{"id":"n21958","layer":"informal","project":"p14","title":"Local minimal \\textttedgeConnectivity \\kappa'(G) (repo has the real one)","kind":"definition","summary":"[Local minimal \\textttedgeConnectivity \\kappa'(G) (repo has the real one)] Corollary 11.5 relat…","labels":["Networks.edgeConnectivity"],"detail_key":"p14"},{"id":"n21959","layer":"informal","project":"p14","title":"Local minimal \\textttvertexConnectivity \\kappa(G) (repo has the real one)","kind":"definition","summary":"[Local minimal \\textttvertexConnectivity \\kappa(G) (repo has the real one)] Corollary 11.7 rela…","labels":["Networks.vertexConnectivity"],"detail_key":"p14"},{"id":"n21960","layer":"informal","project":"p14","title":"Local minimal \\textttInternallyDisjoint (repo has the real one, \\textttTwoConnected.lean:…","kind":"definition","summary":"[Local minimal \\textttInternallyDisjoint (repo has the real one, \\textttTwoConnected.lean:58)]…","labels":["Networks.InternallyDisjoint"],"detail_key":"p14"},{"id":"n21961","layer":"informal","project":"p14","title":"Max number of arc-disjoint directed (x,y)-paths in a unit-ish \\textttNetwork","kind":"definition","summary":"[Max number of arc-disjoint directed (x,y)-paths in a unit-ish \\textttNetwork] For a unit-capac…","labels":["Networks.Network.maxArcDisjointPaths"],"detail_key":"p14"},{"id":"n21962","layer":"informal","project":"p14","title":"Max number of arc-disjoint directed (x,y)-paths in a unit-ish \\textttNetwork","kind":"definition","summary":"[Max number of arc-disjoint directed (x,y)-paths in a unit-ish \\textttNetwork] For a unit-capac…","labels":["Networks.Network.minArcsDestroyingPaths"],"detail_key":"p14"},{"id":"n21963","layer":"informal","project":"p14","title":"Max number of arc-disjoint directed (x,y)-paths in a digraph","kind":"definition","summary":"[Max number of arc-disjoint directed (x,y)-paths in a digraph] The left-hand side of Menger's a…","labels":["Networks.maxArcDisjointDirectedPaths"],"detail_key":"p14"},{"id":"n21964","layer":"informal","project":"p14","title":"Max number of arc-disjoint directed (x,y)-paths in a digraph","kind":"definition","summary":"[Max number of arc-disjoint directed (x,y)-paths in a digraph] The left-hand side of Menger's a…","labels":["Networks.minArcsDestroyingDirectedPaths"],"detail_key":"p14"},{"id":"n21965","layer":"informal","project":"p14","title":"Min number of arcs destroying all directed (x,y)-paths in a digraph","kind":"definition","summary":"[Min number of arcs destroying all directed (x,y)-paths in a digraph] The right-hand side of Me…","labels":["Networks.maxInternallyDisjointDirectedPaths"],"detail_key":"p14"},{"id":"n21966","layer":"informal","project":"p14","title":"Max number of internally-disjoint directed (x,y)-paths in a digraph","kind":"definition","summary":"[Max number of internally-disjoint directed (x,y)-paths in a digraph] The left-hand side of Men…","labels":["Networks.minVerticesDestroyingDirectedPaths"],"detail_key":"p14"},{"id":"n21967","layer":"informal","project":"p14","title":"Max number of edge-disjoint (x,y)-paths in a graph","kind":"definition","summary":"[Max number of edge-disjoint (x,y)-paths in a graph] The left-hand side of Menger's edge theore…","labels":["Networks.maxEdgeDisjointPaths"],"detail_key":"p14"},{"id":"n21968","layer":"informal","project":"p14","title":"Max number of edge-disjoint (x,y)-paths in a graph","kind":"definition","summary":"[Max number of edge-disjoint (x,y)-paths in a graph] The left-hand side of Menger's edge theore…","labels":["Networks.minEdgesDestroyingPaths"],"detail_key":"p14"},{"id":"n21969","layer":"informal","project":"p14","title":"Min number of edges destroying all (x,y)-paths in a graph","kind":"definition","summary":"[Min number of edges destroying all (x,y)-paths in a graph] The right-hand side of Menger's edg…","labels":["Networks.maxInternallyDisjointPaths"],"detail_key":"p14"},{"id":"n21970","layer":"informal","project":"p14","title":"Max number of internally-disjoint (x,y)-paths in a graph","kind":"definition","summary":"[Max number of internally-disjoint (x,y)-paths in a graph] The left-hand side of Menger's verte…","labels":["Networks.minVerticesDestroyingPaths"],"detail_key":"p14"},{"id":"n21971","layer":"informal","project":"p14","title":"Max number of vertex-disjoint S--T paths (Ex 11.4.3)","kind":"definition","summary":"[Max number of vertex-disjoint S--T paths (Ex 11.4.3)] The \"fan\" or set-to-set form of Menger's…","labels":["Networks.maxSTVertexDisjointPaths"],"detail_key":"p14"},{"id":"n21972","layer":"informal","project":"p14","title":"Max number of vertex-disjoint S--T paths (Ex 11.4.3)","kind":"definition","summary":"[Max number of vertex-disjoint S--T paths (Ex 11.4.3)] The \"fan\" or set-to-set form of Menger's…","labels":["Networks.minSTSeparator"],"detail_key":"p14"},{"id":"n21973","layer":"informal","project":"p14","title":"The associated digraph D(G): each edge becomes two opposite arcs (cited as ex 10.3.6)","kind":"definition","summary":"[The associated digraph D(G): each edge becomes two opposite arcs (cited as ex 10.3.6)] Make ev…","labels":["Networks.associatedDigraph"],"detail_key":"p14"},{"id":"n21974","layer":"informal","project":"p14","title":"The vertex-splitting digraph D' of Theorem 11.6, on V \\oplus V: \\textttinl v = v', \\textt…","kind":"definition","summary":"[The vertex-splitting digraph D' of Theorem 11.6, on V \\oplus V: \\textttinl v = v', \\textttinr…","labels":["Networks.splitDigraph"],"detail_key":"p14"},{"id":"n21975","layer":"informal","project":"p14","title":"IsFeasible","kind":"definition","summary":"[IsFeasible] for supplies \\sigma and demands \\textttdem (B\\&M's \\partial). ! MISSING. Honest de…","labels":["Networks.Network.IsFeasible"],"detail_key":"p14"},{"id":"n21976","layer":"informal","project":"p14","title":"(p,q) is realisable by a simple bipartite graph: there is a (0,1)-matrix B with row sums…","kind":"definition","summary":"[(p,q) is realisable by a simple bipartite graph: there is a (0,1)-matrix B with row sums p and…","labels":["Networks.RealisableBipartite"],"detail_key":"p14"},{"id":"n21977","layer":"informal","project":"p14","title":"G is (m,n)-orientable: it admits an orientation in which every indegree is m or n","kind":"definition","summary":"[G is (m,n)-orientable: it admits an orientation in which every indegree is m or n] Make every…","labels":["Networks.IsOrientable"],"detail_key":"p14"},{"id":"n21978","layer":"informal","project":"p14","title":"|(S, \\barS)|: the number of edges of G crossing the cut (S, \\barS) (Ex 11.5.5)","kind":"definition","summary":"[|(S, \\barS)|: the number of edges of G crossing the cut (S, \\barS) (Ex 11.5.5)] The size of th…","labels":["Networks.edgeBoundaryCard"],"detail_key":"p14"},{"id":"n21979","layer":"informal","project":"p14","title":"sum\\_resultant\\_eq\\_fOut\\_sub\\_fIn","kind":"theorem","summary":"[sum\\_resultant\\_eq\\_fOut\\_sub\\_fIn] Summing the \\emphnet outflow over the vertices of S gives…","labels":["Networks.sum_resultant_eq_fOut_sub_fIn"],"detail_key":"p14"},{"id":"n21980","layer":"informal","project":"p14","title":"fOut\\_X\\_sub\\_fIn\\_X\\_eq\\_fIn\\_Y\\_sub\\_fOut\\_Y","kind":"theorem","summary":"[fOut\\_X\\_sub\\_fIn\\_X\\_eq\\_fIn\\_Y\\_sub\\_fOut\\_Y] Whatever leaves the producers must arrive at t…","labels":["Networks.fOut_X_sub_fIn_X_eq_fIn_Y_sub_fOut_Y"],"detail_key":"p14"},{"id":"n21981","layer":"informal","project":"p14","title":"val\\_eq\\_fOut\\_sub\\_fIn","kind":"theorem","summary":"[val\\_eq\\_fOut\\_sub\\_fIn] The value of a flow can be measured across \\emphany cut, not just at…","labels":["Networks.val_eq_fOut_sub_fIn"],"detail_key":"p14"},{"id":"n21982","layer":"informal","project":"p14","title":"val\\_le\\_capOf","kind":"theorem","summary":"[val\\_le\\_capOf] Every unit shipped from x to y must squeeze through the cut, so no flow can ex…","labels":["Networks.val_le_capOf"],"detail_key":"p14"},{"id":"n21983","layer":"informal","project":"p14","title":"val\\_eq\\_capOf\\_iff","kind":"theorem","summary":"[val\\_eq\\_capOf\\_iff] The cut is a genuine bottleneck precisely when it runs at full capacity f…","labels":["Networks.val_eq_capOf_iff"],"detail_key":"p14"},{"id":"n21984","layer":"informal","project":"p14","title":"isMaxFlow\\_and\\_isMinCut\\_of\\_val\\_eq\\_cap","kind":"theorem","summary":"[isMaxFlow\\_and\\_isMinCut\\_of\\_val\\_eq\\_cap] The standard \"weak duality certifies optimality\" a…","labels":["Networks.isMaxFlow_and_isMinCut_of_val_eq_cap"],"detail_key":"p14"},{"id":"n21985","layer":"informal","project":"p14","title":"exists\\_maxFlow","kind":"theorem","summary":"[exists\\_maxFlow] (assumed silently by the book). Nothing deep: finitely many flows, so a best…","labels":["Networks.exists_maxFlow"],"detail_key":"p14"},{"id":"n21986","layer":"informal","project":"p14","title":"exists\\_minCut","kind":"theorem","summary":"[exists\\_minCut] (assumed silently by the book). Finitely many subsets, at least one of them a…","labels":["Networks.exists_minCut"],"detail_key":"p14"},{"id":"n21987","layer":"informal","project":"p14","title":"maxFlow\\_and\\_minCut\\_zero\\_of\\_no\\_path","kind":"theorem","summary":"[maxFlow\\_and\\_minCut\\_zero\\_of\\_no\\_path] If the sink cannot be reached at all, nothing can be…","labels":["Networks.maxFlow_and_minCut_zero_of_no_path"],"detail_key":"p14"},{"id":"n21988","layer":"informal","project":"p14","title":"isMinCut\\_union\\_inter","kind":"theorem","summary":"[isMinCut\\_union\\_inter] The minimum cuts form a lattice. A consequence is that there is a uniq…","labels":["Networks.isMinCut_union_inter"],"detail_key":"p14"},{"id":"n21989","layer":"informal","project":"p14","title":"maxFlow\\_iff\\_no\\_incrementing\\_path","kind":"theorem","summary":"[maxFlow\\_iff\\_no\\_incrementing\\_path] B\\&M stress the analogy: *the r\\^ole played by increment…","labels":["Networks.maxFlow_iff_no_incrementing_path"],"detail_key":"p14"},{"id":"n21990","layer":"informal","project":"p14","title":"exists\\_cut\\_of\\_no\\_incrementing\\_path","kind":"theorem","summary":"[exists\\_cut\\_of\\_no\\_incrementing\\_path] *If N contains no f-incrementing path, then there is…","labels":["Networks.exists_cut_of_no_incrementing_path"],"detail_key":"p14"},{"id":"n21991","layer":"informal","project":"p14","title":"maxFlow\\_min\\_cut","kind":"theorem","summary":"[maxFlow\\_min\\_cut] Theorem 11.1 gave the easy half --- \\textttval f \\le cap K for every flow a…","labels":["Networks.maxFlow_min_cut"],"detail_key":"p14"},{"id":"n21992","layer":"informal","project":"p14","title":"unitCapacity\\_maxFlow\\_eq\\_arcDisjoint","kind":"theorem","summary":"[unitCapacity\\_maxFlow\\_eq\\_arcDisjoint] With unit capacities a flow \\emphis a packing of arc-d…","labels":["Networks.unitCapacity_maxFlow_eq_arcDisjoint"],"detail_key":"p14"},{"id":"n21993","layer":"informal","project":"p14","title":"unitCapacity\\_minCut\\_eq\\_minDestroying","kind":"theorem","summary":"[unitCapacity\\_minCut\\_eq\\_minDestroying] With part (a) and the max-flow min-cut theorem, this…","labels":["Networks.unitCapacity_minCut_eq_minDestroying"],"detail_key":"p14"},{"id":"n21994","layer":"informal","project":"p14","title":"menger\\_arc\\_digraph","kind":"theorem","summary":"[menger\\_arc\\_digraph] The number of independent routes you can run equals the number of links…","labels":["Networks.menger_arc_digraph"],"detail_key":"p14"},{"id":"n21995","layer":"informal","project":"p14","title":"menger\\_edge\\_graph","kind":"theorem","summary":"[menger\\_edge\\_graph] The undirected Menger theorem quoted in \\S3.2 as the edge analogue of Whi…","labels":["Networks.menger_edge_graph"],"detail_key":"p14"},{"id":"n21996","layer":"informal","project":"p14","title":"k\\_edge\\_connected\\_iff\\_edge\\_disjoint\\_paths","kind":"theorem","summary":"[k\\_edge\\_connected\\_iff\\_edge\\_disjoint\\_paths] k-edge-connectedness says no k - 1 edges disco…","labels":["Networks.k_edge_connected_iff_edge_disjoint_paths"],"detail_key":"p14"},{"id":"n21997","layer":"informal","project":"p14","title":"menger\\_vertex\\_digraph","kind":"theorem","summary":"[menger\\_vertex\\_digraph] ! The hypothesis \\texttt\\lnot D.Adj x y is load-bearing: an arc (x, y…","labels":["Networks.menger_vertex_digraph"],"detail_key":"p14"},{"id":"n21998","layer":"informal","project":"p14","title":"menger\\_vertex\\_graph","kind":"theorem","summary":"[menger\\_vertex\\_graph] The most quoted form of Menger's theorem, and the one \\S3.2 announced w…","labels":["Networks.menger_vertex_graph"],"detail_key":"p14"},{"id":"n21999","layer":"informal","project":"p14","title":"k\\_connected\\_iff\\_internally\\_disjoint\\_paths","kind":"theorem","summary":"[k\\_connected\\_iff\\_internally\\_disjoint\\_paths] k-connectivity says no k - 1 vertices disconne…","labels":["Networks.k_connected_iff_internally_disjoint_paths"],"detail_key":"p14"},{"id":"n22000","layer":"informal","project":"p14","title":"dirac\\_k\\_vertices\\_on\\_cycle","kind":"theorem","summary":"[dirac\\_k\\_vertices\\_on\\_cycle] High connectivity forces any prescribed set of k vertices onto…","labels":["Networks.dirac_k_vertices_on_cycle"],"detail_key":"p14"},{"id":"n22001","layer":"informal","project":"p14","title":"gale\\_feasible\\_flow","kind":"theorem","summary":"[gale\\_feasible\\_flow] A feasible flow exists exactly when, for every way of splitting the vert…","labels":["Networks.gale_feasible_flow"],"detail_key":"p14"},{"id":"n22002","layer":"informal","project":"p14","title":"galeRyser\\_realisable","kind":"theorem","summary":"[galeRyser\\_realisable] Equal sums (11.16) are necessary --- both count the edges --- but not s…","labels":["Networks.galeRyser_realisable"],"detail_key":"p14"},{"id":"n22003","layer":"informal","project":"p14","title":"realisable\\_iff\\_reduced","kind":"theorem","summary":"[realisable\\_iff\\_reduced] The bipartite analogue of the Havel--Hakimi reduction for degree seq…","labels":["Networks.realisable_iff_reduced"],"detail_key":"p14"},{"id":"n22004","layer":"informal","project":"p14","title":"isOrientable\\_iff\\_exists\\_partition","kind":"theorem","summary":"[isOrientable\\_iff\\_exists\\_partition] An (m+n)-regular graph is to be oriented so every indegr…","labels":["Networks.isOrientable_iff_exists_partition"],"detail_key":"p14"},{"id":"n22005","layer":"informal","project":"p14","title":"isOrientable\\_pred\\_succ","kind":"theorem","summary":"[isOrientable\\_pred\\_succ] The two permitted indegrees can always be moved one step closer toge…","labels":["Networks.isOrientable_pred_succ"],"detail_key":"p14"},{"id":"n22006","layer":"informal","project":"p14","title":"A walk is internally disjoint from a subgraph H: no \\emphinternal vertex lies in H","kind":"definition","summary":"[A walk is internally disjoint from a subgraph H: no \\emphinternal vertex lies in H] The walk m…","labels":["SimpleGraph.Walk.InternallyDisjointFrom"],"detail_key":"p14"},{"id":"n22007","layer":"informal","project":"p14","title":"B\\&M's relation ~ on E(G) \\ E(H): e_1 ~ e_2 iff joined by a walk internally disjoint from…","kind":"definition","summary":"[B\\&M's relation ~ on E(G) \\ E(H): e_1 ~ e_2 iff joined by a walk internally disjoint from H\\do…","labels":["SimpleGraph.bridgeRel"],"detail_key":"p14"},{"id":"n22008","layer":"informal","project":"p14","title":"The bridge of H in G containing the edge e: the subgraph induced by e's ~-class","kind":"definition","summary":"[The bridge of H in G containing the edge e: the subgraph induced by e's ~-class] A bridge is o…","labels":["SimpleGraph.bridgeOf"],"detail_key":"p14"},{"id":"n22009","layer":"informal","project":"p14","title":"Vertices of attachment of the bridge of e to the cycle traced by c: V(B) \\cap V(C)","kind":"definition","summary":"[Vertices of attachment of the bridge of e to the cycle traced by c: V(B) \\cap V(C)] The points…","labels":["SimpleGraph.attach"],"detail_key":"p14"},{"id":"n22010","layer":"informal","project":"p14","title":"Position of a vertex along a cycle (index into its support)","kind":"definition","summary":"[Position of a vertex along a cycle (index into its support)] Walking around the cycle from its…","labels":["SimpleGraph.Walk.cycleIdx"],"detail_key":"p14"},{"id":"n22011","layer":"informal","project":"p14","title":"Four vertices in cyclic order on c","kind":"definition","summary":"[Four vertices in cyclic order on c] Travelling once round the cycle, the four vertices are met…","labels":["SimpleGraph.Walk.InCyclicOrder"],"detail_key":"p14"},{"id":"n22012","layer":"informal","project":"p14","title":"Forward distance along c from a to x, in steps, wrapping at the base point","kind":"definition","summary":"[Forward distance along c from a to x, in steps, wrapping at the base point] Stand at a and wal…","labels":["SimpleGraph.Walk.cycleDist"],"detail_key":"p14"},{"id":"n22013","layer":"informal","project":"p14","title":"x lies on the closed arc of c running forwards from a to b","kind":"definition","summary":"[x lies on the closed arc of c running forwards from a to b] Walking forwards from a, you meet…","labels":["SimpleGraph.Walk.OnArc"],"detail_key":"p14"},{"id":"n22014","layer":"informal","project":"p14","title":"a and b are consecutive vertices of attachment of the bridge of e: both are attachments,…","kind":"definition","summary":"[a and b are consecutive vertices of attachment of the bridge of e: both are attachments, and n…","labels":["SimpleGraph.ConsecutiveAttach"],"detail_key":"p14"},{"id":"n22015","layer":"informal","project":"p14","title":"Two bridges of a cycle avoid one another: all the attachments of one lie in a single segm…","kind":"definition","summary":"[Two bridges of a cycle avoid one another: all the attachments of one lie in a single segment o…","labels":["SimpleGraph.Avoids"],"detail_key":"p14"},{"id":"n22016","layer":"informal","project":"p14","title":"Two bridges of a cycle overlap: they do not avoid one another","kind":"definition","summary":"[Two bridges of a cycle overlap: they do not avoid one another] Neither bridge's attachments fi…","labels":["SimpleGraph.Overlaps"],"detail_key":"p14"},{"id":"n22017","layer":"informal","project":"p14","title":"Two bridges of a cycle are skew: they have attachments a, a', b, b' occurring in that cyc…","kind":"definition","summary":"[Two bridges of a cycle are skew: they have attachments a, a', b, b' occurring in that cyclic o…","labels":["SimpleGraph.Skew"],"detail_key":"p14"},{"id":"n22018","layer":"informal","project":"p14","title":"A longest cycle","kind":"definition","summary":"[A longest cycle] A cycle no shorter than any other cycle of G. This is the standard extremal d…","labels":["SimpleGraph.Walk.IsLongestCycle"],"detail_key":"p14"},{"id":"n22019","layer":"informal","project":"p14","title":"minDegree\\_le\\_five\\_of\\_card\\_edge\\_le","kind":"theorem","summary":"[minDegree\\_le\\_five\\_of\\_card\\_edge\\_le] A planar graph cannot be everywhere dense: Euler's fo…","labels":["SimpleGraph.minDegree_le_five_of_card_edge_le"],"detail_key":"p14"},{"id":"n22020","layer":"informal","project":"p14","title":"card\\_edgeFinset\\_add\\_card\\_edgeFinset\\_compl","kind":"theorem","summary":"[card\\_edgeFinset\\_add\\_card\\_edgeFinset\\_compl] \\varepsilon(G) + \\varepsilon(G^c) = C(\\nu, 2).…","labels":["SimpleGraph.card_edgeFinset_add_card_edgeFinset_compl"],"detail_key":"p14"},{"id":"n22021","layer":"informal","project":"p14","title":"not\\_both\\_card\\_edge\\_le\\_of\\_eleven\\_le","kind":"theorem","summary":"[not\\_both\\_card\\_edge\\_le\\_of\\_eleven\\_le] Planarity forces sparsity, and a graph and its comp…","labels":["SimpleGraph.not_both_card_edge_le_of_eleven_le"],"detail_key":"p14"},{"id":"n22022","layer":"informal","project":"p14","title":"bridge\\_inter\\_subset\\_cycle","kind":"theorem","summary":"[bridge\\_inter\\_subset\\_cycle] Bridges can only meet on H itself. If a vertex outside H lay in…","labels":["SimpleGraph.bridge_inter_subset_cycle"],"detail_key":"p14"},{"id":"n22023","layer":"informal","project":"p14","title":"bridgeOf\\_connected","kind":"theorem","summary":"[bridgeOf\\_connected] A bridge is one connected lump: its edges all lie in one \\sim-class, and…","labels":["SimpleGraph.bridgeOf_connected"],"detail_key":"p14"},{"id":"n22024","layer":"informal","project":"p14","title":"exists\\_path\\_internallyDisjoint","kind":"theorem","summary":"[exists\\_path\\_internallyDisjoint] Not only is a bridge connected, but the connection can be ma…","labels":["SimpleGraph.exists_path_internallyDisjoint"],"detail_key":"p14"},{"id":"n22025","layer":"informal","project":"p14","title":"bridge\\_tripod\\_of\\_three\\_attachments","kind":"theorem","summary":"[bridge\\_tripod\\_of\\_three\\_attachments] The picture is a tripod: a hub v_0 strictly off the cy…","labels":["SimpleGraph.bridge_tripod_of_three_attachments"],"detail_key":"p14"},{"id":"n22026","layer":"informal","project":"p14","title":"overlap\\_imp\\_skew\\_or\\_equivalent\\_three\\_bridge","kind":"theorem","summary":"[overlap\\_imp\\_skew\\_or\\_equivalent\\_three\\_bridge] Overlapping comes in exactly two flavours.…","labels":["SimpleGraph.overlap_imp_skew_or_equivalent_three_bridge"],"detail_key":"p14"},{"id":"n22027","layer":"informal","project":"p14","title":"exists\\_bridge\\_off\\_longest\\_cycle","kind":"theorem","summary":"[exists\\_bridge\\_off\\_longest\\_cycle] Since G is not hamiltonian the longest cycle misses some…","labels":["SimpleGraph.exists_bridge_off_longest_cycle"],"detail_key":"p14"},{"id":"n22028","layer":"informal","project":"p14","title":"succ\\_attachments\\_not\\_adj","kind":"theorem","summary":"[succ\\_attachments\\_not\\_adj] The successors of a bridge's attachment points form an *independe…","labels":["SimpleGraph.succ_attachments_not_adj"],"detail_key":"p14"},{"id":"n22029","layer":"informal","project":"p14","title":"chvatal\\_erdos\\_hamiltonian","kind":"theorem","summary":"[chvatal\\_erdos\\_hamiltonian] \\alpha is the independence number and \\kappa the connectivity, so…","labels":["SimpleGraph.chvatal_erdos_hamiltonian"],"detail_key":"p14"},{"id":"n22030","layer":"informal","project":"p14","title":"lineGraph\\_colorable\\_three\\_of\\_isHamiltonian\\_of\\_three\\_regular","kind":"theorem","summary":"[lineGraph\\_colorable\\_three\\_of\\_isHamiltonian\\_of\\_three\\_regular] A cubic hamiltonian graph…","labels":["SimpleGraph.lineGraph_colorable_three_of_isHamiltonian_of_three_regular"],"detail_key":"p14"},{"id":"n22031","layer":"informal","project":"p14","title":"An edge cut set: a set of edges whose deletion disconnects G","kind":"definition","summary":"[An edge cut set: a set of edges whose deletion disconnects G] Split the vertices into two none…","labels":["SimpleGraph.IsEdgeCutSet"],"detail_key":"p14"},{"id":"n22032","layer":"informal","project":"p14","title":"A bond (\\S2.2): a minimal (nonempty) edge cut, i.e","kind":"definition","summary":"[A bond (\\S2.2): a minimal (nonempty) edge cut, i.e] An edge cut with nothing to spare --- remo…","labels":["SimpleGraph.IsBond"],"detail_key":"p14"},{"id":"n22033","layer":"informal","project":"p14","title":"The number of spanning trees \\tau(G): the number of tree subgraphs T \\le G on the full ve…","kind":"definition","summary":"[The number of spanning trees \\tau(G): the number of tree subgraphs T \\le G on the full vertex…","labels":["SimpleGraph.numSpanningTrees"],"detail_key":"p14"},{"id":"n22034","layer":"informal","project":"p14","title":"contract","kind":"definition","summary":"[contract] G \\cdot e (\\S2.4). Mathlib has no edge contraction for \\textttSimpleGraph. NOTE: a g…","labels":["SimpleGraph.contract"],"detail_key":"p14"},{"id":"n22035","layer":"informal","project":"p14","title":"tree\\_unique\\_path","kind":"theorem","summary":"[tree\\_unique\\_path] A tree has exactly enough edges to hold the vertices together and not one…","labels":["SimpleGraph.tree_unique_path"],"detail_key":"p14"},{"id":"n22036","layer":"informal","project":"p14","title":"tree\\_card\\_edgeFinset","kind":"theorem","summary":"[tree\\_card\\_edgeFinset] \\nu - 1 edges is exactly the price of connecting \\nu vertices acyclica…","labels":["SimpleGraph.tree_card_edgeFinset"],"detail_key":"p14"},{"id":"n22037","layer":"informal","project":"p14","title":"tree\\_two\\_leaves","kind":"theorem","summary":"[tree\\_two\\_leaves] A degree-one vertex of a tree is a leaf. ! B\\&M offer a more illuminating a…","labels":["SimpleGraph.tree_two_leaves"],"detail_key":"p14"},{"id":"n22038","layer":"informal","project":"p14","title":"cutEdge\\_iff\\_no\\_cycle","kind":"theorem","summary":"[cutEdge\\_iff\\_no\\_cycle] Cut edges are exactly the edges lying on no cycle: an edge on a cycle…","labels":["SimpleGraph.cutEdge_iff_no_cycle"],"detail_key":"p14"},{"id":"n22039","layer":"informal","project":"p14","title":"connected\\_isTree\\_iff\\_forall\\_edge\\_isBridge","kind":"theorem","summary":"[connected\\_isTree\\_iff\\_forall\\_edge\\_isBridge] A tree is exactly a connected graph with no re…","labels":["SimpleGraph.connected_isTree_iff_forall_edge_isBridge"],"detail_key":"p14"},{"id":"n22040","layer":"informal","project":"p14","title":"exists\\_spanningTree","kind":"theorem","summary":"[exists\\_spanningTree] Keep deleting redundant edges --- those lying on cycles --- until none r…","labels":["SimpleGraph.exists_spanningTree"],"detail_key":"p14"},{"id":"n22041","layer":"informal","project":"p14","title":"connected\\_card\\_edgeFinset\\_ge","kind":"theorem","summary":"[connected\\_card\\_edgeFinset\\_ge] \\nu - 1 edges is the minimum price of connecting \\nu vertices…","labels":["SimpleGraph.connected_card_edgeFinset_ge"],"detail_key":"p14"},{"id":"n22042","layer":"informal","project":"p14","title":"spanningTree\\_add\\_edge\\_exists\\_cycle","kind":"theorem","summary":"[spanningTree\\_add\\_edge\\_exists\\_cycle] A spanning tree already offers a unique route between…","labels":["SimpleGraph.spanningTree_add_edge_exists_cycle"],"detail_key":"p14"},{"id":"n22043","layer":"informal","project":"p14","title":"cotree\\_bond","kind":"theorem","summary":"[cotree\\_bond] The exact mirror of theorem 2.5, with \\emphcycle \\to \\emphbond and *spanning tre…","labels":["SimpleGraph.cotree_bond"],"detail_key":"p14"},{"id":"n22044","layer":"informal","project":"p14","title":"tree\\_isCutVertex\\_iff\\_degree","kind":"theorem","summary":"[tree\\_isCutVertex\\_iff\\_degree] In a tree the cut vertices are exactly the internal (non-leaf)…","labels":["SimpleGraph.tree_isCutVertex_iff_degree"],"detail_key":"p14"},{"id":"n22045","layer":"informal","project":"p14","title":"connected\\_two\\_non\\_cutVertices","kind":"theorem","summary":"[connected\\_two\\_non\\_cutVertices] However tangled a connected graph is, there are always at le…","labels":["SimpleGraph.connected_two_non_cutVertices"],"detail_key":"p14"},{"id":"n22046","layer":"informal","project":"p14","title":"numSpanningTrees\\_deletion\\_contraction","kind":"theorem","summary":"[numSpanningTrees\\_deletion\\_contraction] Iterating the recursion reduces any graph to trivial…","labels":["SimpleGraph.numSpanningTrees_deletion_contraction"],"detail_key":"p14"},{"id":"n22047","layer":"informal","project":"p14","title":"cayley","kind":"theorem","summary":"[cayley] ! B\\&M caution that n^n-2 counts \\emphdistinct spanning trees, not non-isomorphic ones…","labels":["SimpleGraph.cayley"],"detail_key":"p14"},{"id":"n22048","layer":"informal","project":"p14","title":"exists\\_min\\_weight\\_spanningTree","kind":"theorem","summary":"[exists\\_min\\_weight\\_spanningTree] Assign each edge a weight and each spanning tree the total;…","labels":["SimpleGraph.exists_min_weight_spanningTree"],"detail_key":"p14"},{"id":"n22049","layer":"informal","project":"p14","title":"G is critical: \\chi(H) < \\chi(G) for every proper subgraph H \\subset G","kind":"definition","summary":"[G is critical: \\chi(H) < \\chi(G) for every proper subgraph H \\subset G] A critical graph is \"m…","labels":["SimpleGraph.IsCritical"],"detail_key":"p14"},{"id":"n22050","layer":"informal","project":"p14","title":"G is k-critical: k-chromatic and critical","kind":"definition","summary":"[G is k-critical: k-chromatic and critical] k-critical graphs are the irreducible witnesses to…","labels":["SimpleGraph.IsKCritical"],"detail_key":"p14"},{"id":"n22051","layer":"informal","project":"p14","title":"G \\cdot uv: contract the edge uv by identifying v into u","kind":"definition","summary":"[G \\cdot uv: contract the edge uv by identifying v into u] Pull the edge uv tight until u and v…","labels":["SimpleGraph.contractEdge"],"detail_key":"p14"},{"id":"n22052","layer":"informal","project":"p14","title":"\\pi_k(G), the number of distinct proper k-colourings","kind":"definition","summary":"[\\pi_k(G), the number of distinct proper k-colourings] Count the proper colourings as \\emphlabe…","labels":["SimpleGraph.numColorings"],"detail_key":"p14"},{"id":"n22053","layer":"informal","project":"p14","title":"A u,v-component: G(C \\cup u,v) for C a component of G - u,v","kind":"definition","summary":"[A u,v-component: G(C \\cup u,v) for C a component of G - u,v] Cut the graph at S and put S back…","labels":["SimpleGraph.uvComponent"],"detail_key":"p14"},{"id":"n22054","layer":"informal","project":"p14","title":"Type 1: every (k-1)-colouring assigns u and v the same colour","kind":"definition","summary":"[Type 1: every (k-1)-colouring assigns u and v the same colour] The component forces u and v to…","labels":["SimpleGraph.Subgraph.IsType1"],"detail_key":"p14"},{"id":"n22055","layer":"informal","project":"p14","title":"Type 2: every (k-1)-colouring assigns u and v different colours","kind":"definition","summary":"[Type 2: every (k-1)-colouring assigns u and v different colours] The component forces u and v…","labels":["SimpleGraph.Subgraph.IsType2"],"detail_key":"p14"},{"id":"n22056","layer":"informal","project":"p14","title":"G contains a subdivision of K_4: four distinct branch vertices joined pairwise by six pat…","kind":"definition","summary":"[G contains a subdivision of K_4: four distinct branch vertices joined pairwise by six paths,\\d…","labels":["SimpleGraph.HasK4Subdivision"],"detail_key":"p14"},{"id":"n22057","layer":"informal","project":"p14","title":"mycielskian","kind":"definition","summary":"[mycielskian] on \\textttV \\oplus V \\oplus Unit: \\textttinl = the old graph, \\textttinr \\circ in…","labels":["SimpleGraph.mycielskian"],"detail_key":"p14"},{"id":"n22058","layer":"informal","project":"p14","title":"The wheel with n spokes: C_n \\lor K_1","kind":"definition","summary":"[The wheel with n spokes: C_n \\lor K_1] A hub joined to every vertex of a rim cycle. As a join…","labels":["SimpleGraph.wheel"],"detail_key":"p14"},{"id":"n22059","layer":"informal","project":"p14","title":"G is uniquely k-colourable: any two k-colourings induce the same partition","kind":"definition","summary":"[G is uniquely k-colourable: any two k-colourings induce the same partition] Colours are arbitr…","labels":["SimpleGraph.UniquelyColorable"],"detail_key":"p14"},{"id":"n22060","layer":"informal","project":"p14","title":"s is a maximal independent set of G within t","kind":"definition","summary":"[s is a maximal independent set of G within t] s is independent, lies inside t, and cannot be e…","labels":["SimpleGraph.IsMaximalIndepSetIn"],"detail_key":"p14"},{"id":"n22061","layer":"informal","project":"p14","title":"A canonical colouring, as a list of colour classes","kind":"definition","summary":"[A canonical colouring, as a list of colour classes] Build the colour classes greedily, each ti…","labels":["SimpleGraph.IsCanonicalColouring"],"detail_key":"p14"},{"id":"n22062","layer":"informal","project":"p14","title":"The Mycielski tower G_2 = K_2, G_3, G_4, \\dots","kind":"definition","summary":"[The Mycielski tower G_2 = K_2, G_3, G_4, \\dots] Iterate Mycielski's construction from a single…","labels":["SimpleGraph.mycielskiTower"],"detail_key":"p14"},{"id":"n22063","layer":"informal","project":"p14","title":"exists\\_isKCritical\\_subgraph","kind":"theorem","summary":"[exists\\_isKCritical\\_subgraph] Start with a k-chromatic graph and keep deleting vertices and e…","labels":["SimpleGraph.exists_isKCritical_subgraph"],"detail_key":"p14"},{"id":"n22064","layer":"informal","project":"p14","title":"connected","kind":"theorem","summary":"[connected] If a graph were disconnected, its chromatic number would be the maximum of those of…","labels":["SimpleGraph.IsCritical.connected"],"detail_key":"p14"},{"id":"n22065","layer":"informal","project":"p14","title":"minDegree\\_ge","kind":"theorem","summary":"[minDegree\\_ge] A vertex of small degree is never an obstacle to colouring, since some colour i…","labels":["SimpleGraph.IsKCritical.minDegree_ge"],"detail_key":"p14"},{"id":"n22066","layer":"informal","project":"p14","title":"card\\_filter\\_degree\\_ge\\_of\\_chromaticNumber","kind":"theorem","summary":"[card\\_filter\\_degree\\_ge\\_of\\_chromaticNumber] Needing k colours is not a global accident ---…","labels":["SimpleGraph.card_filter_degree_ge_of_chromaticNumber"],"detail_key":"p14"},{"id":"n22067","layer":"informal","project":"p14","title":"chromaticNumber\\_le\\_maxDegree\\_add\\_one","kind":"theorem","summary":"[chromaticNumber\\_le\\_maxDegree\\_add\\_one] Colour the vertices one at a time in any order: when…","labels":["SimpleGraph.chromaticNumber_le_maxDegree_add_one"],"detail_key":"p14"},{"id":"n22068","layer":"informal","project":"p14","title":"not\\_isClique\\_of\\_isVertexCut","kind":"theorem","summary":"[not\\_isClique\\_of\\_isVertexCut] A clique cut is too rigid to cause trouble: its vertices are f…","labels":["SimpleGraph.IsCritical.not_isClique_of_isVertexCut"],"detail_key":"p14"},{"id":"n22069","layer":"informal","project":"p14","title":"connected\\_and\\_no\\_cut\\_vertex","kind":"theorem","summary":"[connected\\_and\\_no\\_cut\\_vertex] Criticality forces the graph to hold together tightly: it can…","labels":["SimpleGraph.IsCritical.connected_and_no_cut_vertex"],"detail_key":"p14"},{"id":"n22070","layer":"informal","project":"p14","title":"not\\_adj\\_of\\_isVertexCut\\_pair","kind":"theorem","summary":"[not\\_adj\\_of\\_isVertexCut\\_pair] This is what makes the type 1 / type 2 dichotomy meaningful.…","labels":["SimpleGraph.IsCritical.not_adj_of_isVertexCut_pair"],"detail_key":"p14"},{"id":"n22071","layer":"informal","project":"p14","title":"chromaticNumber\\_ge\\_of\\_card\\_edges","kind":"theorem","summary":"[chromaticNumber\\_ge\\_of\\_card\\_edges] Many edges force many colours: each colour class is edge…","labels":["SimpleGraph.chromaticNumber_ge_of_card_edges"],"detail_key":"p14"},{"id":"n22072","layer":"informal","project":"p14","title":"chromaticNumber\\_le\\_five\\_of\\_odd\\_cycles\\_meet","kind":"theorem","summary":"[chromaticNumber\\_le\\_five\\_of\\_odd\\_cycles\\_meet] Odd cycles are the sole obstruction to 2-col…","labels":["SimpleGraph.chromaticNumber_le_five_of_odd_cycles_meet"],"detail_key":"p14"},{"id":"n22073","layer":"informal","project":"p14","title":"welsh\\_powell","kind":"theorem","summary":"[welsh\\_powell] Colour greedily in order of decreasing degree. When the i-th vertex's turn come…","labels":["SimpleGraph.welsh_powell"],"detail_key":"p14"},{"id":"n22074","layer":"informal","project":"p14","title":"chromaticNumber\\_le\\_ceil\\_sqrt\\_two\\_mul\\_card\\_edges","kind":"theorem","summary":"[chromaticNumber\\_le\\_ceil\\_sqrt\\_two\\_mul\\_card\\_edges] Needing many colours forces many edges…","labels":["SimpleGraph.chromaticNumber_le_ceil_sqrt_two_mul_card_edges"],"detail_key":"p14"},{"id":"n22075","layer":"informal","project":"p14","title":"nordhaus\\_gaddum","kind":"theorem","summary":"[nordhaus\\_gaddum] A graph and its complement cannot both be hard to colour: an edge missing fr…","labels":["SimpleGraph.nordhaus_gaddum"],"detail_key":"p14"},{"id":"n22076","layer":"informal","project":"p14","title":"szekeres\\_wilf","kind":"theorem","summary":"[szekeres\\_wilf] D is the degeneracy of G. Strip off a vertex of degree at most D, repeatedly,…","labels":["SimpleGraph.szekeres_wilf"],"detail_key":"p14"},{"id":"n22077","layer":"informal","project":"p14","title":"gallai\\_two\\_per\\_class","kind":"theorem","summary":"[gallai\\_two\\_per\\_class] The hypothesis provides \\emphsome colouring, possibly using far more…","labels":["SimpleGraph.gallai_two_per_class"],"detail_key":"p14"},{"id":"n22078","layer":"informal","project":"p14","title":"isKCritical\\_three\\_iff\\_odd\\_cycle","kind":"theorem","summary":"[isKCritical\\_three\\_iff\\_odd\\_cycle] An odd cycle is 3-chromatic and minimally so --- delete a…","labels":["SimpleGraph.isKCritical_three_iff_odd_cycle"],"detail_key":"p14"},{"id":"n22079","layer":"informal","project":"p14","title":"isKCritical\\_one\\_iff","kind":"theorem","summary":"[isKCritical\\_one\\_iff] A 1-chromatic graph has no edges and at least one vertex. Criticality f…","labels":["SimpleGraph.isKCritical_one_iff"],"detail_key":"p14"},{"id":"n22080","layer":"informal","project":"p14","title":"isKCritical\\_two\\_iff","kind":"theorem","summary":"[isKCritical\\_two\\_iff] A 2-chromatic graph has at least one edge, and a single edge already ne…","labels":["SimpleGraph.isKCritical_two_iff"],"detail_key":"p14"},{"id":"n22081","layer":"informal","project":"p14","title":"not\\_uniquelyColorable\\_of\\_isVertexCut","kind":"theorem","summary":"[not\\_uniquelyColorable\\_of\\_isVertexCut] This generalises theorem 8.2 from clique cuts to uniq…","labels":["SimpleGraph.IsKCritical.not_uniquelyColorable_of_isVertexCut"],"detail_key":"p14"},{"id":"n22082","layer":"informal","project":"p14","title":"neighborSet\\_not\\_subset","kind":"theorem","summary":"[neighborSet\\_not\\_subset] In a critical graph no vertex is redundant, and a vertex whose neigh…","labels":["SimpleGraph.IsCritical.neighborSet_not_subset"],"detail_key":"p14"},{"id":"n22083","layer":"informal","project":"p14","title":"no\\_isKCritical\\_card\\_eq\\_succ","kind":"theorem","summary":"[no\\_isKCritical\\_card\\_eq\\_succ] So k-critical graphs come in sizes k (the complete graph K_k)…","labels":["SimpleGraph.no_isKCritical_card_eq_succ"],"detail_key":"p14"},{"id":"n22084","layer":"informal","project":"p14","title":"chromaticNumber\\_join","kind":"theorem","summary":"[chromaticNumber\\_join] No colour can be used on both sides of a join, so the two palettes are…","labels":["SimpleGraph.chromaticNumber_join"],"detail_key":"p14"},{"id":"n22085","layer":"informal","project":"p14","title":"join\\_isCritical\\_iff","kind":"theorem","summary":"[join\\_isCritical\\_iff] By part (a) the join's chromatic number is the sum, so a drop on one si…","labels":["SimpleGraph.join_isCritical_iff"],"detail_key":"p14"},{"id":"n22086","layer":"informal","project":"p14","title":"hajos\\_construction","kind":"theorem","summary":"[hajos\\_construction] Glue two k-critical graphs at a single vertex v, delete one edge at v fro…","labels":["SimpleGraph.hajos_construction"],"detail_key":"p14"},{"id":"n22087","layer":"informal","project":"p14","title":"exists\\_isKCritical\\_four","kind":"theorem","summary":"[exists\\_isKCritical\\_four] The excluded case n = 5 is exactly exercise 8.1.9(b): no k-critical…","labels":["SimpleGraph.exists_isKCritical_four"],"detail_key":"p14"},{"id":"n22088","layer":"informal","project":"p14","title":"kainen","kind":"theorem","summary":"[kainen] Colour each side separately; the two colourings may clash across the cut, but there ar…","labels":["SimpleGraph.kainen"],"detail_key":"p14"},{"id":"n22089","layer":"informal","project":"p14","title":"edgeConnectivity\\_ge","kind":"theorem","summary":"[edgeConnectivity\\_ge] Criticality forces robust connectivity: a graph that \\emphminimally need…","labels":["SimpleGraph.IsKCritical.edgeConnectivity_ge"],"detail_key":"p14"},{"id":"n22090","layer":"informal","project":"p14","title":"brooks\\_chromaticNumber\\_le\\_maxDegree","kind":"theorem","summary":"[brooks\\_chromaticNumber\\_le\\_maxDegree] Corollary 8.1.2 gives \\chi \\le \\Delta + 1 for every gr…","labels":["SimpleGraph.brooks_chromaticNumber_le_maxDegree"],"detail_key":"p14"},{"id":"n22091","layer":"informal","project":"p14","title":"brooks\\_iff\\_edge\\_bound","kind":"theorem","summary":"[brooks\\_iff\\_edge\\_bound] Theorem 8.1 already gives 2\\varepsilon \\ge \\nu(k-1) for a k-critical…","labels":["SimpleGraph.brooks_iff_edge_bound"],"detail_key":"p14"},{"id":"n22092","layer":"informal","project":"p14","title":"chromaticIndex\\_le\\_four\\_of\\_maxDegree\\_three","kind":"theorem","summary":"[chromaticIndex\\_le\\_four\\_of\\_maxDegree\\_three] The edge chromatic number \\chi'(G) is the chro…","labels":["SimpleGraph.chromaticIndex_le_four_of_maxDegree_three"],"detail_key":"p14"},{"id":"n22093","layer":"informal","project":"p14","title":"kCritical\\_two\\_vertex\\_cut\\_structure","kind":"theorem","summary":"[kCritical\\_two\\_vertex\\_cut\\_structure] At a 2-vertex cut a critical graph splits into exactly…","labels":["SimpleGraph.kCritical_two_vertex_cut_structure"],"detail_key":"p14"},{"id":"n22094","layer":"informal","project":"p14","title":"kCritical\\_two\\_vertex\\_cut\\_degree\\_sum","kind":"theorem","summary":"[kCritical\\_two\\_vertex\\_cut\\_degree\\_sum] This inequality is precisely what Brooks' theorem us…","labels":["SimpleGraph.kCritical_two_vertex_cut_degree_sum"],"detail_key":"p14"},{"id":"n22095","layer":"informal","project":"p14","title":"fourChromatic\\_hasK4Subdivision","kind":"theorem","summary":"[fourChromatic\\_hasK4Subdivision] This is the case k = 4 of Haj\\'os' conjecture, settled by Dir…","labels":["SimpleGraph.fourChromatic_hasK4Subdivision"],"detail_key":"p14"},{"id":"n22096","layer":"informal","project":"p14","title":"hasK4Subdivision\\_of\\_few\\_low\\_degree","kind":"theorem","summary":"[hasK4Subdivision\\_of\\_few\\_low\\_degree] Essentially-all vertices having degree at least three…","labels":["SimpleGraph.hasK4Subdivision_of_few_low_degree"],"detail_key":"p14"},{"id":"n22097","layer":"informal","project":"p14","title":"hasK4Subdivision\\_of\\_card\\_edges","kind":"theorem","summary":"[hasK4Subdivision\\_of\\_card\\_edges] Enough edges relative to vertices force the graph to be loc…","labels":["SimpleGraph.hasK4Subdivision_of_card_edges"],"detail_key":"p14"},{"id":"n22098","layer":"informal","project":"p14","title":"numColorings\\_add\\_contract","kind":"theorem","summary":"[numColorings\\_add\\_contract] Classify the k-colourings of G - e by whether they give u and v t…","labels":["SimpleGraph.numColorings_add_contract"],"detail_key":"p14"},{"id":"n22099","layer":"informal","project":"p14","title":"exists\\_chromaticPolynomial","kind":"theorem","summary":"[exists\\_chromaticPolynomial] This is what justifies calling \\pi_k(G) the chromatic polynomial.…","labels":["SimpleGraph.exists_chromaticPolynomial"],"detail_key":"p14"},{"id":"n22100","layer":"informal","project":"p14","title":"chromaticPolynomial\\_coeff\\_card\\_sub\\_one","kind":"theorem","summary":"[chromaticPolynomial\\_coeff\\_card\\_sub\\_one] The second coefficient of the chromatic polynomial…","labels":["SimpleGraph.chromaticPolynomial_coeff_card_sub_one"],"detail_key":"p14"},{"id":"n22101","layer":"informal","project":"p14","title":"numColorings\\_of\\_isTree","kind":"theorem","summary":"[numColorings\\_of\\_isTree] Root the tree anywhere and colour outward: the root takes any of the…","labels":["SimpleGraph.numColorings_of_isTree"],"detail_key":"p14"},{"id":"n22102","layer":"informal","project":"p14","title":"numColorings\\_le\\_of\\_connected","kind":"theorem","summary":"[numColorings\\_le\\_of\\_connected] Among connected graphs on \\nu vertices, trees are exactly the…","labels":["SimpleGraph.numColorings_le_of_connected"],"detail_key":"p14"},{"id":"n22103","layer":"informal","project":"p14","title":"numColorings\\_cycleGraph","kind":"theorem","summary":"[numColorings\\_cycleGraph] Sanity checks: at k = 2 the formula gives 1 + (-1)^n, which is 2 for…","labels":["SimpleGraph.numColorings_cycleGraph"],"detail_key":"p14"},{"id":"n22104","layer":"informal","project":"p14","title":"numColorings\\_join\\_singleton","kind":"theorem","summary":"[numColorings\\_join\\_singleton] Choose the apex's colour first --- k ways --- and then G must b…","labels":["SimpleGraph.numColorings_join_singleton"],"detail_key":"p14"},{"id":"n22105","layer":"informal","project":"p14","title":"numColorings\\_wheel","kind":"theorem","summary":"[numColorings\\_wheel] A wheel with n spokes is C_n \\lor K_1 --- a rim cycle plus a hub joined t…","labels":["SimpleGraph.numColorings_wheel"],"detail_key":"p14"},{"id":"n22106","layer":"informal","project":"p14","title":"numColorings\\_eq\\_prod\\_components","kind":"theorem","summary":"[numColorings\\_eq\\_prod\\_components] No edge joins different components, so their colourings ar…","labels":["SimpleGraph.numColorings_eq_prod_components"],"detail_key":"p14"},{"id":"n22107","layer":"informal","project":"p14","title":"numColorings\\_union\\_mul\\_inter","kind":"theorem","summary":"[numColorings\\_union\\_mul\\_inter] A gluing formula. A colouring of the union is a pair of colou…","labels":["SimpleGraph.numColorings_union_mul_inter"],"detail_key":"p14"},{"id":"n22108","layer":"informal","project":"p14","title":"chromaticPolynomial\\_no\\_root\\_gt\\_card","kind":"theorem","summary":"[chromaticPolynomial\\_no\\_root\\_gt\\_card] \\pi_k(G) counts colourings, so it is positive at ever…","labels":["SimpleGraph.chromaticPolynomial_no_root_gt_card"],"detail_key":"p14"},{"id":"n22109","layer":"informal","project":"p14","title":"exists\\_triangleFree\\_chromaticNumber\\_eq","kind":"theorem","summary":"[exists\\_triangleFree\\_chromaticNumber\\_eq] One might expect a graph needing many colours to co…","labels":["SimpleGraph.exists_triangleFree_chromaticNumber_eq"],"detail_key":"p14"},{"id":"n22110","layer":"informal","project":"p14","title":"mycielskiTower\\_isKCritical","kind":"theorem","summary":"[mycielskiTower\\_isKCritical] Theorem 8.7 already shows G_k is k-chromatic; the exercise asks f…","labels":["SimpleGraph.mycielskiTower_isKCritical"],"detail_key":"p14"},{"id":"n22111","layer":"informal","project":"p14","title":"descartes\\_construction","kind":"theorem","summary":"[descartes\\_construction] Mycielski's construction (theorem 8.7) removes triangles but still le…","labels":["SimpleGraph.descartes_construction"],"detail_key":"p14"},{"id":"n22112","layer":"informal","project":"p14","title":"exists\\_chromaticNumber\\_ge\\_girth\\_six","kind":"theorem","summary":"[exists\\_chromaticNumber\\_ge\\_girth\\_six] High chromatic number is compatible not merely with t…","labels":["SimpleGraph.exists_chromaticNumber_ge_girth_six"],"detail_key":"p14"},{"id":"n22113","layer":"informal","project":"p14","title":"exists\\_isCanonicalColouring","kind":"theorem","summary":"[exists\\_isCanonicalColouring] Greedily enlarge the first colour class to a maximal independent…","labels":["SimpleGraph.exists_isCanonicalColouring"],"detail_key":"p14"},{"id":"n22114","layer":"formal","project":"p14","title":"BoolBLR.BLR_accept_prob","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → Real","labels":[],"detail_key":"p14","name":"BoolBLR.BLR_accept_prob","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22115","layer":"formal","project":"p14","title":"BoolBLR.BLR_accept_prob_pm1","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), Eq (BoolBLR.BLR_accept_prob f) (HDiv.hDiv (HAdd…","labels":[],"detail_key":"p14","name":"BoolBLR.BLR_accept_prob_pm1","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22116","layer":"formal","project":"p14","title":"BoolBLR.BLR_completeness","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), BoolBLR.is_linear_bool f → Eq (BoolBLR.BLR_acce…","labels":[],"detail_key":"p14","name":"BoolBLR.BLR_completeness","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22117","layer":"formal","project":"p14","title":"BoolBLR.BLR_soundness","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (ε : Real), BoolBLR.epsilon_far_from_linear f ε…","labels":[],"detail_key":"p14","name":"BoolBLR.BLR_soundness","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22118","layer":"formal","project":"p14","title":"BoolBLR.BLR_soundness_via_fourier","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (ε : Real), BoolBLR.epsilon_far_from_linear f ε…","labels":[],"detail_key":"p14","name":"BoolBLR.BLR_soundness_via_fourier","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22119","layer":"formal","project":"p14","title":"BoolBLR.bool_dist","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → (BoolFourier.hypercube n → Bool) → Real","labels":[],"detail_key":"p14","name":"BoolBLR.bool_dist","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22120","layer":"formal","project":"p14","title":"BoolBLR.epsilon_far_from_linear","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → Real → Prop","labels":[],"detail_key":"p14","name":"BoolBLR.epsilon_far_from_linear","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22121","layer":"formal","project":"p14","title":"BoolBLR.fourier_coeff_le_of_dist_ge","kind":"theorem","summary":"∀ ε : Real n : Nat (f g : BoolFourier.hypercube n → Bool) (S : Finset (Fin n)), Eq (BoolBLR.lif…","labels":[],"detail_key":"p14","name":"BoolBLR.fourier_coeff_le_of_dist_ge","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22122","layer":"formal","project":"p14","title":"BoolBLR.fourier_coeff_le_of_far_from_linear","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (ε : Real), BoolBLR.epsilon_far_from_linear f ε…","labels":[],"detail_key":"p14","name":"BoolBLR.fourier_coeff_le_of_far_from_linear","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22123","layer":"formal","project":"p14","title":"BoolBLR.is_linear_bool","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → Prop","labels":[],"detail_key":"p14","name":"BoolBLR.is_linear_bool","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22124","layer":"formal","project":"p14","title":"BoolBLR.lift_pm1","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → BoolFourier.BoolFun n","labels":[],"detail_key":"p14","name":"BoolBLR.lift_pm1","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22125","layer":"formal","project":"p14","title":"BoolBLR.linear_bool_iff_character","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), Iff (BoolBLR.is_linear_bool f) (Exists fun S =>…","labels":[],"detail_key":"p14","name":"BoolBLR.linear_bool_iff_character","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22126","layer":"formal","project":"p14","title":"BoolBLR.triple_expectation_as_convolution","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n), Eq (BoolFourier.expectation fun x => BoolFourier.expecta…","labels":[],"detail_key":"p14","name":"BoolBLR.triple_expectation_as_convolution","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22127","layer":"formal","project":"p14","title":"BoolBLR.triple_expectation_eq_cube_fourier","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), Eq (BoolFourier.expectation fun x => BoolFourie…","labels":[],"detail_key":"p14","name":"BoolBLR.triple_expectation_eq_cube_fourier","module":"TCSlib.BooleanAnalysis.BLR.BoolBLR"},{"id":"n22128","layer":"formal","project":"p14","title":"BoolFourier.BoolFun","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"BoolFourier.BoolFun","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22129","layer":"formal","project":"p14","title":"BoolFourier.BoolToPM1","kind":"def","summary":"Bool → Real","labels":[],"detail_key":"p14","name":"BoolFourier.BoolToPM1","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22130","layer":"formal","project":"p14","title":"BoolFourier.BoolToPM1_not","kind":"theorem","summary":"∀ (b : Bool), Eq (BoolFourier.BoolToPM1 b.not) (Neg.neg (BoolFourier.BoolToPM1 b))","labels":[],"detail_key":"p14","name":"BoolFourier.BoolToPM1_not","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22131","layer":"formal","project":"p14","title":"BoolFourier.BoolToPM1_sq","kind":"theorem","summary":"∀ (b : Bool), Eq (HMul.hMul (BoolFourier.BoolToPM1 b) (BoolFourier.BoolToPM1 b)) 1","labels":[],"detail_key":"p14","name":"BoolFourier.BoolToPM1_sq","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22132","layer":"formal","project":"p14","title":"BoolFourier.BoolToPM1_xor","kind":"theorem","summary":"∀ (a b : Bool), Eq (BoolFourier.BoolToPM1 (a.xor b)) (HMul.hMul (BoolFourier.BoolToPM1 a) (Bool…","labels":[],"detail_key":"p14","name":"BoolFourier.BoolToPM1_xor","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22133","layer":"formal","project":"p14","title":"BoolFourier.L2_norm_sq","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → Real","labels":[],"detail_key":"p14","name":"BoolFourier.L2_norm_sq","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22134","layer":"formal","project":"p14","title":"BoolFourier.PM1ToBool?","kind":"def","summary":"Real → Option Bool","labels":[],"detail_key":"p14","name":"BoolFourier.PM1ToBool?","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22135","layer":"formal","project":"p14","title":"BoolFourier.avg_BoolToPM1","kind":"theorem","summary":"Eq (HDiv.hDiv (HAdd.hAdd (BoolFourier.BoolToPM1 false) (BoolFourier.BoolToPM1 true)) 2) 0","labels":[],"detail_key":"p14","name":"BoolFourier.avg_BoolToPM1","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22136","layer":"formal","project":"p14","title":"BoolFourier.card_hypercube","kind":"theorem","summary":"∀ (n : Nat), Eq (Fintype.card (BoolFourier.hypercube n)) (HPow.hPow 2 n)","labels":[],"detail_key":"p14","name":"BoolFourier.card_hypercube","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22137","layer":"formal","project":"p14","title":"BoolFourier.char_S","kind":"def","summary":"n : Nat → Finset (Fin n) → BoolFourier.BoolFun n","labels":[],"detail_key":"p14","name":"BoolFourier.char_S","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22138","layer":"formal","project":"p14","title":"BoolFourier.char_S_empty","kind":"theorem","summary":"∀ (n : Nat), Eq (BoolFourier.char_S EmptyCollection.emptyCollection) fun x => 1","labels":[],"detail_key":"p14","name":"BoolFourier.char_S_empty","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22139","layer":"formal","project":"p14","title":"BoolFourier.char_S_of_zero","kind":"theorem","summary":"∀ (n : Nat) (S : Finset (Fin n)), Eq (BoolFourier.char_S S (BoolFourier.zero_vec n)) 1","labels":[],"detail_key":"p14","name":"BoolFourier.char_S_of_zero","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22140","layer":"formal","project":"p14","title":"BoolFourier.char_S_sq","kind":"theorem","summary":"∀ (n : Nat) (S : Finset (Fin n)) (x : BoolFourier.hypercube n), Eq (HMul.hMul (BoolFourier.char…","labels":[],"detail_key":"p14","name":"BoolFourier.char_S_sq","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22141","layer":"formal","project":"p14","title":"BoolFourier.char_S_times_char_T","kind":"theorem","summary":"∀ n : Nat (S T : Finset (Fin n)) (x : BoolFourier.hypercube n), Eq (HMul.hMul (BoolFourier.char…","labels":[],"detail_key":"p14","name":"BoolFourier.char_S_times_char_T","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22142","layer":"formal","project":"p14","title":"BoolFourier.convolution","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → BoolFourier.BoolFun n → BoolFourier.BoolFun n","labels":[],"detail_key":"p14","name":"BoolFourier.convolution","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22143","layer":"formal","project":"p14","title":"BoolFourier.expectation","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → Real","labels":[],"detail_key":"p14","name":"BoolFourier.expectation","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22144","layer":"formal","project":"p14","title":"BoolFourier.expectation_char_empty","kind":"theorem","summary":"∀ (n : Nat), Eq (BoolFourier.expectation (BoolFourier.char_S EmptyCollection.emptyCollection)) 1","labels":[],"detail_key":"p14","name":"BoolFourier.expectation_char_empty","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22145","layer":"formal","project":"p14","title":"BoolFourier.expectation_char_nonempty","kind":"theorem","summary":"∀ n : Nat S : Finset (Fin n), S.Nonempty → Eq (BoolFourier.expectation (BoolFourier.char_S S)) 0","labels":[],"detail_key":"p14","name":"BoolFourier.expectation_char_nonempty","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22146","layer":"formal","project":"p14","title":"BoolFourier.expectation_factorizes","kind":"theorem","summary":"∀ n : Nat (g : Fin n → Bool → Real), Eq (BoolFourier.expectation fun x => Finset.univ.prod fun…","labels":[],"detail_key":"p14","name":"BoolFourier.expectation_factorizes","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22147","layer":"formal","project":"p14","title":"BoolFourier.fourier_coeff","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → Finset (Fin n) → Real","labels":[],"detail_key":"p14","name":"BoolFourier.fourier_coeff","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22148","layer":"formal","project":"p14","title":"BoolFourier.fourier_coeff_char_of_ne","kind":"theorem","summary":"∀ n : Nat S T : Finset (Fin n), Ne S T → Eq (BoolFourier.fourier_coeff (BoolFourier.char_S S) T…","labels":[],"detail_key":"p14","name":"BoolFourier.fourier_coeff_char_of_ne","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22149","layer":"formal","project":"p14","title":"BoolFourier.fourier_coeff_char_self","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), Eq (BoolFourier.fourier_coeff (BoolFourier.char_S S) S) 1","labels":[],"detail_key":"p14","name":"BoolFourier.fourier_coeff_char_self","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22150","layer":"formal","project":"p14","title":"BoolFourier.fourier_coeff_convolution","kind":"theorem","summary":"∀ n : Nat (f g : BoolFourier.BoolFun n) (S : Finset (Fin n)), Eq (BoolFourier.fourier_coeff (Bo…","labels":[],"detail_key":"p14","name":"BoolFourier.fourier_coeff_convolution","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22151","layer":"formal","project":"p14","title":"BoolFourier.fourier_expansion","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n) (x : BoolFourier.hypercube n), Eq (f x) (Finset.univ.sum…","labels":[],"detail_key":"p14","name":"BoolFourier.fourier_expansion","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22152","layer":"formal","project":"p14","title":"BoolFourier.hypercube","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"BoolFourier.hypercube","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22153","layer":"formal","project":"p14","title":"BoolFourier.inner_product","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → BoolFourier.BoolFun n → Real","labels":[],"detail_key":"p14","name":"BoolFourier.inner_product","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22154","layer":"formal","project":"p14","title":"BoolFourier.inner_product_char_nonself","kind":"theorem","summary":"∀ n : Nat S T : Finset (Fin n), Ne S T → Eq (BoolFourier.inner_product (BoolFourier.char_S S) (…","labels":[],"detail_key":"p14","name":"BoolFourier.inner_product_char_nonself","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22155","layer":"formal","project":"p14","title":"BoolFourier.inner_product_char_self","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), Eq (BoolFourier.inner_product (BoolFourier.char_S S) (BoolFouri…","labels":[],"detail_key":"p14","name":"BoolFourier.inner_product_char_self","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22156","layer":"formal","project":"p14","title":"BoolFourier.parseval_identity","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n), Eq (Finset.univ.sum fun S => HPow.hPow (BoolFourier.four…","labels":[],"detail_key":"p14","name":"BoolFourier.parseval_identity","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22157","layer":"formal","project":"p14","title":"BoolFourier.sum_char_S_at_zero","kind":"theorem","summary":"∀ n : Nat, Eq (Finset.univ.sum fun S => BoolFourier.char_S S (BoolFourier.zero_vec n)) ↑(Fintyp…","labels":[],"detail_key":"p14","name":"BoolFourier.sum_char_S_at_zero","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22158","layer":"formal","project":"p14","title":"BoolFourier.sum_char_S_ne_zero","kind":"theorem","summary":"∀ n : Nat x : BoolFourier.hypercube n, Ne x (BoolFourier.zero_vec n) → Eq (Finset.univ.sum fun…","labels":[],"detail_key":"p14","name":"BoolFourier.sum_char_S_ne_zero","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22159","layer":"formal","project":"p14","title":"BoolFourier.xor_vec","kind":"def","summary":"n : Nat → BoolFourier.hypercube n → BoolFourier.hypercube n → BoolFourier.hypercube n","labels":[],"detail_key":"p14","name":"BoolFourier.xor_vec","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22160","layer":"formal","project":"p14","title":"BoolFourier.zero_vec","kind":"def","summary":"(n : Nat) → BoolFourier.hypercube n","labels":[],"detail_key":"p14","name":"BoolFourier.zero_vec","module":"TCSlib.BooleanAnalysis.BLR.BoolFourier"},{"id":"n22161","layer":"formal","project":"p14","title":"LowDegreeTest.RM_test_accept_prob","kind":"def","summary":"n : Nat → Nat → (BoolFourier.hypercube n → Bool) → Real","labels":[],"detail_key":"p14","name":"LowDegreeTest.RM_test_accept_prob","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22162","layer":"formal","project":"p14","title":"LowDegreeTest.RM_test_completeness","kind":"theorem","summary":"∀ n : Nat (d : Nat) (f : BoolFourier.hypercube n → Bool), Membership.mem (LowDegreeTest.ReedMul…","labels":[],"detail_key":"p14","name":"LowDegreeTest.RM_test_completeness","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22163","layer":"formal","project":"p14","title":"LowDegreeTest.RM_test_soundness","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.RM_test_soundness","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22164","layer":"formal","project":"p14","title":"LowDegreeTest.ReedMuller","kind":"def","summary":"Nat → (m : Nat) → Set (BoolFourier.hypercube m → Bool)","labels":[],"detail_key":"p14","name":"LowDegreeTest.ReedMuller","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22165","layer":"formal","project":"p14","title":"LowDegreeTest.ReedMuller_monotone","kind":"theorem","summary":"∀ n : Nat (d : Nat), Subset (LowDegreeTest.ReedMuller d n) (LowDegreeTest.ReedMuller (HAdd.hAdd…","labels":[],"detail_key":"p14","name":"LowDegreeTest.ReedMuller_monotone","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22166","layer":"formal","project":"p14","title":"LowDegreeTest.ReedMuller_one_is_affine","kind":"theorem","summary":"∀ n : Nat f : BoolFourier.hypercube n → Bool, Membership.mem (LowDegreeTest.ReedMuller 1 n) f →…","labels":[],"detail_key":"p14","name":"LowDegreeTest.ReedMuller_one_is_affine","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22167","layer":"formal","project":"p14","title":"LowDegreeTest.abs_fourier_coeff_le_of_far_from_degree","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.abs_fourier_coeff_le_of_far_from_degree","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22168","layer":"formal","project":"p14","title":"LowDegreeTest.boolFun_mul","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → BoolFourier.BoolFun n → BoolFourier.BoolFun n","labels":[],"detail_key":"p14","name":"LowDegreeTest.boolFun_mul","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22169","layer":"formal","project":"p14","title":"LowDegreeTest.card_multi_hypercube","kind":"theorem","summary":"∀ (n k : Nat), Eq (Fintype.card (Fin k → BoolFourier.hypercube n)) (HPow.hPow 2 (HMul.hMul n k))","labels":[],"detail_key":"p14","name":"LowDegreeTest.card_multi_hypercube","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22170","layer":"formal","project":"p14","title":"LowDegreeTest.char_is_degree_le","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)) d : Nat, GE.ge d 1 → LowDegreeTest.is_degree_le_pm1 (BoolFourier…","labels":[],"detail_key":"p14","name":"LowDegreeTest.char_is_degree_le","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22171","layer":"formal","project":"p14","title":"LowDegreeTest.char_is_degree_le_one","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), LowDegreeTest.is_degree_le_pm1 (BoolFourier.char_S S) 1","labels":[],"detail_key":"p14","name":"LowDegreeTest.char_is_degree_le_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22172","layer":"formal","project":"p14","title":"LowDegreeTest.degree_le_one_implies_affine","kind":"theorem","summary":"∀ n : Nat f : BoolFourier.hypercube n → Bool, LowDegreeTest.is_degree_le_bool f 1 → Or (BoolBLR…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_le_one_implies_affine","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22173","layer":"formal","project":"p14","title":"LowDegreeTest.degree_le_succ","kind":"theorem","summary":"∀ n : Nat f : BoolFourier.BoolFun n d : Nat, LowDegreeTest.is_degree_le_pm1 f d → LowDegreeTest…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_le_succ","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22174","layer":"formal","project":"p14","title":"LowDegreeTest.degree_le_zero_iff_constant","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), Iff (LowDegreeTest.is_degree_le_bool f 0) (∀ (x…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_le_zero_iff_constant","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22175","layer":"formal","project":"p14","title":"LowDegreeTest.degree_test_accept_prob","kind":"def","summary":"n : Nat → Nat → (BoolFourier.hypercube n → Bool) → Real","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_test_accept_prob","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22176","layer":"formal","project":"p14","title":"LowDegreeTest.degree_test_accept_prob_eq","kind":"theorem","summary":"∀ n : Nat (d : Nat) (f : BoolFourier.hypercube n → Bool), Eq (LowDegreeTest.degree_test_accept_…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_test_accept_prob_eq","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22177","layer":"formal","project":"p14","title":"LowDegreeTest.degree_test_completeness","kind":"theorem","summary":"∀ n : Nat (d : Nat) (f : BoolFourier.hypercube n → Bool), LowDegreeTest.is_degree_le_bool f d →…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_test_completeness","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22178","layer":"formal","project":"p14","title":"LowDegreeTest.degree_test_qualitative_soundness","kind":"theorem","summary":"∀ n : Nat (d : Nat) (f : BoolFourier.hypercube n → Bool), Not (LowDegreeTest.is_degree_le_bool…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_test_qualitative_soundness","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22179","layer":"formal","project":"p14","title":"LowDegreeTest.degree_test_quantitative_soundness","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.degree_test_quantitative_soundness","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22180","layer":"formal","project":"p14","title":"LowDegreeTest.derivative_distance_lemma","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 2 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.derivative_distance_lemma","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22181","layer":"formal","project":"p14","title":"LowDegreeTest.eps_le_half_of_far","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.eps_le_half_of_far","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22182","layer":"formal","project":"p14","title":"LowDegreeTest.epsilon_far_from_RM","kind":"def","summary":"n : Nat → Nat → (BoolFourier.hypercube n → Bool) → Real → Prop","labels":[],"detail_key":"p14","name":"LowDegreeTest.epsilon_far_from_RM","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22183","layer":"formal","project":"p14","title":"LowDegreeTest.epsilon_far_from_degree","kind":"def","summary":"n : Nat → Nat → (BoolFourier.hypercube n → Bool) → Real → Prop","labels":[],"detail_key":"p14","name":"LowDegreeTest.epsilon_far_from_degree","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22184","layer":"formal","project":"p14","title":"LowDegreeTest.epsilon_far_monotone","kind":"theorem","summary":"∀ n d d' : Nat, LE.le d' d → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.e…","labels":[],"detail_key":"p14","name":"LowDegreeTest.epsilon_far_monotone","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22185","layer":"formal","project":"p14","title":"LowDegreeTest.fourier_coeff_le_of_far_from_degree","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.fourier_coeff_le_of_far_from_degree","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22186","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_U2_eq_L2_conv","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n), Eq (LowDegreeTest.gowers_norm_pow f 2) (BoolFourier.L2_n…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_U2_eq_L2_conv","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22187","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_U2_fourier","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n), Eq (LowDegreeTest.gowers_norm_pow f 2) (Finset.univ.sum…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_U2_fourier","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22188","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_U2_le_of_far","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_U2_le_of_far","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22189","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_eq_one_iff","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (d : Nat), Iff (Eq (LowDegreeTest.gowers_norm_po…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_eq_one_iff","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22190","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_le_of_far","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_le_of_far","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22191","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_le_of_far_d1","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_le_of_far_d1","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22192","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_le_one","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (k : Nat), LE.le (LowDegreeTest.gowers_norm_pow…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_le_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22193","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_mul_degree_le","kind":"theorem","summary":"∀ n d : Nat (f g : BoolFourier.BoolFun n), LowDegreeTest.is_degree_le_pm1 g d → Eq (LowDegreeTe…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_mul_degree_le","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22194","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_norm_pow","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → Nat → Real","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_norm_pow","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22195","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → (k : Nat) → BoolFourier.hypercube n → (Fin k → BoolFourier.hy…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22196","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product_mul","kind":"theorem","summary":"∀ n : Nat (f g : BoolFourier.BoolFun n) (k : Nat) (x : BoolFourier.hypercube n) (hs : Fin k → B…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product_mul","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22197","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product_mul_degree_le","kind":"theorem","summary":"∀ n d : Nat (f g : BoolFourier.BoolFun n), LowDegreeTest.is_degree_le_pm1 g d → ∀ (x : BoolFour…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product_mul_degree_le","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22198","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product_pm1","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool) (k : Nat) (x : BoolFourier.hypercube n) (hs : Fi…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product_pm1","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22199","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product_succ","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n) (k : Nat) (x : BoolFourier.hypercube n) (hs : Fin (HAdd.h…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product_succ","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22200","layer":"formal","project":"p14","title":"LowDegreeTest.gowers_product_zero","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n) (x : BoolFourier.hypercube n) (hs : Fin 0 → BoolFourier.h…","labels":[],"detail_key":"p14","name":"LowDegreeTest.gowers_product_zero","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22201","layer":"formal","project":"p14","title":"LowDegreeTest.is_degree_le_bool","kind":"def","summary":"n : Nat → (BoolFourier.hypercube n → Bool) → Nat → Prop","labels":[],"detail_key":"p14","name":"LowDegreeTest.is_degree_le_bool","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22202","layer":"formal","project":"p14","title":"LowDegreeTest.is_degree_le_pm1","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → Nat → Prop","labels":[],"detail_key":"p14","name":"LowDegreeTest.is_degree_le_pm1","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22203","layer":"formal","project":"p14","title":"LowDegreeTest.linear_is_degree_le_one","kind":"theorem","summary":"∀ n : Nat f : BoolFourier.hypercube n → Bool, BoolBLR.is_linear_bool f → LowDegreeTest.is_degre…","labels":[],"detail_key":"p14","name":"LowDegreeTest.linear_is_degree_le_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22204","layer":"formal","project":"p14","title":"LowDegreeTest.linear_mem_ReedMuller_one","kind":"theorem","summary":"∀ n : Nat f : BoolFourier.hypercube n → Bool, BoolBLR.is_linear_bool f → Membership.mem (LowDeg…","labels":[],"detail_key":"p14","name":"LowDegreeTest.linear_mem_ReedMuller_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22205","layer":"formal","project":"p14","title":"LowDegreeTest.mult_deriv","kind":"def","summary":"n : Nat → BoolFourier.BoolFun n → BoolFourier.hypercube n → BoolFourier.BoolFun n","labels":[],"detail_key":"p14","name":"LowDegreeTest.mult_deriv","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22206","layer":"formal","project":"p14","title":"LowDegreeTest.mult_deriv_eq_gowers_one","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.BoolFun n) (x h : BoolFourier.hypercube n), Eq (LowDegreeTest.mult_d…","labels":[],"detail_key":"p14","name":"LowDegreeTest.mult_deriv_eq_gowers_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22207","layer":"formal","project":"p14","title":"LowDegreeTest.multi_expectation","kind":"def","summary":"n k : Nat → ((Fin k → BoolFourier.hypercube n) → Real) → Real","labels":[],"detail_key":"p14","name":"LowDegreeTest.multi_expectation","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22208","layer":"formal","project":"p14","title":"LowDegreeTest.multi_expectation_const","kind":"theorem","summary":"∀ n k : Nat (c : Real), Eq (LowDegreeTest.multi_expectation fun x => c) c","labels":[],"detail_key":"p14","name":"LowDegreeTest.multi_expectation_const","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22209","layer":"formal","project":"p14","title":"LowDegreeTest.neg_char_is_degree_le","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)) d : Nat, GE.ge d 1 → LowDegreeTest.is_degree_le_pm1 (fun x => Ne…","labels":[],"detail_key":"p14","name":"LowDegreeTest.neg_char_is_degree_le","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22210","layer":"formal","project":"p14","title":"LowDegreeTest.neg_char_is_degree_le_one","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), LowDegreeTest.is_degree_le_pm1 (fun x => Neg.neg (BoolFourier.c…","labels":[],"detail_key":"p14","name":"LowDegreeTest.neg_char_is_degree_le_one","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22211","layer":"formal","project":"p14","title":"LowDegreeTest.neg_fourier_coeff_le_of_far_from_degree","kind":"theorem","summary":"∀ n d : Nat, GE.ge d 1 → ∀ (f : BoolFourier.hypercube n → Bool) (ε : Real), LowDegreeTest.epsil…","labels":[],"detail_key":"p14","name":"LowDegreeTest.neg_fourier_coeff_le_of_far_from_degree","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22212","layer":"formal","project":"p14","title":"LowDegreeTest.one_mem_ReedMuller","kind":"theorem","summary":"∀ n : Nat (d : Nat), Membership.mem (LowDegreeTest.ReedMuller d n) fun x => true","labels":[],"detail_key":"p14","name":"LowDegreeTest.one_mem_ReedMuller","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22213","layer":"formal","project":"p14","title":"LowDegreeTest.parseval_pm1","kind":"theorem","summary":"∀ n : Nat (f : BoolFourier.hypercube n → Bool), Eq (Finset.univ.sum fun S => HPow.hPow (BoolFou…","labels":[],"detail_key":"p14","name":"LowDegreeTest.parseval_pm1","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22214","layer":"formal","project":"p14","title":"LowDegreeTest.sq_one_sub_two_eps_le","kind":"theorem","summary":"∀ ε : Real, LE.le 0 ε → LE.le ε (1 / 2) → LE.le (HPow.hPow (HSub.hSub 1 (HMul.hMul 2 ε)) 2) (HS…","labels":[],"detail_key":"p14","name":"LowDegreeTest.sq_one_sub_two_eps_le","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22215","layer":"formal","project":"p14","title":"LowDegreeTest.xor_vec_assoc","kind":"theorem","summary":"∀ n : Nat (x y z : BoolFourier.hypercube n), Eq (BoolFourier.xor_vec (BoolFourier.xor_vec x y)…","labels":[],"detail_key":"p14","name":"LowDegreeTest.xor_vec_assoc","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22216","layer":"formal","project":"p14","title":"LowDegreeTest.xor_vec_comm","kind":"theorem","summary":"∀ n : Nat (x y : BoolFourier.hypercube n), Eq (BoolFourier.xor_vec x y) (BoolFourier.xor_vec y…","labels":[],"detail_key":"p14","name":"LowDegreeTest.xor_vec_comm","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22217","layer":"formal","project":"p14","title":"LowDegreeTest.xor_vec_self","kind":"theorem","summary":"∀ n : Nat (x : BoolFourier.hypercube n), Eq (BoolFourier.xor_vec x x) (BoolFourier.zero_vec n)","labels":[],"detail_key":"p14","name":"LowDegreeTest.xor_vec_self","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22218","layer":"formal","project":"p14","title":"LowDegreeTest.xor_vec_zero","kind":"theorem","summary":"∀ n : Nat (x : BoolFourier.hypercube n), Eq (BoolFourier.xor_vec x (BoolFourier.zero_vec n)) x","labels":[],"detail_key":"p14","name":"LowDegreeTest.xor_vec_zero","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22219","layer":"formal","project":"p14","title":"LowDegreeTest.zero_mem_ReedMuller","kind":"theorem","summary":"∀ n : Nat (d : Nat), Membership.mem (LowDegreeTest.ReedMuller d n) fun x => false","labels":[],"detail_key":"p14","name":"LowDegreeTest.zero_mem_ReedMuller","module":"TCSlib.BooleanAnalysis.BLR.LowDegree"},{"id":"n22220","layer":"formal","project":"p14","title":"ZkBLR.BLR_accept_prob","kind":"def","summary":"k : Nat → [NeZero k] → n : Nat → (ZkFourier.ZkVec k n → ZMod k) → Real","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_accept_prob","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22221","layer":"formal","project":"p14","title":"ZkBLR.BLR_accept_prob_eq_fourier_sum","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), Eq (ZkBLR.BLR_accept_pr…","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_accept_prob_eq_fourier_sum","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22222","layer":"formal","project":"p14","title":"ZkBLR.BLR_completeness","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), ZkBLR.is_linear f → Eq…","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_completeness","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22223","layer":"formal","project":"p14","title":"ZkBLR.BLR_soundness","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p) (ε : Real), Zk…","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_soundness","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22224","layer":"formal","project":"p14","title":"ZkBLR.BLR_soundness_general","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat, LE.le 2 k → ∀ (f : ZkFourier.ZkVec k n → ZMod k) (ε : Real…","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_soundness_general","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22225","layer":"formal","project":"p14","title":"ZkBLR.BLR_soundness_prime","kind":"theorem","summary":"∀ n p : Nat [inst : Fact (Nat.Prime p)] (f : ZkFourier.ZkVec p n → ZMod p) (ε : Real), ZkBLR.ep…","labels":[],"detail_key":"p14","name":"ZkBLR.BLR_soundness_prime","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22226","layer":"formal","project":"p14","title":"ZkBLR.cube_sum_bound_by_max","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p) (A : Real), (∀…","labels":[],"detail_key":"p14","name":"ZkBLR.cube_sum_bound_by_max","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22227","layer":"formal","project":"p14","title":"ZkBLR.epsilon_far_from_linear","kind":"def","summary":"k : Nat → [NeZero k] → n : Nat → (ZkFourier.ZkVec k n → ZMod k) → Real → Prop","labels":[],"detail_key":"p14","name":"ZkBLR.epsilon_far_from_linear","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22228","layer":"formal","project":"p14","title":"ZkBLR.geom_sum_toOmega_dual","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] (a : ZMod k), Eq (Finset.univ.sum fun j => ZkFourier.toOmega (HMul.…","labels":[],"detail_key":"p14","name":"ZkBLR.geom_sum_toOmega_dual","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22229","layer":"formal","project":"p14","title":"ZkBLR.indicator_eq_char_sum_re","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] (a : ZMod k), Eq (ite (Eq a 0) 1 0) (HMul.hMul (HDiv.hDiv 1 ↑k) (Fi…","labels":[],"detail_key":"p14","name":"ZkBLR.indicator_eq_char_sum_re","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22230","layer":"formal","project":"p14","title":"ZkBLR.lift_omega_j_zero_contribution","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), Eq (Finset.univ.sum fun…","labels":[],"detail_key":"p14","name":"ZkBLR.lift_omega_j_zero_contribution","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22231","layer":"formal","project":"p14","title":"ZkBLR.linear_iff_character","kind":"theorem","summary":"∀ k : Nat [NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), Iff (ZkBLR.is_linear f) (Exist…","labels":[],"detail_key":"p14","name":"ZkBLR.linear_iff_character","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22232","layer":"formal","project":"p14","title":"ZkBLR.linear_normalized","kind":"theorem","summary":"∀ k : Nat [NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), ZkBLR.is_linear f → ZkBLR.norm…","labels":[],"detail_key":"p14","name":"ZkBLR.linear_normalized","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22233","layer":"formal","project":"p14","title":"ZkBLR.normalize","kind":"def","summary":"k n : Nat → (ZkFourier.ZkVec k n → ZMod k) → ZkFourier.ZkVec k n → ZMod k","labels":[],"detail_key":"p14","name":"ZkBLR.normalize","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22234","layer":"formal","project":"p14","title":"ZkBLR.normalize_zero","kind":"theorem","summary":"∀ k : Nat [NeZero k] n : Nat (f : ZkFourier.ZkVec k n → ZMod k), Eq (ZkBLR.normalize f 0) 0","labels":[],"detail_key":"p14","name":"ZkBLR.normalize_zero","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22235","layer":"formal","project":"p14","title":"ZkBLR.parseval_lift_omega","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p), Eq (Finset.un…","labels":[],"detail_key":"p14","name":"ZkBLR.parseval_lift_omega","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22236","layer":"formal","project":"p14","title":"ZkBLR.parseval_lift_omega_j","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (j : ZMod k) (f : ZkFourier.ZkVec k n → ZMod k), Eq (Finset…","labels":[],"detail_key":"p14","name":"ZkBLR.parseval_lift_omega_j","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22237","layer":"formal","project":"p14","title":"ZkBLR.re_fourier_coeff_lift_omega_j_bound","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p) (ε : Real), Zk…","labels":[],"detail_key":"p14","name":"ZkBLR.re_fourier_coeff_lift_omega_j_bound","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22238","layer":"formal","project":"p14","title":"ZkBLR.triple_product_fourier","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (j : ZMod k) (f : ZkFourier.ZkVec k n → ZMod k), Eq (ZkFour…","labels":[],"detail_key":"p14","name":"ZkBLR.triple_product_fourier","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22239","layer":"formal","project":"p14","title":"ZkBLR.weighted_fourier_sum_bound","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p) (ε : Real), Zk…","labels":[],"detail_key":"p14","name":"ZkBLR.weighted_fourier_sum_bound","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22240","layer":"formal","project":"p14","title":"ZkBLR.weighted_fourier_sum_le_one","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (j : ZMod k) (f : ZkFourier.ZkVec k n → ZMod k), LE.le (Fin…","labels":[],"detail_key":"p14","name":"ZkBLR.weighted_fourier_sum_le_one","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22241","layer":"formal","project":"p14","title":"ZkBLR.weighted_sum_lift_omega_j_bound","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] n : Nat (f : ZkFourier.ZkVec p n → ZMod p) (ε : Real), Zk…","labels":[],"detail_key":"p14","name":"ZkBLR.weighted_sum_lift_omega_j_bound","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22242","layer":"formal","project":"p14","title":"ZkBLR.weighted_sum_unit_bound","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat, LE.le 2 k → ∀ (f : ZkFourier.ZkVec k n → ZMod k) (ε : Real…","labels":[],"detail_key":"p14","name":"ZkBLR.weighted_sum_unit_bound","module":"TCSlib.BooleanAnalysis.BLR.ZkBLR"},{"id":"n22243","layer":"formal","project":"p14","title":"ZkFourier.L2_norm_sq","kind":"def","summary":"k : Nat → [NeZero k] → n : Nat → ZkFourier.ZkFun k n → Real","labels":[],"detail_key":"p14","name":"ZkFourier.L2_norm_sq","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22244","layer":"formal","project":"p14","title":"ZkFourier.card_ZkVec","kind":"theorem","summary":"∀ (k : Nat) [inst : NeZero k] (n : Nat), Eq (Fintype.card (ZkFourier.ZkVec k n)) (HPow.hPow k n)","labels":[],"detail_key":"p14","name":"ZkFourier.card_ZkVec","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22245","layer":"formal","project":"p14","title":"ZkFourier.char_s_add","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s x y : ZkFourier.ZkVec k n), Eq (ZkFourier.char_s s (HAdd…","labels":[],"detail_key":"p14","name":"ZkFourier.char_s_add","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22246","layer":"formal","project":"p14","title":"ZkFourier.char_s_conj","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s x : ZkFourier.ZkVec k n), Eq ((starRingEnd Complex) (ZkF…","labels":[],"detail_key":"p14","name":"ZkFourier.char_s_conj","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22247","layer":"formal","project":"p14","title":"ZkFourier.char_s_mul","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s t x : ZkFourier.ZkVec k n), Eq (HMul.hMul (ZkFourier.cha…","labels":[],"detail_key":"p14","name":"ZkFourier.char_s_mul","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22248","layer":"formal","project":"p14","title":"ZkFourier.char_s_ne_zero","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s x : ZkFourier.ZkVec k n), Ne (ZkFourier.char_s s x) 0","labels":[],"detail_key":"p14","name":"ZkFourier.char_s_ne_zero","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22249","layer":"formal","project":"p14","title":"ZkFourier.convolution","kind":"def","summary":"k : Nat → [NeZero k] → n : Nat → ZkFourier.ZkFun k n → ZkFourier.ZkFun k n → ZkFourier.ZkFun k n","labels":[],"detail_key":"p14","name":"ZkFourier.convolution","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22250","layer":"formal","project":"p14","title":"ZkFourier.expectation_char_nontrivial","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat s : ZkFourier.ZkVec k n, Ne s 0 → Eq (ZkFourier.expectation…","labels":[],"detail_key":"p14","name":"ZkFourier.expectation_char_nontrivial","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22251","layer":"formal","project":"p14","title":"ZkFourier.expectation_char_trivial","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat, Eq (ZkFourier.expectation (ZkFourier.char_s 0)) 1","labels":[],"detail_key":"p14","name":"ZkFourier.expectation_char_trivial","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22252","layer":"formal","project":"p14","title":"ZkFourier.fourier_coeff_char_of_ne","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat s t : ZkFourier.ZkVec k n, Ne s t → Eq (ZkFourier.fourier_c…","labels":[],"detail_key":"p14","name":"ZkFourier.fourier_coeff_char_of_ne","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22253","layer":"formal","project":"p14","title":"ZkFourier.fourier_coeff_char_self","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s : ZkFourier.ZkVec k n), Eq (ZkFourier.fourier_coeff (ZkF…","labels":[],"detail_key":"p14","name":"ZkFourier.fourier_coeff_char_self","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22254","layer":"formal","project":"p14","title":"ZkFourier.fourier_coeff_convolution","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f g : ZkFourier.ZkFun k n) (s : ZkFourier.ZkVec k n), Eq (…","labels":[],"detail_key":"p14","name":"ZkFourier.fourier_coeff_convolution","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22255","layer":"formal","project":"p14","title":"ZkFourier.fourier_expansion","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f : ZkFourier.ZkFun k n) (x : ZkFourier.ZkVec k n), Eq (f…","labels":[],"detail_key":"p14","name":"ZkFourier.fourier_expansion","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22256","layer":"formal","project":"p14","title":"ZkFourier.inner_product_char_nonself","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat s t : ZkFourier.ZkVec k n, Ne s t → Eq (ZkFourier.inner_pro…","labels":[],"detail_key":"p14","name":"ZkFourier.inner_product_char_nonself","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22257","layer":"formal","project":"p14","title":"ZkFourier.inner_product_char_self","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s : ZkFourier.ZkVec k n), Eq (ZkFourier.inner_product (ZkF…","labels":[],"detail_key":"p14","name":"ZkFourier.inner_product_char_self","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22258","layer":"formal","project":"p14","title":"ZkFourier.norm_char_s","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (s x : ZkFourier.ZkVec k n), Eq (norm (ZkFourier.char_s s x…","labels":[],"detail_key":"p14","name":"ZkFourier.norm_char_s","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22259","layer":"formal","project":"p14","title":"ZkFourier.parseval_identity","kind":"theorem","summary":"∀ k : Nat [inst : NeZero k] n : Nat (f : ZkFourier.ZkFun k n), Eq (Finset.univ.sum fun s => HPo…","labels":[],"detail_key":"p14","name":"ZkFourier.parseval_identity","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22260","layer":"formal","project":"p14","title":"ZkFourier.zkDot_neg_left","kind":"theorem","summary":"∀ k : Nat [NeZero k] n : Nat (s x : ZkFourier.ZkVec k n), Eq (ZkFourier.zkDot (Neg.neg s) x) (N…","labels":[],"detail_key":"p14","name":"ZkFourier.zkDot_neg_left","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22261","layer":"formal","project":"p14","title":"ZkFourier.zkDot_sub","kind":"theorem","summary":"∀ k : Nat [NeZero k] n : Nat (s t x : ZkFourier.ZkVec k n), Eq (ZkFourier.zkDot (HSub.hSub s t)…","labels":[],"detail_key":"p14","name":"ZkFourier.zkDot_sub","module":"TCSlib.BooleanAnalysis.BLR.ZkFourier"},{"id":"n22262","layer":"formal","project":"p14","title":"BooleanAnalysis.moment_eq_expect","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n) (p : Nat) (P : MeasureTheory.Measure (BooleanAnal…","labels":[],"detail_key":"p14","name":"BooleanAnalysis.moment_eq_expect","module":"TCSlib.BooleanAnalysis.Basic"},{"id":"n22263","layer":"formal","project":"p14","title":"BooleanAnalysis.noiseOp_self_adjoint","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f g : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.innerProduct (B…","labels":[],"detail_key":"p14","name":"BooleanAnalysis.noiseOp_self_adjoint","module":"TCSlib.BooleanAnalysis.Basic"},{"id":"n22264","layer":"formal","project":"p14","title":"BooleanAnalysis.plancherel","kind":"theorem","summary":"∀ n : Nat (f g : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.innerProduct f g) (Finset.…","labels":[],"detail_key":"p14","name":"BooleanAnalysis.plancherel","module":"TCSlib.BooleanAnalysis.Basic"},{"id":"n22265","layer":"formal","project":"p14","title":"Bonami.IsBReasonable","kind":"def","summary":"Ω : Type u_1 → [inst : MeasurableSpace Ω] → (Ω → Real) → MeasureTheory.Measure Ω → Real → Prop","labels":[],"detail_key":"p14","name":"Bonami.IsBReasonable","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22266","layer":"formal","project":"p14","title":"Bonami.avgLast","kind":"def","summary":"n : Nat → BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1) → BooleanAnalysis.BooleanFunc 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LE.le A (HMul.hMul…","labels":[],"detail_key":"p14","name":"Bonami.bonami_algebra","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22270","layer":"formal","project":"p14","title":"Bonami.bonami_expect","kind":"theorem","summary":"∀ n : Nat (k : Nat) (f : BooleanAnalysis.BooleanFunc n), BooleanAnalysis.has_degree_at_most f k…","labels":[],"detail_key":"p14","name":"Bonami.bonami_expect","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22271","layer":"formal","project":"p14","title":"Bonami.bonami_lemma","kind":"theorem","summary":"∀ n : Nat (k : Nat) (f : BooleanAnalysis.BooleanFunc n), BooleanAnalysis.has_degree_at_most f k…","labels":[],"detail_key":"p14","name":"Bonami.bonami_lemma","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22272","layer":"formal","project":"p14","title":"Bonami.degree_avgLast","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (k : Nat), BooleanAnalysis.has_degr…","labels":[],"detail_key":"p14","name":"Bonami.degree_avgLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22273","layer":"formal","project":"p14","title":"Bonami.degree_diffLast","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (k : Nat), BooleanAnalysis.has_degr…","labels":[],"detail_key":"p14","name":"Bonami.degree_diffLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22274","layer":"formal","project":"p14","title":"Bonami.degree_zero_const","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), BooleanAnalysis.has_degree_at_most f 0 → ∀ (x :…","labels":[],"detail_key":"p14","name":"Bonami.degree_zero_const","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22275","layer":"formal","project":"p14","title":"Bonami.degree_zero_fourth_moment","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), BooleanAnalysis.has_degree_at_most f 0 → Eq (Boo…","labels":[],"detail_key":"p14","name":"Bonami.degree_zero_fourth_moment","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22276","layer":"formal","project":"p14","title":"Bonami.diffLast","kind":"def","summary":"n : Nat → BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1) → BooleanAnalysis.BooleanFunc n","labels":[],"detail_key":"p14","name":"Bonami.diffLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22277","layer":"formal","project":"p14","title":"Bonami.expect_cs_sq","kind":"theorem","summary":"∀ n : Nat (g h : BooleanAnalysis.BooleanFunc n), LE.le (HPow.hPow (BooleanAnalysis.expect fun x…","labels":[],"detail_key":"p14","name":"Bonami.expect_cs_sq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22278","layer":"formal","project":"p14","title":"Bonami.expect_fourth_nonneg","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), LE.le 0 (BooleanAnalysis.expect fun x => HPow.hP…","labels":[],"detail_key":"p14","name":"Bonami.expect_fourth_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22279","layer":"formal","project":"p14","title":"Bonami.expect_sq_nonneg","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), LE.le 0 (BooleanAnalysis.expect fun x => HPow.hP…","labels":[],"detail_key":"p14","name":"Bonami.expect_sq_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22280","layer":"formal","project":"p14","title":"Bonami.expect_sq_nonneg_prod","kind":"theorem","summary":"∀ n : Nat (g h : BooleanAnalysis.BooleanFunc n), LE.le 0 (BooleanAnalysis.expect fun x => HMul.…","labels":[],"detail_key":"p14","name":"Bonami.expect_sq_nonneg_prod","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22281","layer":"formal","project":"p14","title":"Bonami.expect_succ_eq","kind":"theorem","summary":"∀ n : Nat (φ : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.expect φ) (HDi…","labels":[],"detail_key":"p14","name":"Bonami.expect_succ_eq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22282","layer":"formal","project":"p14","title":"Bonami.fourierCoeff_avgLast","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (S : Finset (Fin n)), Eq (BooleanAn…","labels":[],"detail_key":"p14","name":"Bonami.fourierCoeff_avgLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22283","layer":"formal","project":"p14","title":"Bonami.fourierCoeff_diffLast","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (S : Finset (Fin n)), Eq (BooleanAn…","labels":[],"detail_key":"p14","name":"Bonami.fourierCoeff_diffLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22284","layer":"formal","project":"p14","title":"Bonami.fourth_moment_decomp","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.expect fun x =…","labels":[],"detail_key":"p14","name":"Bonami.fourth_moment_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22285","layer":"formal","project":"p14","title":"Bonami.fourth_pow_sum","kind":"theorem","summary":"∀ (a b : Real), Eq (HAdd.hAdd (HPow.hPow (HAdd.hAdd a b) 4) (HPow.hPow (HSub.hSub a b) 4)) (HMu…","labels":[],"detail_key":"p14","name":"Bonami.fourth_pow_sum","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22286","layer":"formal","project":"p14","title":"Bonami.min_prob_b_reasonable","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] [Fintype Ω] [DiscreteMeasurableSpace Ω] P : MeasureTh…","labels":[],"detail_key":"p14","name":"Bonami.min_prob_b_reasonable","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22287","layer":"formal","project":"p14","title":"Bonami.moment_eq_expect","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n) (p : Nat) (P : MeasureTheory.Measure (BooleanAnal…","labels":[],"detail_key":"p14","name":"Bonami.moment_eq_expect","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22288","layer":"formal","project":"p14","title":"Bonami.paley_zygmund_ineq","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] μ : MeasureTheory.Measure Ω [MeasureTheory.IsProbabil…","labels":[],"detail_key":"p14","name":"Bonami.paley_zygmund_ineq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22289","layer":"formal","project":"p14","title":"Bonami.restrictLast","kind":"def","summary":"n : Nat → BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1) → Bool → BooleanAnalysis.BooleanFunc n","labels":[],"detail_key":"p14","name":"Bonami.restrictLast","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22290","layer":"formal","project":"p14","title":"Bonami.restrictLast_false_eq","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (x : BooleanAnalysis.BoolCube n), E…","labels":[],"detail_key":"p14","name":"Bonami.restrictLast_false_eq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22291","layer":"formal","project":"p14","title":"Bonami.restrictLast_true_eq","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (x : BooleanAnalysis.BoolCube n), E…","labels":[],"detail_key":"p14","name":"Bonami.restrictLast_true_eq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22292","layer":"formal","project":"p14","title":"Bonami.second_moment_decomp","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.expect fun x =…","labels":[],"detail_key":"p14","name":"Bonami.second_moment_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22293","layer":"formal","project":"p14","title":"Bonami.second_pow_sum","kind":"theorem","summary":"∀ (a b : Real), Eq (HAdd.hAdd (HPow.hPow (HAdd.hAdd a b) 2) (HPow.hPow (HSub.hSub a b) 2)) (HMu…","labels":[],"detail_key":"p14","name":"Bonami.second_pow_sum","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22294","layer":"formal","project":"p14","title":"Bonami.sum_boolCube_succ","kind":"theorem","summary":"∀ n : Nat (φ : BooleanAnalysis.BoolCube (HAdd.hAdd n 1) → Real), Eq (Finset.univ.sum fun x => φ…","labels":[],"detail_key":"p14","name":"Bonami.sum_boolCube_succ","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22295","layer":"formal","project":"p14","title":"Bonami.uniformMeasure","kind":"def","summary":"(n : Nat) → MeasureTheory.Measure (BooleanAnalysis.BoolCube n)","labels":[],"detail_key":"p14","name":"Bonami.uniformMeasure","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22296","layer":"formal","project":"p14","title":"Bonami.uniformMeasure_apply","kind":"theorem","summary":"∀ n : Nat (x : BooleanAnalysis.BoolCube n), Eq ((Bonami.uniformMeasure n) (singleton x)).toReal…","labels":[],"detail_key":"p14","name":"Bonami.uniformMeasure_apply","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22297","layer":"formal","project":"p14","title":"Bonami.uniformWeight_succ","kind":"theorem","summary":"∀ (n : Nat), Eq (BooleanAnalysis.uniformWeight (HAdd.hAdd n 1)) (HDiv.hDiv (BooleanAnalysis.uni…","labels":[],"detail_key":"p14","name":"Bonami.uniformWeight_succ","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Bonami"},{"id":"n22298","layer":"formal","project":"p14","title":"GeneralHypercontractivity.avg_rpow_ge_one","kind":"theorem","summary":"∀ p b : Real, LE.le 1 p → LE.le 0 b → LE.le b 1 → LE.le 1 (HDiv.hDiv (HAdd.hAdd (HPow.hPow (HAd…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.avg_rpow_ge_one","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22299","layer":"formal","project":"p14","title":"GeneralHypercontractivity.bridging_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p u : Real), LE.le 1 p → LE.le p 2 → LE.le 2 u → ∀ (f : BooleanAnalysis.BooleanFunc…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.bridging_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22300","layer":"formal","project":"p14","title":"GeneralHypercontractivity.convex_sym_sum_mono","kind":"theorem","summary":"∀ f : Real → Real, ConvexOn Real (Set.Ici 0) f → ∀ x y : Real, LE.le 0 x → LE.le x y → LE.le y…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.convex_sym_sum_mono","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22301","layer":"formal","project":"p14","title":"GeneralHypercontractivity.corrExpect_mono","kind":"theorem","summary":"∀ n : Nat ρ : Real, LE.le 0 ρ → LE.le ρ 1 → ∀ h h' : BooleanAnalysis.BoolCube n → BooleanAnalys…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.corrExpect_mono","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22302","layer":"formal","project":"p14","title":"GeneralHypercontractivity.expect_succ_eq_iterated","kind":"theorem","summary":"∀ n : Nat (h : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.expect h) (Boo…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.expect_succ_eq_iterated","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22303","layer":"formal","project":"p14","title":"GeneralHypercontractivity.general_one_function_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p u : Real), LE.le 1 p → LE.le p u → LT.lt 1 u → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.general_one_function_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22304","layer":"formal","project":"p14","title":"GeneralHypercontractivity.general_two_function_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p u : Real), LE.le 1 p → LE.le p u → LE.le 2 u → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.general_two_function_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22305","layer":"formal","project":"p14","title":"GeneralHypercontractivity.h_alpha_ineq","kind":"theorem","summary":"∀ r s c t : Real, LE.le 0 r → LE.le r s → LE.le s 1 → Eq c (HDiv.hDiv r s).sqrt → LE.le 0 t → L…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.h_alpha_ineq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22306","layer":"formal","project":"p14","title":"GeneralHypercontractivity.high_norms_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p u : Real), LE.le 2 p → LE.le p u → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → LE.le ρ (…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.high_norms_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22307","layer":"formal","project":"p14","title":"GeneralHypercontractivity.holder_ineq_bool","kind":"theorem","summary":"∀ n : Nat (p : Real), LT.lt 1 p → ∀ (f h : BooleanAnalysis.BooleanFunc n), LE.le (BooleanAnalys…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.holder_ineq_bool","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22308","layer":"formal","project":"p14","title":"GeneralHypercontractivity.hypercontractivity_induction","kind":"theorem","summary":"∀ (p q : Real), LE.le 1 p → LE.le 1 q → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → (∀ (f g : Boolean…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.hypercontractivity_induction","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22309","layer":"formal","project":"p14","title":"GeneralHypercontractivity.innerProduct_noiseOp_eq_weighted_sum","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f g : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.innerProduct f…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.innerProduct_noiseOp_eq_weighted_sum","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22310","layer":"formal","project":"p14","title":"GeneralHypercontractivity.integrated_h_alpha_ineq","kind":"theorem","summary":"∀ p q b : Real, LE.le 1 p → LE.le p q → LE.le q 2 → LE.le 0 b → LE.le b 1 → have ρ := (HDiv.hDi…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.integrated_h_alpha_ineq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22311","layer":"formal","project":"p14","title":"GeneralHypercontractivity.low_norms_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p u : Real), LT.lt 1 p → LE.le p u → LE.le u 2 → ∀ (f : BooleanAnalysis.BooleanFunc…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.low_norms_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22312","layer":"formal","project":"p14","title":"GeneralHypercontractivity.low_norms_one_bit","kind":"theorem","summary":"∀ (p q : Real), LT.lt 1 p → LE.le p q → LE.le q 2 → ∀ (f : BooleanAnalysis.BooleanFunc 1), LE.l…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.low_norms_one_bit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22313","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseKernel","kind":"def","summary":"Real → n : Nat → BooleanAnalysis.BoolCube n → BooleanAnalysis.BoolCube n → Real","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseKernel","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22314","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseKernel_nonneg","kind":"theorem","summary":"∀ n : Nat ρ : Real, LE.le 0 ρ → LE.le ρ 1 → ∀ (x y : BooleanAnalysis.BoolCube n), LE.le 0 (Gene…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseKernel_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22315","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseKernel_snoc","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (x' y' : BooleanAnalysis.BoolCube n) (b b' : Bool), Eq (GeneralHypercontra…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseKernel_snoc","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22316","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseKernel_sum_left","kind":"theorem","summary":"∀ n : Nat ρ : Real, LE.le 0 ρ → LE.le ρ 1 → ∀ (y : BooleanAnalysis.BoolCube n), Eq (Finset.univ…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseKernel_sum_left","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22317","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseKernel_sum_right","kind":"theorem","summary":"∀ n : Nat ρ : Real, LE.le 0 ρ → LE.le ρ 1 → ∀ (x : BooleanAnalysis.BoolCube n), Eq (Finset.univ…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseKernel_sum_right","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22318","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseOp_abs_rpow_le_kernel_avg","kind":"theorem","summary":"∀ n : Nat ρ : Real, LE.le 0 ρ → LE.le ρ 1 → ∀ (s : Real), LE.le 1 s → ∀ (f : BooleanAnalysis.Bo…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseOp_abs_rpow_le_kernel_avg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22319","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noiseOp_eq_kernel_sum","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (g : BooleanAnalysis.BooleanFunc n) (x : BooleanAnalysis.BoolCube n), Eq (…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noiseOp_eq_kernel_sum","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22320","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noise_Lp_contraction_one_bit","kind":"theorem","summary":"∀ (q : Real), LE.le 1 q → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → ∀ (g : BooleanAnalysis.BooleanF…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noise_Lp_contraction_one_bit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22321","layer":"formal","project":"p14","title":"GeneralHypercontractivity.noise_op_norm_dual","kind":"theorem","summary":"∀ n : Nat (p q : Real), LT.lt 1 p → LT.lt 1 q → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → Iff (∀ (f…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.noise_op_norm_dual","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22322","layer":"formal","project":"p14","title":"GeneralHypercontractivity.norm_collapse_clean","kind":"theorem","summary":"∀ n : Nat (p : Real), LE.le 1 p → ∀ (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (Bool…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.norm_collapse_clean","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22323","layer":"formal","project":"p14","title":"GeneralHypercontractivity.norm_collapse_rpow","kind":"theorem","summary":"∀ n : Nat (p : Real), LT.lt 0 p → ∀ (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (Bool…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.norm_collapse_rpow","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22324","layer":"formal","project":"p14","title":"GeneralHypercontractivity.one_bit_norm_slice","kind":"theorem","summary":"∀ n : Nat (p : Real), LT.lt 0 p → ∀ (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (x' : Boo…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.one_bit_norm_slice","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22325","layer":"formal","project":"p14","title":"GeneralHypercontractivity.one_bit_slice_eq_innerProduct","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f g : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (x' y' : BooleanAnalys…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.one_bit_slice_eq_innerProduct","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22326","layer":"formal","project":"p14","title":"GeneralHypercontractivity.one_function_iff_two_function_hypercontractivity","kind":"theorem","summary":"∀ n : Nat (p q : Real), LE.le 1 p → LE.le p q → LE.le 2 q → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.one_function_iff_two_function_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22327","layer":"formal","project":"p14","title":"GeneralHypercontractivity.rpow_ge_one_add_mul_sub","kind":"theorem","summary":"∀ x r : Real, LE.le 0 x → LE.le 1 r → GE.ge (HPow.hPow x r) (HAdd.hAdd 1 (HMul.hMul r (HSub.hSu…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.rpow_ge_one_add_mul_sub","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22328","layer":"formal","project":"p14","title":"GeneralHypercontractivity.rpow_sum_antitone_exponent","kind":"theorem","summary":"∀ p q x : Real, LT.lt 0 x → LT.lt x 1 → LE.le p 0 → LE.le p q → LE.le q 0 → GE.ge (HAdd.hAdd (H…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.rpow_sum_antitone_exponent","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22329","layer":"formal","project":"p14","title":"GeneralHypercontractivity.sum_fourier_kernel","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (x y : BooleanAnalysis.BoolCube n), Eq (Finset.univ.sum fun S => HMul.hMul…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.sum_fourier_kernel","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22330","layer":"formal","project":"p14","title":"GeneralHypercontractivity.trivial_contractivity","kind":"theorem","summary":"∀ n : Nat (s : Real), LE.le 1 s → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → ∀ (f : BooleanAnalysis.…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.trivial_contractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22331","layer":"formal","project":"p14","title":"GeneralHypercontractivity.two_func_hyp_succ","kind":"theorem","summary":"∀ n : Nat (p q : Real), LE.le 1 p → LE.le 1 q → ∀ (ρ : Real), LE.le 0 ρ → LE.le ρ 1 → (∀ (f g :…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.two_func_hyp_succ","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22332","layer":"formal","project":"p14","title":"GeneralHypercontractivity.two_func_hyp_zero","kind":"theorem","summary":"∀ (p q : Real), LE.le 1 p → LE.le 1 q → ∀ (ρ : Real) (f g : BooleanAnalysis.BooleanFunc 0), LE.…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.two_func_hyp_zero","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22333","layer":"formal","project":"p14","title":"GeneralHypercontractivity.two_point_ineq_general_unit","kind":"theorem","summary":"∀ (b p q : Real), LE.le 1 p → LE.le p q → LE.le q 2 → LE.le 0 b → LE.le b 1 → have ρ := (HDiv.h…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.two_point_ineq_general_unit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22334","layer":"formal","project":"p14","title":"GeneralHypercontractivity.weak_two_function_hypercontractivity","kind":"theorem","summary":"∀ (p q : Real), LE.le 1 p → LE.le p 2 → LE.le 1 q → LE.le q 2 → ∀ n : Nat (f g : BooleanAnalysi…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.weak_two_function_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22335","layer":"formal","project":"p14","title":"GeneralHypercontractivity.weak_two_function_hypercontractivity_one_bit","kind":"theorem","summary":"∀ (p q : Real), LE.le 1 p → LE.le p 2 → LE.le 1 q → LE.le q 2 → ∀ (f g : BooleanAnalysis.Boolea…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.weak_two_function_hypercontractivity_one_bit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22336","layer":"formal","project":"p14","title":"GeneralHypercontractivity.weighted_sum_succ_decomp","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (F : BooleanAnalysis.BoolCube (HAdd.hAdd n 1) → BooleanAnalysis.BoolCube (…","labels":[],"detail_key":"p14","name":"GeneralHypercontractivity.weighted_sum_succ_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.General"},{"id":"n22337","layer":"formal","project":"p14","title":"OneBit.cauchy_schwarz_bool","kind":"theorem","summary":"∀ n : Nat (f g : BooleanAnalysis.BooleanFunc n), LE.le (BooleanAnalysis.innerProduct f g) (HMul…","labels":[],"detail_key":"p14","name":"OneBit.cauchy_schwarz_bool","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22338","layer":"formal","project":"p14","title":"OneBit.expect_abs_rpow_one_bit","kind":"theorem","summary":"∀ (p : Real) (f : BooleanAnalysis.BooleanFunc 1), Eq (BooleanAnalysis.expect fun x => HPow.hPow…","labels":[],"detail_key":"p14","name":"OneBit.expect_abs_rpow_one_bit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22339","layer":"formal","project":"p14","title":"OneBit.expect_noiseOp_sq_one_bit","kind":"theorem","summary":"∀ (ρ : Real) (f : BooleanAnalysis.BooleanFunc 1), Eq (BooleanAnalysis.expect fun x => HPow.hPow…","labels":[],"detail_key":"p14","name":"OneBit.expect_noiseOp_sq_one_bit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22340","layer":"formal","project":"p14","title":"OneBit.expect_nonneg_of_nonneg","kind":"theorem","summary":"∀ n : Nat f : BooleanAnalysis.BooleanFunc n, (∀ (x : BooleanAnalysis.BoolCube n), LE.le 0 (f x)…","labels":[],"detail_key":"p14","name":"OneBit.expect_nonneg_of_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22341","layer":"formal","project":"p14","title":"OneBit.holder_sharpness","kind":"theorem","summary":"∀ n : Nat p q : Real, p.HolderConjugate q → ∀ (u : BooleanAnalysis.BooleanFunc n), Exists fun f…","labels":[],"detail_key":"p14","name":"OneBit.holder_sharpness","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22342","layer":"formal","project":"p14","title":"OneBit.lp_norm_mono","kind":"theorem","summary":"∀ n : Nat (r s : Real), LE.le 1 r → LE.le r s → ∀ (f : BooleanAnalysis.BooleanFunc n), LE.le (H…","labels":[],"detail_key":"p14","name":"OneBit.lp_norm_mono","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22343","layer":"formal","project":"p14","title":"OneBit.noise_l2_abs_mono","kind":"theorem","summary":"∀ (a b ρ : Real), LE.le 0 ρ → LE.le ρ 1 → LE.le (HAdd.hAdd (HPow.hPow a 2) (HMul.hMul (HPow.hPo…","labels":[],"detail_key":"p14","name":"OneBit.noise_l2_abs_mono","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22344","layer":"formal","project":"p14","title":"OneBit.noise_operator_duality","kind":"theorem","summary":"∀ p p_conj : Real, p.HolderConjugate p_conj → LE.le 1 p → (∀ (f : BooleanAnalysis.BooleanFunc 1…","labels":[],"detail_key":"p14","name":"OneBit.noise_operator_duality","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22345","layer":"formal","project":"p14","title":"OneBit.one_bit_2q_hypercontractivity","kind":"theorem","summary":"∀ (q : Real), LE.le 2 q → ∀ (g : BooleanAnalysis.BooleanFunc 1), LE.le (HPow.hPow (BooleanAnaly…","labels":[],"detail_key":"p14","name":"OneBit.one_bit_2q_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22346","layer":"formal","project":"p14","title":"OneBit.one_bit_p2_hypercontractivity","kind":"theorem","summary":"∀ (p : Real), LE.le 1 p → LE.le p 2 → ∀ (ρ : Real), LE.le 0 ρ → LE.le (HPow.hPow ρ 2) (HSub.hSu…","labels":[],"detail_key":"p14","name":"OneBit.one_bit_p2_hypercontractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22347","layer":"formal","project":"p14","title":"OneBit.one_bit_val_false","kind":"theorem","summary":"∀ (f : BooleanAnalysis.BooleanFunc 1), Eq (f fun x => false) (HAdd.hAdd (BooleanAnalysis.fourie…","labels":[],"detail_key":"p14","name":"OneBit.one_bit_val_false","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22348","layer":"formal","project":"p14","title":"OneBit.one_bit_val_true","kind":"theorem","summary":"∀ (f : BooleanAnalysis.BooleanFunc 1), Eq (f fun x => true) (HSub.hSub (BooleanAnalysis.fourier…","labels":[],"detail_key":"p14","name":"OneBit.one_bit_val_true","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22349","layer":"formal","project":"p14","title":"OneBit.two_point_ineq","kind":"theorem","summary":"∀ (a b p ρ : Real), LE.le 1 p → LE.le p 2 → LE.le 0 ρ → LE.le (HPow.hPow ρ 2) (HSub.hSub p 1) →…","labels":[],"detail_key":"p14","name":"OneBit.two_point_ineq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22350","layer":"formal","project":"p14","title":"OneBit.two_point_ineq_a_zero","kind":"theorem","summary":"∀ (p : Real), LE.le 1 p → LE.le p 2 → LE.le (HPow.hPow (HSub.hSub p 1) (HDiv.hDiv p 2)) 1","labels":[],"detail_key":"p14","name":"OneBit.two_point_ineq_a_zero","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22351","layer":"formal","project":"p14","title":"OneBit.two_point_ineq_unit","kind":"theorem","summary":"∀ (b p : Real), LE.le 1 p → LE.le p 2 → LE.le 0 b → LE.le b 1 → LE.le (HPow.hPow (HAdd.hAdd 1 (…","labels":[],"detail_key":"p14","name":"OneBit.two_point_ineq_unit","module":"TCSlib.BooleanAnalysis.Hypercontractivity.OneBit"},{"id":"n22352","layer":"formal","project":"p14","title":"SimpleHypercontractivity.binom_coeff_ineq","kind":"theorem","summary":"∀ (k : Nat), LE.le 1 k → ∀ (j : Nat), LE.le j k → LE.le ((HMul.hMul 2 k).choose (HMul.hMul 2 j)…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.binom_coeff_ineq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22353","layer":"formal","project":"p14","title":"SimpleHypercontractivity.card_image_castSucc","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), Eq (Finset.image Fin.castSucc S).card S.card","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.card_image_castSucc","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22354","layer":"formal","project":"p14","title":"SimpleHypercontractivity.card_image_castSucc_union_last","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)), Eq (Union.union (Finset.image Fin.castSucc S) (singleton (Fin.l…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.card_image_castSucc_union_last","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22355","layer":"formal","project":"p14","title":"SimpleHypercontractivity.chiS_snoc_castSucc","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)) (x : BooleanAnalysis.BoolCube n) (b : Bool), Eq (BooleanAnalysis…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.chiS_snoc_castSucc","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22356","layer":"formal","project":"p14","title":"SimpleHypercontractivity.chiS_snoc_with_last","kind":"theorem","summary":"∀ n : Nat (S : Finset (Fin n)) (x : BooleanAnalysis.BoolCube n) (b : Bool), Eq (BooleanAnalysis…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.chiS_snoc_with_last","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22357","layer":"formal","project":"p14","title":"SimpleHypercontractivity.contractivity","kind":"theorem","summary":"∀ n : Nat (ρ : Real), LE.le (HPow.hPow ρ 2) 1 → ∀ (f : BooleanAnalysis.BooleanFunc n), LE.le (B…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.contractivity","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22358","layer":"formal","project":"p14","title":"SimpleHypercontractivity.expect_rpow_abs_nonneg","kind":"theorem","summary":"∀ n : Nat (p : Real) (f : BooleanAnalysis.BooleanFunc n), LE.le 0 (BooleanAnalysis.expect fun x…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.expect_rpow_abs_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22359","layer":"formal","project":"p14","title":"SimpleHypercontractivity.expect_sq_noiseOp_nonneg","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f : BooleanAnalysis.BooleanFunc n), LE.le 0 (BooleanAnalysis.expect fun x…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.expect_sq_noiseOp_nonneg","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22360","layer":"formal","project":"p14","title":"SimpleHypercontractivity.finset_fin_succ_sum_partition","kind":"theorem","summary":"∀ n : Nat (φ : Finset (Fin (HAdd.hAdd n 1)) → Real), Eq (Finset.univ.sum fun S => φ S) (HAdd.hA…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.finset_fin_succ_sum_partition","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22361","layer":"formal","project":"p14","title":"SimpleHypercontractivity.fourth_moment_noise_decomp","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.exp…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.fourth_moment_noise_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22362","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_2","kind":"theorem","summary":"∀ n : Nat (ρ : Real), LE.le (HPow.hPow ρ 2) 1 → ∀ (f : BooleanAnalysis.BooleanFunc n), LE.le (B…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_2","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22363","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_2k","kind":"theorem","summary":"∀ n : Nat (k : Nat), LE.le 1 k → ∀ (ρ : Real), LE.le (HPow.hPow ρ 2) (HDiv.hDiv 1 (HSub.hSub (H…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_2k","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22364","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_2k_rpow","kind":"theorem","summary":"∀ n : Nat (k : Nat), LE.le 1 k → ∀ (ρ : Real), LE.le (HPow.hPow ρ 2) (HDiv.hDiv 1 (HSub.hSub (H…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_2k_rpow","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22365","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_4","kind":"theorem","summary":"∀ n : Nat (ρ : Real), LE.le (HPow.hPow ρ 2) (1 / 3) → ∀ (f : BooleanAnalysis.BooleanFunc n), LE…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_4","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22366","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_6","kind":"theorem","summary":"∀ n : Nat (ρ : Real), LE.le (HPow.hPow ρ 2) (1 / 5) → ∀ (f : BooleanAnalysis.BooleanFunc n), LE…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_6","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22367","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_2_q","kind":"theorem","summary":"∀ n : Nat (q : Nat), LE.le 2 q → Even q → ∀ (ρ : Real), LE.le (HPow.hPow ρ 2) (HDiv.hDiv 1 (HSu…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_2_q","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22368","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_4_div_3_2","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), LE.le (HPow.hPow (BooleanAnalysis.expect fun x =…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_4_div_3_2","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22369","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_algebra'","kind":"theorem","summary":"∀ a b A B C ρ : Real, LE.le 0 a → LE.le 0 b → LE.le 0 B → LE.le A (HPow.hPow a 2) → LE.le B (HP…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_algebra'","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22370","layer":"formal","project":"p14","title":"SimpleHypercontractivity.hypercontractivity_p_2_general","kind":"theorem","summary":"∀ n : Nat ρ p q : Real, LT.lt 1 p → LE.le 2 q → Eq (HAdd.hAdd (HDiv.hDiv 1 p) (HDiv.hDiv 1 q))…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.hypercontractivity_p_2_general","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22371","layer":"formal","project":"p14","title":"SimpleHypercontractivity.innerProduct_eq_expect_sq","kind":"theorem","summary":"∀ n : Nat (f : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.innerProduct f f) (BooleanAn…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.innerProduct_eq_expect_sq","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22372","layer":"formal","project":"p14","title":"SimpleHypercontractivity.innerProduct_le_L43_L4","kind":"theorem","summary":"∀ n : Nat (f g : BooleanAnalysis.BooleanFunc n), LE.le (BooleanAnalysis.innerProduct f g) (HMul…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.innerProduct_le_L43_L4","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22373","layer":"formal","project":"p14","title":"SimpleHypercontractivity.noiseOp_compose","kind":"theorem","summary":"∀ n : Nat (ρ σ : Real) (f : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.noiseOp ρ (Bool…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.noiseOp_compose","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22374","layer":"formal","project":"p14","title":"SimpleHypercontractivity.noiseOp_snoc","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)) (x : BooleanAnalysis.Boo…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.noiseOp_snoc","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22375","layer":"formal","project":"p14","title":"SimpleHypercontractivity.noise_l2_fourier","kind":"theorem","summary":"∀ n : Nat (ρ : Real) (f : BooleanAnalysis.BooleanFunc n), Eq (BooleanAnalysis.innerProduct (Boo…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.noise_l2_fourier","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22376","layer":"formal","project":"p14","title":"SimpleHypercontractivity.noise_qth_moment_decomp","kind":"theorem","summary":"∀ n : Nat (q : Nat) (ρ : Real) (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAn…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.noise_qth_moment_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22377","layer":"formal","project":"p14","title":"SimpleHypercontractivity.qth_moment_decomp","kind":"theorem","summary":"∀ n : Nat (q : Nat) (f : BooleanAnalysis.BooleanFunc (HAdd.hAdd n 1)), Eq (BooleanAnalysis.expe…","labels":[],"detail_key":"p14","name":"SimpleHypercontractivity.qth_moment_decomp","module":"TCSlib.BooleanAnalysis.Hypercontractivity.Simple"},{"id":"n22378","layer":"formal","project":"p14","title":"BernoulliCost.bernoulli_restriction_asymptotic","kind":"theorem","summary":"∀ (p : Real), LT.lt 0 p → LE.le p 1 → ∀ (w s : Nat), LT.lt 0 w → LT.lt 0 s → ∀ (ε : Real), LT.l…","labels":[],"detail_key":"p14","name":"BernoulliCost.bernoulli_restriction_asymptotic","module":"TCSlib.BooleanAnalysis.LMN.BernoulliCost"},{"id":"n22379","layer":"formal","project":"p14","title":"BernoulliCost.exp_neg_eventually_small","kind":"theorem","summary":"∀ (p : Real), LT.lt 0 p → ∀ (ε : Real), LT.lt 0 ε → Exists fun N => ∀ (m : Nat), LE.le N m → LT…","labels":[],"detail_key":"p14","name":"BernoulliCost.exp_neg_eventually_small","module":"TCSlib.BooleanAnalysis.LMN.BernoulliCost"},{"id":"n22380","layer":"formal","project":"p14","title":"BernoulliCost.fixedSizeRestrProb_nonneg","kind":"theorem","summary":"∀ n : Nat (event : SwitchingLemma2.Restriction n → Prop) [inst : DecidablePred event] (k : Nat)…","labels":[],"detail_key":"p14","name":"BernoulliCost.fixedSizeRestrProb_nonneg","module":"TCSlib.BooleanAnalysis.LMN.BernoulliCost"},{"id":"n22381","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_complement","kind":"theorem","summary":"∀ n : Nat (p : Real), LE.le 0 p → LE.le p 1 → ∀ (A : SwitchingLemma2.Restriction n → Prop) [ins…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_complement","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22382","layer":"formal","project":"p14","title":"LMN.cnf_concat_eval","kind":"theorem","summary":"∀ n : Nat (cnfs : List (CNF n)) (x : Fin n → Bool), Eq (CNF.eval (LMN.listConcat cnfs) x) (cnfs…","labels":[],"detail_key":"p14","name":"LMN.cnf_concat_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22383","layer":"formal","project":"p14","title":"LMN.cnf_concat_width_le","kind":"theorem","summary":"∀ n : Nat (cnfs : List (CNF n)) (l : Nat), (∀ (ψ : CNF n), Membership.mem cnfs ψ → LE.le ψ.widt…","labels":[],"detail_key":"p14","name":"LMN.cnf_concat_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22384","layer":"formal","project":"p14","title":"LMN.compression_and_of_cnfs","kind":"theorem","summary":"∀ n : Nat (children : List ((Fin n → Bool) → Bool)) (l : Nat), (∀ (f : (Fin n → Bool) → Bool),…","labels":[],"detail_key":"p14","name":"LMN.compression_and_of_cnfs","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22385","layer":"formal","project":"p14","title":"LMN.compression_or_of_dnfs","kind":"theorem","summary":"∀ n : Nat (children : List ((Fin n → Bool) → Bool)) (l : Nat), (∀ (f : (Fin n → Bool) → Bool),…","labels":[],"detail_key":"p14","name":"LMN.compression_or_of_dnfs","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22386","layer":"formal","project":"p14","title":"LMN.dnf_concat_eval","kind":"theorem","summary":"∀ n : Nat (dnfs : List (DNF n)) (x : Fin n → Bool), Eq (DNF.eval (LMN.listConcat dnfs) x) (dnfs…","labels":[],"detail_key":"p14","name":"LMN.dnf_concat_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22387","layer":"formal","project":"p14","title":"LMN.dnf_concat_width_le","kind":"theorem","summary":"∀ n : Nat (dnfs : List (DNF n)) (l : Nat), (∀ (φ : DNF n), Membership.mem dnfs φ → LE.le φ.widt…","labels":[],"detail_key":"p14","name":"LMN.dnf_concat_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22388","layer":"formal","project":"p14","title":"LMN.listConcat","kind":"def","summary":"α : Type u → List (List α) → List α","labels":[],"detail_key":"p14","name":"LMN.listConcat","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22389","layer":"formal","project":"p14","title":"LMN.one_step_dtDepth_bound","kind":"theorem","summary":"∀ n s₂ : Nat (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).width w) →…","labels":[],"detail_key":"p14","name":"LMN.one_step_dtDepth_bound","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22390","layer":"formal","project":"p14","title":"LMN.one_step_reduction_failure_bound","kind":"theorem","summary":"∀ n s₂ : Nat (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).width w) →…","labels":[],"detail_key":"p14","name":"LMN.one_step_reduction_failure_bound","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22391","layer":"formal","project":"p14","title":"LMN.one_step_reduction_with_compression","kind":"theorem","summary":"∀ n s₂ : Nat (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).width w) →…","labels":[],"detail_key":"p14","name":"LMN.one_step_reduction_with_compression","module":"TCSlib.BooleanAnalysis.LMN.CircuitCompression"},{"id":"n22392","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_congr_fn","kind":"theorem","summary":"∀ n : Nat f g : (Fin n → Bool) → Bool, (∀ (x : Fin n → Bool), Eq (f x) (g x)) → ∀ (p : Real) (t…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_congr_fn","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22393","layer":"formal","project":"p14","title":"LMN.cleanCNF","kind":"def","summary":"n : Nat → CNF n → CNF n","labels":[],"detail_key":"p14","name":"LMN.cleanCNF","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22394","layer":"formal","project":"p14","title":"LMN.cleanCNF_eval","kind":"theorem","summary":"∀ n : Nat (c : CNF n) (x : Fin n → Bool), Eq ((LMN.cleanCNF c).eval x) (c.eval x)","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22395","layer":"formal","project":"p14","title":"LMN.cleanCNF_nodup","kind":"theorem","summary":"∀ n : Nat (c : CNF n) (t : Term n), Membership.mem (LMN.cleanCNF c) t → List.Nodup t","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_nodup","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22396","layer":"formal","project":"p14","title":"LMN.cleanCNF_var_inj","kind":"theorem","summary":"∀ n : Nat (c : CNF n) (t : Term n), Membership.mem (LMN.cleanCNF c) t → ∀ (l₁ : Literal n), Mem…","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_var_inj","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22397","layer":"formal","project":"p14","title":"LMN.cleanCNF_width_le","kind":"theorem","summary":"∀ n : Nat (c : CNF n), LE.le (LMN.cleanCNF c).width c.width","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22398","layer":"formal","project":"p14","title":"LMN.cleanDNF_eval","kind":"theorem","summary":"∀ n : Nat (d : DNF n) (x : Fin n → Bool), Eq ((LMN.cleanDNF d).eval x) (d.eval x)","labels":[],"detail_key":"p14","name":"LMN.cleanDNF_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22399","layer":"formal","project":"p14","title":"LMN.cleanDNF_var_inj","kind":"theorem","summary":"∀ n : Nat (d : DNF n) (t : Term n), Membership.mem (LMN.cleanDNF d) t → ∀ (l₁ : Literal n), Mem…","labels":[],"detail_key":"p14","name":"LMN.cleanDNF_var_inj","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22400","layer":"formal","project":"p14","title":"LMN.cleanDNF_width_le","kind":"theorem","summary":"∀ n : Nat (d : DNF n), LE.le (LMN.cleanDNF d).width d.width","labels":[],"detail_key":"p14","name":"LMN.cleanDNF_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22401","layer":"formal","project":"p14","title":"LMN.contradiction_clause_eval_true","kind":"theorem","summary":"∀ n : Nat (t : Term n) (x : Fin n → Bool), Eq (LMN.termHasContradiction t) true → Eq (CNF.evalC…","labels":[],"detail_key":"p14","name":"LMN.contradiction_clause_eval_true","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22402","layer":"formal","project":"p14","title":"LMN.contradiction_term_eval_false","kind":"theorem","summary":"∀ n : Nat (t : Term n) (x : Fin n → Bool), Eq (LMN.termHasContradiction t) true → Eq (t.eval x)…","labels":[],"detail_key":"p14","name":"LMN.contradiction_term_eval_false","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22403","layer":"formal","project":"p14","title":"LMN.dedupTermVar_preserves_clause_eval","kind":"theorem","summary":"∀ n : Nat (t : Term n) (x : Fin n → Bool), Eq (LMN.termHasContradiction t) false → Eq (CNF.eval…","labels":[],"detail_key":"p14","name":"LMN.dedupTermVar_preserves_clause_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22404","layer":"formal","project":"p14","title":"LMN.dedupTermVar_preserves_term_eval","kind":"theorem","summary":"∀ n : Nat (t : Term n) (x : Fin n → Bool), Eq (LMN.termHasContradiction t) false → Eq ((LMN.ded…","labels":[],"detail_key":"p14","name":"LMN.dedupTermVar_preserves_term_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22405","layer":"formal","project":"p14","title":"LMN.dedupTermVar_var_inj","kind":"theorem","summary":"∀ n : Nat (t : Term n) (l₁ : Literal n), Membership.mem (LMN.dedupTermVar t) l₁ → ∀ (l₂ : Liter…","labels":[],"detail_key":"p14","name":"LMN.dedupTermVar_var_inj","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22406","layer":"formal","project":"p14","title":"LMN.dedupTermVar_width_le","kind":"theorem","summary":"∀ n : Nat (t : Term n), LE.le (List.length (LMN.dedupTermVar t)) (List.length t)","labels":[],"detail_key":"p14","name":"LMN.dedupTermVar_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22407","layer":"formal","project":"p14","title":"LMN.depth1AndToTerm","kind":"def","summary":"n : Nat → BoolCircuit.Circuit n → Term n","labels":[],"detail_key":"p14","name":"LMN.depth1AndToTerm","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22408","layer":"formal","project":"p14","title":"LMN.depth2AndToCNF","kind":"def","summary":"n : Nat → List (BoolCircuit.Circuit n) → CNF n","labels":[],"detail_key":"p14","name":"LMN.depth2AndToCNF","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22409","layer":"formal","project":"p14","title":"LMN.depth2AndToCNF_eval","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)), LE.le (BoolCircuit.Circuit.node true cs).depth 2…","labels":[],"detail_key":"p14","name":"LMN.depth2AndToCNF_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22410","layer":"formal","project":"p14","title":"LMN.depth2AndToCNF_width_le","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)), LE.le (BoolCircuit.Circuit.node true cs).depth 2…","labels":[],"detail_key":"p14","name":"LMN.depth2AndToCNF_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22411","layer":"formal","project":"p14","title":"LMN.depth2OrToDNF","kind":"def","summary":"n : Nat → List (BoolCircuit.Circuit n) → DNF n","labels":[],"detail_key":"p14","name":"LMN.depth2OrToDNF","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22412","layer":"formal","project":"p14","title":"LMN.depth2OrToDNF_eval","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)), LE.le (BoolCircuit.Circuit.node false cs).depth…","labels":[],"detail_key":"p14","name":"LMN.depth2OrToDNF_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22413","layer":"formal","project":"p14","title":"LMN.depth2OrToDNF_width_le","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)), LE.le (BoolCircuit.Circuit.node false cs).depth…","labels":[],"detail_key":"p14","name":"LMN.depth2OrToDNF_width_le","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22414","layer":"formal","project":"p14","title":"LMN.depth_le_one_children_are_lits","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)) (isAnd : Bool), LE.le (BoolCircuit.Circuit.node i…","labels":[],"detail_key":"p14","name":"LMN.depth_le_one_children_are_lits","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22415","layer":"formal","project":"p14","title":"LMN.depth_le_two_children_depth_le_one","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.Circuit n)) (isAnd : Bool), LE.le (BoolCircuit.Circuit.node i…","labels":[],"detail_key":"p14","name":"LMN.depth_le_two_children_depth_le_one","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22416","layer":"formal","project":"p14","title":"LMN.restrictFn_ext'","kind":"theorem","summary":"∀ n : Nat f g : (Fin n → Bool) → Bool, (∀ (x : Fin n → Bool), Eq (f x) (g x)) → ∀ (ρ : Switchin…","labels":[],"detail_key":"p14","name":"LMN.restrictFn_ext'","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22417","layer":"formal","project":"p14","title":"LMN.switching_bernoulli_dtDepth_cnf_general","kind":"theorem","summary":"∀ n : Nat (f : CNF n) (w : Nat), LE.le f.width w → LT.lt 0 w → LT.lt 0 n → ∀ (p : Real), LT.lt…","labels":[],"detail_key":"p14","name":"LMN.switching_bernoulli_dtDepth_cnf_general","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22418","layer":"formal","project":"p14","title":"LMN.switching_bernoulli_dtDepth_dnf_general","kind":"theorem","summary":"∀ n : Nat (f : DNF n) (w : Nat), LE.le f.width w → LT.lt 0 w → LT.lt 0 n → ∀ (p : Real), LT.lt…","labels":[],"detail_key":"p14","name":"LMN.switching_bernoulli_dtDepth_dnf_general","module":"TCSlib.BooleanAnalysis.LMN.CircuitHelpers"},{"id":"n22419","layer":"formal","project":"p14","title":"LMN.Layer2Data","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"LMN.Layer2Data","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22420","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_congr_fn'","kind":"theorem","summary":"∀ n : Nat p : Real t : Nat f g : (Fin n → Bool) → Bool, (∀ (x : Fin n → Bool), Eq (f x) (g x))…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_congr_fn'","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22421","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_dtDepth_mono","kind":"theorem","summary":"∀ n : Nat (p : Real), LE.le 0 p → LE.le p 1 → ∀ (f : (Fin n → Bool) → Bool) (l₁ l₂ : Nat), LE.l…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_dtDepth_mono","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22422","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_list_union_bound","kind":"theorem","summary":"∀ n : Nat (p : Real), LE.le 0 p → LE.le p 1 → ∀ (cs : List (BoolCircuit.Circuit n)) (bad : Bool…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_list_union_bound","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22423","layer":"formal","project":"p14","title":"LMN.circuit_reduction_aux","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d s w l t : Nat), LE.le 2 d → LT.lt 0 s → LT.lt 0 w → LT…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_aux","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22424","layer":"formal","project":"p14","title":"LMN.circuit_reduction_core","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d s w l t : Nat), LE.le 2 d → LT.lt 0 s → LT.lt 0 w → LT…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_core","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22425","layer":"formal","project":"p14","title":"LMN.circuit_reduction_ind","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d s w l t : Nat), LE.le 2 d → LT.lt 0 s → LT.lt 0 w → LT…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_ind","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22426","layer":"formal","project":"p14","title":"LMN.circuit_reduction_ind_base","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s w l t : Nat), LT.lt 0 s → LT.lt 0 w → LT.lt 0 l → LT.l…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_ind_base","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22427","layer":"formal","project":"p14","title":"LMN.circuit_reduction_ind_step","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d s w l t : Nat), LE.le 3 d → LT.lt 0 s → LT.lt 0 w → LT…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_ind_step","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22428","layer":"formal","project":"p14","title":"LMN.composedDelta","kind":"def","summary":"Nat → Real → Nat → Real","labels":[],"detail_key":"p14","name":"LMN.composedDelta","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22429","layer":"formal","project":"p14","title":"LMN.composedDelta_le_one","kind":"theorem","summary":"∀ (w : Nat) (l : Real) (d : Nat), LE.le 1 w → LE.le 1 l → LE.le 2 d → LE.le (LMN.composedDelta…","labels":[],"detail_key":"p14","name":"LMN.composedDelta_le_one","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22430","layer":"formal","project":"p14","title":"LMN.composedDelta_pos","kind":"theorem","summary":"∀ (w : Nat) (l : Real) (d : Nat), LT.lt 0 w → LT.lt 0 l → LT.lt 0 (LMN.composedDelta w l d)","labels":[],"detail_key":"p14","name":"LMN.composedDelta_pos","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22431","layer":"formal","project":"p14","title":"LMN.composedDelta_step","kind":"theorem","summary":"∀ (w l d : Nat), LE.le 3 d → Eq (LMN.composedDelta w (↑l) d) (HMul.hMul (HDiv.hDiv 1 (HMul.hMul…","labels":[],"detail_key":"p14","name":"LMN.composedDelta_step","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22432","layer":"formal","project":"p14","title":"LMN.composedDelta_step_right","kind":"theorem","summary":"∀ (w : Nat) (l : Real) (d : Nat), LE.le 3 d → LT.lt 0 l → Eq (LMN.composedDelta w l d) (HMul.hM…","labels":[],"detail_key":"p14","name":"LMN.composedDelta_step_right","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22433","layer":"formal","project":"p14","title":"LMN.depth2_circuit_switching_bound","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s w : Nat), LT.lt 0 w → LT.lt 0 n → (Exists fun c => And…","labels":[],"detail_key":"p14","name":"LMN.depth2_circuit_switching_bound","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22434","layer":"formal","project":"p14","title":"LMN.full_iterative_bound","kind":"theorem","summary":"∀ (m : Nat) (layerSize : Fin m → Nat) (s : Nat), LE.le (Finset.univ.sum fun i => layerSize i) s…","labels":[],"detail_key":"p14","name":"LMN.full_iterative_bound","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22435","layer":"formal","project":"p14","title":"LMN.normalform_one_step_cnf_replaceability","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (l : Nat), LT.lt 0 n → ∀ (p : Real), LT.lt 0 p → LE.le p (H…","labels":[],"detail_key":"p14","name":"LMN.normalform_one_step_cnf_replaceability","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22436","layer":"formal","project":"p14","title":"LMN.normalform_one_step_switching","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (l : Nat), LT.lt 0 n → ∀ (p : Real), LT.lt 0 p → LE.le p (H…","labels":[],"detail_key":"p14","name":"LMN.normalform_one_step_switching","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22437","layer":"formal","project":"p14","title":"LMN.restrictFn_composeRestr'","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (ρ₁ ρ₂ : SwitchingLemma2.Restriction n), Eq (SwitchingLem…","labels":[],"detail_key":"p14","name":"LMN.restrictFn_composeRestr'","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22438","layer":"formal","project":"p14","title":"LMN.two_stage_bound'","kind":"theorem","summary":"∀ n : Nat (p₁ p₂ : Real), LT.lt 0 p₁ → LE.le p₁ 1 → LT.lt 0 p₂ → LE.le p₂ 1 → ∀ (E : SwitchingL…","labels":[],"detail_key":"p14","name":"LMN.two_stage_bound'","module":"TCSlib.BooleanAnalysis.LMN.CircuitLayerReduction"},{"id":"n22439","layer":"formal","project":"p14","title":"BoolCircuit.Circuit.reidx","kind":"def","summary":"m m' : Nat → BoolCircuit.Circuit m → (Fin m → Fin m') → BoolCircuit.Circuit m'","labels":[],"detail_key":"p14","name":"BoolCircuit.Circuit.reidx","module":"TCSlib.BooleanAnalysis.LMN.CircuitReindex"},{"id":"n22440","layer":"formal","project":"p14","title":"BoolCircuit.Circuit.reidx_depth","kind":"theorem","summary":"∀ m m' : Nat (c : BoolCircuit.Circuit m) (f : Fin m → Fin m'), Eq (c.reidx f).depth c.depth","labels":[],"detail_key":"p14","name":"BoolCircuit.Circuit.reidx_depth","module":"TCSlib.BooleanAnalysis.LMN.CircuitReindex"},{"id":"n22441","layer":"formal","project":"p14","title":"BoolCircuit.Circuit.reidx_eval","kind":"theorem","summary":"∀ m m' : Nat (c : BoolCircuit.Circuit m) (f : Fin m → Fin m') (g : Fin m' → Bool), Eq ((c.reidx…","labels":[],"detail_key":"p14","name":"BoolCircuit.Circuit.reidx_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitReindex"},{"id":"n22442","layer":"formal","project":"p14","title":"LMN.Circuit.depth0_is_lit","kind":"theorem","summary":"∀ m : Nat (c : BoolCircuit.Circuit m), Eq c.depth 0 → Exists fun lr => Eq c (BoolCircuit.Circui…","labels":[],"detail_key":"p14","name":"LMN.Circuit.depth0_is_lit","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22443","layer":"formal","project":"p14","title":"LMN.Circuit.depth1_all_lits","kind":"theorem","summary":"∀ m : Nat (isAnd : Bool) (cs : List (BoolCircuit.Circuit m)), LE.le (BoolCircuit.Circuit.node i…","labels":[],"detail_key":"p14","name":"LMN.Circuit.depth1_all_lits","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22444","layer":"formal","project":"p14","title":"LMN.Circuit.depth1_children_are_lits","kind":"theorem","summary":"∀ m : Nat (isAnd : Bool) (cs : List (BoolCircuit.Circuit m)), LE.le (BoolCircuit.Circuit.node i…","labels":[],"detail_key":"p14","name":"LMN.Circuit.depth1_children_are_lits","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22445","layer":"formal","project":"p14","title":"LMN.Circuit.exists_node_of_depth_ge_one","kind":"theorem","summary":"∀ m : Nat (c : BoolCircuit.Circuit m), LE.le 1 c.depth → Exists fun isAnd => Exists fun cs => E…","labels":[],"detail_key":"p14","name":"LMN.Circuit.exists_node_of_depth_ge_one","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22446","layer":"formal","project":"p14","title":"LMN.absorbOneLevel","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (c_top : BoolCircuit.Circuit data.numGates) (l : Nat), LT.l…","labels":[],"detail_key":"p14","name":"LMN.absorbOneLevel","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22447","layer":"formal","project":"p14","title":"LMN.absorbOneLevel_depth1","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (c_top : BoolCircuit.Circuit data.numGates) (l : Nat), LT.l…","labels":[],"detail_key":"p14","name":"LMN.absorbOneLevel_depth1","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22448","layer":"formal","project":"p14","title":"LMN.absorbOneLevel_general","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (c_top : BoolCircuit.Circuit data.numGates) (l : Nat), LT.l…","labels":[],"detail_key":"p14","name":"LMN.absorbOneLevel_general","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22449","layer":"formal","project":"p14","title":"LMN.and_of_lit_children_cnf","kind":"theorem","summary":"∀ n m : Nat (cs : List (BoolCircuit.Circuit m)) (gates : Fin m → (Fin n → Bool) → Bool) (l : Na…","labels":[],"detail_key":"p14","name":"LMN.and_of_lit_children_cnf","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22450","layer":"formal","project":"p14","title":"LMN.build_literal_circuit","kind":"theorem","summary":"∀ (isAnd : Bool) (k : Nat) (signs : Fin k → Bool), Exists fun c' => And (LE.le c'.depth 1) (∀ (…","labels":[],"detail_key":"p14","name":"LMN.build_literal_circuit","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22451","layer":"formal","project":"p14","title":"LMN.childFunction","kind":"def","summary":"n m : Nat → BoolCircuit.Circuit m → (Fin m → (Fin n → Bool) → Bool) → (Fin n → Bool) → Bool","labels":[],"detail_key":"p14","name":"LMN.childFunction","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22452","layer":"formal","project":"p14","title":"LMN.child_depth_le1_has_signed_dnf","kind":"theorem","summary":"∀ n m : Nat (c_j : BoolCircuit.Circuit m) (gates : Fin m → (Fin n → Bool) → Bool) (l : Nat), LE…","labels":[],"detail_key":"p14","name":"LMN.child_depth_le1_has_signed_dnf","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22453","layer":"formal","project":"p14","title":"LMN.dnfToDualCNF","kind":"def","summary":"n : Nat → DNF n → CNF n","labels":[],"detail_key":"p14","name":"LMN.dnfToDualCNF","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22454","layer":"formal","project":"p14","title":"LMN.dnfToDualCNF_eval","kind":"theorem","summary":"∀ n : Nat (φ : DNF n) (x : Fin n → Bool), Eq ((LMN.dnfToDualCNF φ).eval x) (φ.eval x).not","labels":[],"detail_key":"p14","name":"LMN.dnfToDualCNF_eval","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22455","layer":"formal","project":"p14","title":"LMN.dnfToDualCNF_width","kind":"theorem","summary":"∀ n : Nat (φ : DNF n), Eq (LMN.dnfToDualCNF φ).width φ.width","labels":[],"detail_key":"p14","name":"LMN.dnfToDualCNF_width","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22456","layer":"formal","project":"p14","title":"LMN.exists_circuit_depth_reduction","kind":"theorem","summary":"∀ n m : Nat (c : BoolCircuit.Circuit m) (gates : Fin m → (Fin n → Bool) → Bool) (l : Nat), LT.l…","labels":[],"detail_key":"p14","name":"LMN.exists_circuit_depth_reduction","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22457","layer":"formal","project":"p14","title":"LMN.exists_circuit_depth_reduction_depth1","kind":"theorem","summary":"∀ n m : Nat (c : BoolCircuit.Circuit m) (gates : Fin m → (Fin n → Bool) → Bool) (l : Nat), Eq c…","labels":[],"detail_key":"p14","name":"LMN.exists_circuit_depth_reduction_depth1","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22458","layer":"formal","project":"p14","title":"LMN.exists_circuit_depth_reduction_depth2","kind":"theorem","summary":"∀ n m : Nat (c : BoolCircuit.Circuit m) (gates : Fin m → (Fin n → Bool) → Bool) (l : Nat), Eq c…","labels":[],"detail_key":"p14","name":"LMN.exists_circuit_depth_reduction_depth2","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22459","layer":"formal","project":"p14","title":"LMN.list_child_signed_dnfs","kind":"theorem","summary":"∀ n m : Nat (cs : List (BoolCircuit.Circuit m)) (gates : Fin m → (Fin n → Bool) → Bool) (l : Na…","labels":[],"detail_key":"p14","name":"LMN.list_child_signed_dnfs","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22460","layer":"formal","project":"p14","title":"LMN.node_eval_eq_of_finRange_map","kind":"theorem","summary":"∀ m M : Nat (isAnd : Bool) (cs : List (BoolCircuit.Circuit m)) (new_cs : Fin cs.length → BoolCi…","labels":[],"detail_key":"p14","name":"LMN.node_eval_eq_of_finRange_map","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22461","layer":"formal","project":"p14","title":"LMN.or_of_lit_children_dnf","kind":"theorem","summary":"∀ n m : Nat (cs : List (BoolCircuit.Circuit m)) (gates : Fin m → (Fin n → Bool) → Bool) (l : Na…","labels":[],"detail_key":"p14","name":"LMN.or_of_lit_children_dnf","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22462","layer":"formal","project":"p14","title":"LMN.reduce_children","kind":"theorem","summary":"∀ n m : Nat (cs : List (BoolCircuit.Circuit m)) (gates : Fin m → (Fin n → Bool) → Bool) (l : Na…","labels":[],"detail_key":"p14","name":"LMN.reduce_children","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22463","layer":"formal","project":"p14","title":"LMN.switched_gates_have_dnf_cnf","kind":"theorem","summary":"∀ n : Nat (m : Nat) (gates : Fin m → DNF n) (ρ : SwitchingLemma2.Restriction n) (l : Nat), (∀ (…","labels":[],"detail_key":"p14","name":"LMN.switched_gates_have_dnf_cnf","module":"TCSlib.BooleanAnalysis.LMN.CircuitTreeManip"},{"id":"n22464","layer":"formal","project":"p14","title":"LMN.circuit_depth_zero_is_lit","kind":"theorem","summary":"∀ m : Nat (c : BoolCircuit.Circuit m), Eq c.depth 0 → Exists fun l => Eq c (BoolCircuit.Circuit…","labels":[],"detail_key":"p14","name":"LMN.circuit_depth_zero_is_lit","module":"TCSlib.BooleanAnalysis.LMN.CompressionStep"},{"id":"n22465","layer":"formal","project":"p14","title":"LMN.layer2_composed_bound_base","kind":"theorem","summary":"∀ n : Nat (data : LMN.Layer2Data n) (c_top : BoolCircuit.Circuit data.numGates) (s_rem l t : Na…","labels":[],"detail_key":"p14","name":"LMN.layer2_composed_bound_base","module":"TCSlib.BooleanAnalysis.LMN.CompressionStep"},{"id":"n22466","layer":"formal","project":"p14","title":"LMN.switched_gates_give_new_dnfs","kind":"theorem","summary":"∀ n : Nat (m : Nat) (gates : Fin m → DNF n) (ρ₁ : SwitchingLemma2.Restriction n) (l : Nat), (∀…","labels":[],"detail_key":"p14","name":"LMN.switched_gates_give_new_dnfs","module":"TCSlib.BooleanAnalysis.LMN.CompressionStep"},{"id":"n22467","layer":"formal","project":"p14","title":"DecisionTree.coeffs","kind":"def","summary":"n : Nat → DecisionTree n → Finset (Fin n) → Real","labels":[],"detail_key":"p14","name":"DecisionTree.coeffs","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22468","layer":"formal","project":"p14","title":"DecisionTree.coeffs_eq_zero_of_depth_lt","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (S : Finset (Fin n)), LT.lt T.depth S.card → Eq (T.coeffs S) 0","labels":[],"detail_key":"p14","name":"DecisionTree.coeffs_eq_zero_of_depth_lt","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22469","layer":"formal","project":"p14","title":"DecisionTree.coeffs_granular","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (S : Finset (Fin n)), Exists fun m => Eq (T.coeffs S) (HDiv.hDiv…","labels":[],"detail_key":"p14","name":"DecisionTree.coeffs_granular","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22470","layer":"formal","project":"p14","title":"DecisionTree.coeffs_mul_two_pow_int","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (k : Nat), LE.le T.depth k → ∀ (S : Finset (Fin n)), Exists fun…","labels":[],"detail_key":"p14","name":"DecisionTree.coeffs_mul_two_pow_int","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22471","layer":"formal","project":"p14","title":"DecisionTree.degree_le_depth","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), BooleanAnalysis.has_degree_at_most T.signEval T.depth","labels":[],"detail_key":"p14","name":"DecisionTree.degree_le_depth","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22472","layer":"formal","project":"p14","title":"DecisionTree.degree_le_dtDepth","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool), BooleanAnalysis.has_degree_at_most (fun x => BooleanAnal…","labels":[],"detail_key":"p14","name":"DecisionTree.degree_le_dtDepth","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22473","layer":"formal","project":"p14","title":"DecisionTree.exists_dtree_of_dtDepth","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool), Exists fun T => And (LE.le T.depth (dtDepth f)) (∀ (x :…","labels":[],"detail_key":"p14","name":"DecisionTree.exists_dtree_of_dtDepth","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22474","layer":"formal","project":"p14","title":"DecisionTree.fourierCoeff_granular","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (S : Finset (Fin n)), Exists fun m => Eq (BooleanAnalysis.fourie…","labels":[],"detail_key":"p14","name":"DecisionTree.fourierCoeff_granular","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22475","layer":"formal","project":"p14","title":"DecisionTree.fourierCoeff_signEval","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (S : Finset (Fin n)), Eq (BooleanAnalysis.fourierCoeff T.signEva…","labels":[],"detail_key":"p14","name":"DecisionTree.fourierCoeff_signEval","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22476","layer":"formal","project":"p14","title":"DecisionTree.fourierCoeff_sum_chiS","kind":"theorem","summary":"∀ n : Nat (c : Finset (Fin n) → Real) (T : Finset (Fin n)), Eq (BooleanAnalysis.fourierCoeff (f…","labels":[],"detail_key":"p14","name":"DecisionTree.fourierCoeff_sum_chiS","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22477","layer":"formal","project":"p14","title":"DecisionTree.signEval","kind":"def","summary":"n : Nat → DecisionTree n → BooleanAnalysis.BooleanFunc n","labels":[],"detail_key":"p14","name":"DecisionTree.signEval","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22478","layer":"formal","project":"p14","title":"DecisionTree.signEval_eq_sum_coeffs","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n) (x : BooleanAnalysis.BoolCube n), Eq (T.signEval x) (Finset.univ…","labels":[],"detail_key":"p14","name":"DecisionTree.signEval_eq_sum_coeffs","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22479","layer":"formal","project":"p14","title":"DecisionTree.size","kind":"def","summary":"n : Nat → DecisionTree n → Nat","labels":[],"detail_key":"p14","name":"DecisionTree.size","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22480","layer":"formal","project":"p14","title":"DecisionTree.size_le_two_pow_depth","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), LE.le T.size (HPow.hPow 2 T.depth)","labels":[],"detail_key":"p14","name":"DecisionTree.size_le_two_pow_depth","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22481","layer":"formal","project":"p14","title":"DecisionTree.sparsity_le","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), LE.le (Finset.filter (fun S => Ne (BooleanAnalysis.fourierCoeff…","labels":[],"detail_key":"p14","name":"DecisionTree.sparsity_le","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22482","layer":"formal","project":"p14","title":"DecisionTree.sparsity_le_four_pow","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), LE.le (Finset.filter (fun S => Ne (BooleanAnalysis.fourierCoeff…","labels":[],"detail_key":"p14","name":"DecisionTree.sparsity_le_four_pow","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22483","layer":"formal","project":"p14","title":"DecisionTree.spectral_one_norm_le","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), LE.le (Finset.univ.sum fun S => abs (BooleanAnalysis.fourierCoe…","labels":[],"detail_key":"p14","name":"DecisionTree.spectral_one_norm_le","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22484","layer":"formal","project":"p14","title":"DecisionTree.sum_abs_coeffs_le","kind":"theorem","summary":"∀ n : Nat (T : DecisionTree n), LE.le (Finset.univ.sum fun S => abs (T.coeffs S)) ↑T.size","labels":[],"detail_key":"p14","name":"DecisionTree.sum_abs_coeffs_le","module":"TCSlib.BooleanAnalysis.LMN.DecisionTreeFourier"},{"id":"n22485","layer":"formal","project":"p14","title":"LMN.and_of_gates_has_cnf","kind":"theorem","summary":"∀ n : Nat (s₂ : Nat) (gates : Fin s₂ → DNF n) (l : Nat) (ρ₁ : SwitchingLemma2.Restriction n), (…","labels":[],"detail_key":"p14","name":"LMN.and_of_gates_has_cnf","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22486","layer":"formal","project":"p14","title":"LMN.circuit_reduction_depth3","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s₂ : Nat) (gates : Fin s₂ → DNF n) (w l t : Nat), (∀ (x…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_depth3","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22487","layer":"formal","project":"p14","title":"LMN.circuit_reduction_depth3_le_eps","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s₂ : Nat) (gates : Fin s₂ → DNF n) (w l t : Nat), (∀ (x…","labels":[],"detail_key":"p14","name":"LMN.circuit_reduction_depth3_le_eps","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22488","layer":"formal","project":"p14","title":"LMN.clauseIsTaut_eval_true","kind":"theorem","summary":"∀ n : Nat (c : List (Literal n)), LMN.clauseIsTaut c → ∀ (x : Fin n → Bool), Eq (c.any fun l =>…","labels":[],"detail_key":"p14","name":"LMN.clauseIsTaut_eval_true","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22489","layer":"formal","project":"p14","title":"LMN.cleanCNF_D3","kind":"def","summary":"n : Nat → CNF n → CNF n","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_D3","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22490","layer":"formal","project":"p14","title":"LMN.cleanCNF_D3_eval","kind":"theorem","summary":"∀ n : Nat (ψ : CNF n) (x : Fin n → Bool), Eq ((LMN.cleanCNF_D3 ψ).eval x) (ψ.eval x)","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_D3_eval","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22491","layer":"formal","project":"p14","title":"LMN.cleanCNF_D3_nodup","kind":"theorem","summary":"∀ n : Nat (ψ : CNF n) (c : Term n), Membership.mem (LMN.cleanCNF_D3 ψ) c → List.Nodup c","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_D3_nodup","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22492","layer":"formal","project":"p14","title":"LMN.cleanCNF_D3_var_inj","kind":"theorem","summary":"∀ n : Nat (ψ : CNF n) (c : Term n), Membership.mem (LMN.cleanCNF_D3 ψ) c → ∀ (l₁ : Literal n),…","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_D3_var_inj","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22493","layer":"formal","project":"p14","title":"LMN.cleanCNF_D3_width_le","kind":"theorem","summary":"∀ n : Nat (ψ : CNF n), LE.le (LMN.cleanCNF_D3 ψ).width ψ.width","labels":[],"detail_key":"p14","name":"LMN.cleanCNF_D3_width_le","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22494","layer":"formal","project":"p14","title":"LMN.dedupClauseVars","kind":"def","summary":"n : Nat → List (Literal n) → List (Literal n)","labels":[],"detail_key":"p14","name":"LMN.dedupClauseVars","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22495","layer":"formal","project":"p14","title":"LMN.dedupClauseVars_eval_of_not_taut","kind":"theorem","summary":"∀ n : Nat (c : List (Literal n)), Not (LMN.clauseIsTaut c) → ∀ (x : Fin n → Bool), Eq ((LMN.ded…","labels":[],"detail_key":"p14","name":"LMN.dedupClauseVars_eval_of_not_taut","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22496","layer":"formal","project":"p14","title":"LMN.dedupClauseVars_length_le","kind":"theorem","summary":"∀ n : Nat (c : List (Literal n)), LE.le (LMN.dedupClauseVars c).length c.length","labels":[],"detail_key":"p14","name":"LMN.dedupClauseVars_length_le","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22497","layer":"formal","project":"p14","title":"LMN.dedupClauseVars_nodup","kind":"theorem","summary":"∀ n : Nat (c : List (Literal n)), (LMN.dedupClauseVars c).Nodup","labels":[],"detail_key":"p14","name":"LMN.dedupClauseVars_nodup","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22498","layer":"formal","project":"p14","title":"LMN.dedupClauseVars_var_inj","kind":"theorem","summary":"∀ n : Nat (c : List (Literal n)) (l₁ : Literal n), Membership.mem (LMN.dedupClauseVars c) l₁ →…","labels":[],"detail_key":"p14","name":"LMN.dedupClauseVars_var_inj","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22499","layer":"formal","project":"p14","title":"LMN.depth3_compression","kind":"theorem","summary":"∀ n : Nat (s₂ : Nat) (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).wid…","labels":[],"detail_key":"p14","name":"LMN.depth3_compression","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22500","layer":"formal","project":"p14","title":"LMN.depth3_restricted_has_nice_cnf","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s₂ : Nat) (gates : Fin s₂ → DNF n) (l : Nat) (ρ₁ : Switc…","labels":[],"detail_key":"p14","name":"LMN.depth3_restricted_has_nice_cnf","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22501","layer":"formal","project":"p14","title":"LMN.depth3_second_stage_bound","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s₂ : Nat) (gates : Fin s₂ → DNF n) (l t : Nat) (ρ₁ : Swi…","labels":[],"detail_key":"p14","name":"LMN.depth3_second_stage_bound","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22502","layer":"formal","project":"p14","title":"LMN.depth3_switching_bound","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (s₂ : Nat) (gates : Fin s₂ → DNF n) (w l t : Nat), (∀ (x…","labels":[],"detail_key":"p14","name":"LMN.depth3_switching_bound","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22503","layer":"formal","project":"p14","title":"LMN.dtDepth_congr","kind":"theorem","summary":"∀ n : Nat (f g : (Fin n → Bool) → Bool), (∀ (x : Fin n → Bool), Eq (f x) (g x)) → Eq (dtDepth f…","labels":[],"detail_key":"p14","name":"LMN.dtDepth_congr","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22504","layer":"formal","project":"p14","title":"LMN.dtDepth_le_implies_nice_cnf","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d : Nat), LE.le (dtDepth f) d → Exists fun ψ => And (LE.…","labels":[],"detail_key":"p14","name":"LMN.dtDepth_le_implies_nice_cnf","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22505","layer":"formal","project":"p14","title":"LMN.dtDepth_le_implies_nice_dnf","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (d : Nat), LE.le (dtDepth f) d → Exists fun φ => And (LE.…","labels":[],"detail_key":"p14","name":"LMN.dtDepth_le_implies_nice_dnf","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22506","layer":"formal","project":"p14","title":"LMN.exists_nice_cnf_of_cnf","kind":"theorem","summary":"∀ n : Nat (ψ : CNF n), Exists fun ψ' => And (LE.le ψ'.width ψ.width) (And (∀ (x : Fin n → Bool)…","labels":[],"detail_key":"p14","name":"LMN.exists_nice_cnf_of_cnf","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22507","layer":"formal","project":"p14","title":"LMN.listAnd","kind":"def","summary":"n : Nat → List ((Fin n → Bool) → Bool) → (Fin n → Bool) → Bool","labels":[],"detail_key":"p14","name":"LMN.listAnd","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22508","layer":"formal","project":"p14","title":"LMN.restrictFn_listAnd","kind":"theorem","summary":"∀ n : Nat (fs : List ((Fin n → Bool) → Bool)) (ρ : SwitchingLemma2.Restriction n) (x : Fin n →…","labels":[],"detail_key":"p14","name":"LMN.restrictFn_listAnd","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22509","layer":"formal","project":"p14","title":"LMN.switching_bernoulli_dtDepth_function","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (w : Nat), LE.le (dtDepth f) w → LT.lt 0 w → LT.lt 0 n →…","labels":[],"detail_key":"p14","name":"LMN.switching_bernoulli_dtDepth_function","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22510","layer":"formal","project":"p14","title":"LMN.two_stage_bound","kind":"theorem","summary":"∀ n : Nat (p₁ p₂ : Real), LT.lt 0 p₁ → LE.le p₁ 1 → LT.lt 0 p₂ → LE.le p₂ 1 → ∀ (E : SwitchingL…","labels":[],"detail_key":"p14","name":"LMN.two_stage_bound","module":"TCSlib.BooleanAnalysis.LMN.Depth3Switching"},{"id":"n22511","layer":"formal","project":"p14","title":"LMN.mergeGates_nodup","kind":"theorem","summary":"∀ n m₁ m₂ : Nat (g₁ : Fin m₁ → DNF n) (g₂ : Fin m₂ → DNF n), (∀ (k : Fin m₁) (t : Term n), Memb…","labels":[],"detail_key":"p14","name":"LMN.mergeGates_nodup","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22512","layer":"formal","project":"p14","title":"LMN.mergeGates_varInj","kind":"theorem","summary":"∀ n m₁ m₂ : Nat (g₁ : Fin m₁ → DNF n) (g₂ : Fin m₂ → DNF n), (∀ (k : Fin m₁) (t : Term n), Memb…","labels":[],"detail_key":"p14","name":"LMN.mergeGates_varInj","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22513","layer":"formal","project":"p14","title":"LMN.mergeGates_width","kind":"theorem","summary":"∀ n m₁ m₂ : Nat (g₁ : Fin m₁ → DNF n) (g₂ : Fin m₂ → DNF n) (l : Nat), (∀ (k : Fin m₁), LE.le (…","labels":[],"detail_key":"p14","name":"LMN.mergeGates_width","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22514","layer":"formal","project":"p14","title":"LMN.mergeGates_width_left","kind":"theorem","summary":"∀ n m₁ m₂ : Nat (g₁ : Fin m₁ → DNF n) (g₂ : Fin m₂ → DNF n) (l : Nat), (∀ (k : Fin m₁), LE.le (…","labels":[],"detail_key":"p14","name":"LMN.mergeGates_width_left","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22515","layer":"formal","project":"p14","title":"LMN.mergeGates_width_right","kind":"theorem","summary":"∀ n m₁ m₂ : Nat (g₁ : Fin m₁ → DNF n) (g₂ : Fin m₂ → DNF n) (l : Nat), (∀ (k : Fin m₂), LE.le (…","labels":[],"detail_key":"p14","name":"LMN.mergeGates_width_right","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22516","layer":"formal","project":"p14","title":"LMN.reidx_eval_mergeGates_left","kind":"theorem","summary":"∀ m₁ m₂ : Nat (c : BoolCircuit.Circuit m₁) (g₁ : Fin m₁ → Bool) (g₂ : Fin m₂ → Bool), Eq ((c.re…","labels":[],"detail_key":"p14","name":"LMN.reidx_eval_mergeGates_left","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22517","layer":"formal","project":"p14","title":"LMN.reidx_eval_mergeGates_right","kind":"theorem","summary":"∀ m₁ m₂ : Nat (c : BoolCircuit.Circuit m₂) (g₁ : Fin m₁ → Bool) (g₂ : Fin m₂ → Bool), Eq ((c.re…","labels":[],"detail_key":"p14","name":"LMN.reidx_eval_mergeGates_right","module":"TCSlib.BooleanAnalysis.LMN.GateMerge"},{"id":"n22518","layer":"formal","project":"p14","title":"LMN.all_gates_have_small_cnf","kind":"theorem","summary":"∀ n s : Nat (gates : Fin s → DNF n) (l : Nat) (ρ : SwitchingLemma2.Restriction n), (∀ (i : Fin…","labels":[],"detail_key":"p14","name":"LMN.all_gates_have_small_cnf","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22519","layer":"formal","project":"p14","title":"LMN.layer2_cnf_replaceability_simplified","kind":"theorem","summary":"∀ n s₂ : Nat (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).width w) →…","labels":[],"detail_key":"p14","name":"LMN.layer2_cnf_replaceability_simplified","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22520","layer":"formal","project":"p14","title":"LMN.layer2_cnf_replaceability_union_bound","kind":"theorem","summary":"∀ n s₂ : Nat (gates : Fin s₂ → DNF n) (w l : Nat), (∀ (i : Fin s₂), LE.le (gates i).width w) →…","labels":[],"detail_key":"p14","name":"LMN.layer2_cnf_replaceability_union_bound","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22521","layer":"formal","project":"p14","title":"LMN.restricted_has_small_cnf_of_dtDepth_le","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (ρ : SwitchingLemma2.Restriction n) (l : Nat), LE.le (dtD…","labels":[],"detail_key":"p14","name":"LMN.restricted_has_small_cnf_of_dtDepth_le","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22522","layer":"formal","project":"p14","title":"LMN.restricted_has_small_dnf_of_dtDepth_le","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (ρ : SwitchingLemma2.Restriction n) (l : Nat), LE.le (dtD…","labels":[],"detail_key":"p14","name":"LMN.restricted_has_small_dnf_of_dtDepth_le","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22523","layer":"formal","project":"p14","title":"LMN.switching_bernoulli_gate_to_dnf_from_cnf","kind":"theorem","summary":"∀ n : Nat (g : CNF n) (w l : Nat), LE.le g.width w → LT.lt 0 w → (∀ (c : Term n), Membership.me…","labels":[],"detail_key":"p14","name":"LMN.switching_bernoulli_gate_to_dnf_from_cnf","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22524","layer":"formal","project":"p14","title":"LMN.switching_bernoulli_union_bound","kind":"theorem","summary":"∀ n s : Nat (gates : Fin s → DNF n) (w l : Nat), (∀ (i : Fin s), LE.le (gates i).width w) → LT.…","labels":[],"detail_key":"p14","name":"LMN.switching_bernoulli_union_bound","module":"TCSlib.BooleanAnalysis.LMN.GateSwitching"},{"id":"n22525","layer":"formal","project":"p14","title":"LMN.abstract_iterative_reduction","kind":"theorem","summary":"∀ (m : Nat) (layer_size : Fin m → Nat) (s : Nat), LE.le (Finset.univ.sum fun i => layer_size i)…","labels":[],"detail_key":"p14","name":"LMN.abstract_iterative_reduction","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22526","layer":"formal","project":"p14","title":"LMN.bernoulliRestrProb_union_bound_fin","kind":"theorem","summary":"∀ n : Nat (p : Real), LE.le 0 p → LE.le p 1 → ∀ (m : Nat) (A : Fin m → SwitchingLemma2.Restrict…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_union_bound_fin","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22527","layer":"formal","project":"p14","title":"LMN.iterative_dominant_term_bound","kind":"theorem","summary":"∀ (m : Nat) (layer_size : Fin m → Nat) (s : Nat), LE.le (Finset.univ.sum fun i => layer_size i)…","labels":[],"detail_key":"p14","name":"LMN.iterative_dominant_term_bound","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22528","layer":"formal","project":"p14","title":"LMN.multi_stage_failure_bound","kind":"theorem","summary":"∀ (m : Nat) (layer_size : Fin m → Nat) (s : Nat), LE.le (Finset.univ.sum fun i => layer_size i)…","labels":[],"detail_key":"p14","name":"LMN.multi_stage_failure_bound","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22529","layer":"formal","project":"p14","title":"LMN.subsequent_step_dtDepth","kind":"theorem","summary":"∀ n s_i : Nat (gates : Fin s_i → DNF n) (l : Nat), LT.lt 0 l → (∀ (i : Fin s_i), LE.le (gates i…","labels":[],"detail_key":"p14","name":"LMN.subsequent_step_dtDepth","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22530","layer":"formal","project":"p14","title":"LMN.subsequent_step_reduction","kind":"theorem","summary":"∀ n s_i : Nat (gates : Fin s_i → DNF n) (l : Nat), LT.lt 0 l → (∀ (i : Fin s_i), LE.le (gates i…","labels":[],"detail_key":"p14","name":"LMN.subsequent_step_reduction","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22531","layer":"formal","project":"p14","title":"LMN.two_stage_composed_union_bound","kind":"theorem","summary":"∀ n : Nat (p q : Real), LE.le 0 p → LE.le p 1 → LE.le 0 q → LE.le q 1 → ∀ (A : SwitchingLemma2.…","labels":[],"detail_key":"p14","name":"LMN.two_stage_composed_union_bound","module":"TCSlib.BooleanAnalysis.LMN.IterativeReduction"},{"id":"n22532","layer":"formal","project":"p14","title":"BoolCircuit.Lit.eval_eq_toLiteral_eval","kind":"theorem","summary":"∀ n : Nat (l : BoolCircuit.Lit n) (x : Fin n → Bool), Eq (l.eval x) (l.toLiteral.eval x)","labels":[],"detail_key":"p14","name":"BoolCircuit.Lit.eval_eq_toLiteral_eval","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22533","layer":"formal","project":"p14","title":"BoolCircuit.Lit.toLiteral","kind":"def","summary":"n : Nat → BoolCircuit.Lit n → Literal n","labels":[],"detail_key":"p14","name":"BoolCircuit.Lit.toLiteral","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22534","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.clauseToTerm","kind":"def","summary":"n : Nat → BoolCircuit.NAndCircuit n → Term n","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.clauseToTerm","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22535","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.clauseToTerm_nodup","kind":"theorem","summary":"∀ n : Nat (lits : List (BoolCircuit.Lit n)) (h : (List.map BoolCircuit.Lit.idx lits).Nodup), Li…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.clauseToTerm_nodup","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22536","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.clauseToTerm_var_inj","kind":"theorem","summary":"∀ n : Nat (lits : List (BoolCircuit.Lit n)) (h : (List.map BoolCircuit.Lit.idx lits).Nodup) (l₁…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.clauseToTerm_var_inj","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22537","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.clauseToTerm_width","kind":"theorem","summary":"∀ n : Nat (lits : List (BoolCircuit.Lit n)) (h : (List.map BoolCircuit.Lit.idx lits).Nodup), Eq…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.clauseToTerm_width","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22538","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.node_eval_eq_toCNF_eval","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.NOrCircuit n)) (x : Fin n → Bool), (∀ (c : BoolCircuit.NOrCir…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.node_eval_eq_toCNF_eval","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22539","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.toCNF","kind":"def","summary":"n : Nat → BoolCircuit.NAndCircuit n → CNF n","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.toCNF","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22540","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.toCNF_terms_nodup","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.NOrCircuit n)), (∀ (c : BoolCircuit.NOrCircuit n), Membership…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.toCNF_terms_nodup","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22541","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.toCNF_var_inj","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.NOrCircuit n)), (∀ (c : BoolCircuit.NOrCircuit n), Membership…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.toCNF_var_inj","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22542","layer":"formal","project":"p14","title":"BoolCircuit.NAndCircuit.toCNF_width_bounded","kind":"theorem","summary":"∀ n : Nat (cs : List (BoolCircuit.NOrCircuit n)) (w : Nat), (∀ (c : BoolCircuit.NOrCircuit n),…","labels":[],"detail_key":"p14","name":"BoolCircuit.NAndCircuit.toCNF_width_bounded","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22543","layer":"formal","project":"p14","title":"BoolCircuit.NOrCircuit.clauseToTerm","kind":"def","summary":"n : Nat → BoolCircuit.NOrCircuit n → Term n","labels":[],"detail_key":"p14","name":"BoolCircuit.NOrCircuit.clauseToTerm","module":"TCSlib.BooleanAnalysis.LMN.NormalFormConversion"},{"id":"n22544","layer":"formal","project":"p14","title":"BoolCircuit.NOrCircuit.clauseToTerm_nodup","kind":"theorem","summary":"∀ n : Nat (lits : 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n : Nat (f : (Fin n → Bool) → Bool) (t : Nat) (p₁ p₂ : Real), LT.lt 0 p₁ → LE.le p₁ 1 → LT.lt…","labels":[],"detail_key":"p14","name":"LMN.bernoulliRestrProb_dtDepth_compose_le","module":"TCSlib.BooleanAnalysis.LMN.RestrictionMonotonicity"},{"id":"n22600","layer":"formal","project":"p14","title":"LMN.dtDepth_composeRestr_le","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (ρ₁ ρ₂ : SwitchingLemma2.Restriction n), LE.le (dtDepth (…","labels":[],"detail_key":"p14","name":"LMN.dtDepth_composeRestr_le","module":"TCSlib.BooleanAnalysis.LMN.RestrictionMonotonicity"},{"id":"n22601","layer":"formal","project":"p14","title":"LMN.dtDepth_restrictFn_le'","kind":"theorem","summary":"∀ n : Nat (f : (Fin n → Bool) → Bool) (ρ : SwitchingLemma2.Restriction n), LE.le (dtDepth 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((LMN…","labels":[],"detail_key":"p14","name":"LMN.dtRestrict_eval","module":"TCSlib.BooleanAnalysis.LMN.RestrictionMonotonicity"},{"id":"n22605","layer":"formal","project":"p14","title":"SwitchingBernoulli.switching_bernoulli_dtDepth_cnf","kind":"theorem","summary":"∀ n : Nat (f : CNF n) (w : Nat), LE.le f.width w → LT.lt 0 w → (∀ (c : Term n), Membership.mem…","labels":[],"detail_key":"p14","name":"SwitchingBernoulli.switching_bernoulli_dtDepth_cnf","module":"TCSlib.BooleanAnalysis.LMN.SwitchingBernoulli"},{"id":"n22606","layer":"formal","project":"p14","title":"LMN.iterative_reduction_bound","kind":"theorem","summary":"∀ n : Nat (c : BoolCircuit.Circuit n) (d s w l t : Nat), LE.le c.depth d → LE.le c.size s → LE.…","labels":[],"detail_key":"p14","name":"LMN.iterative_reduction_bound","module":"TCSlib.BooleanAnalysis.LMN"},{"id":"n22607","layer":"formal","project":"p14","title":"LMN.logb_2_div_eps_le_l","kind":"theorem","summary":"∀ (s : Nat), LT.lt 0 s → ∀ (ε : Real), LT.lt 0 ε → 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n : Nat (lits : List (Literal n)) (ρ : SwitchingLemma2.Restriction n) (cont : SwitchingLemma2…","labels":[],"detail_key":"p14","name":"SwitchingLemma2.termSubTree_deepPath_append","module":"TCSlib.BooleanAnalysis.Switching.CanonicalDTree"},{"id":"n22757","layer":"formal","project":"p14","title":"SwitchingLemma2.termSubTree_deepPath_split","kind":"theorem","summary":"∀ n : Nat (lits : List (Literal n)) (ρ : SwitchingLemma2.Restriction n) (cont : SwitchingLemma2…","labels":[],"detail_key":"p14","name":"SwitchingLemma2.termSubTree_deepPath_split","module":"TCSlib.BooleanAnalysis.Switching.CanonicalDTree"},{"id":"n22758","layer":"formal","project":"p14","title":"BoolCircuit.Circuit.depth","kind":"def","summary":"n : Nat → BoolCircuit.Circuit n → Nat","labels":[],"detail_key":"p14","name":"BoolCircuit.Circuit.depth","module":"TCSlib.BooleanAnalysis.Switching.Circuit"},{"id":"n22759","layer":"formal","project":"p14","title":"BoolCircuit.Circuit.litCount","kind":"def","summary":"n : Nat → 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(n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.conditionalSpecialYLaw_singleton","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23023","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.const_mul_n_le_comple…","kind":"theorem","summary":"∀ (n : PNat) (p : 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(i : Fin ↑n) (bits : Fin ↑n → Bool), Eq (Set.preimage (CommunicationComplexity.Fun…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondBits_fiber_eq_iInter","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23026","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondit…","kind":"def","summary":"(n : PNat) → Fin ↑n → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23027","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondit…","kind":"def","summary":"(n : PNat) → Fin ↑n → (Fin ↑n → Bool) → Prod (Fin ↑n → Bool) (Fin ↑n → Bool)","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceConditioningOfCondBits","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23028","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondit…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Function.Injective (CommunicationComplexity.Functions.Disjointness.R…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceConditioningOfCondBits_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23029","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceCondit…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Eq (CommunicationComplexity.Functions.Disjointness.RandomizedLowerBo…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateAliceConditioning_eq_recode_condBits","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23030","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateInput","kind":"def","summary":"(n : PNat) → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.Disj…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateInput","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23031","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateMessage","kind":"def","summary":"(n : PNat) → (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateMessage","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23032","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit","kind":"def","summary":"(n : PNat) → Fin ↑n → Fin ↑n → Bool → CommunicationComplexity.Functions.Disjointness.Randomized…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23033","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit…","kind":"theorem","summary":"∀ (n : PNat) (i k : Fin ↑n) (b : Bool), Eq (ite (LT.lt k i) (CommunicationComplexity.Functions.…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateWithCondBit_spec","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23034","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateXBefore","kind":"def","summary":"(n : PNat) → Fin ↑n → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateXBefore","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23035","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateXSet","kind":"def","summary":"(n : PNat) → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.Disj…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateXSet","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23036","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYBefore","kind":"def","summary":"(n : PNat) → Fin ↑n → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYBefore","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23037","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYGe","kind":"def","summary":"(n : PNat) → Fin ↑n → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYGe","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23038","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYSet","kind":"def","summary":"(n : PNat) → (Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.Disj…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.coordinateYSet","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23039","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_a…","kind":"theorem","summary":"∀ (n : PNat), Filter.Eventually (fun ω => Membership.mem (CommunicationComplexity.Functions.Dis…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_ae_disjointEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23040","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat), MeasureTheory.MeasurePreserving (CommunicationComplexity.Functions.Disjointness.R…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measurePreserving_dualHardSample","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23041","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (z : Prod (Fin ↑n) (Prod (Fin ↑n → CommunicationComplexity.Functions.Disjointness.…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_disjointModel_fiber","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23042","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSampl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_singleton_dualHardSample","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23043","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (bx bY : Bool), Not (And (Eq bx true) (Eq bY true)) → Eq ((CommunicationComplexity…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialBitsEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23044","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Eq ((CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinateEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23045","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n) (coords : Fin ↑n → CommunicationComplexity.Functions.Disjointness.Ran…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinate_disjointCoordinateVector_fiber","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23046","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Eq ((CommunicationComplexity.Functions.Disjointness.RandomizedLowerB…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialCoordinate_preimage_singleton","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23047","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat), Eq ((CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjoint…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialY_false","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23048","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_m…","kind":"theorem","summary":"∀ (n : PNat), Eq ((CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjoint…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCondMeasure_measureReal_specialZeroZero","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23049","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVec…","kind":"theorem","summary":"∀ (n : PNat) ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVector_xBit_of_mem_disjointEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23050","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVec…","kind":"theorem","summary":"∀ (n : PNat) ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointCoordinateVector_yBit_of_mem_disjointEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23051","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointEvent_dualHar…","kind":"theorem","summary":"∀ (n : PNat) (ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSampl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointEvent_dualHardSample","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23052","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"def","summary":"(n : PNat) → MeasureTheory.Measure (CommunicationComplexity.Functions.Disjointness.RandomizedLo…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23053","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat), Filter.Eventually (fun ω => Eq (CommunicationComplexity.Functions.Disjointness.Ra…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_ae_specialY_false","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23054","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_cond_zVariable_eq_cond_inter","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23055","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_integral_xDistance_le_of_integral_sq_le","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23056","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat), MeasureTheory.IsProbabilityMeasure (CommunicationComplexity.Functions.Disjointnes…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_isProbabilityMeasure","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23057","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (b : Bool) (c : Prod (Fin ↑n) (Prod (Fin ↑n → Bool) (Fin ↑n → Bool))), Eq ((Commun…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_measureReal_specialX_inter_coarseConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23058","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (b : Bool), Eq ((CommunicationComplexity.Functions.Disjointness.RandomizedLowerBou…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_measureReal_specialX_singleton","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23059","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_measureReal_zVariable","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23060","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat), Eq (MeasureTheory.Measure.map (CommunicationComplexity.Functions.Disjointness.Ran…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_specialX_law_eq_uniformBool","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23061","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalse…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.disjointSpecialYFalseMeasure_xDistance_bad_le_of_integral_le","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23062","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.distributionalError_i…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.distributionalError_inputDist_eq_protocolErrorEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23063","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue","kind":"def","summary":"(n : PNat) → Prod (Fin ↑n) (Prod (Fin ↑n → Bool) (Fin ↑n → Bool)) → Prod (Fin ↑n) (Prod (Fin ↑n…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23064","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue…","kind":"theorem","summary":"∀ (n : PNat) (c : Prod (Fin ↑n) (Prod (Fin ↑n → Bool) (Fin ↑n → Bool))), Eq (CommunicationCompl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue_dualConditioningValue","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23065","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue…","kind":"theorem","summary":"∀ (n : PNat), Function.Injective (CommunicationComplexity.Functions.Disjointness.RandomizedLowe…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualConditioningValue_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23066","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualHardSample","kind":"def","summary":"(n : PNat) → 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CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSampl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualHardSample_preimage_singleton","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23069","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualHardSample_preima…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualHardSample_preimage_zVariable_dualZValue_inter_specialX","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23070","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocol","kind":"def","summary":"(n : PNat) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolType n…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocol","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23071","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocolTranscrip…","kind":"def","summary":"(n : PNat) → (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocolTranscriptMap","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23072","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocolTranscrip…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualProtocolTranscriptMap_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23073","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualRawZValue","kind":"def","summary":"(n : PNat) → (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualRawZValue","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23074","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualRawZValue_injecti…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualRawZValue_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23075","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualZValue","kind":"def","summary":"(n : PNat) → (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualZValue","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23076","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualZValue_injective","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.dualZValue_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23077","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.entropy_message_le_co…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.entropy_message_le_complexity_mul_log_two_of_measure","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23078","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fiber_volume_factoriz…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fiber_volume_factorization","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23079","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainCondit…","kind":"def","summary":"(n : PNat) → (i : Fin ↑n) → Prod (Fin ↑i → Bool) (Fin ↑n → Bool) → Prod (Fin ↑n → Bool) (Prod (…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainConditioningValue","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23080","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainCondit…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Eq (Function.comp (CommunicationComplexity.Functions.Disjointness.Ra…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainConditioningValue_fixedAliceFullYConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23081","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainCondit…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Function.Injective (CommunicationComplexity.Functions.Disjointness.R…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceChainConditioningValue_injective","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23082","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceConditioning","kind":"def","summary":"(n : PNat) → Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardS…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23083","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceConditionin…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), (MeasureTheory.ae (CommunicationComplexity.Functions.Disjointness.Ra…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceConditioning_ae_eq_coordinateAliceConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23084","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceCrossInfoTe…","kind":"def","summary":"(n : PNat) → Fin ↑n → Real","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceCrossInfoTerm","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23085","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceCrossInfoTe…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), Eq (CommunicationComplexity.Functions.Disjointness.RandomizedLowerBo…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceCrossInfoTerm_eq_zero","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23086","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceFullYCondit…","kind":"def","summary":"(n : PNat) → (i : Fin ↑n) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceFullYConditioning","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23087","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceFullYInfoTe…","kind":"def","summary":"(n : PNat) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolType n…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceFullYInfoTerm","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23088","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceInfoTerm","kind":"def","summary":"(n : PNat) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolType n…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceInfoTerm","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23089","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceInfoTerm_le…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedAliceInfoTerm_le_fixedAliceFullYInfoTerm","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23090","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedXBefore","kind":"def","summary":"(n : PNat) → Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardS…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedXBefore","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23091","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedXBit","kind":"def","summary":"(n : PNat) → Fin ↑n → 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(CommunicationComplexity.Functions.Disjointness.Ra…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedYBefore_ae_eq_coordinateYBefore","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23096","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedYGe","kind":"def","summary":"(n : PNat) → Fin ↑n → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardS…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.fixedYGe","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23097","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX","kind":"def","summary":"(n : PNat) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSample n →…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23098","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX_flipSpec…","kind":"theorem","summary":"∀ (n : PNat) (ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSampl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX_flipSpecialX","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23099","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX_preimage…","kind":"theorem","summary":"∀ (n : PNat) (ω : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.HardSampl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.flipSpecialX_preimage_singleton","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23100","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.floor_div_pow_lt_publ…","kind":"theorem","summary":"∀ (n : PNat), LT.lt (↑(Nat.floor (HDiv.hDiv (↑↑n) (HPow.hPow 2 32)))) (CommunicationComplexity.…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.floor_div_pow_lt_publicCoin_communicationComplexity_disjointness","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23101","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZ","kind":"def","summary":"(n : PNat) → (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZ","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23102","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent","kind":"def","summary":"(n : PNat) → CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolType n…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23103","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarte…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarter_sub_two_mul_le_distributionalError","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23104","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarte…","kind":"theorem","summary":"∀ (n : PNat) (p : CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.ProtocolT…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.goodZEvent_mul_quarter_sub_two_mul_le_protocolErrorEvent","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23105","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_disjoint…","kind":"theorem","summary":"∀ (n : PNat), ProbabilityTheory.IdentDistrib (CommunicationComplexity.Functions.Disjointness.Ra…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_disjointCoordinateVector_uniform","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23106","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_disjoint…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), ProbabilityTheory.IdentDistrib (CommunicationComplexity.Functions.Di…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_disjointCoordinateVector_uniform_cond_specialCoordinate","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23107","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_fixedAli…","kind":"theorem","summary":"∀ (n : PNat) (i : Fin ↑n), ProbabilityTheory.IdentDistrib (fun ω => Prod.mk (CommunicationCompl…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_fixedAliceCrossInfoTriple_uniform","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23108","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_self_com…","kind":"theorem","summary":"∀ Ω : Type u_1 α : Type u_2 [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace α] μ : Measure…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_self_comp_measurePreserving","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23109","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_specialC…","kind":"theorem","summary":"∀ (n : PNat), ProbabilityTheory.IdentDistrib (fun ω => Prod.mk (CommunicationComplexity.Functio…","labels":[],"detail_key":"p14","name":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_specialCoordinate_disjointCoordinateVector_uniform_prod","module":"TCSlib.CommunicationComplexity.NewmanTheorem.FuncDisjointnessLowerBound"},{"id":"n23110","layer":"formal","project":"p14","title":"CommunicationComplexity.Functions.Disjointness.RandomizedLowerBound.identDistrib_uniformP…","kind":"theorem","summary":"∀ (n : PNat), ProbabilityTheory.IdentDistrib Prod.snd id 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(HM…","labels":[],"detail_key":"p14","name":"RSA.rsa_zmod_p","module":"TCSlib.Cryptography.RSA"},{"id":"n23448","layer":"formal","project":"p14","title":"RSA.rsa_zmod_q","kind":"theorem","summary":"∀ p q m e d k : Nat [inst : Fact (Nat.Prime q)], Eq (HMul.hMul e d) (HAdd.hAdd 1 (HMul.hMul (HM…","labels":[],"detail_key":"p14","name":"RSA.rsa_zmod_q","module":"TCSlib.Cryptography.RSA"},{"id":"n23449","layer":"formal","project":"p14","title":"TimeM.bind","kind":"def","summary":"T : Type u_1 → α : Type u_2 → β : Type u_3 → [Add T] → TimeM T α → (α → TimeM T β) → TimeM T β","labels":[],"detail_key":"p14","name":"TimeM.bind","module":"TCSlib.Cryptography.RSA"},{"id":"n23450","layer":"formal","project":"p14","title":"TimeM.pure","kind":"def","summary":"T : Type u_1 → [Zero T] → α : Type u_2 → α → TimeM T α","labels":[],"detail_key":"p14","name":"TimeM.pure","module":"TCSlib.Cryptography.RSA"},{"id":"n23451","layer":"formal","project":"p14","title":"TimeM.ret_bind","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 [inst : Add T] (m : TimeM T α) (f : α → TimeM T β), Eq (bind m f)…","labels":[],"detail_key":"p14","name":"TimeM.ret_bind","module":"TCSlib.Cryptography.RSA"},{"id":"n23452","layer":"formal","project":"p14","title":"TimeM.ret_map","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 (f : α → β) (x : TimeM T α), Eq (Functor.map f x).ret (f x.ret)","labels":[],"detail_key":"p14","name":"TimeM.ret_map","module":"TCSlib.Cryptography.RSA"},{"id":"n23453","layer":"formal","project":"p14","title":"TimeM.ret_pure","kind":"theorem","summary":"∀ T : Type u_1 α : Type u_2 [inst : Zero T] (a : α), Eq (pure a).ret a","labels":[],"detail_key":"p14","name":"TimeM.ret_pure","module":"TCSlib.Cryptography.RSA"},{"id":"n23454","layer":"formal","project":"p14","title":"TimeM.ret_seq","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 [inst : Add T] (f : TimeM T (α → β)) (x : Unit → TimeM T α), Eq (…","labels":[],"detail_key":"p14","name":"TimeM.ret_seq","module":"TCSlib.Cryptography.RSA"},{"id":"n23455","layer":"formal","project":"p14","title":"TimeM.ret_seqLeft","kind":"theorem","summary":"∀ T : Type u_1 β α : Type u_2 [inst : Add T] (x : TimeM T α) (y : Unit → TimeM T β), Eq (SeqLef…","labels":[],"detail_key":"p14","name":"TimeM.ret_seqLeft","module":"TCSlib.Cryptography.RSA"},{"id":"n23456","layer":"formal","project":"p14","title":"TimeM.ret_seqRight","kind":"theorem","summary":"∀ T : Type u_1 β α : Type u_2 (x : TimeM T α) (y : Unit → TimeM T β) [inst : Add T], Eq (SeqRig…","labels":[],"detail_key":"p14","name":"TimeM.ret_seqRight","module":"TCSlib.Cryptography.RSA"},{"id":"n23457","layer":"formal","project":"p14","title":"TimeM.ret_tick","kind":"theorem","summary":"∀ T : Type u_1 (c : T), Eq (TimeM.tick c).ret Unit.unit","labels":[],"detail_key":"p14","name":"TimeM.ret_tick","module":"TCSlib.Cryptography.RSA"},{"id":"n23458","layer":"formal","project":"p14","title":"TimeM.tick","kind":"def","summary":"T : Type u_1 → T → TimeM T PUnit.u_2 + 1","labels":[],"detail_key":"p14","name":"TimeM.tick","module":"TCSlib.Cryptography.RSA"},{"id":"n23459","layer":"formal","project":"p14","title":"TimeM.time_bind","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 [inst : Add T] (m : TimeM T α) (f : α → TimeM T β), Eq (bind m f)…","labels":[],"detail_key":"p14","name":"TimeM.time_bind","module":"TCSlib.Cryptography.RSA"},{"id":"n23460","layer":"formal","project":"p14","title":"TimeM.time_map","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 (f : α → β) (x : TimeM T α), Eq (Functor.map f x).time x.time","labels":[],"detail_key":"p14","name":"TimeM.time_map","module":"TCSlib.Cryptography.RSA"},{"id":"n23461","layer":"formal","project":"p14","title":"TimeM.time_pure","kind":"theorem","summary":"∀ T : Type u_1 α : Type u_2 [inst : Zero T] (a : α), Eq (pure a).time 0","labels":[],"detail_key":"p14","name":"TimeM.time_pure","module":"TCSlib.Cryptography.RSA"},{"id":"n23462","layer":"formal","project":"p14","title":"TimeM.time_seq","kind":"theorem","summary":"∀ T : Type u_1 α β : Type u_2 [inst : Add T] (f : TimeM T (α → β)) (x : Unit → TimeM T α), Eq (…","labels":[],"detail_key":"p14","name":"TimeM.time_seq","module":"TCSlib.Cryptography.RSA"},{"id":"n23463","layer":"formal","project":"p14","title":"TimeM.time_seqLeft","kind":"theorem","summary":"∀ T : Type u_1 β α : Type u_2 [inst : Add T] (x : TimeM T α) (y : Unit → TimeM T β), Eq (SeqLef…","labels":[],"detail_key":"p14","name":"TimeM.time_seqLeft","module":"TCSlib.Cryptography.RSA"},{"id":"n23464","layer":"formal","project":"p14","title":"TimeM.time_seqRight","kind":"theorem","summary":"∀ T : Type u_1 β α : Type u_2 [inst : Add T] (x : TimeM T α) (y : Unit → TimeM T β), Eq (SeqRig…","labels":[],"detail_key":"p14","name":"TimeM.time_seqRight","module":"TCSlib.Cryptography.RSA"},{"id":"n23465","layer":"formal","project":"p14","title":"TimeM.time_tick","kind":"theorem","summary":"∀ T : Type u_1 (c : T), Eq (TimeM.tick c).time c","labels":[],"detail_key":"p14","name":"TimeM.time_tick","module":"TCSlib.Cryptography.RSA"},{"id":"n23466","layer":"formal","project":"p14","title":"insert","kind":"def","summary":"α : Type → [LinearOrder α] → α → List α → TimeM Nat (List α)","labels":[],"detail_key":"p14","name":"insert","module":"TCSlib.Cryptography.RSA"},{"id":"n23467","layer":"formal","project":"p14","title":"insert_perm","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (x : α) (xs : List α), (_root_.insert x xs).ret.Perm (List.co…","labels":[],"detail_key":"p14","name":"insert_perm","module":"TCSlib.Cryptography.RSA"},{"id":"n23468","layer":"formal","project":"p14","title":"insert_sorted","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (x : α) (xs : List α), IsSorted xs → IsSorted (_root_.insert…","labels":[],"detail_key":"p14","name":"insert_sorted","module":"TCSlib.Cryptography.RSA"},{"id":"n23469","layer":"formal","project":"p14","title":"insertionSort","kind":"def","summary":"α : Type → [LinearOrder α] → List α → TimeM Nat (List α)","labels":[],"detail_key":"p14","name":"insertionSort","module":"TCSlib.Cryptography.RSA"},{"id":"n23470","layer":"formal","project":"p14","title":"insertionSort_correct","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), And (IsSorted (insertionSort xs).ret) ((insert…","labels":[],"detail_key":"p14","name":"insertionSort_correct","module":"TCSlib.Cryptography.RSA"},{"id":"n23471","layer":"formal","project":"p14","title":"insertionSort_perm","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), (insertionSort xs).ret.Perm xs","labels":[],"detail_key":"p14","name":"insertionSort_perm","module":"TCSlib.Cryptography.RSA"},{"id":"n23472","layer":"formal","project":"p14","title":"insertionSort_sorted","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), IsSorted (insertionSort xs).ret","labels":[],"detail_key":"p14","name":"insertionSort_sorted","module":"TCSlib.Cryptography.RSA"},{"id":"n23473","layer":"formal","project":"p14","title":"isSorted_iff_isChain","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), Iff (IsSorted xs) (List.IsChain (fun x1 x2 =>…","labels":[],"detail_key":"p14","name":"isSorted_iff_isChain","module":"TCSlib.Cryptography.RSA"},{"id":"n23474","layer":"formal","project":"p14","title":"isSorted_iff_sorted","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), Iff (IsSorted xs) (List.Sorted (fun x1 x2 => L…","labels":[],"detail_key":"p14","name":"isSorted_iff_sorted","module":"TCSlib.Cryptography.RSA"},{"id":"n23475","layer":"formal","project":"p14","title":"length_ret_insert","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (x : α) (xs : List α), Eq (_root_.insert x xs).ret.length (HA…","labels":[],"detail_key":"p14","name":"length_ret_insert","module":"TCSlib.Cryptography.RSA"},{"id":"n23476","layer":"formal","project":"p14","title":"length_ret_insertionSort","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), Eq (insertionSort xs).ret.length xs.length","labels":[],"detail_key":"p14","name":"length_ret_insertionSort","module":"TCSlib.Cryptography.RSA"},{"id":"n23477","layer":"formal","project":"p14","title":"max2","kind":"def","summary":"Nat → Nat → TimeM Nat Nat","labels":[],"detail_key":"p14","name":"max2","module":"TCSlib.Cryptography.RSA"},{"id":"n23478","layer":"formal","project":"p14","title":"median3","kind":"def","summary":"α : Type → [LinearOrder α] → α → α → α → TimeM Nat α","labels":[],"detail_key":"p14","name":"median3","module":"TCSlib.Cryptography.RSA"},{"id":"n23479","layer":"formal","project":"p14","title":"median3_correct","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (a b c : α), isMedian (median3 a b c).ret a b c","labels":[],"detail_key":"p14","name":"median3_correct","module":"TCSlib.Cryptography.RSA"},{"id":"n23480","layer":"formal","project":"p14","title":"median3_time","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (a b c : α), LE.le (median3 a b c).time 3","labels":[],"detail_key":"p14","name":"median3_time","module":"TCSlib.Cryptography.RSA"},{"id":"n23481","layer":"formal","project":"p14","title":"ret_insert","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (x : α) (xs : List α), Eq (_root_.insert x xs).ret (List.orde…","labels":[],"detail_key":"p14","name":"ret_insert","module":"TCSlib.Cryptography.RSA"},{"id":"n23482","layer":"formal","project":"p14","title":"ret_insertionSort","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), Eq (insertionSort xs).ret (List.insertionSort…","labels":[],"detail_key":"p14","name":"ret_insertionSort","module":"TCSlib.Cryptography.RSA"},{"id":"n23483","layer":"formal","project":"p14","title":"timeInsertionSortRec","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p14","name":"timeInsertionSortRec","module":"TCSlib.Cryptography.RSA"},{"id":"n23484","layer":"formal","project":"p14","title":"timeInsertionSortRec_le_sq","kind":"theorem","summary":"∀ (n : Nat), LE.le (timeInsertionSortRec n) (HMul.hMul n n)","labels":[],"detail_key":"p14","name":"timeInsertionSortRec_le_sq","module":"TCSlib.Cryptography.RSA"},{"id":"n23485","layer":"formal","project":"p14","title":"time_insert_le","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (x : α) (xs : List α), LE.le (_root_.insert x xs).time xs.len…","labels":[],"detail_key":"p14","name":"time_insert_le","module":"TCSlib.Cryptography.RSA"},{"id":"n23486","layer":"formal","project":"p14","title":"time_insertionSort_le_rec","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), LE.le (insertionSort xs).time (timeInsertionSo…","labels":[],"detail_key":"p14","name":"time_insertionSort_le_rec","module":"TCSlib.Cryptography.RSA"},{"id":"n23487","layer":"formal","project":"p14","title":"time_insertionSort_le_sq","kind":"theorem","summary":"∀ α : Type [inst : LinearOrder α] (xs : List α), LE.le (insertionSort xs).time (HMul.hMul xs.le…","labels":[],"detail_key":"p14","name":"time_insertionSort_le_sq","module":"TCSlib.Cryptography.RSA"},{"id":"n23488","layer":"formal","project":"p14","title":"Schnorr.extract","kind":"def","summary":"q : Nat → [Fact (Nat.Prime q)] → ZMod q → ZMod q → ZMod q → ZMod q → ZMod q","labels":[],"detail_key":"p14","name":"Schnorr.extract","module":"TCSlib.Cryptography.SchnorrProtocol"},{"id":"n23489","layer":"formal","project":"p14","title":"Schnorr.schnorr_soundness","kind":"theorem","summary":"∀ G : Type u_1 [inst : CommGroup G] q : Nat [inst_1 : Fact (Nat.Prime q)] (g : G), Eq (orderOf…","labels":[],"detail_key":"p14","name":"Schnorr.schnorr_soundness","module":"TCSlib.Cryptography.SchnorrProtocol"},{"id":"n23490","layer":"formal","project":"p14","title":"CodingTheory.Johnson.AdmissibleCode","kind":"def","summary":"(n : Nat) → Nat → Nat → Finset (CodingTheory.Johnson.BitVec n) → Prop","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.AdmissibleCode","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23491","layer":"formal","project":"p14","title":"CodingTheory.Johnson.BitVec","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.BitVec","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23492","layer":"formal","project":"p14","title":"CodingTheory.Johnson.Euc","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.Euc","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23493","layer":"formal","project":"p14","title":"CodingTheory.Johnson.J2","kind":"def","summary":"Nat → Nat → Real","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.J2","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23494","layer":"formal","project":"p14","title":"CodingTheory.Johnson.alpha","kind":"def","summary":"Nat → Nat → Real","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.alpha","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23495","layer":"formal","project":"p14","title":"CodingTheory.Johnson.alpha_lt_one_of_hd1","kind":"theorem","summary":"∀ n d : Nat, LT.lt 0 n → LE.le 1 d → LE.le (HMul.hMul 2 d) n → LT.lt (CodingTheory.Johnson.alph…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.alpha_lt_one_of_hd1","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23496","layer":"formal","project":"p14","title":"CodingTheory.Johnson.alpha_nonneg","kind":"theorem","summary":"∀ (n d : Nat), LE.le 0 (CodingTheory.Johnson.alpha n d)","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.alpha_nonneg","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23497","layer":"formal","project":"p14","title":"CodingTheory.Johnson.alpha_sq","kind":"theorem","summary":"∀ n d : Nat, LT.lt 0 n → LE.le (HMul.hMul 2 d) n → Eq (HPow.hPow (CodingTheory.Johnson.alpha n…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.alpha_sq","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23498","layer":"formal","project":"p14","title":"CodingTheory.Johnson.binary_johnson_card_bound","kind":"theorem","summary":"∀ n d w : Nat, LT.lt 0 n → LE.le 1 d → LE.le (HMul.hMul 2 d) n → ∀ (C : Finset (CodingTheory.Jo…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.binary_johnson_card_bound","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23499","layer":"formal","project":"p14","title":"CodingTheory.Johnson.binary_johnson_card_bound_of_admissible","kind":"theorem","summary":"∀ n d w : Nat, LT.lt 0 n → LE.le 1 d → LE.le (HMul.hMul 2 d) n → ∀ (C : Finset (CodingTheory.Jo…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.binary_johnson_card_bound_of_admissible","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23500","layer":"formal","project":"p14","title":"CodingTheory.Johnson.binary_johnson_card_bound_parametric","kind":"theorem","summary":"∀ n d w : Nat, LT.lt 0 n → ∀ (C : Finset (CodingTheory.Johnson.BitVec n)), (∀ (x : CodingTheory…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.binary_johnson_card_bound_parametric","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23501","layer":"formal","project":"p14","title":"CodingTheory.Johnson.card_filter_add_two","kind":"theorem","summary":"∀ V : Type u_1 [inst : AddGroup V] (S : Finset V) (u : V), Membership.mem S u → LE.le S.card (H…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.card_filter_add_two","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23502","layer":"formal","project":"p14","title":"CodingTheory.Johnson.coord_mul_pmOne","kind":"theorem","summary":"∀ n : Nat (x y : CodingTheory.Johnson.BitVec n) (i : Fin n), Eq (HMul.hMul ((CodingTheory.Johns…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.coord_mul_pmOne","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23503","layer":"formal","project":"p14","title":"CodingTheory.Johnson.finrank_orthogonal_span_singleton","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] [FiniteDimensi…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.finrank_orthogonal_span_singleton","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23504","layer":"formal","project":"p14","title":"CodingTheory.Johnson.hdist","kind":"def","summary":"n : Nat → CodingTheory.Johnson.BitVec n → CodingTheory.Johnson.BitVec n → Nat","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.hdist","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23505","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_ones_ones","kind":"theorem","summary":"∀ n : Nat, Eq (RCLike.wInner 1 CodingTheory.Johnson.ones.ofLp CodingTheory.Johnson.ones.ofLp) ↑n","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_ones_ones","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23506","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_pmOne_ones","kind":"theorem","summary":"∀ n : Nat (x : CodingTheory.Johnson.BitVec n), Eq (RCLike.wInner 1 (CodingTheory.Johnson.pmOne…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_pmOne_ones","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23507","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_pmOne_pmOne","kind":"theorem","summary":"∀ n : Nat (x y : CodingTheory.Johnson.BitVec n), Eq (RCLike.wInner 1 (CodingTheory.Johnson.pmOn…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_pmOne_pmOne","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23508","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_proj_le_zero","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u x y : V), E…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_proj_le_zero","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23509","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_shifted_expand","kind":"theorem","summary":"∀ n : Nat (α : Real) (x y : CodingTheory.Johnson.BitVec n), Eq (RCLike.wInner 1 (CodingTheory.J…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_shifted_expand","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23510","layer":"formal","project":"p14","title":"CodingTheory.Johnson.inner_shifted_le_expr","kind":"theorem","summary":"∀ n d w : Nat α : Real x y : CodingTheory.Johnson.BitVec n, LE.le 0 α → LE.le d (CodingTheory.J…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.inner_shifted_le_expr","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23511","layer":"formal","project":"p14","title":"CodingTheory.Johnson.johnson_arith","kind":"theorem","summary":"∀ n d w : Nat, LT.lt 0 n → LE.le (HMul.hMul 2 d) n → LE.le (↑w) (CodingTheory.Johnson.J2 n d) →…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.johnson_arith","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23512","layer":"formal","project":"p14","title":"CodingTheory.Johnson.norm_normalized_orthProj","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u v : V), Eq…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.norm_normalized_orthProj","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23513","layer":"formal","project":"p14","title":"CodingTheory.Johnson.normalize","kind":"def","summary":"n : Nat → CodingTheory.Johnson.Euc n → CodingTheory.Johnson.Euc n","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.normalize","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23514","layer":"formal","project":"p14","title":"CodingTheory.Johnson.normalized_orthProj_injective","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u v w : V), E…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.normalized_orthProj_injective","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23515","layer":"formal","project":"p14","title":"CodingTheory.Johnson.normalized_orthProj_inner_nonpos","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u v w : V), E…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.normalized_orthProj_inner_nonpos","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23516","layer":"formal","project":"p14","title":"CodingTheory.Johnson.ones","kind":"def","summary":"n : Nat → CodingTheory.Johnson.Euc n","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.ones","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23517","layer":"formal","project":"p14","title":"CodingTheory.Johnson.ones_apply","kind":"theorem","summary":"∀ n : Nat (i : Fin n), Eq (CodingTheory.Johnson.ones.ofLp i) 1","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.ones_apply","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23518","layer":"formal","project":"p14","title":"CodingTheory.Johnson.orthProj","kind":"def","summary":"V : Type u_1 → [inst : NormedAddCommGroup V] → [InnerProductSpace Real V] → V → V → V","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.orthProj","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23519","layer":"formal","project":"p14","title":"CodingTheory.Johnson.orthProj_inner_nonpos","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u v w 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((CodingTheory.Jo…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.pmOne_apply_true","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23525","layer":"formal","project":"p14","title":"CodingTheory.Johnson.proj_inj_on","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u : V) (S : S…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.proj_inj_on","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23526","layer":"formal","project":"p14","title":"CodingTheory.Johnson.proj_nonzero","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] (u x : V), Eq…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.proj_nonzero","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23527","layer":"formal","project":"p14","title":"CodingTheory.Johnson.proj_norm_sq","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup 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Membe…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.rankin_finset_bound","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23530","layer":"formal","project":"p14","title":"CodingTheory.Johnson.shifted","kind":"def","summary":"n : Nat → Real → CodingTheory.Johnson.BitVec n → CodingTheory.Johnson.Euc n","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.shifted","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23531","layer":"formal","project":"p14","title":"CodingTheory.Johnson.shifted_ne_zero_of_alpha_lt_one","kind":"theorem","summary":"∀ n : Nat, LT.lt 0 n → ∀ α : Real, LE.le 0 α → LT.lt α 1 → ∀ (x : CodingTheory.Johnson.BitVec n…","labels":[],"detail_key":"p14","name":"CodingTheory.Johnson.shifted_ne_zero_of_alpha_lt_one","module":"TCSlib.ErrorCorrectingCodes.JohnsonBound"},{"id":"n23532","layer":"formal","project":"p14","title":"CodingTheory.Johnson.wt","kind":"def","summary":"n : Nat → CodingTheory.Johnson.BitVec n → 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(Real.logb 2 ((F…","labels":[],"detail_key":"p14","name":"binomial_tail_entropy_asymptotic","module":"TCSlib.ErrorCorrectingCodes.MRRW"},{"id":"n23537","layer":"formal","project":"p14","title":"cdKernel","kind":"def","summary":"Nat → Nat → Real → Real → Real","labels":[],"detail_key":"p14","name":"cdKernel","module":"TCSlib.ErrorCorrectingCodes.MRRW"},{"id":"n23538","layer":"formal","project":"p14","title":"cdKernel_nonpos_beyond_threshold","kind":"theorem","summary":"∀ (n t : Nat), LE.le (HAdd.hAdd t 1) n → ∀ (a : Real), (∀ (ξ : Real), Eq (Polynomial.eval ξ (kr…","labels":[],"detail_key":"p14","name":"cdKernel_nonpos_beyond_threshold","module":"TCSlib.ErrorCorrectingCodes.MRRW"},{"id":"n23539","layer":"formal","project":"p14","title":"cd_identity","kind":"theorem","summary":"∀ (n t : Nat), LE.le (HAdd.hAdd t 1) n → ∀ (a x : Real), Ne a x → Exists fun c => And (GT.gt 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(Kruskal.WEdge…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges","module":"TCSlib.GraphTheory.Kruskal.Basic"},{"id":"n24202","layer":"formal","project":"p14","title":"Kruskal.processEdges_mem","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (uf : Kruskal.UF n) (acc : List (Kruskal.WEdge n)) (…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_mem","module":"TCSlib.GraphTheory.Kruskal.Basic"},{"id":"n24203","layer":"formal","project":"p14","title":"Kruskal.exchange_take_head","kind":"theorem","summary":"∀ n : Nat base rest : List (Kruskal.WEdge n) g e : Kruskal.WEdge n, Kruskal.Reach (HAppend.hApp…","labels":[],"detail_key":"p14","name":"Kruskal.exchange_take_head","module":"TCSlib.GraphTheory.Kruskal.Exchange"},{"id":"n24204","layer":"formal","project":"p14","title":"Kruskal.exchange_take_head'","kind":"theorem","summary":"∀ n : Nat base rest : List (Kruskal.WEdge n) g e : Kruskal.WEdge n, Kruskal.Reach (HAppend.hApp…","labels":[],"detail_key":"p14","name":"Kruskal.exchange_take_head'","module":"TCSlib.GraphTheory.Kruskal.Exchange"},{"id":"n24205","layer":"formal","project":"p14","title":"Kruskal.exchange_with_base","kind":"theorem","summary":"∀ n : Nat base S : List (Kruskal.WEdge n) e : Kruskal.WEdge n, Kruskal.Reach (HAppend.hAppend b…","labels":[],"detail_key":"p14","name":"Kruskal.exchange_with_base","module":"TCSlib.GraphTheory.Kruskal.Exchange"},{"id":"n24206","layer":"formal","project":"p14","title":"Kruskal.reduce_to_rest","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (e : Kruskal.WEdge n) (rest S : List (Kruskal.WEdge n)), (∀ (x :…","labels":[],"detail_key":"p14","name":"Kruskal.reduce_to_rest","module":"TCSlib.GraphTheory.Kruskal.Exchange"},{"id":"n24207","layer":"formal","project":"p14","title":"Kruskal.uf_exchange","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (base S : List (Kruskal.WEdge n)) (e : Kruskal.WEdge n), (∀ (a b…","labels":[],"detail_key":"p14","name":"Kruskal.uf_exchange","module":"TCSlib.GraphTheory.Kruskal.Exchange"},{"id":"n24208","layer":"formal","project":"p14","title":"Kruskal.kruskal_optimal","kind":"theorem","summary":"∀ n : Nat (E S : List (Kruskal.WEdge n)), (∀ (e : Kruskal.WEdge n), Membership.mem S e → Member…","labels":[],"detail_key":"p14","name":"Kruskal.kruskal_optimal","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24209","layer":"formal","project":"p14","title":"Kruskal.kruskal_spans","kind":"theorem","summary":"∀ n : Nat (E : List (Kruskal.WEdge n)), Kruskal.SpansLike (Kruskal.kruskal n E) E","labels":[],"detail_key":"p14","name":"Kruskal.kruskal_spans","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24210","layer":"formal","project":"p14","title":"Kruskal.processEdges_acc","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (uf : Kruskal.UF n) (acc : List (Kruskal.WEdge n)),…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_acc","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24211","layer":"formal","project":"p14","title":"Kruskal.processEdges_optimal","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (uf : Kruskal.UF n) (base : List (Kruskal.WEdge n)),…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_optimal","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24212","layer":"formal","project":"p14","title":"Kruskal.processEdges_same_partition","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (uf : Kruskal.UF n), (uf.mergeAll (Kruskal.processEd…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_same_partition","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24213","layer":"formal","project":"p14","title":"Kruskal.processEdges_skip","kind":"theorem","summary":"∀ n : Nat e : Kruskal.WEdge n rest : List (Kruskal.WEdge n) uf : Kruskal.UF n, Eq (uf.find e.u)…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_skip","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24214","layer":"formal","project":"p14","title":"Kruskal.processEdges_take_weight","kind":"theorem","summary":"∀ n : Nat e : Kruskal.WEdge n rest : List (Kruskal.WEdge n) uf : Kruskal.UF n, Ne (uf.find e.u)…","labels":[],"detail_key":"p14","name":"Kruskal.processEdges_take_weight","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24215","layer":"formal","project":"p14","title":"Kruskal.ufAfterProcessEdges","kind":"def","summary":"n : Nat → List (Kruskal.WEdge n) → Kruskal.UF n → Kruskal.UF n","labels":[],"detail_key":"p14","name":"Kruskal.ufAfterProcessEdges","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24216","layer":"formal","project":"p14","title":"Kruskal.ufAfterProcessEdges_partition","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (uf : Kruskal.UF n), (Kruskal.ufAfterProcessEdges ed…","labels":[],"detail_key":"p14","name":"Kruskal.ufAfterProcessEdges_partition","module":"TCSlib.GraphTheory.Kruskal.Optimality"},{"id":"n24217","layer":"formal","project":"p14","title":"Kruskal.SpansLike","kind":"def","summary":"n : Nat → List (Kruskal.WEdge n) → List (Kruskal.WEdge n) → Prop","labels":[],"detail_key":"p14","name":"Kruskal.SpansLike","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24218","layer":"formal","project":"p14","title":"Kruskal.reach_lift","kind":"theorem","summary":"∀ n : Nat edges edges' : List (Kruskal.WEdge n), (∀ (a b : Fin n), Kruskal.SymAdj edges a b → K…","labels":[],"detail_key":"p14","name":"Kruskal.reach_lift","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24219","layer":"formal","project":"p14","title":"Kruskal.reach_mem_iff","kind":"theorem","summary":"∀ n : Nat l1 l2 : List (Kruskal.WEdge n), (∀ (x : Kruskal.WEdge n), Iff (Membership.mem l1 x) (…","labels":[],"detail_key":"p14","name":"Kruskal.reach_mem_iff","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24220","layer":"formal","project":"p14","title":"Kruskal.totalWeight","kind":"def","summary":"n : Nat → List (Kruskal.WEdge n) → Nat","labels":[],"detail_key":"p14","name":"Kruskal.totalWeight","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24221","layer":"formal","project":"p14","title":"Kruskal.totalWeight_erase","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (e : Kruskal.WEdge n), Membership.mem edges e → Eq (…","labels":[],"detail_key":"p14","name":"Kruskal.totalWeight_erase","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24222","layer":"formal","project":"p14","title":"Kruskal.totalWeight_nil","kind":"theorem","summary":"∀ n : Nat, Eq (Kruskal.totalWeight List.nil) 0","labels":[],"detail_key":"p14","name":"Kruskal.totalWeight_nil","module":"TCSlib.GraphTheory.Kruskal.Reach"},{"id":"n24223","layer":"formal","project":"p14","title":"Kruskal.UF.SamePartition.rfl","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n), uf.SamePartition uf","labels":[],"detail_key":"p14","name":"Kruskal.UF.SamePartition.rfl","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24224","layer":"formal","project":"p14","title":"Kruskal.UF.SamePartition.symm","kind":"theorem","summary":"∀ n : Nat uf1 uf2 : Kruskal.UF n, uf1.SamePartition uf2 → uf2.SamePartition uf1","labels":[],"detail_key":"p14","name":"Kruskal.UF.SamePartition.symm","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24225","layer":"formal","project":"p14","title":"Kruskal.UF.SamePartition.trans","kind":"theorem","summary":"∀ n : Nat uf1 uf2 uf3 : Kruskal.UF n, uf1.SamePartition uf2 → uf2.SamePartition uf3 → uf1.SameP…","labels":[],"detail_key":"p14","name":"Kruskal.UF.SamePartition.trans","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24226","layer":"formal","project":"p14","title":"Kruskal.erase_append_perm","kind":"theorem","summary":"∀ n : Nat l : List (Kruskal.WEdge n) e : Kruskal.WEdge n, Membership.mem l e → l.Perm (HAppend.…","labels":[],"detail_key":"p14","name":"Kruskal.erase_append_perm","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24227","layer":"formal","project":"p14","title":"Kruskal.mergeAll_append_single","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (l : List (Kruskal.WEdge n)) (e : Kruskal.WEdge n), Eq (uf.mergeA…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_append_single","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24228","layer":"formal","project":"p14","title":"Kruskal.mergeAll_congr","kind":"theorem","summary":"∀ n : Nat uf1 uf2 : Kruskal.UF n (edges : List (Kruskal.WEdge n)), uf1.SamePartition uf2 → (uf1…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_congr","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24229","layer":"formal","project":"p14","title":"Kruskal.mergeAll_cons_eq","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (e : Kruskal.WEdge n) (S : List (Kruskal.WEdge n)), (uf.mergeAll…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_cons_eq","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24230","layer":"formal","project":"p14","title":"Kruskal.mergeAll_iff_reach","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (base edges : List (Kruskal.WEdge n)), (∀ (a b : Fin n), Iff (Eq…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_iff_reach","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24231","layer":"formal","project":"p14","title":"Kruskal.mergeAll_init_iff","kind":"theorem","summary":"∀ n : Nat (edges : List (Kruskal.WEdge n)) (a b : Fin n), Iff (Eq ((Kruskal.UF.init n).mergeAll…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_init_iff","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24232","layer":"formal","project":"p14","title":"Kruskal.mergeAll_perm","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) l1 l2 : List (Kruskal.WEdge n), l1.Perm l2 → (uf.mergeAll l1).Sam…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_perm","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24233","layer":"formal","project":"p14","title":"Kruskal.mergeAll_preserves","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (edges : List (Kruskal.WEdge n)) a b : Fin n, Eq (uf a) (uf b) →…","labels":[],"detail_key":"p14","name":"Kruskal.mergeAll_preserves","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24234","layer":"formal","project":"p14","title":"Kruskal.merge_noop","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (i j : Fin n), Eq (uf i) (uf j) → Eq (uf.merge i j) uf","labels":[],"detail_key":"p14","name":"Kruskal.merge_noop","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24235","layer":"formal","project":"p14","title":"Kruskal.merge_swap_partition","kind":"theorem","summary":"∀ n : Nat (uf : Kruskal.UF n) (a b : Kruskal.WEdge n), ((uf.merge a.u a.v).merge b.u b.v).SameP…","labels":[],"detail_key":"p14","name":"Kruskal.merge_swap_partition","module":"TCSlib.GraphTheory.Kruskal.UnionFind"},{"id":"n24236","layer":"formal","project":"p14","title":"binomial_ratio_lower","kind":"theorem","summary":"∀ (n ell q : Nat), LT.lt 0 n → LT.lt 0 q → LE.le q ell → LE.le (HMul.hMul 2 ell) n → LE.le (HMu…","labels":[],"detail_key":"p14","name":"binomial_ratio_lower","module":"TCSlib.KikuchiLDC.LDC.BackgroundFacts"},{"id":"n24237","layer":"formal","project":"p14","title":"binomial_ratio_upper","kind":"theorem","summary":"∀ (n ell q : Nat), LT.lt 0 n → LT.lt 0 q → LE.le q ell → LE.le (HMul.hMul 2 ell) n → LE.le (HMu…","labels":[],"detail_key":"p14","name":"binomial_ratio_upper","module":"TCSlib.KikuchiLDC.LDC.BackgroundFacts"},{"id":"n24238","layer":"formal","project":"p14","title":"chooseR","kind":"def","summary":"Nat → Nat → Real","labels":[],"detail_key":"p14","name":"chooseR","module":"TCSlib.KikuchiLDC.LDC.BackgroundFacts"},{"id":"n24239","layer":"formal","project":"p14","title":"matrix_khintchine","kind":"theorem","summary":"∀ (k : Nat) (sigma_sq : Real) (d1 d2 : Nat), LT.lt 0 k → LT.lt 0 sigma_sq → LT.lt 0 d1 → LT.lt…","labels":[],"detail_key":"p14","name":"matrix_khintchine","module":"TCSlib.KikuchiLDC.LDC.BackgroundFacts"},{"id":"n24240","layer":"formal","project":"p14","title":"normal_form_reduction","kind":"theorem","summary":"∀ (q k n : Nat) (delta epsilon : Real), LE.le 2 q → LT.lt 0 k → LT.lt 0 n → LT.lt 0 delta → LT.…","labels":[],"detail_key":"p14","name":"normal_form_reduction","module":"TCSlib.KikuchiLDC.LDC.BackgroundFacts"},{"id":"n24241","layer":"formal","project":"p14","title":"heavyPairs","kind":"def","summary":"(n : Nat) → Hypergraph n → Nat → Finset (Finset (Fin n))","labels":[],"detail_key":"p14","name":"heavyPairs","module":"TCSlib.KikuchiLDC.LDC.Decomposition"},{"id":"n24242","layer":"formal","project":"p14","title":"heavy_pair_count_bound","kind":"theorem","summary":"∀ (n : Nat) (H : Hypergraph n) (d : Nat), LT.lt 0 d → H.IsUniform 3 → LE.le (HMul.hMul (heavyPa…","labels":[],"detail_key":"p14","name":"heavy_pair_count_bound","module":"TCSlib.KikuchiLDC.LDC.Decomposition"},{"id":"n24243","layer":"formal","project":"p14","title":"hypergraph_decomposition","kind":"theorem","summary":"∀ (k n : Nat) (L : NormalLDC k n) (d : Nat), LT.lt 0 d → Exists fun H' => And (∀ (i : Fin k), S…","labels":[],"detail_key":"p14","name":"hypergraph_decomposition","module":"TCSlib.KikuchiLDC.LDC.Decomposition"},{"id":"n24244","layer":"formal","project":"p14","title":"DoubleCopy","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"DoubleCopy","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24245","layer":"formal","project":"p14","title":"Hypergraph","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p14","name":"Hypergraph","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24246","layer":"formal","project":"p14","title":"Hypergraph.IsMatching","kind":"def","summary":"n : Nat → Hypergraph n → Prop","labels":[],"detail_key":"p14","name":"Hypergraph.IsMatching","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24247","layer":"formal","project":"p14","title":"Hypergraph.IsUniform","kind":"def","summary":"n : Nat → Hypergraph n → Nat → Prop","labels":[],"detail_key":"p14","name":"Hypergraph.IsUniform","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24248","layer":"formal","project":"p14","title":"Hypergraph.degree","kind":"def","summary":"n : Nat → Hypergraph n → Finset (Fin n) → Nat","labels":[],"detail_key":"p14","name":"Hypergraph.degree","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24249","layer":"formal","project":"p14","title":"Hypergraph.pairDegreeBound","kind":"def","summary":"n : Nat → Hypergraph n → Nat → Prop","labels":[],"detail_key":"p14","name":"Hypergraph.pairDegreeBound","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24250","layer":"formal","project":"p14","title":"IsPMOne","kind":"def","summary":"Int → Prop","labels":[],"detail_key":"p14","name":"IsPMOne","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24251","layer":"formal","project":"p14","title":"NormalLDC","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p14","name":"NormalLDC","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24252","layer":"formal","project":"p14","title":"NormalLDC.combined","kind":"def","summary":"k n : Nat → NormalLDC k n → Hypergraph n","labels":[],"detail_key":"p14","name":"NormalLDC.combined","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24253","layer":"formal","project":"p14","title":"NormalLDC.totalConstraints","kind":"def","summary":"k n : Nat → NormalLDC k n → Nat","labels":[],"detail_key":"p14","name":"NormalLDC.totalConstraints","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24254","layer":"formal","project":"p14","title":"NormalLDC.xorPoly","kind":"def","summary":"k n : Nat → NormalLDC k n → (Fin k → Int) → (Fin n → Int) → Real","labels":[],"detail_key":"p14","name":"NormalLDC.xorPoly","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24255","layer":"formal","project":"p14","title":"NormalLDC.xorVal","kind":"def","summary":"k n : Nat → NormalLDC k n → (Fin k → Int) → Real","labels":[],"detail_key":"p14","name":"NormalLDC.xorVal","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24256","layer":"formal","project":"p14","title":"assignment_prod","kind":"def","summary":"n : Nat → (Fin n → Int) → Finset (Fin n) → Int","labels":[],"detail_key":"p14","name":"assignment_prod","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24257","layer":"formal","project":"p14","title":"copy1","kind":"def","summary":"n : Nat → Fin n → DoubleCopy n","labels":[],"detail_key":"p14","name":"copy1","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24258","layer":"formal","project":"p14","title":"copy2","kind":"def","summary":"n : Nat → Fin n → DoubleCopy n","labels":[],"detail_key":"p14","name":"copy2","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24259","layer":"formal","project":"p14","title":"liftCopy1","kind":"def","summary":"n : Nat → Finset (Fin n) → Finset (DoubleCopy n)","labels":[],"detail_key":"p14","name":"liftCopy1","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24260","layer":"formal","project":"p14","title":"liftCopy2","kind":"def","summary":"n : Nat → Finset (Fin n) → Finset (DoubleCopy n)","labels":[],"detail_key":"p14","name":"liftCopy2","module":"TCSlib.KikuchiLDC.LDC.Defs"},{"id":"n24261","layer":"formal","project":"p14","title":"high_value_observation","kind":"theorem","summary":"∀ (k n : Nat) (L : NormalLDC k n), LT.lt 0 k → LT.lt 0 n → LT.lt 0 L.totalConstraints → LE.le L…","labels":[],"detail_key":"p14","name":"high_value_observation","module":"TCSlib.KikuchiLDC.LDC.HighValue"},{"id":"n24262","layer":"formal","project":"p14","title":"even_q_ldc_lower_bound","kind":"theorem","summary":"∀ (q k n : Nat) (delta epsilon : Real), Dvd.dvd 2 q → LE.le 2 q → LT.lt 0 k → LE.le 2 n → LT.lt…","labels":[],"detail_key":"p14","name":"even_q_ldc_lower_bound","module":"TCSlib.KikuchiLDC.LDC.MainTheorem"},{"id":"n24263","layer":"formal","project":"p14","title":"main_theorem_ldc_lower_bound","kind":"theorem","summary":"∀ (k n : Nat) (delta epsilon : Real), LT.lt 0 k → LE.le 2 n → LT.lt 0 delta → LE.le delta 1 → L…","labels":[],"detail_key":"p14","name":"main_theorem_ldc_lower_bound","module":"TCSlib.KikuchiLDC.LDC.MainTheorem"},{"id":"n24264","layer":"formal","project":"p14","title":"cauchy_schwarz_trick","kind":"theorem","summary":"∀ (n m : Nat) (val_f val_fLR : Real), LT.lt 0 n → LT.lt 0 m → LE.le (HMul.hMul 9 (HPow.hPow val…","labels":[],"detail_key":"p14","name":"cauchy_schwarz_trick","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24265","layer":"formal","project":"p14","title":"nonzero_entry_count","kind":"theorem","summary":"∀ (n k ell : Nat), LE.le 4 n → LT.lt 0 k → LE.le 2 ell → LE.le (HMul.hMul 2 ell) (HMul.hMul 2 n…","labels":[],"detail_key":"p14","name":"nonzero_entry_count","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24266","layer":"formal","project":"p14","title":"row_bound_le_two_d","kind":"theorem","summary":"∀ (d row_nnz : Nat), LE.le row_nnz (HMul.hMul 2 d) → LE.le row_nnz (HMul.hMul 2 d)","labels":[],"detail_key":"p14","name":"row_bound_le_two_d","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24267","layer":"formal","project":"p14","title":"spectral_certificate_bound","kind":"theorem","summary":"∀ (val_fLR N D spectral_norm_A : Real), LT.lt 0 D → LT.lt 0 N → LE.le val_fLR (HMul.hMul (HDiv.…","labels":[],"detail_key":"p14","name":"spectral_certificate_bound","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24268","layer":"formal","project":"p14","title":"spectral_norm_bound","kind":"theorem","summary":"∀ (k d ell n : Nat), LT.lt 0 k → LT.lt 0 d → LE.le 2 n → LT.lt 0 ell → Exists fun C₁ => And (GT…","labels":[],"detail_key":"p14","name":"spectral_norm_bound","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24269","layer":"formal","project":"p14","title":"three_xor_refutation","kind":"theorem","summary":"∀ (k n d : Nat), LT.lt 0 k → LE.le 2 n → LT.lt 0 d → ∀ (m : Nat), LE.le m (HMul.hMul n k) → Exi…","labels":[],"detail_key":"p14","name":"three_xor_refutation","module":"TCSlib.KikuchiLDC.LDC.ThreeXOR"},{"id":"n24270","layer":"formal","project":"p14","title":"twoXORVal","kind":"def","summary":"(k : Nat) → Nat → Nat → Nat → (Fin k → Int) → Real","labels":[],"detail_key":"p14","name":"twoXORVal","module":"TCSlib.KikuchiLDC.LDC.TwoXOR"},{"id":"n24271","layer":"formal","project":"p14","title":"two_xor_refutation","kind":"theorem","summary":"∀ (k n d : Nat), LT.lt 0 k → LE.le 2 n → LT.lt 0 d → Exists fun C₀ => And (GT.gt C₀ 0) (∀ (nP 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: Type u_1 [inst : MeasurableSpace Ω] (A : Ω → Matrix (Fin k) (Fin d) Real), Meas…","labels":[],"detail_key":"p14","name":"measurableSet_badSingle","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Main"},{"id":"n24353","layer":"formal","project":"p14","title":"hasSubgaussianMGF_entry","kind":"theorem","summary":"∀ (k d : Nat) (i : Fin k) (j : Fin d), ProbabilityTheory.HasSubgaussianMGF (fun ω => ω i j) 1 (…","labels":[],"detail_key":"p14","name":"hasSubgaussianMGF_entry","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24354","layer":"formal","project":"p14","title":"hasSubgaussianMGF_id_rademacherReal","kind":"theorem","summary":"ProbabilityTheory.HasSubgaussianMGF id 1 rademacherReal","labels":[],"detail_key":"p14","name":"hasSubgaussianMGF_id_rademacherReal","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24355","layer":"formal","project":"p14","title":"hasSubgaussianMGF_row_proj","kind":"theorem","summary":"∀ (k d : Nat) (hk_pos : LT.lt 0 k) (x : EuclideanSpace Real (Fin d)) (i : Fin k), ProbabilityTh…","labels":[],"detail_key":"p14","name":"hasSubgaussianMGF_row_proj","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24356","layer":"formal","project":"p14","title":"hasSubgaussianMGF_scaled_entry","kind":"theorem","summary":"∀ (k d : Nat) (hk_pos : LT.lt 0 k) (x : EuclideanSpace Real (Fin d)) (i : Fin k) (j : Fin d), P…","labels":[],"detail_key":"p14","name":"hasSubgaussianMGF_scaled_entry","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24357","layer":"formal","project":"p14","title":"integrable_rademacherReal","kind":"theorem","summary":"∀ (f : Real → Real), MeasureTheory.Integrable f rademacherReal","labels":[],"detail_key":"p14","name":"integrable_rademacherReal","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24358","layer":"formal","project":"p14","title":"integral_entry","kind":"theorem","summary":"∀ (k d : Nat) (i : Fin 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((Matr…","labels":[],"detail_key":"p14","name":"radMatrix_proj_indep","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24371","layer":"formal","project":"p14","title":"radMatrix_proj_meas","kind":"theorem","summary":"∀ (k d : Nat) (x : EuclideanSpace Real (Fin d)) (i : Fin k), Measurable fun ω => ((Matrix.toEuc…","labels":[],"detail_key":"p14","name":"radMatrix_proj_meas","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24372","layer":"formal","project":"p14","title":"rademacherReal_mem_Icc","kind":"theorem","summary":"Filter.Eventually (fun y => Membership.mem (Set.Icc (-1) 1) y) (MeasureTheory.ae rademacherReal)","labels":[],"detail_key":"p14","name":"rademacherReal_mem_Icc","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24373","layer":"formal","project":"p14","title":"variance_entry","kind":"theorem","summary":"∀ (k d : Nat) (i : Fin k) (j : Fin d), Eq (ProbabilityTheory.variance (fun ω => ω i j) 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(Pro…","labels":[],"detail_key":"p14","name":"variance_scaled_entry","module":"TCSlib.LearningTheory.JohnsonLindenstrauss.Rademacher"},{"id":"n24377","layer":"formal","project":"p14","title":"Game","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p14","name":"Game","module":"TCSlib.LearningTheory.Minimax.CCE"},{"id":"n24378","layer":"formal","project":"p14","title":"IsApproxCoarseCorrelatedEquilibrium","kind":"def","summary":"M N : Nat → Game M N → JointDistribution M N → Real → Prop","labels":[],"detail_key":"p14","name":"IsApproxCoarseCorrelatedEquilibrium","module":"TCSlib.LearningTheory.Minimax.CCE"},{"id":"n24379","layer":"formal","project":"p14","title":"IsCoarseCorrelatedEquilibrium","kind":"def","summary":"M N : Nat → Game M N → JointDistribution M N → 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Completeness","kind":"theorem","summary":"[Polynomial Equality Testing Completeness] The polynomial equality testing reduction satisfies…","labels":[],"detail_key":"p15"},{"id":"n24575","layer":"informal","project":"p15","title":"Polynomial Equality Testing Soundness","kind":"theorem","summary":"[Polynomial Equality Testing Soundness] The polynomial equality testing reduction satisfies rou…","labels":[],"detail_key":"p15"},{"id":"n24576","layer":"informal","project":"p15","title":"Batching Polynomial Evaluation Claims","kind":"definition","summary":"[Batching Polynomial Evaluation Claims] Consider an n-tuple of values v = (v_1, \\ldots, v_n) \\i…","labels":["def:batching_polynomial_evaluation"],"detail_key":"p15"},{"id":"n24577","layer":"informal","project":"p15","title":"Batching Completeness","kind":"theorem","summary":"[Batching Completeness] The batching polynomial evaluation reduction satisfies perfect complete…","labels":[],"detail_key":"p15"},{"id":"n24578","layer":"informal","project":"p15","title":"Batching Security","kind":"remark","summary":"[Batching Security] The security of this reduction depends on the degree and non-degeneracy pro…","labels":[],"detail_key":"p15"},{"id":"n24579","layer":"informal","project":"p15","title":"SendClaim Oracle Reduction","kind":"definition","summary":"[SendClaim Oracle Reduction] The SendClaim reduction is a one-round protocol for claim transmis…","labels":["def:sendclaim_oracle_reduction"],"detail_key":"p15"},{"id":"n24580","layer":"informal","project":"p15","title":"SendClaim Perfect Completeness","kind":"theorem","summary":"[SendClaim Perfect Completeness] The SendClaim oracle reduction satisfies perfect completeness…","labels":[],"detail_key":"p15"},{"id":"n24581","layer":"informal","project":"p15","title":"SendClaim Development Status","kind":"remark","summary":"[SendClaim Development Status] The SendClaim reduction is currently under active development in…","labels":[],"detail_key":"p15"},{"id":"n24582","layer":"informal","project":"p15","title":"ReduceClaim Reduction","kind":"definition","summary":"[ReduceClaim Reduction] The ReduceClaim reduction is a zero-round protocol that transforms clai…","labels":["def:reduceclaim_reduction"],"detail_key":"p15"},{"id":"n24583","layer":"informal","project":"p15","title":"ReduceClaim Perfect Completeness","kind":"theorem","summary":"[ReduceClaim Perfect Completeness] The ReduceClaim reduction satisfies perfect completeness whe…","labels":[],"detail_key":"p15"},{"id":"n24584","layer":"informal","project":"p15","title":"ReduceClaim Oracle Reduction","kind":"definition","summary":"[ReduceClaim Oracle Reduction] The oracle version additionally handles oracle statements throug…","labels":["def:reduceclaim_oracle_reduction"],"detail_key":"p15"},{"id":"n24585","layer":"informal","project":"p15","title":"ReduceClaim Oracle Completeness","kind":"remark","summary":"[ReduceClaim Oracle Completeness] The oracle version's completeness proof is currently under de…","labels":[],"detail_key":"p15"},{"id":"n24586","layer":"informal","project":"p15","title":"CheckClaim Reduction","kind":"definition","summary":"[CheckClaim Reduction] The CheckClaim reduction is a zero-round protocol that verifies predicat…","labels":["def:checkclaim_reduction"],"detail_key":"p15"},{"id":"n24587","layer":"informal","project":"p15","title":"CheckClaim Perfect Completeness","kind":"theorem","summary":"[CheckClaim Perfect Completeness] The CheckClaim reduction satisfies perfect completeness.","labels":[],"detail_key":"p15"},{"id":"n24588","layer":"informal","project":"p15","title":"CheckClaim Oracle Reduction","kind":"definition","summary":"[CheckClaim Oracle Reduction] The oracle version handles predicates that require oracle access:…","labels":["def:checkclaim_oracle_reduction"],"detail_key":"p15"},{"id":"n24589","layer":"informal","project":"p15","title":"CheckClaim Oracle Perfect Completeness","kind":"theorem","summary":"[CheckClaim Oracle Perfect Completeness] The CheckClaim oracle reduction satisfies perfect comp…","labels":[],"detail_key":"p15"},{"id":"n24590","layer":"informal","project":"p15","title":"CheckClaim Security Analysis","kind":"remark","summary":"[CheckClaim Security Analysis] The round-by-round knowledge soundness proofs for both reduction…","labels":[],"detail_key":"p15"},{"id":"n24591","layer":"informal","project":"p15","title":"Perfect Completeness","kind":"theorem","summary":"[Perfect Completeness] The sum-check protocol satisfies perfect completeness. That is, for any…","labels":["thm:sumcheck_perfect_completeness"],"detail_key":"p15"},{"id":"n24592","layer":"informal","project":"p15","title":"Knowledge Soundness","kind":"theorem","summary":"[Knowledge Soundness] The sum-check protocol satisfies knowledge soundness. The soundness error…","labels":["thm:sumcheck_knowledge_soundness"],"detail_key":"p15"},{"id":"n24593","layer":"informal","project":"p15","title":"Round-by-Round Knowledge Soundness","kind":"theorem","summary":"[Round-by-Round Knowledge Soundness] The sum-check protocol satisfies round-by-round knowledge…","labels":["thm:sumcheck_rbr_knowledge_soundness_standard"],"detail_key":"p15"},{"id":"n24594","layer":"informal","project":"p15","title":"Single Round Protocol","kind":"definition","summary":"[Single Round Protocol] The i-th round of sum-check consists of: \\item Input: A statement conta…","labels":["def:sumcheck_single_round"],"detail_key":"p15"},{"id":"n24595","layer":"informal","project":"p15","title":"Single Round Completeness","kind":"theorem","summary":"[Single Round Completeness] Each individual round of the sum-check protocol is perfectly comple…","labels":["thm:sumcheck_single_round_completeness"],"detail_key":"p15"},{"id":"n24596","layer":"informal","project":"p15","title":"Single Round Soundness","kind":"theorem","summary":"[Single Round Soundness] Each individual round of the sum-check protocol is sound with error pr…","labels":["thm:sumcheck_single_round_soundness"],"detail_key":"p15"},{"id":"n24597","layer":"informal","project":"p15","title":"Round-by-Round Knowledge Soundness","kind":"theorem","summary":"[Round-by-Round Knowledge Soundness] The sum-check protocol satisfies round-by-round knowledge…","labels":["thm:sumcheck_rbr_knowledge_soundness"],"detail_key":"p15"},{"id":"n24598","layer":"informal","project":"p15","title":"def:virtual_sumcheck_round_protocol","kind":"definition","summary":"\\item In this protocol, the context is a pair (p, d), where p is now a \\emphunivariate polynomi…","labels":["def:virtual_sumcheck_round_protocol"],"detail_key":"p15"},{"id":"n24599","layer":"informal","project":"p15","title":"thm:virtual_sumcheck_round_protocol_sound","kind":"theorem","summary":"The virtual sum-check round protocol is sound.","labels":["thm:virtual_sumcheck_round_protocol_sound"],"detail_key":"p15"},{"id":"n24600","layer":"informal","project":"p15","title":"thm:virtual_sumcheck_round_protocol_knowledge_sound","kind":"theorem","summary":"The virtual sum-check round protocol is knowledge sound.","labels":["thm:virtual_sumcheck_round_protocol_knowledge_sound"],"detail_key":"p15"},{"id":"n24601","layer":"informal","project":"p15","title":"State Function","kind":"definition","summary":"[State Function] The state function for the virtual sum-check round protocol is given by: \\item…","labels":["def:virtual_sumcheck_round_protocol_state_function"],"detail_key":"p15"},{"id":"n24602","layer":"informal","project":"p15","title":"thm:virtual_sumcheck_round_protocol_rbr_sound","kind":"theorem","summary":"The virtual sum-check round protocol is round-by-round sound.","labels":["thm:virtual_sumcheck_round_protocol_rbr_sound"],"detail_key":"p15"},{"id":"n24603","layer":"informal","project":"p15","title":"Ring-switching profile","kind":"definition","summary":"[Ring-switching profile] Fix commutative rings B (small) and L (large) with L a B-algebra, and…","labels":["def:ring_switching_profile","eq:rs-rows","eq:rs-cols"],"detail_key":"p15"},{"id":"n24604","layer":"informal","project":"p15","title":"Tensor-product profile","kind":"definition","summary":"[Tensor-product profile] For any fields B = K and L with a chosen degree-2^\\kappa extension bas…","labels":["def:tensor_product_profile"],"detail_key":"p15"},{"id":"n24605","layer":"informal","project":"p15","title":"Multilinear packing","kind":"definition","summary":"[Multilinear packing] With \\ell = \\ell' + \\kappa, a small-field multilinear t in \\ell variables…","labels":["def:pack_mle"],"detail_key":"p15"},{"id":"n24606","layer":"informal","project":"p15","title":"Full ring-switching reduction","kind":"definition","summary":"[Full ring-switching reduction] The composition of the batching phase, the core sum-check inter…","labels":["def:full_ring_switching"],"detail_key":"p15"},{"id":"n24607","layer":"informal","project":"p15","title":"Perfect completeness target","kind":"theorem","summary":"[Perfect completeness target] The full ring-switching oracle reduction is perfectly complete: a…","labels":["thm:ring_switching_completeness"],"detail_key":"p15"},{"id":"n24608","layer":"informal","project":"p15","title":"Round-by-round knowledge soundness target","kind":"theorem","summary":"[Round-by-round knowledge soundness target] When L is an integral domain, the full reduction is…","labels":["thm:ring_switching_soundness"],"detail_key":"p15"},{"id":"n24609","layer":"informal","project":"p15","title":"Binary Tower Field","kind":"definition","summary":"[Binary Tower Field] A binary tower field T_\\iota for \\iota \\in N is defined inductively as the…","labels":["def:binary_tower_field"],"detail_key":"p15"},{"id":"n24610","layer":"informal","project":"p15","title":"Irreducible defining polynomial","kind":"theorem","summary":"[Irreducible defining polynomial] The defining polynomial X_\\iota-1^2+X_\\iota-2 \\cdot X_\\iota-1…","labels":["thm:binary_tower_field_irreducible"],"detail_key":"p15"},{"id":"n24611","layer":"informal","project":"p15","title":"Binary Tower Fields are fields","kind":"theorem","summary":"[Binary Tower Fields are fields] We prove that the binary tower fields are finite fields: \\item…","labels":["thm:binary_tower_fields_are_fields"],"detail_key":"p15"},{"id":"n24612","layer":"informal","project":"p15","title":"Multilinear Bases for Tower Fields","kind":"definition","summary":"[Multilinear Bases for Tower Fields] For any tower field T_\\iota, we define its canonical bases…","labels":["def:multilinear_basis"],"detail_key":"p15"},{"id":"n24613","layer":"informal","project":"p15","title":"Computable Binary Tower Fields","kind":"definition","summary":"[Computable Binary Tower Fields] Building upon the abstract definition of binary tower fields,…","labels":["def:computable_binary_tower_field"],"detail_key":"p15"},{"id":"n24614","layer":"informal","project":"p15","title":"thm:proximity_gap","kind":"theorem","summary":"Let \\code := \\rscode[\\field,\\evaldomain,\\degree] be a Reed Solomon code with rate \\rate:=\\frac\\…","labels":["thm:proximity_gap"],"detail_key":"p15"},{"id":"n24615","layer":"informal","project":"p15","title":"def:quotient","kind":"definition","summary":"Let f:\\evaldomain\\to\\field be a function, S\\subseteq\\field be a set, and \\mathsfAns,\\mathsfFill…","labels":["def:quotient"],"detail_key":"p15"},{"id":"n24616","layer":"informal","project":"p15","title":"def:poly_quotient","kind":"definition","summary":"Let \\hatf\\in\\field^<\\degree[X] be a polynomial and S\\subseteq\\field be a set, let \\hatV_S\\in\\fi…","labels":["def:poly_quotient"],"detail_key":"p15"},{"id":"n24617","layer":"informal","project":"p15","title":"lemma:quotienting","kind":"lemma","summary":"Let f:\\evaldomain\\rightarrow\\field be a function, \\degree\\in\\N be the degree parameter, \\distan…","labels":["lemma:quotienting"],"detail_key":"p15"},{"id":"n24618","layer":"informal","project":"p15","title":"lemma:out_of_domain_smpl","kind":"lemma","summary":"Let f:\\evaldomain\\rightarrow\\field be a function, \\degree\\in\\N be a degree parameter, s\\in\\N be…","labels":["lemma:out_of_domain_smpl"],"detail_key":"p15"},{"id":"n24619","layer":"informal","project":"p15","title":"fact:poly_folding","kind":"lemma","summary":"Given a polynomial \\hatq\\in\\field[X]: \\item For every univariate polynomial \\hatf\\in\\field[X],…","labels":["fact:poly_folding"],"detail_key":"p15"},{"id":"n24620","layer":"informal","project":"p15","title":"def:poly_folding","kind":"definition","summary":"Given a polynomial \\hatf\\in\\field^<\\degree[X], a folding parameter k\\in\\N and r\\in\\field, we de…","labels":["def:poly_folding"],"detail_key":"p15"},{"id":"n24621","layer":"informal","project":"p15","title":"def:fn_folding","kind":"definition","summary":"Let f:\\evaldomain\\rightarrow\\field be a function, k\\in\\N a folding parameter and \\alpha\\in\\fiel…","labels":["def:fn_folding"],"detail_key":"p15"},{"id":"n24622","layer":"informal","project":"p15","title":"lemma:folding","kind":"lemma","summary":"For every function f:\\evaldomain\\rightarrow\\field, degree parameter \\degree\\in\\N, folding param…","labels":["lemma:folding"],"detail_key":"p15"},{"id":"n24623","layer":"informal","project":"p15","title":"fact:geometric_sum","kind":"lemma","summary":"Let \\field be a field, r\\in\\field be a field element, a\\in\\N be a natural number. Then \\[ \\sum_…","labels":["fact:geometric_sum"],"detail_key":"p15"},{"id":"n24624","layer":"informal","project":"p15","title":"def:combine","kind":"definition","summary":"Given target degree \\degree^*\\in\\N, shifting parameter r\\in\\field, functions f_0,\\ldots,f_m-1:\\…","labels":["def:combine"],"detail_key":"p15"},{"id":"n24625","layer":"informal","project":"p15","title":"def:deg_corr","kind":"definition","summary":"Given target degree \\degree^*\\in\\N, shifting parameter r\\in\\field, function f:\\evaldomain\\right…","labels":["def:deg_corr"],"detail_key":"p15"},{"id":"n24626","layer":"informal","project":"p15","title":"lemma:combine","kind":"lemma","summary":"Let \\degree^* be a target degree, f_0,\\ldots,f_m-1:\\evaldomain\\rightarrow\\field be functions, 0…","labels":["lemma:combine"],"detail_key":"p15"},{"id":"n24627","layer":"informal","project":"p15","title":"STIR Main Theorem","kind":"theorem","summary":"[STIR Main Theorem] Consider the following ingrediants: \\item A security parameter \\lambda\\in\\N…","labels":["thm:stir"],"detail_key":"p15"},{"id":"n24628","layer":"informal","project":"p15","title":"lemma:rnd_by_rnd_soundness","kind":"lemma","summary":"Consider (\\field,M,\\degree,k_0,\\ldots,k_M,\\evaldomain_0,\\ldots,\\evaldomain_M,t_0,\\ldots,t_M) an…","labels":["lemma:rnd_by_rnd_soundness"],"detail_key":"p15"},{"id":"n24629","layer":"informal","project":"p15","title":"def:proximity_generator","kind":"definition","summary":"Let \\code\\subseteq \\field^\\evaldomain be a linear code. We say that \\mathsfGen is a proximity g…","labels":["def:proximity_generator"],"detail_key":"p15"},{"id":"n24630","layer":"informal","project":"p15","title":"thm:proximity_gap_whir","kind":"theorem","summary":"Let \\code = \\rscode[\\field,\\evaldomain,m] be a Reed Solomon code with rate \\rate = 2^m/|\\evaldo…","labels":["thm:proximity_gap_whir"],"detail_key":"p15"},{"id":"n24631","layer":"informal","project":"p15","title":"def:gen_mutual_corr_agreement","kind":"definition","summary":"Let \\code be a linear code. We say that \\gen be a proximity generator with mutual correlated ag…","labels":["def:gen_mutual_corr_agreement"],"detail_key":"p15"},{"id":"n24632","layer":"informal","project":"p15","title":"lemma:gen_mutual_corr_agreement","kind":"lemma","summary":"Let \\code be a linear code with minimum distance \\delta_\\code and let \\gen be a proximity gener…","labels":["lemma:gen_mutual_corr_agreement"],"detail_key":"p15"},{"id":"n24633","layer":"informal","project":"p15","title":"Let \\c","kind":"lemma","summary":"Let \\c","labels":[],"detail_key":"p15"},{"id":"n24634","layer":"informal","project":"p15","title":"conjecture:whir","kind":"theorem","summary":"The function \\gen(\\parl; \\alpha) := (1, \\alpha, \\ldots, \\alpha^\\parl - 1) is a proximity genera…","labels":["conjecture:whir"],"detail_key":"p15"},{"id":"n24635","layer":"informal","project":"p15","title":"lemma:","kind":"lemma","summary":"Let \\code \\subseteq \\field^\\evaldomain be a linear code with minimum distance \\delta_\\code, and…","labels":["{lemma:"],"detail_key":"p15"},{"id":"n24636","layer":"informal","project":"p15","title":"def:extract","kind":"definition","summary":"Let \\mathsfextract:\\evaldomain^2^k+1\\rightarrow \\evaldomain^2^k be a function. There exists x \\…","labels":["def:extract"],"detail_key":"p15"},{"id":"n24637","layer":"informal","project":"p15","title":"def:foldf","kind":"definition","summary":"Let f : \\evaldomain^2^k \\to \\field be a function, and \\alpha \\in \\field. We define Fold_f(f, \\a…","labels":["def:foldf"],"detail_key":"p15"},{"id":"n24638","layer":"informal","project":"p15","title":"def:fold_k","kind":"definition","summary":"For k \\leq m and \\vec\\alpha = (\\alpha_0, \\ldots, \\alpha_k-1) \\in \\field^k we define Fold(f, \\ve…","labels":["def:fold_k"],"detail_key":"p15"},{"id":"n24639","layer":"informal","project":"p15","title":"def:fold_k_set","kind":"definition","summary":"For a set S \\subseteq \\field^\\evaldomain we denote Fold_S(S, \\vec\\alpha) := \\Fold_S(f, \\vec\\alp…","labels":["def:fold_k_set"],"detail_key":"p15"},{"id":"n24640","layer":"informal","project":"p15","title":"lemma:fold_fg","kind":"lemma","summary":"Let f","labels":["lemma:fold_fg"],"detail_key":"p15"},{"id":"n24641","layer":"informal","project":"p15","title":"def:block","kind":"definition","summary":"Let \\evaldomain \\subseteq \\field be a smooth evaluation domain and k \\in N be a folding paramet…","labels":["def:block"],"detail_key":"p15"},{"id":"n24642","layer":"informal","project":"p15","title":"def:block_rel_distance","kind":"definition","summary":"Let \\code := \\rscode[\\field, \\evaldomain, m] be a smooth Reed Solomon code and let f, g : \\eval…","labels":["def:block_rel_distance"],"detail_key":"p15"},{"id":"n24643","layer":"informal","project":"p15","title":"def:min_block_rel_distance","kind":"definition","summary":"For S \\subseteq \\field^\\evaldomain, we let \\Delta_r(\\code, i, k, f, S) := \\min_g \\in S \\Delta_r…","labels":["def:min_block_rel_distance"],"detail_key":"p15"},{"id":"n24644","layer":"informal","project":"p15","title":"def:list_close_codewords_block","kind":"definition","summary":"For a smooth Reed Solomon code \\rscode := \\rscode[\\field, \\evaldomain, m], proximity parameter…","labels":["def:list_close_codewords_block"],"detail_key":"p15"},{"id":"n24645","layer":"informal","project":"p15","title":"lemma:block_rel_distance","kind":"lemma","summary":"For any \\code := \\rscode[\\field, \\evaldomain, m], k \\in N, and f, g : \\evaldomain^2^i \\to \\fiel…","labels":["lemma:block_rel_distance"],"detail_key":"p15"},{"id":"n24646","layer":"informal","project":"p15","title":"thm:folding_preserves_listdecoding","kind":"theorem","summary":"Let \\code = \\rscode[\\field, \\evaldomain, m] be a smooth Reed Solomon code and k \\leq m. For 0 \\…","labels":["thm:folding_preserves_listdecoding"],"detail_key":"p15"},{"id":"n24647","layer":"informal","project":"p15","title":"lemma:folding_preserves_listdecoding_base","kind":"lemma","summary":"Let \\code := \\rscode[\\field, \\evaldomain, m] be a Reed Solomon code, and k \\leq m be a paramete…","labels":["lemma:folding_preserves_listdecoding_base"],"detail_key":"p15"},{"id":"n24648","layer":"informal","project":"p15","title":"lemma:folding_preserves_listdecoding_bound","kind":"lemma","summary":"For every \\alpha \\in \\field, Fold_S(\\Lambda_r(\\code, 0, k, f, \\delta), \\alpha) \\subseteq \\Lambd…","labels":["lemma:folding_preserves_listdecoding_bound"],"detail_key":"p15"},{"id":"n24649","layer":"informal","project":"p15","title":"lemma:folding_preserves_listdecoding_base_ne_subset","kind":"lemma","summary":"\\[ \\Pr_\\alpha \\leftarrow \\field \\left[ \\Lambda_r(\\code', 1, k, Fold(f, \\alpha), \\delta) \\not\\su…","labels":["lemma:folding_preserves_listdecoding_base_ne_subset"],"detail_key":"p15"},{"id":"n24650","layer":"informal","project":"p15","title":"lemma:crs_equiv_rs_randpompt_agreement","kind":"lemma","summary":"Let f : \\evaldomain \\rightarrow \\field be a function, m \\in N be a number of variables, s \\in N…","labels":["lemma:crs_equiv_rs_randpompt_agreement"],"detail_key":"p15"},{"id":"n24651","layer":"informal","project":"p15","title":"lemma:out_of_domain_sampling_crs_eq_rs","kind":"lemma","summary":"Let f : \\evaldomain \\rightarrow \\field be a function, m \\in N be a number of variables, s \\in N…","labels":["lemma:out_of_domain_sampling_crs_eq_rs"],"detail_key":"p15"},{"id":"n24652","layer":"informal","project":"p15","title":"thm:","kind":"theorem","summary":"Consider (\\field, M, (k_i, m_i, \\evaldomain_i, t_i)_0 \\leq i \\leq M, \\widehatw_0, \\sigma_0, m,…","labels":["{thm:"],"detail_key":"p15"},{"id":"n24653","layer":"informal","project":"p15","title":"Multilinear Extension","kind":"definition","summary":"[Multilinear Extension]","labels":["def:multilinear_extension"],"detail_key":"p15"},{"id":"n24654","layer":"informal","project":"p15","title":"Multilinear Extension is Unique","kind":"theorem","summary":"[Multilinear Extension is Unique]","labels":["thm:multilinear_extension_unique"],"detail_key":"p15"},{"id":"n24655","layer":"informal","project":"p15","title":"Schwartz-Zippel Lemma","kind":"theorem","summary":"[Schwartz-Zippel Lemma]","labels":["thm:schwartz_zippel"],"detail_key":"p15"},{"id":"n24656","layer":"informal","project":"p15","title":"Computable Univariate Polynomials","kind":"definition","summary":"[Computable Univariate Polynomials]","labels":["def:computable_univariate_polynomials"],"detail_key":"p15"},{"id":"n24657","layer":"informal","project":"p15","title":"Computable Multilinear Polynomials","kind":"definition","summary":"[Computable Multilinear Polynomials]","labels":["def:computable_multilinear_polynomials"],"detail_key":"p15"},{"id":"n24658","layer":"informal","project":"p15","title":"Code Distance","kind":"definition","summary":"[Code Distance]","labels":["def:code_distance"],"detail_key":"p15"},{"id":"n24659","layer":"informal","project":"p15","title":"Distance from a Code","kind":"definition","summary":"[Distance from a Code]","labels":["def:distance_from_code"],"detail_key":"p15"},{"id":"n24660","layer":"informal","project":"p15","title":"Generator Matrix","kind":"definition","summary":"[Generator Matrix]","labels":["def:generator_matrix"],"detail_key":"p15"},{"id":"n24661","layer":"informal","project":"p15","title":"Parity Check Matrix","kind":"definition","summary":"[Parity Check Matrix]","labels":["def:parity_check_matrix"],"detail_key":"p15"},{"id":"n24662","layer":"informal","project":"p15","title":"Code","kind":"definition","summary":"[Code]","labels":["def:code"],"detail_key":"p15"},{"id":"n24663","layer":"informal","project":"p15","title":"Linear Code","kind":"definition","summary":"[Linear Code]","labels":["def:linear_code"],"detail_key":"p15"},{"id":"n24664","layer":"informal","project":"p15","title":"Interleaved Code","kind":"definition","summary":"[Interleaved Code]","labels":["def:interleaved_code"],"detail_key":"p15"},{"id":"n24665","layer":"informal","project":"p15","title":"Reed-Solomon Code","kind":"definition","summary":"[Reed-Solomon Code]","labels":["def:reed_solomon_code"],"detail_key":"p15"},{"id":"n24666","layer":"informal","project":"p15","title":"Smooth Reed-Solomon Code","kind":"definition","summary":"[Smooth Reed-Solomon 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I…","labels":["def:oracle_computation"],"detail_key":"p15"},{"id":"n24675","layer":"informal","project":"p15","title":"Handling Oracle Queries","kind":"definition","summary":"[Handling Oracle Queries] To actually run oracle computations, we need a way to handle (or impl…","labels":["def:handling_oracle_queries"],"detail_key":"p15"},{"id":"n24676","layer":"informal","project":"p15","title":"Probabilistic Semantics of Oracle Computations","kind":"definition","summary":"[Probabilistic Semantics of Oracle Computations] We can view oracle computations as probabilist…","labels":["def:probabilistic_semantics_of_oracle_computations"],"detail_key":"p15"},{"id":"n24677","layer":"informal","project":"p15","title":"Simulating Oracle Queries with Other Oracles","kind":"definition","summary":"[Simulating Oracle Queries with Other Oracles] We can simulate complex oracles using simpler on…","labels":["def:sim_oracle"],"detail_key":"p15"},{"id":"n24678","layer":"informal","project":"p15","title":"Logging \\& Caching Oracle Queries","kind":"definition","summary":"[Logging \\& Caching Oracle Queries] Using the simulation framework, we can add logging and cach…","labels":["def:logging_caching_oracle_queries"],"detail_key":"p15"},{"id":"n24679","layer":"informal","project":"p15","title":"Random Oracle","kind":"definition","summary":"[Random Oracle] A random oracle is implemented as a caching oracle that uses lazy sampling: \\it…","labels":["def:random_oracle"],"detail_key":"p15"},{"id":"n24680","layer":"informal","project":"p15","title":"Fixed subring","kind":"definition","summary":"[Fixed subring] R_q^H := \\x \\in R_q : \\sigma_-1(x) = x \\wedge \\sigma_4k+1(x) = x\\, defined as t…","labels":["def:fixedSubring"],"detail_key":"p15"},{"id":"n24681","layer":"informal","project":"p15","title":"Cardinality, Eq.~7","kind":"theorem","summary":"[Cardinality, Eq.~7] Over R = \\Z_q with q prime and 2k \\mid d, \\;|R_q^H| = q^k.","labels":["thm:card_fixedSubring"],"detail_key":"p15"},{"id":"n24682","layer":"informal","project":"p15","title":"Packing bijection, Theorem~2","kind":"theorem","summary":"[Packing bijection, Theorem~2] The map \\psi : (R_q^H)^d/k \\to R_q is a bijection.","labels":["thm:psi_bijective"],"detail_key":"p15"},{"id":"n24683","layer":"informal","project":"p15","title":"Lemma 5: field isomorphism","kind":"theorem","summary":"[Lemma 5: field isomorphism] Let q be prime with q \\equiv 5 \\pmod 8 and 2 \\cdot 2^\\kappa \\mid 2…","labels":["thm:fixedSubring_equiv_galoisField"],"detail_key":"p15"},{"id":"n24684","layer":"informal","project":"p15","title":"By Theorem~\\refthm:fixedSubring_isField R_q^H is a finite field, and by Theorem~\\refthm:c…","kind":"proof","summary":"By Theorem~\\refthm:fixedSubring_isField R_q^H is a finite field, and by Theorem~\\refthm:card_fi…","labels":[],"detail_key":"p15"},{"id":"n24685","layer":"informal","project":"p15","title":"Finite field by cardinality","kind":"lemma","summary":"[Finite field by cardinality] Two finite fields of equal cardinality are isomorphic; 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Since q \\nmid 2^\\alpha+1 it is s…","kind":"proof","summary":"X^2^\\alpha+1 is the 2^\\alpha+1-th cyclotomic polynomial. 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α…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.exists_irreducible_factorization","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24731","layer":"formal","project":"p15","title":"ArkLib.Lattices.CyclotomicModulus.fixedSubringEquivGaloisField","kind":"theorem","summary":"∀ (q : Nat) [inst : Fact (Nat.Prime q)] [NeZero q] [inst_2 : BEq (ZMod q)] [inst_3 : LawfulBEq…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.fixedSubringEquivGaloisField","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24732","layer":"formal","project":"p15","title":"ArkLib.Lattices.CyclotomicModulus.fixedSubring_isField","kind":"theorem","summary":"∀ (q : Nat) [inst : Fact (Nat.Prime q)] [NeZero q] [inst_2 : BEq (ZMod q)] [inst_3 : LawfulBEq…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.fixedSubring_isField","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24733","layer":"formal","project":"p15","title":"ArkLib.Lattices.CyclotomicModulus.galoisAutₛ_fixed_isUnit","kind":"theorem","summary":"∀ (q : Nat) [inst : Fact (Nat.Prime q)] [NeZero q] [inst_2 : BEq (ZMod q)] [inst_3 : LawfulBEq…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.galoisAutₛ_fixed_isUnit","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24734","layer":"formal","project":"p15","title":"ArkLib.Lattices.CyclotomicModulus.mk_reverse_eq_galoisAutₛ_mul","kind":"theorem","summary":"∀ (q : Nat) [inst : Fact (Nat.Prime q)] [inst_1 : BEq (ZMod q)] [inst_2 : LawfulBEq (ZMod q)] (…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.mk_reverse_eq_galoisAutₛ_mul","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24735","layer":"formal","project":"p15","title":"ArkLib.Lattices.CyclotomicModulus.no_selfReciprocal_factor","kind":"theorem","summary":"∀ (q : Nat) [inst : Fact (Nat.Prime q)] [NeZero q] [inst_2 : BEq (ZMod q)] [LawfulBEq (ZMod q)]…","labels":[],"detail_key":"p15","name":"ArkLib.Lattices.CyclotomicModulus.no_selfReciprocal_factor","module":"ArkLib.Data.Lattices.CyclotomicRing.Subfield.Field"},{"id":"n24736","layer":"formal","project":"p15","title":"MvPolynomial.MLE","kind":"def","summary":"σ : Type u_1 → R : Type u_2 → [inst : CommRing R] → [Fintype σ] → [DecidableEq σ] → ((σ → Fin 2…","labels":[],"detail_key":"p15","name":"MvPolynomial.MLE","module":"ArkLib.Data.MvPolynomial.Multilinear"},{"id":"n24737","layer":"formal","project":"p15","title":"Polynomial.FoldingPolynomial.polyFold","kind":"def","summary":"F 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(oSpec : OracleSpec ι) → (Statement : Type) → ιₛ : Type → (OStatement : ιₛ → Type) →…","labels":[],"detail_key":"p15","name":"SendSingleWitness.oracleReduction","module":"ArkLib.ProofSystem.Component.SendWitness"},{"id":"n24843","layer":"formal","project":"p15","title":"SendSingleWitness.oracleReduction_completeness","kind":"theorem","summary":"∀ ι : Type (oSpec : OracleSpec ι) Statement ιₛ : Type OStatement : ιₛ → Type [inst : (i : ιₛ) →…","labels":[],"detail_key":"p15","name":"SendSingleWitness.oracleReduction_completeness","module":"ArkLib.ProofSystem.Component.SendWitness"},{"id":"n24844","layer":"formal","project":"p15","title":"SendWitness.reduction","kind":"def","summary":"ι : Type → (oSpec : OracleSpec ι) → (Statement Witness : Type) → Reduction oSpec Statement 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[inst_3…","labels":[],"detail_key":"p15","name":"RingSwitching.FullRingSwitching.fullOracleVerifier_rbrKnowledgeSoundness","module":"ArkLib.ProofSystem.RingSwitching.Packing.General"},{"id":"n24849","layer":"formal","project":"p15","title":"RingSwitching.packMLE","kind":"def","summary":"(κ : Nat) → [NeZero κ] → (L : Type) → [inst : CommRing L] → (K : Type) → [inst_1 : CommRing K]…","labels":[],"detail_key":"p15","name":"RingSwitching.packMLE","module":"ArkLib.ProofSystem.RingSwitching.Packing.Prelude"},{"id":"n24850","layer":"formal","project":"p15","title":"RingSwitching.tensorProductProfile","kind":"def","summary":"(κ : Nat) → [NeZero κ] → (K L : Type) → [inst : Field K] → [inst_1 : Field L] → [inst_2 : Algeb…","labels":[],"detail_key":"p15","name":"RingSwitching.tensorProductProfile","module":"ArkLib.ProofSystem.RingSwitching.Packing.Prelude"},{"id":"n24851","layer":"formal","project":"p15","title":"RingSwitching.RingSwitchingProfile","kind":"inductive","summary":"(B : Type u_1) → 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((Finset.range…","labels":[],"detail_key":"p15","name":"Combine.geometric_sum_units","module":"ArkLib.ProofSystem.Stir.Combine"},{"id":"n24855","layer":"formal","project":"p15","title":"StirIOP.stir_main","kind":"theorem","summary":"∀ F : Type [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] M : Nat (secpar : Nat…","labels":[],"detail_key":"p15","name":"StirIOP.stir_main","module":"ArkLib.ProofSystem.Stir.MainThm"},{"id":"n24856","layer":"formal","project":"p15","title":"StirIOP.stir_rbr_soundness","kind":"theorem","summary":"∀ F : Type [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] M : Nat (ι : Fin (HAd…","labels":[],"detail_key":"p15","name":"StirIOP.stir_rbr_soundness","module":"ArkLib.ProofSystem.Stir.MainThm"},{"id":"n24857","layer":"formal","project":"p15","title":"OutOfDomSmpl.out_of_dom_smpl_1","kind":"theorem","summary":"∀ F : Type [inst : Field F] [inst_1 : Fintype F] [inst_2 : DecidableEq F] ι : Type [inst_3 : 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ι…","labels":[],"detail_key":"p15","name":"Quotienting.funcQuotient","module":"ArkLib.ProofSystem.Stir.Quotienting"},{"id":"n24861","layer":"formal","project":"p15","title":"Quotienting.polyQuotient","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [DecidableEq F] → Finset F → Polynomial F → Polynomial F","labels":[],"detail_key":"p15","name":"Quotienting.polyQuotient","module":"ArkLib.ProofSystem.Stir.Quotienting"},{"id":"n24862","layer":"formal","project":"p15","title":"Quotienting.quotienting","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : DecidableEq F] ι : Finset F degree : Nat domain : Fun…","labels":[],"detail_key":"p15","name":"Quotienting.quotienting","module":"ArkLib.ProofSystem.Stir.Quotienting"},{"id":"n24863","layer":"informal","project":"p16","title":"Simple Ring","kind":"definition","summary":"[Simple Ring] A ring R is simple if the only two-sided-ideals of R are 0 and R. An algebra is s…","labels":[],"detail_key":"p16"},{"id":"n24864","layer":"informal","project":"p16","title":"Division rings are simple.","kind":"remark","summary":"Division rings are simple.","labels":[],"detail_key":"p16"},{"id":"n24865","layer":"informal","project":"p16","title":"lem:center-simple-ring","kind":"lemma","summary":"Let A be a simple ring, then centre of A is a field.","labels":["lem:center-simple-ring"],"detail_key":"p16"},{"id":"n24866","layer":"informal","project":"p16","title":"Let 0\\ne x be an element of centre of A. Then I := \\xy | y\\in A\\ is a two-sided-ideal of…","kind":"proof","summary":"Let 0\\ne x be an element of centre of A. Then I := \\xy | y\\in A\\ is a two-sided-ideal of A. Sin…","labels":[],"detail_key":"p16"},{"id":"n24867","layer":"informal","project":"p16","title":"Central Algebras","kind":"definition","summary":"[Central Algebras] Let R be a ring and A an R-algebra, we say A is central if and only if the c…","labels":[],"detail_key":"p16"},{"id":"n24868","layer":"informal","project":"p16","title":"Every commutative ring is a central algebra over itself.","kind":"remark","summary":"Every commutative ring is a central algebra over itself.","labels":[],"detail_key":"p16"},{"id":"n24869","layer":"informal","project":"p16","title":"Simpleness is invariant under ring isomorphism and centrality is invariant under algebra…","kind":"remark","summary":"Simpleness is invariant under ring isomorphism and centrality is invariant under algebra isomor…","labels":[],"detail_key":"p16"},{"id":"n24870","layer":"informal","project":"p16","title":"lem:opp-central","kind":"lemma","summary":"If A is a central R-algerba, A^\\opp is also central. .","labels":["lem:opp-central"],"detail_key":"p16"},{"id":"n24871","layer":"informal","project":"p16","title":"lem:simple-ring-iff","kind":"lemma","summary":"R is a simple ring if and only if any ring homomorphism f : R \\to S either injective or S is th…","labels":["lem:simple-ring-iff"],"detail_key":"p16"},{"id":"n24872","layer":"informal","project":"p16","title":"If R is simple, then the \\ker f is either \\0\\ or R. 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Then \\dim_K\\im f = \\dim_K B - \\dim_K\\ker f = \\dim_K…","labels":[],"detail_key":"p16"},{"id":"n24876","layer":"informal","project":"p16","title":"lem:tensor-central","kind":"lemma","summary":"If A and B are central K-algebras, A\\otimes_KB is a central K-algebra as well.","labels":["lem:tensor-central"],"detail_key":"p16"},{"id":"n24877","layer":"informal","project":"p16","title":"Assume A and B are central algebras, then by~\\crefcor:center-tensor-center Z\\left(A\\otime…","kind":"proof","summary":"Assume A and B are central algebras, then by~\\crefcor:center-tensor-center Z\\left(A\\otimes_RB\\r…","labels":[],"detail_key":"p16"},{"id":"n24878","layer":"informal","project":"p16","title":"thm:tensor-csa","kind":"theorem","summary":"If A is a simple K-algebra and B is a central simple K-algebra, A\\otimes_KB is a central simple…","labels":["thm:tensor-csa"],"detail_key":"p16"},{"id":"n24879","layer":"informal","project":"p16","title":"By~\\creflem:tensor-central, we need to prove A\\otimes_KB is a simple ring. Denote f as th…","kind":"proof","summary":"By~\\creflem:tensor-central, we need to prove A\\otimes_KB is a simple ring. Denote f as the map…","labels":[],"detail_key":"p16"},{"id":"n24880","layer":"informal","project":"p16","title":"cor:csa-basechange","kind":"corollary","summary":"Central simple algebras are stable under base change. That is, if L/K is a field extension and…","labels":["cor:csa-basechange"],"detail_key":"p16"},{"id":"n24881","layer":"informal","project":"p16","title":"By~\\crefthm:tensor-csa, L\\otimes_K D is simple. Let x\\in Z\\left(L\\otimes_KD\\right), by~\\c…","kind":"proof","summary":"By~\\crefthm:tensor-csa, L\\otimes_K D is simple. Let x\\in Z\\left(L\\otimes_KD\\right), by~\\crefcor…","labels":[],"detail_key":"p16"},{"id":"n24882","layer":"informal","project":"p16","title":"thm:tensor-simple-implies-simple","kind":"theorem","summary":"If A\\otimes_K B is a simple ring, then A and B are both simple.","labels":["thm:tensor-simple-implies-simple"],"detail_key":"p16"},{"id":"n24883","layer":"informal","project":"p16","title":"By symmetry, we only prove that A is simple. If A or B is the trivial ring then A\\otimes_…","kind":"proof","summary":"By symmetry, we only prove that A is simple. If A or B is the trivial ring then A\\otimes_K B is…","labels":[],"detail_key":"p16"},{"id":"n24884","layer":"informal","project":"p16","title":"Subfield","kind":"definition","summary":"[Subfield] For any field K and K-algebra A, a subfield B \\subseteq A is a commutative K-subalge…","labels":[],"detail_key":"p16"},{"id":"n24885","layer":"informal","project":"p16","title":"Subfields inherit a natural ordering from subalgebras.","kind":"remark","summary":"Subfields inherit a natural ordering from subalgebras.","labels":[],"detail_key":"p16"},{"id":"n24886","layer":"informal","project":"p16","title":"lem:dim-maximal-subfield","kind":"lemma","summary":"Let k be a maximal subfield of D, \\[ \\dim_KD = \\left(\\dim_Kk\\right)^2. \\]","labels":["lem:dim-maximal-subfield"],"detail_key":"p16"},{"id":"n24887","layer":"informal","project":"p16","title":"By~\\creflem:dim-centralizer","kind":"proof","summary":"By~\\creflem:dim-centralizer","labels":[],"detail_key":"p16"},{"id":"n24888","layer":"informal","project":"p16","title":"lem:tfae-subfield","kind":"lemma","summary":"Suppose L is a subfield of A, the following are equivalent: \\item L = C_A(L) \\item \\dim_KA = \\l…","labels":["lem:tfae-subfield"],"detail_key":"p16"},{"id":"n24889","layer":"informal","project":"p16","title":"We prove the following: \\item ``1. implies 2.'': this is~\\creflem:dim-centralizer. \\item…","kind":"proof","summary":"We prove the following: \\item ``1. implies 2.'': this is~\\creflem:dim-centralizer. \\item ``2. i…","labels":[],"detail_key":"p16"},{"id":"n24890","layer":"informal","project":"p16","title":"con:morita-eqv-functor0","kind":"construction","summary":"If M is an R-module, we have a natural \\Mat_n(R)-module structure on \\widehatM:=M^n given by (m…","labels":["con:morita-eqv-functor0"],"detail_key":"p16"},{"id":"n24891","layer":"informal","project":"p16","title":"Note that all modules are assumed to be left modules; when we need to consider right R-mo…","kind":"remark","summary":"Note that all modules are assumed to be left modules; when we need to consider right R-modules,…","labels":[],"detail_key":"p16"},{"id":"n24892","layer":"informal","project":"p16","title":"con:morita-eqv-functor1","kind":"construction","summary":"If M is a \\Mat_n(R)-module, then \\widetildeM := \\\\delta_ij\\cdot m | m \\in M\\ \\subseteq M is an…","labels":["con:morita-eqv-functor1"],"detail_key":"p16"},{"id":"n24893","layer":"informal","project":"p16","title":"Morita Equivalence","kind":"theorem","summary":"[Morita Equivalence] The functors constructed in \\crefcon:morita-eqv-functor0 and \\crefcon:mori…","labels":["thm:morita"],"detail_key":"p16"},{"id":"n24894","layer":"informal","project":"p16","title":"Let M be an R-module, then the unit \\widetilde\\widehatM \\cong M is given by \\[ x \\mapsto…","kind":"proof","summary":"Let M be an R-module, then the unit \\widetilde\\widehatM \\cong M is given by \\[ x \\mapsto \\sum_j…","labels":[],"detail_key":"p16"},{"id":"n24895","layer":"informal","project":"p16","title":"lem:isomorphic-simple-mod","kind":"lemma","summary":"Let M and N be simple A-modules, then M and N are isomorphic as A-modules.","labels":["lem:isomorphic-simple-mod"],"detail_key":"p16"},{"id":"n24896","layer":"informal","project":"p16","title":"By~\\crefthm:wed-artin-algebra, there exists non-zero n\\in N, k-division algebra D such th…","kind":"proof","summary":"By~\\crefthm:wed-artin-algebra, there exists non-zero n\\in N, k-division algebra D such that A\\c…","labels":[],"detail_key":"p16"},{"id":"n24897","layer":"informal","project":"p16","title":"lem:direct-sum-simple-mod","kind":"lemma","summary":"Let M be an A-module, there exists a simple A-module S such that M is a direct sum of copies of…","labels":["lem:direct-sum-simple-mod"],"detail_key":"p16"},{"id":"n24898","layer":"informal","project":"p16","title":"lem:direct-sum-simple-mod2","kind":"proof","summary":"By~\\crefthm:wed-artin-algebra, there exists non-zero n\\in N, k-division algebra D such that A\\c…","labels":["lem:direct-sum-simple-mod2"],"detail_key":"p16"},{"id":"n24899","layer":"informal","project":"p16","title":"Note that by~\\creflem:isomorphic-simple-mod, any two simple A-module are isomorphic, henc…","kind":"remark","summary":"Note that by~\\creflem:isomorphic-simple-mod, any two simple A-module are isomorphic, hence for…","labels":[],"detail_key":"p16"},{"id":"n24900","layer":"informal","project":"p16","title":"lem:iso-of-dim-eq","kind":"lemma","summary":"Let M and N be two finite A-module with compatible k-action. Then M and N are isomorphic as A-m…","labels":["lem:iso-of-dim-eq"],"detail_key":"p16"},{"id":"n24901","layer":"informal","project":"p16","title":"The forward direction is trivial as an A-linear isomorphism is a k-linear isomorphism as…","kind":"proof","summary":"The forward direction is trivial as an A-linear isomorphism is a k-linear isomorphism as well.…","labels":[],"detail_key":"p16"},{"id":"n24902","layer":"informal","project":"p16","title":"lem:exists-simple-mod","kind":"lemma","summary":"D^n is a simple A-module where the module structure is given by pulling back the \\Mat_n(D)-modu…","labels":["lem:exists-simple-mod"],"detail_key":"p16"},{"id":"n24903","layer":"informal","project":"p16","title":"By~\\crefthm:morita, we have \\MOD_A\\cong\\MOD_D\\cong\\MOD_\\Mat_n(D). Since D is a simple D-m…","kind":"proof","summary":"By~\\crefthm:morita, we have \\MOD_A\\cong\\MOD_D\\cong\\MOD_\\Mat_n(D). Since D is a simple D-module,…","labels":[],"detail_key":"p16"},{"id":"n24904","layer":"informal","project":"p16","title":"Note that any A-linear endomorphism of D^n is \\Mat_n(D)-linear, and vice versa. Thus we h…","kind":"remark","summary":"Note that any A-linear endomorphism of D^n is \\Mat_n(D)-linear, and vice versa. Thus we have \\E…","labels":[],"detail_key":"p16"},{"id":"n24905","layer":"informal","project":"p16","title":"lem:end-vec-iso","kind":"lemma","summary":"\\End_A\\left(D^n\\right) is isomorphic to D^\\opp as k-algebras.","labels":["lem:end-vec-iso"],"detail_key":"p16"},{"id":"n24906","layer":"informal","project":"p16","title":"Indeed, we calculate: \\[ \\End_A\\left(D^n\\right) &\\cong \\End_\\Mat_n(D)\\left(D^n\\right) & \\…","kind":"proof","summary":"Indeed, we calculate: \\[ \\End_A\\left(D^n\\right) &\\cong \\End_\\Mat_n(D)\\left(D^n\\right) & \\\\ &\\co…","labels":[],"detail_key":"p16"},{"id":"n24907","layer":"informal","project":"p16","title":"lem:end-simple-iso","kind":"lemma","summary":"Let M be a simple A-module, then \\End_A M\\cong D^\\opp as k-algebras.","labels":["lem:end-simple-iso"],"detail_key":"p16"},{"id":"n24908","layer":"informal","project":"p16","title":"By~\\crefthm:morita, D^n is simple as A-module; hence by~\\creflem:isomorphic-simple-mod, D…","kind":"proof","summary":"By~\\crefthm:morita, D^n is simple as A-module; hence by~\\creflem:isomorphic-simple-mod, D^n and…","labels":[],"detail_key":"p16"},{"id":"n24909","layer":"informal","project":"p16","title":"In particular, if M is a simple A-module, then \\End_AM is a simple k-algbera.","kind":"remark","summary":"In particular, if M is a simple A-module, then \\End_AM is a simple k-algbera.","labels":[],"detail_key":"p16"},{"id":"n24910","layer":"informal","project":"p16","title":"end_simple_mod_finite","kind":"lemma","summary":"Let M be a simple A-module, then \\End_AM has finite k-dimension.","labels":[],"detail_key":"p16"},{"id":"n24911","layer":"informal","project":"p16","title":"By~\\crefthm:wed, such D and n always exists. Hence we only need to show D^\\opp has finite…","kind":"proof","summary":"By~\\crefthm:wed, such D and n always exists. Hence we only need to show D^\\opp has finite k-dim…","labels":[],"detail_key":"p16"},{"id":"n24912","layer":"informal","project":"p16","title":"Note that for all A-module M, \\End_\\End_AMM is a k-algebra as well, with k \\hookrightarro…","kind":"remark","summary":"Note that for all A-module M, \\End_\\End_AMM is a k-algebra as well, with k \\hookrightarrow \\End…","labels":[],"detail_key":"p16"},{"id":"n24913","layer":"informal","project":"p16","title":"Balanced Module","kind":"definition","summary":"[Balanced Module] For any ring A and A-module M, we say M is a balanced A-module, if the A-line…","labels":["def:balanced-mod"],"detail_key":"p16"},{"id":"n24914","layer":"informal","project":"p16","title":"Balancedness is invariant under linear isomorphism.","kind":"remark","summary":"Balancedness is invariant under linear isomorphism.","labels":[],"detail_key":"p16"},{"id":"n24915","layer":"informal","project":"p16","title":"lem:balanced-self","kind":"lemma","summary":"For any ring A, A is balanced as A-module.","labels":["lem:balanced-self"],"detail_key":"p16"},{"id":"n24916","layer":"informal","project":"p16","title":"If f \\in \\End_\\End_AMA, then the image of f(1) under A \\to\\End_\\End_AA is f again.","kind":"proof","summary":"If f \\in \\End_\\End_AMA, then the image of f(1) under A \\to\\End_\\End_AA is f again.","labels":[],"detail_key":"p16"},{"id":"n24917","layer":"informal","project":"p16","title":"isBalanced_of_simpleMod","kind":"lemma","summary":"Any simple A-module is balanced.","labels":[],"detail_key":"p16"},{"id":"n24918","layer":"informal","project":"p16","title":"Indeed, if M is a simple A-module, then A \\cong \\bigoplus_i\\in\\iota M for some indexing s…","kind":"proof","summary":"Indeed, if M is a simple A-module, then A \\cong \\bigoplus_i\\in\\iota M for some indexing set \\io…","labels":[],"detail_key":"p16"},{"id":"n24919","layer":"informal","project":"p16","title":"lem:iso-end-end","kind":"lemma","summary":"For any simple A-module M, we have A \\cong \\End_\\End_AMM as k-algebras.","labels":["lem:iso-end-end"],"detail_key":"p16"},{"id":"n24920","layer":"informal","project":"p16","title":"The canonical map A \\to \\End_\\End_AMM is both injective and surjective, as M is a balance…","kind":"proof","summary":"The canonical map A \\to \\End_\\End_AMM is both injective and surjective, as M is a balanced A-mo…","labels":[],"detail_key":"p16"},{"id":"n24921","layer":"informal","project":"p16","title":"lem:expand-tensor-in-basis","kind":"lemma","summary":"Let M and N be R-modules such that C_i\\in\\iota is a basis for N, then every elements of x \\in M…","labels":["lem:expand-tensor-in-basis"],"detail_key":"p16"},{"id":"n24922","layer":"informal","project":"p16","title":"Given the basis C, we have R-linear isomorphism N \\cong\\bigoplus_i\\in\\iotaR, hence M\\otim…","kind":"proof","summary":"Given the basis C, we have R-linear isomorphism N \\cong\\bigoplus_i\\in\\iotaR, hence M\\otimes_RN…","labels":[],"detail_key":"p16"},{"id":"n24923","layer":"informal","project":"p16","title":"lem:submodule-tensor-submodule","kind":"lemma","summary":"Let K be a field, M and N be flat K-modules. Suppose p \\subseteq M and q \\subseteq N are K-subm…","labels":["lem:submodule-tensor-submodule"],"detail_key":"p16"},{"id":"n24924","layer":"informal","project":"p16","title":"The hard direction is to show (p \\otimes_R N) \\sqcap (M \\otimes_R q) \\le p \\otimes_R q. C…","kind":"proof","summary":"The hard direction is to show (p \\otimes_R N) \\sqcap (M \\otimes_R q) \\le p \\otimes_R q. Conside…","labels":[],"detail_key":"p16"},{"id":"n24925","layer":"informal","project":"p16","title":"lem:sup-centralizer","kind":"lemma","summary":"Let S, T be two subalgebras of A, then C_A(S\\sqcup T)=C_A(S)\\sqcap C_A(T).","labels":["lem:sup-centralizer"],"detail_key":"p16"},{"id":"n24926","layer":"informal","project":"p16","title":"lem:centralizer-inclusion-tensor","kind":"lemma","summary":"If we assume B is free as R-module, then for any R-subalgebra S, we have that C_A\\otimes_RB\\lef…","labels":["lem:centralizer-inclusion-tensor"],"detail_key":"p16"},{"id":"n24927","layer":"informal","project":"p16","title":"Let w\\in C_A\\otimes_RB\\left(\\im\\left(S\\to A\\otimes_RB\\right)\\right). Since B is free, we…","kind":"proof","summary":"Let w\\in C_A\\otimes_RB\\left(\\im\\left(S\\to A\\otimes_RB\\right)\\right). Since B is free, we choose…","labels":[],"detail_key":"p16"},{"id":"n24928","layer":"informal","project":"p16","title":"A useful special case is when S=A, then since C_A(A)=Z(A), we have C_A\\otimes_RB\\left(\\im…","kind":"remark","summary":"A useful special case is when S=A, then since C_A(A)=Z(A), we have C_A\\otimes_RB\\left(\\im\\left(…","labels":[],"detail_key":"p16"},{"id":"n24929","layer":"informal","project":"p16","title":"cor:centralizer-tensor-centralizer","kind":"corollary","summary":"Assume R is a field. Let S and T be R-subalgebras of A and B respectively. Then C_A\\otimes_RB\\l…","labels":["cor:centralizer-tensor-centralizer"],"detail_key":"p16"},{"id":"n24930","layer":"informal","project":"p16","title":"From~\\creflem:submodule-tensor-submodule, C_A(S)\\otimes_RC_B(T) is equal to \\left(C_A(S)\\…","kind":"proof","summary":"From~\\creflem:submodule-tensor-submodule, C_A(S)\\otimes_RC_B(T) is equal to \\left(C_A(S)\\otimes…","labels":[],"detail_key":"p16"},{"id":"n24931","layer":"informal","project":"p16","title":"cor:center-tensor-center","kind":"corollary","summary":"Assume R is a field. The centre of A\\otimes_R B is Z\\left(A\\right)\\otimes_RZ\\left(B\\right).","labels":["cor:center-tensor-center"],"detail_key":"p16"},{"id":"n24932","layer":"informal","project":"p16","title":"Special case of~\\crefcor:centralizer-tensor-centralizer.","kind":"proof","summary":"Special case of~\\crefcor:centralizer-tensor-centralizer.","labels":[],"detail_key":"p16"},{"id":"n24933","layer":"informal","project":"p16","title":"con:self-tensor-opp-iso-end","kind":"construction","summary":"Let R be a commutative ring and A an R-algebra. Then we have an R-algebra homomorphism A\\otimes…","labels":["con:self-tensor-opp-iso-end"],"detail_key":"p16"},{"id":"n24934","layer":"informal","project":"p16","title":"con:end-vec-iso-matrix","kind":"construction","summary":"Let A be an R-algebra and M an A-module. We have isomorphism \\End_A\\left(M^n\\right)\\cong \\Mat_n…","labels":["con:end-vec-iso-matrix"],"detail_key":"p16"},{"id":"n24935","layer":"informal","project":"p16","title":"con:matrix-matrix","kind":"construction","summary":"Let A be an R-algebra. Then \\Mat_m\\left(\\Mat_n(A)\\right) \\cong \\Mat_mn(A). The trick is to thin…","labels":["con:matrix-matrix"],"detail_key":"p16"},{"id":"n24936","layer":"informal","project":"p16","title":"con:matrix-tensor-matrix","kind":"construction","summary":"Let A, B be R-algebras. Then \\Mat_mn\\left(A\\otimes_RB\\right)\\cong\\Mat_m(A)\\otimes_R\\Mat_nB as K…","labels":["con:matrix-tensor-matrix"],"detail_key":"p16"},{"id":"n24937","layer":"informal","project":"p16","title":"minimal ideal of simple rings","kind":"lemma","summary":"[minimal ideal of simple rings] Let A be a ring and I a non-trivial minimal left ideal of A, th…","labels":["lemma:min-ideal-simple-ring"],"detail_key":"p16"},{"id":"n24938","layer":"informal","project":"p16","title":"Let J\\le I be an A-submodule of I, suppose J is non-trivial, we prove that J=I. Then the…","kind":"proof","summary":"Let J\\le I be an A-submodule of I, suppose J is non-trivial, we prove that J=I. Then the image…","labels":[],"detail_key":"p16"},{"id":"n24939","layer":"informal","project":"p16","title":"Let A be a simple ring and I a non-trivial left ideal. One can write 1 \\in A as \\sum_i=0^…","kind":"lemma","summary":"Let A be a simple ring and I a non-trivial left ideal. One can write 1 \\in A as \\sum_i=0^nx_iy_…","labels":[],"detail_key":"p16"},{"id":"n24940","layer":"informal","project":"p16","title":"Let I' be the two-sided ideal spanned by I. Then since A is a simple ring, I' = A. Thus 1…","kind":"proof","summary":"Let I' be the two-sided ideal spanned by I. Then since A is a simple ring, I' = A. Thus 1 \\in I…","labels":[],"detail_key":"p16"},{"id":"n24941","layer":"informal","project":"p16","title":"Wedderburn_Artin.aux.nxi_ne_zero","kind":"lemma","summary":"The n, x_i and y_i are all non-zero.","labels":[],"detail_key":"p16"},{"id":"n24942","layer":"informal","project":"p16","title":"If n is 0, then 1 = 0 in A, but all simple rings are non-trivial. We argue by contradicti…","kind":"proof","summary":"If n is 0, then 1 = 0 in A, but all simple rings are non-trivial. We argue by contradiction to…","labels":[],"detail_key":"p16"},{"id":"n24943","layer":"informal","project":"p16","title":"Wedderburn","kind":"theorem","summary":"[Wedderburn] Let A be a simple ring and I a non-trivial minimal left ideal. Then there exists a…","labels":["thm:wed"],"detail_key":"p16"},{"id":"n24944","layer":"informal","project":"p16","title":"We continue to write 1 = \\sum_i=0^nx_iy_i in the shortest possible manner. Then we can de…","kind":"proof","summary":"We continue to write 1 = \\sum_i=0^nx_iy_i in the shortest possible manner. Then we can define a…","labels":[],"detail_key":"p16"},{"id":"n24945","layer":"informal","project":"p16","title":"Wedderburn-Artin (Ideal)","kind":"theorem","summary":"[Wedderburn-Artin (Ideal)] Let A be an Artinian simple ring. There exists a non-zero n and an i…","labels":["thm:wed-artin-ideal"],"detail_key":"p16"},{"id":"n24946","layer":"informal","project":"p16","title":"By~\\crefthm:wed, we only need a minimal left ideal. Since A is Artinian, such ideal exist…","kind":"proof","summary":"By~\\crefthm:wed, we only need a minimal left ideal. Since A is Artinian, such ideal exists.","labels":[],"detail_key":"p16"},{"id":"n24947","layer":"informal","project":"p16","title":"Wedderburn-Artin (Algebra)","kind":"theorem","summary":"[Wedderburn-Artin (Algebra)] Let K be a field and B an finite dimensional simple algebra over K…","labels":["thm:wed-artin-algebra"],"detail_key":"p16"},{"id":"n24948","layer":"informal","project":"p16","title":"By~\\crefthm:wed-artin-ideal, we can find a n and a minimal left ideal I such A \\cong I^n…","kind":"proof","summary":"By~\\crefthm:wed-artin-ideal, we can find a n and a minimal left ideal I such A \\cong I^n as A-m…","labels":[],"detail_key":"p16"},{"id":"n24949","layer":"informal","project":"p16","title":"Uniqueness of Wedderburn-Artin theorem","kind":"theorem","summary":"[Uniqueness of Wedderburn-Artin theorem] Let B be a finite-dimensional simple K-algebra. Suppos…","labels":["thm:wed-artin-uniq","thm:wed-artin-unique"],"detail_key":"p16"},{"id":"n24950","layer":"informal","project":"p16","title":"Since D^n is a simple B-module, by~\\creflem:end-simple-iso, we see that \\End_AD^n\\cong D^…","kind":"proof","summary":"Since D^n is a simple B-module, by~\\creflem:end-simple-iso, we see that \\End_AD^n\\cong D^\\opp a…","labels":[],"detail_key":"p16"},{"id":"n24951","layer":"informal","project":"p16","title":"IsMod","kind":"construction","summary":"For any K-algebra homomorphism f : B \\to A, we give M a B \\otimes_K\\End_AM-module structure by…","labels":[],"detail_key":"p16"},{"id":"n24952","layer":"informal","project":"p16","title":"module_inst_findim","kind":"lemma","summary":"Let f : B \\to A be a K-algebra homomorphism, M^f is finitely generated as a B\\otimes_K\\End_AM-m…","labels":[],"detail_key":"p16"},{"id":"n24953","layer":"informal","project":"p16","title":"Since M is a finite A-module and A a finite dimensional K-vector space, M is a finite dim…","kind":"proof","summary":"Since M is a finite A-module and A a finite dimensional K-vector space, M is a finite dimension…","labels":[],"detail_key":"p16"},{"id":"n24954","layer":"informal","project":"p16","title":"Given that B is simple, any k-algebra homomorphism f : B\\to A injective; therefore by fin…","kind":"remark","summary":"Given that B is simple, any k-algebra homomorphism f : B\\to A injective; therefore by finite K-…","labels":[],"detail_key":"p16"},{"id":"n24955","layer":"informal","project":"p16","title":"lem:iso-fg","kind":"lemma","summary":"Let f,g : B \\to A be two K-algebra homomorphisms. Then M^f and M^g are isomorphic as B\\otimes_K…","labels":["lem:iso-fg"],"detail_key":"p16"},{"id":"n24956","layer":"informal","project":"p16","title":"By~\\creflem:iso-of-dim-eq, it is sufficient to prove \\dim_KM^f=\\dim_KM^g. But as K-vector…","kind":"proof","summary":"By~\\creflem:iso-of-dim-eq, it is sufficient to prove \\dim_KM^f=\\dim_KM^g. But as K-vector space…","labels":[],"detail_key":"p16"},{"id":"n24957","layer":"informal","project":"p16","title":"Skolem-Noether","kind":"theorem","summary":"[Skolem-Noether] Let f, g : B \\to A be two K-algebra homomorphism. Then f and g differ only by…","labels":["thm:skolem-noether"],"detail_key":"p16"},{"id":"n24958","layer":"informal","project":"p16","title":"Let M be any simple A-module (which exists by~\\creflem:exists-simple-mod). By~\\creflem:is…","kind":"proof","summary":"Let M be any simple A-module (which exists by~\\creflem:exists-simple-mod). By~\\creflem:iso-fg,…","labels":[],"detail_key":"p16"},{"id":"n24959","layer":"informal","project":"p16","title":"lem:centralizer-mul-left-le","kind":"lemma","summary":"The centralizer of \\McL_A in \\End_FA is smaller than or equal to \\McR_A: \\[ C_\\End_FA\\left(\\McL…","labels":["lem:centralizer-mul-left-le"],"detail_key":"p16"},{"id":"n24960","layer":"informal","project":"p16","title":"Indeed, let x \\in C_\\End_FA\\left(\\McL_A\\right). Recall from~\\crefcon:self-tensor-opp-iso-…","kind":"proof","summary":"Indeed, let x \\in C_\\End_FA\\left(\\McL_A\\right). Recall from~\\crefcon:self-tensor-opp-iso-end th…","labels":[],"detail_key":"p16"},{"id":"n24961","layer":"informal","project":"p16","title":"rem:mul-left-center-linear","kind":"remark","summary":"For any F-algebra B, every element in C_\\End_FB\\left(\\McL_B\\right) is in fact Z(B)-linear. Let…","labels":["rem:mul-left-center-linear"],"detail_key":"p16"},{"id":"n24962","layer":"informal","project":"p16","title":"A is a Z(A)-algebra whose algebra structure is given by Z(A)\\hookrightarrow A. By~\\crefle…","kind":"remark","summary":"A is a Z(A)-algebra whose algebra structure is given by Z(A)\\hookrightarrow A. By~\\creflem:cent…","labels":[],"detail_key":"p16"},{"id":"n24963","layer":"informal","project":"p16","title":"lem:mul-right-iso-opp","kind":"lemma","summary":"As F-algebras, we have \\McR_A\\cong A^\\opp.","labels":["lem:mul-right-iso-opp"],"detail_key":"p16"},{"id":"n24964","layer":"informal","project":"p16","title":"We prove the map A^\\opp \\to \\McR_A is bijective. It is injective because if \\left(\\bullet…","kind":"proof","summary":"We prove the map A^\\opp \\to \\McR_A is bijective. It is injective because if \\left(\\bullet \\cdot…","labels":[],"detail_key":"p16"},{"id":"n24965","layer":"informal","project":"p16","title":"lem:centralizer-mul-left-eq-mul-right","kind":"lemma","summary":"Let B be any simple F-algebra (\\em not\\/ necessarily central). The centralizer of \\McL_B in \\En…","labels":["lem:centralizer-mul-left-eq-mul-right"],"detail_key":"p16"},{"id":"n24966","layer":"informal","project":"p16","title":"It is straightforward to show \\McR_B^F\\le C_\\End_FA\\left(\\McL_B^F\\right). So we only need…","kind":"proof","summary":"It is straightforward to show \\McR_B^F\\le C_\\End_FA\\left(\\McL_B^F\\right). So we only need to pr…","labels":[],"detail_key":"p16"},{"id":"n24967","layer":"informal","project":"p16","title":"Subalgebra.conj","kind":"construction","summary":"Let B be any F-algebra and S\\subseteq B an F-subalgebra. For any x\\in B^\\times, we have that x…","labels":[],"detail_key":"p16"},{"id":"n24968","layer":"informal","project":"p16","title":"lem:centralizer-conj","kind":"lemma","summary":"Let B be any F-algebra, x \\in B^\\times and S\\subseteq B be an F-subalgebra of B, then C_B(xSx^-…","labels":["lem:centralizer-conj"],"detail_key":"p16"},{"id":"n24969","layer":"informal","project":"p16","title":"If a \\in C_B\\left(xSx^-1\\right), then x^-1ax is in C_B(S). Conversely if a is equal to xb…","kind":"proof","summary":"If a \\in C_B\\left(xSx^-1\\right), then x^-1ax is in C_B(S). Conversely if a is equal to xbx^-1 w…","labels":[],"detail_key":"p16"},{"id":"n24970","layer":"informal","project":"p16","title":"For any finite dimensional F-module B, we have isomorphism \\End_FB \\cong \\Mat_\\dim_FBF as…","kind":"remark","summary":"For any finite dimensional F-module B, we have isomorphism \\End_FB \\cong \\Mat_\\dim_FBF as F-alg…","labels":[],"detail_key":"p16"},{"id":"n24971","layer":"informal","project":"p16","title":"lem:tensor-mul-right-simple","kind":"lemma","summary":"Let S\\subseteq A be a simple F-subalgebra, then A \\otimes_F \\McR_S is a simple ring.","labels":["lem:tensor-mul-right-simple"],"detail_key":"p16"},{"id":"n24972","layer":"informal","project":"p16","title":"By~\\creflem:mul-right-iso-opp, we have A\\otimes \\McR_S\\cong A\\otimes S^\\opp as F-algebras…","kind":"proof","summary":"By~\\creflem:mul-right-iso-opp, we have A\\otimes \\McR_S\\cong A\\otimes S^\\opp as F-algebras. The…","labels":[],"detail_key":"p16"},{"id":"n24973","layer":"informal","project":"p16","title":"lem:centralizer-tensor-end-eq-conj-tensor-mul-right","kind":"lemma","summary":"Let S\\subseteq A be a simple F-subalgebra, then there exists an x\\in \\left(A\\otimes_F\\End_FS\\ri…","labels":["lem:centralizer-tensor-end-eq-conj-tensor-mul-right"],"detail_key":"p16"},{"id":"n24974","layer":"informal","project":"p16","title":"By~\\creflem:tensor-central and~\\crefthm:tensor-csa, A \\otimes_FC_A(S) is a central simple…","kind":"proof","summary":"By~\\creflem:tensor-central and~\\crefthm:tensor-csa, A \\otimes_FC_A(S) is a central simple F-alg…","labels":[],"detail_key":"p16"},{"id":"n24975","layer":"informal","project":"p16","title":"lem:simple-centralizer","kind":"lemma","summary":"Let S\\subseteq A be a simple F-subalgebra, then C_A(S) is simple as well.","labels":["lem:simple-centralizer"],"detail_key":"p16"},{"id":"n24976","layer":"informal","project":"p16","title":"By~\\creflem:centralizer-tensor-end-eq-conj-tensor-mul-right, C_A(S) \\otimes_F\\End_FS is i…","kind":"proof","summary":"By~\\creflem:centralizer-tensor-end-eq-conj-tensor-mul-right, C_A(S) \\otimes_F\\End_FS is isomorp…","labels":[],"detail_key":"p16"},{"id":"n24977","layer":"informal","project":"p16","title":"lem:dim-centralizer","kind":"lemma","summary":"Let S\\subseteq A be a simple F-subalgebra. Then \\[ \\dim_FC_A(S)\\cdot\\dim_FS=\\dim_FA. \\]","labels":["lem:dim-centralizer"],"detail_key":"p16"},{"id":"n24978","layer":"informal","project":"p16","title":"By~\\creflem:centralizer-tensor-end-eq-conj-tensor-mul-right, C_A(S) \\otimes_F\\End_FS is i…","kind":"proof","summary":"By~\\creflem:centralizer-tensor-end-eq-conj-tensor-mul-right, C_A(S) \\otimes_F\\End_FS is isomorp…","labels":[],"detail_key":"p16"},{"id":"n24979","layer":"informal","project":"p16","title":"cor:self-tensor-centralizer","kind":"corollary","summary":"Let S \\subseteq A be a central simple F-subalgebra, \\[ A \\cong B \\ox_F C_A(B). \\]","labels":["cor:self-tensor-centralizer"],"detail_key":"p16"},{"id":"n24980","layer":"informal","project":"p16","title":"By~\\creflem:simple-centralizer, C_A(B) is simple and by~\\crefthm:tensor-csa, B \\ox_F C_A(…","kind":"proof","summary":"By~\\creflem:simple-centralizer, C_A(B) is simple and by~\\crefthm:tensor-csa, B \\ox_F C_A(B) is…","labels":[],"detail_key":"p16"},{"id":"n24981","layer":"informal","project":"p16","title":"Double Centralizer","kind":"theorem","summary":"[Double Centralizer] Let S\\subseteq A be a simple F-subalgebra, we have \\[ C_A\\left(C_A(S)\\righ…","labels":["thm:double-centralizer"],"detail_key":"p16"},{"id":"n24982","layer":"informal","project":"p16","title":"It is straightforward that S \\le C_A\\left(C_A(S)\\right). By~\\creflem:simple-centralizer,…","kind":"proof","summary":"It is straightforward that S \\le C_A\\left(C_A(S)\\right). By~\\creflem:simple-centralizer, C_A(S)…","labels":[],"detail_key":"p16"},{"id":"n24983","layer":"informal","project":"p16","title":"By~\\creflem:tensor-central and~\\crefthm:tensor-csa, \\CSA is closed under tensor product,…","kind":"remark","summary":"By~\\creflem:tensor-central and~\\crefthm:tensor-csa, \\CSA is closed under tensor product, that i…","labels":[],"detail_key":"p16"},{"id":"n24984","layer":"informal","project":"p16","title":"Brauer Equivalence","kind":"definition","summary":"[Brauer Equivalence] For any two A, B\\in\\CSA, we say A and B are Brauer equivalent, when there…","labels":["def:br-eqv"],"detail_key":"p16"},{"id":"n24985","layer":"informal","project":"p16","title":"Isomorphic K-algebras are Brauer equivalent.","kind":"remark","summary":"Isomorphic K-algebras are Brauer equivalent.","labels":[],"detail_key":"p16"},{"id":"n24986","layer":"informal","project":"p16","title":"IsBrauerEquivalent.refl","kind":"lemma","summary":"\\sim_\\Br is reflexive.","labels":[],"detail_key":"p16"},{"id":"n24987","layer":"informal","project":"p16","title":"Indeed, A \\cong \\Mat_1(A) as K-algerbas.","kind":"proof","summary":"Indeed, A \\cong \\Mat_1(A) as K-algerbas.","labels":[],"detail_key":"p16"},{"id":"n24988","layer":"informal","project":"p16","title":"IsBrauerEquivalent.symm","kind":"lemma","summary":"\\sim_\\Br is symmetric.","labels":[],"detail_key":"p16"},{"id":"n24989","layer":"informal","project":"p16","title":"Indeed, just exchange m and n.","kind":"proof","summary":"Indeed, just exchange m and n.","labels":[],"detail_key":"p16"},{"id":"n24990","layer":"informal","project":"p16","title":"IsBrauerEquivalent.trans","kind":"lemma","summary":"\\sim_\\Br is transitive.","labels":[],"detail_key":"p16"},{"id":"n24991","layer":"informal","project":"p16","title":"Let A\\sim_\\BrB and B\\sim_\\BrC; that is for some m,n,p, q\\inN_\\ge0 we have \\Mat_n(A)\\cong\\…","kind":"proof","summary":"Let A\\sim_\\BrB and B\\sim_\\BrC; that is for some m,n,p, q\\inN_\\ge0 we have \\Mat_n(A)\\cong\\Mat_m(…","labels":[],"detail_key":"p16"},{"id":"n24992","layer":"informal","project":"p16","title":"lem:br-mul-wd","kind":"lemma","summary":"(\\bullet\\otimes_K\\bullet):\\CSA\\times\\CSA \\to \\CSA descends to a function on \\Br(K).","labels":["lem:br-mul-wd"],"detail_key":"p16"},{"id":"n24993","layer":"informal","project":"p16","title":"We need to prove that for all A, B, C, D \\in \\CSA such that A\\sim_\\Br B and C\\sim_\\BrD, A…","kind":"proof","summary":"We need to prove that for all A, B, C, D \\in \\CSA such that A\\sim_\\Br B and C\\sim_\\BrD, A\\otime…","labels":[],"detail_key":"p16"},{"id":"n24994","layer":"informal","project":"p16","title":"Brauer Group","kind":"construction","summary":"[Brauer Group] \\Br(K) forms a group under [A]_\\sim_\\Br\\cdot[B]_\\sim_\\Br=[A\\otimes_KB]_\\sim_\\Br…","labels":["con:br"],"detail_key":"p16"},{"id":"n24995","layer":"informal","project":"p16","title":"thm:br-triv-alg-closed","kind":"theorem","summary":"If K is algebraically closed, \\Br(K) is trivial; in particular \\br_n(C) is trivial.","labels":["thm:br-triv-alg-closed"],"detail_key":"p16"},{"id":"n24996","layer":"informal","project":"p16","title":"We need to show that every A\\in\\CSA is isomorphic to \\Mat_n(K) for some K when K is algeb…","kind":"proof","summary":"We need to show that every A\\in\\CSA is isomorphic to \\Mat_n(K) for some K when K is algebraical…","labels":[],"detail_key":"p16"},{"id":"n24997","layer":"informal","project":"p16","title":"lem:common-div-alg","kind":"lemma","summary":"Let A, B \\in \\csa_K. There exists a division K-algebra D and non-zero m,n\\inN such that A\\cong\\…","labels":["lem:common-div-alg"],"detail_key":"p16"},{"id":"n24998","layer":"informal","project":"p16","title":"By~\\crefthm:wed-artin-algebra","kind":"proof","summary":"By~\\crefthm:wed-artin-algebra","labels":[],"detail_key":"p16"},{"id":"n24999","layer":"informal","project":"p16","title":"con:br-base-change","kind":"construction","summary":"We will construct a series of isomorphisms (either over K or E) to arrive at the conclusion tha…","labels":["con:br-base-change"],"detail_key":"p16"},{"id":"n25000","layer":"informal","project":"p16","title":"lem:br-base-change-self","kind":"lemma","summary":"\\Br_K^K is identity.","labels":["lem:br-base-change-self"],"detail_key":"p16"},{"id":"n25001","layer":"informal","project":"p16","title":"If A\\in\\CSA, then A\\sim_\\BrK\\ox_KA.","kind":"proof","summary":"If A\\in\\CSA, then A\\sim_\\BrK\\ox_KA.","labels":[],"detail_key":"p16"},{"id":"n25002","layer":"informal","project":"p16","title":"lem:br-base-change-tower","kind":"lemma","summary":"Consider the tower of field extension E/F/K, \\[ \\Br_K^E = \\Br_E^F\\circ\\Br_K^E. \\]","labels":["lem:br-base-change-tower"],"detail_key":"p16"},{"id":"n25003","layer":"informal","project":"p16","title":"BrauerGroupHom.baseChange_idem","kind":"proof","summary":"If A\\in\\CSA_K, then E\\ox_F\\left(F\\ox_KA\\right) is isomorphic to E \\ox_F A as E-algebras.","labels":[],"detail_key":"p16"},{"id":"n25004","layer":"informal","project":"p16","title":"BrauerGroupHom.Br","kind":"corollary","summary":"\\Br forms a functor from category of field to category of abelian groups.","labels":[],"detail_key":"p16"},{"id":"n25005","layer":"informal","project":"p16","title":"This is the categorical version of~\\creflem:br-base-change-self and~\\creflem:br-base-chan…","kind":"proof","summary":"This is the categorical version of~\\creflem:br-base-change-self and~\\creflem:br-base-change-tow…","labels":[],"detail_key":"p16"},{"id":"n25006","layer":"informal","project":"p16","title":"Relative Brauer Group","kind":"definition","summary":"[Relative Brauer Group] Let E/K be a field extension, we define the relative Brauer group \\Br(E…","labels":[],"detail_key":"p16"},{"id":"n25007","layer":"informal","project":"p16","title":"Unpacking the definition of the relative Brauer group, we see that for any A \\in \\CSA_K,…","kind":"remark","summary":"Unpacking the definition of the relative Brauer group, we see that for any A \\in \\CSA_K, if E\\o…","labels":[],"detail_key":"p16"},{"id":"n25008","layer":"informal","project":"p16","title":"Splitting Field","kind":"definition","summary":"[Splitting Field] For any field extension E/K and any K-algebra A, we say E is a splitting fiel…","labels":[],"detail_key":"p16"},{"id":"n25009","layer":"informal","project":"p16","title":"thm:split-iff-mem-relative","kind":"theorem","summary":"Let E/K be a field extension and A\\in\\csa_K, E splits A if and only if [A]_\\sim_\\br\\in\\br(E/K).","labels":["thm:split-iff-mem-relative"],"detail_key":"p16"},{"id":"n25010","layer":"informal","project":"p16","title":"The ``onl","kind":"proof","summary":"The ``onl","labels":[],"detail_key":"p16"},{"id":"n25011","layer":"informal","project":"p16","title":"In light of~\\creflem:br-base-change-tower, if K is algebraic closed then K splits any K-a…","kind":"remark","summary":"In light of~\\creflem:br-base-change-tower, if K is algebraic closed then K splits any K-algebra…","labels":[],"detail_key":"p16"},{"id":"n25012","layer":"informal","project":"p16","title":"If two \\csa_K are Brauer equivalent, in another word, A \\sim_\\br_K B, then E splits A if…","kind":"remark","summary":"If two \\csa_K are Brauer equivalent, in another word, A \\sim_\\br_K B, then E splits A if and on…","labels":[],"detail_key":"p16"},{"id":"n25013","layer":"informal","project":"p16","title":"lem:good-rep-inv","kind":"lemma","summary":"Let A\\in\\csa_F splitted by K. There exists a B\\in\\csa_F such that \\item [A]_\\sim_\\br[B]_\\sim_\\b…","labels":["lem:good-rep-inv"],"detail_key":"p16"},{"id":"n25014","layer":"informal","project":"p16","title":"Since K splits A, we find a non-zero natural number n such that K\\ox_FA\\cong\\mat_nK\\cong\\…","kind":"proof","summary":"Since K splits A, we find a non-zero natural number n such that K\\ox_FA\\cong\\mat_nK\\cong\\End_K\\…","labels":[],"detail_key":"p16"},{"id":"n25015","layer":"informal","project":"p16","title":"cor:good-rep","kind":"corollary","summary":"Let A\\in\\csa_F splitted by K. There exists a B \\in \\csa_F such that \\item [B]_\\sim_\\br=[A]_\\sim…","labels":["cor:good-rep"],"detail_key":"p16"},{"id":"n25016","layer":"informal","project":"p16","title":"Let B and \\iota : K \\hookrightarrow B be as in~\\creflem:good-rep-inv. Consider B^\\opp and…","kind":"proof","summary":"Let B and \\iota : K \\hookrightarrow B be as in~\\creflem:good-rep-inv. Consider B^\\opp and K \\ho…","labels":[],"detail_key":"p16"},{"id":"n25017","layer":"informal","project":"p16","title":"thm:good-rep-iff-split","kind":"theorem","summary":"Let A\\in\\csa_F. K splits A if and only if there exists a B\\in\\csa_F such that \\item [B]_\\sim_\\b…","labels":["thm:good-rep-iff-split"],"detail_key":"p16"},{"id":"n25018","layer":"informal","project":"p16","title":"The ``if'' direction is~\\crefcor:good-rep. For the ``only if'' direction, let B \\in\\csa_F…","kind":"proof","summary":"The ``if'' direction is~\\crefcor:good-rep. For the ``only if'' direction, let B \\in\\csa_F and \\…","labels":[],"detail_key":"p16"},{"id":"n25019","layer":"informal","project":"p16","title":"Good Representation","kind":"definition","summary":"[Good Representation] For any X \\in \\br(F), a K-good representation of X is an A \\in \\csa_F and…","labels":["def:good-rep"],"detail_key":"p16"},{"id":"n25020","layer":"informal","project":"p16","title":"cor:mem-relative-br-iff-good-rep","kind":"corollary","summary":"For any X\\in\\br(F), X\\in\\br(K/F) if and only if X admits a good representation.","labels":["cor:mem-relative-br-iff-good-rep"],"detail_key":"p16"},{"id":"n25021","layer":"informal","project":"p16","title":"Rephrase of~\\crefthm:good-rep-iff-split and~\\crefthm:split-iff-mem-relative.","kind":"proof","summary":"Rephrase of~\\crefthm:good-rep-iff-split and~\\crefthm:split-iff-mem-relative.","labels":[],"detail_key":"p16"},{"id":"n25022","layer":"informal","project":"p16","title":"GoodRep.ιRange","kind":"lemma","summary":"The range \\iota_A(A) is a simple ring.","labels":[],"detail_key":"p16"},{"id":"n25023","layer":"informal","project":"p16","title":"Because K is a simple ring, \\iota_A is injective therefore \\iota_A(A)\\cong K.","kind":"proof","summary":"Because K is a simple ring, \\iota_A is injective therefore \\iota_A(A)\\cong K.","labels":[],"detail_key":"p16"},{"id":"n25024","layer":"informal","project":"p16","title":"lem:centralizer-range-good-rep","kind":"lemma","summary":"C_A\\left(\\iota_A(A)\\right) = \\iota_A(A).","labels":["lem:centralizer-range-good-rep"],"detail_key":"p16"},{"id":"n25025","layer":"informal","project":"p16","title":"In the language of~\\crefsec:subfield, \\iota_A(A) is a subfield of A, hence by~\\creflem:tf…","kind":"proof","summary":"In the language of~\\crefsec:subfield, \\iota_A(A) is a subfield of A, hence by~\\creflem:tfae-sub…","labels":[],"detail_key":"p16"},{"id":"n25026","layer":"informal","project":"p16","title":"con:good-rep-mod","kind":"construction","summary":"We give A a K-module structure by \\em left\\/ multiplication, that is for any c \\in K and a \\in…","labels":["con:good-rep-mod"],"detail_key":"p16"},{"id":"n25027","layer":"informal","project":"p16","title":"lem:good-rep-iso","kind":"lemma","summary":"If A and B are two good representations of X, then A \\cong B as F-algebras.","labels":["lem:good-rep-iso"],"detail_key":"p16"},{"id":"n25028","layer":"informal","project":"p16","title":"By~\\creflem:common-div-alg, we find a division F-algebra D and non-zero natural numbers m…","kind":"proof","summary":"By~\\creflem:common-div-alg, we find a division F-algebra D and non-zero natural numbers m, n su…","labels":[],"detail_key":"p16"},{"id":"n25029","layer":"informal","project":"p16","title":"Since \\gal(K/F) acts on K^\\star, for x\\in K^\\star,we feel free to write \\sigma\\cdot x whe…","kind":"remark","summary":"Since \\gal(K/F) acts on K^\\star, for x\\in K^\\star,we feel free to write \\sigma\\cdot x when it f…","labels":[],"detail_key":"p16"},{"id":"n25030","layer":"informal","project":"p16","title":"Conjugation Factor","kind":"definition","summary":"[Conjugation Factor] With respect to A, a conjugation factor of \\sigma is a unit x_\\sigma \\in A…","labels":["def:conj-factor"],"detail_key":"p16"},{"id":"n25031","layer":"informal","project":"p16","title":"rem:conj-factor-alternative-eq","kind":"remark","summary":"When x_\\sigma is a conjugation factor of \\sigma, the equalities x_\\sigma\\iota_A(c) = x_\\sigma\\i…","labels":["rem:conj-factor-alternative-eq"],"detail_key":"p16"},{"id":"n25032","layer":"informal","project":"p16","title":"con:exists-conj-seq","kind":"construction","summary":"A has a conjugation sequence: let \\sigma \\in \\gal(K/F), we have two F-algebra homomorphisms K \\…","labels":["con:exists-conj-seq"],"detail_key":"p16"},{"id":"n25033","layer":"informal","project":"p16","title":"GoodRep.mul'","kind":"construction","summary":"If x is a conjugation factor of \\sigma and y of \\tau, then xy is a conjugation factor of \\sigma…","labels":[],"detail_key":"p16"},{"id":"n25034","layer":"informal","project":"p16","title":"thm:conj-seq-basis","kind":"theorem","summary":"If x is an A-conjugation sequence, then \\x_\\sigma|\\sigma\\in \\gal(K/F)\\ is an K-linearly indepen…","labels":["thm:conj-seq-basis"],"detail_key":"p16"},{"id":"n25035","layer":"informal","project":"p16","title":"Suppose \\x_\\sigma\\ is linearly dependent. Let J \\subseteq \\gal(K/F) be such that \\x_\\sigm…","kind":"proof","summary":"Suppose \\x_\\sigma\\ is linearly dependent. Let J \\subseteq \\gal(K/F) be such that \\x_\\sigma|\\sig…","labels":[],"detail_key":"p16"},{"id":"n25036","layer":"informal","project":"p16","title":"the Second Group Cohomology","kind":"definition","summary":"[the Second Group Cohomology] Let G be a group and M an abelian group (written multiplicatively…","labels":["def:second-group-coh"],"detail_key":"p16"},{"id":"n25037","layer":"informal","project":"p16","title":"lem:2-cocycle-one-one","kind":"lemma","summary":"If f \\in B^2(G, M) is a 2-cocycle and x\\in G, we have \\[ f(1, x) &= f(1, 1) \\\\ f(x, 1) &= x\\cdo…","labels":["lem:2-cocycle-one-one"],"detail_key":"p16"},{"id":"n25038","layer":"informal","project":"p16","title":"CrossProductAlgebra.a_one_left","kind":"proof","summary":"Indeed: \\[ f(1\\cdot 1, x)f(1, 1) &= (1\\cdot f(1,x))f(1, 1\\cdot x) \\\\ f(1, x)f(1, 1) &= f(1, x)…","labels":[],"detail_key":"p16"},{"id":"n25039","layer":"informal","project":"p16","title":"Twisting Conjugation Factors","kind":"lemma","summary":"[Twisting Conjugation Factors] If x and y are two conjugation factors of \\sigma, then there exi…","labels":["lem:twist-spec1"],"detail_key":"p16"},{"id":"n25040","layer":"informal","project":"p16","title":"The uniqu","kind":"proof","summary":"The uniqu","labels":[],"detail_key":"p16"},{"id":"n25041","layer":"informal","project":"p16","title":"\\twist(x, x) is equal to 1 by uniqueness.","kind":"remark","summary":"\\twist(x, x) is equal to 1 by uniqueness.","labels":[],"detail_key":"p16"},{"id":"n25042","layer":"informal","project":"p16","title":"In fact, \\twist(x, y) is in K^\\star and \\twist(x, y)^-1=\\twist(y, x).","kind":"remark","summary":"In fact, \\twist(x, y) is in K^\\star and \\twist(x, y)^-1=\\twist(y, x).","labels":[],"detail_key":"p16"},{"id":"n25043","layer":"informal","project":"p16","title":"lem:twist-spec2","kind":"lemma","summary":"If x and y are conjugation factors for \\sigma, x=\\iota_A(\\sigma(\\twist_x, y))y.","labels":["lem:twist-spec2"],"detail_key":"p16"},{"id":"n25044","layer":"informal","project":"p16","title":"\\[ x &= x \\iota_A\\left(\\twist_x,y\\right) x^-1 x \\iota_A(\\twist_y,x) \\\\ &= \\iota_A\\left(\\s…","kind":"proof","summary":"\\[ x &= x \\iota_A\\left(\\twist_x,y\\right) x^-1 x \\iota_A(\\twist_y,x) \\\\ &= \\iota_A\\left(\\sigma\\c…","labels":[],"detail_key":"p16"},{"id":"n25045","layer":"informal","project":"p16","title":"Comparing Conjugation Factors","kind":"construction","summary":"[Comparing Conjugation Factors] Let x be a conjugation factor for \\sigma, y for \\tau and z for…","labels":["con:compare-conj-factors"],"detail_key":"p16"},{"id":"n25046","layer":"informal","project":"p16","title":"lem:compare-conj-factor-spec","kind":"lemma","summary":"Let x : \\gal(K/F) \\to A^\\star be a conjugation sequence. We have \\[ \\comp_x_\\rho, x_\\sigma, x_\\…","labels":["lem:compare-conj-factor-spec"],"detail_key":"p16"},{"id":"n25047","layer":"informal","project":"p16","title":"eq:1","kind":"proof","summary":"It is sufficient to make the following calculations: x_\\rho\\,x_\\sigmax_\\tau &= \\iota_A\\left(\\co…","labels":["eq:1","eq:2"],"detail_key":"p16"},{"id":"n25048","layer":"informal","project":"p16","title":"from good representation to 2-cocycle","kind":"construction","summary":"[from good representation to 2-cocycle] Let x be an A-conjugation sequence. We associate with x…","labels":["con:good-rep-to-2-cocycles"],"detail_key":"p16"},{"id":"n25049","layer":"informal","project":"p16","title":"GoodRep.isTwoCocyles","kind":"lemma","summary":"For any A-conjugation sequence x, B^2_x \\in B^2\\left(\\gal(K/F), K^\\star\\right), that is B_x is…","labels":[],"detail_key":"p16"},{"id":"n25050","layer":"informal","project":"p16","title":"We need to prove \\[ B_x(\\rho\\sigma, \\tau)\\, B_x(\\rho,\\sigma) = \\rho\\left(B_x(\\sigma, \\tau…","kind":"proof","summary":"We need to prove \\[ B_x(\\rho\\sigma, \\tau)\\, B_x(\\rho,\\sigma) = \\rho\\left(B_x(\\sigma, \\tau)\\righ…","labels":[],"detail_key":"p16"},{"id":"n25051","layer":"informal","project":"p16","title":"con:good-rep-iso-coeff","kind":"construction","summary":"By~\\creflem:good-rep-iso, A and B are isomorphic as F-algebras, we use e_A,B to denote an arbit…","labels":["con:good-rep-iso-coeff"],"detail_key":"p16"},{"id":"n25052","layer":"informal","project":"p16","title":"lem:iso-conj-coeff-spec2","kind":"lemma","summary":"For any c \\in K, \\sigma\\in\\gal(K/F) and A-conjugation factor x of \\sigma, we have \\[ \\iota_B\\le…","labels":["lem:iso-conj-coeff-spec2"],"detail_key":"p16"},{"id":"n25053","layer":"informal","project":"p16","title":"From~\\crefdef:conj-factor, we have e\\left(\\iota_A(\\sigma \\cdot c)\\right) = e\\left(x\\iota_…","kind":"proof","summary":"From~\\crefdef:conj-factor, we have e\\left(\\iota_A(\\sigma \\cdot c)\\right) = e\\left(x\\iota_A(c)x^…","labels":[],"detail_key":"p16"},{"id":"n25054","layer":"informal","project":"p16","title":"con:push-forward-conj-factor","kind":"construction","summary":"If x is an A-conjugation factor for \\sigma, we can obtain a B-conjugation factor for \\sigma by…","labels":["con:push-forward-conj-factor"],"detail_key":"p16"},{"id":"n25055","layer":"informal","project":"p16","title":"lem:compare-to-2-cocycle-aux","kind":"lemma","summary":"Let x be an A-conjugation sequence and y a B-conjugation sequence. We have \\[ \\comp_y_\\sigma, y…","labels":["lem:compare-to-2-cocycle-aux"],"detail_key":"p16"},{"id":"n25056","layer":"informal","project":"p16","title":"By~\\crefcon:compare-conj-factors, we have y_\\sigmay_\\tau = \\iota_B\\left(\\comp_y_\\sigma,y_…","kind":"proof","summary":"By~\\crefcon:compare-conj-factors, we have y_\\sigmay_\\tau = \\iota_B\\left(\\comp_y_\\sigma,y_\\tau,y…","labels":[],"detail_key":"p16"},{"id":"n25057","layer":"informal","project":"p16","title":"lem:compare-to-2-cocycle","kind":"lemma","summary":"Let x be an A-conjugation sequence and y a B-conjugation sequence. We have \\[ B^2_B,y(\\sigma,\\t…","labels":["lem:compare-to-2-cocycle"],"detail_key":"p16"},{"id":"n25058","layer":"informal","project":"p16","title":"If we unfold~\\crefcon:good-rep-to-2-cocycles, we discover the lemma is saying exactly~\\cr…","kind":"proof","summary":"If we unfold~\\crefcon:good-rep-to-2-cocycles, we discover the lemma is saying exactly~\\creflem:…","labels":[],"detail_key":"p16"},{"id":"n25059","layer":"informal","project":"p16","title":"cor:to-2-cocycle-wd","kind":"corollary","summary":"Let x be an A-conjugation sequence and y a B-conjugation sequence. B^2_A,x and B^2_B,y are 2-co…","labels":["cor:to-2-cocycle-wd"],"detail_key":"p16"},{"id":"n25060","layer":"informal","project":"p16","title":"By~\\crefdef:second-group-coh, we need to find a function f : \\gal(K/F)\\to K^\\star such th…","kind":"proof","summary":"By~\\crefdef:second-group-coh, we need to find a function f : \\gal(K/F)\\to K^\\star such that for…","labels":[],"detail_key":"p16"},{"id":"n25061","layer":"informal","project":"p16","title":"from \\br(K/F) to \\HH^2\\left(\\gal(K/F), K^\\star\\right)","kind":"construction","summary":"[from \\br(K/F) to \\HH^2\\left(\\gal(K/F), K^\\star\\right)] Let X \\in \\br(K/F), by~\\crefcor:mem-rel…","labels":["con:br-to-snd-coh"],"detail_key":"p16"},{"id":"n25062","layer":"informal","project":"p16","title":"Cross product","kind":"construction","summary":"[Cross product] Denote \\cross_\\mathfraka to be \\gal(K/F) \\to K, i.e. functions from \\gal(K/F) t…","labels":["con:cross-product"],"detail_key":"p16"},{"id":"n25063","layer":"informal","project":"p16","title":"When K/F is infinite dimensional, the correct definition of \\cross_\\mathfraka is perhaps…","kind":"remark","summary":"When K/F is infinite dimensional, the correct definition of \\cross_\\mathfraka is perhaps \\bigop…","labels":[],"detail_key":"p16"},{"id":"n25064","layer":"informal","project":"p16","title":"CrossProductAlgebra.algebra","kind":"lemma","summary":"The cross product \\cross_\\mathfraka is a ring with the multiplicative unit \\Delta_\\mathsfid, \\m…","labels":[],"detail_key":"p16"},{"id":"n25065","layer":"informal","project":"p16","title":"We verify the axioms of rings on elements of the form \\Delta_\\sigma, c. Let \\sigma,\\tau,\\…","kind":"proof","summary":"We verify the axioms of rings on elements of the form \\Delta_\\sigma, c. Let \\sigma,\\tau,\\rho\\in…","labels":[],"detail_key":"p16"},{"id":"n25066","layer":"informal","project":"p16","title":"K-embedding","kind":"construction","summary":"[K-embedding] The map \\iota_\\cross_\\mfa: K \\to \\cross_\\mfa defined by \\[ b \\mapsto \\Delta_\\id,…","labels":["con:cross-product-iota"],"detail_key":"p16"},{"id":"n25067","layer":"informal","project":"p16","title":"lem:cross-product-delta-invertible","kind":"lemma","summary":"For every \\sigma \\in \\gal(K/F), \\Delta_\\sigma,1 is invertible.","labels":["lem:cross-product-delta-invertible"],"detail_key":"p16"},{"id":"n25068","layer":"informal","project":"p16","title":"It is sufficient to prove that \\Delta_\\sigma, 1 has a left inverse and right inverse. The…","kind":"proof","summary":"It is sufficient to prove that \\Delta_\\sigma, 1 has a left inverse and right inverse. The left…","labels":[],"detail_key":"p16"},{"id":"n25069","layer":"informal","project":"p16","title":"lem:cross-product-basis-conj","kind":"lemma","summary":"For any c \\in K, we have \\[ \\Delta_\\sigma,1\\,\\iota_\\mfa(c) = \\iota_\\mfa(\\sigma\\cdot c)\\Delta_\\s…","labels":["lem:cross-product-basis-conj"],"detail_key":"p16"},{"id":"n25070","layer":"informal","project":"p16","title":"We calculate \\[ \\Delta_\\sigma,1\\iota_\\mfa(c) &= \\Delta_\\sigma,1\\,\\Delta_\\id,c\\mfa(\\id,\\id…","kind":"proof","summary":"We calculate \\[ \\Delta_\\sigma,1\\iota_\\mfa(c) &= \\Delta_\\sigma,1\\,\\Delta_\\id,c\\mfa(\\id,\\id)^-1\\\\…","labels":[],"detail_key":"p16"},{"id":"n25071","layer":"informal","project":"p16","title":"lem:cross-product-basis-mul","kind":"lemma","summary":"We have \\Delta_\\sigma,1\\,\\Delta_\\tau,1 = \\iota_\\mfa(\\mfa(\\sigma,\\tau))\\Delta_\\sigma\\tau,1 = \\mf…","labels":["lem:cross-product-basis-mul"],"detail_key":"p16"},{"id":"n25072","layer":"informal","project":"p16","title":"The first equality is in~\\crefcon:cross-product-iota. For the second equality, by~\\crefle…","kind":"proof","summary":"The first equality is in~\\crefcon:cross-product-iota. For the second equality, by~\\creflem:cros…","labels":[],"detail_key":"p16"},{"id":"n25073","layer":"informal","project":"p16","title":"lem:cross-product-basis","kind":"lemma","summary":"The set \\\\Delta_\\sigma, 1|\\sigma\\in\\gal(K/F)\\ forms a K-basis for \\cross_\\mfa.","labels":["lem:cross-product-basis"],"detail_key":"p16"},{"id":"n25074","layer":"informal","project":"p16","title":"Suppose some linear combination \\sum_\\sigma\\lambda_\\sigma\\cdot\\Delta_\\sigma, 1 is 0 for s…","kind":"proof","summary":"Suppose some linear combination \\sum_\\sigma\\lambda_\\sigma\\cdot\\Delta_\\sigma, 1 is 0 for some \\l…","labels":[],"detail_key":"p16"},{"id":"n25075","layer":"informal","project":"p16","title":"cor:dim-cross-product","kind":"corollary","summary":"When K/F is a finite dimensional Galois extension, the K-dimension of \\cross_\\mfa is \\dim_FK an…","labels":["cor:dim-cross-product"],"detail_key":"p16"},{"id":"n25076","layer":"informal","project":"p16","title":"Centrality","kind":"theorem","summary":"[Centrality] \\cross_\\mfa is a central F-algebra.","labels":["thm:cross-product-central"],"detail_key":"p16"},{"id":"n25077","layer":"informal","project":"p16","title":"eq:cross-product-central-eqn","kind":"proof","summary":"Let z \\in \\cross_\\mfa that is in the centre. We want to prove that z is in F. We write z as \\su…","labels":["eq:cross-product-central-eqn"],"detail_key":"p16"},{"id":"n25078","layer":"informal","project":"p16","title":"con:cross-product-simple-ring-mod","kind":"construction","summary":"The quotient ring ^\\cross_\\mfa/_I is a \\Pi-module defined by \\pi\\left(\\iota_\\mfa(a)\\right)\\cdot…","labels":["con:cross-product-simple-ring-mod"],"detail_key":"p16"},{"id":"n25079","layer":"informal","project":"p16","title":"lem:cross-product-quotient-basis","kind":"lemma","summary":"If I \\ne \\cross_\\mfa, the set \\\\pi\\left(\\Delta_\\sigma, 1\\right)|\\sigma\\in\\gal(K/F)\\ forms a K-b…","labels":["lem:cross-product-quotient-basis"],"detail_key":"p16"},{"id":"n25080","layer":"informal","project":"p16","title":"It is easy to see that the set spans ^\\cross_\\mfa/_I because \\\\Delta_\\sigma, 1|\\sigma\\in\\…","kind":"proof","summary":"It is easy to see that the set spans ^\\cross_\\mfa/_I because \\\\Delta_\\sigma, 1|\\sigma\\in\\gal(K/…","labels":[],"detail_key":"p16"},{"id":"n25081","layer":"informal","project":"p16","title":"cor:cross-product-quot-iso","kind":"corollary","summary":"If I \\ne \\cross_\\mfa, the quotient ring ^\\cross_\\mfa/_I is isomorphic to \\cross_\\mfa as K-modul…","labels":["cor:cross-product-quot-iso"],"detail_key":"p16"},{"id":"n25082","layer":"informal","project":"p16","title":"Indeed, b","kind":"proof","summary":"Indeed, b","labels":[],"detail_key":"p16"},{"id":"n25083","layer":"informal","project":"p16","title":"Simple Ring","kind":"corollary","summary":"[Simple Ring] \\cross_\\mfa is a simple ring.","labels":["cor:cross-product-simple"],"detail_key":"p16"},{"id":"n25084","layer":"informal","project":"p16","title":"For any two-sided-ideal I that is not equal to \\cross_\\mfa, by~\\crefcor:cross-product-quo…","kind":"proof","summary":"For any two-sided-ideal I that is not equal to \\cross_\\mfa, by~\\crefcor:cross-product-quot-iso,…","labels":[],"detail_key":"p16"},{"id":"n25085","layer":"informal","project":"p16","title":"thm:cross-product-csa","kind":"theorem","summary":"Let K/F be a finite dimensional and Galois field extension and \\mfa be a 2-cocycle in B^2\\left(…","labels":["thm:cross-product-csa"],"detail_key":"p16"},{"id":"n25086","layer":"informal","project":"p16","title":"\\Crefthm:cross-product-central, \\creflem:cross-product-basis and \\crefcor:cross-product-s…","kind":"proof","summary":"\\Crefthm:cross-product-central, \\creflem:cross-product-basis and \\crefcor:cross-product-simple.","labels":[],"detail_key":"p16"},{"id":"n25087","layer":"informal","project":"p16","title":"thm:snd-coh-to-relative-br","kind":"theorem","summary":"If K/F is a finite dimensional and Galois field extension, the function \\cross : \\HH^2\\left(\\ga…","labels":["thm:snd-coh-to-relative-br"],"detail_key":"p16"},{"id":"n25088","layer":"informal","project":"p16","title":"eq:cross-product-cohomologous","kind":"proof","summary":"Let \\mfa and \\mfb be two cohomologous 2-cocycles. By~\\crefdef:second-group-coh, for some \\mfc:…","labels":["eq:cross-product-cohomologous"],"detail_key":"p16"},{"id":"n25089","layer":"informal","project":"p16","title":"lem:relative-br-snd-inverse-1","kind":"lemma","summary":"The composition of \\cross and \\HH^2 is the identity: \\displaystyle \\HH^2\\left(\\gal(K/F), K^\\sta…","labels":["lem:relative-br-snd-inverse-1"],"detail_key":"p16"},{"id":"n25090","layer":"informal","project":"p16","title":"Let \\mfa be any 2-cocycle, by~\\creflem:cross-product-basis-conj, we notice that x : \\sigm…","kind":"proof","summary":"Let \\mfa be any 2-cocycle, by~\\creflem:cross-product-basis-conj, we notice that x : \\sigma \\map…","labels":[],"detail_key":"p16"},{"id":"n25091","layer":"informal","project":"p16","title":"lem:relative-br-snd-inverse-2","kind":"lemma","summary":"The composition of \\HH^2 and \\cross is the identity: \\br(K/F) \\arrowr\\HH^2 \\arrow[bend right =…","labels":["lem:relative-br-snd-inverse-2"],"detail_key":"p16"},{"id":"n25092","layer":"informal","project":"p16","title":"Let X \\in \\br(K/F), A be an arbitrary good representation of X and x be an arbitrary A-co…","kind":"proof","summary":"Let X \\in \\br(K/F), A be an arbitrary good representation of X and x be an arbitrary A-conjugat…","labels":[],"detail_key":"p16"},{"id":"n25093","layer":"informal","project":"p16","title":"cor:relative-br-2nd-coh-bij","kind":"corollary","summary":"For a finite dimensional and Galois extension of field K/F, the relative Brauer group K/F bijec…","labels":["cor:relative-br-2nd-coh-bij"],"detail_key":"p16"},{"id":"n25094","layer":"informal","project":"p16","title":"Exactly~\\creflem:relative-br-snd-inverse-1 and~\\creflem:relative-br-snd-inverse-2.","kind":"proof","summary":"Exactly~\\creflem:relative-br-snd-inverse-1 and~\\creflem:relative-br-snd-inverse-2.","labels":[],"detail_key":"p16"},{"id":"n25095","layer":"informal","project":"p16","title":"thm:map_one","kind":"theorem","summary":"The function \\cross:\\HH^2\\left(\\gal(K/F), K^\\star\\right) \\to \\br(K/F) preserves one, that is \\c…","labels":["thm:map_one"],"detail_key":"p16"},{"id":"n25096","layer":"informal","project":"p16","title":"Since \\\\Delta_\\sigma, 1|\\sigma\\in\\gal(K/F)\\ is a K-basis for \\cross_1 where 1 \\in B^2\\lef…","kind":"proof","summary":"Since \\\\Delta_\\sigma, 1|\\sigma\\in\\gal(K/F)\\ is a K-basis for \\cross_1 where 1 \\in B^2\\left(\\gal…","labels":[],"detail_key":"p16"},{"id":"n25097","layer":"informal","project":"p16","title":"RelativeBrGroup.toSndAddMonoidHom","kind":"corollary","summary":"The function \\HH^2:\\br(K/F) \\to \\HH^2\\left(\\gal(K/F), K^\\star\\right) preserves one, that is \\HH…","labels":[],"detail_key":"p16"},{"id":"n25098","layer":"informal","project":"p16","title":"Apply \\cross then use~\\creflem:relative-br-snd-inverse-2 and~\\crefthm:map_one.","kind":"proof","summary":"Apply \\cross then use~\\creflem:relative-br-snd-inverse-2 and~\\crefthm:map_one.","labels":[],"detail_key":"p16"},{"id":"n25099","layer":"informal","project":"p16","title":"M","kind":"construction","summary":"[M] Consider the quotient module \\[M := ^\\cross_\\mfa\\ox_F\\cross_\\mfb/_\\left\\langle (k\\cdot a)\\o…","labels":[],"detail_key":"p16"},{"id":"n25100","layer":"informal","project":"p16","title":"lem:M-iso-tensor-over-K","kind":"lemma","summary":"M is isomorphic to \\cross_\\mfa\\ox_K\\cross_\\mfb as F-modules.","labels":["lem:M-iso-tensor-over-K"],"detail_key":"p16"},{"id":"n25101","layer":"informal","project":"p16","title":"The map M \\to \\cross_\\mfa\\ox_K\\cross_\\mfb is obtained by descending the obvious F-linear…","kind":"proof","summary":"The map M \\to \\cross_\\mfa\\ox_K\\cross_\\mfb is obtained by descending the obvious F-linear map \\c…","labels":[],"detail_key":"p16"},{"id":"n25102","layer":"informal","project":"p16","title":"cor:dim-finite-M","kind":"corollary","summary":"The F-dimension of M is equal to \\left(\\dim_FK\\right)^3, consequently M is a finitely generated…","labels":["cor:dim-finite-M"],"detail_key":"p16"},{"id":"n25103","layer":"informal","project":"p16","title":"By~\\creflem:M-iso-tensor-over-K, the dimension of M is equal to \\dim_F\\cross_\\mfa\\ox_K\\cr…","kind":"proof","summary":"By~\\creflem:M-iso-tensor-over-K, the dimension of M is equal to \\dim_F\\cross_\\mfa\\ox_K\\cross_\\m…","labels":[],"detail_key":"p16"},{"id":"n25104","layer":"informal","project":"p16","title":"con:cross-product-mul-iso","kind":"construction","summary":"By~\\creflem:direct-sum-simple-mod, there exists some simple \\cross_\\mfc-module S such that \\cro…","labels":["con:cross-product-mul-iso"],"detail_key":"p16"},{"id":"n25105","layer":"informal","project":"p16","title":"cor:dim-eq-1","kind":"corollary","summary":"\\[ \\left(\\dim_FK\\right)^2 = |J|^2\\dim_F\\End_\\cross_\\mfcS. \\]","labels":["cor:dim-eq-1"],"detail_key":"p16"},{"id":"n25106","layer":"informal","project":"p16","title":"They are both equal to \\dim_F\\cross_\\mfc by~\\crefcon:cross-product-mul-iso.","kind":"proof","summary":"They are both equal to \\dim_F\\cross_\\mfc by~\\crefcon:cross-product-mul-iso.","labels":[],"detail_key":"p16"},{"id":"n25107","layer":"informal","project":"p16","title":"cor:dim-eq-2","kind":"corollary","summary":"\\[ |J|\\dim_FS = \\left(\\dim_FK\\right)^2 \\]","labels":["cor:dim-eq-2"],"detail_key":"p16"},{"id":"n25108","layer":"informal","project":"p16","title":"They are all equal to \\dim_F\\cross_\\mfc = \\dim_FS^|J| by~\\crefcon:cross-product-mul-iso.","kind":"proof","summary":"They are all equal to \\dim_F\\cross_\\mfc = \\dim_FS^|J| by~\\crefcon:cross-product-mul-iso.","labels":[],"detail_key":"p16"},{"id":"n25109","layer":"informal","project":"p16","title":"lem:M-iso-simple-pow","kind":"lemma","summary":"There exists a \\cross_\\mfc-linear isomorphism between M and S^|J|\\dim_FK.","labels":["lem:M-iso-simple-pow"],"detail_key":"p16"},{"id":"n25110","layer":"informal","project":"p16","title":"By~\\creflem:iso-of-dim-eq, we only need to show that \\dim_FM = \\dim_FS^|J|\\dim_FK. We alr…","kind":"proof","summary":"By~\\creflem:iso-of-dim-eq, we only need to show that \\dim_FM = \\dim_FS^|J|\\dim_FK. We already h…","labels":[],"detail_key":"p16"},{"id":"n25111","layer":"informal","project":"p16","title":"cor:M-iso-simple-pow","kind":"corollary","summary":"As F-vector spaces, M \\cong S^|J|\\dim_FK.","labels":["cor:M-iso-simple-pow"],"detail_key":"p16"},{"id":"n25112","layer":"informal","project":"p16","title":"Restricting scalars on the \\cross_\\mfc-linear isomorphism in~\\creflem:M-iso-simple-pow","kind":"proof","summary":"Restricting scalars on the \\cross_\\mfc-linear isomorphism in~\\creflem:M-iso-simple-pow","labels":[],"detail_key":"p16"},{"id":"n25113","layer":"informal","project":"p16","title":"cor:end-M-iso-mat","kind":"corollary","summary":"As F-algebras, \\End_\\cross_\\mfcM \\cong \\mat_|J|\\dim_FK(\\End_\\cross_\\mfcS).","labels":["cor:end-M-iso-mat"],"detail_key":"p16"},{"id":"n25114","layer":"informal","project":"p16","title":"From~\\crefcor:M-iso-simple-pow, we have \\End_\\cross_\\mfcM\\cong \\End_\\cross_\\mfc\\left(S^|J…","kind":"proof","summary":"From~\\crefcor:M-iso-simple-pow, we have \\End_\\cross_\\mfcM\\cong \\End_\\cross_\\mfc\\left(S^|J|\\dim_…","labels":[],"detail_key":"p16"},{"id":"n25115","layer":"informal","project":"p16","title":"cor:dim-eq-3","kind":"corollary","summary":"\\[ \\dim_F\\End_\\cross_\\mfcM = \\left(\\dim_FK\\right)^4. \\]","labels":["cor:dim-eq-3"],"detail_key":"p16"},{"id":"n25116","layer":"informal","project":"p16","title":"\\[ \\dim_F\\End_\\cross_\\mfcM &= \\dim_F\\mat_|J|\\dim_FK(\\End_\\cross_\\mfcS) \\\\ &= \\dim_F\\left(…","kind":"proof","summary":"\\[ \\dim_F\\End_\\cross_\\mfcM &= \\dim_F\\mat_|J|\\dim_FK(\\End_\\cross_\\mfcS) \\\\ &= \\dim_F\\left(\\End_\\…","labels":[],"detail_key":"p16"},{"id":"n25117","layer":"informal","project":"p16","title":"thm:cross-mul","kind":"theorem","summary":"The cross product \\cross_\\mfc and the tensor product \\cross_\\mfa\\ox_F\\cross_\\mfb are Brauer equ…","labels":["thm:cross-mul"],"detail_key":"p16"},{"id":"n25118","layer":"informal","project":"p16","title":"We define an F-algebra homomorphism \\phi : \\left(\\cross_\\mfa\\ox_F\\cross_\\mfb\\right)^\\opp…","kind":"proof","summary":"We define an F-algebra homomorphism \\phi : \\left(\\cross_\\mfa\\ox_F\\cross_\\mfb\\right)^\\opp \\to \\E…","labels":[],"detail_key":"p16"},{"id":"n25119","layer":"informal","project":"p16","title":"group isomorphism","kind":"corollary","summary":"[group isomorphism] For a finite dimensional Galois field extension K/F, the relative Brauer gr…","labels":[],"detail_key":"p16"},{"id":"n25120","layer":"informal","project":"p16","title":"In~\\crefcor:relative-br-2nd-coh-bij, we have seen that \\HH^2 and \\cross form a bijection,…","kind":"proof","summary":"In~\\crefcor:relative-br-2nd-coh-bij, we have seen that \\HH^2 and \\cross form a bijection, 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(…","labels":[],"detail_key":"p16","name":"end_simple_mod_of_wedderburn","module":"BrauerGroup.ZeroSevenFourE"},{"id":"n25220","layer":"formal","project":"p16","title":"end_simple_mod_of_wedderburn'","kind":"theorem","summary":"∀ (k : Type u) (A : Type v) [inst : Field k] [inst_1 : Ring A] [inst_2 : Algebra k A] [IsSimple…","labels":[],"detail_key":"p16","name":"end_simple_mod_of_wedderburn'","module":"BrauerGroup.ZeroSevenFourE"},{"id":"n25221","layer":"formal","project":"p16","title":"isBalanced_of_simpleMod","kind":"theorem","summary":"∀ (k : Type u) (A : Type v) [inst : Field k] [inst_1 : Ring A] [inst_2 : Algebra k A] [IsSimple…","labels":[],"detail_key":"p16","name":"isBalanced_of_simpleMod","module":"BrauerGroup.ZeroSevenFourE"},{"id":"n25222","layer":"formal","project":"p16","title":"linearEquiv_iff_finrank_eq_over_simple_ring","kind":"theorem","summary":"∀ (k : Type u) (A : Type v) [inst : Field k] [inst_1 : Ring A] [inst_2 : Algebra k A] 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If f:G\\toR is such that E_x f(x)=0 and \\left\\lv…","labels":["mzi"],"detail_key":"p17"},{"id":"n25226","layer":"informal","project":"p17","title":"Let S be the left-hand side. Since 0=E_y f(y) we have, by the triangle inequality, and H\\…","kind":"proof","summary":"Let S be the left-hand side. Since 0=E_y f(y) we have, by the triangle inequality, and H\\\"older…","labels":[],"detail_key":"p17"},{"id":"n25227","layer":"informal","project":"p17","title":"Complex-valued Marcinkiewicz-Zygmund inequality","kind":"lemma","summary":"[Complex-valued Marcinkiewicz-Zygmund inequality] Let m\\geq 1. If f:G\\toC is such that E_x f(x)…","labels":["mzi_complex"],"detail_key":"p17"},{"id":"n25228","layer":"informal","project":"p17","title":"Test.","kind":"proof","summary":"Test.","labels":[],"detail_key":"p17"},{"id":"n25229","layer":"informal","project":"p17","title":"random_approx_expect","kind":"lemma","summary":"Let \\epsilon>0 and m\\geq 1. Let A\\subseteq G and f:G\\to C. 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It remains to note that \\[\\| 1_A\\circ 1_A\\|_p(\\mu)\\geq…","labels":[],"detail_key":"p17"},{"id":"n25271","layer":"informal","project":"p17","title":"ap_in_ff","kind":"theorem","summary":"If A_1,A_2,S\\subseteq F_q^n are such that A_1 and A_2 both have density at least \\alpha then th…","labels":["ap_in_ff"],"detail_key":"p17"},{"id":"n25272","layer":"informal","project":"p17","title":"(In this proof we write G=F_q^n.) Let k=\\lceil L(\\epsilon\\alpha/4)\\rceil. Note that \\lver…","kind":"proof","summary":"(In this proof we write G=F_q^n.) Let k=\\lceil L(\\epsilon\\alpha/4)\\rceil. Note that \\lvert A_1+…","labels":[],"detail_key":"p17"},{"id":"n25273","layer":"informal","project":"p17","title":"ast_le_circ","kind":"lemma","summary":"For any function f:G\\to C and integer k\\geq 1 \\[\\| f\\ast f\\|_2k\\leq \\| f\\circ f\\|_2k.\\]","labels":["ast_le_circ"],"detail_key":"p17"},{"id":"n25274","layer":"informal","project":"p17","title":"To finish, similar trick to unbalancing.","kind":"proof","summary":"To finish, similar trick to unbalancing.","labels":[],"detail_key":"p17"},{"id":"n25275","layer":"informal","project":"p17","title":"ast_expand","kind":"lemma","summary":"For any function f with \\sum f(x)=1 \\[f\\ast f-1/N = (f-1/N)\\ast (f-1/N).\\]","labels":["ast_expand"],"detail_key":"p17"},{"id":"n25276","layer":"informal","project":"p17","title":"Expand everything out.","kind":"proof","summary":"Expand everything out.","labels":[],"detail_key":"p17"},{"id":"n25277","layer":"informal","project":"p17","title":"circ_expand","kind":"lemma","summary":"For any function f with \\sum f(x)=1 \\[f\\circ f-1/N = (f-1/N)\\circ (f-1/N).\\]","labels":["circ_expand"],"detail_key":"p17"},{"id":"n25278","layer":"informal","project":"p17","title":"Expand everything out.","kind":"proof","summary":"Expand everything out.","labels":[],"detail_key":"p17"},{"id":"n25279","layer":"informal","project":"p17","title":"global_dichotomy","kind":"lemma","summary":"Let \\epsilon >0 and \\mu\\equiv 1/N. If A,C\\subseteq G, where C has density at least \\gamma, and…","labels":["global_dichotomy"],"detail_key":"p17"},{"id":"n25280","layer":"informal","project":"p17","title":"By H\\\"older's inequality, for any p\\geq 1 \\[\\epsilon < \\left\\lvert N\\langle \\mu_A \\ast \\m…","kind":"proof","summary":"By H\\\"older's inequality, for any p\\geq 1 \\[\\epsilon < \\left\\lvert N\\langle \\mu_A \\ast \\mu_A -…","labels":[],"detail_key":"p17"},{"id":"n25281","layer":"informal","project":"p17","title":"di_in_ff","kind":"proposition","summary":"Let \\epsilon \\in (0,1). 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The corresponding Bohr set is defined to be \\[Bohr(\\nu)=\\…","labels":["bohr-set"],"detail_key":"p17"},{"id":"n25288","layer":"informal","project":"p17","title":"bohr-size","kind":"lemma","summary":"If \\rho\\in (0,1) and \\nu:\\widehatG\\to R then \\[\\left\\lvert Bohr(\\rho\\cdot \\nu)\\right\\rvert\\geq…","labels":["bohr-size"],"detail_key":"p17"},{"id":"n25289","layer":"informal","project":"p17","title":"There are at most \\lceil 4/\\rho\\rceil many z_i such that if \\lvert 1-w\\rvert \\leq \\nu(\\ga…","kind":"proof","summary":"There are at most \\lceil 4/\\rho\\rceil many z_i such that if \\lvert 1-w\\rvert \\leq \\nu(\\gamma) t…","labels":[],"detail_key":"p17"},{"id":"n25290","layer":"informal","project":"p17","title":"Regularity","kind":"definition","summary":"[Regularity] We say \\nu:\\widehatG\\to R is regular if, with d=rk(\\nu), for all \\kappa\\inR with \\…","labels":["bohr-reg-def"],"detail_key":"p17"},{"id":"n25291","layer":"informal","project":"p17","title":"bohr-regularity","kind":"lemma","summary":"For any \\nu:\\widehatG\\to R there exists \\rho\\in[\\frac12,1] such that \\rho\\cdot \\nu is regular.","labels":["bohr-regularity"],"detail_key":"p17"},{"id":"n25292","layer":"informal","project":"p17","title":"To do.","kind":"proof","summary":"To do.","labels":[],"detail_key":"p17"},{"id":"n25293","layer":"informal","project":"p17","title":"reg-conv","kind":"lemma","summary":"If B is a regular Bohr set of rank d and \\mu:G\\toR_\\geq 0 is supported on B_\\rho, with \\rho \\in…","labels":["reg-conv"],"detail_key":"p17"},{"id":"n25294","layer":"informal","project":"p17","title":"To do.","kind":"proof","summary":"To do.","labels":[],"detail_key":"p17"},{"id":"n25295","layer":"informal","project":"p17","title":"bohr-majorise","kind":"lemma","summary":"There is a constant c>0 such that the following holds. 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Let \\epsilon>0 and B,B'\\subseteq G be re…","labels":["th-ap-int"],"detail_key":"p17"},{"id":"n25300","layer":"informal","project":"p17","title":"To do.","kind":"proof","summary":"To do.","labels":[],"detail_key":"p17"},{"id":"n25301","layer":"informal","project":"p17","title":"Lp-orth","kind":"proposition","summary":"There is a constant c>0 such that the following holds. Let \\epsilon >0 and p \\geq 2 be an integ…","labels":["Lp-orth"],"detail_key":"p17"},{"id":"n25302","layer":"informal","project":"p17","title":"To do.","kind":"proof","summary":"To do.","labels":[],"detail_key":"p17"},{"id":"n25303","layer":"informal","project":"p17","title":"pos-def-measures","kind":"proposition","summary":"There is a constant c>0 such that the following holds. Let p \\geq 2 be an even integer. 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Re…","labels":[],"detail_key":"p17","name":"largeSpec","module":"APAP.Prereqs.LargeSpec"},{"id":"n25346","layer":"formal","project":"p17","title":"RCLike.marcinkiewicz_zygmund","kind":"theorem","summary":"∀ ι : Type u_1 A : Finset ι m n : Nat 𝕜 : Type u_2 [inst : RCLike 𝕜], Ne m 0 → ∀ (f : ι → 𝕜), (…","labels":[],"detail_key":"p17","name":"RCLike.marcinkiewicz_zygmund","module":"APAP.Prereqs.MarcinkiewiczZygmund"},{"id":"n25347","layer":"formal","project":"p17","title":"Real.marcinkiewicz_zygmund","kind":"theorem","summary":"∀ ι : Type u_1 A : Finset ι m n : Nat, Ne m 0 → ∀ (f : ι → Real), (∀ (i : Fin n), Eq ((Fintype.…","labels":[],"detail_key":"p17","name":"Real.marcinkiewicz_zygmund","module":"APAP.Prereqs.MarcinkiewiczZygmund"},{"id":"n25348","layer":"formal","project":"p17","title":"Real.marcinkiewicz_zygmund'","kind":"theorem","summary":"∀ ι : Type u_1 A : Finset ι n : Nat (m : Nat) (f : ι → Real), (∀ (i : Fin n), Eq ((Fintype.piFi…","labels":[],"detail_key":"p17","name":"Real.marcinkiewicz_zygmund'","module":"APAP.Prereqs.MarcinkiewiczZygmund"},{"id":"n25349","layer":"formal","project":"p17","title":"rudin_exp_ineq","kind":"theorem","summary":"∀ G : Type u_1 [inst : Fintype G] [inst_1 : AddCommGroup G] [inst_2 : MeasurableSpace G] [Discr…","labels":[],"detail_key":"p17","name":"rudin_exp_ineq","module":"APAP.Prereqs.Rudin"},{"id":"n25350","layer":"formal","project":"p17","title":"rudin_ineq","kind":"theorem","summary":"∀ G : Type u_1 [inst : Fintype G] [inst_1 : AddCommGroup G] p : Nat [inst_2 : MeasurableSpace G…","labels":[],"detail_key":"p17","name":"rudin_ineq","module":"APAP.Prereqs.Rudin"},{"id":"n25351","layer":"informal","project":"p18","title":"Forgetting a coordinate","kind":"definition","summary":"[Forgetting a coordinate] For an index i : [d], we define \\forget_i : G^d & \\to G^\\j : [d] \\mid…","labels":["def:forget"],"detail_key":"p18"},{"id":"n25352","layer":"informal","project":"p18","title":"Forbidden pattern","kind":"definition","summary":"[Forbidden pattern] We say a tuple (a_1, \\dots, a_d) : (G^d)^d is a \\bf forbidden pattern with…","labels":["def:forbidden-pattern"],"detail_key":"p18"},{"id":"n25353","layer":"informal","project":"p18","title":"Multidimensional corner","kind":"definition","summary":"[Multidimensional corner] A \\bf multidimensional corner in d dimensions is a tuple of the form…","labels":["def:multicorner"],"detail_key":"p18"},{"id":"n25354","layer":"informal","project":"p18","title":"Corner-free number","kind":"definition","summary":"[Corner-free number] The \\bf d-dimensional corner-free number of G, denoted r_d(G) is the size…","labels":["def:corner-free-num"],"detail_key":"p18"},{"id":"n25355","layer":"informal","project":"p18","title":"Corner-coloring number","kind":"definition","summary":"[Corner-coloring number] The \\bf d-dimensional corner-coloring number of G, denoted \\chi_d(G),…","labels":["def:corner-color-num"],"detail_key":"p18"},{"id":"n25356","layer":"informal","project":"p18","title":"Lower bound on the corner-coloring number","kind":"lemma","summary":"[Lower bound on the corner-coloring number] r_d(G) \\chi_d(G) \\ge N^d","labels":["lem:corner-num-lower"],"detail_key":"p18"},{"id":"n25357","layer":"informal","project":"p18","title":"Find a coloring of G^d in \\chi_d(G) colors without non-trivial monochromatic d-dimensiona…","kind":"proof","summary":"Find a coloring of G^d in \\chi_d(G) colors without non-trivial monochromatic d-dimensional corn…","labels":[],"detail_key":"p18"},{"id":"n25358","layer":"informal","project":"p18","title":"Upper bound on the corner-coloring number","kind":"lemma","summary":"[Upper bound on the corner-coloring number] r_d(G) \\chi_d(G) \\le 2d N^d \\log N","labels":["lem:corner-num-upper"],"detail_key":"p18"},{"id":"n25359","layer":"informal","project":"p18","title":"Find A a corner-free set of density \\alpha = r_d(G)/N^d. If we pick m > d\\log N/\\alpha tr…","kind":"proof","summary":"Find A a corner-free set of density \\alpha = r_d(G)/N^d. If we pick m > d\\log N/\\alpha translat…","labels":[],"detail_key":"p18"},{"id":"n25360","layer":"informal","project":"p18","title":"NOF protocol","kind":"definition","summary":"[NOF protocol] A \\bf NOF protocol P consists of maps \\strat : & [d] \\to G^d - 1 \\to \\leanList \\…","labels":["def:protocol"],"detail_key":"p18"},{"id":"n25361","layer":"informal","project":"p18","title":"NOF broadcast","kind":"definition","summary":"[NOF broadcast] Given a NOF protocol P, the \\bf NOF broadcast on input x : G^d is inductively d…","labels":["def:broadcast"],"detail_key":"p18"},{"id":"n25362","layer":"informal","project":"p18","title":"Length of a broadcast","kind":"lemma","summary":"[Length of a broadcast] For every NOF protocol P, every input x : G^d and every time t, \\broad(…","labels":["lem:length-broadcast"],"detail_key":"p18"},{"id":"n25363","layer":"informal","project":"p18","title":"Induction on t.","kind":"proof","summary":"Induction on t.","labels":[],"detail_key":"p18"},{"id":"n25364","layer":"informal","project":"p18","title":"Valid NOF protocol","kind":"definition","summary":"[Valid NOF protocol] Given a function F : G^d \\to \\Bool, the NOF protocol P is \\bf valid in F a…","labels":["def:valid-protocol"],"detail_key":"p18"},{"id":"n25365","layer":"informal","project":"p18","title":"The trivial protocol","kind":"definition","summary":"[The trivial protocol] For all F, we define the \\bf trivial protocol by making participant i do…","labels":["def:trivial-protocol"],"detail_key":"p18"},{"id":"n25366","layer":"informal","project":"p18","title":"The trivial protocol is valid","kind":"lemma","summary":"[The trivial protocol is valid] For all F, the trivial protocol for F is valid in time d\\left\\l…","labels":["lem:trivial-protocol-valid"],"detail_key":"p18"},{"id":"n25367","layer":"informal","project":"p18","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p18"},{"id":"n25368","layer":"informal","project":"p18","title":"Deterministic complexity of a protocol","kind":"definition","summary":"[Deterministic complexity of a protocol] The \\bf communication complexity of a NOF protocol P f…","labels":["def:det-protocol-complex"],"detail_key":"p18"},{"id":"n25369","layer":"informal","project":"p18","title":"Deterministic complexity of a function","kind":"definition","summary":"[Deterministic complexity of a function] The \\bf deterministic communication complexity of a fu…","labels":["def:det-fun-complex"],"detail_key":"p18"},{"id":"n25370","layer":"informal","project":"p18","title":"Trivial bound on the function complexity","kind":"lemma","summary":"[Trivial bound on the function complexity] The communication complexity of any function F is at…","labels":["lem:trivial-bound-det-fun-complex"],"detail_key":"p18"},{"id":"n25371","layer":"informal","project":"p18","title":"The trivial protocol is a protocol valid in F in time d\\left\\lceil \\log_2 n\\right\\rceil.","kind":"proof","summary":"The trivial protocol is a protocol valid in F in time d\\left\\lceil \\log_2 n\\right\\rceil.","labels":[],"detail_key":"p18"},{"id":"n25372","layer":"informal","project":"p18","title":"The tip of a monochromatic forbidden pattern","kind":"lemma","summary":"[The tip of a monochromatic forbidden pattern] Given P a NOF protocol and a time t, if (a_1, \\d…","labels":["lem:mono-forbidden-pattern-tip"],"detail_key":"p18"},{"id":"n25373","layer":"informal","project":"p18","title":"Induction on t. TODO: Expand","kind":"proof","summary":"Induction on t. TODO: Expand","labels":[],"detail_key":"p18"},{"id":"n25374","layer":"informal","project":"p18","title":"\\eval function","kind":"definition","summary":"[\\eval function] The \\bf \\eval function is defined by \\eval : G^d & \\to \\Bool \\\\ x & \\mapsto 1…","labels":["def:eval"],"detail_key":"p18"},{"id":"n25375","layer":"informal","project":"p18","title":"Forbidden patterns project to multidimensional corners","kind":"lemma","summary":"[Forbidden patterns project to multidimensional corners] If (a_1, \\dots, a_d) is a forbidden pa…","labels":["lem:forbidden-pattern-project-multicorner"],"detail_key":"p18"},{"id":"n25376","layer":"informal","project":"p18","title":"Let v be the tip of (a_1, \\dots, a_d). Then, using that \\sum_k a_j, k = 0 and v_k = a_j,…","kind":"proof","summary":"Let v be the tip of (a_1, \\dots, a_d). Then, using that \\sum_k a_j, k = 0 and v_k = a_j, k for…","labels":[],"detail_key":"p18"},{"id":"n25377","layer":"informal","project":"p18","title":"Monochromatic forbidden patterns are trivial","kind":"lemma","summary":"[Monochromatic forbidden patterns are trivial] Given P a NOF protocol valid in time t for \\eval…","labels":["lem:mono-forbidden-pattern-trivial"],"detail_key":"p18"},{"id":"n25378","layer":"informal","project":"p18","title":"Assume (a_1, \\dots, a_d) is a monochromatic forbidden pattern with tip v, say \\broad(a_i,…","kind":"proof","summary":"Assume (a_1, \\dots, a_d) is a monochromatic forbidden pattern with tip v, say \\broad(a_i, t) =…","labels":[],"detail_key":"p18"},{"id":"n25379","layer":"informal","project":"p18","title":"Lower bound for D(\\eval) in terms of \\chi_d(G)","kind":"theorem","summary":"[Lower bound for D(\\eval) in terms of \\chi_d(G)] D(\\eval) \\ge \\lceil\\log_2\\chi_d(G)\\rceil","labels":["thm:det-fun-complex-eval-ge-corner-color-num"],"detail_key":"p18"},{"id":"n25380","layer":"informal","project":"p18","title":"Let P b","kind":"proof","summary":"Let P b","labels":[],"detail_key":"p18"},{"id":"n25381","layer":"informal","project":"p18","title":"Lower bound for D(\\eval) in terms of r_d(G)","kind":"corollary","summary":"[Lower bound for D(\\eval) in terms of r_d(G)] D(\\eval) \\ge d\\log_2\\frac Nr_d(G)","labels":["cor:det-fun-complex-eval-ge-corner-free-num"],"detail_key":"p18"},{"id":"n25382","layer":"informal","project":"p18","title":"Putting Theorem \\refthm:det-fun-complex-eval-ge-corner-color-num and Lemma \\reflem:corner…","kind":"proof","summary":"Putting Theorem \\refthm:det-fun-complex-eval-ge-corner-color-num and Lemma \\reflem:corner-num-l…","labels":[],"detail_key":"p18"},{"id":"n25383","layer":"informal","project":"p18","title":"The non-monochromatic protocol","kind":"definition","summary":"[The non-monochromatic protocol] Given a colori","labels":["def:non-monochromatic-protocol"],"detail_key":"p18"},{"id":"n25384","layer":"informal","project":"p18","title":"The non-monochromatic protocol is valid","kind":"lemma","summary":"[The non-monochromatic protocol is valid] If c is a coloring such that all monochromatic forbid…","labels":["lem:non-monochromatic-protocol-valid"],"detail_key":"p18"},{"id":"n25385","layer":"informal","project":"p18","title":"We have \\textanswer is 1 \\iff \\text all a_i have the same color \\iff \\text all a_i are eq…","kind":"proof","summary":"We have \\textanswer is 1 \\iff \\text all a_i have the same color \\iff \\text all a_i are equal \\i…","labels":[],"detail_key":"p18"},{"id":"n25386","layer":"informal","project":"p18","title":"Upper bound for D(\\eval) in terms of \\chi_d(G)","kind":"theorem","summary":"[Upper bound for D(\\eval) in terms of \\chi_d(G)] D(\\eval) \\le \\lceil\\log_2\\chi_d(G)\\rceil + d","labels":["thm:det-fun-complex-eval-le-corner-color-num"],"detail_key":"p18"},{"id":"n25387","layer":"informal","project":"p18","title":"Using Lemma \\reflem:forbidden-pattern-project-multicorner, find some coloring c of \\x \\mi…","kind":"proof","summary":"Using Lemma \\reflem:forbidden-pattern-project-multicorner, find some coloring c of \\x \\mid \\sum…","labels":[],"detail_key":"p18"},{"id":"n25388","layer":"informal","project":"p18","title":"Upper bound for D(\\eval) in terms of r_d(G)","kind":"corollary","summary":"[Upper bound for D(\\eval) in terms of r_d(G)] D(\\eval) \\le 2d\\log_2\\frac Nr_d(G)","labels":["cor:det-fun-complex-eval-le-corner-free-num"],"detail_key":"p18"},{"id":"n25389","layer":"informal","project":"p18","title":"Putting Theorem \\refthm:det-fun-complex-eval-le-corner-color-num and Lemma \\reflem:corner…","kind":"proof","summary":"Putting Theorem \\refthm:det-fun-complex-eval-le-corner-color-num and Lemma \\reflem:corner-num-u…","labels":[],"detail_key":"p18"},{"id":"n25390","layer":"informal","project":"p18","title":"Randomised complexity of a protocol","kind":"definition","summary":"[Randomised complexity of a protocol] The \\bf communication complexity of a randomised NOF prot…","labels":["def:rand-protocol-complex"],"detail_key":"p18"},{"id":"n25391","layer":"informal","project":"p18","title":"Randomised complexity of a function","kind":"definition","summary":"[Randomised complexity of a function] The \\bf randomised communication complexity of a function…","labels":["def:rand-complex"],"detail_key":"p18"},{"id":"n25392","layer":"informal","project":"p18","title":"The randomised equality testing protocol for \\eval","kind":"definition","summary":"[The randomised equality testing protocol for \\eval] The \\bf randomised equality testing protoc…","labels":["def:rand-eq-test-protocol-eval"],"detail_key":"p18"},{"id":"n25393","layer":"informal","project":"p18","title":"The randomised equality testing protocol for \\eval is valid","kind":"lemma","summary":"[The randomised equality testing protocol for \\eval is valid] The randomised equality testing p…","labels":["lem:rand-eq-test-protocol-eval-valid"],"detail_key":"p18"},{"id":"n25394","layer":"informal","project":"p18","title":"If \\eval(x) = 1, then the protocol guesses correctly. Else it errors with probability 2^-…","kind":"proof","summary":"If \\eval(x) = 1, then the protocol guesses correctly. Else it errors with probability 2^-\\lceil…","labels":[],"detail_key":"p18"},{"id":"n25395","layer":"informal","project":"p18","title":"The randomised complexity of \\eval is constant","kind":"theorem","summary":"[The randomised complexity of \\eval is constant] R_\\epsilon(\\eval) \\le 2d\\lceil\\log_2\\epsilon^-…","labels":["thm:eval-rand-complexity"],"detail_key":"p18"},{"id":"n25396","layer":"informal","project":"p18","title":"By Lemma \\reflem:rand-eq-test-protocol-eval-valid, the randomised equality testing protoc…","kind":"proof","summary":"By Lemma \\reflem:rand-eq-test-protocol-eval-valid, the randomised equality testing protocol is…","labels":[],"detail_key":"p18"},{"id":"n25397","layer":"formal","project":"p18","title":"NOF.Protocol.trivial","kind":"def","summary":"G : Type u_2 → [AddCommGroup G] → d : Nat → [Fintype G] → LE.le 3 d → ((ZMod d → G) → Bool) → N…","labels":[],"detail_key":"p18","name":"NOF.Protocol.trivial","module":"ChandraFurstLipton.LowerBoundEval"},{"id":"n25398","layer":"formal","project":"p18","title":"NOF.eval","kind":"def","summary":"ι : Type u_1 → G : Type u_2 → [AddCommGroup G] → [DecidableEq G] → [Fintype ι] → (ι → G) → Bool","labels":[],"detail_key":"p18","name":"NOF.eval","module":"ChandraFurstLipton.LowerBoundEval"},{"id":"n25399","layer":"formal","project":"p18","title":"NOF.trivial_of_isForbiddenPattern_of_isValid_eval","kind":"theorem","summary":"∀ G : Type u_2 [inst : AddCommGroup G] [inst_1 : DecidableEq G] d : Nat [inst_2 : NeZero d] P :…","labels":[],"detail_key":"p18","name":"NOF.trivial_of_isForbiddenPattern_of_isValid_eval","module":"ChandraFurstLipton.LowerBoundEval"},{"id":"n25400","layer":"formal","project":"p18","title":"NOF.IsForbiddenPattern","kind":"def","summary":"ι : Type u_1 → G : Type u_2 → (ι → ι → G) → Prop","labels":[],"detail_key":"p18","name":"NOF.IsForbiddenPattern","module":"ChandraFurstLipton.MultidimCorners"},{"id":"n25401","layer":"formal","project":"p18","title":"NOF.IsForbiddenPatternWithTip","kind":"def","summary":"ι : Type u_1 → G : Type u_2 → (ι → ι → G) → (ι → G) → Prop","labels":[],"detail_key":"p18","name":"NOF.IsForbiddenPatternWithTip","module":"ChandraFurstLipton.MultidimCorners"},{"id":"n25402","layer":"formal","project":"p18","title":"NOF.IsMultidimCorner","kind":"inductive","summary":"ι : Type u_1 → G : Type u_2 → [Fintype ι] → [AddCommGroup G] → (ι → ι → G) → (ι → G) → Prop","labels":[],"detail_key":"p18","name":"NOF.IsMultidimCorner","module":"ChandraFurstLipton.MultidimCorners"},{"id":"n25403","layer":"formal","project":"p18","title":"NOF.forget","kind":"def","summary":"ι : Type u_1 → G : Type u_2 → (i : ι) → (ι → G) → (Subtype fun j => Ne j i) → G","labels":[],"detail_key":"p18","name":"NOF.forget","module":"ChandraFurstLipton.MultidimCorners"},{"id":"n25404","layer":"formal","project":"p18","title":"NOF.isMultidimCorner_forget_of_isForbiddenPattern","kind":"theorem","summary":"∀ ι : Type u_1 G : Type u_2 [inst : Fintype ι] [inst_1 : AddCommGroup G] [inst_2 : DecidableEq…","labels":[],"detail_key":"p18","name":"NOF.isMultidimCorner_forget_of_isForbiddenPattern","module":"ChandraFurstLipton.MultidimCorners"},{"id":"n25405","layer":"formal","project":"p18","title":"NOF.IsForbiddenPatternWithTip.broadcast_eq","kind":"theorem","summary":"∀ G : Type u_2 d : Nat P : NOF.Protocol G d a : ZMod d → ZMod d → G v : ZMod d → G B : List Boo…","labels":[],"detail_key":"p18","name":"NOF.IsForbiddenPatternWithTip.broadcast_eq","module":"ChandraFurstLipton.NOFModel"},{"id":"n25406","layer":"formal","project":"p18","title":"NOF.Protocol","kind":"inductive","summary":"Type u_2 → Nat → Type u_2","labels":[],"detail_key":"p18","name":"NOF.Protocol","module":"ChandraFurstLipton.NOFModel"},{"id":"n25407","layer":"formal","project":"p18","title":"NOF.Protocol.IsValid","kind":"def","summary":"G : Type u_2 → d : Nat → NOF.Protocol G d → ((ZMod d → G) → Bool) → Nat → Prop","labels":[],"detail_key":"p18","name":"NOF.Protocol.IsValid","module":"ChandraFurstLipton.NOFModel"},{"id":"n25408","layer":"formal","project":"p18","title":"NOF.Protocol.broadcast","kind":"def","summary":"G : Type u_2 → d : Nat → NOF.Protocol G d → (ZMod d → G) → Nat → List Bool","labels":[],"detail_key":"p18","name":"NOF.Protocol.broadcast","module":"ChandraFurstLipton.NOFModel"},{"id":"n25409","layer":"formal","project":"p18","title":"NOF.Protocol.complexity","kind":"def","summary":"G : Type u_2 → d : Nat → NOF.Protocol G d → ((ZMod d → G) → Bool) → ENat","labels":[],"detail_key":"p18","name":"NOF.Protocol.complexity","module":"ChandraFurstLipton.NOFModel"},{"id":"n25410","layer":"formal","project":"p18","title":"NOF.Protocol.length_broadcast","kind":"theorem","summary":"∀ G : Type u_2 d : Nat (P : NOF.Protocol G d) (x : ZMod d → G) (t : Nat), Eq (P.broadcast x t).…","labels":[],"detail_key":"p18","name":"NOF.Protocol.length_broadcast","module":"ChandraFurstLipton.NOFModel"},{"id":"n25411","layer":"formal","project":"p18","title":"NOF.funComplexity","kind":"def","summary":"G : Type u_2 → d : Nat → ((ZMod d → G) → Bool) → ENat","labels":[],"detail_key":"p18","name":"NOF.funComplexity","module":"ChandraFurstLipton.NOFModel"},{"id":"n25412","layer":"informal","project":"p19","title":"Lebesgue conditional expectation","kind":"definition","summary":"[Lebesgue conditional expectation] The \\bf conditional expectation of a X-measurable function f…","labels":["def:lebesgue-cond-exp"],"detail_key":"p19"},{"id":"n25413","layer":"informal","project":"p19","title":"Characterisation of the Lebesgue conditional expectation","kind":"lemma","summary":"[Characterisation of the Lebesgue conditional expectation] If f : X \\to [0, \\infty] is a X-meas…","labels":["lem:lebesgue-cond-exp-char"],"detail_key":"p19"},{"id":"n25414","layer":"informal","project":"p19","title":"Standard machine.","kind":"proof","summary":"Standard machine.","labels":[],"detail_key":"p19"},{"id":"n25415","layer":"informal","project":"p19","title":"Juxtaposition","kind":"definition","summary":"[Juxtaposition] Let E and S be sets. Let \\Delta\\inP(S), and let \\omega\\in E^S. We define \\juxtO…","labels":["def:juxtaposition"],"detail_key":"p19"},{"id":"n25416","layer":"informal","project":"p19","title":"Cylinder events","kind":"definition","summary":"[Cylinder events] Let (E,E) be a measurable space, and let S be a set. Then, F:P(S)&\\to \\left\\…","labels":["def:cylinder-event"],"detail_key":"p19"},{"id":"n25417","layer":"informal","project":"p19","title":"Kernel","kind":"definition","summary":"[Kernel] Let (X, X) and (Y,Y) be measurable spaces. Then, \\[\\textKer_Y, X:=\\left\\\\pi: X\\times Y…","labels":["def:kernel"],"detail_key":"p19"},{"id":"n25418","layer":"informal","project":"p19","title":"Markov kernel","kind":"definition","summary":"[Markov kernel] Let (X, X) and (Y,Y) be measurable spaces. We say that \\pi\\in\\textKer_Y, X is a…","labels":["def:markov-ker"],"detail_key":"p19"},{"id":"n25419","layer":"informal","project":"p19","title":"Proper kernel","kind":"definition","summary":"[Proper kernel] \\pi is proper iff \\pi(A\\cap B\\mid x)=\\pi(A\\mid x)\\cdot1_B(x) for all A\\in X, B\\…","labels":["def:proper-ker"],"detail_key":"p19"},{"id":"n25420","layer":"informal","project":"p19","title":"Lebesgue integral characterisation of proper kernels","kind":"lemma","summary":"[Lebesgue integral characterisation of proper kernels] If \\pi is proper, then \\[\\int f(x) g(x)\\…","labels":["lem:proper-ker-lintegral"],"detail_key":"p19"},{"id":"n25421","layer":"informal","project":"p19","title":"Standard machine.","kind":"proof","summary":"Standard machine.","labels":[],"detail_key":"p19"},{"id":"n25422","layer":"informal","project":"p19","title":"Integral characterisation of proper kernels","kind":"lemma","summary":"[Integral characterisation of proper kernels] If \\pi is a proper Markov kernel and g is a bound…","labels":["lem:proper-ker-integral"],"detail_key":"p19"},{"id":"n25423","layer":"informal","project":"p19","title":"Standard machine.","kind":"proof","summary":"Standard machine.","labels":[],"detail_key":"p19"},{"id":"n25424","layer":"informal","project":"p19","title":"Conditional expectation kernel","kind":"definition","summary":"[Conditional expectation kernel] Let \\mu\\in\\mathfrakM(X, X). 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Now…","labels":[],"detail_key":"p19"},{"id":"n25438","layer":"informal","project":"p19","title":"Product probability measure","kind":"definition","summary":"[Product probability measure] Let I be a set. 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Prove that G is finitely generated free.","kind":"proof","summary":"Embed M inside its Grothendieck group G. Prove that G is finitely generated free.","labels":[],"detail_key":"p20"},{"id":"n25514","layer":"informal","project":"p20","title":"Affine monoid algebras are domains","kind":"proposition","summary":"[Affine monoid algebras are domains] If R is an integral domain M is an affine monoid, then R[M…","labels":["0-aff-mon-alg-domain"],"detail_key":"p20"},{"id":"n25515","layer":"informal","project":"p20","title":"i : R[M] \\hookrightarrow R[\\Z M] injects into an integral domain so is an integral domain…","kind":"proof","summary":"i : R[M] \\hookrightarrow R[\\Z M] injects into an integral domain so is an integral domain. It's…","labels":[],"detail_key":"p20"},{"id":"n25516","layer":"informal","project":"p20","title":"Irreducible element","kind":"definition","summary":"[Irreducible element] An element x of a monoid M is \\emphirreducible if x = y + z implies y = 0…","labels":["0-irred"],"detail_key":"p20"},{"id":"n25517","layer":"informal","project":"p20","title":"Irreducible elements lie in all sets generating a salient monoid","kind":"proposition","summary":"[Irreducible elements lie in all sets generating a salient monoid] If M is a monoid with a sing…","labels":["0-irred-subset-gen"],"detail_key":"p20"},{"id":"n25518","layer":"informal","project":"p20","title":"Assume p is an irreducible element. Since S generates M, write \\[ p = \\sum_i a_i \\] where…","kind":"proof","summary":"Assume p is an irreducible element. Since S generates M, write \\[ p = \\sum_i a_i \\] where the a…","labels":[],"detail_key":"p20"},{"id":"n25519","layer":"informal","project":"p20","title":"A salient finitely generated monoid has finitely many irreducible elements","kind":"proposition","summary":"[A salient finitely generated monoid has finitely many irreducible elements] If M is a finitely…","labels":["0-irred-finite"],"detail_key":"p20"},{"id":"n25520","layer":"informal","project":"p20","title":"Let S be a finite set generating M. Write I the set of irreducible elements. By Propositi…","kind":"proof","summary":"Let S be a finite set generating M. Write I the set of irreducible elements. By Proposition \\re…","labels":[],"detail_key":"p20"},{"id":"n25521","layer":"informal","project":"p20","title":"A salient finitely generated cancellative monoid is generated by its irreducible elements","kind":"proposition","summary":"[A salient finitely generated cancellative monoid is generated by its irreducible elements] If…","labels":["0-irred-gen"],"detail_key":"p20"},{"id":"n25522","layer":"informal","project":"p20","title":"We do not follow the proof from \\citeCox_2011. Let S be a finite minimal generating set a…","kind":"proof","summary":"We do not follow the proof from \\citeCox_2011. Let S be a finite minimal generating set and ass…","labels":[],"detail_key":"p20"},{"id":"n25523","layer":"informal","project":"p20","title":"Coideal","kind":"definition","summary":"[Coideal] Let R be a commutative ring and (C,\\Delta,\\varepsilon) be a coalgebra over R. An R-su…","labels":["0-coideal"],"detail_key":"p20"},{"id":"n25524","layer":"informal","project":"p20","title":"Quotient coalgebra","kind":"proposition","summary":"[Quotient coalgebra] If C is a coalgebra over R and I is a coideal, the quotient C /I is equipp…","labels":["0-coquot"],"detail_key":"p20"},{"id":"n25525","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25526","layer":"informal","project":"p20","title":"Quotient coalgebra map","kind":"proposition","summary":"[Quotient coalgebra map] If C is a coalgebra over R and I is a coideal, the quotient map C \\to…","labels":["0-coquot-hom"],"detail_key":"p20"},{"id":"n25527","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25528","layer":"informal","project":"p20","title":"Bialgebra ideal","kind":"definition","summary":"[Bialgebra ideal] Let B be a bialgebra over a commutative ring R. A \\emphbialgebra ideal I is a…","labels":["0-biideal"],"detail_key":"p20"},{"id":"n25529","layer":"informal","project":"p20","title":"Quotient bialgebra","kind":"proposition","summary":"[Quotient bialgebra] If B is a bialgebra over R and I is a bialgebra ideal, the quotient B / I…","labels":["0-biquot"],"detail_key":"p20"},{"id":"n25530","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25531","layer":"informal","project":"p20","title":"Quotient bialgebra map","kind":"proposition","summary":"[Quotient bialgebra map] If B is a bialgebra over R and I is a bialgebra ideal, the quotient ma…","labels":["0-biquot-hom"],"detail_key":"p20"},{"id":"n25532","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25533","layer":"informal","project":"p20","title":"Hopf ideal","kind":"definition","summary":"[Hopf ideal] Let A be a Hopf algebra over a commutative ring R. A \\emphHopf ideal I is a bialge…","labels":["0-hopf-ideal"],"detail_key":"p20"},{"id":"n25534","layer":"informal","project":"p20","title":"Quotient Hopf algebra","kind":"proposition","summary":"[Quotient Hopf algebra] If A is a Hopf algebra over R and I is a Hopf ideal, the quotient A / I…","labels":["0-hopf-quot"],"detail_key":"p20"},{"id":"n25535","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25536","layer":"informal","project":"p20","title":"Quotient Hopf algebra map","kind":"proposition","summary":"[Quotient Hopf algebra map] If A is a Hopf algebra over R and I is a Hopf ideal, the quotient m…","labels":["0-hopf-quot-hom"],"detail_key":"p20"},{"id":"n25537","layer":"informal","project":"p20","title":"Follows immediately from Proposition \\ref0-biquot-hom.","kind":"proof","summary":"Follows immediately from Proposition \\ref0-biquot-hom.","labels":[],"detail_key":"p20"},{"id":"n25538","layer":"informal","project":"p20","title":"Freeness of group algebras under an injective hom","kind":"proposition","summary":"[Freeness of group algebras under an injective hom] Let R be a commutative ring. Let G, H be ab…","labels":["0-grp-alg-free"],"detail_key":"p20"},{"id":"n25539","layer":"informal","project":"p20","title":"Pick a section \\sigma : H / f(G) \\to H and the unique map \\varphi : H \\to G such that h =…","kind":"proof","summary":"Pick a section \\sigma : H / f(G) \\to H and the unique map \\varphi : H \\to G such that h = \\sigm…","labels":[],"detail_key":"p20"},{"id":"n25540","layer":"informal","project":"p20","title":"The kernel of a map on direct sums","kind":"proposition","summary":"[The kernel of a map on direct sums] Let G be an abelian group generated by a set S. Let A, B b…","labels":["0-ker-mon-alg"],"detail_key":"p20"},{"id":"n25541","layer":"informal","project":"p20","title":"Write I = \\Span\\gX^a_1 - gX^a_2 | g \\in G, a_1, a_2 \\in A, f(a_1) = f(a_2)\\ for brevity.…","kind":"proof","summary":"Write I = \\Span\\gX^a_1 - gX^a_2 | g \\in G, a_1, a_2 \\in A, f(a_1) = f(a_2)\\ for brevity. Note t…","labels":[],"detail_key":"p20"},{"id":"n25542","layer":"informal","project":"p20","title":"Localising a monoid algebra","kind":"proposition","summary":"[Localising a monoid algebra] Let R be a commutative ring. Let M be a commutative monoid and M'…","labels":["0-loc-mon-alg"],"detail_key":"p20"},{"id":"n25543","layer":"informal","project":"p20","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p20"},{"id":"n25544","layer":"informal","project":"p20","title":"Group-like elements","kind":"definition","summary":"[Group-like elements] An element a of a coalgebra A is \\emphgroup-like if \\eta(a) = 1 and \\Delt…","labels":["0-grp-like"],"detail_key":"p20"},{"id":"n25545","layer":"informal","project":"p20","title":"Group-like elements form a group","kind":"proposition","summary":"[Group-like elements form a group] Group-like elements \\GrpLike A of a bialgebra A form a monoi…","labels":["0-grp-like-grp"],"detail_key":"p20"},{"id":"n25546","layer":"informal","project":"p20","title":"Check that group-like elements are closed under unit, multiplication and inverses.","kind":"proof","summary":"Check that group-like elements are closed under unit, multiplication and inverses.","labels":[],"detail_key":"p20"},{"id":"n25547","layer":"informal","project":"p20","title":"Bialgebra homs preserve group-like elements","kind":"lemma","summary":"[Bialgebra homs preserve group-like elements] Let f : A \\to B be a bi-algebra hom. If a \\in A i…","labels":["0-grp-like-map"],"detail_key":"p20"},{"id":"n25548","layer":"informal","project":"p20","title":"a is a unit, so f(a) is a unit too. Then \\[ f(a) \\otimes f(a) = (f \\otimes f)(\\Delta_A(a)…","kind":"proof","summary":"a is a unit, so f(a) is a unit too. Then \\[ f(a) \\otimes f(a) = (f \\otimes f)(\\Delta_A(a)) = \\D…","labels":[],"detail_key":"p20"},{"id":"n25549","layer":"informal","project":"p20","title":"0-grp-like-grp-alg-of","kind":"lemma","summary":"If R is a commutative semiring, A is a Hopf algebra over R and G is a group, then every element…","labels":["0-grp-like-grp-alg-of"],"detail_key":"p20"},{"id":"n25550","layer":"informal","project":"p20","title":"This is an easy check.","kind":"proof","summary":"This is an easy check.","labels":[],"detail_key":"p20"},{"id":"n25551","layer":"informal","project":"p20","title":"0-grp-like-grp-alg-span","kind":"lemma","summary":"If R is a commutative semiring, A is a Hopf algebra over R and G is a group, then the group-lik…","labels":["0-grp-like-grp-alg-span"],"detail_key":"p20"},{"id":"n25552","layer":"informal","project":"p20","title":"This follows immediately from \\ref0-grp-like-grp-alg-of.","kind":"proof","summary":"This follows immediately from \\ref0-grp-like-grp-alg-of.","labels":[],"detail_key":"p20"},{"id":"n25553","layer":"informal","project":"p20","title":"Independence of group-like elements","kind":"lemma","summary":"[Independence of group-like elements] The group-like elements in a bialgebra A over a domain ar…","labels":["0-grp-like-lin-indep"],"detail_key":"p20"},{"id":"n25554","layer":"informal","project":"p20","title":"Let's prove that any finite set s of group-like elements is linearly independent, by indu…","kind":"proof","summary":"Let's prove that any finite set s of group-like elements is linearly independent, by induction…","labels":[],"detail_key":"p20"},{"id":"n25555","layer":"informal","project":"p20","title":"Group-like elements in a group algebra","kind":"lemma","summary":"[Group-like elements in a group algebra] Let R be a domain. The group-like elements of R[M] are…","labels":["0-grp-like-grp-alg"],"detail_key":"p20"},{"id":"n25556","layer":"informal","project":"p20","title":"See Lemma 12.4 in \\citeMilne_2017.","kind":"proof","summary":"See Lemma 12.4 in \\citeMilne_2017.","labels":[],"detail_key":"p20"},{"id":"n25557","layer":"informal","project":"p20","title":"Galois connection between group algebra and group-like elements","kind":"proposition","summary":"[Galois connection between group algebra and group-like elements] Let R be a domain, G a commut…","labels":["0-grp-alg-grp-like-gc"],"detail_key":"p20"},{"id":"n25558","layer":"informal","project":"p20","title":"If f : G \\to \\GrpLike A is a group hom, then we get R[G] \\to A g \\mapsto f(g) This is cle…","kind":"proof","summary":"If f : G \\to \\GrpLike A is a group hom, then we get R[G] \\to A g \\mapsto f(g) This is clearly a…","labels":[],"detail_key":"p20"},{"id":"n25559","layer":"informal","project":"p20","title":"Quotients by binomial ideals","kind":"proposition","summary":"[Quotients by binomial ideals] Let A be a Hopf algebra, H be a subgroup of \\GrpLike A and \\[ I…","labels":["0-grp-like-quot-hopf"],"detail_key":"p20"},{"id":"n25560","layer":"informal","project":"p20","title":"It suffices to check the conditions of a Hopf ideal on generators. For the comultiplicati…","kind":"proof","summary":"It suffices to check the conditions of a Hopf ideal on generators. For the comultiplication con…","labels":[],"detail_key":"p20"},{"id":"n25561","layer":"informal","project":"p20","title":"Diagonalizable bialgebras","kind":"definition","summary":"[Diagonalizable bialgebras] A bialgebra is called diagonalizable if it is isomorphic to a group…","labels":["0-is-diag-bialg"],"detail_key":"p20"},{"id":"n25562","layer":"informal","project":"p20","title":"0-is-diag-bialg-group-like-span","kind":"lemma","summary":"A diagonalizable bialgebra is spanned by its group-like elements.","labels":["0-is-diag-bialg-group-like-span"],"detail_key":"p20"},{"id":"n25563","layer":"informal","project":"p20","title":"This is true for a group algebra by \\ref0-grp-like-grp-alg-span, and the property of bein…","kind":"proof","summary":"This is true for a group algebra by \\ref0-grp-like-grp-alg-span, and the property of being span…","labels":[],"detail_key":"p20"},{"id":"n25564","layer":"informal","project":"p20","title":"0-bialg-bij-of-span-grp-like","kind":"proposition","summary":"Let A be a bialgebra over a domain R, let G be a subgroup of \\GrpLike(A) (which is a monoid by…","labels":["0-bialg-bij-of-span-grp-like"],"detail_key":"p20"},{"id":"n25565","layer":"informal","project":"p20","title":"This morphism is injective by the linear independence of group-like elements (\\ref0-grp-l…","kind":"proof","summary":"This morphism is injective by the linear independence of group-like elements (\\ref0-grp-like-li…","labels":[],"detail_key":"p20"},{"id":"n25566","layer":"informal","project":"p20","title":"Quotient of a diagonalisable bialgebra is diagonalisable","kind":"proposition","summary":"[Quotient of a diagonalisable bialgebra is diagonalisable] Let R be a domain, G a commutative g…","labels":["0-is-diag-bialg-quot"],"detail_key":"p20"},{"id":"n25567","layer":"informal","project":"p20","title":"Note that R[G] \\xrightarrow f A factors as R[G] \\xrightarrow f R[f(G)] \\xrightarrow \\phi…","kind":"proof","summary":"Note that R[G] \\xrightarrow f A factors as R[G] \\xrightarrow f R[f(G)] \\xrightarrow \\phi A, whe…","labels":[],"detail_key":"p20"},{"id":"n25568","layer":"informal","project":"p20","title":"0-is-diag-bialg-iff-span-group-like","kind":"corollary","summary":"A bialgebra over a domain is diagonalizable if and only if it is spanned by its group-like elem…","labels":["0-is-diag-bialg-iff-span-group-like"],"detail_key":"p20"},{"id":"n25569","layer":"informal","project":"p20","title":"We know that a diagonalizable bialgebra is spanned by its group-like elements by \\ref0-is…","kind":"proof","summary":"We know that a diagonalizable bialgebra is spanned by its group-like elements by \\ref0-is-diag-…","labels":[],"detail_key":"p20"},{"id":"n25570","layer":"informal","project":"p20","title":"The antipode is a antihomomorphism","kind":"proposition","summary":"[The antipode is a antihomomorphism] If A is a R-Hopf algebra, then the antipode map s : A \\to…","labels":["0-antipode-mul"],"detail_key":"p20"},{"id":"n25571","layer":"informal","project":"p20","title":"Any standard reference will have a proof.","kind":"proof","summary":"Any standard reference will have a proof.","labels":[],"detail_key":"p20"},{"id":"n25572","layer":"informal","project":"p20","title":"Bialgebras are comonoid objects in the category of algebras","kind":"proposition","summary":"[Bialgebras are comonoid objects in the category of algebras] The category of R-bialgebras is e…","labels":["0-bialg-equiv-comon-alg"],"detail_key":"p20"},{"id":"n25573","layer":"informal","project":"p20","title":"Turn the arrows around.","kind":"proof","summary":"Turn the arrows around.","labels":[],"detail_key":"p20"},{"id":"n25574","layer":"informal","project":"p20","title":"Hopf algebras are cogroup objects in the category of algebras","kind":"proposition","summary":"[Hopf algebras are cogroup objects in the category of algebras] The category of R-Hopf algebras…","labels":["0-hopf-alg-equiv-cogrp-alg"],"detail_key":"p20"},{"id":"n25575","layer":"informal","project":"p20","title":"Turn the arrows around. Most of the diagrams have been turned around in Proposition \\ref0…","kind":"proof","summary":"Turn the arrows around. Most of the diagrams have been turned around in Proposition \\ref0-bialg…","labels":[],"detail_key":"p20"},{"id":"n25576","layer":"informal","project":"p20","title":"The group algebra functor","kind":"definition","summary":"[The group algebra functor] For a commutative ring R, we have a functor G \\rightsquigarrow R[G]…","labels":["0-grp-alg"],"detail_key":"p20"},{"id":"n25577","layer":"informal","project":"p20","title":"The group algebra functor is fully faithful","kind":"proposition","summary":"[The group algebra functor is fully faithful] Let R be a domain. The functor G \\rightsquigarrow…","labels":["0-full-faithful-grp-alg"],"detail_key":"p20"},{"id":"n25578","layer":"informal","project":"p20","title":"The functor is clearly faithful. Now for the full part, if f : R[G] \\to R[H] is a Hopf al…","kind":"proof","summary":"The functor is clearly faithful. Now for the full part, if f : R[G] \\to R[H] is a Hopf algebra…","labels":[],"detail_key":"p20"},{"id":"n25579","layer":"informal","project":"p20","title":"Convex cone generated by a set","kind":"definition","summary":"[Convex cone generated by a set] For a set S \\subseteq N, the \\bf cone generated by S, aka \\bf…","labels":["1-2-1-cone-hull"],"detail_key":"p20"},{"id":"n25580","layer":"informal","project":"p20","title":"Convex polyhedral cone","kind":"definition","summary":"[Convex polyhedral cone] A \\bf polyhedral cone is a set that can be written as \\Cone(S) for som…","labels":["1-2-1-polyhedral-cone"],"detail_key":"p20"},{"id":"n25581","layer":"informal","project":"p20","title":"Convex hull","kind":"definition","summary":"[Convex hull] For a set S \\subseteq N, the \\bf convex hull of S is \\Conv(S) := \\left\\\\sum_u \\in…","labels":["1-2-2-convex-hull"],"detail_key":"p20"},{"id":"n25582","layer":"informal","project":"p20","title":"Polytope","kind":"definition","summary":"[Polytope] A \\bf polytope is a set that can be written as \\Conv(S) for some finite set S.","labels":["1-2-2-polytope"],"detail_key":"p20"},{"id":"n25583","layer":"informal","project":"p20","title":"Dual cone","kind":"definition","summary":"[Dual cone] Given a polyhedral cone \\sigma \\subseteq N, its \\bf dual cone is defined by \\sigma^…","labels":["1-2-3-dual-cone"],"detail_key":"p20"},{"id":"n25584","layer":"informal","project":"p20","title":"Dual of a polyhedral cone","kind":"proposition","summary":"[Dual of a polyhedral cone] If \\sigma is polyhedral, then its dual \\sigma^\\vee is polyhedral to…","labels":["1-2-4-dual-polyhedral-cone"],"detail_key":"p20"},{"id":"n25585","layer":"informal","project":"p20","title":"Classic, use Fourier-Motzkin eliminiation.","kind":"proof","summary":"Classic, use Fourier-Motzkin eliminiation.","labels":[],"detail_key":"p20"},{"id":"n25586","layer":"informal","project":"p20","title":"Dual cone of a sumset","kind":"proposition","summary":"[Dual cone of a sumset] If \\sigma_1, \\sigma_2 are two cones, then (\\sigma_1 + \\sigma_2)^\\vee =…","labels":["1-2-dual-cone-add"],"detail_key":"p20"},{"id":"n25587","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25588","layer":"informal","project":"p20","title":"Double dual of a polyhedral cone","kind":"proposition","summary":"[Double dual of a polyhedral cone] If \\sigma is polyhedral, then \\sigma^\\vee\\vee = \\sigma.","labels":["1-2-4-double-dual-polyhedral-cone"],"detail_key":"p20"},{"id":"n25589","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25590","layer":"informal","project":"p20","title":"Face of a cone","kind":"definition","summary":"[Face of a cone] If \\sigma is a cone, then a subset of \\sigma is a \\bf face iff it is the inter…","labels":["1-2-5-face"],"detail_key":"p20"},{"id":"n25591","layer":"informal","project":"p20","title":"Edge of a cone","kind":"definition","summary":"[Edge of a cone] A dimension 1 face of a cone is called an \\emphedge.","labels":["1-2-5-edge"],"detail_key":"p20"},{"id":"n25592","layer":"informal","project":"p20","title":"Facet of a cone","kind":"definition","summary":"[Facet of a cone] A codimension 1 face of a cone is called a \\emphfacet.","labels":["1-2-5-facet"],"detail_key":"p20"},{"id":"n25593","layer":"informal","project":"p20","title":"Face of a polyhedral cone","kind":"lemma","summary":"[Face of a polyhedral cone] If \\sigma is a polyhedral cone, then every face of \\sigma is a poly…","labels":["1-2-6-face-polyhedral-cone"],"detail_key":"p20"},{"id":"n25594","layer":"informal","project":"p20","title":"Intersection of faces","kind":"lemma","summary":"[Intersection of faces] If \\sigma is a polyhedral cone, then the intersection of two faces of \\…","labels":["1-2-6-inter-faces"],"detail_key":"p20"},{"id":"n25595","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25596","layer":"informal","project":"p20","title":"Face of a face","kind":"lemma","summary":"[Face of a face] A face of a face of a polyhedral cone \\sigma is again a face of \\sigma.","labels":["1-2-6-face-face"],"detail_key":"p20"},{"id":"n25597","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25598","layer":"informal","project":"p20","title":"1-2-7-face-mem-of-add","kind":"lemma","summary":"Let \\tau be a face of a polyhedral cone \\sigma. If v, w \\in \\sigma and v + w \\in \\tau, then v,…","labels":["1-2-7-face-mem-of-add"],"detail_key":"p20"},{"id":"n25599","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25600","layer":"informal","project":"p20","title":"Dual cone of the intersection of halfspaces","kind":"proposition","summary":"[Dual cone of the intersection of halfspaces] If \\sigma = H_m_1^+ \\cap \\dots \\cap H_m_s^+, then…","labels":["1-2-8-dual-cone-inter-halfspaces"],"detail_key":"p20"},{"id":"n25601","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25602","layer":"informal","project":"p20","title":"Facets of a full dimensional cone","kind":"proposition","summary":"[Facets of a full dimensional cone] If \\sigma is a full dimensional cone, then facets of \\sigma…","labels":["1-2-8-facet-full-dim-cone"],"detail_key":"p20"},{"id":"n25603","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25604","layer":"informal","project":"p20","title":"Intersection of facets containing a face","kind":"proposition","summary":"[Intersection of facets containing a face] Every proper face \\tau \\prec \\sigma of a polyhedral…","labels":["1-2-8-inter-facet"],"detail_key":"p20"},{"id":"n25605","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25606","layer":"informal","project":"p20","title":"Dual face","kind":"definition","summary":"[Dual face] Given a cone \\sigma and a face \\tau \\preceq \\sigma, the \\bf dual face to \\tau is \\t…","labels":["1-2-10-dual-face"],"detail_key":"p20"},{"id":"n25607","layer":"informal","project":"p20","title":"The dual face is a face of the dual","kind":"proposition","summary":"[The dual face is a face of the dual] If \\tau \\preceq \\sigma, then \\tau^* \\preceq \\sigma^\\vee.","labels":["1-2-10-dual-face-face-dual"],"detail_key":"p20"},{"id":"n25608","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25609","layer":"informal","project":"p20","title":"The double dual of a face","kind":"proposition","summary":"[The double dual of a face] If \\tau \\preceq \\sigma, then \\tau^** = \\tau.","labels":["1-2-10-double-dual-face-dual-face"],"detail_key":"p20"},{"id":"n25610","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25611","layer":"informal","project":"p20","title":"The dual of a face is antitone","kind":"proposition","summary":"[The dual of a face is antitone] If \\tau' \\preceq \\tau \\preceq \\sigma, then \\tau' \\preceq \\tau.","labels":["1-2-10-dual-face-antitone"],"detail_key":"p20"},{"id":"n25612","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25613","layer":"informal","project":"p20","title":"The dimension of the dual of a face","kind":"proposition","summary":"[The dimension of the dual of a face] If \\tau \\preceq \\sigma, then \\dim \\tau + \\dim \\tau^* = \\d…","labels":["1-2-10-dim-dual-face"],"detail_key":"p20"},{"id":"n25614","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25615","layer":"informal","project":"p20","title":"Relative interior","kind":"definition","summary":"[Relative interior] The \\bf relative interior, aka \\bf intrinsic interior, of a cone \\sigma is…","labels":["1-2-rel-interior"],"detail_key":"p20"},{"id":"n25616","layer":"informal","project":"p20","title":"The relative interior in terms of the inner product","kind":"lemma","summary":"[The relative interior in terms of the inner product] For a cone \\sigma, u \\in \\Relint(\\sigma)…","labels":["1-2-rel-interior-inner"],"detail_key":"p20"},{"id":"n25617","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25618","layer":"informal","project":"p20","title":"Relative interior of a dual face","kind":"lemma","summary":"[Relative interior of a dual face] If \\tau \\preceq \\sigma and m \\in \\sigma^\\vee, then m \\in \\Re…","labels":["1-2-rel-interior-dual-face"],"detail_key":"p20"},{"id":"n25619","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25620","layer":"informal","project":"p20","title":"Minimal face of a cone","kind":"lemma","summary":"[Minimal face of a cone] If \\sigma is a cone, then W := \\sigma \\cap (-\\sigma) is a subspace. Fu…","labels":["1-2-min-face"],"detail_key":"p20"},{"id":"n25621","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25622","layer":"informal","project":"p20","title":"Salient cones","kind":"definition","summary":"[Salient cones] A cone \\sigma is \\bf salient, aka \\bf pointed or \\bf strongly convex, if \\sigma…","labels":["1-2-12-salient-cone"],"detail_key":"p20"},{"id":"n25623","layer":"informal","project":"p20","title":"Alternative definitions of salient cones","kind":"proposition","summary":"[Alternative definitions of salient cones] The following are equivalent \\item \\sigma is salient…","labels":["1-2-12-salient-cone-tfae"],"detail_key":"p20"},{"id":"n25624","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25625","layer":"informal","project":"p20","title":"Separation lemma","kind":"lemma","summary":"[Separation lemma] Let \\sigma_1, \\sigma_2 be polyhedral cones meeting along a common face \\tau.…","labels":["1-2-13-separation-lemma"],"detail_key":"p20"},{"id":"n25626","layer":"informal","project":"p20","title":"See \\citeCox_2011.","kind":"proof","summary":"See \\citeCox_2011.","labels":[],"detail_key":"p20"},{"id":"n25627","layer":"informal","project":"p20","title":"Rational cone","kind":"definition","summary":"[Rational cone] A cone \\sigma \\subseteq N_\\R is \\bf rational if \\sigma = \\Cone(S) for some fini…","labels":["1-2-14-rat-cone"],"detail_key":"p20"},{"id":"n25628","layer":"informal","project":"p20","title":"Faces of a rational cone","kind":"lemma","summary":"[Faces of a rational cone] If \\tau \\preceq \\sigma is a face of a rational cone, then \\tau itsel…","labels":["1-2-14-face-rat-cone"],"detail_key":"p20"},{"id":"n25629","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25630","layer":"informal","project":"p20","title":"The dual of a rational cone","kind":"lemma","summary":"[The dual of a rational cone] \\sigma^\\vee is a rational cone iff \\sigma is.","labels":["1-2-14-dual-rat-cone"],"detail_key":"p20"},{"id":"n25631","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25632","layer":"informal","project":"p20","title":"Ray generator","kind":"definition","summary":"[Ray generator] If \\rho is an edge of a rational cone \\sigma, then the monoid \\rho \\cap N is ge…","labels":["1-2-ray-gen"],"detail_key":"p20"},{"id":"n25633","layer":"informal","project":"p20","title":"Minimal generators","kind":"definition","summary":"[Minimal generators] The \\bf minimal generators of a rational cone \\sigma are the ray generator…","labels":["1-2-min-gen"],"detail_key":"p20"},{"id":"n25634","layer":"informal","project":"p20","title":"A rational cone is generated by its minimal generators","kind":"lemma","summary":"[A rational cone is generated by its minimal generators] A salient convex rational polyhedral c…","labels":["1-2-15-cone-hull-min-gen"],"detail_key":"p20"},{"id":"n25635","layer":"informal","project":"p20","title":"Classic. See \\citeOda_1988 maybe.","kind":"proof","summary":"Classic. See \\citeOda_1988 maybe.","labels":[],"detail_key":"p20"},{"id":"n25636","layer":"informal","project":"p20","title":"Regular cone","kind":"definition","summary":"[Regular cone] A salient rational polyhedral cone \\sigma is \\bf regular, aka \\bf smooth, if its…","labels":["1-2-16-reg-cone"],"detail_key":"p20"},{"id":"n25637","layer":"informal","project":"p20","title":"Simplicial cone","kind":"definition","summary":"[Simplicial cone] A salient rational polyhedral cone \\sigma is \\bf simplicial if its minimal ge…","labels":["1-2-16-simplicial-cone"],"detail_key":"p20"},{"id":"n25638","layer":"informal","project":"p20","title":"Dual lattice of a cone","kind":"definition","summary":"[Dual lattice of a cone] If \\sigma \\subseteq N_\\R is a polyhedral cone, then the lattice points…","labels":["1-2-17-dual-lat-cone"],"detail_key":"p20"},{"id":"n25639","layer":"informal","project":"p20","title":"Gordan's lemma","kind":"proposition","summary":"[Gordan's lemma] S_\\sigma is finitely generated as a monoid.","labels":["1-2-17-gordan-lemma"],"detail_key":"p20"},{"id":"n25640","layer":"informal","project":"p20","title":"See \\citeCox_2011.","kind":"proof","summary":"See \\citeCox_2011.","labels":[],"detail_key":"p20"},{"id":"n25641","layer":"informal","project":"p20","title":"Affine toric variety of a rational polyhedral cone","kind":"definition","summary":"[Affine toric variety of a rational polyhedral cone] U_\\sigma := \\Spec \\bbC[S_\\sigma] is an aff…","labels":["1-2-18-aff-tor-var-rat-polyhedral-cone"],"detail_key":"p20"},{"id":"n25642","layer":"informal","project":"p20","title":"Dimension of the affine toric variety of a rational polyhedral cone","kind":"theorem","summary":"[Dimension of the affine toric variety of a rational polyhedral cone] \\[ \\dim U_\\sigma = \\dim N…","labels":["1-2-18-dim-aff-tor-var-rat-polyhedral-cone"],"detail_key":"p20"},{"id":"n25643","layer":"informal","project":"p20","title":"See \\citeCox_2011.","kind":"proof","summary":"See \\citeCox_2011.","labels":[],"detail_key":"p20"},{"id":"n25644","layer":"informal","project":"p20","title":"The irreducible elements of the dual lattice of a cone","kind":"proposition","summary":"[The irreducible elements of the dual lattice of a cone] If \\sigma \\subseteq N_\\R is salient of…","labels":["1-2-22-irred-dual-lat"],"detail_key":"p20"},{"id":"n25645","layer":"informal","project":"p20","title":"See \\citeCox_2011.","kind":"proof","summary":"See \\citeCox_2011.","labels":[],"detail_key":"p20"},{"id":"n25646","layer":"informal","project":"p20","title":"Spec as a functor on algebras","kind":"definition","summary":"[Spec as a functor on algebras] Spec is a contravariant functor from the category of R-algebras…","labels":["0-spec-alg"],"detail_key":"p20"},{"id":"n25647","layer":"informal","project":"p20","title":"Spec as a functor on algebras is fully faithful","kind":"proposition","summary":"[Spec as a functor on algebras is fully faithful] Spec is a fully faithful contravariant functo…","labels":["0-full-faithful-spec-alg"],"detail_key":"p20"},{"id":"n25648","layer":"informal","project":"p20","title":"\\Spec : \\Ring \\to \\Sch is a fully faithful contravariant functor which preserves all limi…","kind":"proof","summary":"\\Spec : \\Ring \\to \\Sch is a fully faithful contravariant functor which preserves all limits, he…","labels":[],"detail_key":"p20"},{"id":"n25649","layer":"informal","project":"p20","title":"Spec as a functor on bialgebras","kind":"definition","summary":"[Spec as a functor on bialgebras] Spec is a contravariant functor from the category of R-bialge…","labels":["0-spec-bialg"],"detail_key":"p20"},{"id":"n25650","layer":"informal","project":"p20","title":"Spec as a functor on bialgebras is fully faithful","kind":"proposition","summary":"[Spec as a functor on bialgebras is fully faithful] Spec is a fully faithful contravariant func…","labels":["0-full-faithful-spec-bialg"],"detail_key":"p20"},{"id":"n25651","layer":"informal","project":"p20","title":"\\Spec : \\Ring_R \\to \\Sch_\\Spec R is a fully faithful contravariant functor preserving all…","kind":"proof","summary":"\\Spec : \\Ring_R \\to \\Sch_\\Spec R is a fully faithful contravariant functor preserving all limit…","labels":[],"detail_key":"p20"},{"id":"n25652","layer":"informal","project":"p20","title":"Spec sends cocommutative bialgebras to commutative monoid schemes","kind":"proposition","summary":"[Spec sends cocommutative bialgebras to commutative monoid schemes] If A is a cocommutative bia…","labels":["0-spec-cocomm-bialg"],"detail_key":"p20"},{"id":"n25653","layer":"informal","project":"p20","title":"Diagrams are the same up to identifying \\Spec (A \\otimes A) with \\Spec A \\otimes \\Spec A.","kind":"proof","summary":"Diagrams are the same up to identifying \\Spec (A \\otimes A) with \\Spec A \\otimes \\Spec A.","labels":[],"detail_key":"p20"},{"id":"n25654","layer":"informal","project":"p20","title":"Spec as a functor on Hopf algebras","kind":"definition","summary":"[Spec as a functor on Hopf algebras] Spec is a contravariant functor from the category of R-Hop…","labels":["0-spec-hopf"],"detail_key":"p20"},{"id":"n25655","layer":"informal","project":"p20","title":"Spec as a functor on Hopf algebras is fully faithful","kind":"proposition","summary":"[Spec as a functor on Hopf algebras is fully faithful] Spec is a fully faithful contravariant f…","labels":["0-full-faithful-spec-hopf"],"detail_key":"p20"},{"id":"n25656","layer":"informal","project":"p20","title":"\\Spec : \\Ring_R \\to \\Sch_\\Spec R is a fully faithful contravariant functor preserving all…","kind":"proof","summary":"\\Spec : \\Ring_R \\to \\Sch_\\Spec R is a fully faithful contravariant functor preserving all limit…","labels":[],"detail_key":"p20"},{"id":"n25657","layer":"informal","project":"p20","title":"Essential image of Spec on algebras","kind":"proposition","summary":"[Essential image of Spec on algebras] The essential image of \\Spec : \\Ring_R \\to \\Sch_\\Spec R i…","labels":["0-ess-image-spec-alg"],"detail_key":"p20"},{"id":"n25658","layer":"informal","project":"p20","title":"Direct consequence of Proposition \\ref0-ess-image-over.","kind":"proof","summary":"Direct consequence of Proposition \\ref0-ess-image-over.","labels":[],"detail_key":"p20"},{"id":"n25659","layer":"informal","project":"p20","title":"Essential image of Spec on bialgebras","kind":"proposition","summary":"[Essential image of Spec on bialgebras] The essential image of \\Spec : \\Bialg_R \\to \\GrpSch_\\Sp…","labels":["0-ess-image-spec-bialg"],"detail_key":"p20"},{"id":"n25660","layer":"informal","project":"p20","title":"Direct consequence of Propositions \\ref0-ess-image-grp and \\ref0-ess-image-spec-alg.","kind":"proof","summary":"Direct consequence of Propositions \\ref0-ess-image-grp and \\ref0-ess-image-spec-alg.","labels":[],"detail_key":"p20"},{"id":"n25661","layer":"informal","project":"p20","title":"Essential image of Spec on Hopf algebras","kind":"proposition","summary":"[Essential image of Spec on Hopf algebras] The essential image of \\Spec : \\Hopf_R \\to \\GrpSch_\\…","labels":["0-ess-image-spec-hopf"],"detail_key":"p20"},{"id":"n25662","layer":"informal","project":"p20","title":"Direct consequence of Propositions \\ref0-ess-image-grp and \\ref0-ess-image-spec-alg.","kind":"proof","summary":"Direct consequence of Propositions \\ref0-ess-image-grp and \\ref0-ess-image-spec-alg.","labels":[],"detail_key":"p20"},{"id":"n25663","layer":"informal","project":"p20","title":"The diagonalisable group scheme functor","kind":"definition","summary":"[The diagonalisable group scheme functor] Let G be a commutative group and S a base scheme. The…","labels":["0-diag"],"detail_key":"p20"},{"id":"n25664","layer":"informal","project":"p20","title":"Diagonalisable group schemes","kind":"definition","summary":"[Diagonalisable group schemes] An algebraic group G over \\Spec R is \\bf diagonalisable if it is…","labels":["0-is-diag"],"detail_key":"p20"},{"id":"n25665","layer":"informal","project":"p20","title":"The diagonalisable group scheme torus over \\Spec R","kind":"lemma","summary":"[The diagonalisable group scheme torus over \\Spec R] Let R be a commutative ring and M an abeli…","labels":["0-diag-spec"],"detail_key":"p20"},{"id":"n25666","layer":"informal","project":"p20","title":"Ask any toddler on the street.","kind":"proof","summary":"Ask any toddler on the street.","labels":[],"detail_key":"p20"},{"id":"n25667","layer":"informal","project":"p20","title":"0-diag-iff-grp-like-span","kind":"theorem","summary":"An algebraic group G over a field k is diagonalizable if and only if \\Gamma(G) is spanned by it…","labels":["0-diag-iff-grp-like-span"],"detail_key":"p20"},{"id":"n25668","layer":"informal","project":"p20","title":"See Theorem 12.8 in \\citeMilne_2017.","kind":"proof","summary":"See Theorem 12.8 in \\citeMilne_2017.","labels":[],"detail_key":"p20"},{"id":"n25669","layer":"informal","project":"p20","title":"0-full-faithful-diag","kind":"theorem","summary":"Let R be a domain. The functor D_R(G) := G \\rightsquigarrow \\Spec R[G] from the category of gro…","labels":["0-full-faithful-diag"],"detail_key":"p20"},{"id":"n25670","layer":"informal","project":"p20","title":"Compose Propositions \\ref0-full-faithful-spec-hopf and \\ref0-full-faithful-grp-alg. Also…","kind":"proof","summary":"Compose Propositions \\ref0-full-faithful-spec-hopf and \\ref0-full-faithful-grp-alg. Also see Th…","labels":[],"detail_key":"p20"},{"id":"n25671","layer":"informal","project":"p20","title":"Morphisms between diagonalisable group schemes are affine","kind":"proposition","summary":"[Morphisms between diagonalisable group schemes are affine] Let S be a scheme. Let M, N be comm…","labels":["0-diag-aff-hom"],"detail_key":"p20"},{"id":"n25672","layer":"informal","project":"p20","title":"\\Spec f : \\Spec \\Z[N] \\to \\Spec \\Z[M] is affine, since it's a morphism of affine schemes.…","kind":"proof","summary":"\\Spec f : \\Spec \\Z[N] \\to \\Spec \\Z[M] is affine, since it's a morphism of affine schemes. There…","labels":[],"detail_key":"p20"},{"id":"n25673","layer":"informal","project":"p20","title":"Closed embeddings between diagonalisable group schemes","kind":"proposition","summary":"[Closed embeddings between diagonalisable group schemes] Let S be a scheme. Let M, N be commuta…","labels":["0-diag-closed-emb"],"detail_key":"p20"},{"id":"n25674","layer":"informal","project":"p20","title":"Since f is surjective, the corresponding map \\hat f : \\Z[M] \\to \\Z[N] is surjective too.…","kind":"proof","summary":"Since f is surjective, the corresponding map \\hat f : \\Z[M] \\to \\Z[N] is surjective too. Hence…","labels":[],"detail_key":"p20"},{"id":"n25675","layer":"informal","project":"p20","title":"Faithfully flat morphisms between diagonalisable group schemes","kind":"proposition","summary":"[Faithfully flat morphisms between diagonalisable group schemes] Let S be a scheme. Let G, H be…","labels":["0-diag-faithful-flat"],"detail_key":"p20"},{"id":"n25676","layer":"informal","project":"p20","title":"Since f is injective, \\Z[H] is a free module over \\Z[G] by Proposition \\ref0-grp-alg-free…","kind":"proof","summary":"Since f is injective, \\Z[H] is a free module over \\Z[G] by Proposition \\ref0-grp-alg-free, henc…","labels":[],"detail_key":"p20"},{"id":"n25677","layer":"informal","project":"p20","title":"A subgroup of a diagonalisable group scheme is a diagonalisable group scheme","kind":"proposition","summary":"[A subgroup of a diagonalisable group scheme is a diagonalisable group scheme] Let R be a domai…","labels":["0-subgroup-diag"],"detail_key":"p20"},{"id":"n25678","layer":"informal","project":"p20","title":"H is a closed subscheme of an affine scheme, hence it is affine. By Proposition \\ref0-ess…","kind":"proof","summary":"H is a closed subscheme of an affine scheme, hence it is affine. By Proposition \\ref0-ess-image…","labels":[],"detail_key":"p20"},{"id":"n25679","layer":"informal","project":"p20","title":"Diagonalisable group scheme of a torsion group is disconnected","kind":"proposition","summary":"[Diagonalisable group scheme of a torsion group is disconnected] Let G be an abelian group with…","labels":["0-diag-tors"],"detail_key":"p20"},{"id":"n25680","layer":"informal","project":"p20","title":"Say x \\in G is such that x^n = 1. Then \\[e : R[G] := \\frac 1n \\sum_i = 0^n x^i\\] is such…","kind":"proof","summary":"Say x \\in G is such that x^n = 1. Then \\[e : R[G] := \\frac 1n \\sum_i = 0^n x^i\\] is such that e…","labels":[],"detail_key":"p20"},{"id":"n25681","layer":"informal","project":"p20","title":"The split torus","kind":"definition","summary":"[The split torus] The split torus \\G_m^n over a scheme S is the pullback of \\Spec \\Z[x_1^\\pm 1,…","labels":["0-torus"],"detail_key":"p20"},{"id":"n25682","layer":"informal","project":"p20","title":"Diag is a group isomorphism on hom sets","kind":"lemma","summary":"[Diag is a group isomorphism on hom sets] Let R be a domain. The functor G \\rightsquigarrow \\Sp…","labels":["0-diag-hom"],"detail_key":"p20"},{"id":"n25683","layer":"informal","project":"p20","title":"Toddlers and streets by Lemmas \\ref0-full-faithful-diag and \\ref0-full-faithful-grp-hom.","kind":"proof","summary":"Toddlers and streets by Lemmas \\ref0-full-faithful-diag and \\ref0-full-faithful-grp-hom.","labels":[],"detail_key":"p20"},{"id":"n25684","layer":"informal","project":"p20","title":"Characters of a group scheme","kind":"definition","summary":"[Characters of a group scheme] For a group scheme G over S, the \\bf character lattice of G is \\…","labels":["0-char"],"detail_key":"p20"},{"id":"n25685","layer":"informal","project":"p20","title":"Cocharacters of a group scheme","kind":"definition","summary":"[Cocharacters of a group scheme] For a group scheme G over S, the \\bf cocharacter lattice of G…","labels":["0-cochar"],"detail_key":"p20"},{"id":"n25686","layer":"informal","project":"p20","title":"Character lattice of a diagonalisable group scheme","kind":"proposition","summary":"[Character lattice of a diagonalisable group scheme] Let R be a domain and G be a commutative g…","labels":["0-char-diag"],"detail_key":"p20"},{"id":"n25687","layer":"informal","project":"p20","title":"By Propositions \\ref0-diag-spec and \\ref0-full-faithful-diag in turn, we have \\[ X(G) = \\…","kind":"proof","summary":"By Propositions \\ref0-diag-spec and \\ref0-full-faithful-diag in turn, we have \\[ X(G) = \\Hom_\\m…","labels":[],"detail_key":"p20"},{"id":"n25688","layer":"informal","project":"p20","title":"Cocharacter lattice of a diagonalisable group scheme","kind":"proposition","summary":"[Cocharacter lattice of a diagonalisable group scheme] Let R be a domain and G be a commutative…","labels":["0-cochar-diag"],"detail_key":"p20"},{"id":"n25689","layer":"informal","project":"p20","title":"By Propositions \\ref0-diag-spec and \\ref0-full-faithful-diag in turn, we have \\[ X^*(G) =…","kind":"proof","summary":"By Propositions \\ref0-diag-spec and \\ref0-full-faithful-diag in turn, we have \\[ X^*(G) = \\Hom_…","labels":[],"detail_key":"p20"},{"id":"n25690","layer":"informal","project":"p20","title":"Character lattice of the torus","kind":"proposition","summary":"[Character lattice of the torus] Let G be a torus of dimension n over a domain R. Then X(G) = \\…","labels":["0-char-torus"],"detail_key":"p20"},{"id":"n25691","layer":"informal","project":"p20","title":"Immediate from Propositions \\ref0-diag-spec and \\ref0-char-diag.","kind":"proof","summary":"Immediate from Propositions \\ref0-diag-spec and \\ref0-char-diag.","labels":[],"detail_key":"p20"},{"id":"n25692","layer":"informal","project":"p20","title":"Cocharacter lattice of the torus","kind":"proposition","summary":"[Cocharacter lattice of the torus] Let G be a torus of dimension n over a domain R. Then X^*(G)…","labels":["0-cochar-torus"],"detail_key":"p20"},{"id":"n25693","layer":"informal","project":"p20","title":"Immediate from Propositions \\ref0-diag-spec and \\ref0-cochar-diag.","kind":"proof","summary":"Immediate from Propositions \\ref0-diag-spec and \\ref0-cochar-diag.","labels":[],"detail_key":"p20"},{"id":"n25694","layer":"informal","project":"p20","title":"The character-cocharacter pairing","kind":"definition","summary":"[The character-cocharacter pairing] Let R be a domain and G a group scheme over \\Spec R. Then t…","labels":["0-char-cochar-pairing"],"detail_key":"p20"},{"id":"n25695","layer":"informal","project":"p20","title":"The character-cocharacter pairing is perfect","kind":"proposition","summary":"[The character-cocharacter pairing is perfect] The character-cocharacter pairing on a torus is…","labels":["0-char-cochar-pairing-perfect"],"detail_key":"p20"},{"id":"n25696","layer":"informal","project":"p20","title":"Transfer everything across the isos X(\\G_m^n) = \\Z^n, X*(\\G_m^n) = \\Hom(\\Z^n, \\Z).","kind":"proof","summary":"Transfer everything across the isos X(\\G_m^n) = \\Z^n, X*(\\G_m^n) = \\Hom(\\Z^n, \\Z).","labels":[],"detail_key":"p20"},{"id":"n25697","layer":"informal","project":"p20","title":"The image of a torus is a torus","kind":"proposition","summary":"[The image of a torus is a torus] Let R be a domain. Let T be a split torus over R. Let G be a…","labels":["1-1-1-group-hom-torus"],"detail_key":"p20"},{"id":"n25698","layer":"informal","project":"p20","title":"By fullness of D_R (Proposition \\ref0-full-faithful-diag), it's enough to handle the case…","kind":"proof","summary":"By fullness of D_R (Proposition \\ref0-full-faithful-diag), it's enough to handle the case where…","labels":[],"detail_key":"p20"},{"id":"n25699","layer":"informal","project":"p20","title":"A subgroup of a torus is a torus","kind":"proposition","summary":"[A subgroup of a torus is a torus] Let R be a commutative ring of characteristic zero. Let T be…","labels":["1-1-1-subgroup-torus"],"detail_key":"p20"},{"id":"n25700","layer":"informal","project":"p20","title":"By assumption, write T \\cong D_k[G] for G a free abelian group. By Proposition \\ref0-subg…","kind":"proof","summary":"By assumption, write T \\cong D_k[G] for G a free abelian group. By Proposition \\ref0-subgroup-d…","labels":[],"detail_key":"p20"},{"id":"n25701","layer":"informal","project":"p20","title":"Toric varieties","kind":"definition","summary":"[Toric varieties] Let k be a field. Let T be a torus over k. A \\emphtoric variety structure on…","labels":["5-1-tor-var"],"detail_key":"p20"},{"id":"n25702","layer":"informal","project":"p20","title":"Torus morphisms, torus isomorphisms","kind":"definition","summary":"[Torus morphisms, torus isomorphisms] Let k be a field. Let T_1, T_2 be tori over k. 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Let 0\\to M'\\to M\\to M''\\to 0 be a short exact sequence of finitely…","labels":["char-ideal-additive"],"detail_key":"p21"},{"id":"n29685","layer":"informal","project":"p21","title":"Since localization is exact, for any prime ideal \\fp of A, the 0\\to M_\\fp'\\to M_\\fp\\to M_…","kind":"proof","summary":"Since localization is exact, for any prime ideal \\fp of A, the 0\\to M_\\fp'\\to M_\\fp\\to M_\\fp''\\…","labels":[],"detail_key":"p21"},{"id":"n29686","layer":"informal","project":"p21","title":"pseudo-null","kind":"definition","summary":"Let A be a Noetherian ring. \\rm(i) A finitely generated A-module M is called a \\emphpseudo-null…","labels":["pseudo-null"],"detail_key":"p21"},{"id":"n29687","layer":"informal","project":"p21","title":"We warn the reader that M\\sim N not necessarily implies N\\sim M.","kind":"remark","summary":"We warn the reader that M\\sim N not necessarily implies N\\sim M.","labels":[],"detail_key":"p21"},{"id":"n29688","layer":"informal","project":"p21","title":"pseudo-null-criterion","kind":"prop","summary":"Let A be a Noetherian ring, M be a finitely generated A-module. \\rm(i) If A is of Krull dimensi…","labels":["pseudo-null-criterion"],"detail_key":"p21"},{"id":"n29689","layer":"informal","project":"p21","title":"(i) Clear. (ii) Let \\fm be the maximal ideal of A. If M is finite, then there exists r\\in…","kind":"proof","summary":"(i) Clear. (ii) Let \\fm be the maximal ideal of A. If M is finite, then there exists r\\in\\BN su…","labels":[],"detail_key":"p21"},{"id":"n29690","layer":"informal","project":"p21","title":"pseudo-null-char-ideal","kind":"prop","summary":"Let A be a Noetherian ring, M, N be finitely generated torsion A-modules. \\rm(i) If M is pseudo…","labels":["pseudo-null-char-ideal"],"detail_key":"p21"},{"id":"n29691","layer":"informal","project":"p21","title":"Clear from definition and Proposition \\refchar-ideal-additive.","kind":"proof","summary":"Clear from definition and Proposition \\refchar-ideal-additive.","labels":[],"detail_key":"p21"},{"id":"n29692","layer":"informal","project":"p21","title":"smul-is-pis","kind":"prop","summary":"Let A be a Noetherian ring, M be a finitely generated torsion A-module, a\\in A such that M_\\fp=…","labels":["smul-is-pis"],"detail_key":"p21"},{"id":"n29693","layer":"informal","project":"p21","title":"Let K and L be the kernel and cokernel of the map a:M\\xrightarrow\\times aM, respectively.…","kind":"proof","summary":"Let K and L be the kernel and cokernel of the map a:M\\xrightarrow\\times aM, respectively. Since…","labels":[],"detail_key":"p21"},{"id":"n29694","layer":"informal","project":"p21","title":"ht-1-localization-is-PID","kind":"definition","summary":"A Noetherian ring A is called ``height one localizations are PID'', if for any finitely many he…","labels":["ht-1-localization-is-PID"],"detail_key":"p21"},{"id":"n29695","layer":"informal","project":"p21","title":"pis-iff","kind":"lem","summary":"Let A be a Noetherian ring and let M,N be finitely generated torsion A-modules. Let \\Sigma=\\\\fq…","labels":["pis-iff"],"detail_key":"p21"},{"id":"n29696","layer":"informal","project":"p21","title":"Since the height one support of \\ker(f) and \\coker(f) are contained in \\Sigma, and since…","kind":"proof","summary":"Since the height one support of \\ker(f) and \\coker(f) are contained in \\Sigma, and since S^-1\\k…","labels":[],"detail_key":"p21"},{"id":"n29697","layer":"informal","project":"p21","title":"pseudo-null-iff-localization-eq-zero","kind":"lem","summary":"Let M be a finitely generated torsion A-module, \\Sigma be a finite set of height one prime idea…","labels":["pseudo-null-iff-localization-eq-zero"],"detail_key":"p21"},{"id":"n29698","layer":"informal","project":"p21","title":"``\\Leftarrow'': Clear. ``\\Rightarrow'': For all \\fq\\in\\Sigma, M_\\fq=0 means that \\Ann(M)\\…","kind":"proof","summary":"``\\Leftarrow'': Clear. ``\\Rightarrow'': For all \\fq\\in\\Sigma, M_\\fq=0 means that \\Ann(M)\\not\\su…","labels":[],"detail_key":"p21"},{"id":"n29699","layer":"informal","project":"p21","title":"Structure theorem of finitely generated torsion A-modules","kind":"prop","summary":"[Structure theorem of finitely generated torsion A-modules] Let A be a Noetherian ring whose he…","labels":["structure-thm"],"detail_key":"p21"},{"id":"n29700","layer":"informal","project":"p21","title":"Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\=\\\\fq\\in\\Supp(M)\\mid\\height(\\fq)=1\\ (by Proposition \\refch…","kind":"proof","summary":"Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\=\\\\fq\\in\\Supp(M)\\mid\\height(\\fq)=1\\ (by Proposition \\refchar-ide…","labels":[],"detail_key":"p21"},{"id":"n29701","layer":"informal","project":"p21","title":"pis-symm","kind":"prop","summary":"Let A be a Noetherian ring whose height one localizations are PID (Definition \\refht-1-localiza…","labels":["pis-symm"],"detail_key":"p21"},{"id":"n29702","layer":"informal","project":"p21","title":"(a)\\Rightarrow(b): Clear. (b)\\Rightarrow(a): Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\ =\\\\fq\\in\\Sup…","kind":"proof","summary":"(a)\\Rightarrow(b): Clear. (b)\\Rightarrow(a): Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\ =\\\\fq\\in\\Supp(M)\\c…","labels":[],"detail_key":"p21"},{"id":"n29703","layer":"informal","project":"p21","title":"pis-torsion-oplus-torsion-free","kind":"prop","summary":"Let A be a Noetherian ring whose height one localizations are PID (Definition \\refht-1-localiza…","labels":["pis-torsion-oplus-torsion-free"],"detail_key":"p21"},{"id":"n29704","layer":"informal","project":"p21","title":"Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\=\\\\fq\\in\\Supp(M)\\mid\\height(\\fq)=1\\ (by Proposition \\refch…","kind":"proof","summary":"Let \\Sigma=\\\\fq_1,\\cdots,\\fq_r\\=\\\\fq\\in\\Supp(M)\\mid\\height(\\fq)=1\\ (by Proposition \\refchar-ide…","labels":[],"detail_key":"p21"},{"id":"n29705","layer":"informal","project":"p21","title":"regular-domain-is-good","kind":"prop","summary":"Let A be a Noetherian regular domain (more generally, a Noetherian integrally closed domain). T…","labels":["regular-domain-is-good"],"detail_key":"p21"},{"id":"n29706","layer":"informal","project":"p21","title":"If A is a Noetherian regular domain (more generally, a Noetherian integrally closed domai…","kind":"proof","summary":"If A is a Noetherian regular domain (more generally, a Noetherian integrally closed domain), th…","labels":[],"detail_key":"p21"},{"id":"n29707","layer":"informal","project":"p21","title":"Iwasawa-alg-defn","kind":"definition","summary":"The \\emphIwasawa algebra is defined as the completed group algebra \\Lambda:=\\BZ_p[[\\Gamma]]:=\\v…","labels":["Iwasawa-alg-defn"],"detail_key":"p21"},{"id":"n29708","layer":"informal","project":"p21","title":"linear-map-is-adic-continuous","kind":"prop","summary":"Let A be a ring, \\fa be an ideal of A. Let M,N be two A-modules, and \\varphi:M\\to N be an A-mod…","labels":["linear-map-is-adic-continuous"],"detail_key":"p21"},{"id":"n29709","layer":"informal","project":"p21","title":"Let x\\in M and y:=\\varphi(x)\\in N. Let U be any open neighborhood of y in N. Then there e…","kind":"proof","summary":"Let x\\in M and y:=\\varphi(x)\\in N. Let U be any open neighborhood of y in N. Then there exists…","labels":[],"detail_key":"p21"},{"id":"n29710","layer":"informal","project":"p21","title":"Iwasawa-alg-isom-finite-level","kind":"prop","summary":"For each n\\geq 0, there is an isomorphism of \\BZ_p-algebras \\BZ_p[\\Gamma_n]\\xrightarrow\\sim\\BZ_…","labels":["Iwasawa-alg-isom-finite-level"],"detail_key":"p21"},{"id":"n29711","layer":"informal","project":"p21","title":"We have \\Gamma/\\Gamma^p^n\\cong\\BZ/p^n\\BZ as an abelian group, and the image of \\gamma\\in\\…","kind":"proof","summary":"We have \\Gamma/\\Gamma^p^n\\cong\\BZ/p^n\\BZ as an abelian group, and the image of \\gamma\\in\\Gamma…","labels":[],"detail_key":"p21"},{"id":"n29712","layer":"informal","project":"p21","title":"Iwasawa-alg-isom","kind":"prop","summary":"There is an isomorphism of topological rings \\BZ_p[[T]]\\xrightarrow\\sim\\Lambda sending 1+T to \\…","labels":["Iwasawa-alg-isom"],"detail_key":"p21"},{"id":"n29713","layer":"informal","project":"p21","title":"We prove that there is a natural isomorphism of \\BZ_p-algebras \\varprojlim_n\\BZ_p[T]/\\big…","kind":"proof","summary":"We prove that there is a natural isomorphism of \\BZ_p-algebras \\varprojlim_n\\BZ_p[T]/\\big((1+T)…","labels":[],"detail_key":"p21"},{"id":"n29714","layer":"informal","project":"p21","title":"distinguished-polynomial","kind":"definition","summary":"If (A,\\fm,k) is a local ring, then a polynomial f(X)=\\sum_i=0^na_iX^i\\in A[X] is called a \\emph…","labels":["distinguished-polynomial"],"detail_key":"p21"},{"id":"n29715","layer":"informal","project":"p21","title":"power-series-invertible-iff","kind":"prop","summary":"Let A be a ring, and let f(X)=\\sum_n=0^\\infty a_nX^n\\in A[[X]] be a formal power series. Then f…","labels":["power-series-invertible-iff"],"detail_key":"p21"},{"id":"n29716","layer":"informal","project":"p21","title":"If f(X)\\in A[[X]]^\\times, then there exists g(X)=\\sum_n=0^\\infty b_nX^n\\in A[[X]] such th…","kind":"proof","summary":"If f(X)\\in A[[X]]^\\times, then there exists g(X)=\\sum_n=0^\\infty b_nX^n\\in A[[X]] such that f(X…","labels":[],"detail_key":"p21"},{"id":"n29717","layer":"informal","project":"p21","title":"Weierstrass division","kind":"prop","summary":"[Weierstrass division] Let (A,\\fm,k) be a complete local ring, g(X)=\\sum_i=0^\\infty a_iX^i\\in A…","labels":["weierstrass-division"],"detail_key":"p21"},{"id":"n29718","layer":"informal","project":"p21","title":"Write g(X)=\\sum_i=0^n-1a_iX^i+X^ng_1(X) for some g_1(X)\\in A[[X]]^\\times, and f(X)=\\sum_i…","kind":"proof","summary":"Write g(X)=\\sum_i=0^n-1a_iX^i+X^ng_1(X) for some g_1(X)\\in A[[X]]^\\times, and f(X)=\\sum_i=0^n-1…","labels":[],"detail_key":"p21"},{"id":"n29719","layer":"informal","project":"p21","title":"isom-given-by-weierstrass-division","kind":"cor","summary":"Let (A,\\fm,k) be a complete local ring, g(X)=\\sum_i=0^n a_iX^i\\in A[X] be a polynomial such tha…","labels":["isom-given-by-weierstrass-division"],"detail_key":"p21"},{"id":"n29720","layer":"informal","project":"p21","title":"Let f\\in A[[X]]. Then by Proposition \\refweierstrass-division, we may find a unique forma…","kind":"proof","summary":"Let f\\in A[[X]]. Then by Proposition \\refweierstrass-division, we may find a unique formal powe…","labels":[],"detail_key":"p21"},{"id":"n29721","layer":"informal","project":"p21","title":"Weierstrass preparation theorem","kind":"prop","summary":"[Weierstrass preparation theorem] Let (A,\\fm,k) be a complete local ring. Let g(X)\\in A[[X]]\\se…","labels":["weierstrass-preparation"],"detail_key":"p21"},{"id":"n29722","layer":"informal","project":"p21","title":"Take f(X)=X^n in Proposition \\refweierstrass-division, we obtain q(X)\\in A[[X]] and r(X)\\…","kind":"proof","summary":"Take f(X)=X^n in Proposition \\refweierstrass-division, we obtain q(X)\\in A[[X]] and r(X)\\in A[X…","labels":[],"detail_key":"p21"},{"id":"n29723","layer":"informal","project":"p21","title":"Lambda-is-reg-of-dim-2","kind":"prop","summary":"\\Lambda is a Noetherian regular local ring of Krull dimension 2. (We try to avoid this result b…","labels":["Lambda-is-reg-of-dim-2"],"detail_key":"p21"},{"id":"n29724","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29725","layer":"informal","project":"p21","title":"Lambda-is-UFD","kind":"cor","summary":"\\Lambda is a UFD.","labels":["Lambda-is-UFD"],"detail_key":"p21"},{"id":"n29726","layer":"informal","project":"p21","title":"A corollary of Proposition \\refLambda-is-reg-of-dim-2 and Theorem \\refregular-local-ring-…","kind":"proof","summary":"A corollary of Proposition \\refLambda-is-reg-of-dim-2 and Theorem \\refregular-local-ring-is-UFD…","labels":[],"detail_key":"p21"},{"id":"n29727","layer":"informal","project":"p21","title":"Lambda-ht-1-principal","kind":"cor","summary":"Any height 1 prime \\fp of \\Lambda is principal. Moreover, \\rm(i) If the generator of \\fp is in…","labels":["Lambda-ht-1-principal"],"detail_key":"p21"},{"id":"n29728","layer":"informal","project":"p21","title":"By Proposition \\refUFD-iff-ht-1-principal, any height 1 prime of \\Lambda is principal. ...","kind":"proof","summary":"By Proposition \\refUFD-iff-ht-1-principal, any height 1 prime of \\Lambda is principal. ...","labels":[],"detail_key":"p21"},{"id":"n29729","layer":"informal","project":"p21","title":"Lambda-module-structure","kind":"prop","summary":"If X is a finitely generated torsion \\Lambda-module, then there exists a pseudo-isomorphism X\\t…","labels":["Lambda-module-structure"],"detail_key":"p21"},{"id":"n29730","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29731","layer":"informal","project":"p21","title":"iwasawa-mod-invariants","kind":"definition","summary":"\\rm(i) The \\mu-invariant of X is defined to be \\mu(X):=\\sum_j=1^sn_j, and the \\lambda-invariant…","labels":["iwasawa-mod-invariants"],"detail_key":"p21"},{"id":"n29732","layer":"informal","project":"p21","title":"iwasawa-mod-invariants-2","kind":"prop","summary":"We have \\mu(X)=\\sum_i=0^\\infty\\rank_\\BF_p[[T]]X[p^i+1]/X[p^i], and \\lambda(X)=\\rank_\\BZ_pX/X[p^…","labels":["iwasawa-mod-invariants-2"],"detail_key":"p21"},{"id":"n29733","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29734","layer":"informal","project":"p21","title":"p:rank-growth","kind":"lem","summary":"Suppose X is a finitely generated \\Lambda-module of rank r, then we have \\rank_\\BZ_p\\left(X/(\\g…","labels":["p:rank-growth"],"detail_key":"p21"},{"id":"n29735","layer":"informal","project":"p21","title":"This is left as an exercise.","kind":"proof","summary":"This is left as an exercise.","labels":[],"detail_key":"p21"},{"id":"n29736","layer":"informal","project":"p21","title":"coinvariant-growth","kind":"prop","summary":"If X is a finitely generated \\Lambda-module such that X/(\\gamma^p^n-1)X is finite for all suffi…","labels":["coinvariant-growth"],"detail_key":"p21"},{"id":"n29737","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29738","layer":"informal","project":"p21","title":"\\citeWas97, Proposition 13.19","kind":"prop","summary":"[\\citeWas97, Proposition 13.19] Let e\\geq 0 be an integer. If X is a finitely generated \\Lambda…","labels":["coinvariant-growth-2"],"detail_key":"p21"},{"id":"n29739","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29740","layer":"informal","project":"p21","title":"conjugate-action","kind":"definition","summary":"X has a natural, well-defined \\Gamma-action, given by \\sigma\\bullet x =\\widetilde\\sigma x\\widet…","labels":["conjugate-action"],"detail_key":"p21"},{"id":"n29741","layer":"informal","project":"p21","title":"conjugate-action-cont","kind":"lem","summary":"If G is a topological group, X is a closed abelian normal subgroup, then \\Gamma\\times X\\to X, (…","labels":["conjugate-action-cont"],"detail_key":"p21"},{"id":"n29742","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29743","layer":"informal","project":"p21","title":"\\citeWas97, Lemma 13.14","kind":"lem","summary":"[\\citeWas97, Lemma 13.14] Let G be a compact Hausdorff topological group, X be a closed abelian…","labels":["commutator-eq"],"detail_key":"p21"},{"id":"n29744","layer":"informal","project":"p21","title":"Note that G=IX, G'=\\overline[G,G] and [G,G] is generated by elements of form (\\alpha x)(\\…","kind":"proof","summary":"Note that G=IX, G'=\\overline[G,G] and [G,G] is generated by elements of form (\\alpha x)(\\beta y…","labels":[],"detail_key":"p21"},{"id":"n29745","layer":"informal","project":"p21","title":"p-primary-part","kind":"definition","summary":"If M is a finite abelian group, p is a prime, let M(p) be the p-primary part of M, which is the…","labels":["p-primary-part"],"detail_key":"p21"},{"id":"n29746","layer":"informal","project":"p21","title":"Zp-ext-Ln","kind":"definition","summary":"For each n\\geq 0 let L_n be the maximal unramified abelian extension of K_n of exponent p. Let…","labels":["Zp-ext-Ln"],"detail_key":"p21"},{"id":"n29747","layer":"informal","project":"p21","title":"Zp-ext-An","kind":"definition","summary":"For each n\\geq 0 let A_n:=\\Cl(K_n)(p) be the p-primary part of the class group \\Cl(K_n), viewed…","labels":["Zp-ext-An"],"detail_key":"p21"},{"id":"n29748","layer":"informal","project":"p21","title":"Xn-isom-An","kind":"prop","summary":"There is a canonical isomorphism X_n\\cong A_n.","labels":["Xn-isom-An"],"detail_key":"p21"},{"id":"n29749","layer":"informal","project":"p21","title":"Class field theory.","kind":"proof","summary":"Class field theory.","labels":[],"detail_key":"p21"},{"id":"n29750","layer":"informal","project":"p21","title":"Ln-K-Gal","kind":"lem","summary":"Each L_n is Galois over K.","labels":["Ln-K-Gal"],"detail_key":"p21"},{"id":"n29751","layer":"informal","project":"p21","title":"Since L_n is maximal.","kind":"proof","summary":"Since L_n is maximal.","labels":[],"detail_key":"p21"},{"id":"n29752","layer":"informal","project":"p21","title":"Ln-mono","kind":"lem","summary":"K_n+1L_n\\subset L_n+1. In particular, L_n\\subset L_n+1.","labels":["Ln-mono"],"detail_key":"p21"},{"id":"n29753","layer":"informal","project":"p21","title":"Since K_n+1 and L_n are Galois over K_n, the K_n+1L_n is also Galois over K_n, and the na…","kind":"proof","summary":"Since K_n+1 and L_n are Galois over K_n, the K_n+1L_n is also Galois over K_n, and the natural…","labels":[],"detail_key":"p21"},{"id":"n29754","layer":"informal","project":"p21","title":"Xn-map-eq-An-map","kind":"prop","summary":"For m\\geq n, the natural map X_m\\to X_n, \\sigma\\mapsto\\sigma|_L_n corresponds to the norm map A…","labels":["Xn-map-eq-An-map"],"detail_key":"p21"},{"id":"n29755","layer":"informal","project":"p21","title":"Class field theory.","kind":"proof","summary":"Class field theory.","labels":[],"detail_key":"p21"},{"id":"n29756","layer":"informal","project":"p21","title":"Zp-ext-L-inf","kind":"definition","summary":"Let L_\\infty:=\\bigcup_n\\geq 0L_n. Note that each L_n is Galois over K, so L_\\infty/K is also Ga…","labels":["Zp-ext-L-inf"],"detail_key":"p21"},{"id":"n29757","layer":"informal","project":"p21","title":"X-inf-is-proj-lim","kind":"lem","summary":"There is a natural isomorphism X_\\infty\\xrightarrow\\sim\\varprojlim_n X_n, \\sigma\\mapsto(\\sigma|…","labels":["X-inf-is-proj-lim"],"detail_key":"p21"},{"id":"n29758","layer":"informal","project":"p21","title":"The inverse map maps (\\sigma_n)_n\\geq 0 to x\\mapsto\\sigma_n(x) if x\\in L_n.","kind":"proof","summary":"The inverse map maps (\\sigma_n)_n\\geq 0 to x\\mapsto\\sigma_n(x) if x\\in L_n.","labels":[],"detail_key":"p21"},{"id":"n29759","layer":"informal","project":"p21","title":"X-has-Gamma-action","kind":"lem","summary":"Each X_n has a natural \\Gamma_n-action, and is a \\BZ_p[\\Gamma_n]-module. X_\\infty has a natural…","labels":["X-has-Gamma-action"],"detail_key":"p21"},{"id":"n29760","layer":"informal","project":"p21","title":"The \\Gamma_n-action on X_n is given by Definition \\refconjugate-action. The \\Gamma-action…","kind":"proof","summary":"The \\Gamma_n-action on X_n is given by Definition \\refconjugate-action. The \\Gamma-action on X_…","labels":[],"detail_key":"p21"},{"id":"n29761","layer":"informal","project":"p21","title":"Xn-isom-An-equivariant","kind":"prop","summary":"The isomorphism X_n\\cong A_n preserves \\Gamma_n-action.","labels":["Xn-isom-An-equivariant"],"detail_key":"p21"},{"id":"n29762","layer":"informal","project":"p21","title":"Class field theory.","kind":"proof","summary":"Class field theory.","labels":[],"detail_key":"p21"},{"id":"n29763","layer":"informal","project":"p21","title":"\\citeWas97, Proposition 13.2, \\citeIwa59, Lemma 7.1","kind":"prop","summary":"[\\citeWas97, Proposition 13.2, \\citeIwa59, Lemma 7.1] If K is any number field, K_\\infty/K is a…","labels":["p:unr-outside-p"],"detail_key":"p21"},{"id":"n29764","layer":"informal","project":"p21","title":"Let D_\\fl be the decomposition subgroup of \\fl in \\Gamma:=\\Gal(K_\\infty/K)\\cong\\BZ_p, whi…","kind":"proof","summary":"Let D_\\fl be the decomposition subgroup of \\fl in \\Gamma:=\\Gal(K_\\infty/K)\\cong\\BZ_p, which is…","labels":[],"detail_key":"p21"},{"id":"n29765","layer":"informal","project":"p21","title":"Zp-ext-totally-ramified","kind":"definition","summary":"A \\BZ_p-extension K_\\infty/K is called totally ramified, if all primes which are ramified in K_…","labels":["Zp-ext-totally-ramified"],"detail_key":"p21"},{"id":"n29766","layer":"informal","project":"p21","title":"\\citeWas97, Lemma 13.3","kind":"lem","summary":"[\\citeWas97, Lemma 13.3] If K is any number field, K_\\infty/K is any \\BZ_p-extension, then K_\\i…","labels":["exists-totally-ramified"],"detail_key":"p21"},{"id":"n29767","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29768","layer":"informal","project":"p21","title":"isom-of-totally-ramified","kind":"lem","summary":"Suppose K_\\infty/K is totally ramified (Definition \\refZp-ext-totally-ramified). Then K_n+1\\cap…","labels":["isom-of-totally-ramified"],"detail_key":"p21"},{"id":"n29769","layer":"informal","project":"p21","title":"...","kind":"proof","summary":"...","labels":[],"detail_key":"p21"},{"id":"n29770","layer":"informal","project":"p21","title":"\\citeWas97, Lemma 13.15, 13.17","kind":"thm","summary":"[\\citeWas97, Lemma 13.15, 13.17] Suppose K_\\infty/K is totally ramified (Definition \\refZp-ext-…","labels":["Xn-isom-qout-of-totally-ramified"],"detail_key":"p21"},{"id":"n29771","layer":"informal","project":"p21","title":"The Y\\subset X_\\infty is constructed as follows. Let \\fp_1,\\cdots,\\fp_s be primes of \\CO_…","kind":"proof","summary":"The Y\\subset X_\\infty is constructed as follows. 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Then for every a \\in A and \\eta :…","labels":["prop:abstraction-equal"],"detail_key":"p23"},{"id":"n29876","layer":"informal","project":"p23","title":"By straightforward structural induction on~e.","kind":"proof","summary":"By straightforward structural induction on~e.","labels":[],"detail_key":"p23"},{"id":"n29877","layer":"informal","project":"p23","title":"Identity","kind":"definition","summary":"[Identity] There is the \\emphidentity combinator \\mathsfI \\in A such that I \\, a = a for all a…","labels":["def:combinator-I"],"detail_key":"p23"},{"id":"n29878","layer":"informal","project":"p23","title":"Pairing","kind":"definition","summary":"[Pairing] There are combinators \\mathsfpair, \\mathsffst, \\mathsfsnd \\in A such that, for all a,…","labels":["def:combinator-pairing"],"detail_key":"p23"},{"id":"n29879","layer":"informal","project":"p23","title":"Booleans","kind":"definition","summary":"[Booleans] There are combinators \\mathsfite \\, \\mathsffal, \\mathsftru \\in A such that, for all…","labels":["def:combinator-bool"],"detail_key":"p23"},{"id":"n29880","layer":"informal","project":"p23","title":"Numerals","kind":"definition","summary":"[Numerals] For each n \\in N there is \\overlinen \\in A, as well as combinators \\mathsfsucc, \\mat…","labels":["def:combinator-nat"],"detail_key":"p23"},{"id":"n29881","layer":"informal","project":"p23","title":"def:combinator-Y","kind":"definition","summary":"There is a combinator \\mathsfY \\in A such that, for all a \\in A, \\mathsfY \\, a = a (\\mathsfZ \\,…","labels":["def:combinator-Y"],"detail_key":"p23"},{"id":"n29882","layer":"informal","project":"p23","title":"General recursion","kind":"definition","summary":"[General recursion] There is a combinator \\mathsfZ \\in A such that, for all f, a \\in A, \\mathsf…","labels":["def:combinator-Z"],"detail_key":"p23"},{"id":"n29883","layer":"informal","project":"p23","title":"def:representable-function","kind":"definition","summary":"A partial map f : A \\rightharpoonup A is \\emphrepresented by a \\in A when, for all b \\in B, f(b…","labels":["def:representable-function"],"detail_key":"p23"},{"id":"n29884","layer":"informal","project":"p23","title":"thm:recursive-map-representable","kind":"theorem","summary":"Every general recursive map f : N\\rightharpoonupN is represented in the following sense: there…","labels":["thm:recursive-map-representable"],"detail_key":"p23"},{"id":"n29885","layer":"informal","project":"p23","title":"def:CA","kind":"definition","summary":"A \\emph(total) combinatory algebra (CA) is given by a carrier set A and a \\emphtotal binary ope…","labels":["def:CA"],"detail_key":"p23"},{"id":"n29886","layer":"informal","project":"p23","title":"prop:CAisPCA","kind":"proposition","summary":"Every combinatory algebra is a partial combinatory algebra.","labels":["prop:CAisPCA"],"detail_key":"p23"},{"id":"n29887","layer":"informal","project":"p23","title":"Simply reuse the total application as the partial one.","kind":"proof","summary":"Simply reuse the total application as the partial one.","labels":[],"detail_key":"p23"},{"id":"n29888","layer":"informal","project":"p23","title":"def:FreeCA","kind":"definition","summary":"The \\emphfree combinatory algebra is generated freely by the symbols \\mathsfK and \\mathsfS and…","labels":["def:FreeCA"],"detail_key":"p23"},{"id":"n29889","layer":"informal","project":"p23","title":"def:Listing","kind":"definition","summary":"A \\emphlisting on a set A is a section \\mathsffromList : \\mathsfList\\,A \\to A and a retraction…","labels":["def:Listing"],"detail_key":"p23"},{"id":"n29890","layer":"informal","project":"p23","title":"def:graph-model-application","kind":"definition","summary":"The \\emphapplication - \\cdot - : P\\,A \\times P\\,A \\to P\\,A is defined by S \\cdot T = \\x \\in A \\…","labels":["def:graph-model-application"],"detail_key":"p23"},{"id":"n29891","layer":"informal","project":"p23","title":"thm:graph-model","kind":"theorem","summary":"The set P\\,A with the application as above is a combinatory algebra.","labels":["thm:graph-model"],"detail_key":"p23"},{"id":"n29892","layer":"formal","project":"p23","title":"PartialApplication","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p23","name":"PartialApplication","module":"PartialCombinatoryAlgebras.Basic"},{"id":"n29893","layer":"formal","project":"p23","title":"CA","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p23","name":"CA","module":"PartialCombinatoryAlgebras.CombinatoryAlgebra"},{"id":"n29894","layer":"formal","project":"p23","title":"CA.isPCA","kind":"def","summary":"A : Type → [inst : CA A] → PCA A","labels":[],"detail_key":"p23","name":"CA.isPCA","module":"PartialCombinatoryAlgebras.CombinatoryAlgebra"},{"id":"n29895","layer":"formal","project":"p23","title":"FreeCA","kind":"def","summary":"CA FreeCA.carrier","labels":[],"detail_key":"p23","name":"FreeCA","module":"PartialCombinatoryAlgebras.FreeCombinatoryAlgebra"},{"id":"n29896","layer":"formal","project":"p23","title":"GraphModel.apply","kind":"def","summary":"α : Type → [inst : Listing α] → Set α → Set α → Set α","labels":[],"detail_key":"p23","name":"GraphModel.apply","module":"PartialCombinatoryAlgebras.GraphModel"},{"id":"n29897","layer":"formal","project":"p23","title":"GraphModel.isCA","kind":"def","summary":"α : Type → [inst : Listing α] → CA (Set α)","labels":[],"detail_key":"p23","name":"GraphModel.isCA","module":"PartialCombinatoryAlgebras.GraphModel"},{"id":"n29898","layer":"formal","project":"p23","title":"Listing","kind":"inductive","summary":"Type → Type","labels":[],"detail_key":"p23","name":"Listing","module":"PartialCombinatoryAlgebras.GraphModel"},{"id":"n29899","layer":"formal","project":"p23","title":"PCA","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p23","name":"PCA","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29900","layer":"formal","project":"p23","title":"PCA.Expr","kind":"inductive","summary":"Type u_1 → Type u_2 → Type (max u_1 u_2)","labels":[],"detail_key":"p23","name":"PCA.Expr","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29901","layer":"formal","project":"p23","title":"PCA.abstr","kind":"def","summary":"Γ : Type u → [inst : DecidableEq Γ] → A : Type v → Γ → PCA.Expr Γ A → PCA.Expr Γ A","labels":[],"detail_key":"p23","name":"PCA.abstr","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29902","layer":"formal","project":"p23","title":"PCA.defined","kind":"def","summary":"Γ : Type u → A : Type v → [inst : PCA A] → PCA.Expr Γ A → Prop","labels":[],"detail_key":"p23","name":"PCA.defined","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29903","layer":"formal","project":"p23","title":"PCA.df_abstr","kind":"theorem","summary":"∀ Γ : Type u [inst : DecidableEq Γ] A : Type v [inst_1 : PCA A] (x : Γ) (e : PCA.Expr Γ A), PCA…","labels":[],"detail_key":"p23","name":"PCA.df_abstr","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29904","layer":"formal","project":"p23","title":"PCA.eval","kind":"def","summary":"Γ : Type u → A : Type v → [inst : PCA A] → (Γ → A) → PCA.Expr Γ A → Part A","labels":[],"detail_key":"p23","name":"PCA.eval","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29905","layer":"formal","project":"p23","title":"PCA.eval_abstr","kind":"theorem","summary":"∀ Γ : Type u [inst : DecidableEq Γ] A : Type v [inst_1 : PCA A] (x : Γ) (e : PCA.Expr Γ A) (a :…","labels":[],"detail_key":"p23","name":"PCA.eval_abstr","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29906","layer":"formal","project":"p23","title":"PCA.override","kind":"def","summary":"Γ : Type u → [inst : DecidableEq Γ] → A : Type v → Γ → A → (Γ → A) → Γ → A","labels":[],"detail_key":"p23","name":"PCA.override","module":"PartialCombinatoryAlgebras.PartialCombinatoryAlgebra"},{"id":"n29907","layer":"formal","project":"p23","title":"PCA.I","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.I","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29908","layer":"formal","project":"p23","title":"PCA.Y","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.Y","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29909","layer":"formal","project":"p23","title":"PCA.Z","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.Z","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29910","layer":"formal","project":"p23","title":"PCA.df_Y","kind":"theorem","summary":"∀ A : Type u [inst : PCA A], PCA.Y.Dom","labels":[],"detail_key":"p23","name":"PCA.df_Y","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29911","layer":"formal","project":"p23","title":"PCA.df_Z","kind":"theorem","summary":"∀ A : Type u [inst : PCA A], PCA.Z.Dom","labels":[],"detail_key":"p23","name":"PCA.df_Z","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29912","layer":"formal","project":"p23","title":"PCA.eq_I","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] u : Part A, u.Dom → Eq (HasDot.dot PCA.I u) u","labels":[],"detail_key":"p23","name":"PCA.eq_I","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29913","layer":"formal","project":"p23","title":"PCA.eq_Y","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u : Part A), u.Dom → Eq (HasDot.dot PCA.Y u) (HasDot.dot u (HasDot…","labels":[],"detail_key":"p23","name":"PCA.eq_Y","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29914","layer":"formal","project":"p23","title":"PCA.eq_Z","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u v : Part A), u.Dom → v.Dom → Eq (HasDot.dot (HasDot.dot PCA.Z u)…","labels":[],"detail_key":"p23","name":"PCA.eq_Z","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29915","layer":"formal","project":"p23","title":"PCA.eq_fst_pair","kind":"def","summary":"∀ A : Type u [inst : PCA A] (u v : Part A), u.Dom → v.Dom → Eq (HasDot.dot PCA.fst (HasDot.dot…","labels":[],"detail_key":"p23","name":"PCA.eq_fst_pair","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29916","layer":"formal","project":"p23","title":"PCA.eq_ite_fal","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u v : Part A), u.Dom → v.Dom → Eq (HasDot.dot (HasDot.dot (HasDot.…","labels":[],"detail_key":"p23","name":"PCA.eq_ite_fal","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29917","layer":"formal","project":"p23","title":"PCA.eq_ite_tru","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u v : Part A), u.Dom → v.Dom → Eq (HasDot.dot (HasDot.dot (HasDot.…","labels":[],"detail_key":"p23","name":"PCA.eq_ite_tru","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29918","layer":"formal","project":"p23","title":"PCA.eq_primrec_succ","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u f n : Part A), u.Dom → f.Dom → n.Dom → Eq (HasDot.dot (HasDot.do…","labels":[],"detail_key":"p23","name":"PCA.eq_primrec_succ","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29919","layer":"formal","project":"p23","title":"PCA.eq_primrec_zero","kind":"theorem","summary":"∀ A : Type u [inst : PCA A] (u f : Part A), u.Dom → f.Dom → Eq (HasDot.dot (HasDot.dot (HasDot.…","labels":[],"detail_key":"p23","name":"PCA.eq_primrec_zero","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29920","layer":"formal","project":"p23","title":"PCA.eq_snd_pair","kind":"def","summary":"∀ A : Type u [inst : PCA A] (u v : Part A), u.Dom → v.Dom → Eq (HasDot.dot PCA.snd (HasDot.dot…","labels":[],"detail_key":"p23","name":"PCA.eq_snd_pair","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29921","layer":"formal","project":"p23","title":"PCA.fal","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.fal","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29922","layer":"formal","project":"p23","title":"PCA.fst","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.fst","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29923","layer":"formal","project":"p23","title":"PCA.ite","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.ite","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29924","layer":"formal","project":"p23","title":"PCA.numeral","kind":"def","summary":"A : Type u → [inst : PCA A] → Nat → Part A","labels":[],"detail_key":"p23","name":"PCA.numeral","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29925","layer":"formal","project":"p23","title":"PCA.pair","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.pair","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29926","layer":"formal","project":"p23","title":"PCA.primrec","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.primrec","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29927","layer":"formal","project":"p23","title":"PCA.snd","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.snd","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29928","layer":"formal","project":"p23","title":"PCA.succ","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.succ","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29929","layer":"formal","project":"p23","title":"PCA.tru","kind":"def","summary":"A : Type u → [inst : PCA A] → Part A","labels":[],"detail_key":"p23","name":"PCA.tru","module":"PartialCombinatoryAlgebras.Programming"},{"id":"n29930","layer":"informal","project":"p24","title":"thm:ord_iff_real","kind":"theorem","summary":"A field can be ordered if and only if it is real (that is, -1 is not a sum of squares).","labels":["thm:ord_iff_real"],"detail_key":"p24"},{"id":"n29931","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29932","layer":"informal","project":"p24","title":"lem:unique_ord_cond","kind":"lemma","summary":"Let F be a real field. There is a unique ordering on F if and only if, for each a\\in F, either…","labels":["lem:unique_ord_cond"],"detail_key":"p24"},{"id":"n29933","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29934","layer":"informal","project":"p24","title":"cor:unique_ord_cond_ordered","kind":"corollary","summary":"Let F be an ordered field. Then F has a unique field ordering if and only if every non-negative…","labels":["cor:unique_ord_cond_ordered"],"detail_key":"p24"},{"id":"n29935","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29936","layer":"informal","project":"p24","title":"cor:unique_ord_Q","kind":"corollary","summary":"There is a unique field ordering on Q.","labels":["cor:unique_ord_Q"],"detail_key":"p24"},{"id":"n29937","layer":"informal","project":"p24","title":"We use Corollary \\refcor:unique_ord_cond_ordered. Let x\\inQ be non-negative. Then x=p/q f…","kind":"proof","summary":"We use Corollary \\refcor:unique_ord_cond_ordered. Let x\\inQ be non-negative. Then x=p/q for som…","labels":[],"detail_key":"p24"},{"id":"n29938","layer":"informal","project":"p24","title":"thm:ext_ord_cond","kind":"theorem","summary":"Let F be an ordered field, and let K/F be a field extension. Then there is an ordering on K mak…","labels":["thm:ext_ord_cond"],"detail_key":"p24"},{"id":"n29939","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29940","layer":"informal","project":"p24","title":"lem:ext_ord_functional_suff","kind":"lemma","summary":"Let F be an ordered field, and let K/F be a field extension. Suppose that there is an F-linear…","labels":["lem:ext_ord_functional_suff"],"detail_key":"p24"},{"id":"n29941","layer":"informal","project":"p24","title":"Consider the sum \\sum_i x_i\\alpha_i^2 for some \\alpha_i\\in K and x_i\\in F_\\geq0. By F-lin…","kind":"proof","summary":"Consider the sum \\sum_i x_i\\alpha_i^2 for some \\alpha_i\\in K and x_i\\in F_\\geq0. By F-linearity…","labels":[],"detail_key":"p24"},{"id":"n29942","layer":"informal","project":"p24","title":"cor:ext_ord_to_adj_sqrt","kind":"corollary","summary":"Let F be a ordered field, and suppose a\\in F is non-negative (and not a square). Then there is…","labels":["cor:ext_ord_to_adj_sqrt"],"detail_key":"p24"},{"id":"n29943","layer":"informal","project":"p24","title":"Let \\pi:F(\\sqrta)\\to F be the projection induced by the F-basis \\1,\\sqrta\\. For x,y\\in F,…","kind":"proof","summary":"Let \\pi:F(\\sqrta)\\to F be the projection induced by the F-basis \\1,\\sqrta\\. For x,y\\in F, we ha…","labels":[],"detail_key":"p24"},{"id":"n29944","layer":"informal","project":"p24","title":"lem:ext_ord_odd_deg","kind":"lemma","summary":"Let F be an ordered field, and let K/F be an odd-degree extension. Then there is a field orderi…","labels":["lem:ext_ord_odd_deg"],"detail_key":"p24"},{"id":"n29945","layer":"informal","project":"p24","title":"Since \\ch R=0, we can apply the primitive element theorem. Let K=F(\\alpha) for some \\alph…","kind":"proof","summary":"Since \\ch R=0, we can apply the primitive element theorem. Let K=F(\\alpha) for some \\alpha\\in K…","labels":[],"detail_key":"p24"},{"id":"n29946","layer":"informal","project":"p24","title":"lem:order_fun_field","kind":"lemma","summary":"Let F be an ordered field, and let a\\in F. Then there is a unique ordering on the function fiel…","labels":["lem:order_fun_field"],"detail_key":"p24"},{"id":"n29947","layer":"informal","project":"p24","title":"def:RCF","kind":"definition","summary":"A real closed field F is a real field such that \\item for all x in F, either x or -x is a squar…","labels":["def:RCF"],"detail_key":"p24"},{"id":"n29948","layer":"informal","project":"p24","title":"lem:RCF_ord_ax","kind":"lemma","summary":"An ordered field is real closed if \\item every non-negative element is a square, and \\item ever…","labels":["lem:RCF_ord_ax"],"detail_key":"p24"},{"id":"n29949","layer":"informal","project":"p24","title":"Squares in an ordered field are non-negative.","kind":"proof","summary":"Squares in an ordered field are non-negative.","labels":[],"detail_key":"p24"},{"id":"n29950","layer":"informal","project":"p24","title":"lem:RCF_sumsq_is_sq","kind":"lemma","summary":"Every sum of squares in R is a square in R.","labels":["lem:RCF_sumsq_is_sq"],"detail_key":"p24"},{"id":"n29951","layer":"informal","project":"p24","title":"If the negative of a sum of squares is a square, then, dividing through, -1 is a sum of s…","kind":"proof","summary":"If the negative of a sum of squares is a square, then, dividing through, -1 is a sum of squares…","labels":[],"detail_key":"p24"},{"id":"n29952","layer":"informal","project":"p24","title":"lem:RCF_ord_unique","kind":"lemma","summary":"R has a unique ordering making it an ordered field. In this ordering, the non-negative elements…","labels":["lem:RCF_ord_unique"],"detail_key":"p24"},{"id":"n29953","layer":"informal","project":"p24","title":"Follows directly from Lemmas \\reflem:unique_ord_cond and \\reflem:RCF_sumsq_is_sq.","kind":"proof","summary":"Follows directly from Lemmas \\reflem:unique_ord_cond and \\reflem:RCF_sumsq_is_sq.","labels":[],"detail_key":"p24"},{"id":"n29954","layer":"informal","project":"p24","title":"lem:alg_ext_odd_deg","kind":"lemma","summary":"There is no nontrivial odd-degree finite extension of R.","labels":["lem:alg_ext_odd_deg"],"detail_key":"p24"},{"id":"n29955","layer":"informal","project":"p24","title":"Let K/R be an odd-degree extension of R. By the primitive element theorem, K=R(\\alpha) fo…","kind":"proof","summary":"Let K/R be an odd-degree extension of R. By the primitive element theorem, K=R(\\alpha) for some…","labels":[],"detail_key":"p24"},{"id":"n29956","layer":"informal","project":"p24","title":"lem:ext_deg_2","kind":"lemma","summary":"The field R(i) is the unique quadratic extension of R.","labels":["lem:ext_deg_2"],"detail_key":"p24"},{"id":"n29957","layer":"informal","project":"p24","title":"Let K/R be a quadratic extension. Since \\ch R\\neq2, we have K\\cong R(\\sqrta) for some a\\i…","kind":"proof","summary":"Let K/R be a quadratic extension. Since \\ch R\\neq2, we have K\\cong R(\\sqrta) for some a\\in R. S…","labels":[],"detail_key":"p24"},{"id":"n29958","layer":"informal","project":"p24","title":"lem:Ri_ext_deg_2","kind":"lemma","summary":"There is no quadratic extension of R(i).","labels":["lem:Ri_ext_deg_2"],"detail_key":"p24"},{"id":"n29959","layer":"informal","project":"p24","title":"Since charR(i)\\neq 2, it suffices to show that every element of R(i) is a square. Observe…","kind":"proof","summary":"Since charR(i)\\neq 2, it suffices to show that every element of R(i) is a square. Observe that…","labels":[],"detail_key":"p24"},{"id":"n29960","layer":"informal","project":"p24","title":"thm:FTAlg","kind":"theorem","summary":"The only finite extensions of R are R itself and R(i).","labels":["thm:FTAlg"],"detail_key":"p24"},{"id":"n29961","layer":"informal","project":"p24","title":"By separability, every finite extension of R is contained in a finite Galois extension. S…","kind":"proof","summary":"By separability, every finite extension of R is contained in a finite Galois extension. Since R…","labels":[],"detail_key":"p24"},{"id":"n29962","layer":"informal","project":"p24","title":"cor:FTAlg_alg","kind":"corollary","summary":"The only algebraic extensions of R are R itself and R(i).","labels":["cor:FTAlg_alg"],"detail_key":"p24"},{"id":"n29963","layer":"informal","project":"p24","title":"An infinite algebraic extension contains finite subextensions of arbitrarily large degree.","kind":"proof","summary":"An infinite algebraic extension contains finite subextensions of arbitrarily large degree.","labels":[],"detail_key":"p24"},{"id":"n29964","layer":"informal","project":"p24","title":"cor:RCF_ac","kind":"corollary","summary":"\\barR=R(i).","labels":["cor:RCF_ac"],"detail_key":"p24"},{"id":"n29965","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29966","layer":"informal","project":"p24","title":"lem:sumsq_is_sq","kind":"lemma","summary":"Let F be a field in which -1 is not a square, and suppose every element of F(i) is a square. Th…","labels":["lem:sumsq_is_sq"],"detail_key":"p24"},{"id":"n29967","layer":"informal","project":"p24","title":"Given a,b\\in F, find c,d\\in F such that a+bi=(c+di)^2 in F(i). Then a=c^2-d^2 and b=2cd,…","kind":"proof","summary":"Given a,b\\in F, find c,d\\in F such that a+bi=(c+di)^2 in F(i). Then a=c^2-d^2 and b=2cd, so a^2…","labels":[],"detail_key":"p24"},{"id":"n29968","layer":"informal","project":"p24","title":"lem:FTAlg_converse","kind":"lemma","summary":"Suppose R is a field with unique nontrivial finite extension R(i). Then R is real closed.","labels":["lem:FTAlg_converse"],"detail_key":"p24"},{"id":"n29969","layer":"informal","project":"p24","title":"Since R(i) has no quadratic extensions, we can apply Lemma \\reflem:sumsq_is_sq: sums of s…","kind":"proof","summary":"Since R(i) has no quadratic extensions, we can apply Lemma \\reflem:sumsq_is_sq: sums of squares…","labels":[],"detail_key":"p24"},{"id":"n29970","layer":"informal","project":"p24","title":"lem:FTAlg_converse_alg_closure","kind":"corollary","summary":"Suppose R is a field with nontrivial algebraic closure R(i). Then R is real closed.","labels":["lem:FTAlg_converse_alg_closure"],"detail_key":"p24"},{"id":"n29971","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29972","layer":"informal","project":"p24","title":"lem:RCF_max","kind":"lemma","summary":"R has no nontrivial real algebraic extensions.","labels":["lem:RCF_max"],"detail_key":"p24"},{"id":"n29973","layer":"informal","project":"p24","title":"The field R(i) is not real since -1 is a square in it. We are done by Corollary \\refcor:F…","kind":"proof","summary":"The field R(i) is not real since -1 is a square in it. We are done by Corollary \\refcor:FTAlg_a…","labels":[],"detail_key":"p24"},{"id":"n29974","layer":"informal","project":"p24","title":"lem:RCF_max_ord","kind":"corollary","summary":"R has no nontrivial ordered algebraic extensions (with respect to the unique order).","labels":["lem:RCF_max_ord"],"detail_key":"p24"},{"id":"n29975","layer":"informal","project":"p24","title":"Since ordered fields are real, we are done by Lemma \\reflem:RCF_max.","kind":"proof","summary":"Since ordered fields are real, we are done by Lemma \\reflem:RCF_max.","labels":[],"detail_key":"p24"},{"id":"n29976","layer":"informal","project":"p24","title":"lem:irreds_class","kind":"lemma","summary":"The monic irreducible polynomials over R[X] have form X-c for some c\\in R or (X-a)^2+b^2 for so…","labels":["lem:irreds_class"],"detail_key":"p24"},{"id":"n29977","layer":"informal","project":"p24","title":"Let f\\in R[X] be monic and irreducible. The field R_f=R[X]/(f) is a finite extension of R…","kind":"proof","summary":"Let f\\in R[X] be monic and irreducible. The field R_f=R[X]/(f) is a finite extension of R, so i…","labels":[],"detail_key":"p24"},{"id":"n29978","layer":"informal","project":"p24","title":"lem:IVP_poly","kind":"lemma","summary":"Polynomials over R satisfy the intermediate value property (with respect to the unique order).","labels":["lem:IVP_poly"],"detail_key":"p24"},{"id":"n29979","layer":"informal","project":"p24","title":"We will prove that, for all f\\in R[X] and all a,b\\in R with a\\leq b, if f(a)\\leq 0\\leq f(…","kind":"proof","summary":"We will prove that, for all f\\in R[X] and all a,b\\in R with a\\leq b, if f(a)\\leq 0\\leq f(b), th…","labels":[],"detail_key":"p24"},{"id":"n29980","layer":"informal","project":"p24","title":"thm:IVP_poly_imp_RCF","kind":"theorem","summary":"Let R be an ordered field whose polynomials satisfy the intermediate value property. Then R is…","labels":["thm:IVP_poly_imp_RCF"],"detail_key":"p24"},{"id":"n29981","layer":"informal","project":"p24","title":"We use Lemma \\reflem:RCF_ord_ax. Let a\\in R be non-negative, and consider the polynomial…","kind":"proof","summary":"We use Lemma \\reflem:RCF_ord_ax. Let a\\in R be non-negative, and consider the polynomial f=X^2-…","labels":[],"detail_key":"p24"},{"id":"n29982","layer":"informal","project":"p24","title":"thm:ord_max_imp_RCF","kind":"theorem","summary":"Let R be an ordered field with no nontrivial ordered algebraic extensions. Then R is real close…","labels":["thm:ord_max_imp_RCF"],"detail_key":"p24"},{"id":"n29983","layer":"informal","project":"p24","title":"We use Lemma \\reflem:RCF_ord_ax. Let a\\in R be non-negative, and suppose a is not a squar…","kind":"proof","summary":"We use Lemma \\reflem:RCF_ord_ax. Let a\\in R be non-negative, and suppose a is not a square. By…","labels":[],"detail_key":"p24"},{"id":"n29984","layer":"informal","project":"p24","title":"cor:real_max_imp_RCF","kind":"corollary","summary":"Let R be a real field with no nontrivial real algebraic extensions. Then R is real closed.","labels":["cor:real_max_imp_RCF"],"detail_key":"p24"},{"id":"n29985","layer":"informal","project":"p24","title":"Since R can be ordered and ordered fields are real, we are done by Theorem \\refthm:ord_ma…","kind":"proof","summary":"Since R can be ordered and ordered fields are real, we are done by Theorem \\refthm:ord_max_imp_…","labels":[],"detail_key":"p24"},{"id":"n29986","layer":"informal","project":"p24","title":"lem:ACF_ind_2_RCF","kind":"lemma","summary":"An algebraically closed field of characteristic 0 has an index-2 real closed subfield.","labels":["lem:ACF_ind_2_RCF"],"detail_key":"p24"},{"id":"n29987","layer":"informal","project":"p24","title":"Let C be an algebraically closed field of characteristic 0. Observe that the prime subfie…","kind":"proof","summary":"Let C be an algebraically closed field of characteristic 0. Observe that the prime subfield Q c…","labels":[],"detail_key":"p24"},{"id":"n29988","layer":"informal","project":"p24","title":"thm:RCF_tfae","kind":"theorem","summary":"Let R be a field. TFAE: \\item R is real closed. \\item \\barR=R(i) (and R(i)\\neq R). \\item R is r…","labels":["thm:RCF_tfae"],"detail_key":"p24"},{"id":"n29989","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29990","layer":"informal","project":"p24","title":"thm:RCF_tfae_ord","kind":"theorem","summary":"Let R be an ordered field. TFAE: \\item R is real closed. \\item R is maximal with respect to ord…","labels":["thm:RCF_tfae_ord"],"detail_key":"p24"},{"id":"n29991","layer":"informal","project":"p24","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p24"},{"id":"n29992","layer":"informal","project":"p24","title":"def:real_closure","kind":"definition","summary":"Let F be an ordered field. A real closure of F is a real closed ordered algebraic extension of…","labels":["def:real_closure"],"detail_key":"p24"},{"id":"n29993","layer":"informal","project":"p24","title":"lem:real_closure_exists","kind":"lemma","summary":"Let F be an ordered field. Then F has a real closure.","labels":["lem:real_closure_exists"],"detail_key":"p24"},{"id":"n29994","layer":"informal","project":"p24","title":"Apply Zorn's lemma to ordered algebraic extensions of F. We are done by Theorem \\refthm:o…","kind":"proof","summary":"Apply Zorn's lemma to ordered algebraic extensions of F. We are done by Theorem \\refthm:ord_max…","labels":[],"detail_key":"p24"},{"id":"n29995","layer":"informal","project":"p24","title":"Corollary to Sturm's Theorem","kind":"theorem","summary":"[Corollary to Sturm's Theorem] Let F be an ordered field, and let f be a polynomial over F. The…","labels":["thm:Sturm"],"detail_key":"p24"},{"id":"n29996","layer":"informal","project":"p24","title":"TODO : decide on the generality of the statement of Sturm's Theorem","kind":"proof","summary":"TODO : decide on the generality of the statement of Sturm's Theorem","labels":[],"detail_key":"p24"},{"id":"n29997","layer":"informal","project":"p24","title":"lem:closure_emb_ext_unordered","kind":"lemma","summary":"Let F be an ordered field with a real closure R, and let K/F be a finite ordered extension. The…","labels":["lem:closure_emb_ext_unordered"],"detail_key":"p24"},{"id":"n29998","layer":"informal","project":"p24","title":"By the primitive element theorem, K=F(\\alpha) for some \\alpha\\in K. Let f be the minimal…","kind":"proof","summary":"By the primitive element theorem, K=F(\\alpha) for some \\alpha\\in K. Let f be the minimal polyno…","labels":[],"detail_key":"p24"},{"id":"n29999","layer":"informal","project":"p24","title":"lem:closure_emb_ext","kind":"lemma","summary":"Let F be an ordered field with a real closure R, and let K/F be a finite ordered extension. The…","labels":["lem:closure_emb_ext"],"detail_key":"p24"},{"id":"n30000","layer":"informal","project":"p24","title":"Fix a real closure R' of K (one exists by Lemma \\reflem:real_closure_exists). By the prim…","kind":"proof","summary":"Fix a real closure R' of K (one exists by Lemma \\reflem:real_closure_exists). By the primitive…","labels":[],"detail_key":"p24"},{"id":"n30001","layer":"informal","project":"p24","title":"thm:real_closure_unique","kind":"theorem","summary":"Let F be an ordered field. Then the real closure of F is unique up to unique F-isomorphism.","labels":["thm:real_closure_unique"],"detail_key":"p24"},{"id":"n30002","layer":"informal","project":"p24","title":"Let R_1","kind":"proof","summary":"Let R_1","labels":[],"detail_key":"p24"},{"id":"n30003","layer":"informal","project":"p24","title":"cor:RCF_no_auto","kind":"corollary","summary":"A real closed field has no nontrivial field automorphisms.","labels":["cor:RCF_no_auto"],"detail_key":"p24"},{"id":"n30004","layer":"informal","project":"p24","title":"Let R be a real closed field. By Theorem \\refthm:real_closure_unique, R has no nontrivial…","kind":"proof","summary":"Let R be a real closed field. By Theorem \\refthm:real_closure_unique, R has no nontrivial order…","labels":[],"detail_key":"p24"},{"id":"n30005","layer":"informal","project":"p24","title":"lem:ord_alg_ext_eq_emb","kind":"lemma","summary":"Let F be an ordered field with real closure R, and let K/F be algebraic. Then field orderings o…","labels":["lem:ord_alg_ext_eq_emb"],"detail_key":"p24"},{"id":"n30006","layer":"informal","project":"p24","title":"Fix an ordering on K extending that on F, and let K have real closure R_K (exists by Lemm…","kind":"proof","summary":"Fix an ordering on K extending that on F, and let K have real closure R_K (exists by Lemma \\ref…","labels":[],"detail_key":"p24"},{"id":"n30007","layer":"informal","project":"p24","title":"thm:Artin-Schreier_weak","kind":"theorem","summary":"Let R be a field with \\ch K\\neq2, and suppose [\\barR:R]=2. Then R is real closed.","labels":["thm:Artin-Schreier_weak"],"detail_key":"p24"},{"id":"n30008","layer":"informal","project":"p24","title":"By Lemma \\reflem:FTAlg_converse_alg_closure, it suffices to show that \\barR\\cong R(i). Si…","kind":"proof","summary":"By Lemma \\reflem:FTAlg_converse_alg_closure, it suffices to show that \\barR\\cong R(i). Since \\c…","labels":[],"detail_key":"p24"},{"id":"n30009","layer":"informal","project":"p24","title":"Artin-Schreier Theorem","kind":"theorem","summary":"[Artin-Schreier Theorem] Let R be a field, and suppose \\barR is a finite extension of R. Then R…","labels":["thm:Artin-Schreier"],"detail_key":"p24"},{"id":"n30010","layer":"informal","project":"p24","title":"TODO: this needs a lot more preliminaries eg Artin-Schreier theory, Kummer theory","kind":"proof","summary":"TODO: this needs a lot more preliminaries eg Artin-Schreier theory, Kummer theory","labels":[],"detail_key":"p24"},{"id":"n30011","layer":"informal","project":"p24","title":"cor:ACF_char_p_no_fin_ind","kind":"corollary","summary":"An algebraically closed field of nonzero characteristic has no finite-index subfields.","labels":["cor:ACF_char_p_no_fin_ind"],"detail_key":"p24"},{"id":"n30012","layer":"informal","project":"p24","title":"Ordered fields have characteristic 0.","kind":"proof","summary":"Ordered fields have characteristic 0.","labels":[],"detail_key":"p24"},{"id":"n30013","layer":"informal","project":"p24","title":"thm:Q_alg_unique_fin_ind","kind":"theorem","summary":"The finite-index subfields of \\barQ are isomorphic copies of Q_\\textalg=\\barQ\\capR indexed by \\…","labels":["thm:Q_alg_unique_fin_ind"],"detail_key":"p24"},{"id":"n30014","layer":"informal","project":"p24","title":"Since Q_\\textalg(i)=\\barQ, the field Q_\\textalg is a finite-index subfield. By Theore","kind":"proof","summary":"Since Q_\\textalg(i)=\\barQ, the field Q_\\textalg is a finite-index subfield. By Theore","labels":[],"detail_key":"p24"},{"id":"n30015","layer":"formal","project":"p24","title":"Field.exists_isStrictOrderedRing_iff_isSemireal","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"Field.exists_isStrictOrderedRing_iff_isSemireal","module":"RealClosedField.Algebra.Order.Field.IsSemireal"},{"id":"n30016","layer":"formal","project":"p24","title":"IsSemireal.existsUnique_isStrictOrderedRing_iff","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsSemireal.existsUnique_isStrictOrderedRing_iff","module":"RealClosedField.Algebra.Order.Field.IsSemireal"},{"id":"n30017","layer":"formal","project":"p24","title":"IsSemireal.unique_isStrictOrderedRing","kind":"def","summary":"","labels":[],"detail_key":"p24","name":"IsSemireal.unique_isStrictOrderedRing","module":"RealClosedField.Algebra.Order.Field.IsSemireal"},{"id":"n30018","layer":"formal","project":"p24","title":"IsStrictOrderedRing.unique_isStrictOrderedRing_iff","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsStrictOrderedRing.unique_isStrictOrderedRing_iff","module":"RealClosedField.Algebra.Order.Field.IsSemireal"},{"id":"n30019","layer":"formal","project":"p24","title":"Rat.unique_isStrictOrderedRing","kind":"def","summary":"","labels":[],"detail_key":"p24","name":"Rat.unique_isStrictOrderedRing","module":"RealClosedField.Algebra.Order.Field.IsSemireal"},{"id":"n30020","layer":"formal","project":"p24","title":"Field.exists_isOrderedAlgebra_iff_neg_one_notMem_sup","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"Field.exists_isOrderedAlgebra_iff_neg_one_notMem_sup","module":"RealClosedField.OrderedAlgebra"},{"id":"n30021","layer":"formal","project":"p24","title":"Field.exists_isOrderedAlgebra_of_projection","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"Field.exists_isOrderedAlgebra_of_projection","module":"RealClosedField.OrderedAlgebra"},{"id":"n30022","layer":"formal","project":"p24","title":"adj_sqrt_ordered","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"adj_sqrt_ordered","module":"RealClosedField.OrderedAlgebra"},{"id":"n30023","layer":"formal","project":"p24","title":"odd_deg_ordered","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"odd_deg_ordered","module":"RealClosedField.OrderedAlgebra"},{"id":"n30024","layer":"formal","project":"p24","title":"isSquare_of_isSumSq_of_forall_adjoinRoot_i_isSquare","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"isSquare_of_isSumSq_of_forall_adjoinRoot_i_isSquare","module":"RealClosedField.PrimitiveElement.Quadratic"},{"id":"n30025","layer":"formal","project":"p24","title":"Artin_Schreier","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"Artin_Schreier","module":"RealClosedField.RealClosedField"},{"id":"n30026","layer":"formal","project":"p24","title":"IsRealClosed","kind":"class","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed","module":"RealClosedField.RealClosedField"},{"id":"n30027","layer":"formal","project":"p24","title":"IsRealClosed.TFAE","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.TFAE","module":"RealClosedField.RealClosedField"},{"id":"n30028","layer":"formal","project":"p24","title":"IsRealClosed.TFAE_linearOrderedField","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.TFAE_linearOrderedField","module":"RealClosedField.RealClosedField"},{"id":"n30029","layer":"formal","project":"p24","title":"IsRealClosed._root_.IsSquare.of_isSumSq","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed._root_.IsSquare.of_isSumSq","module":"RealClosedField.RealClosedField"},{"id":"n30030","layer":"formal","project":"p24","title":"IsRealClosed.finite_extension_classify","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.finite_extension_classify","module":"RealClosedField.RealClosedField"},{"id":"n30031","layer":"formal","project":"p24","title":"IsRealClosed.finrank_neq_two_of_isAdjoinRoot_i","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.finrank_neq_two_of_isAdjoinRoot_i","module":"RealClosedField.RealClosedField"},{"id":"n30032","layer":"formal","project":"p24","title":"IsRealClosed.intermediate_value_property","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.intermediate_value_property","module":"RealClosedField.RealClosedField"},{"id":"n30033","layer":"formal","project":"p24","title":"IsRealClosed.irred_poly_classify","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.irred_poly_classify","module":"RealClosedField.RealClosedField"},{"id":"n30034","layer":"formal","project":"p24","title":"IsRealClosed.isAdjoinRoot_i_of_isAlgClosure","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.isAdjoinRoot_i_of_isAlgClosure","module":"RealClosedField.RealClosedField"},{"id":"n30035","layer":"formal","project":"p24","title":"IsRealClosed.isAdjoinRoot_i_of_isQuadraticExtension","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.isAdjoinRoot_i_of_isQuadraticExtension","module":"RealClosedField.RealClosedField"},{"id":"n30036","layer":"formal","project":"p24","title":"IsRealClosed.maximal_isSemireal","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.maximal_isSemireal","module":"RealClosedField.RealClosedField"},{"id":"n30037","layer":"formal","project":"p24","title":"IsRealClosed.nonneg_iff_isSquare","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.nonneg_iff_isSquare","module":"RealClosedField.RealClosedField"},{"id":"n30038","layer":"formal","project":"p24","title":"IsRealClosed.odd_finrank_extension","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.odd_finrank_extension","module":"RealClosedField.RealClosedField"},{"id":"n30039","layer":"formal","project":"p24","title":"IsRealClosed.of_intermediateValueProperty","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_intermediateValueProperty","module":"RealClosedField.RealClosedField"},{"id":"n30040","layer":"formal","project":"p24","title":"IsRealClosed.of_isAdjoinRoot_i_isAlgClosure","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_isAdjoinRoot_i_isAlgClosure","module":"RealClosedField.RealClosedField"},{"id":"n30041","layer":"formal","project":"p24","title":"IsRealClosed.of_isAdjoinRoot_i_or_finrank_eq_one","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_isAdjoinRoot_i_or_finrank_eq_one","module":"RealClosedField.RealClosedField"},{"id":"n30042","layer":"formal","project":"p24","title":"IsRealClosed.of_linearOrderedField","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_linearOrderedField","module":"RealClosedField.RealClosedField"},{"id":"n30043","layer":"formal","project":"p24","title":"IsRealClosed.of_maximal_isOrderedAlgebra","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_maximal_isOrderedAlgebra","module":"RealClosedField.RealClosedField"},{"id":"n30044","layer":"formal","project":"p24","title":"IsRealClosed.of_maximal_isSemireal","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosed.of_maximal_isSemireal","module":"RealClosedField.RealClosedField"},{"id":"n30045","layer":"formal","project":"p24","title":"IsRealClosure","kind":"class","summary":"","labels":[],"detail_key":"p24","name":"IsRealClosure","module":"RealClosedField.RealClosedField"},{"id":"n30046","layer":"formal","project":"p24","title":"weak_Artin_Schreier","kind":"theorem","summary":"","labels":[],"detail_key":"p24","name":"weak_Artin_Schreier","module":"RealClosedField.RealClosedField"},{"id":"n30047","layer":"informal","project":"p25","title":"def:Nlambda","kind":"definition","summary":"For \\lambda>0 define N_\\lambda(X) as the number of triples (a,b,c)\\in N^3 with a+b=c, \\gcd(a,b,…","labels":["def:Nlambda"],"detail_key":"p25"},{"id":"n30048","layer":"informal","project":"p25","title":"lem:deBruijnRadical","kind":"lemma","summary":"For any \\varepsilon>0, we have \\#\\left\\n\\leq x : \\rad(n)\\leq x^\\lambda\\right\\ \\ll_\\varepsilon x…","labels":["lem:deBruijnRadical","eq:bruijn"],"detail_key":"p25"},{"id":"n30049","layer":"informal","project":"p25","title":"eq1","kind":"proof","summary":"It suffices to show that for any integer k\\geq 2 we have |\\n\\leq X:\\,\\, \\rad(n)=k\\|\\ll X^O(1/\\l…","labels":["eq1"],"detail_key":"p25"},{"id":"n30050","layer":"informal","project":"p25","title":"prop:ABCTrivialBound","kind":"proposition","summary":"Let \\lambda>0. Then N_\\lambda(X)=O_\\varepsilon( X^2\\lambda/3+\\varepsilon), for any \\varepsilon>…","labels":["prop:ABCTrivialBound"],"detail_key":"p25"},{"id":"n30051","layer":"informal","project":"p25","title":"Any triple (a,b,c) counted by N_\\lambda(X) must satisfy \\rad(abc)< X^\\lambda, and so we m…","kind":"proof","summary":"Any triple (a,b,c) counted by N_\\lambda(X) must satisfy \\rad(abc)< X^\\lambda, and so we must ha…","labels":[],"detail_key":"p25"},{"id":"n30052","layer":"informal","project":"p25","title":"thm:ABCExceptionalBound","kind":"theorem","summary":"Let \\lambda\\in (0, 1.001) be fixed. Then N_\\lambda(X)=O( X^33/50).","labels":["thm:ABCExceptionalBound"],"detail_key":"p25"},{"id":"n30053","layer":"informal","project":"p25","title":"def:Sabc","kind":"definition","summary":"For \\alpha,\\beta,\\gamma>0, define S_\\alpha,\\beta,\\gamma(X) as the number of (a,b,c)\\in N^3 with…","labels":["def:Sabc"],"detail_key":"p25"},{"id":"n30054","layer":"informal","project":"p25","title":"lem:TrivialBdforS","kind":"lemma","summary":"S_\\alpha,\\beta,\\gamma(X)\\ll_\\varepsilon X^\\min\\\\alpha+\\beta , \\alpha+\\gamma , \\beta+\\gamma\\+\\va…","labels":["lem:TrivialBdforS","eq:trivial"],"detail_key":"p25"},{"id":"n30055","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30056","layer":"informal","project":"p25","title":"def:Sstar","kind":"definition","summary":"Let S^*_\\alpha,\\beta,\\gamma(X) to be the number of (a,b,c)\\in N^3 with \\gcd(a,b,c)=1 and c\\in […","labels":["def:Sstar"],"detail_key":"p25"},{"id":"n30057","layer":"informal","project":"p25","title":"lem:NlambdatoSstar","kind":"lemma","summary":"We have N_\\lambda(X)\\ll (\\log X)^4 \\max_\\substack \\alpha,\\beta,\\gamma>0\\\\ \\alpha+\\beta+\\gamma\\l…","labels":["lem:NlambdatoSstar","eq:step1"],"detail_key":"p25"},{"id":"n30058","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30059","layer":"informal","project":"p25","title":"thm:BdExceptionalBound","kind":"theorem","summary":"There exists \\varepsilon>0 such that for all c\\in Z^3 and X,Y,Z\\in R_>0^d. we have B_d(\\bf c, X…","labels":["thm:BdExceptionalBound"],"detail_key":"p25"},{"id":"n30060","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30061","layer":"informal","project":"p25","title":"Proof of Theorem \\refthm:ABCExceptionalBound","kind":"proof","summary":"[Proof of Theorem \\refthm:ABCExceptionalBound]","labels":[],"detail_key":"p25"},{"id":"n30062","layer":"informal","project":"p25","title":"def:BdDiophantineCount","kind":"definition","summary":"For c\\in Z^3 and X,Y,Z\\in R_>0^d. we have B_d(c,X,Y,Z) := \\#\\left\\(x,y,z)\\in N^3d\\;:\\; l x_i\\si…","labels":["def:BdDiophantineCount","eq:Bk"],"detail_key":"p25"},{"id":"n30063","layer":"informal","project":"p25","title":"lem:DiophantineFactor","kind":"lemma","summary":"Let \\varepsilon\\in (0,1/2), and let 2\\leq n\\leq X be an integer. Then there exists a factorisat…","labels":["lem:DiophantineFactor"],"detail_key":"p25"},{"id":"n30064","layer":"informal","project":"p25","title":"Fix 2\\le n\\le X and let K=2\\lceil \\varepsilon^-1\\rceil, M=\\lfloor \\frac52\\varepsilon^-2\\r…","kind":"proof","summary":"Fix 2\\le n\\le X and let K=2\\lceil \\varepsilon^-1\\rceil, M=\\lfloor \\frac52\\varepsilon^-2\\rfloor.…","labels":[],"detail_key":"p25"},{"id":"n30065","layer":"informal","project":"p25","title":"prop:DiophantineReduction","kind":"proposition","summary":"Let \\alpha,\\beta,\\gamma\\in (0,1] be fixed and let X\\geq 2. For any \\varepsilon>0 there exists a…","labels":["prop:DiophantineReduction","eq:xiyizi_1","eq:xiyizi_2"],"detail_key":"p25"},{"id":"n30066","layer":"informal","project":"p25","title":"Proof of Proposition~\\refprop:DiophantineReduction","kind":"proof","summary":"[Proof of Proposition~\\refprop:DiophantineReduction] We may assume that X is large enough in te…","labels":[],"detail_key":"p25"},{"id":"n30067","layer":"informal","project":"p25","title":"Fourier analysis bound","kind":"proposition","summary":"[Fourier analysis bound] Let d\\geq 1, \\varepsilon>0 and A\\geq 1 be fixed. Let \\[ X_1,\\ldots, X_…","labels":["prop:FourierAnalysis","eq:Delta"],"detail_key":"p25"},{"id":"n30068","layer":"informal","project":"p25","title":"eq:Bc123","kind":"proof","summary":"By the orthogonality of characters, we have B_d(c,X,Y,Z) &\\leq \\int_0^1 \\sum_x_j\\sim X_j\\sum_y_…","labels":["eq:Bc123","eq:prop3.1I1","eq:I2","eq:N","eq:N1","eq:N2","eq:prop3.1I2"],"detail_key":"p25"},{"id":"n30069","layer":"informal","project":"p25","title":"Geometry of numbers bound","kind":"proposition","summary":"[Geometry of numbers bound] Let d\\geq 1 and \\varepsilon>0 be fixed, and let \\[ X_1,\\ldots, X_d,…","labels":["prop:GeometryofNumbers"],"detail_key":"p25"},{"id":"n30070","layer":"informal","project":"p25","title":"Take any sets I,I',I''\\subset[d]. Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) b…","kind":"proof","summary":"Take any sets I,I',I''\\subset[d]. Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) be a tu…","labels":[],"detail_key":"p25"},{"id":"n30071","layer":"informal","project":"p25","title":"thm:HeathBrownGeometry","kind":"theorem","summary":"Let \\gcd(a_1,a_2,a_3)=1. Then for X_1,X_2,X_3>1, we have \\#\\left((x_1,x_2,x_3)\\inZ^3 \\;:\\; \\gcd…","labels":["thm:HeathBrownGeometry"],"detail_key":"p25"},{"id":"n30072","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30073","layer":"informal","project":"p25","title":"lem:EisensteinCrit","kind":"lemma","summary":"Let r\\geq 1 and let g\\in C[x] be a polynomial which has at least one root of multiplicity 1. Th…","labels":["lem:EisensteinCrit"],"detail_key":"p25"},{"id":"n30074","layer":"informal","project":"p25","title":"We may assume a factorisation g(x)=l_1(x)^e_1\\dots l_t(x)^e_t, with pairwise non-proporti…","kind":"proof","summary":"We may assume a factorisation g(x)=l_1(x)^e_1\\dots l_t(x)^e_t, with pairwise non-proportional l…","labels":[],"detail_key":"p25"},{"id":"n30075","layer":"informal","project":"p25","title":"thm:BombieriPilaforDet","kind":"theorem","summary":"Let f(x) be a C^\\infty function on a closed subinterval of [0,N], and suppose thatF (x, f ) = 0…","labels":["thm:BombieriPilaforDet"],"detail_key":"p25"},{"id":"n30076","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30077","layer":"informal","project":"p25","title":"thm:HeathBrownpadicDet","kind":"theorem","summary":"Let F\\in Z[x_1 ,\\ldots, x_n] be an absolutely irreducible polynomial of degree d, and let \\vare…","labels":["thm:HeathBrownpadicDet"],"detail_key":"p25"},{"id":"n30078","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30079","layer":"informal","project":"p25","title":"thm:QuadraticFormsEstimate","kind":"theorem","summary":"Given integers n,a_1,a_2\\neq0, there are at most O_\\varepsilon,D(|na_1a_2 X_1 X_2|^\\varepsilon)…","labels":["thm:QuadraticFormsEstimate"],"detail_key":"p25"},{"id":"n30080","layer":"informal","project":"p25","title":"Suppose n=a_1x_1^2+a_2x_2^2. Then a_1n=a_1^2x_1^2+a_1a_2x_2^2. Let D be the squarefree pa…","kind":"proof","summary":"Suppose n=a_1x_1^2+a_2x_2^2. Then a_1n=a_1^2x_1^2+a_1a_2x_2^2. Let D be the squarefree part of…","labels":[],"detail_key":"p25"},{"id":"n30081","layer":"informal","project":"p25","title":"lem:DivisorBoundQuadField","kind":"lemma","summary":"Let \\varepsilon>0. Let D\\ge1 be a squarefree integer, and set K=Q(\\sqrt-D). Then for all \\alpha…","labels":["lem:DivisorBoundQuadField"],"detail_key":"p25"},{"id":"n30082","layer":"informal","project":"p25","title":"Mimics the proof over the integers using the fundamental theorem of arithmetic, but with…","kind":"proof","summary":"Mimics the proof over the integers using the fundamental theorem of arithmetic, but with ideals…","labels":[],"detail_key":"p25"},{"id":"n30083","layer":"informal","project":"p25","title":"thm:BombierSchmidtforThueEqs","kind":"theorem","summary":"Given integers n,a_1,a_2\\neq0 and p\\ge3, there are at most O(p^1+\\omega(|n|)) many solutions (x…","labels":["thm:BombierSchmidtforThueEqs"],"detail_key":"p25"},{"id":"n30084","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30085","layer":"informal","project":"p25","title":"def:omega","kind":"definition","summary":"Let \\omega(n) denote the number of distinct prime factors of an integer n.","labels":["def:omega"],"detail_key":"p25"},{"id":"n30086","layer":"informal","project":"p25","title":"lem:omegaUpperBound","kind":"lemma","summary":"For any n\\ge2, we have \\omega(n) \\ll \\log(3n)/\\log\\log(3n).","labels":["lem:omegaUpperBound"],"detail_key":"p25"},{"id":"n30087","layer":"informal","project":"p25","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p25"},{"id":"n30088","layer":"informal","project":"p25","title":"lem:MixedDetMethods","kind":"lemma","summary":"Let \\varepsilon>0 and D\\geq 1 and assume that p,q,r\\in [1,D] are integers. Then \\[ N(X,Y,Z)\\ll_…","labels":["lem:MixedDetMethods"],"detail_key":"p25"},{"id":"n30089","layer":"informal","project":"p25","title":"We fix a choice of non-zero integer z\\in [-Z,Z], of which there are O(Z). When z is fixed…","kind":"proof","summary":"We fix a choice of non-zero integer z\\in [-Z,Z], of which there are O(Z). When z is fixed, the…","labels":[],"detail_key":"p25"},{"id":"n30090","layer":"informal","project":"p25","title":"lem:MixedThueMethods","kind":"lemma","summary":"Let \\varepsilon>0 and D\\geq 1 and assume that p=q,r\\in [1,D] are integers. Then \\[ N(X,Y,Z)\\ll_…","labels":["lem:MixedThueMethods"],"detail_key":"p25"},{"id":"n30091","layer":"informal","project":"p25","title":"Suppose now that p=q\\geq 2. Then, for given z\\in Z_\\neq 0, we are left with counting the…","kind":"proof","summary":"Suppose now that p=q\\geq 2. Then, for given z\\in Z_\\neq 0, we are left with counting the number…","labels":[],"detail_key":"p25"},{"id":"n30092","layer":"informal","project":"p25","title":"prop:DeterminantMethod","kind":"proposition","summary":"Let d\\geq 1, and let \\[ X_1,\\ldots, X_d, Y_1,\\ldots, Y_d,Z_1,\\ldots, Z_d\\geq 1. \\] Let c\\in (c_…","labels":["prop:DeterminantMethod"],"detail_key":"p25"},{"id":"n30093","layer":"informal","project":"p25","title":"Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) be a tuple counted by B_d(c,X,Y,Z).…","kind":"proof","summary":"Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) be a tuple counted by B_d(c,X,Y,Z). For a…","labels":[],"detail_key":"p25"},{"id":"n30094","layer":"informal","project":"p25","title":"prop:ThueEquations","kind":"proposition","summary":"Let d\\geq 1, and let \\[ X_1,\\ldots, X_d, Y_1,\\ldots, Y_d,Z_1,\\ldots, Z_d\\geq 1. \\] Let c\\in (c_…","labels":["prop:ThueEquations"],"detail_key":"p25"},{"id":"n30095","layer":"informal","project":"p25","title":"Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) be a tuple counted by B_d(c,X,Y,Z).…","kind":"proof","summary":"Let (x_1,\\ldots, x_d,y_1,\\ldots, y_d,z_1,\\ldots, z_d) be a tuple counted by B_d(c,X,Y,Z). We co…","labels":[],"detail_key":"p25"},{"id":"n30096","layer":"informal","project":"p25","title":"prop:TrivialBoundforSstar","kind":"proposition","summary":"Let \\alpha,\\beta,\\gamma>0, and let \\varepsilon>0 be fixed. Then \\[ S^*_\\alpha,\\beta,\\gamma(X)\\l…","labels":["prop:TrivialBoundforSstar"],"detail_key":"p25"},{"id":"n30097","layer":"informal","project":"p25","title":"This is an immediate consequence of~\\eqrefeq:trivial (with \\varepsilon^2/2 in place of \\v…","kind":"proof","summary":"This is an immediate consequence of~\\eqrefeq:trivial (with \\varepsilon^2/2 in place of \\varepsi…","labels":[],"detail_key":"p25"},{"id":"n30098","layer":"informal","project":"p25","title":"def:aibici","kind":"definition","summary":"Define a_i,b_i,c_i\\in R_\\geq0 via \\[ X_i=X^a_i, \\quad Y_i=X^b_i, \\quad Z_i=X^c_i, \\] for 1\\leq…","labels":["def:aibici"],"detail_key":"p25"},{"id":"n30099","layer":"informal","project":"p25","title":"lem:aibiciconstraints","kind":"lemma","summary":"\\sum_i\\leq d ia_i \\leq 1, \\quad \\sum_i\\leq d ib_i \\leq 1, \\quad 1-\\varepsilon^2\\leq \\sum_i\\leq…","labels":["lem:aibiciconstraints","eq:oat1"],"detail_key":"p25"},{"id":"n30100","layer":"informal","project":"p25","title":"Follows from~\\refprop:DiophantineReduction","kind":"proof","summary":"Follows from~\\refprop:DiophantineReduction","labels":[],"detail_key":"p25"},{"id":"n30101","layer":"informal","project":"p25","title":"lem:Trivialaibici","kind":"lemma","summary":"We have \\sum_i\\leq d (a_i+b_i)\\geq 0.66-\\varepsilon^2,\\quad \\sum_i\\leq d (a_i+c_i)\\geq 0.66-\\va…","labels":["lem:Trivialaibici","eq:oat4","eq:oat3"],"detail_key":"p25"},{"id":"n30102","layer":"informal","project":"p25","title":"Follows from~\\eqrefeq:goat and Proposition~\\refprop:TrivialBoundforSstar.","kind":"proof","summary":"Follows from~\\eqrefeq:goat and Proposition~\\refprop:TrivialBoundforSstar.","labels":[],"detail_key":"p25"},{"id":"n30103","layer":"informal","project":"p25","title":"def:nu","kind":"definition","summary":"It will be convenient to henceforth define \\nu=2\\varepsilon^2+\\frac\\log B_d(c,X,Y,Z)\\log X.","labels":["def:nu"],"detail_key":"p25"},{"id":"n30104","layer":"informal","project":"p25","title":"prop:Eqn4Point5","kind":"proposition","summary":"0.32 - \\delta \\leq \\sum_i\\leq d a_i ,~ \\sum_i\\leq d b_i ,~ \\sum_i\\leq d c_i \\leq 0.34 + \\delta-…","labels":["prop:Eqn4Point5","eq:oat5"],"detail_key":"p25"},{"id":"n30105","layer":"informal","project":"p25","title":"Indeed, suppose that \\sum_i\\leq d c_i> 0.34 + \\delta-\\varepsilon/2. Then~\\eqrefeq:oat3 im…","kind":"proof","summary":"Indeed, suppose that \\sum_i\\leq d c_i> 0.34 + \\delta-\\varepsilon/2. Then~\\eqrefeq:oat3 implies…","labels":[],"detail_key":"p25"},{"id":"n30106","layer":"informal","project":"p25","title":"Fourier bound","kind":"proposition","summary":"[Fourier bound] \\nu & < \\frac12\\Big(1+\\delta +\\sum_i\\leq d \\max(a_i,b_i) - \\max_m>1(a_m,b_m)\\Bi…","labels":["prop:FourierBound"],"detail_key":"p25"},{"id":"n30107","layer":"informal","project":"p25","title":"It follows from Proposition~\\refprop:FourierAnalysis that \\[ \\nu \\leq 3\\varepsilon^2+ \\fr…","kind":"proof","summary":"It follows from Proposition~\\refprop:FourierAnalysis that \\[ \\nu \\leq 3\\varepsilon^2+ \\frac12\\s…","labels":[],"detail_key":"p25"},{"id":"n30108","layer":"informal","project":"p25","title":"Geometry bound","kind":"proposition","summary":"[Geometry bound] \\nu < \\delta + \\min_I,I',I''\\subset [d] \\left( \\max\\left( 1 \\,,\\, \\sum_i\\in I…","labels":["prop:GeometryBound"],"detail_key":"p25"},{"id":"n30109","layer":"informal","project":"p25","title":"eq:Geoalt","kind":"proof","summary":"Applying Proposition~\\refprop:GeometryofNumbers, we obtain \\[ \\nu \\leq 3\\varepsilon^2+ \\min_I,I…","labels":["eq:Geoalt"],"detail_key":"p25"},{"id":"n30110","layer":"informal","project":"p25","title":"Determinant Bound","kind":"proposition","summary":"[Determinant Bound] \\nu < \\min_p,q\\ge1 \\left( 1+\\delta- a_p - b_q +\\min\\left(\\fraca_pq, \\fracb_…","labels":["prop:DeterminantBound"],"detail_key":"p25"},{"id":"n30111","layer":"informal","project":"p25","title":"Proposition~\\refprop:DeterminantMethod implies that \\[ \\nu\\leq 3\\varepsilon^2+ \\sum_i\\leq…","kind":"proof","summary":"Proposition~\\refprop:DeterminantMethod implies that \\[ \\nu\\leq 3\\varepsilon^2+ \\sum_i\\leq d (a_…","labels":[],"detail_key":"p25"},{"id":"n30112","layer":"informal","project":"p25","title":"Thue bound","kind":"proposition","summary":"[Thue bound] \\nu < 1 +\\delta- \\max_p\\ge2\\sum_p\\mid i(a_i+b_i).","labels":["prop:ThueBound"],"detail_key":"p25"},{"id":"n30113","layer":"informal","project":"p25","title":"This easily follows from Proposition~\\refprop:ThueEquations and~\\eqrefeq:oat3.","kind":"proof","summary":"This easily follows from Proposition~\\refprop:ThueEquations and~\\eqrefeq:oat3.","labels":[],"detail_key":"p25"},{"id":"n30114","layer":"informal","project":"p25","title":"def:delta","kind":"definition","summary":"It will be convenient to define constants \\delta_a,\\delta_b,\\delta_c via \\sum_i\\leq d a_i \\, =…","labels":["def:delta","eq:ai1/3"],"detail_key":"p25"},{"id":"n30115","layer":"informal","project":"p25","title":"lem:deltaBounds","kind":"lemma","summary":"We have \\delta_ab, \\delta_ac, \\delta_bc \\leq 0.00\\overline6 + \\varepsilon^2, -0.00\\bar 6 - \\del…","labels":["lem:deltaBounds","eq:robin2","eq:robin1","eq:robin3"],"detail_key":"p25"},{"id":"n30116","layer":"informal","project":"p25","title":"It follows from~\\eqrefeq:oat4 that \\delta_ab, \\delta_ac, \\delta_bc \\leq 0.00\\overline6 +…","kind":"proof","summary":"It follows from~\\eqrefeq:oat4 that \\delta_ab, \\delta_ac, \\delta_bc \\leq 0.00\\overline6 + \\varep…","labels":[],"detail_key":"p25"},{"id":"n30117","layer":"informal","project":"p25","title":"def:s","kind":"definition","summary":"Define s_i := a_i + b_i + c_i.","labels":["def:s"],"detail_key":"p25"},{"id":"n30118","layer":"informal","project":"p25","title":"prop:SpecialThue","kind":"proposition","summary":"To show \\nu \\leq 0.66, it suffices to assume a_j+b_j, \\,a_j+c_j,\\, b_j+c_j < 0.34+\\delta, for e…","labels":["prop:SpecialThue","eq:Thue","eq:Thue2","eq:sThue"],"detail_key":"p25"},{"id":"n30119","layer":"informal","project":"p25","title":"By the Thue bound, we have \\[ \\nu < 1 +\\delta- \\max_p\\ge2\\sum_p\\mid i (a_i+b_i), \\] and s…","kind":"proof","summary":"By the Thue bound, we have \\[ \\nu < 1 +\\delta- \\max_p\\ge2\\sum_p\\mid i (a_i+b_i), \\] and similar…","labels":[],"detail_key":"p25"},{"id":"n30120","layer":"informal","project":"p25","title":"lem:SpecialGeometry","kind":"lemma","summary":"We may assume s_1+s_2\\leq 0.34 +\\delta.","labels":["lem:SpecialGeometry","eq:SpecialGeometry"],"detail_key":"p25"},{"id":"n30121","layer":"informal","project":"p25","title":"If s_1+s_2> 0.34+\\delta then the Geometry bound and~\\eqrefeq:sThue imply that \\nu & \\le \\…","kind":"proof","summary":"If s_1+s_2> 0.34+\\delta then the Geometry bound and~\\eqrefeq:sThue imply that \\nu & \\le \\max\\bi…","labels":[],"detail_key":"p25"},{"id":"n30122","layer":"informal","project":"p25","title":"lem:SubSumGeometry","kind":"lemma","summary":"For any j\\ge3, allow \\tau_j to be an element \\tau_j\\in\\a_j,b_j,c_j, s_j, a_j+b_j,a_j+c_j,b_j+c_…","labels":["lem:SubSumGeometry","eq:taujdef","eq:Geotauj","eq:Geotau3"],"detail_key":"p25"},{"id":"n30123","layer":"informal","project":"p25","title":"The Geometry bound gives \\nu & \\le \\max\\big(1, s_1+2s_2+j\\tau_j\\big)-s_1-s_2-\\tau_j +\\del…","kind":"proof","summary":"The Geometry bound gives \\nu & \\le \\max\\big(1, s_1+2s_2+j\\tau_j\\big)-s_1-s_2-\\tau_j +\\delta\\\\ &…","labels":[],"detail_key":"p25"},{"id":"n30124","layer":"informal","project":"p25","title":"lem:ThreeLowerBounds","kind":"lemma","summary":"We have a_3 \\ge \\frac13-4\\delta_a - 3a_1 - 2a_2, a_3 \\ge \\frac13-\\frac52\\delta_a - 2a_1 - \\frac…","labels":["lem:ThreeLowerBounds","eq:black1","eq:s3>s12","eq:s3>s124"],"detail_key":"p25"},{"id":"n30125","layer":"informal","project":"p25","title":"Note that~\\eqrefeq:ai gives \\sum_i\\ge4a_i \\le \\sum_i\\ge4(i-3)a_i \\le 2a_1+a_2+3\\delta_a.…","kind":"proof","summary":"Note that~\\eqrefeq:ai gives \\sum_i\\ge4a_i \\le \\sum_i\\ge4(i-3)a_i \\le 2a_1+a_2+3\\delta_a. Simila…","labels":[],"detail_key":"p25"},{"id":"n30126","layer":"informal","project":"p25","title":"lem:Case1Basic","kind":"lemma","summary":"If s_2 \\geq 0.3, s_1 \\leq 0.04+\\delta and s_4 < 0.21 +\\frac32\\delta.","labels":["lem:Case1Basic","eq:s1"],"detail_key":"p25"},{"id":"n30127","layer":"informal","project":"p25","title":"Note that~\\eqrefeq:SpecialGeometry gives s_1 \\leq 0.34-s_2+\\delta \\le 0.04+\\delta, and~\\e…","kind":"proof","summary":"Note that~\\eqrefeq:SpecialGeometry gives s_1 \\leq 0.34-s_2+\\delta \\le 0.04+\\delta, and~\\eqrefeq…","labels":[],"detail_key":"p25"},{"id":"n30128","layer":"informal","project":"p25","title":"lem:Subcase1.1","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s_2 \\geq 0.3 and b_3\\le 0.34-s_1-s_2+\\delta.","labels":["lem:Subcase1.1"],"detail_key":"p25"},{"id":"n30129","layer":"informal","project":"p25","title":"Then b_3+c_3\\le 2b_3 &\\le 2(0.34-s_1-s_2+\\delta)\\\\ &\\le 0.68-2s_2 +2\\delta \\\\ &\\leq 0.33…","kind":"proof","summary":"Then b_3+c_3\\le 2b_3 &\\le 2(0.34-s_1-s_2+\\delta)\\\\ &\\le 0.68-2s_2 +2\\delta \\\\ &\\leq 0.33 -\\frac…","labels":[],"detail_key":"p25"},{"id":"n30130","layer":"informal","project":"p25","title":"lem:Subcase1.2","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s_2 \\geq 0.3 and b_3 > 0.34-s_1-s_2+\\delta.","labels":["lem:Subcase1.2"],"detail_key":"p25"},{"id":"n30131","layer":"informal","project":"p25","title":"eq:b3","kind":"proof","summary":"By~\\eqrefeq:Geotau3 we may assume b_3\\geq 0.33 -\\frac12s_2-\\frac12\\delta. By permuting the vari…","labels":["eq:b3","eq:2v-1.2"],"detail_key":"p25"},{"id":"n30132","layer":"informal","project":"p25","title":"lem:Case1","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s_2 \\geq 0.3.","labels":["lem:Case1"],"detail_key":"p25"},{"id":"n30133","layer":"informal","project":"p25","title":"Immediate from Lemmas~\\reflem:Subcase1.1 and~\\reflem:Subcase1.2.","kind":"proof","summary":"Immediate from Lemmas~\\reflem:Subcase1.1 and~\\reflem:Subcase1.2.","labels":[],"detail_key":"p25"},{"id":"n30134","layer":"informal","project":"p25","title":"lem:Case2Basic1","kind":"lemma","summary":"b_3 < 0.17+\\frac\\delta2.","labels":["lem:Case2Basic1","eq:2b3"],"detail_key":"p25"},{"id":"n30135","layer":"informal","project":"p25","title":"It follows from~\\eqrefeq:Thue that 2b_3 \\le a_3 + b_3 < 0.34+\\delta, whence b_3 < 0.17+\\f…","kind":"proof","summary":"It follows from~\\eqrefeq:Thue that 2b_3 \\le a_3 + b_3 < 0.34+\\delta, whence b_3 < 0.17+\\frac\\de…","labels":[],"detail_key":"p25"},{"id":"n30136","layer":"informal","project":"p25","title":"lem:Case2Basic2","kind":"lemma","summary":"a_3 \\geq 0.32 - 4 \\delta_s - s_1 - 2 \\delta.","labels":["lem:Case2Basic2","eq:Case2Basic2"],"detail_key":"p25"},{"id":"n30137","layer":"informal","project":"p25","title":"We have 0.17+\\frac\\delta2 \\leq 0.33-\\fracs_22-\\frac\\delta2 since s_2<0.3 and \\delta\\leq 0…","kind":"proof","summary":"We have 0.17+\\frac\\delta2 \\leq 0.33-\\fracs_22-\\frac\\delta2 since s_2<0.3 and \\delta\\leq 0.001.…","labels":[],"detail_key":"p25"},{"id":"n30138","layer":"informal","project":"p25","title":"lem:Case2","kind":"lemma","summary":"Assuming \\reflem:Subcase2.3, \\reflem:Subcase2.4, \\reflem:Subcase2.5, \\reflem:Subcase2.6, \\nu \\l…","labels":["lem:Case2"],"detail_key":"p25"},{"id":"n30139","layer":"informal","project":"p25","title":"Indeed 2.3, 2.4, 2.6 each define half-planes that cover [0,1]^2\\setminus T, for the close…","kind":"proof","summary":"Indeed 2.3, 2.4, 2.6 each define half-planes that cover [0,1]^2\\setminus T, for the closed tria…","labels":[],"detail_key":"p25"},{"id":"n30140","layer":"informal","project":"p25","title":"lem:Subcase2.1","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s_2<0.3 and a_3\\geq 0.32.","labels":["lem:Subcase2.1"],"detail_key":"p25"},{"id":"n30141","layer":"informal","project":"p25","title":"By~\\eqrefeq:Thue we have b_3,c_3\\le 0.34+\\delta-a_3\\le 0.02+\\delta. Let m_i=\\min(b_i,c_i)…","kind":"proof","summary":"By~\\eqrefeq:Thue we have b_3,c_3\\le 0.34+\\delta-a_3\\le 0.02+\\delta. Let m_i=\\min(b_i,c_i), M_i=…","labels":[],"detail_key":"p25"},{"id":"n30142","layer":"informal","project":"p25","title":"lem:Subcase2.2","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s.2 < 0.3 and b_3+c_3<0.33-\\fracs_22-\\frac\\delta2.","labels":["lem:Subcase2.2"],"detail_key":"p25"},{"id":"n30143","layer":"informal","project":"p25","title":"Then by~\\eqrefeq:Geotau3 we may assume \\tau_3=b_3+c_3<0.34-s_1-s_2+\\delta. By~\\eqrefeq:s3…","kind":"proof","summary":"Then by~\\eqrefeq:Geotau3 we may assume \\tau_3=b_3+c_3<0.34-s_1-s_2+\\delta. By~\\eqrefeq:s3>s124…","labels":[],"detail_key":"p25"},{"id":"n30144","layer":"informal","project":"p25","title":"lem:Subcase2.3","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s.2 < 0.3 and 4s_1+3s_2>0.71.","labels":["lem:Subcase2.3"],"detail_key":"p25"},{"id":"n30145","layer":"informal","project":"p25","title":"Then the inequalities b_3,c_3 \\le 0.34-s_1-s_2+ \\delta give \\[ b_3+c_3<0.68-2(s_1+s_2)+2\\…","kind":"proof","summary":"Then the inequalities b_3,c_3 \\le 0.34-s_1-s_2+ \\delta give \\[ b_3+c_3<0.68-2(s_1+s_2)+2\\delta<…","labels":[],"detail_key":"p25"},{"id":"n30146","layer":"informal","project":"p25","title":"lem:Subcase2.4","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s.2 < 0.3 and 4s_1+s_2<0.4.","labels":["lem:Subcase2.4"],"detail_key":"p25"},{"id":"n30147","layer":"informal","project":"p25","title":"In this case,~\\eqrefeq:Thue and~\\eqrefeq:Case2Basic2 give \\[ b_3,c_3 \\le 0.34-a_3+\\delta…","kind":"proof","summary":"In this case,~\\eqrefeq:Thue and~\\eqrefeq:Case2Basic2 give \\[ b_3,c_3 \\le 0.34-a_3+\\delta \\le 0.…","labels":[],"detail_key":"p25"},{"id":"n30148","layer":"informal","project":"p25","title":"lem:Subcase2.5","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s.2 < 0.3 and 0.066\\leq s_2\\leq 0.204.","labels":["lem:Subcase2.5"],"detail_key":"p25"},{"id":"n30149","layer":"informal","project":"p25","title":"It follows from~\\eqrefeq:robin3 and~\\eqrefeq:Case2Basic2 that \\[ a_3 \\geq 0.32-4\\delta_s-…","kind":"proof","summary":"It follows from~\\eqrefeq:robin3 and~\\eqrefeq:Case2Basic2 that \\[ a_3 \\geq 0.32-4\\delta_s-s_1 -2…","labels":[],"detail_key":"p25"},{"id":"n30150","layer":"informal","project":"p25","title":"lem:Subcase2.6","kind":"lemma","summary":"\\nu \\leq 0.66 in the case s_2<0.3 and 2s_1-s_2>0.025.","labels":["lem:Subcase2.6"],"detail_key":"p25"},{"id":"n30151","layer":"informal","project":"p25","title":"eq:tau3S6","kind":"proof","summary":"In this case we note that the intervals in \\eqrefeq:Geotau3 overlap, since \\delta\\leq 0.001. He…","labels":["eq:tau3S6","eq:nuleqM2"],"detail_key":"p25"},{"id":"n30152","layer":"informal","project":"p25","title":"thm:ABCExponentNu","kind":"theorem","summary":"\\nu \\le 0.66.","labels":["thm:ABCExponentNu"],"detail_key":"p25"},{"id":"n30153","layer":"informal","project":"p25","title":"Immediate from Lemmas~\\reflem:Case1 and~\\reflem:Case2.","kind":"proof","summary":"Immediate from Lemmas~\\reflem:Case1 and~\\reflem:Case2.","labels":[],"detail_key":"p25"},{"id":"n30154","layer":"informal","project":"p25","title":"Proof of Theorem\\refthm:BdExceptionalBound","kind":"proof","summary":"[Proof of Theorem\\refthm:BdExceptionalBound]","labels":[],"detail_key":"p25"},{"id":"n30155","layer":"formal","project":"p25","title":"B","kind":"def","summary":"(d : Nat) → (Fin 3 → Nat) → (Fin d → Nat) → (Fin d → Nat) → (Fin d → Nat) → Nat","labels":[],"detail_key":"p25","name":"B","module":"ABCExceptions.Section2"},{"id":"n30156","layer":"formal","project":"p25","title":"B_finset","kind":"def","summary":"(d : Nat) → (Fin 3 → Nat) → (Fin d → Nat) → (Fin d → Nat) → (Fin d → Nat) → Finset (Prod (Fin d…","labels":[],"detail_key":"p25","name":"B_finset","module":"ABCExceptions.Section2"},{"id":"n30157","layer":"formal","project":"p25","title":"countTriples","kind":"def","summary":"Real → Nat → Nat","labels":[],"detail_key":"p25","name":"countTriples","module":"ABCExceptions.Section2"},{"id":"n30158","layer":"formal","project":"p25","title":"countTriples_isBigO_dyadicSup","kind":"theorem","summary":"∀ (ε : Real), Asymptotics.IsBigO Filter.atTop (fun X => ↑(countTriples ε X)) fun X => HMul.hMul…","labels":[],"detail_key":"p25","name":"countTriples_isBigO_dyadicSup","module":"ABCExceptions.Section2"},{"id":"n30159","layer":"formal","project":"p25","title":"countTriples_le_log_pow_mul_sup","kind":"theorem","summary":"∀ (ε : Real) (X : Nat), LE.le (countTriples ε X) (HMul.hMul (HPow.hPow (HAdd.hAdd (Nat.log 2 X)…","labels":[],"detail_key":"p25","name":"countTriples_le_log_pow_mul_sup","module":"ABCExceptions.Section2"},{"id":"n30160","layer":"formal","project":"p25","title":"exists_nice_factorization'","kind":"theorem","summary":"∀ ε : Real, LT.lt 0 ε → LT.lt ε (1 / 2) → ∀ d : Nat, Eq d (Nat.floor (HMul.hMul 10 (HPow.hPow (…","labels":[],"detail_key":"p25","name":"exists_nice_factorization'","module":"ABCExceptions.Section2"},{"id":"n30161","layer":"formal","project":"p25","title":"refinedCountTriplesStar","kind":"def","summary":"Real → Real → Real → Nat → Nat","labels":[],"detail_key":"p25","name":"refinedCountTriplesStar","module":"ABCExceptions.Section2"},{"id":"n30162","layer":"formal","project":"p25","title":"refinedCountTriplesStar_isBigO_B","kind":"theorem","summary":"∀ α β γ : Real x : Nat, LE.le 2 x → ∀ ε : Real, LT.lt 0 ε → LT.lt ε (1 / 2) → Exists fun s => A…","labels":[],"detail_key":"p25","name":"refinedCountTriplesStar_isBigO_B","module":"ABCExceptions.Section2"},{"id":"n30163","layer":"formal","project":"p25","title":"Bound4Point3","kind":"def","summary":"Nat → Real → (Nat → Real) → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"Bound4Point3","module":"ABCExceptions.Section4"},{"id":"n30164","layer":"formal","project":"p25","title":"Bound4Point4","kind":"def","summary":"Nat → Real → Real → (Nat → Real) → (Nat → Real) → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"Bound4Point4","module":"ABCExceptions.Section4"},{"id":"n30165","layer":"formal","project":"p25","title":"Bound4Point5","kind":"inductive","summary":"Nat → Real → Real → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"Bound4Point5","module":"ABCExceptions.Section4"},{"id":"n30166","layer":"formal","project":"p25","title":"DeterminantBound","kind":"def","summary":"Nat → Real → Real → (Nat → Real) → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"DeterminantBound","module":"ABCExceptions.Section4"},{"id":"n30167","layer":"formal","project":"p25","title":"FourierBound","kind":"def","summary":"Nat → Real → Real → (Nat → Real) → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"FourierBound","module":"ABCExceptions.Section4"},{"id":"n30168","layer":"formal","project":"p25","title":"GeometryBound","kind":"def","summary":"Nat → Real → Real → (Nat → Real) → (Nat → Real) → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"GeometryBound","module":"ABCExceptions.Section4"},{"id":"n30169","layer":"formal","project":"p25","title":"S","kind":"def","summary":"(Nat → Real) → (Nat → Real) → (Nat → Real) → Nat → Real","labels":[],"detail_key":"p25","name":"S","module":"ABCExceptions.Section4"},{"id":"n30170","layer":"formal","project":"p25","title":"baseAssumptions","kind":"inductive","summary":"Nat → (Nat → Real) → Prop","labels":[],"detail_key":"p25","name":"baseAssumptions","module":"ABCExceptions.Section4"},{"id":"n30171","layer":"formal","project":"p25","title":"bound_4_point_10_lower","kind":"theorem","summary":"∀ d : Nat δ ε : Real a b c : Nat → Real, LT.lt 0 ε → Bound4Point4 d δ ε a b c → LT.lt (Neg.neg…","labels":[],"detail_key":"p25","name":"bound_4_point_10_lower","module":"ABCExceptions.Section4"},{"id":"n30172","layer":"formal","project":"p25","title":"bound_4_point_10_upper","kind":"theorem","summary":"∀ d : Nat ε : Real a b c : Nat → Real, LT.lt 0 ε → LE.le ε (2 / 3) → Bound4Point3 d ε a b → Bou…","labels":[],"detail_key":"p25","name":"bound_4_point_10_upper","module":"ABCExceptions.Section4"},{"id":"n30173","layer":"formal","project":"p25","title":"bound_4_point_12","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b : Nat → Real, baseAssumptions d a → baseAssumptions d b → ThueBound d…","labels":[],"detail_key":"p25","name":"bound_4_point_12","module":"ABCExceptions.Section4"},{"id":"n30174","layer":"formal","project":"p25","title":"bound_4_point_13","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b : Nat → Real, baseAssumptions d a → baseAssumptions d b → ThueBound d…","labels":[],"detail_key":"p25","name":"bound_4_point_13","module":"ABCExceptions.Section4"},{"id":"n30175","layer":"formal","project":"p25","title":"bound_4_point_14_general","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssump…","labels":[],"detail_key":"p25","name":"bound_4_point_14_general","module":"ABCExceptions.Section4"},{"id":"n30176","layer":"formal","project":"p25","title":"bound_4_point_14_two_four","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssump…","labels":[],"detail_key":"p25","name":"bound_4_point_14_two_four","module":"ABCExceptions.Section4"},{"id":"n30177","layer":"formal","project":"p25","title":"bound_4_point_15","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"bound_4_point_15","module":"ABCExceptions.Section4"},{"id":"n30178","layer":"formal","project":"p25","title":"bound_4_point_18","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, Bound4Point4 d δ ε a b c → GeometryBound d ε ν a b c…","labels":[],"detail_key":"p25","name":"bound_4_point_18","module":"ABCExceptions.Section4"},{"id":"n30179","layer":"formal","project":"p25","title":"bound_4_point_19_first","kind":"theorem","summary":"∀ d : Nat a : Nat → Real, baseAssumptions d a → LE.le 3 d → LE.le (HSub.hSub (HSub.hSub (HSub.h…","labels":[],"detail_key":"p25","name":"bound_4_point_19_first","module":"ABCExceptions.Section4"},{"id":"n30180","layer":"formal","project":"p25","title":"bound_4_point_19_second","kind":"theorem","summary":"∀ d : Nat a : Nat → Real, baseAssumptions d a → LE.le 4 d → LE.le (HSub.hSub (HSub.hSub (HSub.h…","labels":[],"detail_key":"p25","name":"bound_4_point_19_second","module":"ABCExceptions.Section4"},{"id":"n30181","layer":"formal","project":"p25","title":"bound_4_point_20","kind":"theorem","summary":"∀ d : Nat a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssumptions d c →…","labels":[],"detail_key":"p25","name":"bound_4_point_20","module":"ABCExceptions.Section4"},{"id":"n30182","layer":"formal","project":"p25","title":"bound_4_point_21","kind":"theorem","summary":"∀ d : Nat a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssumptions d c →…","labels":[],"detail_key":"p25","name":"bound_4_point_21","module":"ABCExceptions.Section4"},{"id":"n30183","layer":"formal","project":"p25","title":"bound_4_point_22","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"bound_4_point_22","module":"ABCExceptions.Section4"},{"id":"n30184","layer":"formal","project":"p25","title":"bound_4_point_24","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b : Nat → Real, baseAssumptions d a → baseAssumptions d b → FourierBound…","labels":[],"detail_key":"p25","name":"bound_4_point_24","module":"ABCExceptions.Section4"},{"id":"n30185","layer":"formal","project":"p25","title":"bound_4_point_25","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b : Nat → Real, baseAssumptions d a → baseAssumptions d b → ThueBound d…","labels":[],"detail_key":"p25","name":"bound_4_point_25","module":"ABCExceptions.Section4"},{"id":"n30186","layer":"formal","project":"p25","title":"bound_4_point_8","kind":"theorem","summary":"∀ d : Nat ε : Real a b : Nat → Real, Bound4Point3 d ε a b → LE.le (HAdd.hAdd (δ_ d a) (δ_ d b))…","labels":[],"detail_key":"p25","name":"bound_4_point_8","module":"ABCExceptions.Section4"},{"id":"n30187","layer":"formal","project":"p25","title":"bound_4_point_9_lower","kind":"theorem","summary":"∀ d : Nat δ ε : Real, LT.lt 0 ε → ∀ (f : Nat → Real), Bound4Point5 d δ ε f → LE.le (HSub.hSub (…","labels":[],"detail_key":"p25","name":"bound_4_point_9_lower","module":"ABCExceptions.Section4"},{"id":"n30188","layer":"formal","project":"p25","title":"bound_4_point_9_upper","kind":"theorem","summary":"∀ d : Nat δ ε : Real, LT.lt 0 ε → ∀ (f : Nat → Real), Bound4Point5 d δ ε f → LE.le (δ_ d f) (HA…","labels":[],"detail_key":"p25","name":"bound_4_point_9_upper","module":"ABCExceptions.Section4"},{"id":"n30189","layer":"formal","project":"p25","title":"case_1","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_1","module":"ABCExceptions.Section4"},{"id":"n30190","layer":"formal","project":"p25","title":"case_1_helper","kind":"theorem","summary":"∀ d : Nat δ ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssump…","labels":[],"detail_key":"p25","name":"case_1_helper","module":"ABCExceptions.Section4"},{"id":"n30191","layer":"formal","project":"p25","title":"case_2","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2","module":"ABCExceptions.Section4"},{"id":"n30192","layer":"formal","project":"p25","title":"case_2_subcase_1","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_1","module":"ABCExceptions.Section4"},{"id":"n30193","layer":"formal","project":"p25","title":"case_2_subcase_2","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_2","module":"ABCExceptions.Section4"},{"id":"n30194","layer":"formal","project":"p25","title":"case_2_subcase_3","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_3","module":"ABCExceptions.Section4"},{"id":"n30195","layer":"formal","project":"p25","title":"case_2_subcase_4","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_4","module":"ABCExceptions.Section4"},{"id":"n30196","layer":"formal","project":"p25","title":"case_2_subcase_5","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_5","module":"ABCExceptions.Section4"},{"id":"n30197","layer":"formal","project":"p25","title":"case_2_subcase_6","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"case_2_subcase_6","module":"ABCExceptions.Section4"},{"id":"n30198","layer":"formal","project":"p25","title":"delta_s","kind":"def","summary":"Nat → (Nat → Real) → (Nat → Real) → (Nat → Real) → Real","labels":[],"detail_key":"p25","name":"delta_s","module":"ABCExceptions.Section4"},{"id":"n30199","layer":"formal","project":"p25","title":"subcase_1_point_1","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"subcase_1_point_1","module":"ABCExceptions.Section4"},{"id":"n30200","layer":"formal","project":"p25","title":"subcase_1_point_2","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"subcase_1_point_2","module":"ABCExceptions.Section4"},{"id":"n30201","layer":"formal","project":"p25","title":"thm_4_point_3","kind":"theorem","summary":"∀ d : Nat δ ε ν : Real a b c : Nat → Real, baseAssumptions d a → baseAssumptions d b → baseAssu…","labels":[],"detail_key":"p25","name":"thm_4_point_3","module":"ABCExceptions.Section4"},{"id":"n30202","layer":"formal","project":"p25","title":"δ_","kind":"def","summary":"Nat → (Nat → Real) → Real","labels":[],"detail_key":"p25","name":"δ_","module":"ABCExceptions.Section4"},{"id":"n30203","layer":"informal","project":"p26","title":"\\textttid\\_mul\\_geom\\_sum","kind":"definition","summary":"[\\textttid\\_mul\\_geom\\_sum] The expression \\sum_k=1^n k v^k.","labels":["id_mul_geom_sum"],"detail_key":"p26"},{"id":"n30204","layer":"informal","project":"p26","title":"\\textttIa","kind":"definition","summary":"[\\textttIa] Present value of an increasing annuity: \\sum_k=1^n k v^k where v=1/(1+i).","labels":["Ia"],"detail_key":"p26"},{"id":"n30205","layer":"informal","project":"p26","title":"\\textttgeom\\_sum","kind":"definition","summary":"[\\textttgeom\\_sum] geom\\_sum is the function g : N \\to R \\to R given by g(n,v)= \\sum_k=1^n v ^…","labels":["geom_sum"],"detail_key":"p26"},{"id":"n30206","layer":"informal","project":"p26","title":"\\textttbond\\_price\\_sum","kind":"definition","summary":"[\\textttbond\\_price\\_sum] The bond price sum is r \\cdot \\textgeom\\_sum (n, v) + v ^ n.","labels":["bond_price_sum"],"detail_key":"p26"},{"id":"n30207","layer":"informal","project":"p26","title":"\\textttbond\\_price","kind":"definition","summary":"[\\textttbond\\_price] The bond price is the bond price sum with v=1/(1+i).","labels":["bond_price"],"detail_key":"p26"},{"id":"n30208","layer":"informal","project":"p26","title":"\\textttD","kind":"definition","summary":"[\\textttD] Macaulay duration:\\\\ \\verb!(r * Ia n i + n * (1+i)!^-n\\verb!) / bond_price n i r!","labels":["D"],"detail_key":"p26"},{"id":"n30209","layer":"informal","project":"p26","title":"\\textttid\\_mul\\_geom\\_sum\\_formula","kind":"lemma","summary":"[\\textttid\\_mul\\_geom\\_sum\\_formula] Given x\\in R with x \\ne 1 and n\\in N, we have \\[ \\sum_k=1^…","labels":["id_mul_geom_sum_formula"],"detail_key":"p26"},{"id":"n30210","layer":"informal","project":"p26","title":"A well-known formula from undergraduate mathematics.","kind":"proof","summary":"A well-known formula from undergraduate mathematics.","labels":[],"detail_key":"p26"},{"id":"n30211","layer":"informal","project":"p26","title":"\\textttannuity.a","kind":"definition","summary":"[\\textttannuity.a] The present value of annuity function a : N \\to R \\to R is defined by \\[ a_\\…","labels":["a"],"detail_key":"p26"},{"id":"n30212","layer":"informal","project":"p26","title":"\\textttannuity.a\\_formula","kind":"definition","summary":"[\\textttannuity.a\\_formula] The expression for the present value of annuity that is well-define…","labels":["a_formula"],"detail_key":"p26"},{"id":"n30213","layer":"informal","project":"p26","title":"\\texttta\\_eq\\_a\\_formula","kind":"lemma","summary":"[\\texttta\\_eq\\_a\\_formula] The value of the expression for the present value of annuity that is…","labels":["a_eq_a_formula"],"detail_key":"p26"},{"id":"n30214","layer":"informal","project":"p26","title":"Apply the formula \\sum_i=0^n-1 x^i = \\fracx^n - 1x - 1 to x being the discount factor (1+…","kind":"proof","summary":"Apply the formula \\sum_i=0^n-1 x^i = \\fracx^n - 1x - 1 to x being the discount factor (1+i)^-1.","labels":[],"detail_key":"p26"},{"id":"n30215","layer":"informal","project":"p26","title":"\\textttCPT\\_PV","kind":"definition","summary":"[\\textttCPT\\_PV] Defines a function to compute PV, as\\\\ \\verb! - PMT * (annuity.a N (IY / 100))…","labels":["CPT_PV"],"detail_key":"p26"},{"id":"n30216","layer":"informal","project":"p26","title":"\\textttCPT\\_PMT","kind":"definition","summary":"[\\textttCPT\\_PMT] Defines a function to compute PMT, as\\\\ \\verb!(-PV - FV * (1 + IY / 100)!^-N\\…","labels":["CPT_PMT"],"detail_key":"p26"},{"id":"n30217","layer":"informal","project":"p26","title":"\\textttCPT\\_IY","kind":"definition","summary":"[\\textttCPT\\_IY] Defines a function to compute IY.","labels":["CPT_IY"],"detail_key":"p26"},{"id":"n30218","layer":"informal","project":"p26","title":"\\textttCPT\\_N","kind":"definition","summary":"[\\textttCPT\\_N] Defines a function to compute N, as\\\\ \\verb! (log ((PV * (IY / 100) + PMT) / (P…","labels":["CPT_N"],"detail_key":"p26"},{"id":"n30219","layer":"informal","project":"p26","title":"[\\textttCPT\\_FV]","kind":"definition","summary":"[[\\textttCPT\\_FV]] Defines a function to compute FV, as\\\\ \\verb!(- PV - PMT * (annuity.a N (IY…","labels":["CPT_FV"],"detail_key":"p26"},{"id":"n30220","layer":"informal","project":"p26","title":"\\textttPMT\\_eq\\_CPT\\_PMT","kind":"lemma","summary":"[\\textttPMT\\_eq\\_CPT\\_PMT] If \\IY > -100 and \\N \\ne 0 then \\PMT=\\CPTPMT.","labels":["PMT_eq_CPT_PMT"],"detail_key":"p26"},{"id":"n30221","layer":"informal","project":"p26","title":"An exercise in algebraic manipulation.","kind":"proof","summary":"An exercise in algebraic manipulation.","labels":[],"detail_key":"p26"},{"id":"n30222","layer":"informal","project":"p26","title":"\\textttIY\\_eq\\_CPT\\_IY","kind":"lemma","summary":"[\\textttIY\\_eq\\_CPT\\_IY] If \\N \\ne 0, \\PMT \\ge 0, 0 \\le \\PV + \\PMT \\cdot \\N + \\FV, \\PV < 0, \\FV…","labels":["IY_eq_CPT_IY"],"detail_key":"p26"},{"id":"n30223","layer":"informal","project":"p26","title":"Use the Intermediate Value Theorem and strict monotonicity in the Annuity Equation as a f…","kind":"proof","summary":"Use the Intermediate Value Theorem and strict monotonicity in the Annuity Equation as a functio…","labels":[],"detail_key":"p26"},{"id":"n30224","layer":"informal","project":"p26","title":"\\textttPV\\_eq\\_CPT\\_PV","kind":"lemma","summary":"[\\textttPV\\_eq\\_CPT\\_PV] Assume the annuity equation holds for \\IY, \\PMT, \\PV, \\FV and \\N. Then…","labels":["PV_eq_CPT_PV"],"detail_key":"p26"},{"id":"n30225","layer":"informal","project":"p26","title":"Another exercise in algebraic manipulation.","kind":"proof","summary":"Another exercise in algebraic manipulation.","labels":[],"detail_key":"p26"},{"id":"n30226","layer":"informal","project":"p26","title":"\\textttFV\\_eq\\_CPT\\_FV","kind":"lemma","summary":"[\\textttFV\\_eq\\_CPT\\_FV] Assume the annuity equation holds for \\IY, \\PMT, \\PV, \\FV and \\N, and…","labels":["FV_eq_CPT_FV"],"detail_key":"p26"},{"id":"n30227","layer":"informal","project":"p26","title":"Another exercise in algebraic manipulation.","kind":"proof","summary":"Another exercise in algebraic manipulation.","labels":[],"detail_key":"p26"},{"id":"n30228","layer":"informal","project":"p26","title":"\\textttN\\_eq\\_CPT\\_N","kind":"lemma","summary":"[\\textttN\\_eq\\_CPT\\_N] Assume the annuity equation holds for \\IY, \\PMT, \\PV, \\FV and \\N. Assume…","labels":["N_eq_CPT_N"],"detail_key":"p26"},{"id":"n30229","layer":"informal","project":"p26","title":"An exercise is algebraic manipulation.","kind":"proof","summary":"An exercise is algebraic manipulation.","labels":[],"detail_key":"p26"},{"id":"n30230","layer":"informal","project":"p26","title":"\\textttannuity\\_equation\\_unique\\_solvability","kind":"theorem","summary":"[\\textttannuity\\_equation\\_unique\\_solvability] Given real numbers \\IY>0, \\PMT\\ge 0, \\PV<0, \\FV…","labels":["annuity_equation_unique_solvability"],"detail_key":"p26"},{"id":"n30231","layer":"informal","project":"p26","title":"We combine the results of Lemmas \\refPMT_eq_CPT_PMT, \\refIY_eq_CPT_IY, \\refPV_eq_CPT_PV,…","kind":"proof","summary":"We combine the results of Lemmas \\refPMT_eq_CPT_PMT, \\refIY_eq_CPT_IY, \\refPV_eq_CPT_PV, \\refFV…","labels":[],"detail_key":"p26"},{"id":"n30232","layer":"informal","project":"p26","title":"\\textttAriMagic.exists\\_pos\\_root\\_of\\_limits","kind":"lemma","summary":"[\\textttAriMagic.exists\\_pos\\_root\\_of\\_limits] If a continuous function starts positive at 0 a…","labels":["AriMagic.exists_pos_root_of_limits"],"detail_key":"p26"},{"id":"n30233","layer":"informal","project":"p26","title":"A simple application of the Intermediate Value Theorem, formalized by Aristotle.","kind":"proof","summary":"A simple application of the Intermediate Value Theorem, formalized by Aristotle.","labels":[],"detail_key":"p26"},{"id":"n30234","layer":"informal","project":"p26","title":"\\textttduration\\_equation","kind":"definition","summary":"[\\textttduration\\_equation] Defined to equal \\verb!d * annuity.bond_price n i r! \\\\ \\verb!- (r…","labels":["duration_equation"],"detail_key":"p26"},{"id":"n30235","layer":"informal","project":"p26","title":"\\textttAriMagic.unique\\_solution\\_n","kind":"lemma","summary":"[\\textttAriMagic.unique\\_solution\\_n] Given D>0, i>0, r>0, with D < 1 + 1/i, there is a unique…","labels":["AriMagic.unique_solution_n"],"detail_key":"p26"},{"id":"n30236","layer":"informal","project":"p26","title":"This is a subtle result, outsourced to Aristotle.","kind":"proof","summary":"This is a subtle result, outsourced to Aristotle.","labels":[],"detail_key":"p26"},{"id":"n30237","layer":"informal","project":"p26","title":"\\texttteq\\_CPT\\_I\\_of\\_D","kind":"theorem","summary":"[\\texttteq\\_CPT\\_I\\_of\\_D] If D>1, n \\ge 2, and r > 0 then we can uniquely compute the yield ra…","labels":["eq_CPT_I_of_D"],"detail_key":"p26"},{"id":"n30238","layer":"informal","project":"p26","title":"This proof was outsourced to Aristotle.","kind":"proof","summary":"This proof was outsourced to Aristotle.","labels":[],"detail_key":"p26"},{"id":"n30239","layer":"informal","project":"p26","title":"\\texttteq\\_CPT\\_N\\_of\\_D","kind":"theorem","summary":"[\\texttteq\\_CPT\\_N\\_of\\_D] Given n\\in N with n>1 and i,d,r positive real numbers, with d < 1 +…","labels":["eq_CPT_N_of_D"],"detail_key":"p26"},{"id":"n30240","layer":"informal","project":"p26","title":"This result is simply a wrapper for Lemma \\refAriMagic.unique_solution_n.","kind":"proof","summary":"This result is simply a wrapper for Lemma \\refAriMagic.unique_solution_n.","labels":[],"detail_key":"p26"},{"id":"n30241","layer":"informal","project":"p26","title":"\\textttchan\\_tse\\_exe\\_1\\_36","kind":"lemma","summary":"[\\textttchan\\_tse\\_exe\\_1\\_36] Assume that \\delta(t)=\\frac110(1+t)^3 and A(0)=100, that a(t)\\ne…","labels":["chan_tse_exe_1_36"],"detail_key":"p26"},{"id":"n30242","layer":"formal","project":"p26","title":"AriMagic.exists_pos_root_of_limits","kind":"theorem","summary":"∀ f : Real → Real L : Real, ContinuousOn f (Set.Ici 0) → LT.lt 0 (f 0) → Filter.Tendsto f Filte…","labels":[],"detail_key":"p26","name":"AriMagic.exists_pos_root_of_limits","module":"Interest.AristotleMagic"},{"id":"n30243","layer":"formal","project":"p26","title":"AriMagic.unique_solution_n","kind":"theorem","summary":"∀ i d r : Real, LT.lt 0 d → LT.lt 0 i → LT.lt 0 r → LT.lt d (HAdd.hAdd 1 (HDiv.hDiv 1 i)) → Exi…","labels":[],"detail_key":"p26","name":"AriMagic.unique_solution_n","module":"Interest.AristotleMagic"},{"id":"n30244","layer":"formal","project":"p26","title":"annuity.Ia","kind":"def","summary":"Nat → Real → Real","labels":[],"detail_key":"p26","name":"annuity.Ia","module":"Interest.NFM"},{"id":"n30245","layer":"formal","project":"p26","title":"annuity.bond_price","kind":"def","summary":"Nat → Real → Real → Real","labels":[],"detail_key":"p26","name":"annuity.bond_price","module":"Interest.NFM"},{"id":"n30246","layer":"formal","project":"p26","title":"annuity.bond_price_sum","kind":"def","summary":"Nat → Real → Real → Real","labels":[],"detail_key":"p26","name":"annuity.bond_price_sum","module":"Interest.NFM"},{"id":"n30247","layer":"formal","project":"p26","title":"annuity.duration_equation","kind":"def","summary":"Nat → Real → Real → Real → Prop","labels":[],"detail_key":"p26","name":"annuity.duration_equation","module":"Interest.NFM"},{"id":"n30248","layer":"formal","project":"p26","title":"annuity.geom_sum","kind":"def","summary":"Nat → Real → Real","labels":[],"detail_key":"p26","name":"annuity.geom_sum","module":"Interest.NFM"},{"id":"n30249","layer":"formal","project":"p26","title":"annuity.id_mul_geom_sum","kind":"def","summary":"Nat → Real → Real","labels":[],"detail_key":"p26","name":"annuity.id_mul_geom_sum","module":"Interest.NFM"},{"id":"n30250","layer":"formal","project":"p26","title":"annuity.id_mul_geom_sum_formula","kind":"theorem","summary":"∀ (x : Real), Ne x 1 → ∀ (n : Nat), Eq (annuity.id_mul_geom_sum n x) (HDiv.hDiv (HMul.hMul x (H…","labels":[],"detail_key":"p26","name":"annuity.id_mul_geom_sum_formula","module":"Interest.NFM"},{"id":"n30251","layer":"formal","project":"p26","title":"annuity_equation_unique_solvability","kind":"theorem","summary":"∀ IY PMT PV FV : Real N : Nat (hann : annuity_equation IY PMT PV FV N) (hPMT : GE.ge PMT 0) (hP…","labels":[],"detail_key":"p26","name":"annuity_equation_unique_solvability","module":"Interest.NFM"},{"id":"n30252","layer":"formal","project":"p26","title":"D","kind":"def","summary":"Nat → Real → Real → Real","labels":[],"detail_key":"p26","name":"D","module":"Interest.NFM_duration"},{"id":"n30253","layer":"formal","project":"p26","title":"eq_CPT_I_of_D","kind":"theorem","summary":"∀ n : Nat, GE.ge n 2 → ∀ i : Real d r : Real, Membership.mem (Set.Ioo 1 ↑n) d → GT.gt r 0 → Exi…","labels":[],"detail_key":"p26","name":"eq_CPT_I_of_D","module":"Interest.NFM_duration"},{"id":"n30254","layer":"formal","project":"p26","title":"eq_CPT_N_of_D","kind":"theorem","summary":"∀ n : Nat, GT.gt (↑n) 0 → ∀ i d r : Real (hd : LT.lt 0 d) (hi : LT.lt 0 i) (hr : LT.lt 0 r) (hd…","labels":[],"detail_key":"p26","name":"eq_CPT_N_of_D","module":"Interest.NFM_duration"},{"id":"n30255","layer":"informal","project":"p27","title":"Partial derivative","kind":"definition","summary":"[Partial derivative] The partial derivative function. For \\(f : \\Vectm \\to \\Vectn\\), \\(\\pdiv\\,…","labels":["ax:pdiv"],"detail_key":"p27"},{"id":"n30256","layer":"informal","project":"p27","title":"Chain rule","kind":"theorem","summary":"[Chain rule] \\Assume \\item \\(f : \\Vectm \\to \\Vectn\\) is differentiable at \\(x\\) \\hfill[\\texttth…","labels":["ax:pdiv_comp"],"detail_key":"p27"},{"id":"n30257","layer":"informal","project":"p27","title":"\\emphSketch: reduce to Mathlib's chain rule for \\(\\fderiv\\), then decompose over the stan…","kind":"proof","summary":"\\emphSketch: reduce to Mathlib's chain rule for \\(\\fderiv\\), then decompose over the standard b…","labels":[],"detail_key":"p27"},{"id":"n30258","layer":"informal","project":"p27","title":"Sum rule","kind":"theorem","summary":"[Sum rule] \\Assume \\item \\(f, g : \\Vectm \\to \\Vectn\\) are both differentiable at \\(x\\) \\hfill[\\…","labels":["ax:pdiv_add"],"detail_key":"p27"},{"id":"n30259","layer":"informal","project":"p27","title":"\\item \\(\\pdiv(f + g)\\, x\\, i\\, j = \\fderiv_\\R\\,(f + g)\\, x\\, (\\basisVec_i)\\, j\\). \\pfDefi…","kind":"proof","summary":"\\item \\(\\pdiv(f + g)\\, x\\, i\\, j = \\fderiv_\\R\\,(f + g)\\, x\\, (\\basisVec_i)\\, j\\). \\pfDefinition…","labels":[],"detail_key":"p27"},{"id":"n30260","layer":"informal","project":"p27","title":"Product rule","kind":"theorem","summary":"[Product rule] \\Assume \\item \\(f, g : \\Vectm \\to \\Vectn\\) are both differentiable at \\(x\\) \\hfi…","labels":["ax:pdiv_mul"],"detail_key":"p27"},{"id":"n30261","layer":"informal","project":"p27","title":"\\item \\(\\pdiv(f \\odot g)\\, x\\, i\\, j = \\fderiv_\\R\\,(f \\cdot g)\\, x\\, (\\basisVec_i)\\, j\\).…","kind":"proof","summary":"\\item \\(\\pdiv(f \\odot g)\\, x\\, i\\, j = \\fderiv_\\R\\,(f \\cdot g)\\, x\\, (\\basisVec_i)\\, j\\). \\pfDe…","labels":[],"detail_key":"p27"},{"id":"n30262","layer":"informal","project":"p27","title":"Identity Jacobian","kind":"theorem","summary":"[Identity Jacobian] \\(\\pdiv(id)\\, x\\, i\\, j = \\delta_ij\\).","labels":["ax:pdiv_id"],"detail_key":"p27"},{"id":"n30263","layer":"informal","project":"p27","title":"Mechanical; see \\leandocrefProofs.pdiv\\_id.","kind":"proof","summary":"Mechanical; see \\leandocrefProofs.pdiv\\_id.","labels":[],"detail_key":"p27"},{"id":"n30264","layer":"informal","project":"p27","title":"Constant has zero Jacobian","kind":"theorem","summary":"[Constant has zero Jacobian]","labels":["ax:pdiv_const"],"detail_key":"p27"},{"id":"n30265","layer":"informal","project":"p27","title":"Mechanical; see \\leandocrefProofs.pdiv\\_const.","kind":"proof","summary":"Mechanical; see \\leandocrefProofs.pdiv\\_const.","labels":[],"detail_key":"p27"},{"id":"n30266","layer":"informal","project":"p27","title":"Gather / reindex Jacobian","kind":"theorem","summary":"[Gather / reindex Jacobian] Covers permutations, reshapes, slicing. Generalizes \\textttpdiv\\_id.","labels":["ax:pdiv_reindex"],"detail_key":"p27"},{"id":"n30267","layer":"informal","project":"p27","title":"Mechanical; see \\leandocrefProofs.pdiv\\_reindex.","kind":"proof","summary":"Mechanical; see \\leandocrefProofs.pdiv\\_reindex.","labels":[],"detail_key":"p27"},{"id":"n30268","layer":"informal","project":"p27","title":"Finite-sum rule","kind":"theorem","summary":"[Finite-sum rule] \\Assume \\item \\(S\\) is a finite index set with a function \\(f_s : \\Vectm \\to…","labels":["thm:pdiv_finset_sum"],"detail_key":"p27"},{"id":"n30269","layer":"informal","project":"p27","title":"\\emphSketch: induction on \\(S\\); the two-summand sum rule does each inductive step. \\item…","kind":"proof","summary":"\\emphSketch: induction on \\(S\\); the two-summand sum rule does each inductive step. \\item \\Case…","labels":[],"detail_key":"p27"},{"id":"n30270","layer":"informal","project":"p27","title":"VJP record","kind":"definition","summary":"[VJP record] For \\(f : \\Vectm \\to \\Vectn\\), \\(\\HasVJP\\, f\\) bundles a backward function \\(B : \\…","labels":["def:hasvjp"],"detail_key":"p27"},{"id":"n30271","layer":"informal","project":"p27","title":"VJP chain rule","kind":"theorem","summary":"[VJP chain rule] \\Assume \\item \\(B_f\\) is a correct backward function for \\(f\\) (\\(\\HasVJP\\, f\\…","labels":["thm:vjp_comp"],"detail_key":"p27"},{"id":"n30272","layer":"informal","project":"p27","title":"\\emphSketch: the composite backward is ``run \\(B_g\\), feed the result to \\(B_f\\)''; corre…","kind":"proof","summary":"\\emphSketch: the composite backward is ``run \\(B_g\\), feed the result to \\(B_f\\)''; correctness…","labels":[],"detail_key":"p27"},{"id":"n30273","layer":"informal","project":"p27","title":"Additive fan-in VJP","kind":"theorem","summary":"[Additive fan-in VJP] Used for residual connections. \\Assume \\item \\(B_f\\) is a correct backwar…","labels":["thm:biPath_has_vjp"],"detail_key":"p27"},{"id":"n30274","layer":"informal","project":"p27","title":"\\item Define \\(B(x, dy)_i := B_f(x, dy)_i + B_g(x, dy)_i\\). \\item \\Suffices for all \\(x\\)…","kind":"proof","summary":"\\item Define \\(B(x, dy)_i := B_f(x, dy)_i + B_g(x, dy)_i\\). \\item \\Suffices for all \\(x\\), \\(dy…","labels":[],"detail_key":"p27"},{"id":"n30275","layer":"informal","project":"p27","title":"Multiplicative fan-in VJP","kind":"theorem","summary":"[Multiplicative fan-in VJP] Used for Squeeze-and-Excitation. \\Assume \\item \\(B_f\\) is a correct…","labels":["thm:elemwiseProduct_has_vjp"],"detail_key":"p27"},{"id":"n30276","layer":"informal","project":"p27","title":"\\item Define \\(B(x, dy)_i := B_f\\bigl(x,\\; g(x) \\odot dy\\bigr)_i + B_g\\bigl(x,\\; f(x) \\od…","kind":"proof","summary":"\\item Define \\(B(x, dy)_i := B_f\\bigl(x,\\; g(x) \\odot dy\\bigr)_i + B_g\\bigl(x,\\; f(x) \\odot dy\\…","labels":[],"detail_key":"p27"},{"id":"n30277","layer":"informal","project":"p27","title":"Identity VJP","kind":"theorem","summary":"[Identity VJP]","labels":["thm:identity_has_vjp"],"detail_key":"p27"},{"id":"n30278","layer":"informal","project":"p27","title":"Mechanical; see \\leandocrefProofs.identity\\_has\\_vjp.","kind":"proof","summary":"Mechanical; see \\leandocrefProofs.identity\\_has\\_vjp.","labels":[],"detail_key":"p27"},{"id":"n30279","layer":"informal","project":"p27","title":"Dense Jacobian wrt input","kind":"theorem","summary":"[Dense Jacobian wrt input] For the dense layer \\(dense(W, b)\\, x = \\lambda j.\\; \\bigl(\\sum_i x_…","labels":["ax:pdiv_dense"],"detail_key":"p27"},{"id":"n30280","layer":"informal","project":"p27","title":"\\emphSketch: factor the layer as (finite sum of bilinear summands) + constant, distribute…","kind":"proof","summary":"\\emphSketch: factor the layer as (finite sum of bilinear summands) + constant, distribute \\(\\pd…","labels":[],"detail_key":"p27"},{"id":"n30281","layer":"informal","project":"p27","title":"Dense Jacobian wrt weight","kind":"theorem","summary":"[Dense Jacobian wrt weight] The symmetric counterpart of Theorem~\\refax:pdiv_dense, differentia…","labels":["ax:pdiv_dense_W"],"detail_key":"p27"},{"id":"n30282","layer":"informal","project":"p27","title":"\\emphSketch: same skeleton as Theorem~\\refax:pdiv_dense --- split, distribute, product ru…","kind":"proof","summary":"\\emphSketch: same skeleton as Theorem~\\refax:pdiv_dense --- split, distribute, product rule, co…","labels":[],"detail_key":"p27"},{"id":"n30283","layer":"informal","project":"p27","title":"ReLU Jacobian (guarded subgradient)","kind":"theorem","summary":"[ReLU Jacobian (guarded subgradient)] \\Assume \\item \\(x\\) is a smooth point of \\(ReLU\\): every…","labels":["ax:pdiv_relu"],"detail_key":"p27"},{"id":"n30284","layer":"informal","project":"p27","title":"\\emphSketch: near a smooth point, ReLU \\emphis a fixed linear map (each coordinate is com…","kind":"proof","summary":"\\emphSketch: near a smooth point, ReLU \\emphis a fixed linear map (each coordinate is committed…","labels":[],"detail_key":"p27"},{"id":"n30285","layer":"informal","project":"p27","title":"Softmax cross-entropy gradient","kind":"theorem","summary":"[Softmax cross-entropy gradient] Write \\(p := softmax(z)\\) and \\(CE(z, \\ell) := -\\log p_\\ell\\)…","labels":["ax:softmaxCE_grad"],"detail_key":"p27"},{"id":"n30286","layer":"informal","project":"p27","title":"\\emphSketch: chain rule on \\(-\\log \\circ (z \\mapsto p_\\ell)\\), with the softmax Jacobian…","kind":"proof","summary":"\\emphSketch: chain rule on \\(-\\log \\circ (z \\mapsto p_\\ell)\\), with the softmax Jacobian (prove…","labels":[],"detail_key":"p27"},{"id":"n30287","layer":"informal","project":"p27","title":"Dense VJP","kind":"theorem","summary":"[Dense VJP] \\(\\HasVJP\\, \\bigl(dense(W, b)\\bigr)\\): the input-gradient backward of a dense layer…","labels":["thm:dense_has_vjp"],"detail_key":"p27"},{"id":"n30288","layer":"informal","project":"p27","title":"\\item Define \\(B(x, dy) := W\\, dy\\), i.e.\\ \\(B(x, dy)_i = \\sum_j W_ij\\, dy_j\\). \\item \\Su…","kind":"proof","summary":"\\item Define \\(B(x, dy) := W\\, dy\\), i.e.\\ \\(B(x, dy)_i = \\sum_j W_ij\\, dy_j\\). \\item \\Suffices…","labels":[],"detail_key":"p27"},{"id":"n30289","layer":"informal","project":"p27","title":"Dense weight gradient is the outer product","kind":"theorem","summary":"[Dense weight gradient is the outer product] \\(dW = x \\otimes dy\\), with \\(F\\) and \\(\\varphi\\)…","labels":["thm:dense_weight_grad_correct"],"detail_key":"p27"},{"id":"n30290","layer":"informal","project":"p27","title":"\\item Each summand: \\(\\pdiv F\\, (flatten\\, W)\\, \\varphi(i, j)\\, k \\cdot dy_k = (\\textif k…","kind":"proof","summary":"\\item Each summand: \\(\\pdiv F\\, (flatten\\, W)\\, \\varphi(i, j)\\, k \\cdot dy_k = (\\textif k = j \\…","labels":[],"detail_key":"p27"},{"id":"n30291","layer":"informal","project":"p27","title":"Dense bias gradient is identity","kind":"theorem","summary":"[Dense bias gradient is identity] \\(db = dy\\). \\Prove for all \\(i\\): \\[ dy_i = \\sum_j \\pdiv\\big…","labels":["thm:dense_bias_grad_correct"],"detail_key":"p27"},{"id":"n30292","layer":"informal","project":"p27","title":"\\item \\(\\pdiv\\bigl(b' \\mapsto dense(W, b')\\, x\\bigr)\\, b\\, i\\, j = \\delta_ij\\). \\pfAs a f…","kind":"proof","summary":"\\item \\(\\pdiv\\bigl(b' \\mapsto dense(W, b')\\, x\\bigr)\\, b\\, i\\, j = \\delta_ij\\). \\pfAs a functio…","labels":[],"detail_key":"p27"},{"id":"n30293","layer":"informal","project":"p27","title":"ReLU VJP","kind":"definition","summary":"[ReLU VJP] \\textttnoncomputable def over the canonical pdiv-derived witness; \\textttHasVJP.corr…","labels":["thm:relu_has_vjp"],"detail_key":"p27"},{"id":"n30294","layer":"informal","project":"p27","title":"MLP composition VJP","kind":"definition","summary":"[MLP composition VJP] \\textttnoncomputable def over the canonical pdiv-derived witness; same sh…","labels":["ax:mlp_has_vjp"],"detail_key":"p27"},{"id":"n30295","layer":"informal","project":"p27","title":"Conv2d forward","kind":"definition","summary":"[Conv2d forward] Concrete \\(\\sum_c,kh,kw\\) cross-correlation with SAME padding. Codegen emits \\…","labels":["ax:conv2d"],"detail_key":"p27"},{"id":"n30296","layer":"informal","project":"p27","title":"3D VJP record","kind":"definition","summary":"[3D VJP record] The rank-3 analogue of Definition~\\refdef:hasvjp. For \\(f : Tensor3 \\to Tensor3…","labels":["def:hasvjp3"],"detail_key":"p27"},{"id":"n30297","layer":"informal","project":"p27","title":"Conv2d input VJP","kind":"theorem","summary":"[Conv2d input VJP] \\(\\HasVJPthree\\, \\bigl(conv2d(W, b)\\bigr)\\): the input backward of a convolu…","labels":["ax:conv2d_has_vjp3"],"detail_key":"p27"},{"id":"n30298","layer":"informal","project":"p27","title":"\\emphSketch: same split--distribute--product-rule--collapse skeleton as the dense Jacobia…","kind":"proof","summary":"\\emphSketch: same split--distribute--product-rule--collapse skeleton as the dense Jacobians (Th…","labels":[],"detail_key":"p27"},{"id":"n30299","layer":"informal","project":"p27","title":"Conv2d weight VJP","kind":"theorem","summary":"[Conv2d weight VJP] The transpose-trick formula. As with the dense weight Jacobian (Theorem~\\re…","labels":["ax:conv2d_weight_grad_has_vjp"],"detail_key":"p27"},{"id":"n30300","layer":"informal","project":"p27","title":"\\emphSketch: the dual of Theorem~\\refax:conv2d_has_vjp3 --- the same affine decomposition…","kind":"proof","summary":"\\emphSketch: the dual of Theorem~\\refax:conv2d_has_vjp3 --- the same affine decomposition, but…","labels":[],"detail_key":"p27"},{"id":"n30301","layer":"informal","project":"p27","title":"Conv2d bias VJP","kind":"theorem","summary":"[Conv2d bias VJP] Sum the cotangent over spatial dims per channel. \\Prove \\(\\HasVJP\\, \\bigl(b \\…","labels":["ax:conv2d_bias_grad_has_vjp"],"detail_key":"p27"},{"id":"n30302","layer":"informal","project":"p27","title":"\\item Define \\(B(b, dy)_o := \\sum_h_i, w_i dy_flat(o, h_i, w_i)\\). \\item \\Suffices for al…","kind":"proof","summary":"\\item Define \\(B(b, dy)_o := \\sum_h_i, w_i dy_flat(o, h_i, w_i)\\). \\item \\Suffices for all \\(b\\…","labels":[],"detail_key":"p27"},{"id":"n30303","layer":"informal","project":"p27","title":"MaxPool 2x2 stride 2 forward","kind":"definition","summary":"[MaxPool 2x2 stride 2 forward] Concrete four-way max over 2x2 windows.","labels":["ax:maxPool2"],"detail_key":"p27"},{"id":"n30304","layer":"informal","project":"p27","title":"MaxPool input VJP","kind":"definition","summary":"[MaxPool input VJP] \\textttnoncomputable def over the canonical pdiv-derived witness. \\textttHa…","labels":["ax:maxPool2_has_vjp3"],"detail_key":"p27"},{"id":"n30305","layer":"informal","project":"p27","title":"BN affine step Jacobian","kind":"theorem","summary":"[BN affine step Jacobian] For \\(bnAffine(\\gamma, \\beta)\\, v = \\lambda i.\\; \\gamma v_i + \\beta\\)…","labels":["ax:pdiv_bnAffine"],"detail_key":"p27"},{"id":"n30306","layer":"informal","project":"p27","title":"\\item \\(\\pdiv\\bigl(bnAffine(\\gamma, \\beta)\\bigr)\\, v\\, i\\, j = \\pdiv(y \\mapsto \\gamma \\cd…","kind":"proof","summary":"\\item \\(\\pdiv\\bigl(bnAffine(\\gamma, \\beta)\\bigr)\\, v\\, i\\, j = \\pdiv(y \\mapsto \\gamma \\cdot y)\\…","labels":[],"detail_key":"p27"},{"id":"n30307","layer":"informal","project":"p27","title":"BN centering Jacobian","kind":"theorem","summary":"[BN centering Jacobian] For \\(bnCentered\\, x = \\lambda j.\\; x_j - \\mu(x)\\), where \\(\\mu(x) = \\f…","labels":["ax:pdiv_bnCentered"],"detail_key":"p27"},{"id":"n30308","layer":"informal","project":"p27","title":"\\item \\(\\pdiv(bnCentered)\\, x\\, i\\, j = \\delta_ij + \\pdiv\\bigl(y \\mapsto -\\tfrac1n\\textst…","kind":"proof","summary":"\\item \\(\\pdiv(bnCentered)\\, x\\, i\\, j = \\delta_ij + \\pdiv\\bigl(y \\mapsto -\\tfrac1n\\textstyle\\su…","labels":[],"detail_key":"p27"},{"id":"n30309","layer":"informal","project":"p27","title":"BN inverse-stddev broadcast smoothness","kind":"theorem","summary":"[BN inverse-stddev broadcast smoothness] \\Assume \\item \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\va…","labels":["ax:bnIstdBroadcast_diff"],"detail_key":"p27"},{"id":"n30310","layer":"informal","project":"p27","title":"\\item \\(\\sigma^2(x) + \\varepsilon > 0\\) for every \\(x\\), in particular \\(\\neq 0\\). \\pf\\(\\…","kind":"proof","summary":"\\item \\(\\sigma^2(x) + \\varepsilon > 0\\) for every \\(x\\), in particular \\(\\neq 0\\). \\pf\\(\\sigma^…","labels":[],"detail_key":"p27"},{"id":"n30311","layer":"informal","project":"p27","title":"BN inverse-stddev broadcast Jacobian","kind":"theorem","summary":"[BN inverse-stddev broadcast Jacobian] \\Assume \\item \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\vare…","labels":["ax:pdiv_bnIstdBroadcast"],"detail_key":"p27"},{"id":"n30312","layer":"informal","project":"p27","title":"\\emphSketch: the one genuinely analytic Jacobian of the chapter --- a Fréchet-derivative…","kind":"proof","summary":"\\emphSketch: the one genuinely analytic Jacobian of the chapter --- a Fréchet-derivative chain…","labels":[],"detail_key":"p27"},{"id":"n30313","layer":"informal","project":"p27","title":"BN normalize 3-term VJP","kind":"theorem","summary":"[BN normalize 3-term VJP] \\Assume \\item \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\varepsilon\\)] \\Pr…","labels":["thm:bnNormalize_has_vjp"],"detail_key":"p27"},{"id":"n30314","layer":"informal","project":"p27","title":"\\emphSketch: product rule on \\(\\hatx = (x - \\mu) \\cdot s\\) merges the two elementary Jaco…","kind":"proof","summary":"\\emphSketch: product rule on \\(\\hatx = (x - \\mu) \\cdot s\\) merges the two elementary Jacobians…","labels":[],"detail_key":"p27"},{"id":"n30315","layer":"informal","project":"p27","title":"BN affine VJP","kind":"theorem","summary":"[BN affine VJP] \\(\\HasVJP\\, \\bigl(bnAffine(\\gamma, \\beta)\\bigr)\\): each input feeds one output…","labels":["thm:bnAffine_has_vjp"],"detail_key":"p27"},{"id":"n30316","layer":"informal","project":"p27","title":"\\item Define \\(B(v, dy)_i := \\gamma \\cdot dy_i\\). \\item \\Suffices for all \\(v\\), \\(dy\\),…","kind":"proof","summary":"\\item Define \\(B(v, dy)_i := \\gamma \\cdot dy_i\\). \\item \\Suffices for all \\(v\\), \\(dy\\), \\(i\\):…","labels":[],"detail_key":"p27"},{"id":"n30317","layer":"informal","project":"p27","title":"Full BN VJP","kind":"theorem","summary":"[Full BN VJP] \\Assume \\item \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\varepsilon\\)] \\Prove \\(\\HasVJ…","labels":["thm:bn_has_vjp"],"detail_key":"p27"},{"id":"n30318","layer":"informal","project":"p27","title":"\\item \\(bnForward = bnAffine \\circ bnNormalize\\). \\pfDefinitional (\\textttbnForward\\_eq\\_…","kind":"proof","summary":"\\item \\(bnForward = bnAffine \\circ bnNormalize\\). \\pfDefinitional (\\textttbnForward\\_eq\\_compos…","labels":[],"detail_key":"p27"},{"id":"n30319","layer":"informal","project":"p27","title":"Residual block VJP","kind":"theorem","summary":"[Residual block VJP] \\Assume \\item \\(B_f\\) is a correct backward function for \\(f\\) (\\(\\HasVJP\\…","labels":["thm:residual_has_vjp"],"detail_key":"p27"},{"id":"n30320","layer":"informal","project":"p27","title":"\\item \\(residual\\, f = biPath\\, f\\, id\\). \\pfDefinitional. \\item \\Qed \\pfInstantiate the…","kind":"proof","summary":"\\item \\(residual\\, f = biPath\\, f\\, id\\). \\pfDefinitional. \\item \\Qed \\pfInstantiate the additi…","labels":[],"detail_key":"p27"},{"id":"n30321","layer":"informal","project":"p27","title":"Depthwise conv forward","kind":"definition","summary":"[Depthwise conv forward] Concrete per-channel cross-correlation.","labels":["ax:depthwiseConv2d"],"detail_key":"p27"},{"id":"n30322","layer":"informal","project":"p27","title":"Depthwise input VJP","kind":"theorem","summary":"[Depthwise input VJP] \\Prove \\(\\HasVJPthree\\, \\bigl(depthwiseConv2d(W, b)\\bigr)\\), with the per…","labels":["ax:depthwise_has_vjp3"],"detail_key":"p27"},{"id":"n30323","layer":"informal","project":"p27","title":"\\emphSketch: the conv2d input VJP (Theorem~\\refax:conv2d_has_vjp3) with one fewer \\(\\Sigm…","kind":"proof","summary":"\\emphSketch: the conv2d input VJP (Theorem~\\refax:conv2d_has_vjp3) with one fewer \\(\\Sigma\\) le…","labels":[],"detail_key":"p27"},{"id":"n30324","layer":"informal","project":"p27","title":"Depthwise weight VJP","kind":"theorem","summary":"[Depthwise weight VJP] \\textttDepthwiseKernel is definitionally \\(\\mathsfTensor3\\), so \\(\\HasVJ…","labels":["ax:depthwise_weight_grad_has_vjp3"],"detail_key":"p27"},{"id":"n30325","layer":"informal","project":"p27","title":"\\item Define \\(B(W, dy)_c, k_h, k_w\\) as the formula above. \\item \\Suffices \\(B\\) matches…","kind":"proof","summary":"\\item Define \\(B(W, dy)_c, k_h, k_w\\) as the formula above. \\item \\Suffices \\(B\\) matches the \\…","labels":[],"detail_key":"p27"},{"id":"n30326","layer":"informal","project":"p27","title":"Depthwise bias VJP","kind":"theorem","summary":"[Depthwise bias VJP] \\Prove \\(\\HasVJP\\, \\bigl(b \\mapsto flatten (depthwiseConv2d(W, b)\\, x)\\big…","labels":["ax:depthwise_bias_grad_has_vjp"],"detail_key":"p27"},{"id":"n30327","layer":"informal","project":"p27","title":"Same three steps as the conv2d bias VJP (Theorem~\\refax:conv2d_bias_grad_has_vjp), with i…","kind":"proof","summary":"Same three steps as the conv2d bias VJP (Theorem~\\refax:conv2d_bias_grad_has_vjp), with input c…","labels":[],"detail_key":"p27"},{"id":"n30328","layer":"informal","project":"p27","title":"SE block VJP","kind":"theorem","summary":"[SE block VJP] \\Assume \\item \\(B_g\\) is a correct backward function for the gate (\\(\\HasVJP\\, g…","labels":["thm:seBlock_has_vjp"],"detail_key":"p27"},{"id":"n30329","layer":"informal","project":"p27","title":"\\item \\(seBlock\\, gate = elemwiseProduct\\, id\\, gate\\). \\pfDefinitional. \\item \\Qed \\pfIn…","kind":"proof","summary":"\\item \\(seBlock\\, gate = elemwiseProduct\\, id\\, gate\\). \\pfDefinitional. \\item \\Qed \\pfInstanti…","labels":[],"detail_key":"p27"},{"id":"n30330","layer":"informal","project":"p27","title":"GELU scalar function","kind":"definition","summary":"[GELU scalar function] The tanh approximation the codegen actually emits (not the exact \\(x \\cd…","labels":["ax:geluScalar"],"detail_key":"p27"},{"id":"n30331","layer":"informal","project":"p27","title":"GELU scalar derivative","kind":"definition","summary":"[GELU scalar derivative] Defined as Mathlib's \\(deriv\\, geluScalar\\) --- so the connection to t…","labels":["ax:geluScalarDeriv"],"detail_key":"p27"},{"id":"n30332","layer":"informal","project":"p27","title":"GELU Jacobian","kind":"theorem","summary":"[GELU Jacobian] \\Prove \\(\\pdiv(gelu)\\, x\\, i\\, j = \\delta_ij \\cdot geluScalar'(x_i)\\) --- the d…","labels":["ax:pdiv_gelu"],"detail_key":"p27"},{"id":"n30333","layer":"informal","project":"p27","title":"\\emphSketch: the template for every elementwise activation with a smooth scalar function…","kind":"proof","summary":"\\emphSketch: the template for every elementwise activation with a smooth scalar function --- ex…","labels":[],"detail_key":"p27"},{"id":"n30334","layer":"informal","project":"p27","title":"LayerNorm VJP","kind":"theorem","summary":"[LayerNorm VJP] \\Assume \\item \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\varepsilon\\)] \\Prove \\(\\Has…","labels":["thm:layerNorm_has_vjp"],"detail_key":"p27"},{"id":"n30335","layer":"informal","project":"p27","title":"\\item \\(layerNormForward = bnForward\\), definitionally: on a single feature vector, Layer…","kind":"proof","summary":"\\item \\(layerNormForward = bnForward\\), definitionally: on a single feature vector, LayerNorm \\…","labels":[],"detail_key":"p27"},{"id":"n30336","layer":"informal","project":"p27","title":"GELU VJP","kind":"theorem","summary":"[GELU VJP] \\(\\HasVJP\\, (gelu)\\), with backward \\(B(x, dy)_i = dy_i \\cdot geluScalar'(x_i)\\).","labels":["thm:gelu_has_vjp"],"detail_key":"p27"},{"id":"n30337","layer":"informal","project":"p27","title":"\\item \\Suffices \\(B(x, dy)_i = \\sum_j \\pdiv(gelu)\\, x\\, i\\, j \\cdot dy_j\\). \\pfDefinition…","kind":"proof","summary":"\\item \\Suffices \\(B(x, dy)_i = \\sum_j \\pdiv(gelu)\\, x\\, i\\, j \\cdot dy_j\\). \\pfDefinition~\\refd…","labels":[],"detail_key":"p27"},{"id":"n30338","layer":"informal","project":"p27","title":"Matrix VJP record","kind":"definition","summary":"[Matrix VJP record] The rank-2 analogue of Definition~\\refdef:hasvjp. Define \\(\\pdivMat f\\, A\\,…","labels":["def:hasvjpmat"],"detail_key":"p27"},{"id":"n30339","layer":"informal","project":"p27","title":"Row-independence for matrices","kind":"theorem","summary":"[Row-independence for matrices] \\Assume \\item \\(g : \\Vectn \\to \\Vectp\\) is differentiable every…","labels":["ax:pdivMat_rowIndep"],"detail_key":"p27"},{"id":"n30340","layer":"informal","project":"p27","title":"\\item Coordinate factoring: the \\((k, l)\\) output coordinate of the flattened rowwise map…","kind":"proof","summary":"\\item Coordinate factoring: the \\((k, l)\\) output coordinate of the flattened rowwise map is \\(…","labels":[],"detail_key":"p27"},{"id":"n30341","layer":"informal","project":"p27","title":"Matrix-level chain rule","kind":"theorem","summary":"[Matrix-level chain rule] \\Assume \\item \\(B_F\\), \\(B_G\\) are correct backward functions (\\(\\Has…","labels":["thm:vjpMat_comp"],"detail_key":"p27"},{"id":"n30342","layer":"informal","project":"p27","title":"\\emphSketch: the VJP chain rule (Theorem~\\refthm:vjp_comp) transcribed to rank-2 indices;…","kind":"proof","summary":"\\emphSketch: the VJP chain rule (Theorem~\\refthm:vjp_comp) transcribed to rank-2 indices; every…","labels":[],"detail_key":"p27"},{"id":"n30343","layer":"informal","project":"p27","title":"Matrix-level additive fan-in","kind":"theorem","summary":"[Matrix-level additive fan-in] Rank-2 transcription of Theorem~\\refthm:biPath_has_vjp: given \\(…","labels":["thm:biPathMat_has_vjp"],"detail_key":"p27"},{"id":"n30344","layer":"informal","project":"p27","title":"\\item \\Suffices the summed backward matches the \\(\\pdivMat\\) contraction. \\pfDefinition~\\…","kind":"proof","summary":"\\item \\Suffices the summed backward matches the \\(\\pdivMat\\) contraction. \\pfDefinition~\\refdef…","labels":[],"detail_key":"p27"},{"id":"n30345","layer":"informal","project":"p27","title":"Matrix-level identity","kind":"theorem","summary":"[Matrix-level identity] \\(\\HasVJPMat\\, (id)\\), backward \\(B(A, dY) = dY\\).","labels":["thm:identityMat_has_vjp"],"detail_key":"p27"},{"id":"n30346","layer":"informal","project":"p27","title":"\\item \\(\\pdivMat(id)\\, A\\, (i,j)\\, (k,l) = [i = k \\wedge j = l]\\). \\pfFlatten \\(\\circ\\) u…","kind":"proof","summary":"\\item \\(\\pdivMat(id)\\, A\\, (i,j)\\, (k,l) = [i = k \\wedge j = l]\\). \\pfFlatten \\(\\circ\\) unflatt…","labels":[],"detail_key":"p27"},{"id":"n30347","layer":"informal","project":"p27","title":"Scalar-scale Jacobian","kind":"theorem","summary":"[Scalar-scale Jacobian] \\Prove \\(\\pdivMat(M \\mapsto s \\cdot M)\\, A\\, (i,j)\\, (k,l) = [i = k \\we…","labels":["thm:pdivMat_scalarScale"],"detail_key":"p27"},{"id":"n30348","layer":"informal","project":"p27","title":"\\item The flattened scalar-scale map reduces to \\(v \\mapsto s \\cdot v\\). \\pfFlatten/unfla…","kind":"proof","summary":"\\item The flattened scalar-scale map reduces to \\(v \\mapsto s \\cdot v\\). \\pfFlatten/unflatten r…","labels":[],"detail_key":"p27"},{"id":"n30349","layer":"informal","project":"p27","title":"Transpose Jacobian","kind":"theorem","summary":"[Transpose Jacobian] \\Prove \\(\\pdivMat(transpose)\\, A\\, (i,j)\\, (k,l) = [j = k \\wedge i = l]\\).","labels":["thm:pdivMat_transpose"],"detail_key":"p27"},{"id":"n30350","layer":"informal","project":"p27","title":"\\item The flattened transpose is a pure gather: at output index \\(idx\\), it reads \\(v\\) a…","kind":"proof","summary":"\\item The flattened transpose is a pure gather: at output index \\(idx\\), it reads \\(v\\) at the…","labels":[],"detail_key":"p27"},{"id":"n30351","layer":"informal","project":"p27","title":"Matmul Jacobian, left factor fixed","kind":"theorem","summary":"[Matmul Jacobian, left factor fixed] \\Prove \\(\\pdivMat(B' \\mapsto C \\cdot B')\\, B\\, (i,j)\\, (k,…","labels":["thm:pdivMat_matmul_left_const"],"detail_key":"p27"},{"id":"n30352","layer":"informal","project":"p27","title":"\\item The flattened map at output index \\(idx\\) is \\(v \\mapsto \\sum_s C_k(idx), s \\cdot v…","kind":"proof","summary":"\\item The flattened map at output index \\(idx\\) is \\(v \\mapsto \\sum_s C_k(idx), s \\cdot v_flat(…","labels":[],"detail_key":"p27"},{"id":"n30353","layer":"informal","project":"p27","title":"Matmul Jacobian, right factor fixed","kind":"theorem","summary":"[Matmul Jacobian, right factor fixed] \\Prove \\(\\pdivMat(A' \\mapsto A' \\cdot D)\\, A\\, (i,j)\\, (k…","labels":["thm:pdivMat_matmul_right_const"],"detail_key":"p27"},{"id":"n30354","layer":"informal","project":"p27","title":"Mirror image of Theorem~\\refthm:pdivMat_matmul_left_const: the flattened map at output \\(…","kind":"proof","summary":"Mirror image of Theorem~\\refthm:pdivMat_matmul_left_const: the flattened map at output \\(idx\\)…","labels":[],"detail_key":"p27"},{"id":"n30355","layer":"informal","project":"p27","title":"Matmul VJP, left factor fixed","kind":"theorem","summary":"[Matmul VJP, left factor fixed] \\(\\HasVJPMat\\, (B' \\mapsto C \\cdot B')\\), backward \\(dB = C^T \\…","labels":["thm:matmul_left_const_has_vjp"],"detail_key":"p27"},{"id":"n30356","layer":"informal","project":"p27","title":"\\item Define \\(B(B', dY)_ij := \\sum_k C_ki\\, dY_kj\\) (that is, \\(C^T \\cdot dY\\)). \\item \\…","kind":"proof","summary":"\\item Define \\(B(B', dY)_ij := \\sum_k C_ki\\, dY_kj\\) (that is, \\(C^T \\cdot dY\\)). \\item \\Suffic…","labels":[],"detail_key":"p27"},{"id":"n30357","layer":"informal","project":"p27","title":"Matmul VJP, right factor fixed","kind":"theorem","summary":"[Matmul VJP, right factor fixed] \\(\\HasVJPMat\\, (A' \\mapsto A' \\cdot D)\\), backward \\(dA = dY \\…","labels":["thm:matmul_right_const_has_vjp"],"detail_key":"p27"},{"id":"n30358","layer":"informal","project":"p27","title":"\\item Define \\(B(A', dY)_ij := \\sum_l dY_il\\, D_jl\\) (that is, \\(dY \\cdot D^T\\)). \\item \\…","kind":"proof","summary":"\\item Define \\(B(A', dY)_ij := \\sum_l dY_il\\, D_jl\\) (that is, \\(dY \\cdot D^T\\)). \\item \\Suffic…","labels":[],"detail_key":"p27"},{"id":"n30359","layer":"informal","project":"p27","title":"Scalar-scale VJP","kind":"theorem","summary":"[Scalar-scale VJP] \\(\\HasVJPMat\\, (M \\mapsto s \\cdot M)\\), backward \\(dA = s \\cdot dY\\).","labels":["thm:scalarScale_has_vjp"],"detail_key":"p27"},{"id":"n30360","layer":"informal","project":"p27","title":"By Definition~\\refdef:hasvjpmat it suffices to contract the Jacobian (Theorem~\\refthm:pdi…","kind":"proof","summary":"By Definition~\\refdef:hasvjpmat it suffices to contract the Jacobian (Theorem~\\refthm:pdivMat_s…","labels":[],"detail_key":"p27"},{"id":"n30361","layer":"informal","project":"p27","title":"Transpose VJP","kind":"theorem","summary":"[Transpose VJP] \\(\\HasVJPMat\\, (transpose)\\), backward \\(dA = dY^T\\).","labels":["thm:transpose_has_vjp"],"detail_key":"p27"},{"id":"n30362","layer":"informal","project":"p27","title":"By Definition~\\refdef:hasvjpmat it suffices to contract the Jacobian (Theorem~\\refthm:pdi…","kind":"proof","summary":"By Definition~\\refdef:hasvjpmat it suffices to contract the Jacobian (Theorem~\\refthm:pdivMat_t…","labels":[],"detail_key":"p27"},{"id":"n30363","layer":"informal","project":"p27","title":"Row-wise lifting of any \\(\\HasVJP\\)","kind":"theorem","summary":"[Row-wise lifting of any \\(\\HasVJP\\)] \\Assume \\item \\(B_g\\) is a correct backward for \\(g : \\Ve…","labels":["thm:rowwise_has_vjp_mat"],"detail_key":"p27"},{"id":"n30364","layer":"informal","project":"p27","title":"\\item \\Suffices \\(B\\) matches the \\(\\pdivMat\\) contraction. \\pfDefinition~\\refdef:hasvjpm…","kind":"proof","summary":"\\item \\Suffices \\(B\\) matches the \\(\\pdivMat\\) contraction. \\pfDefinition~\\refdef:hasvjpmat. \\i…","labels":[],"detail_key":"p27"},{"id":"n30365","layer":"informal","project":"p27","title":"3D chain rule","kind":"theorem","summary":"[3D chain rule] \\Prove for flattened-differentiable \\(f\\), \\(g\\): \\(\\pdiv_3(g \\circ f)\\) is the…","labels":["thm:pdiv3_comp"],"detail_key":"p27"},{"id":"n30366","layer":"informal","project":"p27","title":"\\item \\(\\pdiv_3\\) is \\(\\pdiv\\) of the flattened map (Definition~\\refdef:hasvjp3), and the…","kind":"proof","summary":"\\item \\(\\pdiv_3\\) is \\(\\pdiv\\) of the flattened map (Definition~\\refdef:hasvjp3), and the flatt…","labels":[],"detail_key":"p27"},{"id":"n30367","layer":"informal","project":"p27","title":"3D VJP chain rule","kind":"theorem","summary":"[3D VJP chain rule] Rank-3 transcription of Theorem~\\refthm:vjp_comp: composite backward \\(B(x,…","labels":["thm:vjp3_comp"],"detail_key":"p27"},{"id":"n30368","layer":"informal","project":"p27","title":"\\item \\Suffices the composite backward matches the \\(\\pdiv_3\\) triple-sum contraction. \\p…","kind":"proof","summary":"\\item \\Suffices the composite backward matches the \\(\\pdiv_3\\) triple-sum contraction. \\pfDefin…","labels":[],"detail_key":"p27"},{"id":"n30369","layer":"informal","project":"p27","title":"3D additive fan-in","kind":"theorem","summary":"[3D additive fan-in] Rank-3 transcription of Theorem~\\refthm:biPath_has_vjp: backward is the su…","labels":["thm:biPath3_has_vjp"],"detail_key":"p27"},{"id":"n30370","layer":"informal","project":"p27","title":"By Definition~\\refdef:hasvjp3 it suffices to match the triple-sum contraction: expand bot…","kind":"proof","summary":"By Definition~\\refdef:hasvjp3 it suffices to match the triple-sum contraction: expand both \\tex…","labels":[],"detail_key":"p27"},{"id":"n30371","layer":"informal","project":"p27","title":"Softmax Jacobian","kind":"theorem","summary":"[Softmax Jacobian] Writing \\(p := softmax(z)\\): \\Prove \\(\\pdiv(softmax)\\, z\\, i\\, j = p_j\\, (\\d…","labels":["ax:pdiv_softmax"],"detail_key":"p27"},{"id":"n30372","layer":"informal","project":"p27","title":"\\item \\(\\pdiv(softmax)\\, z\\, i\\, j = \\fderiv_\\R\\,\\bigl(z' \\mapsto e^z'_j \\cdot (\\textstyl…","kind":"proof","summary":"\\item \\(\\pdiv(softmax)\\, z\\, i\\, j = \\fderiv_\\R\\,\\bigl(z' \\mapsto e^z'_j \\cdot (\\textstyle\\sum_…","labels":[],"detail_key":"p27"},{"id":"n30373","layer":"informal","project":"p27","title":"Standalone softmax VJP","kind":"theorem","summary":"[Standalone softmax VJP] \\(\\HasVJP\\, (softmax)\\), with the closed-form \\(O(c)\\) backward \\[ B(z…","labels":["thm:softmax_has_vjp"],"detail_key":"p27"},{"id":"n30374","layer":"informal","project":"p27","title":"\\item \\Suffices \\(B(z, dy)_i = \\sum_j \\pdiv(softmax)\\, z\\, i\\, j \\cdot dy_j\\). \\pfDefinit…","kind":"proof","summary":"\\item \\Suffices \\(B(z, dy)_i = \\sum_j \\pdiv(softmax)\\, z\\, i\\, j \\cdot dy_j\\). \\pfDefinition~\\r…","labels":[],"detail_key":"p27"},{"id":"n30375","layer":"informal","project":"p27","title":"Row-wise softmax VJP on a matrix","kind":"theorem","summary":"[Row-wise softmax VJP on a matrix] \\(\\HasVJPMat\\, (rowSoftmax)\\): rows are independent, so the…","labels":["thm:rowSoftmax_has_vjp_mat"],"detail_key":"p27"},{"id":"n30376","layer":"informal","project":"p27","title":"The rowwise-lifting argument (Theorem~\\refthm:rowwise_has_vjp_mat) instantiated at \\(g =…","kind":"proof","summary":"The rowwise-lifting argument (Theorem~\\refthm:rowwise_has_vjp_mat) instantiated at \\(g = softma…","labels":[],"detail_key":"p27"},{"id":"n30377","layer":"informal","project":"p27","title":"SDPA backward wrt Q","kind":"theorem","summary":"[SDPA backward wrt Q] \\Prove for fixed \\(K, V\\): \\(sdpa\\_back\\_Q\\) matches the \\(\\pdivMat\\) con…","labels":["thm:sdpa_back_Q_correct"],"detail_key":"p27"},{"id":"n30378","layer":"informal","project":"p27","title":"\\item The forward is a four-link chain: \\[ Q' \\;\\mapsto\\; Q' K^T \\;\\mapsto\\; \\tfrac1\\sqrt…","kind":"proof","summary":"\\item The forward is a four-link chain: \\[ Q' \\;\\mapsto\\; Q' K^T \\;\\mapsto\\; \\tfrac1\\sqrtd \\cdo…","labels":[],"detail_key":"p27"},{"id":"n30379","layer":"informal","project":"p27","title":"SDPA backward wrt K","kind":"theorem","summary":"[SDPA backward wrt K] \\Prove for fixed \\(Q, V\\): \\(sdpa\\_back\\_K\\) matches the \\(\\pdivMat\\) con…","labels":["thm:sdpa_back_K_correct"],"detail_key":"p27"},{"id":"n30380","layer":"informal","project":"p27","title":"Same shape as Theorem~\\refthm:sdpa_back_Q_correct, with a leading transpose link: \\item T…","kind":"proof","summary":"Same shape as Theorem~\\refthm:sdpa_back_Q_correct, with a leading transpose link: \\item The for…","labels":[],"detail_key":"p27"},{"id":"n30381","layer":"informal","project":"p27","title":"SDPA backward wrt V","kind":"theorem","summary":"[SDPA backward wrt V] \\Prove for fixed \\(Q, K\\): \\(sdpa\\_back\\_V\\) matches the \\(\\pdivMat\\) con…","labels":["thm:sdpa_back_V_correct"],"detail_key":"p27"},{"id":"n30382","layer":"informal","project":"p27","title":"\\item \\(V'\\) enters only through the final matmul: \\(sdpa(Q, K, V') = W \\cdot V'\\) with \\…","kind":"proof","summary":"\\item \\(V'\\) enters only through the final matmul: \\(sdpa(Q, K, V') = W \\cdot V'\\) with \\(W :=…","labels":[],"detail_key":"p27"},{"id":"n30383","layer":"informal","project":"p27","title":"Multi-head SDPA VJP","kind":"theorem","summary":"[Multi-head SDPA VJP] \\Prove \\(\\HasVJPMat\\, (mhsa\\_layer)\\).","labels":["ax:mhsa_has_vjp_mat"],"detail_key":"p27"},{"id":"n30384","layer":"informal","project":"p27","title":"\\emphSketch: column-stacking --- each head touches only its own column slab, so the singl…","kind":"proof","summary":"\\emphSketch: column-stacking --- each head touches only its own column slab, so the single-head…","labels":[],"detail_key":"p27"},{"id":"n30385","layer":"informal","project":"p27","title":"MHSA layer smoothness","kind":"theorem","summary":"[MHSA layer smoothness] \\(\\Differentiable\\) sibling of Theorem~\\refax:mhsa_has_vjp_mat.","labels":["ax:mhsa_layer_flat_diff"],"detail_key":"p27"},{"id":"n30386","layer":"informal","project":"p27","title":"Factor as in Theorem~\\refax:mhsa_has_vjp_mat step 1; each factor's flattened form is diff…","kind":"proof","summary":"Factor as in Theorem~\\refax:mhsa_has_vjp_mat step 1; each factor's flattened form is differenti…","labels":[],"detail_key":"p27"},{"id":"n30387","layer":"informal","project":"p27","title":"Patch-embedding VJP","kind":"theorem","summary":"[Patch-embedding VJP] The concrete \\textttdef unfolds to conv-with-stride + CLS prepend + posit…","labels":["ax:patchEmbed_flat_has_vjp"],"detail_key":"p27"},{"id":"n30388","layer":"informal","project":"p27","title":"\\emphSketch: the conv2d input-VJP recipe (Theorem~\\refax:conv2d_has_vjp3) reapplied to th…","kind":"proof","summary":"\\emphSketch: the conv2d input-VJP recipe (Theorem~\\refax:conv2d_has_vjp3) reapplied to the stri…","labels":[],"detail_key":"p27"},{"id":"n30389","layer":"informal","project":"p27","title":"Patch-embedding smoothness","kind":"theorem","summary":"[Patch-embedding smoothness] \\(\\Differentiable\\) sibling of Theorem~\\refax:patchEmbed_flat_has_…","labels":["ax:patchEmbed_flat_diff"],"detail_key":"p27"},{"id":"n30390","layer":"informal","project":"p27","title":"The decomposition of Theorem~\\refax:patchEmbed_flat_has_vjp step 3 exhibits the flattened…","kind":"proof","summary":"The decomposition of Theorem~\\refax:patchEmbed_flat_has_vjp step 3 exhibits the flattened forwa…","labels":[],"detail_key":"p27"},{"id":"n30391","layer":"informal","project":"p27","title":"Per-token LayerNorm lifted to a matrix","kind":"theorem","summary":"[Per-token LayerNorm lifted to a matrix]","labels":["thm:layerNorm_per_token_has_vjp_mat"],"detail_key":"p27"},{"id":"n30392","layer":"informal","project":"p27","title":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = layerNormForw…","kind":"proof","summary":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = layerNormForward\\),…","labels":[],"detail_key":"p27"},{"id":"n30393","layer":"informal","project":"p27","title":"Per-token dense lifted to a matrix","kind":"theorem","summary":"[Per-token dense lifted to a matrix]","labels":["thm:dense_per_token_has_vjp_mat"],"detail_key":"p27"},{"id":"n30394","layer":"informal","project":"p27","title":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = dense(W, b)\\)…","kind":"proof","summary":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = dense(W, b)\\), with…","labels":[],"detail_key":"p27"},{"id":"n30395","layer":"informal","project":"p27","title":"Per-token GELU lifted to a matrix","kind":"theorem","summary":"[Per-token GELU lifted to a matrix]","labels":["thm:gelu_per_token_has_vjp_mat"],"detail_key":"p27"},{"id":"n30396","layer":"informal","project":"p27","title":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = gelu\\), with…","kind":"proof","summary":"Instantiate the rowwise lift (Theorem~\\refthm:rowwise_has_vjp_mat) at \\(g = gelu\\), with Theore…","labels":[],"detail_key":"p27"},{"id":"n30397","layer":"informal","project":"p27","title":"Row-wise softmax smoothness","kind":"theorem","summary":"[Row-wise softmax smoothness] \\Prove the flattened \\(rowSoftmax\\) is \\(\\Differentiable\\).","labels":["ax:rowSoftmax_flat_diff"],"detail_key":"p27"},{"id":"n30398","layer":"informal","project":"p27","title":"\\item \\Case \\(n = 0\\): the codomain is \\(0\\)-dimensional; the map is constant. \\pfTrivial…","kind":"proof","summary":"\\item \\Case \\(n = 0\\): the codomain is \\(0\\)-dimensional; the map is constant. \\pfTrivial. \\ite…","labels":[],"detail_key":"p27"},{"id":"n30399","layer":"informal","project":"p27","title":"Transformer MLP sublayer VJP","kind":"theorem","summary":"[Transformer MLP sublayer VJP] \\(\\HasVJPMat\\) of \\(dense_2 \\circ gelu \\circ dense_1\\), all per-…","labels":["thm:transformerMlp_has_vjp_mat"],"detail_key":"p27"},{"id":"n30400","layer":"informal","project":"p27","title":"\\item Inner composition \\(gelu \\circ dense_1\\): matrix chain rule (Theorem~\\refthm:vjpMat…","kind":"proof","summary":"\\item Inner composition \\(gelu \\circ dense_1\\): matrix chain rule (Theorem~\\refthm:vjpMat_comp)…","labels":[],"detail_key":"p27"},{"id":"n30401","layer":"informal","project":"p27","title":"Transformer attention sublayer with residual VJP","kind":"theorem","summary":"[Transformer attention sublayer with residual VJP] \\Assume \\(\\varepsilon > 0\\) \\hfill[\\texttth\\…","labels":["thm:transformerAttnSublayer_has_vjp_mat"],"detail_key":"p27"},{"id":"n30402","layer":"informal","project":"p27","title":"\\item Inner arm \\(MHSA \\circ LN_1\\): matrix chain rule (Theorem~\\refthm:vjpMat_comp) on T…","kind":"proof","summary":"\\item Inner arm \\(MHSA \\circ LN_1\\): matrix chain rule (Theorem~\\refthm:vjpMat_comp) on Theorem…","labels":[],"detail_key":"p27"},{"id":"n30403","layer":"informal","project":"p27","title":"Transformer MLP sublayer with residual VJP","kind":"theorem","summary":"[Transformer MLP sublayer with residual VJP] \\Assume \\(\\varepsilon > 0\\) \\hfill[\\texttth\\(\\vare…","labels":["thm:transformerMlpSublayer_has_vjp_mat"],"detail_key":"p27"},{"id":"n30404","layer":"informal","project":"p27","title":"Identical structure to Theorem~\\refthm:transformerAttnSublayer_has_vjp_mat: chain \\(MLP \\…","kind":"proof","summary":"Identical structure to Theorem~\\refthm:transformerAttnSublayer_has_vjp_mat: chain \\(MLP \\circ L…","labels":[],"detail_key":"p27"},{"id":"n30405","layer":"informal","project":"p27","title":"Transformer block VJP","kind":"theorem","summary":"[Transformer block VJP] \\Prove \\(\\HasVJPMat\\) of one pre-norm encoder block \\(z = x + MHSA(LN_1…","labels":["thm:transformerBlock_has_vjp_mat"],"detail_key":"p27"},{"id":"n30406","layer":"informal","project":"p27","title":"One matrix chain rule (Theorem~\\refthm:vjpMat_comp) glueing the attention sublayer (Theor…","kind":"proof","summary":"One matrix chain rule (Theorem~\\refthm:vjpMat_comp) glueing the attention sublayer (Theorem~\\re…","labels":[],"detail_key":"p27"},{"id":"n30407","layer":"informal","project":"p27","title":"Transformer tower, any depth","kind":"theorem","summary":"[Transformer tower, any depth] \\Prove \\(\\HasVJPMat\\) of the \\(k\\)-block tower, for every \\(k\\)…","labels":["thm:transformerTower_has_vjp_mat"],"detail_key":"p27"},{"id":"n30408","layer":"informal","project":"p27","title":"Induction on \\(k\\): \\item \\Case \\(k = 0\\): the tower is the identity. \\pfTheorem~\\refthm:…","kind":"proof","summary":"Induction on \\(k\\): \\item \\Case \\(k = 0\\): the tower is the identity. \\pfTheorem~\\refthm:identi…","labels":[],"detail_key":"p27"},{"id":"n30409","layer":"informal","project":"p27","title":"ViT body: the grand finale","kind":"theorem","summary":"[ViT body: the grand finale] \\Prove the full ViT transformer backbone \\(finalLN \\circ transform…","labels":["thm:vit_body_has_vjp_mat"],"detail_key":"p27"},{"id":"n30410","layer":"informal","project":"p27","title":"One final matrix chain rule (Theorem~\\refthm:vjpMat_comp) glueing the \\(k\\)-block tower (…","kind":"proof","summary":"One final matrix chain rule (Theorem~\\refthm:vjpMat_comp) glueing the \\(k\\)-block tower (Theore…","labels":[],"detail_key":"p27"},{"id":"n30411","layer":"informal","project":"p27","title":"Inception module","kind":"definition","summary":"[Inception module] The GoogLeNet parallel-branch module (Szegedy et al.\\ 2014). Four branches c…","labels":["layer:inceptionModule"],"detail_key":"p27"},{"id":"n30412","layer":"informal","project":"p27","title":"DenseNet dense block","kind":"definition","summary":"[DenseNet dense block] DenseNet's bundled dense block (Huang et al.\\ 2017). \\textttnLayers BN-R…","labels":["layer:denseBlock"],"detail_key":"p27"},{"id":"n30413","layer":"informal","project":"p27","title":"DenseNet transition layer","kind":"definition","summary":"[DenseNet transition layer] Inter-block downsample for DenseNet. BN + 1\\times1 conv (\\textttic…","labels":["layer:transitionLayer"],"detail_key":"p27"},{"id":"n30414","layer":"informal","project":"p27","title":"ShuffleNet v1 stage","kind":"definition","summary":"[ShuffleNet v1 stage] Grouped 1\\times1 conv + channel-shuffle permutation + 3\\times3 depthwise…","labels":["layer:shuffleBlock"],"detail_key":"p27"},{"id":"n30415","layer":"informal","project":"p27","title":"ShuffleNet v2 stage","kind":"definition","summary":"[ShuffleNet v2 stage] \\textttnUnits v2 units (Ma et al.\\ 2018). Basic unit (stride 1): channel-…","labels":["layer:shuffleV2Block"],"detail_key":"p27"},{"id":"n30416","layer":"informal","project":"p27","title":"MobileViT block","kind":"definition","summary":"[MobileViT block] Hybrid local-conv + patch-level transformer (Mehta \\& Rastegari 2022). Local…","labels":["layer:mobileVitBlock"],"detail_key":"p27"},{"id":"n30417","layer":"informal","project":"p27","title":"Swin Transformer stage","kind":"definition","summary":"[Swin Transformer stage] Windowed multi-head self-attention at fixed spatial resolution (Liu et…","labels":["layer:swinStage"],"detail_key":"p27"},{"id":"n30418","layer":"informal","project":"p27","title":"Patch merging","kind":"definition","summary":"[Patch merging] Swin's 2\\times2 spatial downsample + linear channel projection (\\textttinDim \\t…","labels":["layer:patchMerging"],"detail_key":"p27"},{"id":"n30419","layer":"informal","project":"p27","title":"Darknet residual block","kind":"definition","summary":"[Darknet residual block] YOLOv3's Darknet-53 residual stack (Redmon \\& Farhadi 2018). \\textttnB…","labels":["layer:darknetBlock"],"detail_key":"p27"},{"id":"n30420","layer":"informal","project":"p27","title":"Cross-Stage Partial block","kind":"definition","summary":"[Cross-Stage Partial block] CSP (Wang et al.\\ 2019), used by YOLOv4 onward. 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Measura…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.conditionallyIID_of_contractable","module":"Exchangeability.DeFinetti.TheoremViaMartingale"},{"id":"n30605","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.deFinetti_RyllNardzewski_equivalence","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] [StandardBorelSpace Ω] α : Type u_2 [inst_2 : Measura…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.deFinetti_RyllNardzewski_equivalence","module":"Exchangeability.DeFinetti.TheoremViaMartingale"},{"id":"n30606","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.blockAvg","kind":"def","summary":"α : Type u_1 → (m : Nat) → Nat → Fin m → (α → Real) → (Nat → α) → 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[MeasureTheor…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.integral_prod_eq_integral_blockAvg","module":"Exchangeability.DeFinetti.ViaKoopman.BlockAverage"},{"id":"n30609","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.condexp_product_factorization_contractable","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure (PathSpace α) [MeasureTheor…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.condexp_product_factorization_contractable","module":"Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization"},{"id":"n30610","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.product_blockAvg_L1_convergence","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure (PathSpace α) [MeasureTheor…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.product_blockAvg_L1_convergence","module":"Exchangeability.DeFinetti.ViaKoopman.ContractableFactorization"},{"id":"n30611","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.condexp_lag_constant_from_exchangeability","kind":"theorem","summary":"∀ α : Type u_2 [inst : MeasurableSpace α] [StandardBorelSpace α] μ : MeasureTheory.Measure (Nat…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.condexp_lag_constant_from_exchangeability","module":"Exchangeability.DeFinetti.ViaKoopman.InfraLagConstancy"},{"id":"n30612","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.birkhoffAverage_tendsto_condexp","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure (PathSpace α) [inst_1 : Mea…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.birkhoffAverage_tendsto_condexp","module":"Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation"},{"id":"n30613","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaKoopman.condexpL2_koopman_comm","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure (PathSpace α) [inst_1 : Mea…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaKoopman.condexpL2_koopman_comm","module":"Exchangeability.DeFinetti.ViaKoopman.KoopmanCommutation"},{"id":"n30614","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.indicator_product_bridge_contractable","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure (PathSpace α) [inst_1 : Mea…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.indicator_product_bridge_contractable","module":"Exchangeability.DeFinetti.ViaKoopman"},{"id":"n30615","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.contractable_covariance_structure","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] μ : MeasureTheory.Measure Ω [MeasureTheory.IsProbabil…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.contractable_covariance_structure","module":"Exchangeability.DeFinetti.ViaL2.BlockAverages.Covariance"},{"id":"n30616","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.blockAvg","kind":"def","summary":"Ω : Type u_1 → α : Type u_2 → (α → Real) → (Nat → Ω → α) → Nat → Nat → Ω → 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[MeasureTheory.IsProbabil…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.cesaro_to_condexp_L1","module":"Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L1"},{"id":"n30619","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.cesaro_to_condexp_L2","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] μ : MeasureTheory.Measure Ω [MeasureTheory.IsProbabil…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.cesaro_to_condexp_L2","module":"Exchangeability.DeFinetti.ViaL2.CesaroConvergence.L2"},{"id":"n30620","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.directing_measure_ae_eq_condExpKernel_map","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] μ : MeasureTheory.Mea…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.directing_measure_ae_eq_condExpKernel_map","module":"Exchangeability.DeFinetti.ViaL2.DirectingMeasureBridge"},{"id":"n30621","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.directing_measure_isProbabilityMeasure","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] μ : MeasureTheory.Measure Ω [inst_1 : MeasureTheory.I…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.directing_measure_isProbabilityMeasure","module":"Exchangeability.DeFinetti.ViaL2.DirectingMeasureCore"},{"id":"n30622","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.directing_measure_integral_eq_condExp","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] [StandardBorelSpace Ω] μ : MeasureTheory.Measure Ω [i…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.directing_measure_integral_eq_condExp","module":"Exchangeability.DeFinetti.ViaL2.DirectingMeasureIntegral"},{"id":"n30623","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaL2.weighted_sums_converge_L1","kind":"theorem","summary":"∀ Ω : Type u_1 [inst : MeasurableSpace Ω] μ : MeasureTheory.Measure Ω [MeasureTheory.IsProbabil…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaL2.weighted_sums_converge_L1","module":"Exchangeability.DeFinetti.ViaL2.MainConvergence"},{"id":"n30624","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaMartingale.condexp_convergence","kind":"theorem","summary":"∀ Ω : Type u_1 α : Type u_2 [inst : MeasurableSpace Ω] [inst_1 : MeasurableSpace α] μ : 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[M…","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaMartingale.finite_product_formula","module":"Exchangeability.DeFinetti.ViaMartingale.FiniteProduct"},{"id":"n30633","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaMartingale.futureFiltration","kind":"def","summary":"Ω : Type u_1 → α : Type u_2 → [MeasurableSpace α] → (Nat → Ω → α) → Nat → MeasurableSpace Ω","labels":[],"detail_key":"p28","name":"Exchangeability.DeFinetti.ViaMartingale.futureFiltration","module":"Exchangeability.DeFinetti.ViaMartingale.FutureFiltration"},{"id":"n30634","layer":"formal","project":"p28","title":"Exchangeability.DeFinetti.ViaMartingale.futureFiltration_antitone","kind":"theorem","summary":"∀ Ω : Type u_3 α : Type u_4 [inst : MeasurableSpace α] (X : Nat → Ω → α), Antitone 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UnweightedGraph…","labels":[],"detail_key":"p29","name":"UnweightedGraph.Isoperimetry.connected_iff_cheeger_pos","module":"ExpanderGraphs.chapter2"},{"id":"n30685","layer":"formal","project":"p29","title":"UnweightedGraph.Isoperimetry.edgeBoundaryComplement","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 [DecidableEq α] [Fintype α] [Fintype β] G : UnweightedGraph α β [De…","labels":[],"detail_key":"p29","name":"UnweightedGraph.Isoperimetry.edgeBoundaryComplement","module":"ExpanderGraphs.chapter2"},{"id":"n30686","layer":"formal","project":"p29","title":"UnweightedGraph.Isoperimetry.hG","kind":"def","summary":"α : Type u_1 → β : Type u_2 → [Fintype α] → G : UnweightedGraph α β → Set α → Real","labels":[],"detail_key":"p29","name":"UnweightedGraph.Isoperimetry.hG","module":"ExpanderGraphs.chapter2"},{"id":"n30687","layer":"formal","project":"p29","title":"UnweightedGraph.Isoperimetry.self_connection_eq_boundary","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 [DecidableEq α] [Fintype α] [Fintype β] G : UnweightedGraph α β [De…","labels":[],"detail_key":"p29","name":"UnweightedGraph.Isoperimetry.self_connection_eq_boundary","module":"ExpanderGraphs.chapter2"},{"id":"n30688","layer":"formal","project":"p29","title":"UnweightedGraph.Isoperimetry.two_cheegerV_ge_first_eigval","kind":"theorem","summary":"True","labels":[],"detail_key":"p29","name":"UnweightedGraph.Isoperimetry.two_cheegerV_ge_first_eigval","module":"ExpanderGraphs.chapter2"},{"id":"n30689","layer":"formal","project":"p29","title":"UnweightedGraph.edgeConnection","kind":"def","summary":"α : Type u_1 → β : Type u_2 → G : UnweightedGraph α β → Set α → Set α → Set β","labels":[],"detail_key":"p29","name":"UnweightedGraph.edgeConnection","module":"ExpanderGraphs.chapter2"},{"id":"n30690","layer":"informal","project":"p29","title":"Weighted Graph","kind":"definition","summary":"[Weighted Graph] We extend the definition of a general graph to that of a weighted graph by 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0.","labels":["def:IsIsolated"],"detail_key":"p29"},{"id":"n30695","layer":"informal","project":"p29","title":"def:NonTrivial","kind":"definition","summary":"A graph is said to be nontrivial if it contains at least one edge.","labels":["def:NonTrivial"],"detail_key":"p29"},{"id":"n30696","layer":"informal","project":"p29","title":"def:Adjacency","kind":"definition","summary":"The adjacency matrix of a graph G, denoted A is defined as follows, \\[ A(u,v)= 1 & \\textif u \\t…","labels":["def:Adjacency"],"detail_key":"p29"},{"id":"n30697","layer":"informal","project":"p29","title":"def:L","kind":"definition","summary":"Consider the matrix L, defined as follows \\[ L(u,v)= d_v & \\textif\\ u=v, \\\\ -1 & \\textif\\ u \\te…","labels":["def:L"],"detail_key":"p29"},{"id":"n30698","layer":"informal","project":"p29","title":"Laplacian Matrix","kind":"definition","summary":"[Laplacian Matrix] Consider the matrix L, defined as follows \\[ L(u,v) = \\left\\ cl 1 & \\textif…","labels":["def:Laplacian"],"detail_key":"p29"},{"id":"n30699","layer":"informal","project":"p29","title":"def:T_sqrt","kind":"definition","summary":"Let T denote the diagonal matrix with the (u,v)-th entry having value d_v.","labels":["def:T_sqrt"],"detail_key":"p29"},{"id":"n30700","layer":"informal","project":"p29","title":"def:T_inv_sqrt","kind":"definition","summary":"We also define T^-1/2 with the convention that T^-1/2 = 0 when d_v = 0.","labels":["def:T_inv_sqrt"],"detail_key":"p29"},{"id":"n30701","layer":"informal","project":"p29","title":"Laplacian symmetric normalization","kind":"lemma","summary":"[Laplacian symmetric normalization] \\[ L = T^-1/2 \\text L \\text T^-1/2 \\]","labels":["lem:Lap_symmetric_normalization"],"detail_key":"p29"},{"id":"n30702","layer":"informal","project":"p29","title":"def:LapOperator","kind":"definition","summary":"The Laplacian of a graph can be viewed as an operator on the space of functions g : V(G) \\right…","labels":["def:LapOperator"],"detail_key":"p29"},{"id":"n30703","layer":"informal","project":"p29","title":"def:LapOpFormula","kind":"definition","summary":"The Laplacian Operator satisfies \\[ Lg(u) = \\frac1\\sqrtd_u \\sum_\\substackv \\\\ u \\sim v \\left( \\…","labels":["def:LapOpFormula"],"detail_key":"p29"},{"id":"n30704","layer":"informal","project":"p29","title":"lem:LapOfRegGraph","kind":"lemma","summary":"When G is k-regular, \\[ L = I - \\frac1k A, \\]","labels":["lem:LapOfRegGraph"],"detail_key":"p29"},{"id":"n30705","layer":"informal","project":"p29","title":"lem:LapOfNotIsolatedGraph","kind":"lemma","summary":"For a general graph without isolated vertices, we have \\[ L = I - T^-1/2 A T^-1/2. \\]","labels":["lem:LapOfNotIsolatedGraph"],"detail_key":"p29"},{"id":"n30706","layer":"informal","project":"p29","title":"Boundary Operator","kind":"definition","summary":"[Boundary Operator] Let S denote the matrix whose rows are indexed by the vertices and whose co…","labels":["def:S"],"detail_key":"p29"},{"id":"n30707","layer":"informal","project":"p29","title":"lem:LSS","kind":"lemma","summary":"We note that L can be written as \\[ L = SS^*. \\]","labels":["lem:LSS"],"detail_key":"p29"},{"id":"n30708","layer":"informal","project":"p29","title":"lem:LapHermitian","kind":"lemma","summary":"The Matrix L is hermitian.","labels":["lem:LapHermitian"],"detail_key":"p29"},{"id":"n30709","layer":"informal","project":"p29","title":"By spectral theorem.","kind":"proof","summary":"By spectral theorem.","labels":[],"detail_key":"p29"},{"id":"n30710","layer":"informal","project":"p29","title":"def:lapEigvals","kind":"definition","summary":"Following \\reflem:LapHermitian, the eigenvalues of L are all real and non-negative.","labels":["def:lapEigvals"],"detail_key":"p29"},{"id":"n30711","layer":"informal","project":"p29","title":"def:DirichletSum","kind":"definition","summary":"The Dirichlet sum of a graph G for a function f : \\alpha \\to R is defined as \\[ \\sum_u \\sim v(f…","labels":["def:DirichletSum"],"detail_key":"p29"},{"id":"n30712","layer":"informal","project":"p29","title":"Let \\tau denote the constant function which assigns the value 1 on each vertex.","kind":"definition","summary":"Let \\tau denote the constant function which assigns the value 1 on each vertex.","labels":[],"detail_key":"p29"},{"id":"n30713","layer":"informal","project":"p29","title":"thm:zero_mem_lapSpectrum","kind":"theorem","summary":"T^1/2 * \\tau is an eigenfunction of L with eigenvalue 0.","labels":["thm:zero_mem_lapSpectrum"],"detail_key":"p29"},{"id":"n30714","layer":"informal","project":"p29","title":"lem:eigval_sum_le_n","kind":"lemma","summary":"The sum of the eigenvalues of the graph is at most its number of vertices, that is \\[ \\sum_i\\la…","labels":["lem:eigval_sum_le_n"],"detail_key":"p29"},{"id":"n30715","layer":"informal","project":"p29","title":"Follows from considering the trace of L.","kind":"proof","summary":"Follows from considering the trace of L.","labels":[],"detail_key":"p29"},{"id":"n30716","layer":"informal","project":"p29","title":"lem:eigval_sum_eq_n_iff_no_isolation","kind":"lemma","summary":"In the previous lemma, the equality holds if and only if G has no isolated vertices.","labels":["lem:eigval_sum_eq_n_iff_no_isolation"],"detail_key":"p29"},{"id":"n30717","layer":"informal","project":"p29","title":"lem:second_eigval_le_div","kind":"lemma","summary":"When n>1, we obtain the following upper bound on the second eigenvalue: \\[ \\lambda_1 \\le \\fracn…","labels":["lem:second_eigval_le_div"],"detail_key":"p29"},{"id":"n30718","layer":"informal","project":"p29","title":"Walk","kind":"definition","summary":"[Walk] A walk is a sequence of adjacent vertices. For vertices u, v \\in V, the walk between the…","labels":["def:Walk"],"detail_key":"p29"},{"id":"n30719","layer":"informal","project":"p29","title":"Reachable","kind":"definition","summary":"[Reachable] Two vertices u and v are said to be reachable if there is a walk between them.","labels":["def:Reachable"],"detail_key":"p29"},{"id":"n30720","layer":"informal","project":"p29","title":"Preconnected","kind":"definition","summary":"[Preconnected] A graph is preconnected if every pair of vertices is reachable from one another.","labels":["def:Preconnected"],"detail_key":"p29"},{"id":"n30721","layer":"informal","project":"p29","title":"Connected","kind":"definition","summary":"[Connected] A graph is connected if it is preconnected and contains at least one vertex.","labels":["def:Connected"],"detail_key":"p29"},{"id":"n30722","layer":"informal","project":"p29","title":"Edge Boundary","kind":"definition","summary":"[Edge Boundary] For a subset of vertices S \\subseteq V, the edge boundary \\partial S consists o…","labels":["def:edgeBoundary"],"detail_key":"p29"},{"id":"n30723","layer":"informal","project":"p29","title":"lem:walk_crossing","kind":"lemma","summary":"If there is a walk going from a vertex u \\in S to a vertex v \\in S^c, then there is at least on…","labels":["lem:walk_crossing"],"detail_key":"p29"},{"id":"n30724","layer":"informal","project":"p29","title":"lem:connected_non_empty_edge_boundary","kind":"lemma","summary":"In a connected graph, every set of vertices S different from the empty set and the universal se…","labels":["lem:connected_non_empty_edge_boundary"],"detail_key":"p29"},{"id":"n30725","layer":"informal","project":"p29","title":"Edge Connection","kind":"definition","summary":"[Edge Connection] For two sets of vertices A and B, E(A, B) denotes the set of edges with one e…","labels":["def:edgeConnection"],"detail_key":"p29"},{"id":"n30726","layer":"informal","project":"p29","title":"lem:edgeBoundaryComplement","kind":"lemma","summary":"For any set of vertices S, the edge boundary of S is equal to the edge boundary of its compleme…","labels":["lem:edgeBoundaryComplement"],"detail_key":"p29"},{"id":"n30727","layer":"informal","project":"p29","title":"lem:self_connection_eq_boundary","kind":"lemma","summary":"The edge boundary of a set S is equal to the edge connection between S and S^c: \\partial S = E(…","labels":["lem:self_connection_eq_boundary"],"detail_key":"p29"},{"id":"n30728","layer":"informal","project":"p29","title":"def:hG","kind":"definition","summary":"For a vertex set S, we define h_G(S) as the ratio of the size of its edge boundary to the minim…","labels":["def:hG"],"detail_key":"p29"},{"id":"n30729","layer":"informal","project":"p29","title":"Cheeger Constant","kind":"definition","summary":"[Cheeger Constant] The Cheeger constant of a graph G, denoted h_G, is defined as the minimum of…","labels":["def:cheeger"],"detail_key":"p29"},{"id":"n30730","layer":"informal","project":"p29","title":"lem:cheeger_mul_volume_le_volume_frontier","kind":"lemma","summary":"For any set of vertices S satisfying vol(S) \\le vol(S^c) and 0 < \\min(vol(S), vol(S^c)), we hav…","labels":["lem:cheeger_mul_volume_le_volume_frontier"],"detail_key":"p29"},{"id":"n30731","layer":"informal","project":"p29","title":"lem:connected_iff_cheeger_pos","kind":"lemma","summary":"A graph G is connected if and only if its Cheeger constant is strictly positive: G \\text is con…","labels":["lem:connected_iff_cheeger_pos"],"detail_key":"p29"},{"id":"n30732","layer":"informal","project":"p29","title":"lem:two_cheegerV_ge_first_eigval","kind":"lemma","summary":"We derive a simple upper bound for the first non-trivial eigenvalue \\lambda_1 in terms of the C…","labels":["lem:two_cheegerV_ge_first_eigval"],"detail_key":"p29"},{"id":"n30733","layer":"informal","project":"p29","title":"Normalized adjacency matrix","kind":"definition","summary":"[Normalized adjacency matrix] The normalized adjacency matrix of G is M = \\frac1d A, where A is…","labels":["def:norm_adj_matrix"],"detail_key":"p29"},{"id":"n30734","layer":"informal","project":"p29","title":"lem:norm_adj_matrix_isHermitian","kind":"lemma","summary":"M is symmetric, hence Hermitian, so the spectral theorem provides an orthonormal basis of eigen…","labels":["lem:norm_adj_matrix_isHermitian"],"detail_key":"p29"},{"id":"n30735","layer":"informal","project":"p29","title":"lem:norm_adj_matrix_mulVec_const","kind":"lemma","summary":"Every constant function is an eigenvector of M with eigenvalue 1: each vertex has exactly d nei…","labels":["lem:norm_adj_matrix_mulVec_const"],"detail_key":"p29"},{"id":"n30736","layer":"informal","project":"p29","title":"thm:eq_const_of_mulVec_eq","kind":"theorem","summary":"Let G be connected and d-regular. Then every eigenvector of M with eigenvalue 1 is constant.","labels":["thm:eq_const_of_mulVec_eq"],"detail_key":"p29"},{"id":"n30737","layer":"informal","project":"p29","title":"Let f satisfy Mf = f and let m = f(v_0) be the maximum value of f, attained at some verte…","kind":"proof","summary":"Let f satisfy Mf = f and let m = f(v_0) be the maximum value of f, attained at some vertex v_0…","labels":[],"detail_key":"p29"},{"id":"n30738","layer":"informal","project":"p29","title":"lem:eigval_eq_one_unique","kind":"lemma","summary":"On a connected d-regular graph the eigenvalue 1 is simple: two distinct basis indices cannot bo…","labels":["lem:eigval_eq_one_unique"],"detail_key":"p29"},{"id":"n30739","layer":"informal","project":"p29","title":"Expander","kind":"definition","summary":"[Expander] G is a \\gamma-expander when every eigenvalue of M other than the top eigenvalue 1 is…","labels":["def:expander"],"detail_key":"p29"},{"id":"n30740","layer":"informal","project":"p29","title":"def:e_AB","kind":"definition","summary":"For vertex sets A, B \\subseteq V, e(A, B) = |\\(a,b) \\in E(G) \\;:\\; a \\in A, b \\in B\\| is the nu…","labels":["def:e_AB"],"detail_key":"p29"},{"id":"n30741","layer":"informal","project":"p29","title":"Expander mixing lemma","kind":"theorem","summary":"[Expander mixing lemma] Let G be a connected d-regular \\gamma-expander on n vertices and let A,…","labels":["thm:expander_mixing_lemma"],"detail_key":"p29"},{"id":"n30742","layer":"informal","project":"p29","title":"Let 1_A, 1_B : V \\to R be the indicator functions of A and B, and write their expansions…","kind":"proof","summary":"Let 1_A, 1_B : V \\to R be the indicator functions of A and B, and write their expansions in the…","labels":[],"detail_key":"p29"},{"id":"n30743","layer":"informal","project":"p30","title":"simplicial set","kind":"defn","summary":"[simplicial set] A simplicial set is a presheaf on the simplex category.","labels":["defn:simplicial-set"],"detail_key":"p30"},{"id":"n30744","layer":"informal","project":"p30","title":"homotopic 1-simplices in a quasi-category","kind":"lemma","summary":"[homotopic 1-simplices in a quasi-category] Parallel 1-sim\\-plices f and g in a quasi-category…","labels":["lem:qcat-1-simplex-htpy"],"detail_key":"p30"},{"id":"n30745","layer":"informal","project":"p30","title":"A lengthy exercise in low-dimensional horn filling.","kind":"proof","summary":"A lengthy exercise in low-dimensional horn filling.","labels":[],"detail_key":"p30"},{"id":"n30746","layer":"informal","project":"p30","title":"lem:1-simplex-htpy-in-homotopy-cat","kind":"lem","summary":"Homotopic 1-simplices in a simplicial set represent the same arrow in the homotopy category.","labels":["lem:1-simplex-htpy-in-homotopy-cat"],"detail_key":"p30"},{"id":"n30747","layer":"informal","project":"p30","title":"This should be relatively straightforward.","kind":"proof","summary":"This should be relatively straightforward.","labels":[],"detail_key":"p30"},{"id":"n30748","layer":"informal","project":"p30","title":"lem:ho-preserves-small-products","kind":"lemma","summary":"The functor \\mathord\\mathsfh\\colon \\mathordsSet\\to \\mathordCat preserves finite products.","labels":["lem:ho-preserves-small-products"],"detail_key":"p30"},{"id":"n30749","layer":"informal","project":"p30","title":"We have a canonical comparison functor from the homotopy category of the products to the…","kind":"proof","summary":"We have a canonical comparison functor from the homotopy category of the products to the produc…","labels":[],"detail_key":"p30"},{"id":"n30750","layer":"informal","project":"p30","title":"isomorphism in a quasi-category","kind":"definition","summary":"[isomorphism in a quasi-category] A 1-simplex in a quasi-category is an isomorphism\\footnoteJoy…","labels":["defn:isomorphism"],"detail_key":"p30"},{"id":"n30751","layer":"informal","project":"p30","title":"prop:coherent-iso","kind":"proposition","summary":"An arrow f in a quasi-category A is an isomorphism if and only if it extends to a homotopy cohe…","labels":["prop:coherent-iso"],"detail_key":"p30"},{"id":"n30752","layer":"informal","project":"p30","title":"lem:qcat-htpy-cat-equiv","kind":"lemma","summary":"If f \\colon A \\to B is an equivalence of quasi-categories, then the functor \\mathord\\mathsfhf \\…","labels":["lem:qcat-htpy-cat-equiv"],"detail_key":"p30"},{"id":"n30753","layer":"informal","project":"p30","title":"defn:n-arrow","kind":"definition","summary":"For each n \\geq 0, an n-simplex in \\mathordA(x,y) is referred to as an n-arrow from x to y.","labels":["defn:n-arrow"],"detail_key":"p30"},{"id":"n30754","layer":"informal","project":"p30","title":"lem:cat-of-n-arrows","kind":"lemma","summary":"For any simplicial category \\mathordA and n \\geq 0, the n-arrows assemble into the arrows of an…","labels":["lem:cat-of-n-arrows"],"detail_key":"p30"},{"id":"n30755","layer":"informal","project":"p30","title":"The category of n-arrows is easy to construct directly. Alternatively, this result can be…","kind":"proof","summary":"The category of n-arrows is easy to construct directly. Alternatively, this result can be prove…","labels":[],"detail_key":"p30"},{"id":"n30756","layer":"informal","project":"p30","title":"lem:cotensor-associativity","kind":"lemma","summary":"When a simplicial category has cotensors, cotensors are associative: given A \\in \\mathordA and…","labels":["lem:cotensor-associativity"],"detail_key":"p30"},{"id":"n30757","layer":"informal","project":"p30","title":"By the enriched Yoneda lemma, these objects represent the same simplicial functors \\matho…","kind":"proof","summary":"By the enriched Yoneda lemma, these objects represent the same simplicial functors \\mathordA^\\m…","labels":[],"detail_key":"p30"},{"id":"n30758","layer":"informal","project":"p30","title":"defn:equivalence","kind":"definition","summary":"In an \\infty-cosmos \\mathordK, a morphism f\\colon A \\to B is an equivalence just when the induc…","labels":["defn:equivalence"],"detail_key":"p30"},{"id":"n30759","layer":"informal","project":"p30","title":"defn:trivial-fibration","kind":"definition","summary":"In an \\infty-cosmos \\mathordK, a morphism f\\colon A \\to B is a trivial fibration just when f is…","labels":["defn:trivial-fibration"],"detail_key":"p30"},{"id":"n30760","layer":"informal","project":"p30","title":"defn:underlying-cat-of-cosmos","kind":"definition","summary":"The underlying category \\mathordK_0 of an \\infty-cosmos \\mathordK is the category whose objects…","labels":["defn:underlying-cat-of-cosmos"],"detail_key":"p30"},{"id":"n30761","layer":"informal","project":"p30","title":"trivial fibrations and conical limits","kind":"lemma","summary":"[trivial fibrations and conical limits] The trivial fibrations in an \\infty-cosmos define a sub…","labels":["lem:trivial-fib-conical"],"detail_key":"p30"},{"id":"n30762","layer":"informal","project":"p30","title":"We know i","kind":"proof","summary":"We know i","labels":[],"detail_key":"p30"},{"id":"n30763","layer":"informal","project":"p30","title":"trivial fibrations and cotensors","kind":"lemma","summary":"[trivial fibrations and cotensors] In an \\infty-cosmos, the Leibniz cotensors of any trivial fi…","labels":["lem:trivial-fib-leibniz"],"detail_key":"p30"},{"id":"n30764","layer":"informal","project":"p30","title":"We know in","kind":"proof","summary":"We know in","labels":[],"detail_key":"p30"},{"id":"n30765","layer":"informal","project":"p30","title":"representable trivial fibrations","kind":"lemma","summary":"[representable trivial fibrations] If E \\mathrel\\ooalign\\xrightarrow[\\mkern4mu]\\smash\\mathlower…","labels":["lem:trivial-fib-representability"],"detail_key":"p30"},{"id":"n30766","layer":"informal","project":"p30","title":"By axiom \\refdefn:cosmos\\refitm:cosmos-isofib and the definition of the trivial fibration…","kind":"proof","summary":"By axiom \\refdefn:cosmos\\refitm:cosmos-isofib and the definition of the trivial fibrations in a…","labels":[],"detail_key":"p30"},{"id":"n30767","layer":"informal","project":"p30","title":"equivalences are homotopy equivalences","kind":"lemma","summary":"[equivalences are homotopy equivalences] A map f \\colon A \\to B between \\infty-categories in an…","labels":["lem:equiv-htpy-equiv"],"detail_key":"p30"},{"id":"n30768","layer":"informal","project":"p30","title":"By hypothesis, if f \\colon A \\to B defines an equivalence in the \\infty-cosmos \\mathordK…","kind":"proof","summary":"By hypothesis, if f \\colon A \\to B defines an equivalence in the \\infty-cosmos \\mathordK then t…","labels":[],"detail_key":"p30"},{"id":"n30769","layer":"informal","project":"p30","title":"lem:equivalence-2-of-3","kind":"lemma","summary":"The equivalences in an \\infty-cosmos are closed under retracts and satisfy the 2-of-3 property:…","labels":["lem:equivalence-2-of-3"],"detail_key":"p30"},{"id":"n30770","layer":"informal","project":"p30","title":"Let f \\colon A \\xrightarrow\\smash\\mathlower0.8\\simB be an equivalence equipped with the d…","kind":"proof","summary":"Let f \\colon A \\xrightarrow\\smash\\mathlower0.8\\simB be an equivalence equipped with the data de…","labels":[],"detail_key":"p30"},{"id":"n30771","layer":"informal","project":"p30","title":"trivial fibrations split","kind":"lem","summary":"[trivial fibrations split] Every trivial fibration admits a section & E \\arrow[d, two heads, \"\\…","labels":["lem:split-triv-fib"],"detail_key":"p30"},{"id":"n30772","layer":"informal","project":"p30","title":"If p \\colon E \\mathrel\\ooalign\\xrightarrow[\\mkern4mu]\\smash\\mathlower0.8\\sim\\mkern4mu\\cr…","kind":"proof","summary":"If p \\colon E \\mathrel\\ooalign\\xrightarrow[\\mkern4mu]\\smash\\mathlower0.8\\sim\\mkern4mu\\cr \\hidew…","labels":[],"detail_key":"p30"},{"id":"n30773","layer":"informal","project":"p30","title":"Brown factorization lemma","kind":"lemma","summary":"[Brown factorization lemma] Any functor f\\colon A \\to B in an \\infty-cosmos may be factored as…","labels":["lem:brown-fact"],"detail_key":"p30"},{"id":"n30774","layer":"informal","project":"p30","title":"The displayed factorization is constructed by the pullback of an isofibration formed by t…","kind":"proof","summary":"The displayed factorization is constructed by the pullback of an isofibration formed by the sim…","labels":[],"detail_key":"p30"},{"id":"n30775","layer":"informal","project":"p30","title":"prop:change-of-base-adjunction","kind":"proposition","summary":"Any adjunction between cartesian closed categories whose left adjoint preserves finite products…","labels":["prop:change-of-base-adjunction"],"detail_key":"p30"},{"id":"n30776","layer":"informal","project":"p30","title":"Of course r","kind":"proof","summary":"Of course r","labels":[],"detail_key":"p30"},{"id":"n30777","layer":"informal","project":"p30","title":"cor:free-underlying-2-adj","kind":"corollary","summary":"For any cartesian closed category \\mathordV with coproducts, the underlying category constructi…","labels":["cor:free-underlying-2-adj"],"detail_key":"p30"},{"id":"n30778","layer":"informal","project":"p30","title":"lem:enriched-cob-adjunction","kind":"lemma","summary":"Any adjunction comprised of finite-product-preserving functors between cartesian closed categor…","labels":["lem:enriched-cob-adjunction"],"detail_key":"p30"},{"id":"n30779","layer":"informal","project":"p30","title":"The internal action U_a,b \\colon U\\mathordW(a,b) \\to \\mathordV(Ua,Ub) of the \\mathordV-fu…","kind":"proof","summary":"The internal action U_a,b \\colon U\\mathordW(a,b) \\to \\mathordV(Ua,Ub) of the \\mathordV-functor…","labels":[],"detail_key":"p30"},{"id":"n30780","layer":"informal","project":"p30","title":"prop:cob-adjunction","kind":"proposition","summary":"Given an adjunction between cartesian closed categories [column sep=large] \\mathordV\\arrow[r, b…","labels":["prop:cob-adjunction"],"detail_key":"p30"},{"id":"n30781","layer":"informal","project":"p30","title":"Suppose \\mathordC admits cotensors as a \\mathordW-category. To verify that U_*\\mathordC a…","kind":"proof","summary":"Suppose \\mathordC admits cotensors as a \\mathordW-category. To verify that U_*\\mathordC admits…","labels":[],"detail_key":"p30"},{"id":"n30782","layer":"informal","project":"p30","title":"lem:pro-normal-monoidal","kind":"lemma","summary":"The change-of-base 2-functor induced by a finite-product-preserving functor T \\colon \\mathordV\\…","labels":["lem:pro-normal-monoidal"],"detail_key":"p30"},{"id":"n30783","layer":"informal","project":"p30","title":"The displayed function defines the component at v \\in \\mathordV of the unique monoidal na…","kind":"proof","summary":"The displayed function defines the component at v \\in \\mathordV of the unique monoidal natural…","labels":[],"detail_key":"p30"},{"id":"n30784","layer":"informal","project":"p30","title":"lem:right-of-monoidal-adj-ucats","kind":"lemma","summary":"Consider a finite-product-preserving adjunction between cartesian closed categories: [column se…","labels":["lem:right-of-monoidal-adj-ucats"],"detail_key":"p30"},{"id":"n30785","layer":"informal","project":"p30","title":"Let \\mathordC be a \\mathordW category. Then the hom-set in the underlying category of U_*…","kind":"proof","summary":"Let \\mathordC be a \\mathordW category. Then the hom-set in the underlying category of U_*\\matho…","labels":[],"detail_key":"p30"},{"id":"n30786","layer":"informal","project":"p30","title":"ex:nerve-ho-change-of-base","kind":"example","summary":"Both adjoints of the adjunction [column sep=large] sSet \\arrow[r, bend left=20, start anchor=10…","labels":["ex:nerve-ho-change-of-base"],"detail_key":"p30"},{"id":"n30787","layer":"informal","project":"p30","title":"the \\infty-cosmos of quasi-categories","kind":"proposition","summary":"[the \\infty-cosmos of quasi-categories] The full subcategory \\mathordQCat\\subset\\mathordsSet of…","labels":["prop:qcat-cosmos"],"detail_key":"p30"},{"id":"n30788","layer":"informal","project":"p30","title":"The proof requires myriad combinatorial results about the class of isofibrations between…","kind":"proof","summary":"The proof requires myriad combinatorial results about the class of isofibrations between quasi-…","labels":[],"detail_key":"p30"},{"id":"n30789","layer":"informal","project":"p30","title":"isofibrations of categories","kind":"definition","summary":"[isofibrations of categories] An isofibration between categories is a functor f \\colon A \\fib B…","labels":["defn:cat-isofibration"],"detail_key":"p30"},{"id":"n30790","layer":"informal","project":"p30","title":"the \\infty-cosmos of categories","kind":"proposition","summary":"[the \\infty-cosmos of categories] The category \\mathordCat of 1-cat\\-e\\-go\\-ri\\-es defines an \\…","labels":["prop:cat-cosmos"],"detail_key":"p30"},{"id":"n30791","layer":"informal","project":"p30","title":"It is well-known that the 2-category of categories is complete (and in fact also cocomple…","kind":"proof","summary":"It is well-known that the 2-category of categories is complete (and in fact also cocomplete) as…","labels":[],"detail_key":"p30"},{"id":"n30792","layer":"informal","project":"p30","title":"the \\infty-cosmos of Kan complexes","kind":"proposition","summary":"[the \\infty-cosmos of Kan complexes] The category \\mathordKan of Kan complexes defines an \\inft…","labels":["prop:kan-cosmos"],"detail_key":"p30"},{"id":"n30793","layer":"informal","project":"p30","title":"sliced \\infty-cosmoi","kind":"proposition","summary":"[sliced \\infty-cosmoi] For any \\infty-cosmos \\mathordK and any \\infty-cat\\-e\\-gory B \\in \\matho…","labels":["prop:sliced-cosmoi","itm:sliced-objects","itm:sliced-functor-space","itm:sliced-isofibrations","itm:sliced-products","itm:sliced-pullbacks","itm:sliced-cotensors","itm:fibered-equivalence"],"detail_key":"p30"},{"id":"n30794","layer":"informal","project":"p30","title":"The functor spaces are quasi-categories since axiom \\refdefn:cosmos\\refitm:cosmos-isofib…","kind":"proof","summary":"The functor spaces are quasi-categories since axiom \\refdefn:cosmos\\refitm:cosmos-isofib assert…","labels":[],"detail_key":"p30"},{"id":"n30795","layer":"informal","project":"p30","title":"cartesian closed \\infty-cosmoi","kind":"definition","summary":"[cartesian closed \\infty-cosmoi] An \\infty-cosmos \\mathordK is cartesian closed if the product…","labels":["defn:closed-cosmos"],"detail_key":"p30"},{"id":"n30796","layer":"informal","project":"p30","title":"prop:2-cat-as-cat-enriched","kind":"proposition","summary":"There is an equivalence between categories enriched in categories and strict bicategories. In p…","labels":["prop:2-cat-as-cat-enriched"],"detail_key":"p30"},{"id":"n30797","layer":"informal","project":"p30","title":"homotopy 2-category","kind":"definition","summary":"[homotopy 2-category] Let \\mathordK be an \\infty-cosmos. Its homotopy 2-category is the 2-categ…","labels":["defn:homotopy-2-cat"],"detail_key":"p30"},{"id":"n30798","layer":"informal","project":"p30","title":"underlying category of a 2-category","kind":"definition","summary":"[underlying category of a 2-category] The underlying category of a 2-category is defined by sim…","labels":["defn:underlying-cat-of-2cat"],"detail_key":"p30"},{"id":"n30799","layer":"informal","project":"p30","title":"lem:underlying-cat-iso","kind":"lemma","summary":"The underlying category of the homotopy 2-category of an \\infty-cosmos is isomorphic to the und…","labels":["lem:underlying-cat-iso"],"detail_key":"p30"},{"id":"n30800","layer":"informal","project":"p30","title":"lem:invertible-2-cell","kind":"lemma","summary":"\\quad \\item Every 2-cell A \\arrow[r, start anchor=15, end anchor=165, bend left, \"f\"] \\arrow[r,…","labels":["lem:invertible-2-cell","itm:2-cell-as-functor","itm:2-iso-as-functor"],"detail_key":"p30"},{"id":"n30801","layer":"informal","project":"p30","title":"The stateme","kind":"proof","summary":"The stateme","labels":[],"detail_key":"p30"},{"id":"n30802","layer":"informal","project":"p30","title":"cartesian (closure)","kind":"proposition","summary":"[cartesian (closure)] \\quad \\item The homotopy 2-category of any \\infty-cosmos has 2-categorica…","labels":["prop:htpy-2-cat-closure","itm:cart-closed"],"detail_key":"p30"},{"id":"n30803","layer":"informal","project":"p30","title":"While the functor \\mathord\\mathsfh\\colon\\mathordsSet\\to \\mathordCat only preserves finite…","kind":"proof","summary":"While the functor \\mathord\\mathsfh\\colon\\mathordsSet\\to \\mathordCat only preserves finite produ…","labels":[],"detail_key":"p30"},{"id":"n30804","layer":"informal","project":"p30","title":"equivalence","kind":"definition","summary":"[equivalence] An equivalence in a 2-category is given by \\item a pair of objects A and B; \\item…","labels":["defn:2-cat-equivalence"],"detail_key":"p30"},{"id":"n30805","layer":"informal","project":"p30","title":"equivalences are equivalences","kind":"theorem","summary":"[equivalences are equivalences] In any \\infty-cosmos \\mathordK, the following are equivalent an…","labels":["thm:equiv-are-equiv","itm:rep-equiv","itm:2cat-equiv","itm:htpy-equiv"],"detail_key":"p30"},{"id":"n30806","layer":"informal","project":"p30","title":"For \\refitm:rep-equiv\\Rightarrow\\refitm:2cat-equiv, if the induced map f_* \\colon \\mathor…","kind":"proof","summary":"For \\refitm:rep-equiv\\Rightarrow\\refitm:2cat-equiv, if the induced map f_* \\colon \\mathord\\math…","labels":[],"detail_key":"p30"},{"id":"n30807","layer":"informal","project":"p30","title":"cor:equiv-invar-fun","kind":"corollary","summary":"Equivalences of \\infty-categories A' \\xrightarrow\\smash\\mathlower0.8\\simA and B \\xrightarrow\\sm…","labels":["cor:equiv-invar-fun"],"detail_key":"p30"},{"id":"n30808","layer":"informal","project":"p30","title":"The represe","kind":"proof","summary":"The represe","labels":[],"detail_key":"p30"},{"id":"n30809","layer":"informal","project":"p30","title":"isofibrations are isofibrations","kind":"proposition","summary":"[isofibrations are isofibrations] An isofibration p \\colon E \\fib B in an \\infty-cosmos \\mathor…","labels":["prop:isofib-define-isofib"],"detail_key":"p30"},{"id":"n30810","layer":"informal","project":"p30","title":"The universal property of the statement says that the functor \\[p_* \\colon \\mathord\\maths…","kind":"proof","summary":"The universal property of the statement says that the functor \\[p_* \\colon \\mathord\\mathsfhFun(…","labels":[],"detail_key":"p30"},{"id":"n30811","layer":"informal","project":"p30","title":"homotopy category of an \\infty-category","kind":"definition","summary":"[homotopy category of an \\infty-category] The homotopy category of an \\infty-category A in an \\…","labels":["defn:htpy-cat-of-infinity-cat"],"detail_key":"p30"},{"id":"n30812","layer":"informal","project":"p31","title":"def:fourier-transform","kind":"definition","summary":"Let f\\in L^1(V,E). Its \\emphFourier transform (w.r.t. L) is the function Ff=\\widehat f:W\\to E g…","labels":["def:fourier-transform"],"detail_key":"p31"},{"id":"n30813","layer":"informal","project":"p31","title":"lem:fourier-bounded","kind":"lemma","summary":"Let f\\in L^1(V,E). Then its Fourier transform \\widehat f is well-defined and bounded. In partic…","labels":["lem:fourier-bounded"],"detail_key":"p31"},{"id":"n30814","layer":"informal","project":"p31","title":"Omitted.","kind":"proof","summary":"Omitted.","labels":[],"detail_key":"p31"},{"id":"n30815","layer":"informal","project":"p31","title":"lem:fourier-cont","kind":"lemma","summary":"Let f\\in L^1(V,E). Then \\widehat f is continuous.","labels":["lem:fourier-cont"],"detail_key":"p31"},{"id":"n30816","layer":"informal","project":"p31","title":"Omitted.","kind":"proof","summary":"Omitted.","labels":[],"detail_key":"p31"},{"id":"n30817","layer":"informal","project":"p31","title":"Multiplication formula","kind":"lemma","summary":"[Multiplication formula] Let f,g\\in L^1(V,E). Then \\int_WM(\\widehat f(w),g(w))\\,d\\nu(w)=\\int_VM…","labels":["lem:fourier-multiplication"],"detail_key":"p31"},{"id":"n30818","layer":"informal","project":"p31","title":"Omitted.","kind":"proof","summary":"Omitted.","labels":[],"detail_key":"p31"},{"id":"n30819","layer":"informal","project":"p31","title":"lem:fourier-prop","kind":"lemma","summary":"Lef f,g\\in L^1(V,E), t\\in\\R and a,b\\in\\C. The Fourier transform satisfies the following element…","labels":["lem:fourier-prop"],"detail_key":"p31"},{"id":"n30820","layer":"informal","project":"p31","title":"Omitted.","kind":"proof","summary":"Omitted.","labels":[],"detail_key":"p31"},{"id":"n30821","layer":"informal","project":"p31","title":"lem:fourier-gaussian","kind":"lemma","summary":"Let x\\in V and \\delta>0. Define the \\emphmodulated Gaussian u_x,\\delta(y):V\\to\\C,\\quad y\\mapsto…","labels":["lem:fourier-gaussian"],"detail_key":"p31"},{"id":"n30822","layer":"informal","project":"p31","title":"By choosing an orthonormal basis, wlog we may assume V=\\R^n. First note \\widehatu_x,\\delt…","kind":"proof","summary":"By choosing an orthonormal basis, wlog we may assume V=\\R^n. First note \\widehatu_x,\\delta(z-x)…","labels":[],"detail_key":"p31"},{"id":"n30823","layer":"informal","project":"p31","title":"lem:weierstrass-kernel","kind":"lemma","summary":"Let K_\\delta(v)=\\delta^-n/2e^-\\pi|v|^2/\\delta as in \\Creflem:fourier-gaussian. This is a \\emphg…","labels":["lem:weierstrass-kernel"],"detail_key":"p31"},{"id":"n30824","layer":"informal","project":"p31","title":"By choosing an orthonormal basis, wlog we may assume V=\\R^n. Then these are all straight-…","kind":"proof","summary":"By choosing an orthonormal basis, wlog we may assume V=\\R^n. Then these are all straight-forwar…","labels":[],"detail_key":"p31"},{"id":"n30825","layer":"informal","project":"p31","title":"thm:kernel-approximation","kind":"theorem","summary":"Let f:V\\to E be integrable. Let K_\\delta be the Weierstrass kernel from \\Creflem:weierstrass-ke…","labels":["thm:kernel-approximation"],"detail_key":"p31"},{"id":"n30826","layer":"informal","project":"p31","title":"Again we may assume V=\\R^n. Consider the difference \\Delta_\\delta(x):=(K_\\delta\\ast f)(x)…","kind":"proof","summary":"Again we may assume V=\\R^n. Consider the difference \\Delta_\\delta(x):=(K_\\delta\\ast f)(x)-f(x)=…","labels":[],"detail_key":"p31"},{"id":"n30827","layer":"informal","project":"p31","title":"One can drop the continuity assumption and still get pointwise convergence almost everywh…","kind":"remark","summary":"One can drop the continuity assumption and still get pointwise convergence almost everywhere. T…","labels":[],"detail_key":"p31"},{"id":"n30828","layer":"informal","project":"p31","title":"Inversion formula","kind":"theorem","summary":"[Inversion formula] Let f:V\\to E be integrable and continuous. Assume \\widehat f is integrable…","labels":["thm:fourier-inversion"],"detail_key":"p31"},{"id":"n30829","layer":"informal","project":"p31","title":"Apply the multiplication formula \\Creflem:fourier-multiplication to u_x,\\delta and f, and…","kind":"proof","summary":"Apply the multiplication formula \\Creflem:fourier-multiplication to u_x,\\delta and f, and concl…","labels":[],"detail_key":"p31"},{"id":"n30830","layer":"informal","project":"p31","title":"Note that both assumptions are necessary, since F^-1 Ff is continuous, and only defined i…","kind":"remark","summary":"Note that both assumptions are necessary, since F^-1 Ff is continuous, and only defined if Ff i…","labels":[],"detail_key":"p31"},{"id":"n30831","layer":"informal","project":"p31","title":"Inversion formula, L^1-version","kind":"theorem","summary":"[Inversion formula, L^1-version] Let f\\in L^1(V,E). If \\widehat f\\in L^1(V,E), then F^-1 Ff=f.","labels":["thm:fourier-inversion-L1"],"detail_key":"p31"},{"id":"n30832","layer":"informal","project":"p31","title":"Similar to \\Crefthm:fourier-inversion.","kind":"proof","summary":"Similar to \\Crefthm:fourier-inversion.","labels":[],"detail_key":"p31"},{"id":"n30833","layer":"informal","project":"p31","title":"Plancherel's Theorem","kind":"theorem","summary":"[Plancherel's Theorem] Suppose that f : V \\to E is in L^1(V,E)\\cap L^2(V,E) and let \\widehatf b…","labels":["thm:plancherel"],"detail_key":"p31"},{"id":"n30834","layer":"informal","project":"p31","title":"Let g(x)=f(-x) and apply the multiplication formula \\Creflem:fourier-multiplication to f\\…","kind":"proof","summary":"Let g(x)=f(-x) and apply the multiplication formula \\Creflem:fourier-multiplication to f\\ast g…","labels":[],"detail_key":"p31"},{"id":"n30835","layer":"informal","project":"p31","title":"lem:L12-dense","kind":"lemma","summary":"L^1(V,E)\\cap L^2(V,E) is dense in L^2(V,E).","labels":["lem:L12-dense"],"detail_key":"p31"},{"id":"n30836","layer":"informal","project":"p31","title":"It is well-known that the space of compactly supported continuous functions is dense in e…","kind":"proof","summary":"It is well-known that the space of compactly supported continuous functions is dense in every L…","labels":[],"detail_key":"p31"},{"id":"n30837","layer":"informal","project":"p31","title":"lem:fourier12-cauchy","kind":"lemma","summary":"Let f\\in L^2(V,E) and (f_n)_n\\subset L^1(V,E)\\cap L^2(V,E) a sequence with f_n\\xrightarrow[L^2]…","labels":["lem:fourier12-cauchy"],"detail_key":"p31"},{"id":"n30838","layer":"informal","project":"p31","title":"\\|\\widehat f_n-\\widehat f_m\\|_2=\\|\\widehatf_n-f_m\\|_2\\overset\\textPlancherel=\\|f_n-f_m\\|_…","kind":"proof","summary":"\\|\\widehat f_n-\\widehat f_m\\|_2=\\|\\widehatf_n-f_m\\|_2\\overset\\textPlancherel=\\|f_n-f_m\\|_2 goes…","labels":[],"detail_key":"p31"},{"id":"n30839","layer":"informal","project":"p31","title":"def:fourier-L2","kind":"definition","summary":"Let f\\in L^2(V,E) and take a sequence (f_n)_n\\subset L^1(V,E)\\cap L^2(V,E) with f_n\\xrightarrow…","labels":["def:fourier-L2"],"detail_key":"p31"},{"id":"n30840","layer":"informal","project":"p31","title":"lem:fourier2-welldef","kind":"lemma","summary":"This is well-defined: By \\Creflem:fourier12-cauchy, the limit exists. Further it does not depen…","labels":["lem:fourier2-welldef"],"detail_key":"p31"},{"id":"n30841","layer":"informal","project":"p31","title":"Let (g_n)_n be another sequence approximating f. Then \\|\\widehat f_n-\\widehat g_n\\|_2=\\|f…","kind":"proof","summary":"Let (g_n)_n be another sequence approximating f. Then \\|\\widehat f_n-\\widehat g_n\\|_2=\\|f_n-g_n…","labels":[],"detail_key":"p31"},{"id":"n30842","layer":"informal","project":"p31","title":"def:invFourier-L2","kind":"definition","summary":"Define analogously F^-1f:=\\check f:=\\lim_n\\to\\infty\\check f_n, if f_n\\xrightarrow[L^2]f\\in L^2(…","labels":["def:invFourier-L2"],"detail_key":"p31"},{"id":"n30843","layer":"informal","project":"p31","title":"thm:fourier2-properties","kind":"corollary","summary":"Plancherel's Theorem, the inversion formula, and the properties of \\Creflem:fourier-prop hold f…","labels":["thm:fourier2-properties"],"detail_key":"p31"},{"id":"n30844","layer":"informal","project":"p31","title":"All of these follow immediately from the definition and the observation, that all operati…","kind":"proof","summary":"All of these follow immediately from the definition and the observation, that all operations (n…","labels":[],"detail_key":"p31"},{"id":"n30845","layer":"informal","project":"p31","title":"thm:fourier-is-l2-linear","kind":"corollary","summary":"The Fourier transform induces a continuous linear map L^2(V,E) \\to L^2(V,E).","labels":["thm:fourier-is-l2-linear"],"detail_key":"p31"},{"id":"n30846","layer":"informal","project":"p31","title":"This follows immediately from \\Crefthm:fourier2-properties: Linearity from the L^2-versio…","kind":"proof","summary":"This follows immediately from \\Crefthm:fourier2-properties: Linearity from the L^2-version of \\…","labels":[],"detail_key":"p31"},{"id":"n30847","layer":"informal","project":"p31","title":"thm:maximum_modulus","kind":"theorem","summary":"Let U be a connected open set in a complex normed space E. Let f:E\\to F be a function that comp…","labels":["thm:maximum_modulus"],"detail_key":"p31"},{"id":"n30848","layer":"informal","project":"p31","title":"Already formalized in Mathlib, along with several variants.","kind":"proof","summary":"Already formalized in Mathlib, along with several variants.","labels":[],"detail_key":"p31"},{"id":"n30849","layer":"informal","project":"p31","title":"lem:threelines","kind":"lemma","summary":"Let S be the strip S:=\\z \\in \\C \\ | \\ 0 < Re \\, z < 1 \\. Let f: \\overlineS \\to \\C be a function…","labels":["lem:threelines"],"detail_key":"p31"},{"id":"n30850","layer":"informal","project":"p31","title":"~\\\\ If |\\phi| is constant, everything holds trivially by setting M_0 and M_1 to be the va…","kind":"proof","summary":"~\\\\ If |\\phi| is constant, everything holds trivially by setting M_0 and M_1 to be the value of…","labels":[],"detail_key":"p31"},{"id":"n30851","layer":"informal","project":"p31","title":"lem:snorm_eq_sSup_snorm","kind":"lemma","summary":"Let p and q be real conjugate exponents. Let f be measurable. Then \\[ \\|f\\|_L^q = \\sup_\\|g\\|_L^…","labels":["lem:snorm_eq_sSup_snorm"],"detail_key":"p31"},{"id":"n30852","layer":"informal","project":"p31","title":"That \\sup_\\|g\\|_L^p \\leq 1, \\ g \\text simple \\| fg \\|_L^1 \\le \\|f\\|_L^q follows from Höld…","kind":"proof","summary":"That \\sup_\\|g\\|_L^p \\leq 1, \\ g \\text simple \\| fg \\|_L^1 \\le \\|f\\|_L^q follows from Hölder's i…","labels":[],"detail_key":"p31"},{"id":"n30853","layer":"informal","project":"p31","title":"lem:snormEssSup_eq_sSup_snorm","kind":"lemma","summary":"Let f be measurable and the measure \\mu be \\sigma-finite. Then \\[ \\|f\\|_L^\\infty = \\sup_\\|g\\|_L…","labels":["lem:snormEssSup_eq_sSup_snorm"],"detail_key":"p31"},{"id":"n30854","layer":"informal","project":"p31","title":"That \\sup_\\|g\\|_L^1 \\leq 1, \\ g \\text simple \\| fg \\|_L^1 \\le \\|f\\|_L^\\infty follows from…","kind":"proof","summary":"That \\sup_\\|g\\|_L^1 \\leq 1, \\ g \\text simple \\| fg \\|_L^1 \\le \\|f\\|_L^\\infty follows from Hölde…","labels":[],"detail_key":"p31"},{"id":"n30855","layer":"informal","project":"p31","title":"lem:hoelder'","kind":"lemma","summary":"Let (X, \\mu) be a measure space and 0< p_0 <p_1\\leq \\infty, 0 \\le p \\le \\infty. Assume we have…","labels":["lem:hoelder'"],"detail_key":"p31"},{"id":"n30856","layer":"informal","project":"p31","title":"This is just a version of Hölder's inequality, but in order to apply it, we should rule o…","kind":"proof","summary":"This is just a version of Hölder's inequality, but in order to apply it, we should rule out som…","labels":[],"detail_key":"p31"},{"id":"n30857","layer":"informal","project":"p31","title":"thm:riesz_interpolation","kind":"theorem","summary":"Let (X, \\mu) and (Y, \\nu) be measure spaces and consider all L^p spaces to be complex valued.\\\\…","labels":["thm:riesz_interpolation"],"detail_key":"p31"},{"id":"n30858","layer":"informal","project":"p31","title":"For a valid choice of p,q, note that we both need to show Tf is in L^q and a bound on the…","kind":"proof","summary":"For a valid choice of p,q, note that we both need to show Tf is in L^q and a bound on the L^q n…","labels":[],"detail_key":"p31"},{"id":"n30859","layer":"informal","project":"p31","title":"lem:hausdorff_young","kind":"lemma","summary":"Let X=[0,2\\pi] with normalized Lebesgue measure \\fracd\\theta2\\pi and let Y=\\bbz with counting m…","labels":["lem:hausdorff_young"],"detail_key":"p31"},{"id":"n30860","layer":"informal","project":"p31","title":"Observe that we may simply regard T as an operator L^1([0, 2\\pi]) \\to L^\\infty(\\bbz) sinc…","kind":"proof","summary":"Observe that we may simply regard T as an operator L^1([0, 2\\pi]) \\to L^\\infty(\\bbz) since L^2(…","labels":[],"detail_key":"p31"},{"id":"n30861","layer":"informal","project":"p31","title":"lem:hausdorff_young_dual","kind":"lemma","summary":"For 1\\leq p \\leq 2 and q conjugate exponent to p, we have \\[ ||T' \\a_n\\||_L^q \\leq ||\\a_n\\||_L^…","labels":["lem:hausdorff_young_dual"],"detail_key":"p31"},{"id":"n30862","layer":"informal","project":"p31","title":"This is similar to the previous corollary. Parseval's identity gives the case p_0=q_0=2.\\…","kind":"proof","summary":"This is similar to the previous corollary. Parseval's identity gives the case p_0=q_0=2.\\\\ For…","labels":[],"detail_key":"p31"},{"id":"n30863","layer":"informal","project":"p31","title":"\\cD(\\Omega) = C_c^\\infty(\\Omega) is the set of test functions together with a topology de…","kind":"definition","summary":"\\cD(\\Omega) = C_c^\\infty(\\Omega) is the set of test functions together with a topology determin…","labels":[],"detail_key":"p31"},{"id":"n30864","layer":"informal","project":"p31","title":"A notion of converging sequence on a set X is \\item The constant function on x converges…","kind":"remark","summary":"A notion of converging sequence on a set X is \\item The constant function on x converges to x \\…","labels":[],"detail_key":"p31"},{"id":"n30865","layer":"informal","project":"p31","title":"Every locally integrable function f \\in L^1_loc(\\Omega) gives us a distribution \\Lambda f…","kind":"example","summary":"Every locally integrable function f \\in L^1_loc(\\Omega) gives us a distribution \\Lambda f \\in D…","labels":[],"detail_key":"p31"},{"id":"n30866","layer":"informal","project":"p31","title":"Let \\mu be a Radon measure on \\Omega (or more generally a signed Borel measure which is f…","kind":"example","summary":"Let \\mu be a Radon measure on \\Omega (or more generally a signed Borel measure which is finite…","labels":[],"detail_key":"p31"},{"id":"n30867","layer":"informal","project":"p31","title":"As Borel sets are \\mu-measurable, every continuous function is \\mu-measurable. Let K := \\…","kind":"proof","summary":"As Borel sets are \\mu-measurable, every continuous function is \\mu-measurable. Let K := \\Supp \\…","labels":[],"detail_key":"p31"},{"id":"n30868","layer":"informal","project":"p31","title":"Dirac-\\delta","kind":"example","summary":"[Dirac-\\delta] We have \\delta \\in D'(\\Omega) given by \\[ \\delta(\\phi) := \\phi(0) \\]","labels":[],"detail_key":"p31"},{"id":"n30869","layer":"informal","project":"p31","title":"For \\phi \\in D define \\phi^R \\in D as \\phi^R(x) = \\phi(-x) and for x \\in \\R^d we have the…","kind":"notation","summary":"For \\phi \\in D define \\phi^R \\in D as \\phi^R(x) = \\phi(-x) and for x \\in \\R^d we have the shift…","labels":[],"detail_key":"p31"},{"id":"n30870","layer":"informal","project":"p31","title":"ex:conv","kind":"example","summary":"For f \\in L^1_loc(\\Omega), g \\in D(\\Omega) we have \\[ (f * g)(x) = \\Lambda f (\\tau_x (\\psi^R))…","labels":["ex:conv"],"detail_key":"p31"},{"id":"n30871","layer":"informal","project":"p31","title":"Let F \\in \\cD'(\\Omega) , \\psi \\in D. The following two distributions coincide: \\item The…","kind":"proposition","summary":"Let F \\in \\cD'(\\Omega) , \\psi \\in D. The following two distributions coincide: \\item The distri…","labels":[],"detail_key":"p31"},{"id":"n30872","layer":"informal","project":"p31","title":"\\zeta := \\psi^R The function x \\mapsto F(\\tau_x(\\psi^R)) is smooth : \\item It is continuo…","kind":"proof","summary":"\\zeta := \\psi^R The function x \\mapsto F(\\tau_x(\\psi^R)) is smooth : \\item It is continuous: If…","labels":[],"detail_key":"p31"},{"id":"n30873","layer":"informal","project":"p31","title":"lemma:distribCommInt","kind":"lemma","summary":"Let \\phi \\in C_c^\\infty(\\Omega \\times \\Omega). Then for any F \\in D'(\\Omega) we have \\[ F \\left…","labels":["lemma:distribCommInt"],"detail_key":"p31"},{"id":"n30874","layer":"informal","project":"p31","title":"Consider S_\\varepsilon \\in D defined by \\[S_\\varepsilon^\\phi(y) = \\varepsilon^d \\sum_n \\i…","kind":"proof","summary":"Consider S_\\varepsilon \\in D defined by \\[S_\\varepsilon^\\phi(y) = \\varepsilon^d \\sum_n \\in \\Z^d…","labels":[],"detail_key":"p31"},{"id":"n30875","layer":"informal","project":"p31","title":"From the description \\refex:conv: Writing \\Lambda f * g is unambiguous.","kind":"example","summary":"From the description \\refex:conv: Writing \\Lambda f * g is unambiguous.","labels":[],"detail_key":"p31"},{"id":"n30876","layer":"informal","project":"p31","title":"We have \\delta_0 * f = \\Lambda f for all f \\in D.","kind":"example","summary":"We have \\delta_0 * f = \\Lambda f for all f \\in D.","labels":[],"detail_key":"p31"},{"id":"n30877","layer":"informal","project":"p31","title":"lemma:cont","kind":"lemma","summary":"Convolution F * \\psi is continuous in both variables.","labels":["lemma:cont"],"detail_key":"p31"},{"id":"n30878","layer":"informal","project":"p31","title":"Continuity in the distribution variable is clear by pointwise convergence. \\\\ For the con…","kind":"proof","summary":"Continuity in the distribution variable is clear by pointwise convergence. \\\\ For the continuit…","labels":[],"detail_key":"p31"},{"id":"n30879","layer":"informal","project":"p31","title":"proposition:deltaInClosure","kind":"proposition","summary":"There exists a sequence \\psi_n \\in C_c^\\infty(\\Omega) such that \\Lambda \\psi_n \\to \\delta_0 in…","labels":["proposition:deltaInClosure"],"detail_key":"p31"},{"id":"n30880","layer":"informal","project":"p31","title":"Fix some \\psi \\in D with \\int \\psi(x) \\dm x = 1. Define \\psi_n (x) := n^d \\psi(nx). Then…","kind":"proof","summary":"Fix some \\psi \\in D with \\int \\psi(x) \\dm x = 1. Define \\psi_n (x) := n^d \\psi(nx). Then \\[ (\\L…","labels":[],"detail_key":"p31"},{"id":"n30881","layer":"informal","project":"p31","title":"Let f,g \\in L^1_loc(\\Omega) such that \\Lambda_f = \\Lambda_g. Then f = g almost everywhere.","kind":"corollary","summary":"Let f,g \\in L^1_loc(\\Omega) such that \\Lambda_f = \\Lambda_g. Then f = g almost everywhere.","labels":[],"detail_key":"p31"},{"id":"n30882","layer":"informal","project":"p31","title":"We have 0 = \\Lambda(f-g)(\\tau_\\bullet \\psi_n^R) = \\psi_n * (f - g) \\to \\delta_0 * (f-g) =…","kind":"proof","summary":"We have 0 = \\Lambda(f-g)(\\tau_\\bullet \\psi_n^R) = \\psi_n * (f - g) \\to \\delta_0 * (f-g) = f-g i…","labels":[],"detail_key":"p31"},{"id":"n30883","layer":"informal","project":"p31","title":"\\delta is not a function! I.e. its not of the form \\Lambda f for some f \\in L^1_loc","kind":"example","summary":"\\delta is not a function! I.e. its not of the form \\Lambda f for some f \\in L^1_loc","labels":[],"detail_key":"p31"},{"id":"n30884","layer":"informal","project":"p31","title":"We have \\Delta|_\\Omega \\setminus \\0\\= \\Lambda 0 so if it would be a function, then it wou…","kind":"proof","summary":"We have \\Delta|_\\Omega \\setminus \\0\\= \\Lambda 0 so if it would be a function, then it would be…","labels":[],"detail_key":"p31"},{"id":"n30885","layer":"informal","project":"p31","title":"Then C^\\infty(\\R^d) is dense in \\cD'(\\R^d).","kind":"corollary","summary":"Then C^\\infty(\\R^d) is dense in \\cD'(\\R^d).","labels":[],"detail_key":"p31"},{"id":"n30886","layer":"informal","project":"p31","title":"We know by \\refproposition:deltaInClosure that there exists \\Lambda \\psi_n \\to \\delta_0 i…","kind":"proof","summary":"We know by \\refproposition:deltaInClosure that there exists \\Lambda \\psi_n \\to \\delta_0 in \\cD'…","labels":[],"detail_key":"p31"},{"id":"n30887","layer":"informal","project":"p31","title":"For a multiindex \\alpha and a distribution F define the distribution \\[\\partial^\\alpha F…","kind":"definition","summary":"For a multiindex \\alpha and a distribution F define the distribution \\[\\partial^\\alpha F (\\phi)…","labels":[],"detail_key":"p31"},{"id":"n30888","layer":"informal","project":"p31","title":"let f \\in L^1_loc(\\Omega). If there exists some f' \\in L^1_loc(\\Omega), such that \\Lambda…","kind":"remark","summary":"let f \\in L^1_loc(\\Omega). If there exists some f' \\in L^1_loc(\\Omega), such that \\Lambda f' =…","labels":[],"detail_key":"p31"},{"id":"n30889","layer":"informal","project":"p31","title":"For F \\in D', \\phi \\in D , We have \\[\\partial^\\alpha (F * \\phi) = (\\partial^\\alpha F) * \\…","kind":"proposition","summary":"For F \\in D', \\phi \\in D , We have \\[\\partial^\\alpha (F * \\phi) = (\\partial^\\alpha F) * \\phi =…","labels":[],"detail_key":"p31"},{"id":"n30890","layer":"informal","project":"p31","title":"First note, that holds in the case where F is a test function, so that we have ordinary c…","kind":"proof","summary":"First note, that holds in the case where F is a test function, so that we have ordinary convolu…","labels":[],"detail_key":"p31"},{"id":"n30891","layer":"informal","project":"p31","title":"Consider the increasing sequence of norms on C^\\infty(\\Omega) defined by \\[ \\|\\phi\\|_N =…","kind":"definition","summary":"Consider the increasing sequence of norms on C^\\infty(\\Omega) defined by \\[ \\|\\phi\\|_N = \\sup \\…","labels":[],"detail_key":"p31"},{"id":"n30892","layer":"informal","project":"p31","title":"We have a continuous inclusions \\cD \\subset \\cS, hence \\cS' \\subset \\cD'(\\R^d). \\\\ Moreov…","kind":"lemma","summary":"We have a continuous inclusions \\cD \\subset \\cS, hence \\cS' \\subset \\cD'(\\R^d). \\\\ Moreover, th…","labels":[],"detail_key":"p31"},{"id":"n30893","layer":"informal","project":"p31","title":"lemma:slowlyInc","kind":"lemma","summary":"Let f \\in L^1_loc(\\R^d) such that there exists N \\ge 0 with \\[ \\int_|x| < R |f(x)| \\dm x = O(R^…","labels":["lemma:slowlyInc"],"detail_key":"p31"},{"id":"n30894","layer":"informal","project":"p31","title":"This condition holds for functions in L^p(\\R^d) for p \\in [1,\\infty]","kind":"example","summary":"This condition holds for functions in L^p(\\R^d) for p \\in [1,\\infty]","labels":[],"detail_key":"p31"},{"id":"n30895","layer":"informal","project":"p31","title":"If F \\in","kind":"lemma","summary":"If F \\in","labels":[],"detail_key":"p31"},{"id":"n30896","layer":"informal","project":"p31","title":"\\item All \\partial^\\alpha F are tempered. \\item Let \\psi \\in C^\\infty be slowly increasin…","kind":"lemma","summary":"\\item All \\partial^\\alpha F are tempered. \\item Let \\psi \\in C^\\infty be slowly increasing, i.e…","labels":[],"detail_key":"p31"},{"id":"n30897","layer":"informal","project":"p31","title":"If \\psi \\in \\cS, then F(\\tau_\\bullet (\\psi^R)) is slowly increasing. The other formulatio…","kind":"example","summary":"If \\psi \\in \\cS, then F(\\tau_\\bullet (\\psi^R)) is slowly increasing. The other formulation is s…","labels":[],"detail_key":"p31"},{"id":"n30898","layer":"informal","project":"p31","title":"The fourier transformation is a continuous bijection \\cS &\\to \\cS \\\\ \\phi &\\mapsto \\hat \\…","kind":"definition","summary":"The fourier transformation is a continuous bijection \\cS &\\to \\cS \\\\ \\phi &\\mapsto \\hat \\phi =…","labels":[],"detail_key":"p31"},{"id":"n30899","layer":"informal","project":"p31","title":"We have \\[\\Lambda_\\hat \\psi(\\phi) = \\int_\\R^d \\hat \\psi (x) \\phi(x) \\dm x = \\int_\\R^d \\ps…","kind":"lemma","summary":"We have \\[\\Lambda_\\hat \\psi(\\phi) = \\int_\\R^d \\hat \\psi (x) \\phi(x) \\dm x = \\int_\\R^d \\psi(x) \\…","labels":[],"detail_key":"p31"},{"id":"n30900","layer":"informal","project":"p31","title":"Define \\[ \\hat F (\\phi) = F (\\hat \\phi) \\] and similarly for the inverse transform f \\map…","kind":"definition","summary":"Define \\[ \\hat F (\\phi) = F (\\hat \\phi) \\] and similarly for the inverse transform f \\mapsto \\c…","labels":[],"detail_key":"p31"},{"id":"n30901","layer":"informal","project":"p31","title":"If 1 \\in S denotes the constant function at 1, then \\[ \\widehat \\delta = \\Lambda_1 \\] bec…","kind":"example","summary":"If 1 \\in S denotes the constant function at 1, then \\[ \\widehat \\delta = \\Lambda_1 \\] because \\…","labels":[],"detail_key":"p31"},{"id":"n30902","layer":"informal","project":"p31","title":"A fundamental solution of L is a distribution F such that L(F) = \\delta","kind":"definition","summary":"A fundamental solution of L is a distribution F such that L(F) = \\delta","labels":[],"detail_key":"p31"},{"id":"n30903","layer":"informal","project":"p31","title":"The operator \\[ f \\mapsto T(f) := F * f \\] defines an inverse to L","kind":"lemma","summary":"The operator \\[ f \\mapsto T(f) := F * f \\] defines an inverse to L","labels":[],"detail_key":"p31"},{"id":"n30904","layer":"informal","project":"p31","title":"because \\[ \\partial^\\alpha (F * f) = (\\partial^\\alpha F) * f = F * (\\partial^\\alpha f) \\]…","kind":"proof","summary":"because \\[ \\partial^\\alpha (F * f) = (\\partial^\\alpha F) * f = F * (\\partial^\\alpha f) \\] Summi…","labels":[],"detail_key":"p31"},{"id":"n30905","layer":"informal","project":"p31","title":"The characteristic polynomial of L is \\[P(\\xi) = \\sum_|\\alpha|\\le m a_\\alpha (2 \\pi i \\xi…","kind":"definition","summary":"The characteristic polynomial of L is \\[P(\\xi) = \\sum_|\\alpha|\\le m a_\\alpha (2 \\pi i \\xi)^\\alp…","labels":[],"detail_key":"p31"},{"id":"n30906","layer":"informal","project":"p31","title":"For \\lambda > -d, Let H_\\lambda be the tempered distribution associated to |x|^\\lambda \\i…","kind":"theorem","summary":"For \\lambda > -d, Let H_\\lambda be the tempered distribution associated to |x|^\\lambda \\in L^1_…","labels":[],"detail_key":"p31"},{"id":"n30907","layer":"informal","project":"p31","title":"If d = 2, the function F := 1/(2\\pi) \\log |x| \\in L^1_loc is a fundamental solution of \\D…","kind":"proposition","summary":"If d = 2, the function F := 1/(2\\pi) \\log |x| \\in L^1_loc is a fundamental solution of \\Delta","labels":[],"detail_key":"p31"},{"id":"n30908","layer":"informal","project":"p31","title":"Sketch. One can actually compute \\hat F = -1 / (4\\pi^2) \\left [\\frac1|x|^2 \\right] - c' \\…","kind":"proof","summary":"Sketch. One can actually compute \\hat F = -1 / (4\\pi^2) \\left [\\frac1|x|^2 \\right] - c' \\delta…","labels":[],"detail_key":"p31"},{"id":"n30909","layer":"informal","project":"p31","title":"If \\phi \\in C^\\infty(\\Omega) slowly increasing (i.e. all derivatives are bounded by polyn…","kind":"notation","summary":"If \\phi \\in C^\\infty(\\Omega) slowly increasing (i.e. all derivatives are bounded by polynomials…","labels":[],"detail_key":"p31"},{"id":"n30910","layer":"informal","project":"p31","title":"remark:approxId","kind":"remark","summary":"\\cH_t \\to \\delta in \\cS' as t \\to 0 and \\int_\\R^d \\cH_t(x) \\dm x = 1 for all t","labels":["remark:approxId"],"detail_key":"p31"},{"id":"n30911","layer":"informal","project":"p31","title":"F is a fundamental solution of L = \\frac\\partial\\partial t - \\Delta_x.","kind":"theorem","summary":"F is a fundamental solution of L = \\frac\\partial\\partial t - \\Delta_x.","labels":[],"detail_key":"p31"},{"id":"n30912","layer":"informal","project":"p31","title":"Denote L' = - \\frac\\partial\\partial t - \\Delta_x, then we have to see the last equation \\…","kind":"proof","summary":"Denote L' = - \\frac\\partial\\partial t - \\Delta_x, then we have to see the last equation \\[LF (\\…","labels":[],"detail_key":"p31"},{"id":"n30913","layer":"formal","project":"p31","title":"DiffContOnCl.norm_le_pow_mul_pow","kind":"theorem","summary":"∀ a b : Real f : Complex → Complex, LT.lt a b → DiffContOnCl Complex f (setOf fun z => Membersh…","labels":[],"detail_key":"p31","name":"DiffContOnCl.norm_le_pow_mul_pow","module":"BonnAnalysis.ComplexInterpolation"},{"id":"n30914","layer":"formal","project":"p31","title":"MeasureTheory.snormEssSup_eq_sSup_snorm","kind":"theorem","summary":"∀ α : Type u_1 m : MeasurableSpace α μ : MeasureTheory.Measure α [inst : MeasureTheory.SigmaFin…","labels":[],"detail_key":"p31","name":"MeasureTheory.snormEssSup_eq_sSup_snorm","module":"BonnAnalysis.ComplexInterpolation"},{"id":"n30915","layer":"formal","project":"p31","title":"MeasureTheory.snorm_eq_sSup_snorm","kind":"theorem","summary":"∀ α : Type u_1 m : MeasurableSpace α μ : MeasureTheory.Measure α (p q : NNReal), p.IsConjExpone…","labels":[],"detail_key":"p31","name":"MeasureTheory.snorm_eq_sSup_snorm","module":"BonnAnalysis.ComplexInterpolation"},{"id":"n30916","layer":"formal","project":"p31","title":"MeasureTheory.fourierIntegralL2","kind":"def","summary":"V : Type u_1 → [inst : NormedAddCommGroup V] → [inst_1 : InnerProductSpace Real V] → [inst_2 :…","labels":[],"detail_key":"p31","name":"MeasureTheory.fourierIntegralL2","module":"BonnAnalysis.Plancherel"},{"id":"n30917","layer":"formal","project":"p31","title":"MeasureTheory.fourier_conj","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] [inst_2 : Meas…","labels":[],"detail_key":"p31","name":"MeasureTheory.fourier_conj","module":"BonnAnalysis.Plancherel"},{"id":"n30918","layer":"formal","project":"p31","title":"MeasureTheory.fourier_convolution","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] [inst_2 : Meas…","labels":[],"detail_key":"p31","name":"MeasureTheory.fourier_convolution","module":"BonnAnalysis.Plancherel"},{"id":"n30919","layer":"formal","project":"p31","title":"MeasureTheory.memℒp_fourierIntegral","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] [inst_2 : Meas…","labels":[],"detail_key":"p31","name":"MeasureTheory.memℒp_fourierIntegral","module":"BonnAnalysis.Plancherel"},{"id":"n30920","layer":"formal","project":"p31","title":"MeasureTheory.snorm_fourierIntegral","kind":"theorem","summary":"∀ V : Type u_1 [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace Real V] [inst_2 : Meas…","labels":[],"detail_key":"p31","name":"MeasureTheory.snorm_fourierIntegral","module":"BonnAnalysis.Plancherel"},{"id":"n30921","layer":"formal","project":"p32","title":"antichain_operator","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"antichain_operator","module":"Carleson.Antichain.AntichainOperator"},{"id":"n30922","layer":"formal","project":"p32","title":"dens1_antichain","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"dens1_antichain","module":"Carleson.Antichain.AntichainOperator"},{"id":"n30923","layer":"formal","project":"p32","title":"Antichain.Ep_inter_G_inter_Ip'_subset_E2","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.Ep_inter_G_inter_Ip'_subset_E2","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30924","layer":"formal","project":"p32","title":"Antichain.global_antichain_density","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.global_antichain_density","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30925","layer":"formal","project":"p32","title":"Antichain.local_antichain_density","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.local_antichain_density","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30926","layer":"formal","project":"p32","title":"Antichain.stack_density","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.stack_density","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30927","layer":"formal","project":"p32","title":"Antichain.tile_count","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.tile_count","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30928","layer":"formal","project":"p32","title":"Antichain.tile_reach","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Antichain.tile_reach","module":"Carleson.Antichain.AntichainTileCount"},{"id":"n30929","layer":"formal","project":"p32","title":"dens2_antichain","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"dens2_antichain","module":"Carleson.Antichain.Basic"},{"id":"n30930","layer":"formal","project":"p32","title":"maximal_bound_antichain","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"maximal_bound_antichain","module":"Carleson.Antichain.Basic"},{"id":"n30931","layer":"formal","project":"p32","title":"tile_disjointness","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"tile_disjointness","module":"Carleson.Antichain.Basic"},{"id":"n30932","layer":"formal","project":"p32","title":"Tile.correlation","kind":"def","summary":"X : Type u_1 → a : Nat → q : Real → K : X → X → Complex → σ₁ σ₂ : X → Int → F G : Set X → [inst…","labels":[],"detail_key":"p32","name":"Tile.correlation","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30933","layer":"formal","project":"p32","title":"Tile.correlation_kernel_bound","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.correlation_kernel_bound","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30934","layer":"formal","project":"p32","title":"Tile.correlation_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.correlation_le","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30935","layer":"formal","project":"p32","title":"Tile.correlation_zero_of_ne_subset","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.correlation_zero_of_ne_subset","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30936","layer":"formal","project":"p32","title":"Tile.mem_ball_of_correlation_ne_zero","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.mem_ball_of_correlation_ne_zero","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30937","layer":"formal","project":"p32","title":"Tile.range_support","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.range_support","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30938","layer":"formal","project":"p32","title":"Tile.uncertainty","kind":"theorem","summary":"∀ X : Type u_1 a : Nat q : Real K : X → X → Complex σ₁ σ₂ : X → Int F G : Set X [inst : MetricS…","labels":[],"detail_key":"p32","name":"Tile.uncertainty","module":"Carleson.Antichain.TileCorrelation"},{"id":"n30939","layer":"formal","project":"p32","title":"close_smooth_approx_periodic","kind":"theorem","summary":"∀ T : Real f : Real → Complex, UniformContinuous f → Function.Periodic f T → ∀ ε : Real, GT.gt…","labels":[],"detail_key":"p32","name":"close_smooth_approx_periodic","module":"Carleson.Classical.Approximation"},{"id":"n30940","layer":"formal","project":"p32","title":"fourierConv_ofTwiceDifferentiable","kind":"theorem","summary":"∀ f : Real → Complex, Function.Periodic f (HMul.hMul 2 Real.pi) → ContDiff Real 2 f → ∀ ε : Rea…","labels":[],"detail_key":"p32","name":"fourierConv_ofTwiceDifferentiable","module":"Carleson.Classical.Approximation"},{"id":"n30941","layer":"formal","project":"p32","title":"lower_secant_bound'","kind":"theorem","summary":"∀ η x : Real, LE.le η (abs x) → LE.le (abs x) (HSub.hSub (HMul.hMul 2 Real.pi) η) → LE.le (HMul…","labels":[],"detail_key":"p32","name":"lower_secant_bound'","module":"Carleson.Classical.Basic"},{"id":"n30942","layer":"formal","project":"p32","title":"exceptional_set_carleson","kind":"theorem","summary":"∀ f : Real → Complex, Function.Periodic f (HMul.hMul 2 Real.pi) → ∀ q : ENNReal, LT.lt 1 q → Me…","labels":[],"detail_key":"p32","name":"exceptional_set_carleson","module":"Carleson.Classical.CarlesonHunt"},{"id":"n30943","layer":"formal","project":"p32","title":"frequency_ball_doubling","kind":"theorem","summary":"∀ x₁ x₂ r : Real f g : Θ Real, LE.le (dist f g) (HMul.hMul 2 (dist f g))","labels":[],"detail_key":"p32","name":"frequency_ball_doubling","module":"Carleson.Classical.CarlesonOnTheRealLineBasic"},{"id":"n30944","layer":"formal","project":"p32","title":"frequency_ball_growth","kind":"theorem","summary":"∀ x₁ x₂ r : Real f g : Θ Real, LE.le (HMul.hMul 2 (dist f g)) (dist f 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(norm…","labels":[],"detail_key":"p32","name":"Hilbert_kernel_regularity","module":"Carleson.Classical.HilbertKernel"},{"id":"n30960","layer":"formal","project":"p32","title":"Hilbert_strong_2_2","kind":"theorem","summary":"∀ ⦃r : Real⦄, LT.lt 0 r → MeasureTheory.HasBoundedStrongType (czOperator K r) 2 2 MeasureTheory…","labels":[],"detail_key":"p32","name":"Hilbert_strong_2_2","module":"Carleson.Classical.HilbertStrongType"},{"id":"n30961","layer":"formal","project":"p32","title":"approxHilbertTransform_eq_dirichletApprox","kind":"theorem","summary":"∀ f : Real → Complex, MeasureTheory.MemLp f Top.top MeasureTheory.volume → ∀ n : Nat x : Real,…","labels":[],"detail_key":"p32","name":"approxHilbertTransform_eq_dirichletApprox","module":"Carleson.Classical.HilbertStrongType"},{"id":"n30962","layer":"formal","project":"p32","title":"continuous_dirichletApprox","kind":"theorem","summary":"∀ n : Nat, Continuous (dirichletApprox 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: MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"eLpNorm_czRemainder_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31140","layer":"formal","project":"p32","title":"encard_czBall3_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"encard_czBall3_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31141","layer":"formal","project":"p32","title":"enorm_czApproximation_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"enorm_czApproximation_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31142","layer":"formal","project":"p32","title":"enorm_czApproximation_le_infinite","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"enorm_czApproximation_le_infinite","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31143","layer":"formal","project":"p32","title":"estimate_bad","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"estimate_bad","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31144","layer":"formal","project":"p32","title":"estimate_bad_partial","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"estimate_bad_partial","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31145","layer":"formal","project":"p32","title":"estimate_good","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"estimate_good","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31146","layer":"formal","project":"p32","title":"integral_czRemainder","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"integral_czRemainder","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31147","layer":"formal","project":"p32","title":"integral_czRemainder'","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"integral_czRemainder'","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31148","layer":"formal","project":"p32","title":"lebesgue_differentiation","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"lebesgue_differentiation","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31149","layer":"formal","project":"p32","title":"maximal_theorem","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"maximal_theorem","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31150","layer":"formal","project":"p32","title":"support_czRemainder'_subset","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"support_czRemainder'_subset","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31151","layer":"formal","project":"p32","title":"tsum_czRemainder'","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"tsum_czRemainder'","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31152","layer":"formal","project":"p32","title":"tsum_eLpNorm_czRemainder'_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"tsum_eLpNorm_czRemainder'_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31153","layer":"formal","project":"p32","title":"tsum_eLpNorm_czRemainder_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"tsum_eLpNorm_czRemainder_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31154","layer":"formal","project":"p32","title":"tsum_volume_czBall3_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"tsum_volume_czBall3_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31155","layer":"formal","project":"p32","title":"volume_univ_le","kind":"theorem","summary":"∀ X : Type u_1 a : Nat [inst : MetricSpace X] [inst_1 : MeasureTheory.DoublingMeasure X ↑(defau…","labels":[],"detail_key":"p32","name":"volume_univ_le","module":"Carleson.TwoSidedCarleson.WeakCalderonZygmund"},{"id":"n31156","layer":"informal","project":"p32","title":"classical Carleson","kind":"theorem","summary":"[classical Carleson] Let f be a 2\\pi-periodic complex-valued continuous function on R. Then for…","labels":["classical-carleson","eq:fourier-limit"],"detail_key":"p32"},{"id":"n31157","layer":"informal","project":"p32","title":"metric space Carleson","kind":"theorem","summary":"[metric space Carleson] For all integers a \\ge 4 and real numbers 1<q\\le 2 the following holds.…","labels":["metric-space-Carleson","nontanbound","resweak"],"detail_key":"p32"},{"id":"n31158","layer":"informal","project":"p32","title":"linearised metric Carleson","kind":"theorem","summary":"[linearised metric Carleson] For all integers a \\ge 4 and real numbers 1<q\\le 2 the following h…","labels":["linearised-metric-Carleson","linnontanbound","linresweak"],"detail_key":"p32"},{"id":"n31159","layer":"informal","project":"p32","title":"The value of the constant factor 2^443a^3 in Theorems~\\refmetric-space-Carleson and~\\refl…","kind":"remark","summary":"The value of the constant factor 2^443a^3 in Theorems~\\refmetric-space-Carleson and~\\reflineari…","labels":[],"detail_key":"p32"},{"id":"n31160","layer":"informal","project":"p32","title":"finitary Carleson","kind":"proposition","summary":"[finitary Carleson] Let \\sigma_1,\\sigma_2\\colon X\\to Z be measurable functions with finite rang…","labels":["finitary-Carleson","eq-linearized"],"detail_key":"p32"},{"id":"n31161","layer":"informal","project":"p32","title":"discrete Carleson","kind":"proposition","summary":"[discrete Carleson] Let (D, c, s) be a grid structure and (\\mathfrak P,I,\\Omega,Q,c,s) a tile s…","labels":["discrete-Carleson","defineep","definetp","disclesssim"],"detail_key":"p32"},{"id":"n31162","layer":"informal","project":"p32","title":"antichain operator","kind":"proposition","summary":"[antichain operator] For any antichain \\mathfrakA and for all f:X\\to \\C with |f|\\le 1_F and all…","labels":["antichain-operator","eq-antiprop"],"detail_key":"p32"},{"id":"n31163","layer":"informal","project":"p32","title":"forest operator","kind":"proposition","summary":"[forest operator] For any n\\ge 0 and any n-forest (\\mathfrak U,\\mathfrak T) we have for all f,g…","labels":["forest-operator"],"detail_key":"p32"},{"id":"n31164","layer":"informal","project":"p32","title":"Holder van der Corput","kind":"proposition","summary":"[Holder van der Corput] Let z\\in X and R>0 and set B=B(z,R). Let \\varphi: X \\to C be supported…","labels":["Holder-van-der-Corput","eq-vdc-cond-tau-2"],"detail_key":"p32"},{"id":"n31165","layer":"informal","project":"p32","title":"Hardy--Littlewood","kind":"proposition","summary":"[Hardy--Littlewood] Let B be a finite collection of balls in X. If for some \\lambda>0 and some…","labels":["Hardy-Littlewood","eq-ball-assumption","eq-besico","eq-hlm","eq-ball-av","eq-hlm-2"],"detail_key":"p32"},{"id":"n31166","layer":"informal","project":"p32","title":"ball metric entropy","kind":"lemma","summary":"[ball metric entropy] Let B' \\subset X be a ball. Let r > 0, \\vartheta\\in \\Theta and k \\in N. S…","labels":["ball-metric-entropy"],"detail_key":"p32"},{"id":"n31167","layer":"informal","project":"p32","title":"By applying property \\eqrefthirddb k times, we obtain a collection Z' \\subset \\Theta with…","kind":"proof","summary":"By applying property \\eqrefthirddb k times, we obtain a collection Z' \\subset \\Theta with |Z'|…","labels":[],"detail_key":"p32"},{"id":"n31168","layer":"informal","project":"p32","title":"monotone cube metrics","kind":"lemma","summary":"[monotone cube metrics] Let (D, c, s) be a grid structure. Denote for cubes I \\in D I^\\circ :=…","labels":["monotone-cube-metrics"],"detail_key":"p32"},{"id":"n31169","layer":"informal","project":"p32","title":"eq-dIJ-est","kind":"proof","summary":"If s(I) \\ge s(J) then \\eqrefdyadicproperty and the assumption I\\subset J imply I = J. Then the…","labels":["eq-dIJ-est"],"detail_key":"p32"},{"id":"n31170","layer":"informal","project":"p32","title":"kernel summand","kind":"lemma","summary":"[kernel summand] Let -S\\le s\\le S and x,y,y'\\in X. If K_s(x,y)\\neq 0, then we have \\frac14 D^s-…","labels":["kernel-summand","supp-Ks","eq-Ks-size","eq-Ks-smooth"],"detail_key":"p32"},{"id":"n31171","layer":"informal","project":"p32","title":"eqkernel-size-Ks","kind":"proof","summary":"By Definition \\eqrefdefks, the function K_s is the product of K with a function which is suppor…","labels":["eqkernel-size-Ks","eq-Ks-aux"],"detail_key":"p32"},{"id":"n31172","layer":"informal","project":"p32","title":"int continuous","kind":"lemma","summary":"[int continuous] Let f be a measurable function with |f| \\le 1. Then the function \\[ G: X \\time…","labels":["int-continuous"],"detail_key":"p32"},{"id":"n31173","layer":"informal","project":"p32","title":"Measurability in x follows from joint measurability of \\[ K(x,y) 1_B(x,R_2) \\setminus B(x…","kind":"proof","summary":"Measurability in x follows from joint measurability of \\[ K(x,y) 1_B(x,R_2) \\setminus B(x,R_1)(…","labels":[],"detail_key":"p32"},{"id":"n31174","layer":"informal","project":"p32","title":"Proof of \\Crefmetric-space-Carleson","kind":"proof","summary":"[Proof of \\Crefmetric-space-Carleson] Let Borel sets F, G in X be given. Let a Borel function f…","labels":[],"detail_key":"p32"},{"id":"n31175","layer":"informal","project":"p32","title":"Proof of \\Creflinearised-metric-Carleson","kind":"proof","summary":"[Proof of \\Creflinearised-metric-Carleson] Let Borel sets F, G in X with finite measure be give…","labels":[],"detail_key":"p32"},{"id":"n31176","layer":"informal","project":"p32","title":"R truncation","kind":"lemma","summary":"[R truncation] Let F, G be Borel sets in X. Let f:X\\to \\C be a Borel function with |f|\\le 1_F.…","labels":["R-truncation","Rcut","TRR"],"detail_key":"p32"},{"id":"n31177","layer":"informal","project":"p32","title":"KKs","kind":"proof","summary":"Let F,G,f as in the lemma be given. Let R\\in 2^N be given. By replacing G with G\\cap B(o,R) if…","labels":["KKs","KsrhoKs","middles","boundarys"],"detail_key":"p32"},{"id":"n31178","layer":"informal","project":"p32","title":"S truncation","kind":"lemma","summary":"[S truncation] Let F, G be bounded Borel sets in X. Let f:X\\to \\C be a Borel function with |f|\\…","labels":["S-truncation","Scut","Tss"],"detail_key":"p32"},{"id":"n31179","layer":"informal","project":"p32","title":"Proof of \\CrefS-truncation","kind":"proof","summary":"[Proof of \\CrefS-truncation] We reduce \\CrefS-truncation to \\Creflinearized-truncation. For eac…","labels":[],"detail_key":"p32"},{"id":"n31180","layer":"informal","project":"p32","title":"linearized truncation","kind":"lemma","summary":"[linearized truncation] Let \\sigma_1,\\sigma_2\\colon X\\to Z be measurable functions with finite…","labels":["linearized-truncation","Sqlin","middles1"],"detail_key":"p32"},{"id":"n31181","layer":"informal","project":"p32","title":"Proof of \\Creflinearized-truncation","kind":"proof","summary":"[Proof of \\Creflinearized-truncation] Fix \\sigma_1, \\sigma_2 and Q as in the lemma. Applying \\C…","labels":["Sqcut2","Sqcut3"],"detail_key":"p32"},{"id":"n31182","layer":"informal","project":"p32","title":"grid existence","kind":"lemma","summary":"[grid existence] There exists a grid structure (D, c,s).","labels":["grid-existence"],"detail_key":"p32"},{"id":"n31183","layer":"informal","project":"p32","title":"tile structure","kind":"lemma","summary":"[tile structure] For a given grid structure (D, c,s), there exists a tile structure (\\mathfrak…","labels":["tile-structure"],"detail_key":"p32"},{"id":"n31184","layer":"informal","project":"p32","title":"tile sum operator","kind":"lemma","summary":"[tile sum operator] We have for all x\\in G\\setminus G' \\sum_\\mathfrak p\\in \\mathfrak PT_\\mathfr…","labels":["tile-sum-operator","eq-sump"],"detail_key":"p32"},{"id":"n31185","layer":"informal","project":"p32","title":"outsump","kind":"proof","summary":"Fix x\\in G\\setminus G'. Sorting the tiles \\mathfrak p on the left-hand-side of \\eqrefeq-sump by…","labels":["outsump","insump"],"detail_key":"p32"},{"id":"n31186","layer":"informal","project":"p32","title":"Proof of \\Creffinitary-Carleson","kind":"proof","summary":"[Proof of \\Creffinitary-Carleson] We now estimate with \\Creftile-sum-operator and \\Crefdiscrete…","labels":[],"detail_key":"p32"},{"id":"n31187","layer":"informal","project":"p32","title":"counting balls","kind":"lemma","summary":"[counting balls] Let -S\\le k\\le S. Consider Y\\subset X such that for any y\\in Y, we have y\\in B…","labels":["counting-balls","ybinb","eq-disj-yballs","boundY"],"detail_key":"p32"},{"id":"n31188","layer":"informal","project":"p32","title":"jballs","kind":"proof","summary":"Let k and Y be given. By applying the doubling property \\eqrefdoublingx inductively, we have fo…","labels":["jballs"],"detail_key":"p32"},{"id":"n31189","layer":"informal","project":"p32","title":"cover big ball","kind":"lemma","summary":"[cover big ball] For each -S\\le k\\le S, the ball B(o, 4D^S-D^k) is contained in the union of th…","labels":["cover-big-ball"],"detail_key":"p32"},{"id":"n31190","layer":"informal","project":"p32","title":"Let x be any point of B(o, 4D^S-D^k). By maximality of |Y_k|, the ball B(x, D^k) intersec…","kind":"proof","summary":"Let x be any point of B(o, 4D^S-D^k). By maximality of |Y_k|, the ball B(x, D^k) intersects one…","labels":[],"detail_key":"p32"},{"id":"n31191","layer":"informal","project":"p32","title":"basic grid structure","kind":"lemma","summary":"[basic grid structure] For each -S\\le k\\le S and 1\\le j\\le 3 the following holds. If j\\neq 2 an…","labels":["basic-grid-structure","disji","unioni","squeezedyadic"],"detail_key":"p32"},{"id":"n31192","layer":"informal","project":"p32","title":"We prove these statements simultaneously by induction on the ordered set of pairs (y,k).…","kind":"proof","summary":"We prove these statements simultaneously by induction on the ordered set of pairs (y,k). Let -S…","labels":[],"detail_key":"p32"},{"id":"n31193","layer":"informal","project":"p32","title":"cover by cubes","kind":"lemma","summary":"[cover by cubes] Let -S\\le l\\le k\\le S and y\\in Y_k. We have I_3(y,k)\\subset \\bigcup_y'\\in Y_l…","labels":["cover-by-cubes","3coverdyadic"],"detail_key":"p32"},{"id":"n31194","layer":"informal","project":"p32","title":"Let -S\\le l\\le k\\le S and y\\in Y_k. If l=k, the inclusion \\eqref3coverdyadic is true from…","kind":"proof","summary":"Let -S\\le l\\le k\\le S and y\\in Y_k. If l=k, the inclusion \\eqref3coverdyadic is true from the d…","labels":[],"detail_key":"p32"},{"id":"n31195","layer":"informal","project":"p32","title":"dyadic property","kind":"lemma","summary":"[dyadic property] Let -S\\le l\\le k\\le S and y\\in Y_k and y'\\in Y_l with I_3(y',l)\\cap I_3(y,k)\\…","labels":["dyadic-property","3dyadicproperty"],"detail_key":"p32"},{"id":"n31196","layer":"informal","project":"p32","title":"wyclaim","kind":"proof","summary":"Let l,k,y,y' be as in the lemma. Pick x\\in I_3(y',l)\\cap I_3(y,k). Assume first l=k. By \\eqrefd…","labels":["wyclaim"],"detail_key":"p32"},{"id":"n31197","layer":"informal","project":"p32","title":"transitive boundary","kind":"lemma","summary":"[transitive boundary] Assume -S\\le k''< k'< k\\le S and y''\\in Y_k'', y'\\in Y_k', y\\in Y_k. Assu…","labels":["transitive-boundary"],"detail_key":"p32"},{"id":"n31198","layer":"informal","project":"p32","title":"yppxp","kind":"proof","summary":"As x\\in I_3(y'',k'')\\cap I_3(y',k') and k''< k', we have by \\Crefdyadic-property that I_3(y'',k…","labels":["yppxp"],"detail_key":"p32"},{"id":"n31199","layer":"informal","project":"p32","title":"small boundary","kind":"lemma","summary":"[small boundary] Let K = 2^4a+1. For each -S+K\\le k\\le S and y\\in Y_k we have \\sum_z\\in Y_k-K:…","labels":["small-boundary","new-small-boundary"],"detail_key":"p32"},{"id":"n31200","layer":"informal","project":"p32","title":"4.31","kind":"proof","summary":"Let K be as in the lemma. Let -S+K\\le k\\le S and y\\in Y_k. Pick k' so that k-K\\le k'\\le k. For…","labels":["4.31","scalecompare","sumcompare","sumcompare1","bulbul"],"detail_key":"p32"},{"id":"n31201","layer":"informal","project":"p32","title":"smaller boundary","kind":"lemma","summary":"[smaller boundary] Let K = 2^4a+1 and let n\\ge 0 be an integer. Then for each -S+nK\\le k\\le S w…","labels":["smaller-boundary","very-new-small"],"detail_key":"p32"},{"id":"n31202","layer":"informal","project":"p32","title":"We prove this by induction on n. If n=0, both sides of \\eqrefvery-new-small are equal to…","kind":"proof","summary":"We prove this by induction on n. If n=0, both sides of \\eqrefvery-new-small are equal to \\mu(I_…","labels":[],"detail_key":"p32"},{"id":"n31203","layer":"informal","project":"p32","title":"boundary measure","kind":"lemma","summary":"[boundary measure] For each -S\\le k\\le S and y\\in Y_k and 0<t<1 with tD^k\\ge D^-S we have \\mu(\\…","labels":["boundary-measure","old-small-boundary"],"detail_key":"p32"},{"id":"n31204","layer":"informal","project":"p32","title":"eq-n-size","kind":"proof","summary":"Let x\\in I_3(y,k) with \\rho(x, X \\setminus I_3(y,k)) \\leq t D^k. Let K = 2^4a+1 as in \\Crefsmal…","labels":["eq-n-size"],"detail_key":"p32"},{"id":"n31205","layer":"informal","project":"p32","title":"Proof of \\Crefgrid-existence","kind":"proof","summary":"[Proof of \\Crefgrid-existence] We first show that (\\tildeD,c,s) satisfies properties \\eqrefcove…","labels":[],"detail_key":"p32"},{"id":"n31206","layer":"informal","project":"p32","title":"frequency ball cover","kind":"lemma","summary":"[frequency ball cover] For each I \\in D, we have Q(X) \\subset \\bigcup_z \\in Z(I) B_I^\\circ(z, 0…","labels":["frequency-ball-cover","eq-tile-cover"],"detail_key":"p32"},{"id":"n31207","layer":"informal","project":"p32","title":"Let \\theta\\in \\bigcup_\\vartheta\\in Q(X) B_I^\\circ(\\vartheta, 1). By maximality of Z(I), t…","kind":"proof","summary":"Let \\theta\\in \\bigcup_\\vartheta\\in Q(X) B_I^\\circ(\\vartheta, 1). By maximality of Z(I), there m…","labels":[],"detail_key":"p32"},{"id":"n31208","layer":"informal","project":"p32","title":"disjoint frequency cubes","kind":"lemma","summary":"[disjoint frequency cubes] For each I \\in D, and \\mathfrak p_1, \\mathfrak p_2\\in \\mathfrak P(I)…","labels":["disjoint-frequency-cubes"],"detail_key":"p32"},{"id":"n31209","layer":"informal","project":"p32","title":"By the definition of the map I, we have \\mathfrak P(I) = \\(I, z) \\, : \\, z \\in Z(I)\\\\,. B…","kind":"proof","summary":"By the definition of the map I, we have \\mathfrak P(I) = \\(I, z) \\, : \\, z \\in Z(I)\\\\,. By \\eqr…","labels":[],"detail_key":"p32"},{"id":"n31210","layer":"informal","project":"p32","title":"frequency cube cover","kind":"lemma","summary":"[frequency cube cover] For each I \\in D, it holds that \\bigcup_z \\in Z(I) B_I^\\circ(z, 0.7)\\sub…","labels":["frequency-cube-cover","eq-omega1-cover","eq-omega1-incl"],"detail_key":"p32"},{"id":"n31211","layer":"informal","project":"p32","title":"For \\eqrefeq-omega1-incl let \\mathfrak p= (I, z). The second inclusion in \\eqrefeq-omega1…","kind":"proof","summary":"For \\eqrefeq-omega1-incl let \\mathfrak p= (I, z). The second inclusion in \\eqrefeq-omega1-incl…","labels":[],"detail_key":"p32"},{"id":"n31212","layer":"informal","project":"p32","title":"Proof of \\Creftile-structure","kind":"proof","summary":"[Proof of \\Creftile-structure] First, we prove \\eqrefeq-freq-comp-ball. If I =I_0, then \\eqrefe…","labels":[],"detail_key":"p32"},{"id":"n31213","layer":"informal","project":"p32","title":"exceptional set","kind":"lemma","summary":"[exceptional set] We have \\mu(G')\\le 2^-1\\mu(G)\\, .","labels":["exceptional-set"],"detail_key":"p32"},{"id":"n31214","layer":"informal","project":"p32","title":"forest union","kind":"lemma","summary":"[forest union] Let \\mathfrak P_1 =\\bigcup_k\\ge 0\\bigcup_n\\ge k \\bigcup_0\\le j\\le 2n+3\\mathfrak…","labels":["forest-union","disclesssim1"],"detail_key":"p32"},{"id":"n31215","layer":"informal","project":"p32","title":"forest complement","kind":"lemma","summary":"[forest complement] Let \\mathfrak P_2 =\\mathfrak P\\setminus \\mathfrak P_1\\,. For all f:X\\to \\C…","labels":["forest-complement","disclesssim2"],"detail_key":"p32"},{"id":"n31216","layer":"informal","project":"p32","title":"Proof of \\Crefdiscrete-Carleson","kind":"proof","summary":"[Proof of \\Crefdiscrete-Carleson] \\Crefdiscrete-Carleson follows by applying the triangle inequ…","labels":[],"detail_key":"p32"},{"id":"n31217","layer":"informal","project":"p32","title":"first exception","kind":"lemma","summary":"[first exception] We have \\mu(G_1)\\le 2^-5\\mu(G)\\, .","labels":["first-exception"],"detail_key":"p32"},{"id":"n31218","layer":"informal","project":"p32","title":"Let K = 2^2a+5\\frac\\mu(F)\\mu(G)\\,. For each \\mathfrak p\\in \\mathfrak P_F,G pick a r(\\math…","kind":"proof","summary":"Let K = 2^2a+5\\frac\\mu(F)\\mu(G)\\,. For each \\mathfrak p\\in \\mathfrak P_F,G pick a r(\\mathfrak p…","labels":[],"detail_key":"p32"},{"id":"n31219","layer":"informal","project":"p32","title":"dense cover","kind":"lemma","summary":"[dense cover] For each k\\ge 0, the union of all dyadic cubes in C(G,k) has measure at most 2^k+…","labels":["dense-cover"],"detail_key":"p32"},{"id":"n31220","layer":"informal","project":"p32","title":"cbymstar","kind":"proof","summary":"The union of dyadic cubes in C(G,k) is contained the union of elements of the set M(k) of all d…","labels":["cbymstar"],"detail_key":"p32"},{"id":"n31221","layer":"informal","project":"p32","title":"pairwise disjoint","kind":"lemma","summary":"[pairwise disjoint] If \\mathfrak p, \\mathfrak p' \\in \\mathfrakM(k,n) and E_1(\\mathfrak p)\\cap E…","labels":["pairwise-disjoint","eintersect"],"detail_key":"p32"},{"id":"n31222","layer":"informal","project":"p32","title":"Let \\mathfrak p,\\mathfrak p' be as in the lemma. By definition of E_1, we have E_1(\\mathf…","kind":"proof","summary":"Let \\mathfrak p,\\mathfrak p' be as in the lemma. By definition of E_1, we have E_1(\\mathfrak p)…","labels":[],"detail_key":"p32"},{"id":"n31223","layer":"informal","project":"p32","title":"dyadic union","kind":"lemma","summary":"[dyadic union] For each x\\in A(\\lambda,k,n), there is a dyadic cube I that contains x and is a…","labels":["dyadic-union"],"detail_key":"p32"},{"id":"n31224","layer":"informal","project":"p32","title":"Fix k,n,\\lambda,x as in the lemma such that x\\in A(\\lambda,k,n). Let M be the set of dyad…","kind":"proof","summary":"Fix k,n,\\lambda,x as in the lemma such that x\\in A(\\lambda,k,n). Let M be the set of dyadic cub…","labels":[],"detail_key":"p32"},{"id":"n31225","layer":"informal","project":"p32","title":"John Nirenberg","kind":"lemma","summary":"[John Nirenberg] For all integers k,n,\\lambda\\ge 0, we have \\mu(A(\\lambda,k,n)) \\le 2^k+1-\\lamb…","labels":["John-Nirenberg","alambdameasure"],"detail_key":"p32"},{"id":"n31226","layer":"informal","project":"p32","title":"suminout","kind":"proof","summary":"Fix k,n as in the lemma and suppress notation to write A(\\lambda) for A(\\lambda,k,n). We prove…","labels":["suminout","mnkonl","mnkintl"],"detail_key":"p32"},{"id":"n31227","layer":"informal","project":"p32","title":"second exception","kind":"lemma","summary":"[second exception] We have \\mu(G_2)\\le 2^-2 \\mu(G)\\, .","labels":["second-exception"],"detail_key":"p32"},{"id":"n31228","layer":"informal","project":"p32","title":"We use \\CrefJohn-Nirenberg and sum twice a geometric series to obtain \\sum_0\\le k\\sum_k\\l…","kind":"proof","summary":"We use \\CrefJohn-Nirenberg and sum twice a geometric series to obtain \\sum_0\\le k\\sum_k\\le n \\m…","labels":[],"detail_key":"p32"},{"id":"n31229","layer":"informal","project":"p32","title":"top tiles","kind":"lemma","summary":"[top tiles] We have \\sum_\\mathfrakm \\in \\mathfrakM(k,n) \\mu(I(\\mathfrakm))\\le 2^n+k+3\\mu(G).","labels":["top-tiles","eq-musum"],"detail_key":"p32"},{"id":"n31230","layer":"informal","project":"p32","title":"We write the left-hand side of \\eqrefeq-musum \\int \\sum_\\mathfrakm \\in \\mathfrakM(k,n) 1_…","kind":"proof","summary":"We write the left-hand side of \\eqrefeq-musum \\int \\sum_\\mathfrakm \\in \\mathfrakM(k,n) 1_I(\\mat…","labels":[],"detail_key":"p32"},{"id":"n31231","layer":"informal","project":"p32","title":"tree count","kind":"lemma","summary":"[tree count] Let k,n,j\\ge 0. We have for every x\\in X \\sum_\\mathfrak u\\in \\mathfrak U_1(k,n,j)…","labels":["tree-count"],"detail_key":"p32"},{"id":"n31232","layer":"informal","project":"p32","title":"ubymsum","kind":"proof","summary":"Let x\\in X. For each \\mathfrak u\\in \\mathfrak U_1(k,n,j) with x\\in I(\\mathfrak u), as \\mathfrak…","labels":["ubymsum","usumbymsum","dby2"],"detail_key":"p32"},{"id":"n31233","layer":"informal","project":"p32","title":"boundary exception","kind":"lemma","summary":"[boundary exception] Let L(\\mathfrak u) be as defined in \\eqrefeq-L-def. We have for each \\math…","labels":["boundary-exception"],"detail_key":"p32"},{"id":"n31234","layer":"informal","project":"p32","title":"Let \\mathfrak u\\in \\mathfrak U_1(k,n,l). Let I \\in L(\\mathfrak u). Then we have s(I) = s(…","kind":"proof","summary":"Let \\mathfrak u\\in \\mathfrak U_1(k,n,l). Let I \\in L(\\mathfrak u). Then we have s(I) = s(\\mathf…","labels":[],"detail_key":"p32"},{"id":"n31235","layer":"informal","project":"p32","title":"third exception","kind":"lemma","summary":"[third exception] We have \\mu(G_3)\\le 2^-4 \\mu(G)\\, .","labels":["third-exception"],"detail_key":"p32"},{"id":"n31236","layer":"informal","project":"p32","title":"As each \\mathfrak p\\in \\mathfrak L_4(k,n,j) is contained in \\cupL(\\mathfrak u) for some \\…","kind":"proof","summary":"As each \\mathfrak p\\in \\mathfrak L_4(k,n,j) is contained in \\cupL(\\mathfrak u) for some \\mathfr…","labels":[],"detail_key":"p32"},{"id":"n31237","layer":"informal","project":"p32","title":"Proof of \\Crefexceptional-set","kind":"proof","summary":"[Proof of \\Crefexceptional-set] Adding up the bounds in Lemmas \\reffirst-exception, \\refsecond-…","labels":[],"detail_key":"p32"},{"id":"n31238","layer":"informal","project":"p32","title":"wiggle order 1","kind":"lemma","summary":"[wiggle order 1] If n\\mathfrak p\\lesssim m\\mathfrak p' and n' \\ge n and m \\ge m' then n'\\mathfr…","labels":["wiggle-order-1"],"detail_key":"p32"},{"id":"n31239","layer":"informal","project":"p32","title":"This follows immediately from the definition \\eqrefwiggleorder of \\lesssim and the two in…","kind":"proof","summary":"This follows immediately from the definition \\eqrefwiggleorder of \\lesssim and the two inclusio…","labels":[],"detail_key":"p32"},{"id":"n31240","layer":"informal","project":"p32","title":"wiggle order 2","kind":"lemma","summary":"[wiggle order 2] Let n, m \\ge 1 and k > 0. If \\mathfrak p, \\mathfrak p' \\in \\mathfrak P with I(…","labels":["wiggle-order-2","eq-wiggle1","eq-wiggle2"],"detail_key":"p32"},{"id":"n31241","layer":"informal","project":"p32","title":"The assumption \\eqrefeq-wiggle1 together with the definition \\eqrefwiggleorder of \\lesssi…","kind":"proof","summary":"The assumption \\eqrefeq-wiggle1 together with the definition \\eqrefwiggleorder of \\lesssim impl…","labels":[],"detail_key":"p32"},{"id":"n31242","layer":"informal","project":"p32","title":"wiggle order 3","kind":"lemma","summary":"[wiggle order 3] The following implications hold for all \\mathfrak q, \\mathfrak q' \\in \\mathfra…","labels":["wiggle-order-3","eq-sc1","eq-sc2","eq-sc3"],"detail_key":"p32"},{"id":"n31243","layer":"informal","project":"p32","title":"\\eqrefeq-sc2 and \\eqrefeq-sc3 are easy consequences of \\Crefwiggle-order-1, \\Crefwiggle-o…","kind":"proof","summary":"\\eqrefeq-sc2 and \\eqrefeq-sc3 are easy consequences of \\Crefwiggle-order-1, \\Crefwiggle-order-2…","labels":[],"detail_key":"p32"},{"id":"n31244","layer":"informal","project":"p32","title":"P convex","kind":"lemma","summary":"[P convex] For each k, the collection \\mathfrak P(k) is convex.","labels":["P-convex"],"detail_key":"p32"},{"id":"n31245","layer":"informal","project":"p32","title":"Suppose that \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' and \\mathfrak p, \\mathfrak p''…","kind":"proof","summary":"Suppose that \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' and \\mathfrak p, \\mathfrak p'' \\in \\…","labels":[],"detail_key":"p32"},{"id":"n31246","layer":"informal","project":"p32","title":"C convex","kind":"lemma","summary":"[C convex] For each k,n, the collection \\mathfrak C(k,n) is convex.","labels":["C-convex"],"detail_key":"p32"},{"id":"n31247","layer":"informal","project":"p32","title":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\ma…","kind":"proof","summary":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31248","layer":"informal","project":"p32","title":"C1 convex","kind":"lemma","summary":"[C1 convex] For each k,n,j, the collection \\mathfrak C_1(k,n,j) is convex.","labels":["C1-convex"],"detail_key":"p32"},{"id":"n31249","layer":"informal","project":"p32","title":"Let \\mathfrak p\\le\\mathfrak p'\\le\\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathf…","kind":"proof","summary":"Let \\mathfrak p\\le\\mathfrak p'\\le\\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathfrak C_…","labels":[],"detail_key":"p32"},{"id":"n31250","layer":"informal","project":"p32","title":"C2 convex","kind":"lemma","summary":"[C2 convex] For each k,n,j, the collection \\mathfrak C_2(k,n,j) is convex.","labels":["C2-convex"],"detail_key":"p32"},{"id":"n31251","layer":"informal","project":"p32","title":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\ma…","kind":"proof","summary":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31252","layer":"informal","project":"p32","title":"C3 convex","kind":"lemma","summary":"[C3 convex] For each k,n,j, the collection \\mathfrak C_3(k,n,j) is convex.","labels":["C3-convex"],"detail_key":"p32"},{"id":"n31253","layer":"informal","project":"p32","title":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\ma…","kind":"proof","summary":"Let \\mathfrak p\\le \\mathfrak p' \\le \\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31254","layer":"informal","project":"p32","title":"C4 convex","kind":"lemma","summary":"[C4 convex] For each k,n,j, the collection \\mathfrak C_4(k,n,j) is convex.","labels":["C4-convex"],"detail_key":"p32"},{"id":"n31255","layer":"informal","project":"p32","title":"The proof is entirely analogous to \\CrefC2-convex, substituting \\mathfrak C_4 for \\mathfr…","kind":"proof","summary":"The proof is entirely analogous to \\CrefC2-convex, substituting \\mathfrak C_4 for \\mathfrak C_2…","labels":[],"detail_key":"p32"},{"id":"n31256","layer":"informal","project":"p32","title":"C5 convex","kind":"lemma","summary":"[C5 convex] For each k,n,j, the collection \\mathfrak C_5(k,n,j) is convex.","labels":["C5-convex"],"detail_key":"p32"},{"id":"n31257","layer":"informal","project":"p32","title":"Let \\mathfrak p\\le \\mathfrak p' \\le\\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mat…","kind":"proof","summary":"Let \\mathfrak p\\le \\mathfrak p' \\le\\mathfrak p'' with \\mathfrak p, \\mathfrak p'' \\in \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31258","layer":"informal","project":"p32","title":"dens compare","kind":"lemma","summary":"[dens compare] We have for every k\\ge 0 and \\mathfrak P'\\subset \\mathfrak P(k) \\dens_1(\\mathfra…","labels":["dens-compare"],"detail_key":"p32"},{"id":"n31259","layer":"informal","project":"p32","title":"mugj","kind":"proof","summary":"It suffices to show that for all \\mathfrak p'\\in \\mathfrak P' and \\lambda\\ge 2 and \\mathfrak p\\…","labels":["mugj"],"detail_key":"p32"},{"id":"n31260","layer":"informal","project":"p32","title":"C dens1","kind":"lemma","summary":"[C dens1] For each set \\mathfrakA \\subset \\mathfrakC(k,n), we have \\dens_1(\\mathfrakA) \\le 2^4a…","labels":["C-dens1"],"detail_key":"p32"},{"id":"n31261","layer":"informal","project":"p32","title":"We have by \\Crefdens-compare that \\dens_1(\\mathfrakA) \\le \\dens_k'(\\mathfrakA). Since \\ma…","kind":"proof","summary":"We have by \\Crefdens-compare that \\dens_1(\\mathfrakA) \\le \\dens_k'(\\mathfrakA). Since \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31262","layer":"informal","project":"p32","title":"relation geometry","kind":"lemma","summary":"[relation geometry] If \\mathfrak u\\sim \\mathfrak u', then I(u) = I(u') and B_\\mathfrak u(Q(\\mat…","labels":["relation-geometry"],"detail_key":"p32"},{"id":"n31263","layer":"informal","project":"p32","title":"eq-Fefferman-trick0","kind":"proof","summary":"Let \\mathfrak u, \\mathfrak u' \\in \\mathfrak U_2(k,n,j) with \\mathfrak u\\sim \\mathfrak u'. If \\m…","labels":["eq-Fefferman-trick0"],"detail_key":"p32"},{"id":"n31264","layer":"informal","project":"p32","title":"equivalence relation","kind":"lemma","summary":"[equivalence relation] For each k,n,j, the relation \\sim on \\mathfrak U_2(k,n,j) is an equivale…","labels":["equivalence-relation"],"detail_key":"p32"},{"id":"n31265","layer":"informal","project":"p32","title":"eq-rel1","kind":"proof","summary":"Reflexivity holds by definition. For transitivity, suppose that \\mathfrak u, \\mathfrak u', \\mat…","labels":["eq-rel1"],"detail_key":"p32"},{"id":"n31266","layer":"informal","project":"p32","title":"C6 forest","kind":"lemma","summary":"[C6 forest] We have \\mathfrak C_6(k,n,j)=\\bigcup_\\mathfrak u\\in \\mathfrak U_3(k,n,j)\\mathfrakT_…","labels":["C6-forest"],"detail_key":"p32"},{"id":"n31267","layer":"informal","project":"p32","title":"Let \\mathfrak p\\in \\mathfrak C_6(k,n,j). By \\eqr","kind":"proof","summary":"Let \\mathfrak p\\in \\mathfrak C_6(k,n,j). By \\eqr","labels":[],"detail_key":"p32"},{"id":"n31268","layer":"informal","project":"p32","title":"forest geometry","kind":"lemma","summary":"[forest geometry] For each \\mathfrak u\\in \\mathfrak U_3(k,n,j), the set \\mathfrakT_2(\\mathfrak…","labels":["forest-geometry"],"detail_key":"p32"},{"id":"n31269","layer":"informal","project":"p32","title":"Let \\mathfrak p\\in \\mathfrakT_2(\\mathfrak u). By \\eqrefdefinesv, there exists \\mathfrak u…","kind":"proof","summary":"Let \\mathfrak p\\in \\mathfrakT_2(\\mathfrak u). By \\eqrefdefinesv, there exists \\mathfrak u' \\sim…","labels":[],"detail_key":"p32"},{"id":"n31270","layer":"informal","project":"p32","title":"forest convex","kind":"lemma","summary":"[forest convex] For each \\mathfrak u\\in \\mathfrak U_3(k,n,j), the set \\mathfrakT_2(\\mathfrak u)…","labels":["forest-convex"],"detail_key":"p32"},{"id":"n31271","layer":"informal","project":"p32","title":"Let \\mathfrak p, \\mathfrak p'' \\in \\mathfrakT_2(\\mathfrak u) and \\mathfrak p' \\in \\mathfr…","kind":"proof","summary":"Let \\mathfrak p, \\mathfrak p'' \\in \\mathfrakT_2(\\mathfrak u) and \\mathfrak p' \\in \\mathfrak P w…","labels":[],"detail_key":"p32"},{"id":"n31272","layer":"informal","project":"p32","title":"forest separation","kind":"lemma","summary":"[forest separation] For each \\mathfrak u,\\mathfrak u'\\in \\mathfrak U_3(k,n,j) with \\mathfrak u\\…","labels":["forest-separation"],"detail_key":"p32"},{"id":"n31273","layer":"informal","project":"p32","title":"By the definition \\eqrefeq-C2-def of \\mathfrak C_2(k,n,j), there exists a tile \\mathfrak…","kind":"proof","summary":"By the definition \\eqrefeq-C2-def of \\mathfrak C_2(k,n,j), there exists a tile \\mathfrak p' \\in…","labels":[],"detail_key":"p32"},{"id":"n31274","layer":"informal","project":"p32","title":"forest inner","kind":"lemma","summary":"[forest inner] For each \\mathfrak u\\in \\mathfrak U_3(k,n,j) and each \\mathfrak p\\in \\mathfrakT_…","labels":["forest-inner"],"detail_key":"p32"},{"id":"n31275","layer":"informal","project":"p32","title":"Let \\mathfrak p\\in \\mathfrakT_2(\\mathfrak u). Then \\mathfrak p\\in \\mathfrak C_4(k,n,j), h…","kind":"proof","summary":"Let \\mathfrak p\\in \\mathfrakT_2(\\mathfrak u). Then \\mathfrak p\\in \\mathfrak C_4(k,n,j), hence t…","labels":[],"detail_key":"p32"},{"id":"n31276","layer":"informal","project":"p32","title":"forest stacking","kind":"lemma","summary":"[forest stacking] It holds for k\\le n that \\sum_\\mathfrak u\\in \\mathfrak U_3(k,n,j) 1_I(\\mathfr…","labels":["forest-stacking"],"detail_key":"p32"},{"id":"n31277","layer":"informal","project":"p32","title":"Suppose that a point x is contained in more than (4n + 12)2^n cubes I(\\mathfrak u) with \\…","kind":"proof","summary":"Suppose that a point x is contained in more than (4n + 12)2^n cubes I(\\mathfrak u) with \\mathfr…","labels":[],"detail_key":"p32"},{"id":"n31278","layer":"informal","project":"p32","title":"Proof of \\Crefforest-union","kind":"proof","summary":"[Proof of \\Crefforest-union] We first fix k,n, j. By \\eqrefdefinetp and \\eqrefdefineep, we have…","labels":[],"detail_key":"p32"},{"id":"n31279","layer":"informal","project":"p32","title":"antichain decomposition","kind":"lemma","summary":"[antichain decomposition] We have that &\\quad \\mathfrak P_2 \\cap \\mathfrak P_G \\setminus G'\\\\ &…","labels":["antichain-decomposition","eq-fp'-decomposition"],"detail_key":"p32"},{"id":"n31280","layer":"informal","project":"p32","title":"Let \\mathfrak p\\in \\mathfrak P_2 \\cap \\mathfrak P_G \\setminus G'. Clearly, for every cube…","kind":"proof","summary":"Let \\mathfrak p\\in \\mathfrak P_2 \\cap \\mathfrak P_G \\setminus G'. Clearly, for every cube J = I…","labels":[],"detail_key":"p32"},{"id":"n31281","layer":"informal","project":"p32","title":"L0 antichain","kind":"lemma","summary":"[L0 antichain] We have that \\mathfrak L_0(k,n) = \\dot\\bigcup_0 \\le l < n \\mathfrak L_0(k,n,l)\\,…","labels":["L0-antichain"],"detail_key":"p32"},{"id":"n31282","layer":"informal","project":"p32","title":"eq-p'","kind":"proof","summary":"It suffices to show that \\mathfrak L_0(k,n) contains no chain of length n + 1. Suppose that we…","labels":["eq-p'","eq-O-bound"],"detail_key":"p32"},{"id":"n31283","layer":"informal","project":"p32","title":"L2 antichain","kind":"lemma","summary":"[L2 antichain] Each of the sets \\mathfrak L_2(k,n,j) is an antichain.","labels":["L2-antichain"],"detail_key":"p32"},{"id":"n31284","layer":"informal","project":"p32","title":"Suppose that there are \\mathfrak p_0, \\mathfrak p_1 \\in \\mathfrak L_2(k,n,j) with \\mathfr…","kind":"proof","summary":"Suppose that there are \\mathfrak p_0, \\mathfrak p_1 \\in \\mathfrak L_2(k,n,j) with \\mathfrak p_0…","labels":[],"detail_key":"p32"},{"id":"n31285","layer":"informal","project":"p32","title":"L1 L3 antichain","kind":"lemma","summary":"[L1 L3 antichain] Each of the sets \\mathfrak L_1(k,n,j,l) and \\mathfrak L_3(k,n,j,l) is an anti…","labels":["L1-L3-antichain"],"detail_key":"p32"},{"id":"n31286","layer":"informal","project":"p32","title":"By its","kind":"proof","summary":"By its","labels":[],"detail_key":"p32"},{"id":"n31287","layer":"informal","project":"p32","title":"Proof of \\Crefforest-complement","kind":"proof","summary":"[Proof of \\Crefforest-complement] If \\mathfrak p\\not\\in \\mathfrak P_G \\setminus G', then \\mu(I(…","labels":[],"detail_key":"p32"},{"id":"n31288","layer":"informal","project":"p32","title":"tile disjointness","kind":"lemma","summary":"[tile disjointness] Let \\mathfrak p,\\mathfrak p'\\in \\mathfrakA. If there exists an x\\in X with…","labels":["tile-disjointness"],"detail_key":"p32"},{"id":"n31289","layer":"informal","project":"p32","title":"Let \\mathfrak p,\\mathfrak p' and x be given. Assume without loss of generality that s(\\ma…","kind":"proof","summary":"Let \\mathfrak p,\\mathfrak p' and x be given. Assume without loss of generality that s(\\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31290","layer":"informal","project":"p32","title":"maximal bound antichain","kind":"lemma","summary":"[maximal bound antichain] Let x\\in X. Then | \\sum_\\mathfrak p\\in \\mathfrakAT_\\mathfrak p f(x)|\\…","labels":["maximal-bound-antichain","hlmbound"],"detail_key":"p32"},{"id":"n31291","layer":"informal","project":"p32","title":"eqtttt0","kind":"proof","summary":"Fix x\\in X. By \\Creftile-disjointness, there is at most one \\mathfrak p\\in \\mathfrakA such that…","labels":["eqtttt0","supp-Ks1"],"detail_key":"p32"},{"id":"n31292","layer":"informal","project":"p32","title":"dens2 antichain","kind":"lemma","summary":"[dens2 antichain] We have that \\left|\\int \\overlineg(x) \\sum_\\mathfrak p\\in \\mathfrakA T_\\mathf…","labels":["dens2-antichain","eqttt9"],"detail_key":"p32"},{"id":"n31293","layer":"informal","project":"p32","title":"eqttt1","kind":"proof","summary":"We have f=1_Ff. Using H\\\"older's inequality, we obtain for each x\\in B' and each B'\\in B using…","labels":["eqttt1","eqttt2"],"detail_key":"p32"},{"id":"n31294","layer":"informal","project":"p32","title":"dens1 antichain","kind":"lemma","summary":"[dens1 antichain] Set p:=4a^4. We have \\left|\\int \\overlineg(x) \\sum_\\mathfrak p\\in \\mathfrakA…","labels":["dens1-antichain","eqttt3"],"detail_key":"p32"},{"id":"n31295","layer":"informal","project":"p32","title":"eq-tstarwritten","kind":"proof","summary":"We write for the expression inside the absolute values on the left-hand side of \\eqrefeqttt3 \\s…","labels":["eq-tstarwritten","eqtts1","eqtts2","eq-Dp-definition","eqttt4","eqtts3","def-hp","eqttt5","eqttt5b","eqtts4","eqtts4a"],"detail_key":"p32"},{"id":"n31296","layer":"informal","project":"p32","title":"tile correlation","kind":"lemma","summary":"[tile correlation] Let \\mathfrak p, \\mathfrak p'\\in \\mathfrak P with s(\\mathfrak p')\\leq s(\\mat…","labels":["tile-correlation","eq-basic-TT*-est"],"detail_key":"p32"},{"id":"n31297","layer":"informal","project":"p32","title":"antichain tile count","kind":"lemma","summary":"[antichain tile count] Set p:=4a^4. For every \\vartheta\\in\\Theta and every antichain \\mathfrakA…","labels":["antichain-tile-count","eq-antichain-Lp"],"detail_key":"p32"},{"id":"n31298","layer":"informal","project":"p32","title":"Proof of \\Crefantichain-operator","kind":"proof","summary":"[Proof of \\Crefantichain-operator] We have \\left(\\frac 1\\tildeq -\\frac 12\\right) (2-q)= \\frac 1…","labels":["eqttt8"],"detail_key":"p32"},{"id":"n31299","layer":"informal","project":"p32","title":"correlation kernel bound","kind":"lemma","summary":"[correlation kernel bound] Let -S\\le s_1\\le s_2\\le S and let x_1,x_2\\in X. Define \\varphi(y) :=…","labels":["correlation-kernel-bound","eqt10","eqt11"],"detail_key":"p32"},{"id":"n31300","layer":"informal","project":"p32","title":"suppart","kind":"proof","summary":"If \\varphi(y) is not zero, then K_s_1(x_1, y) is not zero and thus \\eqrefsupp-Ks gives \\eqrefeq…","labels":["suppart","holderpart"],"detail_key":"p32"},{"id":"n31301","layer":"informal","project":"p32","title":"tile range support","kind":"lemma","summary":"[tile range support] For each \\mathfrak p\\in \\mathfrak P, and each y\\in X, we have that T_\\math…","labels":["tile-range-support","tstargnot0","ynotfar"],"detail_key":"p32"},{"id":"n31302","layer":"informal","project":"p32","title":"Fix \\mathfrak p and y with \\eqreftstargnot0. Then there exists x\\in E(\\mathfrak p) with \\…","kind":"proof","summary":"Fix \\mathfrak p and y with \\eqreftstargnot0. Then there exists x\\in E(\\mathfrak p) with \\overli…","labels":[],"detail_key":"p32"},{"id":"n31303","layer":"informal","project":"p32","title":"tile-uncertainty","kind":"lemma","summary":"Let \\mathfrak p_1, \\mathfrak p_2\\in \\mathfrak P with B(c(\\mathfrak p_1),5D^s(\\mathfrak p_1)) \\c…","labels":["tile-uncertainty","tgeo"],"detail_key":"p32"},{"id":"n31304","layer":"informal","project":"p32","title":"dponetwo","kind":"proof","summary":"Let i\\in \\1,2\\. By Definition \\eqrefdefineep of E, we have Q(x_i)\\in \\Omega(\\mathfrak p_i) With…","labels":["dponetwo","tgeo0.5","tgeo1","tgeo1.5","tgeo2"],"detail_key":"p32"},{"id":"n31305","layer":"informal","project":"p32","title":"Proof of \\Creftile-correlation","kind":"proof","summary":"[Proof of \\Creftile-correlation] We begin with \\eqrefeq-basic-TT*-est. By Lemma \\reftile-range-…","labels":["intersec5B","tstartstar","tstartstar'","eqa1","eqa1.5","eqa2"],"detail_key":"p32"},{"id":"n31306","layer":"informal","project":"p32","title":"tile reach","kind":"lemma","summary":"[tile reach] Let \\vartheta\\in \\Theta and N\\ge0 be an integer. Let \\mathfrak p, \\mathfrak p'\\in…","labels":["tile-reach","eqassumedismfa","eqassumedismfap","lp'lp''"],"detail_key":"p32"},{"id":"n31307","layer":"informal","project":"p32","title":"eqdistqpqp","kind":"proof","summary":"By \\Crefmonotone-cube-metrics, we have d_\\mathfrak p(Q(\\mathfrak p'),\\vartheta) \\le d_\\mathfrak…","labels":["eqdistqpqp","ageo1"],"detail_key":"p32"},{"id":"n31308","layer":"informal","project":"p32","title":"stack density","kind":"lemma","summary":"[stack density] Let \\vartheta\\in \\Theta, N\\ge 0 and L\\in D. Then \\sum_\\mathfrak p\\in\\mathfrakA_…","labels":["stack-density","eqanti-1"],"detail_key":"p32"},{"id":"n31309","layer":"informal","project":"p32","title":"eqanti-3","kind":"proof","summary":"Let \\vartheta,N,L be given and set \\mathfrakA':=\\\\mathfrak p\\in\\mathfrakA_\\vartheta,N:I(\\mathfr…","labels":["eqanti-3","eqanti-4"],"detail_key":"p32"},{"id":"n31310","layer":"informal","project":"p32","title":"local antichain density","kind":"lemma","summary":"[local antichain density] Let \\vartheta\\in\\Theta and N be an integer. Let \\mathfrak p_\\vartheta…","labels":["local-antichain-density","eqanti-0.5"],"detail_key":"p32"},{"id":"n31311","layer":"informal","project":"p32","title":"Let \\mathfrak p be any tile in \\mathfrakA_\\vartheta,N with s(\\mathfrak p_\\vartheta)<s(\\ma…","kind":"proof","summary":"Let \\mathfrak p be any tile in \\mathfrakA_\\vartheta,N with s(\\mathfrak p_\\vartheta)<s(\\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31312","layer":"informal","project":"p32","title":"global antichain density","kind":"lemma","summary":"[global antichain density] Let \\vartheta\\in Q(X) and let N\\ge 0 be an integer. Then we have \\su…","labels":["global-antichain-density","eqanti00"],"detail_key":"p32"},{"id":"n31313","layer":"informal","project":"p32","title":"eq_minus_S","kind":"proof","summary":"Fix \\vartheta and N. Let \\mathfrakA' be the set of \\mathfrak p\\in\\mathfrakA_\\vartheta,N such th…","labels":["eq_minus_S","eqdecAprime","eqanti0","eqanti1","eqanti2","equanti1.5","pmfadens","eqanti0.5","eqanti3.14"],"detail_key":"p32"},{"id":"n31314","layer":"informal","project":"p32","title":"Proof of \\Crefantichain-tile-count","kind":"proof","summary":"[Proof of \\Crefantichain-tile-count] Using that \\mathfrakA is the disjoint union of the \\mathfr…","labels":["eqanti23","eqanti21"],"detail_key":"p32"},{"id":"n31315","layer":"informal","project":"p32","title":"convex scales","kind":"lemma","summary":"[convex scales] For each \\mathfrak u\\in \\mathfrak U, we have \\sigma(\\mathfrak u, x) = Z \\cap [\\…","labels":["convex-scales"],"detail_key":"p32"},{"id":"n31316","layer":"informal","project":"p32","title":"Let s","kind":"proof","summary":"Let s","labels":[],"detail_key":"p32"},{"id":"n31317","layer":"informal","project":"p32","title":"dyadic partitions","kind":"lemma","summary":"[dyadic partitions] For each \\mathfrakS \\subset \\mathfrak P, we have \\bigcup_I \\in D I = \\dot\\b…","labels":["dyadic-partitions","eq-J-partition","eq-L-partition"],"detail_key":"p32"},{"id":"n31318","layer":"informal","project":"p32","title":"Since J(\\mathfrakS) is the set of inclusion maximal cubes in J_0(\\mathfrakS), cubes in J(…","kind":"proof","summary":"Since J(\\mathfrakS) is the set of inclusion maximal cubes in J_0(\\mathfrakS), cubes in J(\\mathf…","labels":[],"detail_key":"p32"},{"id":"n31319","layer":"informal","project":"p32","title":"pointwise tree estimate","kind":"lemma","summary":"[pointwise tree estimate] Let \\mathfrak u\\in \\mathfrak U and L \\in L(\\mathfrak T(\\mathfrak u)).…","labels":["pointwise-tree-estimate","eq-LJ-ptwise"],"detail_key":"p32"},{"id":"n31320","layer":"informal","project":"p32","title":"eq-term-A","kind":"proof","summary":"By \\eqrefdefinetp, if T_\\mathfrak p[ e(-Q(\\mathfrak u))f](x) \\ne 0, then x \\in E(\\mathfrak p).…","labels":["eq-term-A","eq-term-B","eq-term-C"],"detail_key":"p32"},{"id":"n31321","layer":"informal","project":"p32","title":"first tree pointwise","kind":"lemma","summary":"[first tree pointwise] For all \\mathfrak u\\in \\mathfrak U, all L \\in L(\\mathfrak T(\\mathfrak u)…","labels":["first-tree-pointwise"],"detail_key":"p32"},{"id":"n31322","layer":"informal","project":"p32","title":"Let s \\in \\sigma(\\mathfrak u,x). If x, y \\in X are such that K_s(x,y)\\neq 0, then, by \\eq…","kind":"proof","summary":"Let s \\in \\sigma(\\mathfrak u,x). If x, y \\in X are such that K_s(x,y)\\neq 0, then, by \\eqrefsup…","labels":[],"detail_key":"p32"},{"id":"n31323","layer":"informal","project":"p32","title":"second tree pointwise","kind":"lemma","summary":"[second tree pointwise] For all \\mathfrak u\\in \\mathfrak U, all L \\in L(\\mathfrak T(\\mathfrak u…","labels":["second-tree-pointwise"],"detail_key":"p32"},{"id":"n31324","layer":"informal","project":"p32","title":"Let s_1 = \\underline\\sigma(\\mathfrak u, x). By definition, there exists a tile \\mathfrak…","kind":"proof","summary":"Let s_1 = \\underline\\sigma(\\mathfrak u, x). By definition, there exists a tile \\mathfrak p\\in \\…","labels":[],"detail_key":"p32"},{"id":"n31325","layer":"informal","project":"p32","title":"third tree pointwise","kind":"lemma","summary":"[third tree pointwise] For all \\mathfrak u\\in \\mathfrak U, all L \\in L(\\mathfrak T(\\mathfrak u)…","labels":["third-tree-pointwise"],"detail_key":"p32"},{"id":"n31326","layer":"informal","project":"p32","title":"eq-canc-comp","kind":"proof","summary":"We have for J \\in J(\\mathfrak T(\\mathfrak u)): \\int_J K_s(x,y)(1 - P_J(\\mathfrak T(\\mathfrak u)…","labels":["eq-canc-comp"],"detail_key":"p32"},{"id":"n31327","layer":"informal","project":"p32","title":"tree projection estimate","kind":"lemma","summary":"[tree projection estimate] Let \\mathfrak u\\in \\mathfrak U. Then we have for all f, g bounded wi…","labels":["tree-projection-estimate","eq-tree-est"],"detail_key":"p32"},{"id":"n31328","layer":"informal","project":"p32","title":"nontangential operator bound","kind":"lemma","summary":"[nontangential operator bound] For all bounded f with bounded support and all \\vartheta\\in \\The…","labels":["nontangential-operator-bound"],"detail_key":"p32"},{"id":"n31329","layer":"informal","project":"p32","title":"boundary operator bound","kind":"lemma","summary":"[boundary operator bound] For all \\mathfrak u\\in \\mathfrak U and all bounded functions f with b…","labels":["boundary-operator-bound","eq-S-bound"],"detail_key":"p32"},{"id":"n31330","layer":"informal","project":"p32","title":"Proof of \\Creftree-projection-estimate","kind":"proof","summary":"[Proof of \\Creftree-projection-estimate] Let L \\in L(\\mathfrak T(\\mathfrak u)). Let b(x') denot…","labels":[],"detail_key":"p32"},{"id":"n31331","layer":"informal","project":"p32","title":"Proof of \\Crefnontangential-operator-bound","kind":"proof","summary":"[Proof of \\Crefnontangential-operator-bound] Fix s_1, s_2. By \\eqrefeq-psisum we have for all x…","labels":["eq-sharp-trunc-term","eq-lower-bound-term","eq-upper-bound-term","pf-nontangential-operator-bound-imeq"],"detail_key":"p32"},{"id":"n31332","layer":"informal","project":"p32","title":"boundary overlap","kind":"lemma","summary":"[boundary overlap] For every cube I \\in D, there exist at most 2^9a cubes J \\in D with s(J) = s…","labels":["boundary-overlap"],"detail_key":"p32"},{"id":"n31333","layer":"informal","project":"p32","title":"Suppose that B(c(I), 16 D^s(I)) \\cap B(c(J), 16 D^s(J)) \\ne \\emptyset and s(I) = s(J). Th…","kind":"proof","summary":"Suppose that B(c(I), 16 D^s(I)) \\cap B(c(J), 16 D^s(J)) \\ne \\emptyset and s(I) = s(J). Then B(c…","labels":[],"detail_key":"p32"},{"id":"n31334","layer":"informal","project":"p32","title":"Proof of \\Crefboundary-operator-bound","kind":"proof","summary":"[Proof of \\Crefboundary-operator-bound] Note that by definition, S_1,\\mathfrak uf is a finite s…","labels":["eq-boundary-operator-bound-1"],"detail_key":"p32"},{"id":"n31335","layer":"informal","project":"p32","title":"densities tree bound","kind":"lemma","summary":"[densities tree bound] Let \\mathfrak u\\in \\mathfrak U. Then for all bounded f with bounded supp…","labels":["densities-tree-bound","eq-cor-tree-est","eq-cor-tree-est-F"],"detail_key":"p32"},{"id":"n31336","layer":"informal","project":"p32","title":"local dens1 tree bound","kind":"lemma","summary":"[local dens1 tree bound] Let \\mathfrak u\\in \\mathfrak U and L \\in L(\\mathfrak T(\\mathfrak u)).…","labels":["local-dens1-tree-bound","eq-1density-estimate-tree"],"detail_key":"p32"},{"id":"n31337","layer":"informal","project":"p32","title":"local dens2 tree bound","kind":"lemma","summary":"[local dens2 tree bound] Let \\mathfrak u\\in \\mathfrak U and J \\in J(\\mathfrak T(\\mathfrak u)).…","labels":["local-dens2-tree-bound"],"detail_key":"p32"},{"id":"n31338","layer":"informal","project":"p32","title":"Proof of \\Crefdensities-tree-bound","kind":"proof","summary":"[Proof of \\Crefdensities-tree-bound] Denote E(\\mathfrak u) = \\bigcup_\\mathfrak p\\in \\mathfrak T…","labels":["eq-both-factors-tree","eq-factor-L-tree","eq-cor-tree-proof","eq-factor-J-tree"],"detail_key":"p32"},{"id":"n31339","layer":"informal","project":"p32","title":"Proof of \\Creflocal-dens1-tree-bound","kind":"proof","summary":"[Proof of \\Creflocal-dens1-tree-bound] If the set on the right hand side is empty, then \\eqrefe…","labels":[],"detail_key":"p32"},{"id":"n31340","layer":"informal","project":"p32","title":"Proof of \\Creflocal-dens2-tree-bound","kind":"proof","summary":"[Proof of \\Creflocal-dens2-tree-bound] We prove the inequality with the constant 2^201a^3 repla…","labels":["measure-comparison"],"detail_key":"p32"},{"id":"n31341","layer":"informal","project":"p32","title":"adjoint tile support","kind":"lemma","summary":"[adjoint tile support] For each \\mathfrak p\\in \\mathfrak P, we have T_\\mathfrak p^* g = 1_B(c(\\…","labels":["adjoint-tile-support"],"detail_key":"p32"},{"id":"n31342","layer":"informal","project":"p32","title":"By \\eqrefforest1, E(\\mathfrak p) \\subset I(\\mathfrak p) \\subset I(\\mathfrak u). Thus by \\…","kind":"proof","summary":"By \\eqrefforest1, E(\\mathfrak p) \\subset I(\\mathfrak p) \\subset I(\\mathfrak u). Thus by \\eqrefd…","labels":[],"detail_key":"p32"},{"id":"n31343","layer":"informal","project":"p32","title":"adjoint tree estimate","kind":"lemma","summary":"[adjoint tree estimate] For all bounded g supported on G we have that \\left\\| \\sum_\\mathfrak p\\…","labels":["adjoint-tree-estimate"],"detail_key":"p32"},{"id":"n31344","layer":"informal","project":"p32","title":"eq-adjoint-bound","kind":"proof","summary":"By \\Crefdensities-tree-bound, we have for all bounded f and g with |g| \\le 1_G that \\left| \\int…","labels":["eq-adjoint-bound"],"detail_key":"p32"},{"id":"n31345","layer":"informal","project":"p32","title":"adjoint tree control","kind":"lemma","summary":"[adjoint tree control] We have for all \\mathfrak u\\in \\mathfrak U and all bounded g supported o…","labels":["adjoint-tree-control"],"detail_key":"p32"},{"id":"n31346","layer":"informal","project":"p32","title":"This follows immediately from Minkowski's inequality, \\CrefHardy-Littlewood and \\Crefadjo…","kind":"proof","summary":"This follows immediately from Minkowski's inequality, \\CrefHardy-Littlewood and \\Crefadjoint-tr…","labels":[],"detail_key":"p32"},{"id":"n31347","layer":"informal","project":"p32","title":"correlation separated trees","kind":"lemma","summary":"[correlation separated trees] For any \\mathfrak u_1 \\ne \\mathfrak u_2 \\in \\mathfrak U and all b…","labels":["correlation-separated-trees","eq-lhs-sep-tree","eq-rhs-sep-tree"],"detail_key":"p32"},{"id":"n31348","layer":"informal","project":"p32","title":"Proof of \\Crefcorrelation-separated-trees","kind":"proof","summary":"[Proof of \\Crefcorrelation-separated-trees] By \\Crefadjoint-tile-support and \\eqrefdyadicproper…","labels":["def-Tree-S-set"],"detail_key":"p32"},{"id":"n31349","layer":"informal","project":"p32","title":"correlation distant tree parts","kind":"lemma","summary":"[correlation distant tree parts] We have for all \\mathfrak u_1 \\ne \\mathfrak u_2 \\in \\mathfrak…","labels":["correlation-distant-tree-parts","eq-lhs-big-sep-tree","eq-rhs-big-sep-tree"],"detail_key":"p32"},{"id":"n31350","layer":"informal","project":"p32","title":"correlation near tree parts","kind":"lemma","summary":"[correlation near tree parts] We have for all \\mathfrak u_1 \\ne \\mathfrak u_2 \\in \\mathfrak U w…","labels":["correlation-near-tree-parts","eq-lhs-small-sep-tree","eq-rhs-small-sep-tree"],"detail_key":"p32"},{"id":"n31351","layer":"informal","project":"p32","title":"overlap implies distance","kind":"lemma","summary":"[overlap implies distance] Let \\mathfrak u","labels":["overlap-implies-distance"],"detail_key":"p32"},{"id":"n31352","layer":"informal","project":"p32","title":"Suppose first that \\mathfrak p\\in \\mathfrak T(\\mathfrak u_1). Then I(\\mathfrak p) \\subset…","kind":"proof","summary":"Suppose first that \\mathfrak p\\in \\mathfrak T(\\mathfrak u_1). Then I(\\mathfrak p) \\subset I(\\ma…","labels":[],"detail_key":"p32"},{"id":"n31353","layer":"informal","project":"p32","title":"dyadic partition 1","kind":"lemma","summary":"[dyadic partition 1] We have that I(\\mathfrak u_1) = \\dot\\bigcup_J \\in J' J\\,.","labels":["dyadic-partition-1"],"detail_key":"p32"},{"id":"n31354","layer":"informal","project":"p32","title":"By \\Cre","kind":"proof","summary":"By \\Cre","labels":[],"detail_key":"p32"},{"id":"n31355","layer":"informal","project":"p32","title":"Lipschitz partition unity","kind":"lemma","summary":"[Lipschitz partition unity] There exists a family of functions \\chi_J, J \\in J' such that 1_I(\\…","labels":["Lipschitz-partition-unity","eq-pao-1","eq-pao-2","eq-pao-3"],"detail_key":"p32"},{"id":"n31356","layer":"informal","project":"p32","title":"moderate scale change","kind":"lemma","summary":"[moderate scale change] If J, J' \\in J' with B(J) \\cap B(J') \\ne \\emptyset\\,, then |s(J) - s(J'…","labels":["moderate-scale-change"],"detail_key":"p32"},{"id":"n31357","layer":"informal","project":"p32","title":"Proof of \\CrefLipschitz-partition-unity","kind":"proof","summary":"[Proof of \\CrefLipschitz-partition-unity] For each cube J \\in J let \\tilde\\chi_J(y) = 1_I(\\math…","labels":[],"detail_key":"p32"},{"id":"n31358","layer":"informal","project":"p32","title":"Proof of \\Crefmoderate-scale-change","kind":"proof","summary":"[Proof of \\Crefmoderate-scale-change] Suppose that s(J') < s(J) - 1. Then s(J) > -S. Thus, by t…","labels":["eq-tile-incl-1","tile-incl-2"],"detail_key":"p32"},{"id":"n31359","layer":"informal","project":"p32","title":"Holder correlation tree","kind":"lemma","summary":"[Holder correlation tree] We have for all J \\in J' that \\|h_J\\|_C^\\tau(B(c(J), 16D^s(J))) \\le 2…","labels":["Holder-correlation-tree","hHolder"],"detail_key":"p32"},{"id":"n31360","layer":"informal","project":"p32","title":"Holder correlation tile","kind":"lemma","summary":"[Holder correlation tile] Let \\mathfrak u\\in \\mathfrak U and \\mathfrak p\\in \\mathfrak T(\\mathfr…","labels":["Holder-correlation-tile","T*Holder2"],"detail_key":"p32"},{"id":"n31361","layer":"informal","project":"p32","title":"T*Holder1b","kind":"proof","summary":"By \\eqrefdefinetp*, we have |e(Q(\\mathfrak u)(y)) T_\\mathfrak p^* g(y) - e(Q(\\mathfrak u)(y'))…","labels":["T*Holder1b","T*Holder1","eq-lem-tile-Holder-comp","eq-lem-Tile-holder-im1","eq-lem-Tile-holder-im2"],"detail_key":"p32"},{"id":"n31362","layer":"informal","project":"p32","title":"limited scale impact","kind":"lemma","summary":"[limited scale impact] Let \\mathfrak p\\in \\mathfrak T(\\mathfrak u_2) \\setminus \\mathfrakS, J \\i…","labels":["limited-scale-impact"],"detail_key":"p32"},{"id":"n31363","layer":"informal","project":"p32","title":"For the first estimate, assume that s(\\mathfrak p) < s(J), then in particular s(\\mathfrak…","kind":"proof","summary":"For the first estimate, assume that s(\\mathfrak p) < s(J), then in particular s(\\mathfrak p) \\l…","labels":[],"detail_key":"p32"},{"id":"n31364","layer":"informal","project":"p32","title":"local tree control","kind":"lemma","summary":"[local tree control] For all J \\in J' and all bounded g with bounded support \\sup_B^\\circ(J) |T…","labels":["local-tree-control"],"detail_key":"p32"},{"id":"n31365","layer":"informal","project":"p32","title":"eq-sep-tree-aux-3","kind":"proof","summary":"By the triangle inequality and since T_\\mathfrak p^* g = 1_B(c(\\mathfrak p), 5D^s(\\mathfrak p))…","labels":["eq-sep-tree-aux-3"],"detail_key":"p32"},{"id":"n31366","layer":"informal","project":"p32","title":"scales impacting interval","kind":"lemma","summary":"[scales impacting interval] Let \\mathfrak C= \\mathfrak T(\\mathfrak u_1) or \\mathfrak C= \\mathfr…","labels":["scales-impacting-interval"],"detail_key":"p32"},{"id":"n31367","layer":"informal","project":"p32","title":"By \\Cre","kind":"proof","summary":"By \\Cre","labels":[],"detail_key":"p32"},{"id":"n31368","layer":"informal","project":"p32","title":"global tree control 1","kind":"lemma","summary":"[global tree control 1] Let \\mathfrak C_1 = \\mathfrak T(\\mathfrak u_1) and \\mathfrak C_2 = \\mat…","labels":["global-tree-control-1","TreeUB","TreeHolder"],"detail_key":"p32"},{"id":"n31369","layer":"informal","project":"p32","title":"eq-C-Lip","kind":"proof","summary":"Note that \\eqrefTreeUB follows from \\eqrefTreeHolder, since for y'\\in B^\\circ(J), by the triang…","labels":["eq-C-Lip"],"detail_key":"p32"},{"id":"n31370","layer":"informal","project":"p32","title":"global tree control 2","kind":"lemma","summary":"[global tree control 2] We have for all J \\in J' and all bounded g with bounded support \\sup_B'…","labels":["global-tree-control-2"],"detail_key":"p32"},{"id":"n31371","layer":"informal","project":"p32","title":"By \\Crefglobal-tree-control-1 \\sup_B'(J) |T^*_\\mathfrak T(\\mathfrak u_2) \\cap \\mathfrakS…","kind":"proof","summary":"By \\Crefglobal-tree-control-1 \\sup_B'(J) |T^*_\\mathfrak T(\\mathfrak u_2) \\cap \\mathfrakS g| \\le…","labels":[],"detail_key":"p32"},{"id":"n31372","layer":"informal","project":"p32","title":"Proof of \\CrefHolder-correlation-tree","kind":"proof","summary":"[Proof of \\CrefHolder-correlation-tree] Let P be the product on the right hand side of \\eqrefhH…","labels":["eq-h-Lip-1","eq-h-Lip-2","eq-h-Lip-3"],"detail_key":"p32"},{"id":"n31373","layer":"informal","project":"p32","title":"lower oscillation bound","kind":"lemma","summary":"[lower oscillation bound] For all J \\in J', we have that d_B(J)(Q(\\mathfrak u_1), Q(\\mathfrak u…","labels":["lower-oscillation-bound"],"detail_key":"p32"},{"id":"n31374","layer":"informal","project":"p32","title":"Since \\emptyset \\ne \\mathfrak T(\\mathfrak u_1) \\subset \\mathfrakS by \\Crefoverlap-implies…","kind":"proof","summary":"Since \\emptyset \\ne \\mathfrak T(\\mathfrak u_1) \\subset \\mathfrakS by \\Crefoverlap-implies-dista…","labels":[],"detail_key":"p32"},{"id":"n31375","layer":"informal","project":"p32","title":"Proof of \\Crefcorrelation-distant-tree-parts","kind":"proof","summary":"[Proof of \\Crefcorrelation-distant-tree-parts] We have \\eqrefeq-lhs-big-sep-tree = \\left| \\int_…","labels":["eq-big-sep-1"],"detail_key":"p32"},{"id":"n31376","layer":"informal","project":"p32","title":"dyadic partition 2","kind":"lemma","summary":"[dyadic partition 2] We have I(\\mathfrak u_1) = \\dot\\bigcup_J \\in J' J\\,.","labels":["dyadic-partition-2"],"detail_key":"p32"},{"id":"n31377","layer":"informal","project":"p32","title":"By \\Cre","kind":"proof","summary":"By \\Cre","labels":[],"detail_key":"p32"},{"id":"n31378","layer":"informal","project":"p32","title":"bound for tree projection","kind":"lemma","summary":"[bound for tree projection] We have for all bounded f with bounded support \\|P_J'|T_\\mathfrak T…","labels":["bound-for-tree-projection"],"detail_key":"p32"},{"id":"n31379","layer":"informal","project":"p32","title":"Proof of \\Crefcorrelation-near-tree-parts","kind":"proof","summary":"[Proof of \\Crefcorrelation-near-tree-parts] By \\Creftree-projection-estimate and \\Crefadjoint-t…","labels":[],"detail_key":"p32"},{"id":"n31380","layer":"informal","project":"p32","title":"thin scale impact","kind":"lemma","summary":"[thin scale impact] If \\mathfrak p\\in \\mathfrak T(\\mathfrak u_2) \\setminus \\mathfrakS and J \\in…","labels":["thin-scale-impact"],"detail_key":"p32"},{"id":"n31381","layer":"informal","project":"p32","title":"Suppose that s(\\mathfrak p) > s(J) + 2 -\\fracZn202a^3 =: s(J) - s_1. Then, we have s_1 +…","kind":"proof","summary":"Suppose that s(\\mathfrak p) > s(J) + 2 -\\fracZn202a^3 =: s(J) - s_1. Then, we have s_1 + 2 \\ge…","labels":[],"detail_key":"p32"},{"id":"n31382","layer":"informal","project":"p32","title":"square function count","kind":"lemma","summary":"[square function count] For each J \\in J' and all s, we have \\frac1\\mu(J) \\int_J \\Bigg(\\sum_\\su…","labels":["square-function-count"],"detail_key":"p32"},{"id":"n31383","layer":"informal","project":"p32","title":"eq-sep-small-incl","kind":"proof","summary":"Since J \\in J' we have J \\subset I(\\mathfrak u_1). Thus, if B(I) \\cap J \\ne \\emptyset then B(I)…","labels":["eq-sep-small-incl","eq-sep-small-bound"],"detail_key":"p32"},{"id":"n31384","layer":"informal","project":"p32","title":"Proof of \\Crefbound-for-tree-projection","kind":"proof","summary":"[Proof of \\Crefbound-for-tree-projection] Expanding the definition of P_J', we have \\|P_J'|T_\\m…","labels":["eq-sep-tree-small-1","eq-sep-tree-small-2"],"detail_key":"p32"},{"id":"n31385","layer":"informal","project":"p32","title":"forest row decomposition","kind":"lemma","summary":"[forest row decomposition] Let (\\mathfrak U, \\mathfrak T) be an n-forest. Then there exists a d…","labels":["forest-row-decomposition"],"detail_key":"p32"},{"id":"n31386","layer":"informal","project":"p32","title":"Define recursively \\mathfrak U_j to be a maximal disjoint set of tiles \\mathfrak u in \\ma…","kind":"proof","summary":"Define recursively \\mathfrak U_j to be a maximal disjoint set of tiles \\mathfrak u in \\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31387","layer":"informal","project":"p32","title":"row bound","kind":"lemma","summary":"[row bound] For each 1 \\le j \\le 2^n and each bounded g supported on G we have \\left\\| T_\\mathf…","labels":["row-bound","eq-row-bound-1","eq-row-bound-2"],"detail_key":"p32"},{"id":"n31388","layer":"informal","project":"p32","title":"Since for each j the top cubes I(\\mathfrak u), \\mathfrak u\\in \\mathfrak U_j are disjoint,…","kind":"proof","summary":"Since for each j the top cubes I(\\mathfrak u), \\mathfrak u\\in \\mathfrak U_j are disjoint, we ha…","labels":[],"detail_key":"p32"},{"id":"n31389","layer":"informal","project":"p32","title":"row correlation","kind":"lemma","summary":"[row correlation] For all 1 \\le j,j' \\le 2^n with j\\ne j' and for all bounded g_1, g_2 supporte…","labels":["row-correlation"],"detail_key":"p32"},{"id":"n31390","layer":"informal","project":"p32","title":"eq-S2uu'","kind":"proof","summary":"We have by \\Crefadjoint-tile-support and the triangle inequality that \\left| \\int T_\\mathfrakR_…","labels":["eq-S2uu'"],"detail_key":"p32"},{"id":"n31391","layer":"informal","project":"p32","title":"disjoint row support","kind":"lemma","summary":"[disjoint row support] The sets E_j, 1 \\le j \\le 2^n are pairwise disjoint.","labels":["disjoint-row-support"],"detail_key":"p32"},{"id":"n31392","layer":"informal","project":"p32","title":"Suppose that \\mathfrak p\\in \\mathfrak T(\\mathfrak u) and \\mathfrak p' \\in \\mathfrak T(\\ma…","kind":"proof","summary":"Suppose that \\mathfrak p\\in \\mathfrak T(\\mathfrak u) and \\mathfrak p' \\in \\mathfrak T(\\mathfrak…","labels":[],"detail_key":"p32"},{"id":"n31393","layer":"informal","project":"p32","title":"Proof of \\Crefforest-operator","kind":"proof","summary":"[Proof of \\Crefforest-operator] By \\eqrefdefinetp*, we have for each j T_\\mathfrakR_j^*g = \\sum…","labels":["eq-forest-bound-1","eq-forest-bound-2"],"detail_key":"p32"},{"id":"n31394","layer":"informal","project":"p32","title":"Lipschitz Holder approximation","kind":"lemma","summary":"[Lipschitz Holder approximation] Let z\\in X and R>0. Let \\varphi: X \\to C be a function support…","labels":["Lipschitz-Holder-approximation","eq-firstt","eq-secondt"],"detail_key":"p32"},{"id":"n31395","layer":"informal","project":"p32","title":"eql01","kind":"proof","summary":"Define for x,y\\in X the Lipschitz and thus measurable function L(x,y) := \\max\\0, 1 - \\frac\\rho(…","labels":["eql01","eql30","2nt1","eql32","eql1","eql2","eql3","eql4","eql5","eql42","eql52","eql10","eql21","eql22","eql23","eql224","eql225","eql226"],"detail_key":"p32"},{"id":"n31396","layer":"informal","project":"p32","title":"Proof of \\CrefHolder-van-der-Corput","kind":"proof","summary":"[Proof of \\CrefHolder-van-der-Corput] Let z\\in X and R>0 and set B=B(z,R). Let \\varphi be given…","labels":["eql69","eql60","eql61","eql62","eql63","eql64","eql65","eql66"],"detail_key":"p32"},{"id":"n31397","layer":"informal","project":"p32","title":"layer cake representation","kind":"lemma","summary":"[layer cake representation] Let 1\\le p< \\infty. Then for any measurable function u:X\\to [0,\\inf…","labels":["layer-cake-representation","eq-layercake"],"detail_key":"p32"},{"id":"n31398","layer":"informal","project":"p32","title":"The left-hand side of \\eqrefeq-layercake is by definition \\int_X u(x)^p \\, d\\mu(x)\\, . Wr…","kind":"proof","summary":"The left-hand side of \\eqrefeq-layercake is by definition \\int_X u(x)^p \\, d\\mu(x)\\, . Writing…","labels":[],"detail_key":"p32"},{"id":"n31399","layer":"informal","project":"p32","title":"covering separable space","kind":"lemma","summary":"[covering separable space] For each r > 0, there exists a countable collection C(r) \\subset X o…","labels":["covering-separable-space"],"detail_key":"p32"},{"id":"n31400","layer":"informal","project":"p32","title":"It clearly suffices to construct finite collections C(r,k) such that B(o, r2^k) \\subset \\…","kind":"proof","summary":"It clearly suffices to construct finite collections C(r,k) such that B(o, r2^k) \\subset \\bigcup…","labels":[],"detail_key":"p32"},{"id":"n31401","layer":"informal","project":"p32","title":"Proof of \\CrefHardy-Littlewood","kind":"proof","summary":"[Proof of \\CrefHardy-Littlewood] Let the collection B be given. We first show \\eqrefeq-besico.…","labels":["eqbes1","3rone","3rtwo","eqbes2","eqbesi11","eqbesi10","eqbesi12","eqbesi13","eqbesi14","eqbesi15"],"detail_key":"p32"},{"id":"n31402","layer":"informal","project":"p32","title":"two-sided metric space Carleson","kind":"theorem","summary":"[two-sided metric space Carleson] For all integers a \\ge 4 and real numbers 1<q\\le 2 the follow…","labels":["two-sided-metric-space-Carleson","two-sided-Hr-bound-assumption","two-sided-resweak"],"detail_key":"p32"},{"id":"n31403","layer":"informal","project":"p32","title":"nontangential-from-simple","kind":"lemma","summary":"[nontangential-from-simple] Assume \\eqreftwo-sided-Hr-bound-assumption holds. Then, for every b…","labels":["nontangential-from-simple","concretetstarbound"],"detail_key":"p32"},{"id":"n31404","layer":"informal","project":"p32","title":"Proof of \\Creftwo-sided-metric-space-Carleson","kind":"proof","summary":"[Proof of \\Creftwo-sided-metric-space-Carleson] Let 1<q\\le 2 be a real number. Let \\Theta be a…","labels":["original-operator-assumption","modified-operator-assumption"],"detail_key":"p32"},{"id":"n31405","layer":"informal","project":"p32","title":"Calderon-Zygmund Weak (1, 1)","kind":"lemma","summary":"[Calderon-Zygmund Weak (1, 1)] Let f:X\\to\\C be a bounded measurable function supported on a set…","labels":["calderon-zygmund-weak-1-1","eq-strong-2-2-assumption","eq-weak-1-1"],"detail_key":"p32"},{"id":"n31406","layer":"informal","project":"p32","title":"geometric series estimate","kind":"lemma","summary":"[geometric series estimate] For all real numbers x\\ge 4, \\sum_n=0^\\infty 2^-\\fracnx \\le 2^x.","labels":["geometric-series-estimate"],"detail_key":"p32"},{"id":"n31407","layer":"informal","project":"p32","title":"By convexity, for all 0\\le\\lambda\\le1 2^\\lambda(-\\frac14) \\le \\lambda 2^-\\frac14 + (1-\\la…","kind":"proof","summary":"By convexity, for all 0\\le\\lambda\\le1 2^\\lambda(-\\frac14) \\le \\lambda 2^-\\frac14 + (1-\\lambda)2…","labels":[],"detail_key":"p32"},{"id":"n31408","layer":"informal","project":"p32","title":"estimate x shift","kind":"lemma","summary":"[estimate x shift] Let 0<r and x\\in X. Let g:X\\to\\C be a bounded measurable function supported…","labels":["estimate-x-shift"],"detail_key":"p32"},{"id":"n31409","layer":"informal","project":"p32","title":"xx'difference","kind":"proof","summary":"By definition, \\left| T_r g(x) - T_r g(x') \\right| =\\left|\\int_r\\le\\rho(x,y) K(x,y) g(y) \\,d\\mu…","labels":["xx'difference","firstxx'","secondxx'","thirdxx'","firstxx'b","firstxx'c","thirdxx'b","secondxx'b","secondxx'c","secondxx'd"],"detail_key":"p32"},{"id":"n31410","layer":"informal","project":"p32","title":"Cotlar control","kind":"lemma","summary":"[Cotlar control] Let 0<r\\le R and x\\in X. Let g:X\\to\\C be a bounded measurable function support…","labels":["Cotlar-control","eq-cotlar-control"],"detail_key":"p32"},{"id":"n31411","layer":"informal","project":"p32","title":"eqcotlar0","kind":"proof","summary":"Let x and x' be given with \\rho(x,x')\\le\\frac R4. By \\Crefestimate-x-shift, we estimate the lef…","labels":["eqcotlar0","eqcotlar-1","eqcotlar1","eqcotlar5","eqcotlar2"],"detail_key":"p32"},{"id":"n31412","layer":"informal","project":"p32","title":"Cotlar sets","kind":"lemma","summary":"[Cotlar sets] Assume that \\eqreftwo-sided-Hr-bound-assumption holds. Let 0<r\\le R and x\\in X. L…","labels":["Cotlar-sets","first-cotlar-exception","second-cotlar-exception"],"detail_key":"p32"},{"id":"n31413","layer":"informal","project":"p32","title":"Let r, R, x and g be given. If M(T_rg)(x)=0, then T_rg is zero almost everywhere and the…","kind":"proof","summary":"Let r, R, x and g be given. If M(T_rg)(x)=0, then T_rg is zero almost everywhere and the estima…","labels":[],"detail_key":"p32"},{"id":"n31414","layer":"informal","project":"p32","title":"Cotlar estimate","kind":"lemma","summary":"[Cotlar estimate] Assume that \\eqreftwo-sided-Hr-bound-assumption holds. Let 0<r\\le R and x\\in…","labels":["Cotlar-estimate","eq-cotlar-estimate"],"detail_key":"p32"},{"id":"n31415","layer":"informal","project":"p32","title":"By \\CrefCotlar-sets, the set of all x'\\in B(x,\\frac R4) such that at least one of the con…","kind":"proof","summary":"By \\CrefCotlar-sets, the set of all x'\\in B(x,\\frac R4) such that at least one of the condition…","labels":[],"detail_key":"p32"},{"id":"n31416","layer":"informal","project":"p32","title":"simple nontangential operator","kind":"lemma","summary":"[simple nontangential operator] Assume that \\eqreftwo-sided-Hr-bound-assumption holds. For ever…","labels":["simple-nontangential-operator","trzerobound"],"detail_key":"p32"},{"id":"n31417","layer":"informal","project":"p32","title":"For a fixed \\lambda we note that (rewriting x'\\in B(x,R) as x\\in B(x',R)): \\x : T_*^r g(x…","kind":"proof","summary":"For a fixed \\lambda we note that (rewriting x'\\in B(x,R) as x\\in B(x',R)): \\x : T_*^r g(x) > \\l…","labels":[],"detail_key":"p32"},{"id":"n31418","layer":"informal","project":"p32","title":"small annulus","kind":"lemma","summary":"[small annulus] Let f:X\\to\\C be a bounded measurable function supported on a set of finite meas…","labels":["small-annulus"],"detail_key":"p32"},{"id":"n31419","layer":"informal","project":"p32","title":"We only prove the second inequality, the first one is analogous. Note that the integrand…","kind":"proof","summary":"We only prove the second inequality, the first one is analogous. Note that the integrand is bou…","labels":[],"detail_key":"p32"},{"id":"n31420","layer":"informal","project":"p32","title":"nontangential operator boundary","kind":"lemma","summary":"[nontangential operator boundary] Let f:X\\to\\C be a bounded measurable function supported on a…","labels":["nontangential-operator-boundary","tang-unm-op-eq"],"detail_key":"p32"},{"id":"n31421","layer":"informal","project":"p32","title":"eq-without-suprema-1","kind":"proof","summary":"We show two inequalities. Let \\epsilon>0. Let R_1<R_2 and x'\\in B(x,R_1). Then for small enough…","labels":["eq-without-suprema-1","eq-diff-small-1","eq-other-1","eq-without-suprema-2","eq-diff-small-2","eq-other-2"],"detail_key":"p32"},{"id":"n31422","layer":"informal","project":"p32","title":"Proof of \\Crefnontangential-from-simple","kind":"proof","summary":"[Proof of \\Crefnontangential-from-simple] Fix g as in the Lemma. Applying \\Crefsimple-nontangen…","labels":["tzerobound","eq-simpler--nontangential","concretetstartriangle"],"detail_key":"p32"},{"id":"n31423","layer":"informal","project":"p32","title":"Maximal theorem","kind":"lemma","summary":"[Maximal theorem] Let f: X \\to \\C be bounded, measurable, supported on a set of finite measure,…","labels":["maximal-theorem","maximal-theorem-equation"],"detail_key":"p32"},{"id":"n31424","layer":"informal","project":"p32","title":"maximal-theorem-a","kind":"proof","summary":"By definition, for each x\\in X with Mf(x) > \\alpha, there exists a ball B_x such that x\\in B_x…","labels":["maximal-theorem-a"],"detail_key":"p32"},{"id":"n31425","layer":"informal","project":"p32","title":"Lebesgue differentiation","kind":"lemma","summary":"[Lebesgue differentiation] Let f be a bounded measurable function supported on a set of finite…","labels":["Lebesgue-differentiation"],"detail_key":"p32"},{"id":"n31426","layer":"informal","project":"p32","title":"This follows from the Lebesgue differentiation theorem, which is already formalized in Le…","kind":"proof","summary":"This follows from the Lebesgue differentiation theorem, which is already formalized in Lean.","labels":[],"detail_key":"p32"},{"id":"n31427","layer":"informal","project":"p32","title":"Disjoint family countable","kind":"lemma","summary":"[Disjoint family countable] In a doubling metric measure space (X,\\rho,\\mu, a), every disjoint…","labels":["disjoint-family-countable"],"detail_key":"p32"},{"id":"n31428","layer":"informal","project":"p32","title":"Choose an arbitrary x\\in X as reference point. For q, Q\\inQ_+, let J_q,Q denote the set o…","kind":"proof","summary":"Choose an arbitrary x\\in X as reference point. For q, Q\\inQ_+, let J_q,Q denote the set of all…","labels":[],"detail_key":"p32"},{"id":"n31429","layer":"informal","project":"p32","title":"Ball covering","kind":"lemma","summary":"[Ball covering] Given an open set O\\ne X, there exists a countable family of balls B_j = B(x_j,…","labels":["ball-covering","balls-disjoint","balls-covering","enlarged-balls-intersect-complement"],"detail_key":"p32"},{"id":"n31430","layer":"informal","project":"p32","title":"control-distance-growth","kind":"proof","summary":"Define for x\\in O, \\delta(x):= \\sup \\\\delta\\inR: B(x,\\delta)\\subset O\\. Since O is open, and O\\…","labels":["control-distance-growth","control-distance-growth-b","control-distance-growth-c"],"detail_key":"p32"},{"id":"n31431","layer":"informal","project":"p32","title":"Calderon Zygmund decomposition","kind":"lemma","summary":"[Calderon Zygmund decomposition] Let f be a bounded, a.e. measurable function supported on a se…","labels":["Calderon-Zygmund-decomposition","eq-gb-dec","eq-g-max","eq-g-L1-norm","eq-supp-bj","eq-bad-mean-zero","eq-bj-L1","eq-bset-length-sum","eq-b-L1"],"detail_key":"p32"},{"id":"n31432","layer":"informal","project":"p32","title":"eq-g-def","kind":"proof","summary":"Let E_\\alpha:=\\x\\in X: Mf(x)>\\alpha\\. Then E_\\alpha is open. Assume first that E_\\alpha \\ne X.…","labels":["eq-g-def","large-ball-estimate","eq-bj-int"],"detail_key":"p32"},{"id":"n31433","layer":"informal","project":"p32","title":"Estimate good","kind":"lemma","summary":"[Estimate good] \\mu\\left(\\x\\in X: |T_r g(x)|>\\alpha/2\\\\right) \\le \\frac2^2a^3+3a+2c\\alpha \\int…","labels":["estimate-good"],"detail_key":"p32"},{"id":"n31434","layer":"informal","project":"p32","title":"eq-Hr-g","kind":"proof","summary":"We estimate using monotonicity of the integral \\mu\\left(\\x\\in X: |T_r g(x)|>\\alpha/2\\\\right)\\le…","labels":["eq-Hr-g"],"detail_key":"p32"},{"id":"n31435","layer":"informal","project":"p32","title":"Estimate bad partial","kind":"lemma","summary":"[Estimate bad partial] Let x\\in X\\setminus\\Omega. Then |T_rb(x)| \\le 3F(x)+\\alpha/8, where F(x)…","labels":["estimate-bad-partial"],"detail_key":"p32"},{"id":"n31436","layer":"informal","project":"p32","title":"eq-b-dec-1","kind":"proof","summary":"We decompose the index set J into the following disjoint sets: J_1(x)&:=\\j\\,: r+3r_j \\le \\rho(x…","labels":["eq-b-dec-1","eq-b-dec-2","eq-b-dec-3","eq-Om-cj","eq-J1-diff-est","eq-dj-est","eq-J2-diff-est","eq-J2-diff-est-2","eq-J2-union-subset","eq-J2-diff-est-4"],"detail_key":"p32"},{"id":"n31437","layer":"informal","project":"p32","title":"Estimate F set","kind":"lemma","summary":"[Estimate F set] For F as defined in \\Crefestimate-bad-partial, we have \\mu(\\x\\in X\\setminus\\Om…","labels":["estimate-F-set","eq-F-X-minus-Omega"],"detail_key":"p32"},{"id":"n31438","layer":"informal","project":"p32","title":"eq-F-est-1","kind":"proof","summary":"We estimate \\mu(\\x\\in X\\setminus\\Omega&: F(x)> \\alpha/8\\) \\le \\frac8\\alpha \\int_X\\setminus \\Ome…","labels":["eq-F-est-1"],"detail_key":"p32"},{"id":"n31439","layer":"informal","project":"p32","title":"Estimate bad","kind":"lemma","summary":"[Estimate bad] We have \\mu\\left(\\x\\in X: |T_r b(x)|>\\alpha/2\\\\right) \\le \\frac\\frac2^5ac + 2^a^…","labels":["estimate-bad"],"detail_key":"p32"},{"id":"n31440","layer":"informal","project":"p32","title":"eq-set-dec-2","kind":"proof","summary":"We estimate \\mu\\left(\\x\\in X: |T_r b(x)|>\\alpha/2\\\\right) \\le \\mu (\\Omega) + \\mu\\left(\\x\\in X\\s…","labels":["eq-set-dec-2","eq-omega-bd","eq-set-dec-3"],"detail_key":"p32"},{"id":"n31441","layer":"informal","project":"p32","title":"Proof of \\Crefcalderon-zygmund-weak-1-1","kind":"proof","summary":"[Proof of \\Crefcalderon-zygmund-weak-1-1] It follows by the triangle inequality and subadditivi…","labels":["eq-set-dec-1"],"detail_key":"p32"},{"id":"n31442","layer":"informal","project":"p32","title":"smooth approximation","kind":"lemma","summary":"[smooth approximation] The function f_0 is 2\\pi-periodic. The function f_0 is smooth (and there…","labels":["smooth-approximation","eq-ffzero"],"detail_key":"p32"},{"id":"n31443","layer":"informal","project":"p32","title":"Periodicity follows directly from the definitions. The other properties are part of the L…","kind":"proof","summary":"Periodicity follows directly from the definitions. The other properties are part of the Lean li…","labels":[],"detail_key":"p32"},{"id":"n31444","layer":"informal","project":"p32","title":"convergence for smooth","kind":"lemma","summary":"[convergence for smooth] There exists some N_0 \\in N such that for all N>N_0 and x\\in [0,2\\pi]…","labels":["convergence-for-smooth"],"detail_key":"p32"},{"id":"n31445","layer":"informal","project":"p32","title":"control approximation effect","kind":"lemma","summary":"[control approximation effect] There is a set E \\subset R with Lebesgue measure |E|\\le \\epsilon…","labels":["control-approximation-effect","eq-max-partial-sum-diff"],"detail_key":"p32"},{"id":"n31446","layer":"informal","project":"p32","title":"classical Carleson with exceptional sets","kind":"theorem","summary":"[classical Carleson with exceptional sets] Let f be a 2\\pi-periodic complex-valued continuous f…","labels":["exceptional-set-carleson","aeconv"],"detail_key":"p32"},{"id":"n31447","layer":"informal","project":"p32","title":"epsilonthird","kind":"proof","summary":"Let N_0 be as in \\Crefconvergence-for-smooth. For every x\\in [0, 2\\pi) \\setminus E\\, , and ever…","labels":["epsilonthird"],"detail_key":"p32"},{"id":"n31448","layer":"informal","project":"p32","title":"Proof of \\Crefclassical-carleson","kind":"proof","summary":"[Proof of \\Crefclassical-carleson] By applying \\Crefexceptional-set-carleson with a sequence of…","labels":[],"detail_key":"p32"},{"id":"n31449","layer":"informal","project":"p32","title":"real Carleson","kind":"lemma","summary":"[real Carleson] Let F,G be Borel subsets of R with finite measure. Let f be a bounded measurabl…","labels":["real-Carleson","define-T-carleson"],"detail_key":"p32"},{"id":"n31450","layer":"informal","project":"p32","title":"Hilbert strong 2 2","kind":"lemma","summary":"[Hilbert strong 2 2] Let 0<r. Let f be a bounded, measurable function on R. Then \\|H_rf\\|_2\\leq…","labels":["Hilbert-strong-2-2","eq-Hr-L2-bound","def-H-r"],"detail_key":"p32"},{"id":"n31451","layer":"informal","project":"p32","title":"van der Corput","kind":"lemma","summary":"[van der Corput] Let \\alpha\\le\\beta be real numbers. Let g:R\\to \\C be a measurable function and…","labels":["van-der-Corput"],"detail_key":"p32"},{"id":"n31452","layer":"informal","project":"p32","title":"Dirichlet kernel","kind":"lemma","summary":"[Dirichlet kernel] We have for every 2\\pi-periodic bounded measurable f and every N\\ge 0 S_Nf(x…","labels":["Dirichlet-kernel","eqksumexp","eqksumhil"],"detail_key":"p32"},{"id":"n31453","layer":"informal","project":"p32","title":"eq-expsum","kind":"proof","summary":"We have by definitions and interchanging sum and integral S_Nf(x)=\\sum_n=-N^N \\widehatf_n e^inx…","labels":["eq-expsum"],"detail_key":"p32"},{"id":"n31454","layer":"informal","project":"p32","title":"lower secant bound","kind":"lemma","summary":"[lower secant bound] Let \\eta>0 and -2\\pi +\\eta \\le x\\le 2\\pi-\\eta with |x|\\ge \\eta. Then |1-e^…","labels":["lower-secant-bound"],"detail_key":"p32"},{"id":"n31455","layer":"informal","project":"p32","title":"We have |1 - e^ix| = \\sqrt(1 - \\cos(x))^2 + \\sin^2(x) \\ge |\\sin(x)|\\,. If 0 \\le x \\le \\fr…","kind":"proof","summary":"We have |1 - e^ix| = \\sqrt(1 - \\cos(x))^2 + \\sin^2(x) \\ge |\\sin(x)|\\,. If 0 \\le x \\le \\frac\\pi2…","labels":[],"detail_key":"p32"},{"id":"n31456","layer":"informal","project":"p32","title":"spectral projection bound","kind":"lemma","summary":"[spectral projection bound] Let f be a bounded 2\\pi-periodic measurable function. Then, for all…","labels":["spectral-projection-bound","snbound"],"detail_key":"p32"},{"id":"n31457","layer":"informal","project":"p32","title":"Hilbert kernel bound","kind":"lemma","summary":"[Hilbert kernel bound] For x,y\\in R with x\\neq y we have |\\kappa(x-y)|\\le 2^2(2|x-y|)^-1\\, .","labels":["Hilbert-kernel-bound","eqcarl30"],"detail_key":"p32"},{"id":"n31458","layer":"informal","project":"p32","title":"eqcarl31","kind":"proof","summary":"Fix x\\neq y. If \\kappa(x-y) is zero, then \\eqrefeqcarl30 is evident. Assume \\kappa(x-y) is not…","labels":["eqcarl31","eqcarl311"],"detail_key":"p32"},{"id":"n31459","layer":"informal","project":"p32","title":"Hilbert kernel regularity","kind":"lemma","summary":"[Hilbert kernel regularity] For x,y,y'\\in R with x\\neq y,y' and 2|y-y'|\\le |x-y|\\, , we have |\\…","labels":["Hilbert-kernel-regularity","eq-close-hoelder","eqcarl301"],"detail_key":"p32"},{"id":"n31460","layer":"informal","project":"p32","title":"Upon replacing y by y-x and y' by y'-x on the left-hand side of \\eqrefeq-close-hoelder, w…","kind":"proof","summary":"Upon replacing y by y-x and y' by y'-x on the left-hand side of \\eqrefeq-close-hoelder, we can…","labels":[],"detail_key":"p32"},{"id":"n31461","layer":"informal","project":"p32","title":"fourier-coeff-derivative","kind":"lemma","summary":"Let f:R\\to \\C be 2\\pi-periodic and continuously differentiable, and let n \\in Z\\setminus \\0\\. T…","labels":["fourier-coeff-derivative"],"detail_key":"p32"},{"id":"n31462","layer":"informal","project":"p32","title":"This is part of the Lean library.","kind":"proof","summary":"This is part of the Lean library.","labels":[],"detail_key":"p32"},{"id":"n31463","layer":"informal","project":"p32","title":"convergence-of-coeffs-summable","kind":"lemma","summary":"Let f:R\\to \\C such that \\sum_n\\in Z |\\widehatf_n| < \\infty. Then \\sup_x\\in [0,2\\pi] |f(x) - S_N…","labels":["convergence-of-coeffs-summable"],"detail_key":"p32"},{"id":"n31464","layer":"informal","project":"p32","title":"This is part of the Lean library.","kind":"proof","summary":"This is part of the Lean library.","labels":[],"detail_key":"p32"},{"id":"n31465","layer":"informal","project":"p32","title":"convergence-for-twice-contdiff","kind":"lemma","summary":"Let f:R\\to \\C be 2\\pi-periodic and twice continuously differentiable. Then \\sup_x\\in [0,2\\pi] |…","labels":["convergence-for-twice-contdiff"],"detail_key":"p32"},{"id":"n31466","layer":"informal","project":"p32","title":"By \\Crefconvergence-of-coeffs-summable, it suffices to show that the Fourier coefficients…","kind":"proof","summary":"By \\Crefconvergence-of-coeffs-summable, it suffices to show that the Fourier coefficients \\wide…","labels":[],"detail_key":"p32"},{"id":"n31467","layer":"informal","project":"p32","title":"Proof of \\Crefconvergence-for-smooth","kind":"proof","summary":"[Proof of \\Crefconvergence-for-smooth] \\Crefconvergence-for-smooth now follows directly from th…","labels":[],"detail_key":"p32"},{"id":"n31468","layer":"informal","project":"p32","title":"modulated averaged projection","kind":"lemma","summary":"[modulated averaged projection] We have for every bounded measurable 2\\pi-periodic function g \\…","labels":["modulated-averaged-projection","lnbound"],"detail_key":"p32"},{"id":"n31469","layer":"informal","project":"p32","title":"mnbound","kind":"proof","summary":"We have \\|M_ng\\|_L^2[0, 2\\pi]^2=\\int_0^2\\pi |e^inxg(x)|^2\\, dx =\\int_0^2\\pi |g(x)|^2\\, dx=\\|g\\|…","labels":["mnbound"],"detail_key":"p32"},{"id":"n31470","layer":"informal","project":"p32","title":"periodic domain shift","kind":"lemma","summary":"[periodic domain shift] Let f be a bounded 2\\pi-periodic function. We have for any 0 \\le x\\le 2…","labels":["periodic-domain-shift"],"detail_key":"p32"},{"id":"n31471","layer":"informal","project":"p32","title":"eqhil9","kind":"proof","summary":"We have by periodicity and change of variables \\int_-x^0 f(y)\\, dy=\\int_-x^0 f(y+2\\pi)\\, dy= \\i…","labels":["eqhil9"],"detail_key":"p32"},{"id":"n31472","layer":"informal","project":"p32","title":"Young convolution","kind":"lemma","summary":"[Young convolution] Let f and g be two bounded non-negative measurable 2\\pi-periodic functions…","labels":["Young-convolution","eqyoung"],"detail_key":"p32"},{"id":"n31473","layer":"informal","project":"p32","title":"eqhil4","kind":"proof","summary":"Using Fubini and \\Crefperiodic-domain-shift, we observe \\int_0^2\\pi\\int_0^2\\pif(y)^2g(x-y)\\, dy…","labels":["eqhil4"],"detail_key":"p32"},{"id":"n31474","layer":"informal","project":"p32","title":"integrable bump convolution","kind":"lemma","summary":"[integrable bump convolution] Let g,f be bounded measurable 2\\pi-periodic functions. Let 0<r<\\p…","labels":["integrable-bump-convolution","ebump1"],"detail_key":"p32"},{"id":"n31475","layer":"informal","project":"p32","title":"From monotonicity of the integral and \\eqrefebump1, \\|g\\|_L^1[0, 2\\pi] \\le \\int_0^2\\pik_r…","kind":"proof","summary":"From monotonicity of the integral and \\eqrefebump1, \\|g\\|_L^1[0, 2\\pi] \\le \\int_0^2\\pik_r(x)\\,…","labels":[],"detail_key":"p32"},{"id":"n31476","layer":"informal","project":"p32","title":"Dirichlet approximation","kind":"lemma","summary":"[Dirichlet approximation] Let 0<r<1. Let N be the smallest integer larger than \\frac 1r. There…","labels":["Dirichlet-approximation","lthroughlprime","eqdifflhil"],"detail_key":"p32"},{"id":"n31477","layer":"informal","project":"p32","title":"eqhil13","kind":"proof","summary":"We have by definition and \\CrefDirichlet-kernel L_Ng(x)= \\frac 1N\\sum_n=0^N-1 \\int_0^2\\pi e^-i(…","labels":["eqhil13","eqdiffzero","eqhil3","eq-L'L''","eq-diffzero2","eqhil12","eqhil11"],"detail_key":"p32"},{"id":"n31478","layer":"informal","project":"p32","title":"Proof of \\CrefHilbert-strong-2-2","kind":"proof","summary":"[Proof of \\CrefHilbert-strong-2-2] We first show that if f is supported in [1, 4], then \\|H_r f…","labels":["eq-Hr-short-support","eq-Hr-short-support-2"],"detail_key":"p32"},{"id":"n31479","layer":"informal","project":"p32","title":"Proof of \\Crefvan-der-Corput","kind":"proof","summary":"[Proof of \\Crefvan-der-Corput] Let g be a Lipschitz continuous function as in the lemma. Assume…","labels":[],"detail_key":"p32"},{"id":"n31480","layer":"informal","project":"p32","title":"Proof of \\Crefspectral-projection-bound","kind":"proof","summary":"[Proof of \\Crefspectral-projection-bound] The functions e_n:x\\mapsto e^inx form an orthonormal…","labels":[],"detail_key":"p32"},{"id":"n31481","layer":"informal","project":"p32","title":"Dirichlet kernel - Hilbert kernel relation","kind":"lemma","summary":"[Dirichlet kernel - Hilbert kernel relation] For all N\\inZ and x\\in [-\\pi,\\pi] \\setminus \\0\\, \\…","labels":["Dirichlet-Hilbert"],"detail_key":"p32"},{"id":"n31482","layer":"informal","project":"p32","title":"Let N\\inZ and x\\in [-\\pi,\\pi] \\setminus \\0\\. With \\CrefDirichlet-kernel, we obtain K_N(x)…","kind":"proof","summary":"Let N\\inZ and x\\in [-\\pi,\\pi] \\setminus \\0\\. With \\CrefDirichlet-kernel, we obtain K_N(x) - (e^…","labels":[],"detail_key":"p32"},{"id":"n31483","layer":"informal","project":"p32","title":"partial Fourier sum bound","kind":"lemma","summary":"[partial Fourier sum bound] Let g:R\\to\\C be a measurable 2\\pi-periodic function such that for s…","labels":["partial-Fourier-sum-bound"],"detail_key":"p32"},{"id":"n31484","layer":"informal","project":"p32","title":"eq-diff-integrable","kind":"proof","summary":"Let x\\in [0,2\\pi] and N>0. We have with \\CrefDirichlet-kernel |S_N g(x)| = \\frac12\\pi \\left| \\i…","labels":["eq-diff-integrable","eq-diff-singular","eq-Dirichlet-Hilbert"],"detail_key":"p32"},{"id":"n31485","layer":"informal","project":"p32","title":"real Carleson operator measurable","kind":"lemma","summary":"[real Carleson operator measurable] Let f be a bounded measurable function on R. Then Tf as def…","labels":["real-Carleson-operator-measurable"],"detail_key":"p32"},{"id":"n31486","layer":"informal","project":"p32","title":"Since a countable supremum of measurable functions is measurable, it suffices to show tha…","kind":"proof","summary":"Since a countable supremum of measurable functions is measurable, it suffices to show that for…","labels":[],"detail_key":"p32"},{"id":"n31487","layer":"informal","project":"p32","title":"partial Fourier sums of small","kind":"lemma","summary":"[partial Fourier sums of small] Let g:R\\to\\C be a measurable 2\\pi-periodic function such that f…","labels":["partial-Fourier-sums-of-small","g-small","S-Ng-small","C-epsilon-def"],"detail_key":"p32"},{"id":"n31488","layer":"informal","project":"p32","title":"Define E := \\x \\in [0, 2\\pi] \\ : \\ \\sup_N > 0 |S_N g (x)| > C_\\epsilon \\delta \\ \\,. Then…","kind":"proof","summary":"Define E := \\x \\in [0, 2\\pi] \\ : \\ \\sup_N > 0 |S_N g (x)| > C_\\epsilon \\delta \\ \\,. Then \\eqref…","labels":[],"detail_key":"p32"},{"id":"n31489","layer":"informal","project":"p32","title":"Proof of \\Crefcontrol-approximation-effect","kind":"proof","summary":"[Proof of \\Crefcontrol-approximation-effect] \\Crefcontrol-approximation-effect follows directly…","labels":[],"detail_key":"p32"},{"id":"n31490","layer":"informal","project":"p32","title":"real line metric","kind":"lemma","summary":"[real line metric] The space (R,\\rho) is a complete locally compact metric space.","labels":["real-line-metric"],"detail_key":"p32"},{"id":"n31491","layer":"informal","project":"p32","title":"This is part of the Lean library.","kind":"proof","summary":"This is part of the Lean library.","labels":[],"detail_key":"p32"},{"id":"n31492","layer":"informal","project":"p32","title":"real line ball","kind":"lemma","summary":"[real line ball] For x\\in R and R>0, the ball B(x,R) is the interval (x-R,x+R)","labels":["real-line-ball"],"detail_key":"p32"},{"id":"n31493","layer":"informal","project":"p32","title":"Let x'\\in B(x,R). By definition of the ball, |x'-x|<R. It follows that x'-x<R and x-x'<R.…","kind":"proof","summary":"Let x'\\in B(x,R). By definition of the ball, |x'-x|<R. It follows that x'-x<R and x-x'<R. It fo…","labels":[],"detail_key":"p32"},{"id":"n31494","layer":"informal","project":"p32","title":"real line measure","kind":"lemma","summary":"[real line measure] The measure \\mu is a sigma-finite non-zero Radon-Borel measure on R.","labels":["real-line-measure"],"detail_key":"p32"},{"id":"n31495","layer":"informal","project":"p32","title":"This is part of the Lean library.","kind":"proof","summary":"This is part of the Lean library.","labels":[],"detail_key":"p32"},{"id":"n31496","layer":"informal","project":"p32","title":"real line ball measure","kind":"lemma","summary":"[real line ball measure] We have for every x\\in R and R>0 \\mu(B(x,R))=2R\\, .","labels":["real-line-ball-measure"],"detail_key":"p32"},{"id":"n31497","layer":"informal","project":"p32","title":"We have with \\Crefreal-line-ball \\mu(B(x,R))=\\mu((x-R,x+R))=2R\\, .","kind":"proof","summary":"We have with \\Crefreal-line-ball \\mu(B(x,R))=\\mu((x-R,x+R))=2R\\, .","labels":[],"detail_key":"p32"},{"id":"n31498","layer":"informal","project":"p32","title":"real line doubling","kind":"lemma","summary":"[real line doubling] We have for every x\\in R and R>0 \\mu(B(x,2R))=2\\mu(B(x,R))\\, .","labels":["real-line-doubling"],"detail_key":"p32"},{"id":"n31499","layer":"informal","project":"p32","title":"We have with \\Crefreal-line-ball-measure \\mu(B(x,2R)=4R=2\\mu(B(x,R))\\, . This proves the…","kind":"proof","summary":"We have with \\Crefreal-line-ball-measure \\mu(B(x,2R)=4R=2\\mu(B(x,R))\\, . This proves the lemma.","labels":[],"detail_key":"p32"},{"id":"n31500","layer":"informal","project":"p32","title":"frequency metric","kind":"lemma","summary":"[frequency metric] For every R > 0 and x \\in X, the function d_B(x,R) is a metric on \\Theta.","labels":["frequency-metric"],"detail_key":"p32"},{"id":"n31501","layer":"informal","project":"p32","title":"This follows immediately from the fact that the standard metric on Z is a metric.","kind":"proof","summary":"This follows immediately from the fact that the standard metric on Z is a metric.","labels":[],"detail_key":"p32"},{"id":"n31502","layer":"informal","project":"p32","title":"oscillation control","kind":"lemma","summary":"[oscillation control] For every R > 0 and x \\in X, and for all n, m \\in Z, we have \\sup_y,y'\\in…","labels":["oscillation-control","eqcarl2"],"detail_key":"p32"},{"id":"n31503","layer":"informal","project":"p32","title":"The right hand side of \\eqrefeqcarl2 equals \\sup_y,y'\\in B(x,R)|(n-m)(y-x)-(n-m)(y'-x)|\\,…","kind":"proof","summary":"The right hand side of \\eqrefeqcarl2 equals \\sup_y,y'\\in B(x,R)|(n-m)(y-x)-(n-m)(y'-x)|\\,. The…","labels":[],"detail_key":"p32"},{"id":"n31504","layer":"informal","project":"p32","title":"frequency monotone","kind":"lemma","summary":"[frequency monotone] For any x, x' \\in X and R, R' > 0 with B(x,R) \\subset B(x, R'), and for an…","labels":["frequency-monotone"],"detail_key":"p32"},{"id":"n31505","layer":"informal","project":"p32","title":"This follows immediately from the definition \\eqrefeqcarl4 and R \\le R'.","kind":"proof","summary":"This follows immediately from the definition \\eqrefeqcarl4 and R \\le R'.","labels":[],"detail_key":"p32"},{"id":"n31506","layer":"informal","project":"p32","title":"frequency ball doubling","kind":"lemma","summary":"[frequency ball doubling] For any x,x'\\in R and R>0 with x\\in B(x',2R) and any n,m\\in Z, we hav…","labels":["frequency-ball-doubling","firstdb1"],"detail_key":"p32"},{"id":"n31507","layer":"informal","project":"p32","title":"With \\eqrefeqcarl4, both sides of \\eqreffirstdb1 are equal to 4R|n-m|. This proves the le…","kind":"proof","summary":"With \\eqrefeqcarl4, both sides of \\eqreffirstdb1 are equal to 4R|n-m|. This proves the lemma.","labels":[],"detail_key":"p32"},{"id":"n31508","layer":"informal","project":"p32","title":"frequency ball growth","kind":"lemma","summary":"[frequency ball growth] For any x,x'\\in R and R>0 with B(x,R)\\subset B(x',2R) and any n,m\\in Z,…","labels":["frequency-ball-growth","seconddb1"],"detail_key":"p32"},{"id":"n31509","layer":"informal","project":"p32","title":"With \\eqrefeqcarl4, both sides of \\eqreffirstdb1 are equal to 4R|n-m|. This proves the le…","kind":"proof","summary":"With \\eqrefeqcarl4, both sides of \\eqreffirstdb1 are equal to 4R|n-m|. This proves the lemma.","labels":[],"detail_key":"p32"},{"id":"n31510","layer":"informal","project":"p32","title":"integer ball cover","kind":"lemma","summary":"[integer ball cover] For every x\\in R and R>0 and every n\\in Z and R'>0, there exist m_1, m_2,…","labels":["integer-ball-cover","eqcarl5"],"detail_key":"p32"},{"id":"n31511","layer":"informal","project":"p32","title":"eqcarl6","kind":"proof","summary":"Let m_1 be the largest integer smaller than or equal to n- \\frac R'2R. Let m_2=n. Let m_3 be th…","labels":["eqcarl6"],"detail_key":"p32"},{"id":"n31512","layer":"informal","project":"p32","title":"real van der Corput","kind":"lemma","summary":"[real van der Corput] For any x\\in R and R>0 and any function \\varphi: X\\to \\C supported on B'=…","labels":["real-van-der-Corput","eq-vdc-cond1"],"detail_key":"p32"},{"id":"n31513","layer":"informal","project":"p32","title":"eq-vdc-cond2","kind":"proof","summary":"Set n'=n-m. Then we have to prove \\left|\\int_x-R^x+R e^in'y\\varphi(y) dy\\right|\\le 4\\pi R\\|\\var…","labels":["eq-vdc-cond2"],"detail_key":"p32"},{"id":"n31514","layer":"informal","project":"p32","title":"Proof of \\Crefreal-Carleson","kind":"proof","summary":"[Proof of \\Crefreal-Carleson] The preceding chain of lemmas establishes that \\Theta is a cancel…","labels":[],"detail_key":"p32"},{"id":"n31515","layer":"formal","project":"p33","title":"CategoryTheory.FactorizationSystem","kind":"inductive","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → CategoryTheory.MorphismProperty C → Cate…","labels":[],"detail_key":"p33","name":"CategoryTheory.FactorizationSystem","module":"FactorizationSystems.Basic"},{"id":"n31516","layer":"formal","project":"p33","title":"CategoryTheory.FactorizationSystemSlice","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → X : C → L R : CategoryTheory.MorphismPro…","labels":[],"detail_key":"p33","name":"CategoryTheory.FactorizationSystemSlice","module":"FactorizationSystems.Basic"},{"id":"n31517","layer":"formal","project":"p33","title":"CategoryTheory.MorphismPropertySlice","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → CategoryTheory.MorphismProperty C → (X :…","labels":[],"detail_key":"p33","name":"CategoryTheory.MorphismPropertySlice","module":"FactorizationSystems.Basic"},{"id":"n31518","layer":"formal","project":"p33","title":"CategoryTheory.left_cancellation_right_class","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] L R : CategoryTheory.MorphismProperty C, C…","labels":[],"detail_key":"p33","name":"CategoryTheory.left_cancellation_right_class","module":"FactorizationSystems.Basic"},{"id":"n31519","layer":"formal","project":"p33","title":"CategoryTheory.left_right_intersection_iso","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] L R : CategoryTheory.MorphismProperty C, C…","labels":[],"detail_key":"p33","name":"CategoryTheory.left_right_intersection_iso","module":"FactorizationSystems.Basic"},{"id":"n31520","layer":"formal","project":"p33","title":"CategoryTheory.right_cancellation_left_class","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] L R : CategoryTheory.MorphismProperty C, C…","labels":[],"detail_key":"p33","name":"CategoryTheory.right_cancellation_left_class","module":"FactorizationSystems.Basic"},{"id":"n31521","layer":"formal","project":"p33","title":"CategoryTheory.FactorizationSystem_characterization","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → (L R : CategoryTheory.MorphismProperty C…","labels":[],"detail_key":"p33","name":"CategoryTheory.FactorizationSystem_characterization","module":"FactorizationSystems.Characterization"},{"id":"n31522","layer":"formal","project":"p33","title":"CategoryTheory.orthogonal_class","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → CategoryTheory.MorphismProperty C → Cate…","labels":[],"detail_key":"p33","name":"CategoryTheory.orthogonal_class","module":"FactorizationSystems.Characterization"},{"id":"n31523","layer":"formal","project":"p33","title":"CategoryTheory.Arrow.base_change_r_ort_complement","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] [inst_1 : CategoryTheory.Limits.HasPullbac…","labels":[],"detail_key":"p33","name":"CategoryTheory.Arrow.base_change_r_ort_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31524","layer":"formal","project":"p33","title":"CategoryTheory.Arrow.contains_isos_right_ort_complement","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] (L : CategoryTheory.MorphismProperty C), C…","labels":[],"detail_key":"p33","name":"CategoryTheory.Arrow.contains_isos_right_ort_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31525","layer":"formal","project":"p33","title":"CategoryTheory.Arrow.is_closed_under_comp_r_ort_complement","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] (W : CategoryTheory.MorphismProperty C) X…","labels":[],"detail_key":"p33","name":"CategoryTheory.Arrow.is_closed_under_comp_r_ort_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31526","layer":"formal","project":"p33","title":"CategoryTheory.Arrow.is_closed_under_limits_r_ort_complement","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → (W : CategoryTheory.MorphismProperty C)…","labels":[],"detail_key":"p33","name":"CategoryTheory.Arrow.is_closed_under_limits_r_ort_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31527","layer":"formal","project":"p33","title":"CategoryTheory.Arrow.left_cancellation_r_ort_complement","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] (W : CategoryTheory.MorphismProperty C) X…","labels":[],"detail_key":"p33","name":"CategoryTheory.Arrow.left_cancellation_r_ort_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31528","layer":"formal","project":"p33","title":"CategoryTheory.left_orthogonal_complement","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → CategoryTheory.MorphismProperty C → Cate…","labels":[],"detail_key":"p33","name":"CategoryTheory.left_orthogonal_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31529","layer":"formal","project":"p33","title":"CategoryTheory.right_orthogonal_complement","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → CategoryTheory.MorphismProperty C → Cate…","labels":[],"detail_key":"p33","name":"CategoryTheory.right_orthogonal_complement","module":"FactorizationSystems.OrthogonalComplements"},{"id":"n31530","layer":"formal","project":"p33","title":"CategoryTheory.hom_orthogonal_implies_orthogonal","kind":"def","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → A B X Y : C → l : Quiver.Hom A B → r : Q…","labels":[],"detail_key":"p33","name":"CategoryTheory.hom_orthogonal_implies_orthogonal","module":"FactorizationSystems.Orthogonality"},{"id":"n31531","layer":"formal","project":"p33","title":"CategoryTheory.orthogonal","kind":"inductive","summary":"C : Type u → [inst : CategoryTheory.Category.v, u C] → A B X Y : C → Quiver.Hom A B → Quiver.Ho…","labels":[],"detail_key":"p33","name":"CategoryTheory.orthogonal","module":"FactorizationSystems.Orthogonality"},{"id":"n31532","layer":"formal","project":"p33","title":"CategoryTheory.orthogonal_implies_hom_orthogonal","kind":"theorem","summary":"∀ C : Type u [inst : CategoryTheory.Category.v, u C] A B X Y : C (l : Quiver.Hom A B) (r : Quiv…","labels":[],"detail_key":"p33","name":"CategoryTheory.orthogonal_implies_hom_orthogonal","module":"FactorizationSystems.Orthogonality"},{"id":"n31533","layer":"informal","project":"p33","title":"CategoryTheory.FactorizationSystem","kind":"definition","summary":"A factorization system in a category C consists of two classes of morphisms (L,R), such that bo…","labels":[],"detail_key":"p33"},{"id":"n31534","layer":"informal","project":"p33","title":"CategoryTheory.MorphismPropertySlice","kind":"definition","summary":"If W is a class of morphisms in a category C and X is an object in C, we define a class of morp…","labels":[],"detail_key":"p33"},{"id":"n31535","layer":"informal","project":"p33","title":"CategoryTheory.FactorizationSystemSlice","kind":"lemma","summary":"If (L,R) is a factorization system in a category C and X is an object in C, then (L/X,R/X) is a…","labels":[],"detail_key":"p33"},{"id":"n31536","layer":"informal","project":"p33","title":"CategoryTheory.left_right_intersection_iso","kind":"lemma","summary":"If (L,R) is a factorization system in a category C, then the intersection of L and R is precise…","labels":[],"detail_key":"p33"},{"id":"n31537","layer":"informal","project":"p33","title":"CategoryTheory.left_cancellation_right_class","kind":"lemma","summary":"If (L,R) is a factorization system in a category C, then R has the left cancellation property a…","labels":[],"detail_key":"p33"},{"id":"n31538","layer":"informal","project":"p33","title":"(\\textEpi,\\textMono) is a factorization system in \\textSet.","kind":"lemma","summary":"(\\textEpi,\\textMono) is a factorization system in \\textSet.","labels":[],"detail_key":"p33"},{"id":"n31539","layer":"informal","project":"p33","title":"CategoryTheory.orthogonal","kind":"definition","summary":"Given two morphisms l : A \\to B and r : X \\to Y in C, we say that l is left-orthogonal to g, or…","labels":[],"detail_key":"p33"},{"id":"n31540","layer":"informal","project":"p33","title":"CategoryTheory.hom_orthogonal_implies_orthogonal","kind":"lemma","summary":"Given l and r as above, l is left-orthogonal to r iff the square \\emphHom(B,X) & \\emphHom(A,X)…","labels":[],"detail_key":"p33"},{"id":"n31541","layer":"informal","project":"p33","title":"CategoryTheory.left_orthogonal_complement","kind":"definition","summary":"Let W be a class of morphisms in a category C. The left orthogonal complement of W, denoted ^\\b…","labels":[],"detail_key":"p33"},{"id":"n31542","layer":"informal","project":"p33","title":"CategoryTheory.Arrow.base_change_r_ort_complement","kind":"lemma","summary":"For every class of morphisms W, W^\\bot contains isomorphisms and is closed under limits, compos…","labels":[],"detail_key":"p33"},{"id":"n31543","layer":"informal","project":"p33","title":"CategoryTheory.FactorizationSystem_characterization","kind":"theorem","summary":"Given two classes of maps L,R in a category C, there exists a (L,R)-factorization system on C i…","labels":[],"detail_key":"p33"},{"id":"n31544","layer":"informal","project":"p34","title":"c1_c2_c3_norms","kind":"lemma","summary":"\\| C_1 \\| = 1, 98 \\over 100 < \\| C_2 \\| < 99 \\over 100, and 98 \\over 100 < \\| C_3 \\| < 99 \\over…","labels":["c1_c2_c3_norms"],"detail_key":"p34"},{"id":"n31545","layer":"informal","project":"p34","title":"Trivial arithmetic.","kind":"proof","summary":"Trivial arithmetic.","labels":[],"detail_key":"p34"},{"id":"n31546","layer":"informal","project":"p34","title":"def:noperthedron","kind":"definition","summary":"The Noperthedron is polyhedron given by the vertex set \\[C_30 \\cdot C_1 \\cup C_30 \\cdot C_2 \\cu…","labels":["def:noperthedron"],"detail_key":"p34"},{"id":"n31547","layer":"informal","project":"p34","title":"lemma:nopert_verts_norm_le_one","kind":"lemma","summary":"The norm of any vertex in the Noperthedron is no more than 1.","labels":["lemma:nopert_verts_norm_le_one"],"detail_key":"p34"},{"id":"n31548","layer":"informal","project":"p34","title":"Evident from definitions.","kind":"proof","summary":"Evident from definitions.","labels":[],"detail_key":"p34"},{"id":"n31549","layer":"informal","project":"p34","title":"def:pointsymmetric","kind":"definition","summary":"A set S \\subseteq \\R^3 is \\em point-symmetric if x \\in S implies -x \\in S.","labels":["def:pointsymmetric"],"detail_key":"p34"},{"id":"n31550","layer":"informal","project":"p34","title":"lemma:nopert_point_symmetric","kind":"lemma","summary":"The noperthedron is point-symmetric.","labels":["lemma:nopert_point_symmetric"],"detail_key":"p34"},{"id":"n31551","layer":"informal","project":"p34","title":"Follows from Lemma~\\reflemma:pointsymmetrization_is_pointsym.","kind":"proof","summary":"Follows from Lemma~\\reflemma:pointsymmetrization_is_pointsym.","labels":[],"detail_key":"p34"},{"id":"n31552","layer":"informal","project":"p34","title":"lem:symmetries","kind":"lemma","summary":"Let \\PPP = \\NOP, then for all \\theta, \\varphi, \\alpha \\in \\R, the following three identities ho…","labels":["lem:symmetries"],"detail_key":"p34"},{"id":"n31553","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 7.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 7.","labels":[],"detail_key":"p34"},{"id":"n31554","layer":"informal","project":"p34","title":"cor:rupert_tightening","kind":"corollary","summary":"If the noperthedron is Rupert, then there exists a solution with \\theta_1,\\theta_2&\\in[0,2\\pi/1…","labels":["cor:rupert_tightening"],"detail_key":"p34"},{"id":"n31555","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 8.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 8.","labels":[],"detail_key":"p34"},{"id":"n31556","layer":"informal","project":"p34","title":"lem:RaRalpha","kind":"lemma","summary":"For any \\alpha, \\theta,\\varphi \\in \\R and a \\in \\x,y,z\\ one has \\| R(\\alpha)\\| = \\| R_a(\\alpha)…","labels":["lem:RaRalpha"],"detail_key":"p34"},{"id":"n31557","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 9.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 9.","labels":[],"detail_key":"p34"},{"id":"n31558","layer":"informal","project":"p34","title":"lem:RaRa","kind":"lemma","summary":"Let \\epsilon>0, |\\alpha-\\overline\\alpha|\\leq\\varepsilon and a \\in \\x,y,z\\ then \\|R_a(\\alpha)-R_…","labels":["lem:RaRa"],"detail_key":"p34"},{"id":"n31559","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 10.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 10.","labels":[],"detail_key":"p34"},{"id":"n31560","layer":"informal","project":"p34","title":"lem:RxRy","kind":"lemma","summary":"For any \\alpha,\\beta\\in R one has \\[ \\|R_x(\\alpha)R_y(\\beta)-\\id\\| \\leq \\sqrt\\alpha^2+\\beta^2 \\…","labels":["lem:RxRy"],"detail_key":"p34"},{"id":"n31561","layer":"informal","project":"p34","title":"The composition R_d(\\alpha)R_d'(\\beta) is a rotation, i.e.\\ conjugate to some R_z(\\gamma)…","kind":"proof","summary":"The composition R_d(\\alpha)R_d'(\\beta) is a rotation, i.e.\\ conjugate to some R_z(\\gamma) by an…","labels":[],"detail_key":"p34"},{"id":"n31562","layer":"informal","project":"p34","title":"lem:sqrt2","kind":"lemma","summary":"Let \\epsilon>0 and |\\theta-\\overline\\theta|,|\\varphi-\\overline\\varphi| \\leq \\varepsilon then \\|…","labels":["lem:sqrt2"],"detail_key":"p34"},{"id":"n31563","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 13.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 13.","labels":[],"detail_key":"p34"},{"id":"n31564","layer":"informal","project":"p34","title":"lem:XPgt0","kind":"lemma","summary":"Let P \\in \\R^3 with \\|P\\| \\leq 1. Further, let \\epsilon>0 and \\overline\\theta,\\overline\\varphi,…","labels":["lem:XPgt0"],"detail_key":"p34"},{"id":"n31565","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 14.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 14.","labels":[],"detail_key":"p34"},{"id":"n31566","layer":"informal","project":"p34","title":"lem:MPgtr","kind":"lemma","summary":"Let P \\in \\R^3 with \\|P\\| \\leq 1. Further, let \\epsilon, r>0 and \\overline\\theta,\\overline\\varp…","labels":["lem:MPgtr"],"detail_key":"p34"},{"id":"n31567","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 15. Corrigendum: the triangle inquality only im…","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 15. Corrigendum: the triangle inquality only implies…","labels":[],"detail_key":"p34"},{"id":"n31568","layer":"informal","project":"p34","title":"lem:sqrt5","kind":"lemma","summary":"Let \\epsilon>0 and |\\theta-\\overline\\theta|,|\\varphi-\\overline\\varphi|,|\\alpha-\\overline\\alpha|…","labels":["lem:sqrt5"],"detail_key":"p34"},{"id":"n31569","layer":"informal","project":"p34","title":"Write R_z and R_y for rotations about the third and second axes. After dropping the norm-…","kind":"proof","summary":"Write R_z and R_y for rotations about the third and second axes. After dropping the norm-one pr…","labels":[],"detail_key":"p34"},{"id":"n31570","layer":"informal","project":"p34","title":"Rupert Polyhedron iff Rupert Set","kind":"theorem","summary":"[Rupert Polyhedron iff Rupert Set] The following are equivalent: \\item The convex polyhedron wi…","labels":["thm:rupert_iff_rupert_set"],"detail_key":"p34"},{"id":"n31571","layer":"informal","project":"p34","title":"TODO: import this from the other repo","kind":"proof","summary":"TODO: import this from the other repo","labels":[],"detail_key":"p34"},{"id":"n31572","layer":"informal","project":"p34","title":"thm:pose_of_matrix_pose","kind":"theorem","summary":"Given a pose with zero offset, there exists a 5-parameter pose that is equivalent to it.","labels":["thm:pose_of_matrix_pose"],"detail_key":"p34"},{"id":"n31573","layer":"informal","project":"p34","title":"By putting the pose into a canonical form as a Z rotation followed by a Y followed by a Z.","kind":"proof","summary":"By putting the pose into a canonical form as a Z rotation followed by a Y followed by a Z.","labels":[],"detail_key":"p34"},{"id":"n31574","layer":"informal","project":"p34","title":"thm:rupert_implies_rot_rupert","kind":"theorem","summary":"If a set is point symmetric and convex, then it being Rupert implies it being purely rotational…","labels":["thm:rupert_implies_rot_rupert"],"detail_key":"p34"},{"id":"n31575","layer":"informal","project":"p34","title":"TODO: informalize proof","kind":"proof","summary":"TODO: informalize proof","labels":[],"detail_key":"p34"},{"id":"n31576","layer":"informal","project":"p34","title":"lem:hullscalarprod","kind":"lemma","summary":"Suppose V = V_1, \\ldots, V_m \\subseteq \\R^n be a finite sequence of points. Suppose \\hull V is…","labels":["lem:hullscalarprod"],"detail_key":"p34"},{"id":"n31577","layer":"informal","project":"p34","title":"This is a mild generalization of \\citepolyhedron.without.rupert, Lemma 18. Since S \\in \\h…","kind":"proof","summary":"This is a mild generalization of \\citepolyhedron.without.rupert, Lemma 18. Since S \\in \\hull V,…","labels":[],"detail_key":"p34"},{"id":"n31578","layer":"informal","project":"p34","title":"lem:leq1","kind":"lemma","summary":"Let S \\in \\R^3, let w \\in \\R^2 be a unit vector, and set f(x_1,x_2,x_3) = \\langle R(x_3) M(x_1,…","labels":["lem:leq1"],"detail_key":"p34"},{"id":"n31579","layer":"informal","project":"p34","title":"The second-partial bound is \\citepolyhedron.without.rupert, Lemma 19. For the third parti…","kind":"proof","summary":"The second-partial bound is \\citepolyhedron.without.rupert, Lemma 19. For the third partials (a…","labels":[],"detail_key":"p34"},{"id":"n31580","layer":"informal","project":"p34","title":"lem:n2","kind":"lemma","summary":"Let f:\\R^n\\to \\R be a C^3-function, let \\varepsilon_1,\\dots,\\varepsilon_n \\geq 0, and let x_1,\\…","labels":["lem:n2"],"detail_key":"p34"},{"id":"n31581","layer":"informal","project":"p34","title":"This strengthens \\citepolyhedron.without.rupert, Lemma 20, by one Taylor order and per-ax…","kind":"proof","summary":"This strengthens \\citepolyhedron.without.rupert, Lemma 20, by one Taylor order and per-axis rad…","labels":[],"detail_key":"p34"},{"id":"n31582","layer":"informal","project":"p34","title":"lem:rotation_derivatives","kind":"lemma","summary":"The partial derivatives of all relevant rotations, projections, and inner products used in the…","labels":["lem:rotation_derivatives"],"detail_key":"p34"},{"id":"n31583","layer":"informal","project":"p34","title":"By basic properties of derivatives.","kind":"proof","summary":"By basic properties of derivatives.","labels":[],"detail_key":"p34"},{"id":"n31584","layer":"informal","project":"p34","title":"Global Theorem","kind":"theorem","summary":"[Global Theorem] Let \\PPP be a pointsymmetric convex polyhedron with radius \\rho =1 and let S \\…","labels":["thm:global"],"detail_key":"p34"},{"id":"n31585","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Section~4.2.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Section~4.2.","labels":[],"detail_key":"p34"},{"id":"n31586","layer":"informal","project":"p34","title":"lem:pythagoras","kind":"lemma","summary":"For any P \\in R^3 one has \\big\\|M(\\theta, \\phi) P\\big\\|^2=\\|P\\|^2-\\langle X(\\theta,\\varphi),P\\r…","labels":["lem:pythagoras"],"detail_key":"p34"},{"id":"n31587","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 21.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 21.","labels":[],"detail_key":"p34"},{"id":"n31588","layer":"informal","project":"p34","title":"def:spanp","kind":"definition","summary":"Given v_1, \\dots, v_n \\in \\R^n write span^+(v_1,\\dots,v_n) for the set (simplicial cone) in \\R^…","labels":["def:spanp"],"detail_key":"p34"},{"id":"n31589","layer":"informal","project":"p34","title":"lem:langles","kind":"lemma","summary":"Let V_1,V_2,V_3,Y,Z \\in \\R^3 with \\|Y \\|=\\| Z \\| and Z \\in span^+(V_1,V_2,V_3). Then there exis…","labels":["lem:langles"],"detail_key":"p34"},{"id":"n31590","layer":"informal","project":"p34","title":"Write Z=\\sum_i c_iV_i with all c_i>0. If every claimed inequality failed, then \\langle Z,…","kind":"proof","summary":"Write Z=\\sum_i c_iV_i with all c_i>0. If every claimed inequality failed, then \\langle Z,Z\\rang…","labels":[],"detail_key":"p34"},{"id":"n31591","layer":"informal","project":"p34","title":"lem:scalarprodbars","kind":"lemma","summary":"For A,\\overlineA,B,\\overlineB\\in \\R^m\\times n and P_1,P_2\\in \\R^n it holds that \\[ |\\langle AP_…","labels":["lem:scalarprodbars"],"detail_key":"p34"},{"id":"n31592","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 24.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 24.","labels":[],"detail_key":"p34"},{"id":"n31593","layer":"informal","project":"p34","title":"lem:absscalar","kind":"lemma","summary":"For A,B\\in \\R^m\\times n and P_1,P_2\\in \\R^n one has |\\langle AP_1,AP_2\\rangle-\\langle BP_1,BP_2…","labels":["lem:absscalar"],"detail_key":"p34"},{"id":"n31594","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 25.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 25.","labels":[],"detail_key":"p34"},{"id":"n31595","layer":"informal","project":"p34","title":"lem:origintriangle","kind":"lemma","summary":"Let A,B,C\\in R^2 be such that \\langle R(\\pi/2) A,B\\rangle, \\langle R(\\pi/2) B,C\\rangle, \\langle…","labels":["lem:origintriangle"],"detail_key":"p34"},{"id":"n31596","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 26.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 26.","labels":[],"detail_key":"p34"},{"id":"n31597","layer":"informal","project":"p34","title":"def:eps-spanning","kind":"definition","summary":"Let \\theta, \\varphi \\in R, \\varepsilon > 0, and set M := M(\\theta, \\varphi). Three points P_1,…","labels":["def:eps-spanning"],"detail_key":"p34"},{"id":"n31598","layer":"informal","project":"p34","title":"lem:eps-spanning","kind":"lemma","summary":"Let P_1, P_2, P_3 \\in \\R^3 with \\|P_1\\|,\\|P_2\\|,\\|P_3\\| \\leq 1 be \\epsilon-spanning for (\\overl…","labels":["lem:eps-spanning"],"detail_key":"p34"},{"id":"n31599","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 28.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 28.","labels":[],"detail_key":"p34"},{"id":"n31600","layer":"informal","project":"p34","title":"lem:inCirc","kind":"lemma","summary":"Let P, Q \\in \\R^3 with \\|P\\|, \\|Q\\| \\leq 1. Let \\epsilon>0 and \\overline\\theta_1,\\overline\\varp…","labels":["lem:inCirc"],"detail_key":"p34"},{"id":"n31601","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 30, which states that both points lie in the di…","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 30, which states that both points lie in the disc of…","labels":[],"detail_key":"p34"},{"id":"n31602","layer":"informal","project":"p34","title":"def:LMD","kind":"definition","summary":"Let \\PP \\subset \\R^2 be a convex polygon and Q \\in \\PP one of its vertices. Define \\Sect_\\delta…","labels":["def:LMD"],"detail_key":"p34"},{"id":"n31603","layer":"informal","project":"p34","title":"lem:LMD","kind":"lemma","summary":"Let \\PP be a convex polygon and Q \\in \\PP be one of its vertices. Let \\delta > 0. Assume that f…","labels":["lem:LMD"],"detail_key":"p34"},{"id":"n31604","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 32: the disc of radius 2\\delta around Q contain…","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 32: the disc of radius 2\\delta around Q contains the…","labels":[],"detail_key":"p34"},{"id":"n31605","layer":"informal","project":"p34","title":"lem:coss","kind":"lemma","summary":"Let \\epsilon>0 and \\theta,\\overline\\theta, \\phi, \\overline\\varphi\\in \\R with |\\theta - \\overlin…","labels":["lem:coss"],"detail_key":"p34"},{"id":"n31606","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 33.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 33.","labels":[],"detail_key":"p34"},{"id":"n31607","layer":"informal","project":"p34","title":"lem:congruent","kind":"lemma","summary":"Let P_1,P_2,P_3, Q_1,Q_2,Q_3 \\in \\R^3. Define the 3 \\times 3 matrices P \\coloneqq (P_1|P_2|P_3)…","labels":["lem:congruent"],"detail_key":"p34"},{"id":"n31608","layer":"informal","project":"p34","title":"An isometry preserves inner products, so congruence implies equality of the Gram matrices…","kind":"proof","summary":"An isometry preserves inner products, so congruence implies equality of the Gram matrices. Conv…","labels":[],"detail_key":"p34"},{"id":"n31609","layer":"informal","project":"p34","title":"Local Theorem","kind":"theorem","summary":"[Local Theorem] Let \\PPP be a polyhedron with radius \\rho=1 and P_1, P_2, P_3, Q_1, Q_2, Q_3 \\i…","labels":["thm:local"],"detail_key":"p34"},{"id":"n31610","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Theorem 36.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Theorem 36.","labels":[],"detail_key":"p34"},{"id":"n31611","layer":"informal","project":"p34","title":"dfn:sin_cos_approx","kind":"definition","summary":"We define the two functions \\sin_Q, \\cos_Q: \\R \\to \\R by: \\sin_Q(x) & \\coloneqq x-\\fracx^33+\\fr…","labels":["dfn:sin_cos_approx"],"detail_key":"p34"},{"id":"n31612","layer":"informal","project":"p34","title":"lem:sin27cos26","kind":"lemma","summary":"\\[ |\\sin_Q(x)-\\sin(x)|\\leq \\frac|x|^2727! \\quad \\textand \\quad |\\cos_Q(x)-\\cos(x)|\\leq \\frac|x|…","labels":["lem:sin27cos26"],"detail_key":"p34"},{"id":"n31613","layer":"informal","project":"p34","title":"Appeal to Taylor series bounds, using the fact that all absolute values of higher derivat…","kind":"proof","summary":"Appeal to Taylor series bounds, using the fact that all absolute values of higher derivatives o…","labels":[],"detail_key":"p34"},{"id":"n31614","layer":"informal","project":"p34","title":"lem:kappa7","kind":"lemma","summary":"For every x\\in [-4,4] it holds that \\[ |\\sin_Q(x)-\\sin(x)| \\leq \\frac\\kappa7 \\quad \\textand \\qu…","labels":["lem:kappa7"],"detail_key":"p34"},{"id":"n31615","layer":"informal","project":"p34","title":"Straightforward numerical calculation from Lemma~\\reflem:sin27cos26.","kind":"proof","summary":"Straightforward numerical calculation from Lemma~\\reflem:sin27cos26.","labels":[],"detail_key":"p34"},{"id":"n31616","layer":"informal","project":"p34","title":"lem:A_le_deltamn","kind":"lemma","summary":"Let A = (a_i,j)_1 \\leq i \\leq m,\\ 1 \\leq j \\leq n \\in \\R^m \\times n and \\delta >0. Assume that…","labels":["lem:A_le_deltamn"],"detail_key":"p34"},{"id":"n31617","layer":"informal","project":"p34","title":"For any v\\in \\R^n we have \\|Av\\|^2 &=\\sum_i=1^m \\left(\\sum_j=1^na_i,jv_j\\right)^2 \\leq \\s…","kind":"proof","summary":"For any v\\in \\R^n we have \\|Av\\|^2 &=\\sum_i=1^m \\left(\\sum_j=1^na_i,jv_j\\right)^2 \\leq \\sum_i=1…","labels":[],"detail_key":"p34"},{"id":"n31618","layer":"informal","project":"p34","title":"lem:dist_le_kappa","kind":"lemma","summary":"Let A(x,y) be an m\\times n matrix with 1 \\leq m,n\\leq 3 such that every entry is of the form a_…","labels":["lem:dist_le_kappa"],"detail_key":"p34"},{"id":"n31619","layer":"informal","project":"p34","title":"We've replaced the assumption a_i(z)\\in \\0,1,-1,\\pm\\sin(z),\\pm\\cos(z)\\ in \\citepolyhedron…","kind":"proof","summary":"We've replaced the assumption a_i(z)\\in \\0,1,-1,\\pm\\sin(z),\\pm\\cos(z)\\ in \\citepolyhedron.witho…","labels":[],"detail_key":"p34"},{"id":"n31620","layer":"informal","project":"p34","title":"corr:kappa1kappa","kind":"corollary","summary":"Let \\alpha,\\theta,\\phi\\in [-4,4]. Then it holds that \\|R(\\alpha)-R_\\Q(\\alpha)\\|, \\|R'(\\alpha)-R…","labels":["corr:kappa1kappa"],"detail_key":"p34"},{"id":"n31621","layer":"informal","project":"p34","title":"The first statement is a direct application of \\creflem:dist_le_kappa and the second stat…","kind":"proof","summary":"The first statement is a direct application of \\creflem:dist_le_kappa and the second statement…","labels":[],"detail_key":"p34"},{"id":"n31622","layer":"informal","project":"p34","title":"lem:A1AnB1Bn","kind":"lemma","summary":"For 1 \\leq i \\leq n let (A_i,B_i) be pairs of real matrices, such that for each i the dimension…","labels":["lem:A1AnB1Bn"],"detail_key":"p34"},{"id":"n31623","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 42.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 42.","labels":[],"detail_key":"p34"},{"id":"n31624","layer":"informal","project":"p34","title":"lem:boundskappa","kind":"lemma","summary":"Let \\alpha, \\theta, \\phi \\in [-4,4], P\\in \\R^3 with \\|P\\| \\leq 1 and let \\widetildeP be a \\kapp…","labels":["lem:boundskappa","eq:boundskappa1","eq:boundskappa4"],"detail_key":"p34"},{"id":"n31625","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 44.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 44.","labels":[],"detail_key":"p34"},{"id":"n31626","layer":"informal","project":"p34","title":"Rational Global Theorem","kind":"theorem","summary":"[Rational Global Theorem] Let \\PPP be a pointsymmetric convex polyhedron with radius \\rho =1 an…","labels":["thm:global_rational"],"detail_key":"p34"},{"id":"n31627","layer":"informal","project":"p34","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p34"},{"id":"n31628","layer":"informal","project":"p34","title":"def:ekspanning","kind":"definition","summary":"Let \\theta, \\phi \\in \\Q \\cap [-4,4] and M_\\Q \\coloneqq M_\\Q(\\theta, \\phi). Three points \\wideti…","labels":["def:ekspanning"],"detail_key":"p34"},{"id":"n31629","layer":"informal","project":"p34","title":"lem:ekspanningespanning","kind":"lemma","summary":"Let P_1, P_2, P_3 \\in \\R^3 with \\|P_i\\| \\leq 1 and \\widetildeP_1, \\widetildeP_2, \\widetildeP_3…","labels":["lem:ekspanningespanning"],"detail_key":"p34"},{"id":"n31630","layer":"informal","project":"p34","title":"See \\citepolyhedron.without.rupert, Lemma 46.","kind":"proof","summary":"See \\citepolyhedron.without.rupert, Lemma 46.","labels":[],"detail_key":"p34"},{"id":"n31631","layer":"informal","project":"p34","title":"lem:boundskappa3","kind":"lemma","summary":"Let P \\in \\R^3 with \\|P\\|\\leq 1 and \\widetildeP a \\kappa-rational approximation of P. 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Formalized as \\textt…","kind":"proof","summary":"By exhibiting the table and running the validity checking algorithm. Formalized as \\textttNoper…","labels":[],"detail_key":"p34"},{"id":"n31642","layer":"informal","project":"p34","title":"thm:solution_global","kind":"theorem","summary":"If a global node in the solution tree is valid, then there is no Rupert solution for its interv…","labels":["thm:solution_global"],"detail_key":"p34"},{"id":"n31643","layer":"informal","project":"p34","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p34"},{"id":"n31644","layer":"informal","project":"p34","title":"thm:solution_local","kind":"theorem","summary":"If a local node in the solution tree is valid, then there is no Rupert solution for its interva…","labels":["thm:solution_local"],"detail_key":"p34"},{"id":"n31645","layer":"informal","project":"p34","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p34"},{"id":"n31646","layer":"informal","project":"p34","title":"thm:row_valid_imp_not_rupert_ix","kind":"theorem","summary":"If we have a valid solution table, and in particular its ith row is valid, then there is no Rup…","labels":["thm:row_valid_imp_not_rupert_ix"],"detail_key":"p34"},{"id":"n31647","layer":"informal","project":"p34","title":"By strong induction on the number of rows left in the table following the ith. This is be…","kind":"proof","summary":"By strong induction on the number of rows left in the table following the ith. This is because…","labels":[],"detail_key":"p34"},{"id":"n31648","layer":"informal","project":"p34","title":"thm:row_valid_imp_not_rupert","kind":"corollary","summary":"If we have a valid solution table, then there is no Rupert solution of the interval of its zero…","labels":["thm:row_valid_imp_not_rupert"],"detail_key":"p34"},{"id":"n31649","layer":"informal","project":"p34","title":"Immediate special case of \\crefthm:row_valid_imp_not_rupert_ix.","kind":"proof","summary":"Immediate special case of \\crefthm:row_valid_imp_not_rupert_ix.","labels":[],"detail_key":"p34"},{"id":"n31650","layer":"informal","project":"p34","title":"thm:no_nopert_tight_pose","kind":"theorem","summary":"There does not in fact exist a noperthedron Rupert solution with \\theta_1,\\theta_2&\\in[0,2\\pi/1…","labels":["thm:no_nopert_tight_pose"],"detail_key":"p34"},{"id":"n31651","layer":"informal","project":"p34","title":"By \\refthm:exists_solution_table, there is a valid solution table containing a valid row…","kind":"proof","summary":"By \\refthm:exists_solution_table, there is a valid solution table containing a valid row whose…","labels":[],"detail_key":"p34"},{"id":"n31652","layer":"informal","project":"p34","title":"thm:no_nopert_pose","kind":"theorem","summary":"There is no 5-parameter pose that makes the noperthedron have the Rupert property.","labels":["thm:no_nopert_pose"],"detail_key":"p34"},{"id":"n31653","layer":"informal","project":"p34","title":"Theorem~\\refthm:no_nopert_tight_pose says there is no tight pose that makes the noperthed…","kind":"proof","summary":"Theorem~\\refthm:no_nopert_tight_pose says there is no tight pose that makes the noperthedron Ru…","labels":[],"detail_key":"p34"},{"id":"n31654","layer":"informal","project":"p34","title":"thm:no_nopert_rot_pose","kind":"theorem","summary":"There is no purely rotational pose that makes the noperthedron have the Rupert property.","labels":["thm:no_nopert_rot_pose"],"detail_key":"p34"},{"id":"n31655","layer":"informal","project":"p34","title":"Suppose there were a purely rotational pose. Then convert that to an equivalent 5-paramet…","kind":"proof","summary":"Suppose there were a purely rotational pose. Then convert that to an equivalent 5-parameter pos…","labels":[],"detail_key":"p34"},{"id":"n31656","layer":"informal","project":"p34","title":"thm:no_nopert_matrix_pose","kind":"theorem","summary":"There is no pose that makes the noperthedron have the Rupert property.","labels":["thm:no_nopert_matrix_pose"],"detail_key":"p34"},{"id":"n31657","layer":"informal","project":"p34","title":"By Theorem~\\refthm:rupert_implies_rot_rupert, we need only show that the noperthedron is…","kind":"proof","summary":"By Theorem~\\refthm:rupert_implies_rot_rupert, we need only show that the noperthedron is points…","labels":[],"detail_key":"p34"},{"id":"n31658","layer":"informal","project":"p34","title":"thm:nopert_not_rupert_set","kind":"theorem","summary":"The noperthedron is not a Rupert set.","labels":["thm:nopert_not_rupert_set"],"detail_key":"p34"},{"id":"n31659","layer":"informal","project":"p34","title":"By Theorem~\\refthm:no_nopert_matrix_pose, there is no pose that makes the noperthedron a…","kind":"proof","summary":"By Theorem~\\refthm:no_nopert_matrix_pose, there is no pose that makes the noperthedron a Rupert…","labels":[],"detail_key":"p34"},{"id":"n31660","layer":"informal","project":"p34","title":"thm:nopert_not_rupert","kind":"theorem","summary":"The noperthedron is not a Rupert polyhedron.","labels":["thm:nopert_not_rupert"],"detail_key":"p34"},{"id":"n31661","layer":"informal","project":"p34","title":"By Theorem~\\refthm:rupert_iff_rupert_set it suffices to show that the convex hull of the…","kind":"proof","summary":"By Theorem~\\refthm:rupert_iff_rupert_set it suffices to show that the convex hull of the nopert…","labels":[],"detail_key":"p34"},{"id":"n31662","layer":"formal","project":"p34","title":"Bounding.norm_RxL_sub_RxL_eq","kind":"theorem","summary":"∀ α α_ : Real, Eq (norm (HSub.hSub (RxL α) (RxL α_))) (norm (HSub.hSub (rotR α) (rotR 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(…","labels":[],"detail_key":"p34","name":"Bounding.norm_rotR_sub_rotR_lt","module":"Noperthedron.Bounding.BoundingUtil"},{"id":"n31666","layer":"formal","project":"p34","title":"Bounding.Rx_norm_one","kind":"theorem","summary":"∀ (α : Real), Eq (norm (RxL α)) 1","labels":[],"detail_key":"p34","name":"Bounding.Rx_norm_one","module":"Noperthedron.Bounding.OpNorm"},{"id":"n31667","layer":"formal","project":"p34","title":"Bounding.Ry_norm_one","kind":"theorem","summary":"∀ (α : Real), Eq (norm (RyL α)) 1","labels":[],"detail_key":"p34","name":"Bounding.Ry_norm_one","module":"Noperthedron.Bounding.OpNorm"},{"id":"n31668","layer":"formal","project":"p34","title":"Bounding.Rz_norm_one","kind":"theorem","summary":"∀ (α : Real), Eq (norm (RzL α)) 1","labels":[],"detail_key":"p34","name":"Bounding.Rz_norm_one","module":"Noperthedron.Bounding.OpNorm"},{"id":"n31669","layer":"formal","project":"p34","title":"Bounding.rotM_norm_one","kind":"theorem","summary":"∀ (θ φ : Real), Eq (norm (rotM θ φ)) 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(α : Real), Eq (norm (rotR α)) 1","labels":[],"detail_key":"p34","name":"Bounding.rotR_norm_one","module":"Noperthedron.Bounding.OpNorm"},{"id":"n31674","layer":"formal","project":"p34","title":"Bounding.lemma12","kind":"theorem","summary":"∀ d d' : Fin 3 α β : Real, Ne d d' → LE.le (norm (HSub.hSub (((rot3 d) α).comp ((rot3 d') β)) 1…","labels":[],"detail_key":"p34","name":"Bounding.lemma12","module":"Noperthedron.Bounding.SmallConsecutiveRotations"},{"id":"n31675","layer":"formal","project":"p34","title":"Bounding.lemma12_lt_of_ne","kind":"theorem","summary":"∀ d d' : Fin 3 α β : Real, Ne d d' → Not (And (Eq α 0) (Eq β 0)) → LT.lt (norm (HSub.hSub (((ro…","labels":[],"detail_key":"p34","name":"Bounding.lemma12_lt_of_ne","module":"Noperthedron.Bounding.SmallConsecutiveRotations"},{"id":"n31676","layer":"formal","project":"p34","title":"Bounding.XPgt0","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) ε θ θ_ φ φ_ : Real, LE.le (norm P) 1 → LT.lt 0 ε → LE.le (abs…","labels":[],"detail_key":"p34","name":"Bounding.XPgt0","module":"Noperthedron.Bounding"},{"id":"n31677","layer":"formal","project":"p34","title":"Bounding.norm_M_apply_gt","kind":"theorem","summary":"∀ ε r θ θ_ φ φ_ : Real P : EuclideanSpace Real (Fin 3), LE.le (norm P) 1 → LT.lt 0 ε → LE.le (a…","labels":[],"detail_key":"p34","name":"Bounding.norm_M_apply_gt","module":"Noperthedron.Bounding"},{"id":"n31678","layer":"formal","project":"p34","title":"Bounding.norm_M_sub_lt","kind":"theorem","summary":"∀ ε θ θ_ φ φ_ : Real, LT.lt 0 ε → LE.le (abs (HSub.hSub θ θ_)) ε → LE.le (abs (HSub.hSub φ φ_))…","labels":[],"detail_key":"p34","name":"Bounding.norm_M_sub_lt","module":"Noperthedron.Bounding"},{"id":"n31679","layer":"formal","project":"p34","title":"Bounding.norm_RM_sub_RM_le","kind":"theorem","summary":"∀ ε θ θ_ φ φ_ α α_ : Real, LT.lt 0 ε → LE.le (abs (HSub.hSub θ θ_)) ε → LE.le (abs (HSub.hSub 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n…","labels":[],"detail_key":"p34","name":"GlobalTheorem.bounded_partials_control_difference2","module":"Noperthedron.Global.BoundedPartialsControlDifference"},{"id":"n31686","layer":"formal","project":"p34","title":"GlobalTheorem.rotation_partials_exist","kind":"theorem","summary":"∀ S : EuclideanSpace Real (Fin 3) w : EuclideanSpace Real (Fin 2), ContDiff Real 3 (GlobalTheor…","labels":[],"detail_key":"p34","name":"GlobalTheorem.rotation_partials_exist","module":"Noperthedron.Global.Definitions"},{"id":"n31687","layer":"formal","project":"p34","title":"GlobalTheorem.rotation_partials_exist_outer","kind":"theorem","summary":"∀ S : EuclideanSpace Real (Fin 3) w : EuclideanSpace Real (Fin 2), ContDiff Real 3 (GlobalTheor…","labels":[],"detail_key":"p34","name":"GlobalTheorem.rotation_partials_exist_outer","module":"Noperthedron.Global.Definitions"},{"id":"n31688","layer":"formal","project":"p34","title":"GlobalTheorem.rotation_third_partials_bounded","kind":"theorem","summary":"∀ (S : EuclideanSpace Real (Fin 3)) w : EuclideanSpace Real (Fin 2), Eq (norm w) 1 → GlobalTheo…","labels":[],"detail_key":"p34","name":"GlobalTheorem.rotation_third_partials_bounded","module":"Noperthedron.Global.RotationPartials.SecondPartialInner"},{"id":"n31689","layer":"formal","project":"p34","title":"GlobalTheorem.global_theorem","kind":"theorem","summary":"∀ ι : Type [inst : Fintype ι] [inst_1 : Nonempty ι] (pbar : Pose Real) (εα εθ₁ εφ₁ εθ₂ εφ₂ : Re…","labels":[],"detail_key":"p34","name":"GlobalTheorem.global_theorem","module":"Noperthedron.Global"},{"id":"n31690","layer":"formal","project":"p34","title":"GlobalTheorem.hull_scalar_prod","kind":"theorem","summary":"∀ n : Nat (V : Finset (E n)) (Vne : V.Nonempty) (S : E n), Membership.mem ((convexHull Real) ↑V…","labels":[],"detail_key":"p34","name":"GlobalTheorem.hull_scalar_prod","module":"Noperthedron.Global"},{"id":"n31691","layer":"formal","project":"p34","title":"GlobalTheorem.partials_helper0","kind":"theorem","summary":"∀ (pbar : Pose Real) (S : EuclideanSpace Real (Fin 3)) (w : EuclideanSpace Real (Fin 2)), Eq (G…","labels":[],"detail_key":"p34","name":"GlobalTheorem.partials_helper0","module":"Noperthedron.Global"},{"id":"n31692","layer":"formal","project":"p34","title":"GlobalTheorem.partials_helper1","kind":"theorem","summary":"∀ (pbar : Pose Real) (S : EuclideanSpace Real (Fin 3)) (w : EuclideanSpace Real (Fin 2)), Eq (G…","labels":[],"detail_key":"p34","name":"GlobalTheorem.partials_helper1","module":"Noperthedron.Global"},{"id":"n31693","layer":"formal","project":"p34","title":"GlobalTheorem.partials_helper2","kind":"theorem","summary":"∀ (pbar : Pose Real) (S : EuclideanSpace Real (Fin 3)) (w : EuclideanSpace Real (Fin 2)), Eq (G…","labels":[],"detail_key":"p34","name":"GlobalTheorem.partials_helper2","module":"Noperthedron.Global"},{"id":"n31694","layer":"formal","project":"p34","title":"GlobalTheorem.partials_helper3","kind":"theorem","summary":"∀ (pbar : Pose Real) (P : EuclideanSpace Real 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(E…","labels":[],"detail_key":"p34","name":"Local.abs_sub_inner_bars_le","module":"Noperthedron.Local.Prelims"},{"id":"n31704","layer":"formal","project":"p34","title":"Local.abs_sub_inner_le","kind":"theorem","summary":"∀ m n : Nat (A B : ContinuousLinearMap (RingHom.id Real) (EuclideanSpace Real (Fin m)) (Euclide…","labels":[],"detail_key":"p34","name":"Local.abs_sub_inner_le","module":"Noperthedron.Local.Prelims"},{"id":"n31705","layer":"formal","project":"p34","title":"Local.pythagoras","kind":"theorem","summary":"∀ θ φ : Real (P : EuclideanSpace Real (Fin 3)), Eq (HPow.hPow (norm ((rotM θ φ) P)) 2) (HSub.hS…","labels":[],"detail_key":"p34","name":"Local.pythagoras","module":"Noperthedron.Local.Prelims"},{"id":"n31706","layer":"formal","project":"p34","title":"Local.Spanp","kind":"def","summary":"n : Nat → (Fin n → EuclideanSpace Real (Fin n)) → Set (EuclideanSpace Real (Fin n))","labels":[],"detail_key":"p34","name":"Local.Spanp","module":"Noperthedron.Local.Spanp"},{"id":"n31707","layer":"formal","project":"p34","title":"Local.langles","kind":"theorem","summary":"∀ Y Z : EuclideanSpace Real (Fin 3) V : Fin 3 → EuclideanSpace Real (Fin 3), Eq (norm Y) (norm…","labels":[],"detail_key":"p34","name":"Local.langles","module":"Noperthedron.Local.Spanp"},{"id":"n31708","layer":"formal","project":"p34","title":"Local.inCirc","kind":"theorem","summary":"∀ δ ε θ₁ θ₁_ θ₂ θ₂_ φ₁ φ₁_ φ₂ φ₂_ α α_ : Real P Q : EuclideanSpace Real (Fin 3), LE.le (norm P)…","labels":[],"detail_key":"p34","name":"Local.inCirc","module":"Noperthedron.Local"},{"id":"n31709","layer":"formal","project":"p34","title":"Local.local_theorem","kind":"theorem","summary":"∀ ι : Type [inst : Fintype ι] [inst_1 : Nonempty ι] (poly : GoodPoly ι) (p_ : Pose Real) (ε : R…","labels":[],"detail_key":"p34","name":"Local.local_theorem","module":"Noperthedron.Local"},{"id":"n31710","layer":"formal","project":"p34","title":"Noperthedron.no_nopert_matrix_pose","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (Exists fun p => RupertPose p Noperthedron.exa…","labels":[],"detail_key":"p34","name":"Noperthedron.no_nopert_matrix_pose","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31711","layer":"formal","project":"p34","title":"Noperthedron.no_nopert_pose","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (Exists fun v => RupertPose v Noperthedron.exa…","labels":[],"detail_key":"p34","name":"Noperthedron.no_nopert_pose","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31712","layer":"formal","project":"p34","title":"Noperthedron.no_nopert_rot_pose","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (Exists fun p => RupertPose p.zeroOffset Noper…","labels":[],"detail_key":"p34","name":"Noperthedron.no_nopert_rot_pose","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31713","layer":"formal","project":"p34","title":"Noperthedron.no_nopert_tight_pose","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (Exists fun v => And (tightInterval.contains v…","labels":[],"detail_key":"p34","name":"Noperthedron.no_nopert_tight_pose","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31714","layer":"formal","project":"p34","title":"Noperthedron.nopert_not_rupert","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (IsRupert Noperthedron.exactVerts)","labels":[],"detail_key":"p34","name":"Noperthedron.nopert_not_rupert","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31715","layer":"formal","project":"p34","title":"Noperthedron.nopert_not_rupert_set","kind":"theorem","summary":"∀ (vtab : Noperthedron.Solution.ValidTable), Not (IsRupertSet Noperthedron.exactPolyhedron.hull)","labels":[],"detail_key":"p34","name":"Noperthedron.nopert_not_rupert_set","module":"Noperthedron.NoperthedronIsNotRupert"},{"id":"n31716","layer":"formal","project":"p34","title":"PointSym","kind":"def","summary":"n : Nat → Set (EuclideanSpace Real (Fin n)) → Prop","labels":[],"detail_key":"p34","name":"PointSym","module":"Noperthedron.PointSym"},{"id":"n31717","layer":"formal","project":"p34","title":"RationalApprox.norm_matrix_actual_approx_le_kappa","kind":"theorem","summary":"∀ m n : Subtype fun x => Membership.mem (Finset.Icc 1 3) x (A : Matrix (Fin ↑m) (Fin ↑n) Ration…","labels":[],"detail_key":"p34","name":"RationalApprox.norm_matrix_actual_approx_le_kappa","module":"Noperthedron.RationalApprox.ApproximableMatrices"},{"id":"n31718","layer":"formal","project":"p34","title":"RationalApprox.cosℚ","kind":"def","summary":"k : Type → [inst : Field k] → [inst_1 : LinearOrder k] → [FloorRing k] → k → k","labels":[],"detail_key":"p34","name":"RationalApprox.cosℚ","module":"Noperthedron.RationalApprox.Basic"},{"id":"n31719","layer":"formal","project":"p34","title":"RationalApprox.sinℚ","kind":"def","summary":"k : Type → [inst : Field k] → [inst_1 : LinearOrder k] → [FloorRing k] → k → k","labels":[],"detail_key":"p34","name":"RationalApprox.sinℚ","module":"Noperthedron.RationalApprox.Basic"},{"id":"n31720","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa3_X","kind":"theorem","summary":"∀ P P_ : EuclideanSpace Real (Fin 3) θ φ : ↑(Set.Icc (-4) 4), LE.le (norm P) 1 → LE.le (norm (H…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa3_X","module":"Noperthedron.RationalApprox.BoundsKappa3"},{"id":"n31721","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa4","kind":"theorem","summary":"∀ (P Q : EuclideanSpace Real (Fin 3)) (P_ Q_ : Fin 3 → Rat) (p : Pose Rat) (hθBound : Membershi…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa4","module":"Noperthedron.RationalApprox.BoundsKappa4"},{"id":"n31722","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_M","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_M","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31723","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_Mθ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_Mθ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31724","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_Mθθ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_Mθθ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31725","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_Mθφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_Mθφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31726","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_Mφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_Mφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31727","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_Mφφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, LE.…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_Mφφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31728","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_R'M","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_R'M","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31729","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_R'Mθ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_R'Mθ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31730","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_R'Mφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_R'Mφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31731","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RM","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RM","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31732","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RMθ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RMθ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31733","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RMθθ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RMθθ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31734","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RMθφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RMθφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31735","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RMφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RMφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31736","layer":"formal","project":"p34","title":"RationalApprox.bounds_kappa_RMφφ","kind":"theorem","summary":"∀ P : EuclideanSpace Real (Fin 3) P_ : Fin 3 → Rat α θ φ : ↑(Set.Icc (-4) 4) w : Fin 2 → Rat, L…","labels":[],"detail_key":"p34","name":"RationalApprox.bounds_kappa_RMφφ","module":"Noperthedron.RationalApprox.BoundsKappa"},{"id":"n31737","layer":"formal","project":"p34","title":"RationalApprox.ek_spanning_imp_e_spanning","kind":"theorem","summary":"∀ (P P' : Local.Triangle), RationalApprox.κApproxTri P P' → (∀ (i : Fin 3), LE.le (norm (P i))…","labels":[],"detail_key":"p34","name":"RationalApprox.ek_spanning_imp_e_spanning","module":"Noperthedron.RationalApprox.EpsKapSpanning"},{"id":"n31738","layer":"formal","project":"p34","title":"RationalApprox.norm_le_delta_sqrt_dims","kind":"theorem","summary":"∀ m n : Nat δ : Real (A : Matrix (Fin m) (Fin n) Real), LT.lt 0 δ → (∀ (i : Fin m) (j : Fin n),…","labels":[],"detail_key":"p34","name":"RationalApprox.norm_le_delta_sqrt_dims","module":"Noperthedron.RationalApprox.Lemma39"},{"id":"n31739","layer":"formal","project":"p34","title":"RationalApprox.norm_sub_le_prod","kind":"theorem","summary":"∀ n m : Nat (mv : RationalApprox.MatVec n m) (κ : Real), GT.gt κ 0 → mv.DiffBoundedBy κ → LE.le…","labels":[],"detail_key":"p34","name":"RationalApprox.norm_sub_le_prod","module":"Noperthedron.RationalApprox.Lemma42"},{"id":"n31740","layer":"formal","project":"p34","title":"RationalApprox.M_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.M_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31741","layer":"formal","project":"p34","title":"RationalApprox.Mθ_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθ_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31742","layer":"formal","project":"p34","title":"RationalApprox.Mθθ_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθθ_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31743","layer":"formal","project":"p34","title":"RationalApprox.Mθθℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθθℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31744","layer":"formal","project":"p34","title":"RationalApprox.Mθφ_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθφ_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31745","layer":"formal","project":"p34","title":"RationalApprox.Mθφℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθφℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31746","layer":"formal","project":"p34","title":"RationalApprox.Mθℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mθℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31747","layer":"formal","project":"p34","title":"RationalApprox.Mφ_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.Mφ_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31748","layer":"formal","project":"p34","title":"RationalApprox.Mφφ_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.Mφφ_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31749","layer":"formal","project":"p34","title":"RationalApprox.Mφφℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mφφℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31750","layer":"formal","project":"p34","title":"RationalApprox.Mφℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mφℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31751","layer":"formal","project":"p34","title":"RationalApprox.Mℚ_norm_bounded","kind":"theorem","summary":"∀ θ φ : Real, Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (no…","labels":[],"detail_key":"p34","name":"RationalApprox.Mℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31752","layer":"formal","project":"p34","title":"RationalApprox.R'_difference_norm_bounded","kind":"theorem","summary":"∀ (α : Real), Membership.mem (Set.Icc (-4) 4) α → LE.le (norm (HSub.hSub (rotR' α) (RationalApp…","labels":[],"detail_key":"p34","name":"RationalApprox.R'_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31753","layer":"formal","project":"p34","title":"RationalApprox.R'ℚ_norm_bounded","kind":"theorem","summary":"∀ (α : Real), Membership.mem (Set.Icc (-4) 4) α → LE.le (norm (RationalApprox.rotR'ℚℝ α)) (HAdd…","labels":[],"detail_key":"p34","name":"RationalApprox.R'ℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31754","layer":"formal","project":"p34","title":"RationalApprox.R_difference_norm_bounded","kind":"theorem","summary":"∀ (α : Real), Membership.mem (Set.Icc (-4) 4) α → LE.le (norm (HSub.hSub (rotR α) (RationalAppr…","labels":[],"detail_key":"p34","name":"RationalApprox.R_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31755","layer":"formal","project":"p34","title":"RationalApprox.Rℚ_norm_bounded","kind":"theorem","summary":"∀ (α : Real), Membership.mem (Set.Icc (-4) 4) α → LE.le (norm (RationalApprox.rotRℚℝ α)) (HAdd.…","labels":[],"detail_key":"p34","name":"RationalApprox.Rℚ_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31756","layer":"formal","project":"p34","title":"RationalApprox.X_difference_norm_bounded","kind":"theorem","summary":"∀ (θ φ : Real), Membership.mem (Set.Icc (-4) 4) θ → Membership.mem (Set.Icc (-4) 4) φ → LE.le (…","labels":[],"detail_key":"p34","name":"RationalApprox.X_difference_norm_bounded","module":"Noperthedron.RationalApprox.MatrixBounds"},{"id":"n31757","layer":"formal","project":"p34","title":"RationalApprox.GlobalTheorem.rational_global","kind":"theorem","summary":"∀ ι : Type [inst : Fintype ι] [inst_1 : Nonempty ι] (p : Pose Rat) (εα εθ₁ εφ₁ εθ₂ εφ₂ : Rat),…","labels":[],"detail_key":"p34","name":"RationalApprox.GlobalTheorem.rational_global","module":"Noperthedron.RationalApprox.RationalGlobal"},{"id":"n31758","layer":"formal","project":"p34","title":"RationalApprox.LocalTheorem.rational_local","kind":"theorem","summary":"∀ ι : Type [inst : Fintype ι] [inst_1 : DecidableEq ι] [inst_2 : Nonempty ι] (poly : GoodPoly ι…","labels":[],"detail_key":"p34","name":"RationalApprox.LocalTheorem.rational_local","module":"Noperthedron.RationalApprox.RationalLocal"},{"id":"n31759","layer":"formal","project":"p34","title":"RationalApprox.cosℚ_approx","kind":"theorem","summary":"∀ (x : Real), LE.le (abs (HSub.hSub (Real.cos x) (RationalApprox.cosℚ x))) (HAdd.hAdd (HDiv.hDi…","labels":[],"detail_key":"p34","name":"RationalApprox.cosℚ_approx","module":"Noperthedron.RationalApprox.TrigLemmas"},{"id":"n31760","layer":"formal","project":"p34","title":"RationalApprox.cosℚ_approx'","kind":"theorem","summary":"∀ (x : Real), Membership.mem (Set.Icc (-4) 4) x → LE.le (abs (HSub.hSub (Real.cos x) (RationalA…","labels":[],"detail_key":"p34","name":"RationalApprox.cosℚ_approx'","module":"Noperthedron.RationalApprox.TrigLemmas"},{"id":"n31761","layer":"formal","project":"p34","title":"RationalApprox.sinℚ_approx","kind":"theorem","summary":"∀ (x : Real), LE.le (abs (HSub.hSub (Real.sin x) (RationalApprox.sinℚ x))) (HAdd.hAdd (HDiv.hDi…","labels":[],"detail_key":"p34","name":"RationalApprox.sinℚ_approx","module":"Noperthedron.RationalApprox.TrigLemmas"},{"id":"n31762","layer":"formal","project":"p34","title":"RationalApprox.sinℚ_approx'","kind":"theorem","summary":"∀ (x : Real), Membership.mem (Set.Icc (-4) 4) x → LE.le (abs (HSub.hSub (Real.sin x) (RationalA…","labels":[],"detail_key":"p34","name":"RationalApprox.sinℚ_approx'","module":"Noperthedron.RationalApprox.TrigLemmas"},{"id":"n31763","layer":"formal","project":"p34","title":"rupert_iff_rupert_set","kind":"theorem","summary":"∀ (v : Finset (EuclideanSpace Real (Fin 3))), Iff (IsRupert v) (IsRupertSet ((convexHull Real)…","labels":[],"detail_key":"p34","name":"rupert_iff_rupert_set","module":"Noperthedron.Rupert.Equivalences.RupertEquivRupertSet"},{"id":"n31764","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_global_imp_no_rupert","kind":"theorem","summary":"∀ (row : Noperthedron.Solution.Row), row.ValidGlobal → Not (Exists fun q => And (Membership.mem…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_global_imp_no_rupert","module":"Noperthedron.SolutionTable.Global"},{"id":"n31765","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_local_imp_no_rupert","kind":"theorem","summary":"∀ (row : Noperthedron.Solution.Row), row.ValidLocal → Not (Exists fun q => And (Membership.mem…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_local_imp_no_rupert","module":"Noperthedron.SolutionTable.Local"},{"id":"n31766","layer":"formal","project":"p34","title":"Noperthedron.Solution.Row.valid_imp_not_rupert","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat), Noperthedron.Solution.RowsValidAt get s…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.Row.valid_imp_not_rupert","module":"Noperthedron.SolutionTable"},{"id":"n31767","layer":"formal","project":"p34","title":"Noperthedron.Solution.Row.valid_imp_not_rupert_ix","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat), Noperthedron.Solution.RowsValidAt get s…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.Row.valid_imp_not_rupert_ix","module":"Noperthedron.SolutionTable"},{"id":"n31768","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_full_split_imp_no_rupert","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat) (row : Noperthedron.Solution.Row), Noper…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_full_split_imp_no_rupert","module":"Noperthedron.SolutionTable"},{"id":"n31769","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_param_split_imp_no_rupert","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat) (row : Noperthedron.Solution.Row) (p : N…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_param_split_imp_no_rupert","module":"Noperthedron.SolutionTable"},{"id":"n31770","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_single_param_split_imp_no_rupert","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat) (row : Noperthedron.Solution.Row), Noper…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_single_param_split_imp_no_rupert","module":"Noperthedron.SolutionTable"},{"id":"n31771","layer":"formal","project":"p34","title":"Noperthedron.Solution.valid_split_imp_no_rupert","kind":"theorem","summary":"∀ (get : Nat → Noperthedron.Solution.Row) (size : Nat) (row : Noperthedron.Solution.Row), Noper…","labels":[],"detail_key":"p34","name":"Noperthedron.Solution.valid_split_imp_no_rupert","module":"Noperthedron.SolutionTable"},{"id":"n31772","layer":"formal","project":"p34","title":"Noperthedron.Tightening.lemma7_1","kind":"theorem","summary":"∀ (θ φ : Real), Eq (Set.image (⇑(rotM (HAdd.hAdd θ (HMul.hMul (2 / 15) Real.pi)) φ)) Noperthedr…","labels":[],"detail_key":"p34","name":"Noperthedron.Tightening.lemma7_1","module":"Noperthedron.Tightening"},{"id":"n31773","layer":"formal","project":"p34","title":"Noperthedron.Tightening.lemma7_2","kind":"theorem","summary":"∀ (θ φ α : Real), Eq (Set.image (Function.comp ⇑(rotR (HAdd.hAdd α Real.pi)) ⇑(rotM θ φ)) Noper…","labels":[],"detail_key":"p34","name":"Noperthedron.Tightening.lemma7_2","module":"Noperthedron.Tightening"},{"id":"n31774","layer":"formal","project":"p34","title":"Noperthedron.Tightening.lemma7_3","kind":"theorem","summary":"∀ (θ φ : Real), Eq (Set.image (⇑(flip_y.comp (rotM θ φ))) Noperthedron.exactPolyhedron.hull) (S…","labels":[],"detail_key":"p34","name":"Noperthedron.Tightening.lemma7_3","module":"Noperthedron.Tightening"},{"id":"n31775","layer":"formal","project":"p34","title":"Noperthedron.Tightening.rupert_tightening","kind":"theorem","summary":"∀ (p : Pose Real), RupertPose p Noperthedron.exactPolyhedron.hull → Exists fun p' => And (tight…","labels":[],"detail_key":"p34","name":"Noperthedron.Tightening.rupert_tightening","module":"Noperthedron.Tightening"},{"id":"n31776","layer":"formal","project":"p34","title":"Noperthedron.c1_norm_one","kind":"theorem","summary":"Eq (norm Noperthedron.C1R) 1","labels":[],"detail_key":"p34","name":"Noperthedron.c1_norm_one","module":"Noperthedron.Vertices.Exact"},{"id":"n31777","layer":"formal","project":"p34","title":"Noperthedron.c2_norm_bound","kind":"theorem","summary":"Membership.mem (Set.Ioo (98 / 100) (99 / 100)) (norm Noperthedron.C2R)","labels":[],"detail_key":"p34","name":"Noperthedron.c2_norm_bound","module":"Noperthedron.Vertices.Exact"},{"id":"n31778","layer":"formal","project":"p34","title":"Noperthedron.c3_norm_bound","kind":"theorem","summary":"Membership.mem (Set.Ioo (98 / 100) (99 / 100)) (norm Noperthedron.C3R)","labels":[],"detail_key":"p34","name":"Noperthedron.c3_norm_bound","module":"Noperthedron.Vertices.Exact"},{"id":"n31779","layer":"formal","project":"p34","title":"Noperthedron.exactPoly_point_symmetric","kind":"theorem","summary":"PointSym Noperthedron.exactPoly.hull","labels":[],"detail_key":"p34","name":"Noperthedron.exactPoly_point_symmetric","module":"Noperthedron.Vertices.Exact"},{"id":"n31780","layer":"formal","project":"p34","title":"Noperthedron.exactVertex","kind":"def","summary":"Noperthedron.VertexIndex → EuclideanSpace Real (Fin 3)","labels":[],"detail_key":"p34","name":"Noperthedron.exactVertex","module":"Noperthedron.Vertices.Exact"},{"id":"n31781","layer":"formal","project":"p34","title":"Noperthedron.exactVertex_norm_le_one","kind":"theorem","summary":"∀ (j : Noperthedron.VertexIndex), LE.le (norm (Noperthedron.exactVertex j)) 1","labels":[],"detail_key":"p34","name":"Noperthedron.exactVertex_norm_le_one","module":"Noperthedron.Vertices.Exact"},{"id":"n31782","layer":"informal","project":"p35","title":"Let M be a representation of a finite group G over the ring Z, and suppose that for all s…","kind":"theorem","summary":"Let M be a representation of a finite group G over the ring Z, and suppose that for all subgrou…","labels":[],"detail_key":"p35"},{"id":"n31783","layer":"informal","project":"p35","title":"def:inflation_restriction_functors","kind":"definition","summary":"\\mathlibok The map M \\mapsto M \\downarrow S defines a functor res: Rep(R,G) \\to Rep(R,S). If S…","labels":["def:inflation_restriction_functors"],"detail_key":"p35"},{"id":"n31784","layer":"informal","project":"p35","title":"eg:H0","kind":"example","summary":"For example H^0(G,M) is the kernel of the map d^0 : M \\to (G \\to M). Since (d^0m)(g) = g \\bulle…","labels":["eg:H0"],"detail_key":"p35"},{"id":"n31785","layer":"informal","project":"p35","title":"eg:H1_trivial_iso_Hom","kind":"example","summary":"Suppose M is a trivial representation of G. Then the map d^0 is zero, so H^1(G,M) is the kernel…","labels":["eg:H1_trivial_iso_Hom"],"detail_key":"p35"},{"id":"n31786","layer":"informal","project":"p35","title":"lem:Hn_unit","kind":"lemma","summary":"If G is the trivial group then for all n>0, H^n(G,M)\\cong 0.","labels":["lem:Hn_unit"],"detail_key":"p35"},{"id":"n31787","layer":"informal","project":"p35","title":"Each of the modules C^n(G,M) may be identified with M, and the coboundary maps reduce to…","kind":"proof","summary":"Each of the modules C^n(G,M) may be identified with M, and the coboundary maps reduce to altern…","labels":[],"detail_key":"p35"},{"id":"n31788","layer":"informal","project":"p35","title":"The functions d^i : (G^i \\to M) \\to (G^i+1 \\to M) do not depend on the ring R. Consequent…","kind":"remark","summary":"The functions d^i : (G^i \\to M) \\to (G^i+1 \\to M) do not depend on the ring R. Consequently, th…","labels":[],"detail_key":"p35"},{"id":"n31789","layer":"informal","project":"p35","title":"At this point it's worth stressing one particular aspect of this theory. Many of the proo…","kind":"remark","summary":"At this point it's worth stressing one particular aspect of this theory. Many of the proofs inv…","labels":[],"detail_key":"p35"},{"id":"n31790","layer":"informal","project":"p35","title":"def:restriction_map","kind":"definition","summary":"If S is a subgroup of G, then we write H^n(S,M) for the cohomology groups of the restricted rep…","labels":["def:restriction_map"],"detail_key":"p35"},{"id":"n31791","layer":"informal","project":"p35","title":"def:inflation_map","kind":"definition","summary":"If S is a normal subgroup of G, then we write H^n(G/S,M^S) for the cohomology groups of the rep…","labels":["def:inflation_map"],"detail_key":"p35"},{"id":"n31792","layer":"informal","project":"p35","title":"lem:cochainsFunctor_exact","kind":"lemma","summary":"\\mathlibok The functor taking M to C^\\bullet (G,M) is exact. I.e. if 0 \\to A \\to B \\to C \\to 0…","labels":["lem:cochainsFunctor_exact"],"detail_key":"p35"},{"id":"n31793","layer":"informal","project":"p35","title":"This is already in Mathlib. \\mathlibok","kind":"proof","summary":"This is already in Mathlib. \\mathlibok","labels":[],"detail_key":"p35"},{"id":"n31794","layer":"informal","project":"p35","title":"def:cohomology_long_exact_sequence","kind":"definition","summary":"\\mathlibok Given a short exact sequence 0 \\to A \\stackrelf\\to B \\stackrelg\\to C \\to 0 in Rep(R,…","labels":["def:cohomology_long_exact_sequence"],"detail_key":"p35"},{"id":"n31795","layer":"informal","project":"p35","title":"lem:inflation_restriction_naturality","kind":"lemma","summary":"Let S be a subgroup of G and suppose we have a short exact sequence 0 \\to A \\to B \\to C \\to 0 i…","labels":["lem:inflation_restriction_naturality"],"detail_key":"p35"},{"id":"n31796","layer":"informal","project":"p35","title":"Most of the commuting squares have already been proved if inflation and restriction are d…","kind":"proof","summary":"Most of the commuting squares have already been proved if inflation and restriction are defined…","labels":[],"detail_key":"p35"},{"id":"n31797","layer":"informal","project":"p35","title":"def:group_homology","kind":"definition","summary":"\\mathlibok There is also a chain complex of R-modules: \\[ \\cdots \\stackreld_2\\to C_2(G,M) \\stac…","labels":["def:group_homology"],"detail_key":"p35"},{"id":"n31798","layer":"informal","project":"p35","title":"eg:homology_0","kind":"example","summary":"We'll sometimes write single(g,m) for the function with value m at g and value zero elsewhere.…","labels":["eg:homology_0"],"detail_key":"p35"},{"id":"n31799","layer":"informal","project":"p35","title":"eg:homology_unit","kind":"example","summary":"In the case that G=1 is the trivial group then for all n > 0 we have H_n(1,M). This follows in…","labels":["eg:homology_unit"],"detail_key":"p35"},{"id":"n31800","layer":"informal","project":"p35","title":"lem:homology_1_trivial","kind":"lemma","summary":"\\mathlibok If M is a trivial representation of G then H_1(G,M) \\cong G^ab \\otimes ZM. The isomo…","labels":["lem:homology_1_trivial"],"detail_key":"p35"},{"id":"n31801","layer":"informal","project":"p35","title":"\\mathlibok Since M is trivial, the map d_0 : C_1(G,M) \\to C_0(G,M) is zero, so every 1-ch…","kind":"proof","summary":"\\mathlibok Since M is trivial, the map d_0 : C_1(G,M) \\to C_0(G,M) is zero, so every 1-chain is…","labels":[],"detail_key":"p35"},{"id":"n31802","layer":"informal","project":"p35","title":"def:norm","kind":"definition","summary":"\\mathlibok Let G be a finite group and M a representation of G over a commutative ring R. There…","labels":["def:norm"],"detail_key":"p35"},{"id":"n31803","layer":"informal","project":"p35","title":"lem:norm_comm","kind":"lemma","summary":"For any g \\in G and m \\in M we have g \\bullet N_G (m) = N_G (m) and N_G (g \\bullet m) = N_G (m).","labels":["lem:norm_comm"],"detail_key":"p35"},{"id":"n31804","layer":"informal","project":"p35","title":"These equalities follow by reindexing the sums defining N_G (m): \\[ g \\bullet \\sum_x \\in…","kind":"proof","summary":"These equalities follow by reindexing the sums defining N_G (m): \\[ g \\bullet \\sum_x \\in G x \\b…","labels":[],"detail_key":"p35"},{"id":"n31805","layer":"informal","project":"p35","title":"lem:norm_comp_d","kind":"lemma","summary":"\\mathlibok The composition d^0 \\circ N_G is zero.","labels":["lem:norm_comp_d"],"detail_key":"p35"},{"id":"n31806","layer":"informal","project":"p35","title":"The map d^0 : M \\to (G \\to M) is given by (d^0 m)(g) = m - g\\bullet m. Using this formula…","kind":"proof","summary":"The map d^0 : M \\to (G \\to M) is given by (d^0 m)(g) = m - g\\bullet m. Using this formula, we o…","labels":[],"detail_key":"p35"},{"id":"n31807","layer":"informal","project":"p35","title":"lem:d_comp_norm","kind":"lemma","summary":"\\mathlibok The composition N_G \\circ d_0 is zero.","labels":["lem:d_comp_norm"],"detail_key":"p35"},{"id":"n31808","layer":"informal","project":"p35","title":"Since the elements single(g,m) span C_1(G,M), it's sufficient to check that these are all…","kind":"proof","summary":"Since the elements single(g,m) span C_1(G,M), it's sufficient to check that these are all mappe…","labels":[],"detail_key":"p35"},{"id":"n31809","layer":"informal","project":"p35","title":"lem:norm_naturality","kind":"lemma","summary":"\\mathlibok For every map f : A \\to B in Rep(R,G) we have a commuting square: \\[ rcl A & \\stackr…","labels":["lem:norm_naturality"],"detail_key":"p35"},{"id":"n31810","layer":"informal","project":"p35","title":"For m \\in M we have \\[ f(N_G (m)) = f\\left( \\sum_g \\in G g \\bullet m\\right) = \\sum_g \\in…","kind":"proof","summary":"For m \\in M we have \\[ f(N_G (m)) = f\\left( \\sum_g \\in G g \\bullet m\\right) = \\sum_g \\in G g \\b…","labels":[],"detail_key":"p35"},{"id":"n31811","layer":"informal","project":"p35","title":"def:Tate_cohomology","kind":"definition","summary":"\\mathlibok Recall that we have a cochain complex C^n(G,M), indexed by n \\in N, whose zeroth ter…","labels":["def:Tate_cohomology"],"detail_key":"p35"},{"id":"n31812","layer":"informal","project":"p35","title":"lem:Tate_cohomology_is_cohomology_or_homology","kind":"lemma","summary":"Let G be a finite group and M a representation of G. \\item The zeroth Tate cohomology H^0_Tate(…","labels":["lem:Tate_cohomology_is_cohomology_or_homology"],"detail_key":"p35"},{"id":"n31813","layer":"informal","project":"p35","title":"This result is clear from the definition for n > 0 and n < -1. We'll discuss the two rema…","kind":"proof","summary":"This result is clear from the definition for n > 0 and n < -1. We'll discuss the two remaining…","labels":[],"detail_key":"p35"},{"id":"n31814","layer":"informal","project":"p35","title":"def:Tate_long_exact_sequence","kind":"definition","summary":"\\mathlibok Since the functors C^\\bullet(G,-) and C_\\bullet(G,-) are both exact, it follows that…","labels":["def:Tate_long_exact_sequence"],"detail_key":"p35"},{"id":"n31815","layer":"informal","project":"p35","title":"def:trivial_cohomology","kind":"definition","summary":"Let M be a representation of G over a ring R. \\item M is said to have \\emphtrivial cohomology i…","labels":["def:trivial_cohomology"],"detail_key":"p35"},{"id":"n31816","layer":"informal","project":"p35","title":"Shapiro's Lemma","kind":"lemma","summary":"[Shapiro's Lemma] \\mathlibok Let S be a subgroup of G. Then there are isomorphisms for all n \\g…","labels":["lem:Shapiro"],"detail_key":"p35"},{"id":"n31817","layer":"informal","project":"p35","title":"This is already in Mathlib. \\mathlibok","kind":"proof","summary":"This is already in Mathlib. \\mathlibok","labels":[],"detail_key":"p35"},{"id":"n31818","layer":"informal","project":"p35","title":"def:induced","kind":"definition","summary":"Let G be a group, R a commutative ring and A an R-module. \\item There is a representation of G…","labels":["def:induced"],"detail_key":"p35"},{"id":"n31819","layer":"informal","project":"p35","title":"lem:coind₁_trivial_cohomology","kind":"lemma","summary":"The representation coind_1(G,A) has trivial cohomology.","labels":["lem:coind₁_trivial_cohomology"],"detail_key":"p35"},{"id":"n31820","layer":"informal","project":"p35","title":"Let S be a subgroup of G and let C be a set of representatives for the cosets cS of S in…","kind":"proof","summary":"Let S be a subgroup of G and let C be a set of representatives for the cosets cS of S in G. The…","labels":[],"detail_key":"p35"},{"id":"n31821","layer":"informal","project":"p35","title":"lem:coind₁_invariants","kind":"lemma","summary":"Let S be a normal subgroup of G. Then coind_1(G,A)^S is isomorphic to coind_1(G/S,A). In partic…","labels":["lem:coind₁_invariants"],"detail_key":"p35"},{"id":"n31822","layer":"informal","project":"p35","title":"Let f : G \\to A. Then f is in the subspace coind_1(G,A)^S if f is constant on cosets of S…","kind":"proof","summary":"Let f : G \\to A. Then f is in the subspace coind_1(G,A)^S if f is constant on cosets of S, i.e.…","labels":[],"detail_key":"p35"},{"id":"n31823","layer":"informal","project":"p35","title":"lem:ind₁_trivial_homology","kind":"lemma","summary":"The representation ind_1(G,A) has trivial homology.","labels":["lem:ind₁_trivial_homology"],"detail_key":"p35"},{"id":"n31824","layer":"informal","project":"p35","title":"The restriction of ind_1(G,A) to a subgroup S is isomorphic to ind_1(S, C \\to_0 A), where…","kind":"proof","summary":"The restriction of ind_1(G,A) to a subgroup S is isomorphic to ind_1(S, C \\to_0 A), where C is…","labels":[],"detail_key":"p35"},{"id":"n31825","layer":"informal","project":"p35","title":"def:ind₁_to_coind₁","kind":"definition","summary":"There is a morphism of representations ind_1(G,A) \\to coind_1(G,A), which takes a finitely supp…","labels":["def:ind₁_to_coind₁"],"detail_key":"p35"},{"id":"n31826","layer":"informal","project":"p35","title":"lem:induced_trivial_Tate","kind":"lemma","summary":"If the group G is finite then ind_1(G,A) and coind_1(G,A) have trivial Tate cohomology.","labels":["lem:induced_trivial_Tate"],"detail_key":"p35"},{"id":"n31827","layer":"informal","project":"p35","title":"These representations are isomorphic, so it's sufficient to prove that ind_1(G,A) has tri…","kind":"proof","summary":"These representations are isomorphic, so it's sufficient to prove that ind_1(G,A) has trivial T…","labels":[],"detail_key":"p35"},{"id":"n31828","layer":"informal","project":"p35","title":"def:coind₁'","kind":"definition","summary":"Let G be a group and M a representation of G over a commutative ring R. There is a representati…","labels":["def:coind₁'"],"detail_key":"p35"},{"id":"n31829","layer":"informal","project":"p35","title":"lem:coind₁'_iso_coind₁","kind":"lemma","summary":"The representations coind_1'(M) and coind_1(G,M) of G are isomorphic. More precisely there is a…","labels":["lem:coind₁'_iso_coind₁"],"detail_key":"p35"},{"id":"n31830","layer":"informal","project":"p35","title":"The map f \\mapsto (x \\mapsto x \\bullet f(x)) is an isomorphism from coind_1'(M) to coind_…","kind":"proof","summary":"The map f \\mapsto (x \\mapsto x \\bullet f(x)) is an isomorphism from coind_1'(M) to coind_1(G,M).","labels":[],"detail_key":"p35"},{"id":"n31831","layer":"informal","project":"p35","title":"cor:coind₁'_trivial_cohomology","kind":"corollary","summary":"The representation coind_1'(M) has trivial cohomology.","labels":["cor:coind₁'_trivial_cohomology"],"detail_key":"p35"},{"id":"n31832","layer":"informal","project":"p35","title":"This follows directly from Lemmas \\reflem:coind₁'_iso_coind₁ and \\reflem:coind₁_trivial_c…","kind":"proof","summary":"This follows directly from Lemmas \\reflem:coind₁'_iso_coind₁ and \\reflem:coind₁_trivial_cohomol…","labels":[],"detail_key":"p35"},{"id":"n31833","layer":"informal","project":"p35","title":"cor:coind₁'_invariants_trivial_cohomology","kind":"corollary","summary":"Let S be a normal subgroup of G. Then coind_1'(M)^S has trivial cohomology as a representation…","labels":["cor:coind₁'_invariants_trivial_cohomology"],"detail_key":"p35"},{"id":"n31834","layer":"informal","project":"p35","title":"We've seen in Lemma \\reflem:coind₁'_iso_coind₁ that coind_1'(M) is isomorphic to coind_1(…","kind":"proof","summary":"We've seen in Lemma \\reflem:coind₁'_iso_coind₁ that coind_1'(M) is isomorphic to coind_1(M). Ap…","labels":[],"detail_key":"p35"},{"id":"n31835","layer":"informal","project":"p35","title":"def:up","kind":"definition","summary":"There is an injective morphism M \\to coind_1'(M) which takes a vector m \\in M to the constant f…","labels":["def:up"],"detail_key":"p35"},{"id":"n31836","layer":"informal","project":"p35","title":"cor:up_iso","kind":"corollary","summary":"Let S be any subgroup of G and let n \\ge 1. Then the connecting map from the long exact sequenc…","labels":["cor:up_iso"],"detail_key":"p35"},{"id":"n31837","layer":"informal","project":"p35","title":"We have already shown in Corollary \\refcor:coind₁'_trivial_cohomology that coind_1'(M) ha…","kind":"proof","summary":"We have already shown in Corollary \\refcor:coind₁'_trivial_cohomology that coind_1'(M) has triv…","labels":[],"detail_key":"p35"},{"id":"n31838","layer":"informal","project":"p35","title":"def:ind'","kind":"definition","summary":"There is a representation ind_1' (M) on the R-module of finitely supported functions G \\to_0 M.…","labels":["def:ind'"],"detail_key":"p35"},{"id":"n31839","layer":"informal","project":"p35","title":"lem:ind₁'_iso_ind₁","kind":"lemma","summary":"The representations ind_1'(M) and ind_1(G,M) are isomorphic; more precisely the functors ind_1'…","labels":["lem:ind₁'_iso_ind₁"],"detail_key":"p35"},{"id":"n31840","layer":"informal","project":"p35","title":"The data of the isomorphism is contained in the lean file; the isomorphism takes f : G \\t…","kind":"proof","summary":"The data of the isomorphism is contained in the lean file; the isomorphism takes f : G \\to_0 M…","labels":[],"detail_key":"p35"},{"id":"n31841","layer":"informal","project":"p35","title":"cor:ind₁'_trivial_homology","kind":"corollary","summary":"The representation ind_1'(M) has trivial homology.","labels":["cor:ind₁'_trivial_homology"],"detail_key":"p35"},{"id":"n31842","layer":"informal","project":"p35","title":"We've shown that ind_1'(M) is isomorphic to ind_1(M), which is already known to have triv…","kind":"proof","summary":"We've shown that ind_1'(M) is isomorphic to ind_1(M), which is already known to have trivial ho…","labels":[],"detail_key":"p35"},{"id":"n31843","layer":"informal","project":"p35","title":"def:down","kind":"definition","summary":"For any representation M, there is a surjective morphism ind_1'(M) \\to M, which takes a finitel…","labels":["def:down"],"detail_key":"p35"},{"id":"n31844","layer":"informal","project":"p35","title":"lem:induced'_trivial_Tate","kind":"lemma","summary":"If M is a representation of a finite group G then the representations ind_1'(M) and coind_1'(M)…","labels":["lem:induced'_trivial_Tate"],"detail_key":"p35"},{"id":"n31845","layer":"informal","project":"p35","title":"This follows from \\reflem:induced_trivial_Tate together with the isomorphisms \\refdef:ind…","kind":"proof","summary":"This follows from \\reflem:induced_trivial_Tate together with the isomorphisms \\refdef:ind₁_to_c…","labels":[],"detail_key":"p35"},{"id":"n31846","layer":"informal","project":"p35","title":"cor:Tate_up_down_isos","kind":"corollary","summary":"If the group G is finite then for every subgroup S of G and every n \\in Z we have isomorphisms…","labels":["cor:Tate_up_down_isos"],"detail_key":"p35"},{"id":"n31847","layer":"informal","project":"p35","title":"These are the connecting homomorphisms from the short exact sequences linking up(M) and d…","kind":"proof","summary":"These are the connecting homomorphisms from the short exact sequences linking up(M) and down(M)…","labels":[],"detail_key":"p35"},{"id":"n31848","layer":"informal","project":"p35","title":"def:augmentation_module","kind":"definition","summary":"As an example we consider the case of the trivial representation R. The induced representation…","labels":["def:augmentation_module"],"detail_key":"p35"},{"id":"n31849","layer":"informal","project":"p35","title":"lem:Tate_-1_aug","kind":"lemma","summary":"Let S be a subgroup of a finite group G. Then there is an isomorphism of R-modules \\[ S^ab \\oti…","labels":["lem:Tate_-1_aug"],"detail_key":"p35"},{"id":"n31850","layer":"informal","project":"p35","title":"Recall that by \\reflem:homology_1_trivial, \\reflem:Tate_cohomology_is_cohomology_or_homol…","kind":"proof","summary":"Recall that by \\reflem:homology_1_trivial, \\reflem:Tate_cohomology_is_cohomology_or_homology we…","labels":[],"detail_key":"p35"},{"id":"n31851","layer":"informal","project":"p35","title":"thm:inflation_restriction_sequence","kind":"theorem","summary":"Let S be a normal subgroup of a group G and let n be a positive integer. Assume that for all na…","labels":["thm:inflation_restriction_sequence"],"detail_key":"p35"},{"id":"n31852","layer":"informal","project":"p35","title":"This is already in Mathlib for n=1. Assume the result is true for some n\\ge 1; we will pr…","kind":"proof","summary":"This is already in Mathlib for n=1. Assume the result is true for some n\\ge 1; we will prove it…","labels":[],"detail_key":"p35"},{"id":"n31853","layer":"informal","project":"p35","title":"def:corestriction","kind":"definition","summary":"Let S be a subgroup of finite index in G and let \\r_i\\ be a set of representatives for the left…","labels":["def:corestriction"],"detail_key":"p35"},{"id":"n31854","layer":"informal","project":"p35","title":"lem:cor_comp_rest","kind":"lemma","summary":"For all \\sigma \\in H^n(G,M) we have \\(cor(rest(\\sigma)) = [G:S] \\cdot \\sigma\\).","labels":["lem:cor_comp_rest"],"detail_key":"p35"},{"id":"n31855","layer":"informal","project":"p35","title":"We'll prove the result by induction on n. In the case n = 0, this follows from the relati…","kind":"proof","summary":"We'll prove the result by induction on n. In the case n = 0, this follows from the relation for…","labels":[],"detail_key":"p35"},{"id":"n31856","layer":"informal","project":"p35","title":"cor:cohomology_G-torsion","kind":"corollary","summary":"If M is a representation of a finite group G then for all n \\in Z and all \\sigma \\in H^n_Tate(G…","labels":["cor:cohomology_G-torsion"],"detail_key":"p35"},{"id":"n31857","layer":"informal","project":"p35","title":"By dimension-shifting (\\refcor:Tate_up_down_isos) it's enough to prove the result for n >…","kind":"proof","summary":"By dimension-shifting (\\refcor:Tate_up_down_isos) it's enough to prove the result for n > 0, in…","labels":[],"detail_key":"p35"},{"id":"n31858","layer":"informal","project":"p35","title":"cor:cohomology_sub_Sylow","kind":"corollary","summary":"Let M be a representation of a finite group G and let S_p be a Sylow p-subgroup of G for some p…","labels":["cor:cohomology_sub_Sylow"],"detail_key":"p35"},{"id":"n31859","layer":"informal","project":"p35","title":"By dimension-shifting it's enough to prove the result for n > 0, in which case Tate cohom…","kind":"proof","summary":"By dimension-shifting it's enough to prove the result for n > 0, in which case Tate cohomology…","labels":[],"detail_key":"p35"},{"id":"n31860","layer":"informal","project":"p35","title":"lem:map2_image","kind":"lemma","summary":"The image of map_2 : ind_1'(M) \\to ind_1'(M) is precisely the set of functions G \\to_0 M which…","labels":["lem:map2_image"],"detail_key":"p35"},{"id":"n31861","layer":"informal","project":"p35","title":"It's clear that the values of map_2(f) sum to 0, so the image of map_2 is contained in th…","kind":"proof","summary":"It's clear that the values of map_2(f) sum to 0, so the image of map_2 is contained in the kern…","labels":[],"detail_key":"p35"},{"id":"n31862","layer":"informal","project":"p35","title":"def:up_iso_down","kind":"definition","summary":"We have a commutative square with vertical isomorphisms: \\[ ind_1'(M) & \\stackrelmap_2\\to & ind…","labels":["def:up_iso_down"],"detail_key":"p35"},{"id":"n31863","layer":"informal","project":"p35","title":"cor:periodic_cohomology","kind":"corollary","summary":"Let G be a finite cyclic group. For all n > 0 and all representations M we have an isomorphism…","labels":["cor:periodic_cohomology"],"detail_key":"p35"},{"id":"n31864","layer":"informal","project":"p35","title":"By the dimension-shifting isomorphisms we have H^n(G,M) \\cong H^n+1(G,down(M)) \\cong H^n+…","kind":"proof","summary":"By the dimension-shifting isomorphisms we have H^n(G,M) \\cong H^n+1(G,down(M)) \\cong H^n+1(G,up…","labels":[],"detail_key":"p35"},{"id":"n31865","layer":"informal","project":"p35","title":"lem:H2_cyclic_Z","kind":"lemma","summary":"Let G be a finite cyclic group of order n. Then H^1(G, Z) \\cong 0 and H^2(G, Z) \\cong Z/n Z.","labels":["lem:H2_cyclic_Z"],"detail_key":"p35"},{"id":"n31866","layer":"informal","project":"p35","title":"Since the module Z is trivial, we have H^1(G, Z)\\cong Hom(G, Z) \\cong 0. It follows from…","kind":"proof","summary":"Since the module Z is trivial, we have H^1(G, Z)\\cong Hom(G, Z) \\cong 0. It follows from \\refle…","labels":[],"detail_key":"p35"},{"id":"n31867","layer":"informal","project":"p35","title":"def:local_inv","kind":"definition","summary":"Let G be a finite cyclic group of order n generated by an element gen. Then the map inv_G : H^2…","labels":["def:local_inv"],"detail_key":"p35"},{"id":"n31868","layer":"informal","project":"p35","title":"lem:local_inv_iso","kind":"lemma","summary":"Let G be a finite cyclic group of order n generated by an element gen. The local invariant inv_…","labels":["lem:local_inv_iso"],"detail_key":"p35"},{"id":"n31869","layer":"informal","project":"p35","title":"It is easy to check that the formula for inv_ Z defines a homomorphism H^2(G, Z) \\to Z/ n…","kind":"proof","summary":"It is easy to check that the formula for inv_ Z defines a homomorphism H^2(G, Z) \\to Z/ n Z (i.…","labels":[],"detail_key":"p35"},{"id":"n31870","layer":"informal","project":"p35","title":"def:herbrand_quotient","kind":"definition","summary":"Let G be a finite cyclic group and M a representation of G. Recall that there are isomorphisms…","labels":["def:herbrand_quotient"],"detail_key":"p35"},{"id":"n31871","layer":"informal","project":"p35","title":"eg:herbrand_Z","kind":"example","summary":"If G is a cyclic group and Z has the trivial action of G then h(G, Z) = |G|. This follow immedi…","labels":["eg:herbrand_Z"],"detail_key":"p35"},{"id":"n31872","layer":"informal","project":"p35","title":"lem:herbrand_finite","kind":"lemma","summary":"If M is finite then h(G,M)=1.","labels":["lem:herbrand_finite"],"detail_key":"p35"},{"id":"n31873","layer":"informal","project":"p35","title":"Let gen be a generator of G. Recall that H^0_Tate(G,M) \\cong M^G / N_GM. Also, we can wri…","kind":"proof","summary":"Let gen be a generator of G. Recall that H^0_Tate(G,M) \\cong M^G / N_GM. Also, we can write M^G…","labels":[],"detail_key":"p35"},{"id":"n31874","layer":"informal","project":"p35","title":"lem:herbrand_ses","kind":"lemma","summary":"Suppose we have a short exact sequence of representations of a finite cyclic group G: \\[ 0 \\to…","labels":["lem:herbrand_ses"],"detail_key":"p35"},{"id":"n31875","layer":"informal","project":"p35","title":"It follows from the long exact sequence that if two of the representations A,B,C have fin…","kind":"proof","summary":"It follows from the long exact sequence that if two of the representations A,B,C have finite co…","labels":[],"detail_key":"p35"},{"id":"n31876","layer":"informal","project":"p35","title":"thm:triviality_criterion_solvable","kind":"theorem","summary":"Let M be a representation of a finite solvable group G (note that the solvability condition wil…","labels":["thm:triviality_criterion_solvable"],"detail_key":"p35"},{"id":"n31877","layer":"informal","project":"p35","title":"We must prove that H^n(S,M) = 0 for all S and all n > 0. We'll prove this by induction on…","kind":"proof","summary":"We must prove that H^n(S,M) = 0 for all S and all n > 0. We'll prove this by induction on S. Th…","labels":[],"detail_key":"p35"},{"id":"n31878","layer":"informal","project":"p35","title":"thm:triviality_criterion","kind":"theorem","summary":"Let M be a representation of a finite group G (no longer assumed to be solvable). Suppose we ha…","labels":["thm:triviality_criterion"],"detail_key":"p35"},{"id":"n31879","layer":"informal","project":"p35","title":"Let S be a subgroup of G. Fix a prime number p and let S_p be a Sylow p-subgroup of S. Si…","kind":"proof","summary":"Let S be a subgroup of G. Fix a prime number p and let S_p be a Sylow p-subgroup of S. Since S_…","labels":[],"detail_key":"p35"},{"id":"n31880","layer":"informal","project":"p35","title":"cor:up_and_down_trivial_cohomology","kind":"corollary","summary":"If M is a representation of a finite group G and M has trivial cohomology then up(M) and down(M…","labels":["cor:up_and_down_trivial_cohomology"],"detail_key":"p35"},{"id":"n31881","layer":"informal","project":"p35","title":"For each subgroup S of G we have \\[ H^1(S,up(M)) \\cong H^2(S,M) \\cong 0, \\qquad H^2(S,up(…","kind":"proof","summary":"For each subgroup S of G we have \\[ H^1(S,up(M)) \\cong H^2(S,M) \\cong 0, \\qquad H^2(S,up(M)) \\c…","labels":[],"detail_key":"p35"},{"id":"n31882","layer":"informal","project":"p35","title":"thm:trivial_cohomology_implies_trivial_Tate","kind":"theorem","summary":"Let M be a representation of a finite group G, and assume that M has trivial cohomology. Then M…","labels":["thm:trivial_cohomology_implies_trivial_Tate"],"detail_key":"p35"},{"id":"n31883","layer":"informal","project":"p35","title":"Fix an integer n and choose a natural number m such that m + n > 0. By \\refcor:up_and_dow…","kind":"proof","summary":"Fix an integer n and choose a natural number m such that m + n > 0. By \\refcor:up_and_down_triv…","labels":[],"detail_key":"p35"},{"id":"n31884","layer":"informal","project":"p35","title":"def:splitting_module","kind":"definition","summary":"The splitting module of \\sigma' is the R-module M \\times aug(R,G), with the action of an elemen…","labels":["def:splitting_module"],"detail_key":"p35"},{"id":"n31885","layer":"informal","project":"p35","title":"lem:splits_in_splitting_module","kind":"lemma","summary":"The image of \\sigma' in H^2(G,split(\\sigma)) is zero.","labels":["lem:splits_in_splitting_module"],"detail_key":"p35"},{"id":"n31886","layer":"informal","project":"p35","title":"We can check that the cocycle \\sigma' is the coboundary of the 1-cochain \\tau : G \\to spl…","kind":"proof","summary":"We can check that the cocycle \\sigma' is the coboundary of the 1-cochain \\tau : G \\to split(\\si…","labels":[],"detail_key":"p35"},{"id":"n31887","layer":"informal","project":"p35","title":"def:fundamental_class","kind":"definition","summary":"In this section G is a finite group and R is a commutative ring. If M is a representation of G…","labels":["def:fundamental_class"],"detail_key":"p35"},{"id":"n31888","layer":"informal","project":"p35","title":"\\item Given G and R, there can be several nonisomorphic M which are finite class formatio…","kind":"remark","summary":"\\item Given G and R, there can be several nonisomorphic M which are finite class formations. Fo…","labels":[],"detail_key":"p35"},{"id":"n31889","layer":"informal","project":"p35","title":"If G is a finite cyclic group then the trivial representation Z (or more precisely the tr…","kind":"example","summary":"If G is a finite cyclic group then the trivial representation Z (or more precisely the triple (…","labels":[],"detail_key":"p35"},{"id":"n31890","layer":"informal","project":"p35","title":"lem:linear_injective_of_surjective","kind":"lemma","summary":"\\mathlibok Let I be an ideal of a commutative ring R and let f:R/I \\to R/I be a surjective R-li…","labels":["lem:linear_injective_of_surjective"],"detail_key":"p35"},{"id":"n31891","layer":"informal","project":"p35","title":"\\mathlibok (This lemma is already in Mathlib.) Without loss of generality I=0, since such…","kind":"proof","summary":"\\mathlibok (This lemma is already in Mathlib.) Without loss of generality I=0, since such a map…","labels":[],"detail_key":"p35"},{"id":"n31892","layer":"informal","project":"p35","title":"In the case R= Z and I\\not=0, which is the only case we shall use in applications, the le…","kind":"remark","summary":"In the case R= Z and I\\not=0, which is the only case we shall use in applications, the lemma ca…","labels":[],"detail_key":"p35"},{"id":"n31893","layer":"informal","project":"p35","title":"lem:restriction_fundamental_class_generates","kind":"lemma","summary":"Let \\sigma \\in H^2(G,M) be a fundamental class. Then the restriction of \\sigma to any subgroup…","labels":["lem:restriction_fundamental_class_generates"],"detail_key":"p35"},{"id":"n31894","layer":"informal","project":"p35","title":"The restriction and corestriction maps are R-linear maps \\[ H^2(G,M) \\stackrelrest\\to H^2…","kind":"proof","summary":"The restriction and corestriction maps are R-linear maps \\[ H^2(G,M) \\stackrelrest\\to H^2(S,M)…","labels":[],"detail_key":"p35"},{"id":"n31895","layer":"informal","project":"p35","title":"Again when R= Z the above proof can be made much simpler.","kind":"remark","summary":"Again when R= Z the above proof can be made much simpler.","labels":[],"detail_key":"p35"},{"id":"n31896","layer":"informal","project":"p35","title":"thm:splitting_module_trivial","kind":"theorem","summary":"Let \\sigma' be a 2-cocycle representing a fundamental class in H^2(G,M). Then split(\\sigma') ha…","labels":["thm:splitting_module_trivial"],"detail_key":"p35"},{"id":"n31897","layer":"informal","project":"p35","title":"By \\refthm:triviality_criterion, it's enough to prove for every subgroup S of G that H^1(…","kind":"proof","summary":"By \\refthm:triviality_criterion, it's enough to prove for every subgroup S of G that H^1(S,spli…","labels":[],"detail_key":"p35"},{"id":"n31898","layer":"informal","project":"p35","title":"def:reciprocity_iso","kind":"definition","summary":"The theorem implies that we have isomorphisms for all n\\in Z (which depend on \\sigma): \\[ H^n_T…","labels":["def:reciprocity_iso"],"detail_key":"p35"},{"id":"n31899","layer":"informal","project":"p35","title":"lem:reciprocity_formula","kind":"lemma","summary":"The reciprocity isomorphism for a fundamental class \\sigma \\in H^2(G,M) is given by \\[ reciproc…","labels":["lem:reciprocity_formula"],"detail_key":"p35"},{"id":"n31900","layer":"informal","project":"p35","title":"We have a diagram with exact rows. Note that C^0(G,M) and C^-1(G,M) are both M and the ve…","kind":"proof","summary":"We have a diagram with exact rows. Note that C^0(G,M) and C^-1(G,M) are both M and the vertical…","labels":[],"detail_key":"p35"},{"id":"n31901","layer":"informal","project":"p35","title":"lem:subgroup_compatibility","kind":"lemma","summary":"Let (R,G,M) be a finite class formation with a fundamental class \\sigma_G and let S be a subgro…","labels":["lem:subgroup_compatibility"],"detail_key":"p35"},{"id":"n31902","layer":"informal","project":"p35","title":"The fact that (R, S, M \\downarrow S) is a finite class formation is a tautology. The fact…","kind":"proof","summary":"The fact that (R, S, M \\downarrow S) is a finite class formation is a tautology. The fact that…","labels":[],"detail_key":"p35"},{"id":"n31903","layer":"informal","project":"p35","title":"def:norm_submodule","kind":"definition","summary":"For a finite subgroup S of G, we shall call N_G/SM^S the \\emphnorm submodule corresponding to S…","labels":["def:norm_submodule"],"detail_key":"p35"},{"id":"n31904","layer":"informal","project":"p35","title":"cor:norm_submodule_mono","kind":"corollary","summary":"Let ( Z,G,M) be a finite class formation and let S_1 and S_2 be two subgroups of G. Then S_1 G'…","labels":["cor:norm_submodule_mono"],"detail_key":"p35"},{"id":"n31905","layer":"informal","project":"p35","title":"Norm Limitation Theorem","kind":"corollary","summary":"[Norm Limitation Theorem] Let (R,G,M) be a finite class formation. Then \\[ N_G M = N_G/G' (M^G'…","labels":["cor:norm_limitiation"],"detail_key":"p35"},{"id":"n31906","layer":"informal","project":"p35","title":"cor:abelian_norm_submodule_mono","kind":"corollary","summary":"Suppose ( Z,G,M) is a finite class formation with G abelian. Let S_1 and S_2 be subgroups of G.…","labels":["cor:abelian_norm_submodule_mono"],"detail_key":"p35"},{"id":"n31907","layer":"informal","project":"p35","title":"thm:hilbert_90","kind":"theorem","summary":"\\mathlibok Let l/k be a finite Galois extension of fields. Then H^1(l/k, l^\\times) \\cong 0.","labels":["thm:hilbert_90"],"detail_key":"p35"},{"id":"n31908","layer":"informal","project":"p35","title":"\\mathlibok This is already in Mathlib.","kind":"proof","summary":"\\mathlibok This is already in Mathlib.","labels":[],"detail_key":"p35"},{"id":"n31909","layer":"informal","project":"p35","title":"thm:additive_field_trivial","kind":"theorem","summary":"Let l/k be a finite Galois extension of fields. Then there is an isomorphism of Gal(l/k)-repres…","labels":["thm:additive_field_trivial"],"detail_key":"p35"},{"id":"n31910","layer":"informal","project":"p35","title":"Recall from Galois theory that there is a normal basis for l over k, i.e. a basis of the…","kind":"proof","summary":"Recall from Galois theory that there is a normal basis for l over k, i.e. a basis of the form \\…","labels":[],"detail_key":"p35"},{"id":"n31911","layer":"informal","project":"p35","title":"lem:serre_approx","kind":"lemma","summary":"Let G be a group and let M be a G-module. Say we have a decreasing sequence M=M_0\\supseteq M_1\\…","labels":["lem:serre_approx"],"detail_key":"p35"},{"id":"n31912","layer":"informal","project":"p35","title":"Let f_0 be a q+1-cocycle with values in M=M_0. Since H^q+1(G,M/M_1)=0 we can find a q-coc…","kind":"proof","summary":"Let f_0 be a q+1-cocycle with values in M=M_0. Since H^q+1(G,M/M_1)=0 we can find a q-cochain \\…","labels":[],"detail_key":"p35"},{"id":"n31913","layer":"informal","project":"p35","title":"What the proof above seems to use is: (1) group cohomology of M is computed as cohomology…","kind":"remark","summary":"What the proof above seems to use is: (1) group cohomology of M is computed as cohomology of a…","labels":[],"detail_key":"p35"},{"id":"n31914","layer":"informal","project":"p35","title":"lem:serre_cor","kind":"lemma","summary":"Let l/k be a finite Galois extension of local fields. There is a Galois-invariant open subgroup…","labels":["lem:serre_cor"],"detail_key":"p35"},{"id":"n31915","layer":"informal","project":"p35","title":"We know that l\\cong k[G] by the normal basis theorem so we may choose \\alpha\\in L such th…","kind":"proof","summary":"We know that l\\cong k[G] by the normal basis theorem so we may choose \\alpha\\in L such that \\g\\…","labels":[],"detail_key":"p35"},{"id":"n31916","layer":"informal","project":"p35","title":"lem:herbrand_local_units","kind":"lemma","summary":"If l/k is a cyclic extension then h(l/k, O_l^\\times) = 1.","labels":["lem:herbrand_local_units"],"detail_key":"p35"},{"id":"n31917","layer":"informal","project":"p35","title":"Choose a subgroup M\\subseteq O_l^\\times as in Lemma~\\reflem:serre_cor. Since O_l^\\times /…","kind":"proof","summary":"Choose a subgroup M\\subseteq O_l^\\times as in Lemma~\\reflem:serre_cor. Since O_l^\\times / M is…","labels":[],"detail_key":"p35"},{"id":"n31918","layer":"informal","project":"p35","title":"lem:herbrand_local_l*","kind":"lemma","summary":"If l/k is a cyclic extension of local fields then h(l/k, l^\\times)= [l:k].","labels":["lem:herbrand_local_l*"],"detail_key":"p35"},{"id":"n31919","layer":"informal","project":"p35","title":"We have a short exact sequence of representations \\[ 0 \\to O_l^\\times \\to l^\\times \\to Z\\…","kind":"proof","summary":"We have a short exact sequence of representations \\[ 0 \\to O_l^\\times \\to l^\\times \\to Z\\to 0,…","labels":[],"detail_key":"p35"},{"id":"n31920","layer":"informal","project":"p35","title":"lem:local_H2_l*","kind":"lemma","summary":"If l/k is a cyclic extension of local fields then |H^2(l/k,l^\\times)| = [l:k].","labels":["lem:local_H2_l*"],"detail_key":"p35"},{"id":"n31921","layer":"informal","project":"p35","title":"The follows from \\reflem:herbrand_local_l* and \\refthm:hilbert_90.","kind":"proof","summary":"The follows from \\reflem:herbrand_local_l* and \\refthm:hilbert_90.","labels":[],"detail_key":"p35"},{"id":"n31922","layer":"informal","project":"p35","title":"lem:local_H2_upper_bound","kind":"theorem","summary":"Let l/k be a Galois extension of local fields. Then |H^2(l/k,l^\\times)| \\le [l:k].","labels":["lem:local_H2_upper_bound"],"detail_key":"p35"},{"id":"n31923","layer":"informal","project":"p35","title":"Let p be prime number dividing the degree [l:k] and let k_p be the fixed field of a Sylow…","kind":"proof","summary":"Let p be prime number dividing the degree [l:k] and let k_p be the fixed field of a Sylow p-sub…","labels":[],"detail_key":"p35"},{"id":"n31924","layer":"informal","project":"p35","title":"lem:finite_field_trivial","kind":"lemma","summary":"The Galois modules F_l and F_l^\\times have trivial cohomology.","labels":["lem:finite_field_trivial"],"detail_key":"p35"},{"id":"n31925","layer":"informal","project":"p35","title":"In the case of l, this follows from \\refthm:additive_field_trivial as Gal(l/k) may be ide…","kind":"proof","summary":"In the case of l, this follows from \\refthm:additive_field_trivial as Gal(l/k) may be identifie…","labels":[],"detail_key":"p35"},{"id":"n31926","layer":"informal","project":"p35","title":"lem:unramified_additive_trivial","kind":"lemma","summary":"If l/k is unramified then there is a normal basis for O_l over O_k. Hence there is an isomorphi…","labels":["lem:unramified_additive_trivial"],"detail_key":"p35"},{"id":"n31927","layer":"informal","project":"p35","title":"By \\refthm:additive_field_trivial we may choose x_0 \\in F_l such that \\g \\bullet x_0 :g \\…","kind":"proof","summary":"By \\refthm:additive_field_trivial we may choose x_0 \\in F_l such that \\g \\bullet x_0 :g \\in Gal…","labels":[],"detail_key":"p35"},{"id":"n31928","layer":"informal","project":"p35","title":"lem:unramified_units_trivial","kind":"lemma","summary":"If l/k is unramified then O_l^\\times has trivial group cohomology.","labels":["lem:unramified_units_trivial"],"detail_key":"p35"},{"id":"n31929","layer":"informal","project":"p35","title":"Because replacing Gal(l/k) with a subgroup is the same as replacing k with an extension,…","kind":"proof","summary":"Because replacing Gal(l/k) with a subgroup is the same as replacing k with an extension, it suf…","labels":[],"detail_key":"p35"},{"id":"n31930","layer":"informal","project":"p35","title":"cor:cohomology_unramified_iso_cohomology_Z","kind":"corollary","summary":"Let l/k be an unramified extension of local fields. Then there are isomorphisms \\[ H^\\bullet_Ta…","labels":["cor:cohomology_unramified_iso_cohomology_Z"],"detail_key":"p35"},{"id":"n31931","layer":"informal","project":"p35","title":"This follows from the long exact sequence using \\reflem:unramified_units_trivial.","kind":"proof","summary":"This follows from the long exact sequence using \\reflem:unramified_units_trivial.","labels":[],"detail_key":"p35"},{"id":"n31932","layer":"informal","project":"p35","title":"lem:unramified_fundamental_class","kind":"lemma","summary":"Let l/k be an unramified cyclic extension of local fields. Then H^2(l/k,l^\\times) is cyclic of…","labels":["lem:unramified_fundamental_class"],"detail_key":"p35"},{"id":"n31933","layer":"informal","project":"p35","title":"This follows from \\refcor:cohomology_unramified_iso_cohomology_Z together with the descri…","kind":"proof","summary":"This follows from \\refcor:cohomology_unramified_iso_cohomology_Z together with the description…","labels":[],"detail_key":"p35"},{"id":"n31934","layer":"informal","project":"p35","title":"lem:local_unramified_reciprocity","kind":"lemma","summary":"Let l/k be a finite unramified extension of local fields and let F_k be the Frobenius element i…","labels":["lem:local_unramified_reciprocity"],"detail_key":"p35"},{"id":"n31935","layer":"informal","project":"p35","title":"This follows from \\reflem:reciprocity_formula and \\reflem:unramified_fundamental_class.","kind":"proof","summary":"This follows from \\reflem:reciprocity_formula and \\reflem:unramified_fundamental_class.","labels":[],"detail_key":"p35"},{"id":"n31936","layer":"informal","project":"p35","title":"def:unramified_local_inv","kind":"definition","summary":"The \\emphlocal invariant inv_l/k : H^2(l/k, l^\\times) \\cong \\frac1[l:k] Z/ Z is the composition…","labels":["def:unramified_local_inv"],"detail_key":"p35"},{"id":"n31937","layer":"informal","project":"p35","title":"lem:local_inv_fundamental_class","kind":"lemma","summary":"The fundamental class \\sigma_l/k has local invariant \\frac1[l:k].","labels":["lem:local_inv_fundamental_class"],"detail_key":"p35"},{"id":"n31938","layer":"informal","project":"p35","title":"Corollary of inv_Gal(l/k)(\\sigma_Gal(l/k)) = 1 from \\reflem:local_inv_iso.","kind":"proof","summary":"Corollary of inv_Gal(l/k)(\\sigma_Gal(l/k)) = 1 from \\reflem:local_inv_iso.","labels":[],"detail_key":"p35"},{"id":"n31939","layer":"informal","project":"p35","title":"lem:local_unram_rest","kind":"lemma","summary":"Let m / l / k be an unramified tower of extensions of local fields Then the restriction to m/l…","labels":["lem:local_unram_rest"],"detail_key":"p35"},{"id":"n31940","layer":"informal","project":"p35","title":"Up to cohomology, \\sigma_l/k does not depend on the choice of uniformizer, so we may assu…","kind":"proof","summary":"Up to cohomology, \\sigma_l/k does not depend on the choice of uniformizer, so we may assume \\pi…","labels":[],"detail_key":"p35"},{"id":"n31941","layer":"informal","project":"p35","title":"lem:local_unram_inv_infl","kind":"lemma","summary":"Let m / l / k be a tower of unramified extensions of local fields and let infl: H^2(l/k,l^\\time…","labels":["lem:local_unram_inv_infl"],"detail_key":"p35"},{"id":"n31942","layer":"informal","project":"p35","title":"We shall write F_m/k and F_l/k for the Frobenius elements in Gal(m/k) and Gal(l/k) respec…","kind":"proof","summary":"We shall write F_m/k and F_l/k for the Frobenius elements in Gal(m/k) and Gal(l/k) respectively…","labels":[],"detail_key":"p35"},{"id":"n31943","layer":"informal","project":"p35","title":"thm:local_fund_class","kind":"theorem","summary":"For every finite Galois extension l/k of local fields, ( Z,Gal(l/k),l^\\times) is a finite class…","labels":["thm:local_fund_class"],"detail_key":"p35"},{"id":"n31944","layer":"informal","project":"p35","title":"From \\refthm:hilbert_90 we know that H^1(l/k,l^\\times) \\cong 0. From \\reflem:local_H2_upp…","kind":"proof","summary":"From \\refthm:hilbert_90 we know that H^1(l/k,l^\\times) \\cong 0. From \\reflem:local_H2_upper_bou…","labels":[],"detail_key":"p35"},{"id":"n31945","layer":"informal","project":"p35","title":"thm:local_norm_limitation","kind":"theorem","summary":"Let l/k be a finite Galois extension of local fields and let l^ab be the maximal subfield of l…","labels":["thm:local_norm_limitation"],"detail_key":"p35"},{"id":"n31946","layer":"informal","project":"p35","title":"This follows from \\refcor:norm_limitiation and \\refthm:local_fund_class.","kind":"proof","summary":"This follows from \\refcor:norm_limitiation and \\refthm:local_fund_class.","labels":[],"detail_key":"p35"},{"id":"n31947","layer":"informal","project":"p35","title":"lem:local_abelian_classification","kind":"theorem","summary":"Let l_1 and l_2 be two abelian extensions of k contained in a field m. Then l_1 \\subseteq l_2 i…","labels":["lem:local_abelian_classification"],"detail_key":"p35"},{"id":"n31948","layer":"informal","project":"p35","title":"This follows from \\refcor:norm_submodule_mono.","kind":"proof","summary":"This follows from \\refcor:norm_submodule_mono.","labels":[],"detail_key":"p35"},{"id":"n31949","layer":"informal","project":"p35","title":"lem:norm_composite_intersection","kind":"lemma","summary":"Let l/k be a finite abelian extension and let m_1 and m_2 be two intermediate fields between k…","labels":["lem:norm_composite_intersection"],"detail_key":"p35"},{"id":"n31950","layer":"informal","project":"p35","title":"Clearly if x \\in N_l/k(l^\\times) then x is a norm from both m_1 and m_2. Suppose converse…","kind":"proof","summary":"Clearly if x \\in N_l/k(l^\\times) then x is a norm from both m_1 and m_2. Suppose conversely tha…","labels":[],"detail_key":"p35"},{"id":"n31951","layer":"informal","project":"p35","title":"lem:local_unramified_norms","kind":"lemma","summary":"Let l/k be an unramified extension of local fields of degree f. Then N(l^\\times) = \\pi_k^f Z \\t…","labels":["lem:local_unramified_norms"],"detail_key":"p35"},{"id":"n31952","layer":"informal","project":"p35","title":"By \\reflem:unramified_units_trivial we have H^0_Tate(l/k, O_l^\\times) \\cong 0, which impl…","kind":"proof","summary":"By \\reflem:unramified_units_trivial we have H^0_Tate(l/k, O_l^\\times) \\cong 0, which implies th…","labels":[],"detail_key":"p35"},{"id":"n31953","layer":"informal","project":"p35","title":"lem:local_image_inertia","kind":"lemma","summary":"Let l/k be a finite abelian extension of local fields and let I \\subseteq Gal(l/k) be the inert…","labels":["lem:local_image_inertia"],"detail_key":"p35"},{"id":"n31954","layer":"informal","project":"p35","title":"Let m be the fixed field of I. By \\reflem:subgroup_compatibility the image of I in k^\\tim…","kind":"proof","summary":"Let m be the fixed field of I. By \\reflem:subgroup_compatibility the image of I in k^\\times / N…","labels":[],"detail_key":"p35"},{"id":"n31955","layer":"informal","project":"p35","title":"lem:local_isomorphism","kind":"lemma","summary":"Assume l/k is a finite Galois extension of characteristic zero local fields, and let P be the m…","labels":["lem:local_isomorphism"],"detail_key":"p35"},{"id":"n31956","layer":"informal","project":"p35","title":"Choose n large enough so that \\exp(x) converges for all x \\in P^n, and such that all the…","kind":"proof","summary":"Choose n large enough so that \\exp(x) converges for all x \\in P^n, and such that all the terms…","labels":[],"detail_key":"p35"},{"id":"n31957","layer":"informal","project":"p35","title":"lem:local_cyclotomic_norms","kind":"lemma","summary":"Let l = Q_p(\\zeta) where \\zeta is a primitive p^n-th root of unity for some n > 0. Then N(l^\\ti…","labels":["lem:local_cyclotomic_norms"],"detail_key":"p35"},{"id":"n31958","layer":"informal","project":"p35","title":"By Eisenstein's criterion, the cyclotomic polynomial \\Phi_p^n(X) = \\fracX^p^n-1X^p^n-1-1…","kind":"proof","summary":"By Eisenstein's criterion, the cyclotomic polynomial \\Phi_p^n(X) = \\fracX^p^n-1X^p^n-1-1 is irr…","labels":[],"detail_key":"p35"},{"id":"n31959","layer":"informal","project":"p35","title":"thm:local_Kronecker_Weber","kind":"theorem","summary":"Let l/ Q_p be a finite abelian extension. Then l is isomorphic to a subfield of a cyclotomic ex…","labels":["thm:local_Kronecker_Weber"],"detail_key":"p35"},{"id":"n31960","layer":"informal","project":"p35","title":"The subgroup N_l/ Q_p(l^\\times) is open in Q_p^\\times, so it must contain a subgroup of t…","kind":"proof","summary":"The subgroup N_l/ Q_p(l^\\times) is open in Q_p^\\times, so it must contain a subgroup of the for…","labels":[],"detail_key":"p35"},{"id":"n31961","layer":"informal","project":"p35","title":"thm:Kronecker_Weber","kind":"theorem","summary":"Let l/ Q be a finite abelian extension. Then there exists a natural number n such that l is iso…","labels":["thm:Kronecker_Weber"],"detail_key":"p35"},{"id":"n31962","layer":"informal","project":"p35","title":"Let S be a the set of primes which ramify in l. For each p \\in S we let \\hat p be a prime…","kind":"proof","summary":"Let S be a the set of primes which ramify in l. For each p \\in S we let \\hat p be a prime of l…","labels":[],"detail_key":"p35"},{"id":"n31963","layer":"informal","project":"p35","title":"lem:idele_class_invariants","kind":"lemma","summary":"Let l/k be a finite Galois extension of number fields (or even global fields). The map Cl_k \\to…","labels":["lem:idele_class_invariants"],"detail_key":"p35"},{"id":"n31964","layer":"informal","project":"p35","title":"We have a short exact sequence \\[ 0 \\to l^\\times \\to A_l^\\times \\to Cl_l \\to 0. \\] Taking…","kind":"proof","summary":"We have a short exact sequence \\[ 0 \\to l^\\times \\to A_l^\\times \\to Cl_l \\to 0. \\] Taking Gal(l…","labels":[],"detail_key":"p35"},{"id":"n31965","layer":"informal","project":"p35","title":"lem:semi-local_iso_coind","kind":"lemma","summary":"Then there are isomorphisms \\[ \\prod_w | v l_w^\\times \\cong coind_D_\\hat v^Gal(l/k) l_\\hat v^\\t…","labels":["lem:semi-local_iso_coind"],"detail_key":"p35"},{"id":"n31966","layer":"informal","project":"p35","title":"We'll write x for an element of \\prod_w | v l_w^\\times and x_w for it's component in l_w.…","kind":"proof","summary":"We'll write x for an element of \\prod_w | v l_w^\\times and x_w for it's component in l_w. Defin…","labels":[],"detail_key":"p35"},{"id":"n31967","layer":"informal","project":"p35","title":"lem:cohomology_S-ideles_decomp","kind":"lemma","summary":"There are isomorphisms for all n > 0 \\[ H^n(l/k, A_S,l^\\times) \\cong \\prod_v \\in S H^n(l_\\hat v…","labels":["lem:cohomology_S-ideles_decomp"],"detail_key":"p35"},{"id":"n31968","layer":"informal","project":"p35","title":"We note that by \\reflem:semi-local_iso_coind we have \\[ A_S,l^\\times \\cong \\prod_v \\in S…","kind":"proof","summary":"We note that by \\reflem:semi-local_iso_coind we have \\[ A_S,l^\\times \\cong \\prod_v \\in S coind_…","labels":[],"detail_key":"p35"},{"id":"n31969","layer":"informal","project":"p35","title":"lem:herbrand_S-ideles","kind":"lemma","summary":"If l/k is a cyclic extension then we have \\[ h(l/k, A_S,l ^\\times ) = \\prod_v \\in S |D_\\hat v|.…","labels":["lem:herbrand_S-ideles"],"detail_key":"p35"},{"id":"n31970","layer":"informal","project":"p35","title":"This follows from \\reflem:cohomology_S-ideles_decomp and \\reflem:herbrand_local_l*.","kind":"proof","summary":"This follows from \\reflem:cohomology_S-ideles_decomp and \\reflem:herbrand_local_l*.","labels":[],"detail_key":"p35"},{"id":"n31971","layer":"informal","project":"p35","title":"lem:herbrand_L_S","kind":"lemma","summary":"If l/k is a cyclic extension then h(l/k,L_S) = \\prod_v \\in S |D_\\hat v|.","labels":["lem:herbrand_L_S"],"detail_key":"p35"},{"id":"n31972","layer":"informal","project":"p35","title":"This follows from Shapiro's lemma (\\reflem:Shapiro) together with the calculation of the…","kind":"proof","summary":"This follows from Shapiro's lemma (\\reflem:Shapiro) together with the calculation of the cohomo…","labels":[],"detail_key":"p35"},{"id":"n31973","layer":"informal","project":"p35","title":"lem:herbrand_log_lattice","kind":"lemma","summary":"Let l/k be cyclic and let M be any Galois-invariant lattice in V_S. Then h(l/k,M) = \\prod_v \\in…","labels":["lem:herbrand_log_lattice"],"detail_key":"p35"},{"id":"n31974","layer":"informal","project":"p35","title":"The representations M \\otimes Q and L_S \\otimes Q have the same character (this is just t…","kind":"proof","summary":"The representations M \\otimes Q and L_S \\otimes Q have the same character (this is just the cha…","labels":[],"detail_key":"p35"},{"id":"n31975","layer":"informal","project":"p35","title":"thm:Dirichlet_unit_theorem","kind":"theorem","summary":"\\log_S( O_S^\\times) has zero intersection with Span(1,1,\\ldots,1). The direct sum of these subr…","labels":["thm:Dirichlet_unit_theorem"],"detail_key":"p35"},{"id":"n31976","layer":"informal","project":"p35","title":"An equivalent statement is already in Mathlib as \\textttNumberField.Units.dirichletUnitTh…","kind":"proof","summary":"An equivalent statement is already in Mathlib as \\textttNumberField.Units.dirichletUnitTheorem.…","labels":[],"detail_key":"p35"},{"id":"n31977","layer":"informal","project":"p35","title":"lem:herbrand_S-units","kind":"corollary","summary":"Let l/k be a cyclic extension. Then \\[ h(l/k, O_l,S^\\times) = \\frac\\prod_v\\in S |D_\\hat v|[l:k]…","labels":["lem:herbrand_S-units"],"detail_key":"p35"},{"id":"n31978","layer":"informal","project":"p35","title":"Since \\log_S has finite kernel, the Herbrand quotient of O_l,S^\\times is equal to that of…","kind":"proof","summary":"Since \\log_S has finite kernel, the Herbrand quotient of O_l,S^\\times is equal to that of \\log_…","labels":[],"detail_key":"p35"},{"id":"n31979","layer":"informal","project":"p35","title":"lem:herbrand_idele_class_group","kind":"corollary","summary":"If l/k is cyclic then h(l/k,Cl_l) = [l:k].","labels":["lem:herbrand_idele_class_group"],"detail_key":"p35"},{"id":"n31980","layer":"informal","project":"p35","title":"Our choice of S implies Cl_l \\cong A_l,S^\\times / O_l,S^\\times. We have calculated the He…","kind":"proof","summary":"Our choice of S implies Cl_l \\cong A_l,S^\\times / O_l,S^\\times. We have calculated the Herbrand…","labels":[],"detail_key":"p35"},{"id":"n31981","layer":"informal","project":"p35","title":"def:Dirichlet_density","kind":"definition","summary":"Let M be a set of primes of O_k. We'll say that M has a \\emphDirichlet density c \\in R if \\[ \\s…","labels":["def:Dirichlet_density"],"detail_key":"p35"},{"id":"n31982","layer":"informal","project":"p35","title":"lem:Dirichlet_density_union","kind":"lemma","summary":"Suppose M_1 and M_2 are disjoint sets of primes of O_l. If two of the sets M_1, M_2, M_1 \\cup M…","labels":["lem:Dirichlet_density_union"],"detail_key":"p35"},{"id":"n31983","layer":"informal","project":"p35","title":"This is trivial.","kind":"proof","summary":"This is trivial.","labels":[],"detail_key":"p35"},{"id":"n31984","layer":"informal","project":"p35","title":"lem:Dirichlet_density_top","kind":"lemma","summary":"The set of all primes of O_k has Dirichlet density 1.","labels":["lem:Dirichlet_density_top"],"detail_key":"p35"},{"id":"n31985","layer":"informal","project":"p35","title":"Let P be a prime. For s > 1 we have \\[ \\left|N(P)^-s - \\log\\left( \\frac11-N(P)^-s\\right)\\…","kind":"proof","summary":"Let P be a prime. For s > 1 we have \\[ \\left|N(P)^-s - \\log\\left( \\frac11-N(P)^-s\\right)\\right|…","labels":[],"detail_key":"p35"},{"id":"n31986","layer":"informal","project":"p35","title":"lem:Dirichlet_density_degree_one","kind":"lemma","summary":"The set of primes of O_l of degree one has Dirichlet density 1.","labels":["lem:Dirichlet_density_degree_one"],"detail_key":"p35"},{"id":"n31987","layer":"informal","project":"p35","title":"Let M be the set of primes of degree larger than one. It's sufficient to prove that M has…","kind":"proof","summary":"Let M be the set of primes of degree larger than one. It's sufficient to prove that M has Diric…","labels":[],"detail_key":"p35"},{"id":"n31988","layer":"informal","project":"p35","title":"lem:Dirichlet_density_split","kind":"lemma","summary":"Let l/k be a finite Galois extension of number fields. Then the set of degree 1 primes of k whi…","labels":["lem:Dirichlet_density_split"],"detail_key":"p35"},{"id":"n31989","layer":"informal","project":"p35","title":"Let M_k be the set of degree 1 primes of k and M_l the set of degree 1 primes of l. Every…","kind":"proof","summary":"Let M_k be the set of degree 1 primes of k and M_l the set of degree 1 primes of l. Every prime…","labels":[],"detail_key":"p35"},{"id":"n31990","layer":"informal","project":"p35","title":"lem:H0_idele_class_group_finite","kind":"lemma","summary":"H^0_Tate(l/k,Cl_l) is finite.","labels":["lem:H0_idele_class_group_finite"],"detail_key":"p35"},{"id":"n31991","layer":"informal","project":"p35","title":"We have Cl_k / N(Cl_l) & \\cong A_k,S^\\times / O_k,S^\\times N( A_l,S^\\times) \\\\ & \\cong \\l…","kind":"proof","summary":"We have Cl_k / N(Cl_l) & \\cong A_k,S^\\times / O_k,S^\\times N( A_l,S^\\times) \\\\ & \\cong \\left(\\p…","labels":[],"detail_key":"p35"},{"id":"n31992","layer":"informal","project":"p35","title":"def:L-function","kind":"definition","summary":"Let \\chi : H^0_Tate(l/k,Cl_l) \\to C^\\times be a character. For a non-zero ideal I of O_k,S, we…","labels":["def:L-function"],"detail_key":"p35"},{"id":"n31993","layer":"informal","project":"p35","title":"Weak lemma","kind":"lemma","summary":"[Weak lemma] If \\chi is a non-trivial character then L(s,\\chi) is bounded on the interval (1,2).","labels":["lem:L-function_bound"],"detail_key":"p35"},{"id":"n31994","layer":"informal","project":"p35","title":"lem:density_bound","kind":"lemma","summary":"Let M be a set of primes of O_S whose image in H^0_Tate(l/k,Cl_l) is zero. There exists a real…","labels":["lem:density_bound"],"detail_key":"p35"},{"id":"n31995","layer":"informal","project":"p35","title":"Let s > 1. All the series in the following calculation converge absolutely in this region…","kind":"proof","summary":"Let s > 1. All the series in the following calculation converge absolutely in this region. The…","labels":[],"detail_key":"p35"},{"id":"n31996","layer":"informal","project":"p35","title":"In fact the density of the set M in this lemma is precisely \\frac1|H^0_Tate(l/k,Cl_l)|. T…","kind":"remark","summary":"In fact the density of the set M in this lemma is precisely \\frac1|H^0_Tate(l/k,Cl_l)|. This ca…","labels":[],"detail_key":"p35"},{"id":"n31997","layer":"informal","project":"p35","title":"thm:first_inequality","kind":"theorem","summary":"For any finite Galois extension l/k be have \\[ |H^0_Tate(l/k, Cl_l) | \\le [l : k]. \\]","labels":["thm:first_inequality"],"detail_key":"p35"},{"id":"n31998","layer":"informal","project":"p35","title":"Let M_1 be the set of degree 1 primes of O_k,S which split in l and let M_2 be the set of…","kind":"proof","summary":"Let M_1 be the set of degree 1 primes of O_k,S which split in l and let M_2 be the set of prime…","labels":[],"detail_key":"p35"},{"id":"n31999","layer":"informal","project":"p35","title":"cor:H1_H2_cyclic_idele_class","kind":"corollary","summary":"If l/k is cyclic then |H^2(l/k, Cl_l)| = [l:k] and H^1(l/k, Cl_l) = 0.","labels":["cor:H1_H2_cyclic_idele_class"],"detail_key":"p35"},{"id":"n32000","layer":"informal","project":"p35","title":"This follows immediately from (a) the first inequality, (b) the periodicity of the cohomo…","kind":"proof","summary":"This follows immediately from (a) the first inequality, (b) the periodicity of the cohomology f…","labels":[],"detail_key":"p35"},{"id":"n32001","layer":"informal","project":"p35","title":"thm:global_cohomology_bound","kind":"theorem","summary":"If l/k is any finite Galois extension then H^1(l/k, Cl_l) \\cong 0 and |H^2(l/k, Cl_l)| \\le [l:k…","labels":["thm:global_cohomology_bound"],"detail_key":"p35"},{"id":"n32002","layer":"informal","project":"p35","title":"For each prime number p dividing [l:k] we let k_p be the fixed field of a Sylow p-subgrou…","kind":"proof","summary":"For each prime number p dividing [l:k] we let k_p be the fixed field of a Sylow p-subgroup S_p…","labels":[],"detail_key":"p35"},{"id":"n32003","layer":"formal","project":"p35","title":"Rep.aug","kind":"def","summary":"(R G : Type) → [inst : CommRing R] → [inst_1 : Group G] → Rep.0, 0, 0 R G","labels":[],"detail_key":"p35","name":"Rep.aug","module":"ClassFieldTheory.Cohomology.AugmentationModule"},{"id":"n32004","layer":"formal","project":"p35","title":"Rep.herbrandQuotient","kind":"def","summary":"R G : Type → [inst : CommRing R] → [inst_1 : Group G] → Rep.0, 0, 0 R G → Rat","labels":[],"detail_key":"p35","name":"Rep.herbrandQuotient","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.Defs"},{"id":"n32005","layer":"formal","project":"p35","title":"Rep.herbrandQuotient_of_finite","kind":"theorem","summary":"∀ R G : Type [inst : CommRing R] [inst_1 : Group G] [IsCyclic G] [Finite G] (M : Rep.0, 0, 0 R…","labels":[],"detail_key":"p35","name":"Rep.herbrandQuotient_of_finite","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.Finite"},{"id":"n32006","layer":"formal","project":"p35","title":"Representation.herbrandQuotient_of_finite","kind":"theorem","summary":"∀ R G A : Type [inst : CommRing R] [inst_1 : Group G] [inst_2 : IsCyclic G] [inst_3 : AddCommGr…","labels":[],"detail_key":"p35","name":"Representation.herbrandQuotient_of_finite","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.Finite"},{"id":"n32007","layer":"formal","project":"p35","title":"Rep.herbrandQuotient_eq_of_shortExact","kind":"theorem","summary":"∀ R G : Type [inst : CommRing R] [inst_1 : Group G] [Finite G] [IsCyclic G] S : CategoryTheory.…","labels":[],"detail_key":"p35","name":"Rep.herbrandQuotient_eq_of_shortExact","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.SES"},{"id":"n32008","layer":"formal","project":"p35","title":"Rep.herbrandQuotient_ne_zero_of_shortExact₁","kind":"theorem","summary":"∀ R G : Type [inst : CommRing R] [inst_1 : Group G] [Finite G] [IsCyclic G] S : CategoryTheory.…","labels":[],"detail_key":"p35","name":"Rep.herbrandQuotient_ne_zero_of_shortExact₁","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.SES"},{"id":"n32009","layer":"formal","project":"p35","title":"Rep.herbrandQuotient_ne_zero_of_shortExact₂","kind":"theorem","summary":"∀ R G : Type [inst : CommRing R] [inst_1 : Group G] [Finite G] [IsCyclic G] S : CategoryTheory.…","labels":[],"detail_key":"p35","name":"Rep.herbrandQuotient_ne_zero_of_shortExact₂","module":"ClassFieldTheory.Cohomology.FiniteCyclic.HerbrandQuotient.SES"},{"id":"n32010","layer":"formal","project":"p35","title":"Rep.herbrandQuotient_ne_zero_of_shortExact₃","kind":"theorem","summary":"∀ R G : Type [inst : CommRing R] [inst_1 : Group G] [Finite G] [IsCyclic G] S : 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Int","labels":[],"detail_key":"p35","name":"carry","module":"ClassFieldTheory.Cohomology.LocalInv"},{"id":"n32073","layer":"formal","project":"p35","title":"carryCocycle","kind":"def","summary":"(n : Nat) → Subtype fun x => Membership.mem (groupCohomology.cocycles₂ (Rep.trivial Int (Multip…","labels":[],"detail_key":"p35","name":"carryCocycle","module":"ClassFieldTheory.Cohomology.LocalInv"},{"id":"n32074","layer":"formal","project":"p35","title":"localInv","kind":"def","summary":"(n : Nat) → [NeZero n] → AddEquiv (↑(groupCohomology.H2 (Rep.trivial Int (Multiplicative (ZMod…","labels":[],"detail_key":"p35","name":"localInv","module":"ClassFieldTheory.Cohomology.LocalInv"},{"id":"n32075","layer":"formal","project":"p35","title":"localInvIso","kind":"def","summary":"(n : Nat) → [NeZero n] → CategoryTheory.Iso (groupCohomology.H2 (Rep.trivial Int (Multiplicativ…","labels":[],"detail_key":"p35","name":"localInvIso","module":"ClassFieldTheory.Cohomology.LocalInv"},{"id":"n32076","layer":"formal","project":"p35","title":"localInv_symm_apply","kind":"theorem","summary":"∀ (n : Nat) [inst : NeZero n] (i : ZMod n), Eq ((localInv n).symm i) (HSMul.hSMul i.val ((Modul…","labels":[],"detail_key":"p35","name":"localInv_symm_apply","module":"ClassFieldTheory.Cohomology.LocalInv"},{"id":"n32077","layer":"formal","project":"p35","title":"Rep.split.FiniteClassFormation","kind":"inductive","summary":"R : Type → [inst : CommRing R] → G : Type → [inst_1 : Group G] → M : Rep.0, 0, 0 R G → ↑(groupC…","labels":[],"detail_key":"p35","name":"Rep.split.FiniteClassFormation","module":"ClassFieldTheory.Cohomology.SplittingModule"},{"id":"n32078","layer":"formal","project":"p35","title":"Rep.split.reciprocityIso","kind":"def","summary":"G : Type → [inst : Group G] → [inst_1 : Fintype G] → (N : Rep.0, 0, 0 Int G) → (τ : ↑(groupCoho…","labels":[],"detail_key":"p35","name":"Rep.split.reciprocityIso","module":"ClassFieldTheory.Cohomology.SplittingModule"},{"id":"n32079","layer":"formal","project":"p35","title":"TateCohomology.negOneIso","kind":"def","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → [inst_2 : Fintype G] → (M : Rep R G)…","labels":[],"detail_key":"p35","name":"TateCohomology.negOneIso","module":"ClassFieldTheory.Cohomology.TateCohomology"},{"id":"n32080","layer":"formal","project":"p35","title":"TateCohomology.negOneIsoOfIsTrivial","kind":"def","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → [inst_2 : Fintype G] → (M : Rep R G)…","labels":[],"detail_key":"p35","name":"TateCohomology.negOneIsoOfIsTrivial","module":"ClassFieldTheory.Cohomology.TateCohomology"},{"id":"n32081","layer":"formal","project":"p35","title":"TateCohomology.zeroIso","kind":"def","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → [inst_2 : Fintype G] → (M : Rep R G)…","labels":[],"detail_key":"p35","name":"TateCohomology.zeroIso","module":"ClassFieldTheory.Cohomology.TateCohomology"},{"id":"n32082","layer":"formal","project":"p35","title":"TateCohomology.zeroIsoOfIsTrivial","kind":"def","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → [inst_2 : Fintype G] → (M : Rep R G)…","labels":[],"detail_key":"p35","name":"TateCohomology.zeroIsoOfIsTrivial","module":"ClassFieldTheory.Cohomology.TateCohomology"},{"id":"n32083","layer":"formal","project":"p35","title":"Rep.TrivialCohomology","kind":"inductive","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → Rep R G → Prop","labels":[],"detail_key":"p35","name":"Rep.TrivialCohomology","module":"ClassFieldTheory.Cohomology.TrivialCohomology"},{"id":"n32084","layer":"formal","project":"p35","title":"Rep.TrivialHomology","kind":"inductive","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → Rep R G → Prop","labels":[],"detail_key":"p35","name":"Rep.TrivialHomology","module":"ClassFieldTheory.Cohomology.TrivialCohomology"},{"id":"n32085","layer":"formal","project":"p35","title":"Rep.TrivialTateCohomology","kind":"inductive","summary":"R G : Type u → [inst : CommRing R] → [inst_1 : Group G] → [Finite G] → Rep R G → Prop","labels":[],"detail_key":"p35","name":"Rep.TrivialTateCohomology","module":"ClassFieldTheory.Cohomology.TrivialCohomology"},{"id":"n32086","layer":"formal","project":"p35","title":"Rep.instTrivialCohomologyOfSubsingleton","kind":"theorem","summary":"∀ R G : Type u [inst : CommRing R] [inst_1 : Group G] [Subsingleton G] M : Rep R G, M.TrivialCo…","labels":[],"detail_key":"p35","name":"Rep.instTrivialCohomologyOfSubsingleton","module":"ClassFieldTheory.Cohomology.TrivialCohomology"},{"id":"n32087","layer":"formal","project":"p35","title":"Rep.dimensionShift.down_trivialCohomology","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [Finite G] (M : Rep.0, 0, 0 R G) [M.…","labels":[],"detail_key":"p35","name":"Rep.dimensionShift.down_trivialCohomology","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32088","layer":"formal","project":"p35","title":"Rep.dimensionShift.up_trivialCohomology","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [Finite G] (M : Rep.0, 0, 0 R G) [M.…","labels":[],"detail_key":"p35","name":"Rep.dimensionShift.up_trivialCohomology","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32089","layer":"formal","project":"p35","title":"Rep.tateCohomology_of_trivialCohomology","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [inst_2 : Fintype G] (M : Rep.0, 0,…","labels":[],"detail_key":"p35","name":"Rep.tateCohomology_of_trivialCohomology","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32090","layer":"formal","project":"p35","title":"Rep.trivialHomology_of_trivialCohomology","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [Finite G] (M : Rep.0, 0, 0 R G) [M.…","labels":[],"detail_key":"p35","name":"Rep.trivialHomology_of_trivialCohomology","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32091","layer":"formal","project":"p35","title":"groupCohomology.trivialCohomology_of_even_of_odd","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [Finite G] (M : Rep.0, 0, 0 R G) (n…","labels":[],"detail_key":"p35","name":"groupCohomology.trivialCohomology_of_even_of_odd","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32092","layer":"formal","project":"p35","title":"groupCohomology.trivialCohomology_of_even_of_odd_of_solvable","kind":"theorem","summary":"∀ R : Type [inst : CommRing R] G : Type [inst_1 : Group G] [Finite G] [Group.IsSolvable G] (M :…","labels":[],"detail_key":"p35","name":"groupCohomology.trivialCohomology_of_even_of_odd_of_solvable","module":"ClassFieldTheory.Cohomology.TrivialityCriterion"},{"id":"n32093","layer":"formal","project":"p35","title":"Representation.norm","kind":"def","summary":"","labels":[],"detail_key":"p35","name":"Representation.norm","module":"obsolete.HerbrandQuotient_old"},{"id":"n32094","layer":"informal","project":"p36","title":"def:cdf","kind":"definition","summary":"A function F \\colon R\\to R is a \\emphcumulative distribution function (c.d.f.) if \\item[(i)] x…","labels":["def:cdf"],"detail_key":"p36"},{"id":"n32095","layer":"informal","project":"p36","title":"lem:cdf-of-random-var","kind":"lemma","summary":"If X is a real-valued random variable, then the function F \\colon R\\to R given by F(x) = \\maths…","labels":["lem:cdf-of-random-var"],"detail_key":"p36"},{"id":"n32096","layer":"informal","project":"p36","title":"Property (1.) in Definition~\\refdef:cdf is obvious (by monotonicity of measures) and prop…","kind":"proof","summary":"Property (1.) in Definition~\\refdef:cdf is obvious (by monotonicity of measures) and properties…","labels":[],"detail_key":"p36"},{"id":"n32097","layer":"informal","project":"p36","title":"def:degenerate-cdf","kind":"definition","summary":"A c.d.f. F is said to be \\emphdegenerate if for every x \\in R we have either F(x) = 0 or F(x) =…","labels":["def:degenerate-cdf"],"detail_key":"p36"},{"id":"n32098","layer":"informal","project":"p36","title":"lem:degenerate-cdf-iff-exists-jump","kind":"lemma","summary":"F is a degenerate c.d.f. if and only if there exists a x_0 \\in R such that F(x) = 0 & \\text for…","labels":["lem:degenerate-cdf-iff-exists-jump"],"detail_key":"p36"},{"id":"n32099","layer":"informal","project":"p36","title":"The ``if'' direction is clear. To prove the ``only if'' direction, assume that F is a deg…","kind":"proof","summary":"The ``if'' direction is clear. To prove the ``only if'' direction, assume that F is a degenerat…","labels":[],"detail_key":"p36"},{"id":"n32100","layer":"informal","project":"p36","title":"lem:delta-has-degenerate-cdf","kind":"lemma","summary":"The c.d.f. of Dirac delta mass \\delta_x_0 at x_0 \\in R is degenerate.","labels":["lem:delta-has-degenerate-cdf"],"detail_key":"p36"},{"id":"n32101","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32102","layer":"informal","project":"p36","title":"lem:degenerate-cdf-is-delta","kind":"lemma","summary":"If a c.d.f. F is degenerate, then it is the c.d.f. of a Dirac delta mass \\delta_x_0 at some poi…","labels":["lem:degenerate-cdf-is-delta"],"detail_key":"p36"},{"id":"n32103","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32104","layer":"informal","project":"p36","title":"lem:cdf-of-max-two","kind":"lemma","summary":"Let X and Y be two independent real-valued random variables with respective cumulative distribu…","labels":["lem:cdf-of-max-two"],"detail_key":"p36"},{"id":"n32105","layer":"informal","project":"p36","title":"Fix x \\in R. Note that \\max (X, Y) \\le x if and only if both X \\le x and Y \\le x. Calcula…","kind":"proof","summary":"Fix x \\in R. Note that \\max (X, Y) \\le x if and only if both X \\le x and Y \\le x. Calculate, us…","labels":[],"detail_key":"p36"},{"id":"n32106","layer":"informal","project":"p36","title":"lem:cdf-of-max-many","kind":"lemma","summary":"Let X_0, X_1, \\ldots, X_n-1 be independent identically distributed real-valued random variables…","labels":["lem:cdf-of-max-many"],"detail_key":"p36"},{"id":"n32107","layer":"informal","project":"p36","title":"Induction on~n using \\reflem:cdf-of-max-two.","kind":"proof","summary":"Induction on~n using \\reflem:cdf-of-max-two.","labels":[],"detail_key":"p36"},{"id":"n32108","layer":"informal","project":"p36","title":"lem:cdf-of-affine-max-many","kind":"lemma","summary":"Let X_0, X_1, \\ldots, X_n-1 be independent identically distributed real-valued random variables…","labels":["lem:cdf-of-affine-max-many"],"detail_key":"p36"},{"id":"n32109","layer":"informal","project":"p36","title":"Use \\reflem:cdf-of-max-many and do a change of variables.","kind":"proof","summary":"Use \\reflem:cdf-of-max-many and do a change of variables.","labels":[],"detail_key":"p36"},{"id":"n32110","layer":"informal","project":"p36","title":"def:oriented-affine-isomorphism","kind":"definition","summary":"The collection of all transformations R\\to R of the form x \\mapsto a x + b, where a>0, b \\in R,…","labels":["def:oriented-affine-isomorphism"],"detail_key":"p36"},{"id":"n32111","layer":"informal","project":"p36","title":"def:oriented-affine-transform-of-cdf","kind":"definition","summary":"The action of an orientation preserving affine isomorphism A \\in Aff^+_R on a cumulative distri…","labels":["def:oriented-affine-transform-of-cdf"],"detail_key":"p36"},{"id":"n32112","layer":"informal","project":"p36","title":"lem:oriented-affine-action-on-cdf","kind":"lemma","summary":"The actions of orientation preserving affine isomorphisms on a cumulative distribution function…","labels":["lem:oriented-affine-action-on-cdf"],"detail_key":"p36"},{"id":"n32113","layer":"informal","project":"p36","title":"Direct calculations.","kind":"proof","summary":"Direct calculations.","labels":[],"detail_key":"p36"},{"id":"n32114","layer":"informal","project":"p36","title":"lem:degenerate-cdf-transform","kind":"lemma","summary":"Let F be a cumulative distribution function and A \\in Aff^+_R an orientation preserving affine…","labels":["lem:degenerate-cdf-transform"],"detail_key":"p36"},{"id":"n32115","layer":"informal","project":"p36","title":"Straightforward from the definitions.","kind":"proof","summary":"Straightforward from the definitions.","labels":[],"detail_key":"p36"},{"id":"n32116","layer":"informal","project":"p36","title":"lem:cdf-continuity-pt-transform","kind":"lemma","summary":"Let F be a cumulative distribution function, and A \\in Aff^+_R an orientation preserving affine…","labels":["lem:cdf-continuity-pt-transform"],"detail_key":"p36"},{"id":"n32117","layer":"informal","project":"p36","title":"Straightforward.","kind":"proof","summary":"Straightforward.","labels":[],"detail_key":"p36"},{"id":"n32118","layer":"informal","project":"p36","title":"Continuity points of c.d.f.s are those which carry no point mass","kind":"lemma","summary":"[Continuity points of c.d.f.s are those which carry no point mass] Let F be cumulative distribu…","labels":["lem:cdf-continuity-pt-iff-measure-singleton"],"detail_key":"p36"},{"id":"n32119","layer":"informal","project":"p36","title":"A c.d.f. is always continuous from the right. Continuity of F from the left at x means th…","kind":"proof","summary":"A c.d.f. is always continuous from the right. Continuity of F from the left at x means that for…","labels":[],"detail_key":"p36"},{"id":"n32120","layer":"informal","project":"p36","title":"A pair of nontrivial continuity points of nondegenerate c.d.f.","kind":"lemma","summary":"[A pair of nontrivial continuity points of nondegenerate c.d.f.] Let G be a nondegenerate c.d.f…","labels":["lem:exists-two-nontrivial-continuity-pts-cdf"],"detail_key":"p36"},{"id":"n32121","layer":"informal","project":"p36","title":"Since G is nondegenerate, there exists some x_0 \\in R such that 0 < G(x_0) < 1. Since G i…","kind":"proof","summary":"Since G is nondegenerate, there exists some x_0 \\in R such that 0 < G(x_0) < 1. Since G is cont…","labels":[],"detail_key":"p36"},{"id":"n32122","layer":"informal","project":"p36","title":"Equality of c.d.f.s on a dense set suffices","kind":"lemma","summary":"[Equality of c.d.f.s on a dense set suffices] Suppose that F,G are two c.d.f.s and S \\subseteq…","labels":["lem:cdf-equal-on-dense"],"detail_key":"p36"},{"id":"n32123","layer":"informal","project":"p36","title":"We must prove that for any x \\in R we have F(x) = G(x). But by right-continuity of c.d.f.…","kind":"proof","summary":"We must prove that for any x \\in R we have F(x) = G(x). But by right-continuity of c.d.f.s, den…","labels":[],"detail_key":"p36"},{"id":"n32124","layer":"informal","project":"p36","title":"def:affine-transform-topology","kind":"definition","summary":"[] We equip the space Aff^+_R of orientation-preserving affine isomorphisms with the topology o…","labels":["def:affine-transform-topology"],"detail_key":"p36"},{"id":"n32125","layer":"informal","project":"p36","title":"The coefficients of affine map depend continuously on the map","kind":"lemma","summary":"[The coefficients of affine map depend continuously on the map] The coefficients a and b of an…","labels":["lem:affine-coefficients-continuous"],"detail_key":"p36"},{"id":"n32126","layer":"informal","project":"p36","title":"We may first write a = A(1) - A(0) and b = A(0). These depend continuously on A, since th…","kind":"proof","summary":"We may first write a = A(1) - A(0) and b = A(0). These depend continuously on A, since the eval…","labels":[],"detail_key":"p36"},{"id":"n32127","layer":"informal","project":"p36","title":"Metrizability of the topology on oriented affine isomorphisms","kind":"lemma","summary":"[Metrizability of the topology on oriented affine isomorphisms] The topology of pointwise conve…","labels":["lem:affine-metrizable"],"detail_key":"p36"},{"id":"n32128","layer":"informal","project":"p36","title":"The essential claim is that the function cfs \\colon Aff^+_R\\to (0,+\\infty) \\times R obtai…","kind":"proof","summary":"The essential claim is that the function cfs \\colon Aff^+_R\\to (0,+\\infty) \\times R obtained by…","labels":[],"detail_key":"p36"},{"id":"n32129","layer":"informal","project":"p36","title":"Inversion of orientation preserving affine isomorphisms is continuous","kind":"lemma","summary":"[Inversion of orientation preserving affine isomorphisms is continuous] The map A \\mapsto A^-1…","labels":["lem:affine-inversion-continuous"],"detail_key":"p36"},{"id":"n32130","layer":"informal","project":"p36","title":"Calculate and use Lemma~\\reflem:affine-coefficients-continuous.","kind":"proof","summary":"Calculate and use Lemma~\\reflem:affine-coefficients-continuous.","labels":[],"detail_key":"p36"},{"id":"n32131","layer":"informal","project":"p36","title":"The action of oriented affine transforms on c.d.f.s is continuous","kind":"lemma","summary":"[The action of oriented affine transforms on c.d.f.s is continuous] The action A . F of A \\in A…","labels":["lem:action-on-cdf-continuous"],"detail_key":"p36"},{"id":"n32132","layer":"informal","project":"p36","title":"The spaces are metrizable, so it suffices to check sequential continuity. Suppose that A_…","kind":"proof","summary":"The spaces are metrizable, so it suffices to check sequential continuity. Suppose that A_n \\to…","labels":[],"detail_key":"p36"},{"id":"n32133","layer":"informal","project":"p36","title":"def:extr-val-distr","kind":"definition","summary":"A c.d.f. G is said to be an \\emphextreme value distribution if G is nondegenerate and there exi…","labels":["def:extr-val-distr"],"detail_key":"p36"},{"id":"n32134","layer":"informal","project":"p36","title":"lem:extr-val-distr-transform","kind":"lemma","summary":"Let G be an extreme value distribution and and A \\in Aff^+_R an orientation preserving affine i…","labels":["lem:extr-val-distr-transform"],"detail_key":"p36"},{"id":"n32135","layer":"informal","project":"p36","title":"Straightforward using Lemmas~\\reflem:cdf-continuity-pt-transform and~\\reflem:degenerate-c…","kind":"proof","summary":"Straightforward using Lemmas~\\reflem:cdf-continuity-pt-transform and~\\reflem:degenerate-cdf-tra…","labels":[],"detail_key":"p36"},{"id":"n32136","layer":"informal","project":"p36","title":"def:std-Gumbel-cdf","kind":"definition","summary":"\\emphThe standard Gumbel distribution is the c.d.f. \\Lambda given by \\Lambda(x) = \\exp\\big(-\\ex…","labels":["def:std-Gumbel-cdf"],"detail_key":"p36"},{"id":"n32137","layer":"informal","project":"p36","title":"def:std-Weibull-cdf","kind":"definition","summary":"\\emphThe standard (reverse) Weibull distribution of parameter \\alpha > 0 is the c.d.f. \\Psi_\\al…","labels":["def:std-Weibull-cdf"],"detail_key":"p36"},{"id":"n32138","layer":"informal","project":"p36","title":"def:std-Frechet-cdf","kind":"definition","summary":"\\emphThe standard Fr\\'echet distribution of parameter \\alpha > 0 is the c.d.f. \\Phi_\\alpha give…","labels":["def:std-Frechet-cdf"],"detail_key":"p36"},{"id":"n32139","layer":"informal","project":"p36","title":"thm:Gumbel-is-extr-val-distr","kind":"theorem","summary":"The standard Gumbel distribution \\Lambda is an extreme value distribution.","labels":["thm:Gumbel-is-extr-val-distr"],"detail_key":"p36"},{"id":"n32140","layer":"informal","project":"p36","title":"Set A_n(x) = x - \\log(n) for n \\in N. Then A_n^-1(x) = x + \\log(n) and for any n \\ge 1 an…","kind":"proof","summary":"Set A_n(x) = x - \\log(n) for n \\in N. Then A_n^-1(x) = x + \\log(n) and for any n \\ge 1 and x \\i…","labels":[],"detail_key":"p36"},{"id":"n32141","layer":"informal","project":"p36","title":"thm:Weibull-is-extr-val-distr","kind":"theorem","summary":"For any \\alpha > 0, the standard Weibull distribution \\Psi_\\alpha is an extreme value distribut…","labels":["thm:Weibull-is-extr-val-distr"],"detail_key":"p36"},{"id":"n32142","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32143","layer":"informal","project":"p36","title":"thm:Frechet-is-extr-val-distr","kind":"theorem","summary":"For any \\alpha > 0, the standard Fr\\'echet distribution \\Phi_\\alpha is an extreme value distrib…","labels":["thm:Frechet-is-extr-val-distr"],"detail_key":"p36"},{"id":"n32144","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32145","layer":"informal","project":"p36","title":"Logarithmic version of the limit relation","kind":"lemma","summary":"[Logarithmic version of the limit relation] Let F and G be c.d.f.s, and (A_n)_n \\in N a sequenc…","labels":["lem:log-ev-limit"],"detail_key":"p36"},{"id":"n32146","layer":"informal","project":"p36","title":"Recall that (A_n.F)(x) = F (A_n^-1(x)). Then just take logarithms (and use continuity) to…","kind":"proof","summary":"Recall that (A_n.F)(x) = F (A_n^-1(x)). Then just take logarithms (and use continuity) to get f…","labels":[],"detail_key":"p36"},{"id":"n32147","layer":"informal","project":"p36","title":"Relation implies F tending to one","kind":"lemma","summary":"[Relation implies F tending to one] Let F and G be c.d.f.s, and (A_n)_n \\in N a sequence of ori…","labels":["lem:ev-limit-cdf-affine-tendsto-one"],"detail_key":"p36"},{"id":"n32148","layer":"informal","project":"p36","title":"Otherwise \\big( F (A_n^-1(x)) \\big)^n would have 0 \\ne G(x) as an accumulation point, con…","kind":"proof","summary":"Otherwise \\big( F (A_n^-1(x)) \\big)^n would have 0 \\ne G(x) as an accumulation point, contradic…","labels":[],"detail_key":"p36"},{"id":"n32149","layer":"informal","project":"p36","title":"Taylor expansion limit modification","kind":"lemma","summary":"[Taylor expansion limit modification] Let S \\subset R be a subset with 0 \\in S, and let f_1, f_…","labels":["lem:modify-limit-taylor"],"detail_key":"p36"},{"id":"n32150","layer":"informal","project":"p36","title":"This is in principle straightforward: the assumptions are first checked to imply that \\li…","kind":"proof","summary":"This is in principle straightforward: the assumptions are first checked to imply that \\lim_n \\t…","labels":[],"detail_key":"p36"},{"id":"n32151","layer":"informal","project":"p36","title":"Taylored version of the limit relation","kind":"lemma","summary":"[Taylored version of the limit relation] Let F and G be c.d.f.s, and (A_n)_n \\in N a sequence o…","labels":["lem:taylored-ev-limit"],"detail_key":"p36"},{"id":"n32152","layer":"informal","project":"p36","title":"Both implications (ii) \\, \\Rightarrow \\, (iii) and (iii) \\, \\Rightarrow \\, (ii) are prove…","kind":"proof","summary":"Both implications (ii) \\, \\Rightarrow \\, (iii) and (iii) \\, \\Rightarrow \\, (ii) are proven simi…","labels":[],"detail_key":"p36"},{"id":"n32153","layer":"informal","project":"p36","title":"Inverted Taylored version of the limit relation","kind":"lemma","summary":"[Inverted Taylored version of the limit relation] Let F and G be c.d.f.s, and (A_n)_n \\in N a s…","labels":["lem:inv-taylored-ev-limit"],"detail_key":"p36"},{"id":"n32154","layer":"informal","project":"p36","title":"By assumption G(x) \\in (0,1) we have -\\log G(x) > 0. The implication (iii)~\\,\\Rightarrow…","kind":"proof","summary":"By assumption G(x) \\in (0,1) we have -\\log G(x) > 0. The implication (iii)~\\,\\Rightarrow \\,~(iv…","labels":[],"detail_key":"p36"},{"id":"n32155","layer":"informal","project":"p36","title":"Transformed version of the limit relation","kind":"lemma","summary":"[Transformed version of the limit relation] Let F and G be c.d.f.s, and (A_n)_n \\in N a sequenc…","labels":["lem:transformed-ev-limit"],"detail_key":"p36"},{"id":"n32156","layer":"informal","project":"p36","title":"This is in principle straightforward, although certain cases need to be checked separatel…","kind":"proof","summary":"This is in principle straightforward, although certain cases need to be checked separately and…","labels":[],"detail_key":"p36"},{"id":"n32157","layer":"informal","project":"p36","title":"Equivalent versions of the limit relation","kind":"theorem","summary":"[Equivalent versions of the limit relation] Let F and G be c.d.f.s, and (A_n)_n \\in N a sequenc…","labels":["thm:tfae-ev-limit"],"detail_key":"p36"},{"id":"n32158","layer":"informal","project":"p36","title":"This is just a combination of earlier results.","kind":"proof","summary":"This is just a combination of earlier results.","labels":[],"detail_key":"p36"},{"id":"n32159","layer":"informal","project":"p36","title":"Second order differential equation for Q","kind":"lemma","summary":"[Second order differential equation for Q] Suppose that Q \\colon R\\to R is differentiable and s…","labels":["lem:ode-of-order-two-for-Q"],"detail_key":"p36"},{"id":"n32160","layer":"informal","project":"p36","title":"Note first that the equation implies (rearranging and dividing by h), for any s and h \\ne…","kind":"proof","summary":"Note first that the equation implies (rearranging and dividing by h), for any s and h \\ne 0, \\f…","labels":[],"detail_key":"p36"},{"id":"n32161","layer":"informal","project":"p36","title":"Solution for Q","kind":"lemma","summary":"[Solution for Q] Suppose that Q \\colon R\\to R is twice continuously differentiable and Q' is po…","labels":["lem:solve-Q","{eq:"],"detail_key":"p36"},{"id":"n32162","layer":"informal","project":"p36","title":"Since Q is differentiable and Q'(s)>0 for any s \\in R, we can write \\eqrefeq: Q eqn dd as…","kind":"proof","summary":"Since Q is differentiable and Q'(s)>0 for any s \\in R, we can write \\eqrefeq: Q eqn dd as \\frac…","labels":[],"detail_key":"p36"},{"id":"n32163","layer":"informal","project":"p36","title":"Monotone functions are a.e. differentiable","kind":"theorem","summary":"[Monotone functions are a.e. differentiable] \\mathlibok If f : R\\to R is nondecreasing, then th…","labels":["thm:monotone-ae-differentiable"],"detail_key":"p36"},{"id":"n32164","layer":"informal","project":"p36","title":"\\mathlibok (The proof is already in Mathlib.)","kind":"proof","summary":"\\mathlibok (The proof is already in Mathlib.)","labels":[],"detail_key":"p36"},{"id":"n32165","layer":"informal","project":"p36","title":"Solution for E","kind":"lemma","summary":"[Solution for E] Suppose that E \\colon (0,\\infty) \\to R is nondecreasing and nonconstant functi…","labels":["lem:solve-E"],"detail_key":"p36"},{"id":"n32166","layer":"informal","project":"p36","title":"Denote H(s) = E(e^s) for s \\in R. Then H(0) = E(1) = 0 and H is also nondecreasing and no…","kind":"proof","summary":"Denote H(s) = E(e^s) for s \\in R. Then H(0) = E(1) = 0 and H is also nondecreasing and nonconst…","labels":[],"detail_key":"p36"},{"id":"n32167","layer":"informal","project":"p36","title":"Weak convergence of probability measures","kind":"definition","summary":"[Weak convergence of probability measures] \\mathlibok A sequence (\\mu_n)_n \\in N of Borel proba…","labels":["def:convergence-in-distribution"],"detail_key":"p36"},{"id":"n32168","layer":"informal","project":"p36","title":"Monotone real functions have only countably many points of discontinuity","kind":"lemma","summary":"[Monotone real functions have only countably many points of discontinuity] \\mathlibok A monoton…","labels":["lem:monotone-discontinuities-countable"],"detail_key":"p36"},{"id":"n32169","layer":"informal","project":"p36","title":"\\mathlibok (The proof should already be in Mathlib.)","kind":"proof","summary":"\\mathlibok (The proof should already be in Mathlib.)","labels":[],"detail_key":"p36"},{"id":"n32170","layer":"informal","project":"p36","title":"Tightness of a cumulative distribution function","kind":"lemma","summary":"[Tightness of a cumulative distribution function] Let F be a cumulative distribution function.…","labels":["lem:cdf-tight"],"detail_key":"p36"},{"id":"n32171","layer":"informal","project":"p36","title":"Cumulative distribution functions satisfy F(x) \\downarrow 0 as x \\downarrow - \\infty and…","kind":"proof","summary":"Cumulative distribution functions satisfy F(x) \\downarrow 0 as x \\downarrow - \\infty and F(x) \\…","labels":[],"detail_key":"p36"},{"id":"n32172","layer":"informal","project":"p36","title":"Subdivision with small mesh and within dense set","kind":"lemma","summary":"[Subdivision with small mesh and within dense set] Let D \\subset R be a dense set and a,b \\in D…","labels":["lem:subdivision-dense"],"detail_key":"p36"},{"id":"n32173","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32174","layer":"informal","project":"p36","title":"Subdivision for continuous function approximation","kind":"lemma","summary":"[Subdivision for continuous function approximation] Let D \\subset R be a dense set, let f \\colo…","labels":["lem:continuous-function-approximation-subdivision"],"detail_key":"p36"},{"id":"n32175","layer":"informal","project":"p36","title":"On the compact interval [a,b] \\subset R, the continuous function~f is uniformly continuou…","kind":"proof","summary":"On the compact interval [a,b] \\subset R, the continuous function~f is uniformly continuous, so…","labels":[],"detail_key":"p36"},{"id":"n32176","layer":"informal","project":"p36","title":"Simple function integral as linear combination of cdf differences","kind":"lemma","summary":"[Simple function integral as linear combination of cdf differences] Let a = c_0 < c_1 < \\cdots…","labels":["lem:simple-integral-cdf-difference"],"detail_key":"p36"},{"id":"n32177","layer":"informal","project":"p36","title":"\\int_R h \\, d\\mu= \\; & \\int_R \\Big( \\sum_j=1^k \\alpha_j \\, I_(c_j-1,c_j](x) \\Big) \\, d\\mu…","kind":"proof","summary":"\\int_R h \\, d\\mu= \\; & \\int_R \\Big( \\sum_j=1^k \\alpha_j \\, I_(c_j-1,c_j](x) \\Big) \\, d\\mu(x) \\\\…","labels":[],"detail_key":"p36"},{"id":"n32178","layer":"informal","project":"p36","title":"One of the portmanteau implications","kind":"lemma","summary":"[One of the portmanteau implications] \\mathlibok Weak convergence of probability measures impli…","labels":["lem:portmanteau-convergence-for-borel"],"detail_key":"p36"},{"id":"n32179","layer":"informal","project":"p36","title":"\\mathlibok (The proof is in Mathlib.)","kind":"proof","summary":"\\mathlibok (The proof is in Mathlib.)","labels":[],"detail_key":"p36"},{"id":"n32180","layer":"informal","project":"p36","title":"Sufficient condition for convergence in distribution with cdfs","kind":"theorem","summary":"[Sufficient condition for convergence in distribution with cdfs] Let F and F_n, n \\in N, be cum…","labels":["thm:convergence-in-distribution-with-cdf"],"detail_key":"p36"},{"id":"n32181","layer":"informal","project":"p36","title":"Let D \\subset R denote the set of continuity points of F. By Lemma~\\reflem:monotone-disco…","kind":"proof","summary":"Let D \\subset R denote the set of continuity points of F. By Lemma~\\reflem:monotone-discontinui…","labels":[],"detail_key":"p36"},{"id":"n32182","layer":"informal","project":"p36","title":"Necessary condition for convergence in distribution with cdfs","kind":"lemma","summary":"[Necessary condition for convergence in distribution with cdfs] Let \\mu and \\mu_n, n \\in N, be…","labels":["lem:cdf-convergence-from-convergence-in-distribution"],"detail_key":"p36"},{"id":"n32183","layer":"informal","project":"p36","title":"Let x \\in R be a continuity point of F. Then we have \\mu [\\left\\ x \\right\\] = 0, by Lemma…","kind":"proof","summary":"Let x \\in R be a continuity point of F. Then we have \\mu [\\left\\ x \\right\\] = 0, by Lemma~\\refl…","labels":[],"detail_key":"p36"},{"id":"n32184","layer":"informal","project":"p36","title":"Extended cumulative distribution function","kind":"definition","summary":"[Extended cumulative distribution function] The extension \\widetildeF of a c.d.f. F is the func…","labels":["def:cdf-extend"],"detail_key":"p36"},{"id":"n32185","layer":"informal","project":"p36","title":"Continuity points of extended c.d.f.","kind":"lemma","summary":"[Continuity points of extended c.d.f.] The extension \\widetildeF of a c.d.f. F is continuous at…","labels":["lem:cdf-extend-continuity-pts"],"detail_key":"p36"},{"id":"n32186","layer":"informal","project":"p36","title":"Since \\lim_x \\to +\\infty F(x) = 1 by properties of c.d.f.s and \\widetildeF(+\\infty) = 1 b…","kind":"proof","summary":"Since \\lim_x \\to +\\infty F(x) = 1 by properties of c.d.f.s and \\widetildeF(+\\infty) = 1 by defi…","labels":[],"detail_key":"p36"},{"id":"n32187","layer":"informal","project":"p36","title":"One over one minus cumulative distribution function","kind":"definition","summary":"[One over one minus cumulative distribution function] The transform \\frac11-\\widetildeF of a c.…","labels":["def:one-div-one-sub-cdf"],"detail_key":"p36"},{"id":"n32188","layer":"informal","project":"p36","title":"Continuity points of one over one minus c.d.f.","kind":"lemma","summary":"[Continuity points of one over one minus c.d.f.] The transform \\frac11-\\widetildeF of a c.d.f.…","labels":["lem:one-div-one-sub-cdf-continuity-pts"],"detail_key":"p36"},{"id":"n32189","layer":"informal","project":"p36","title":"Since \\lim_x \\to +\\infty \\widetildeF(x) = \\widetildeF(+\\infty) = 1 by Lemma~\\reflem:cdf-e…","kind":"proof","summary":"Since \\lim_x \\to +\\infty \\widetildeF(x) = \\widetildeF(+\\infty) = 1 by Lemma~\\reflem:cdf-extend-…","labels":[],"detail_key":"p36"},{"id":"n32190","layer":"informal","project":"p36","title":"One over negative logarithm cumulative distribution function","kind":"definition","summary":"[One over negative logarithm cumulative distribution function] The transform \\frac1\\widetilde\\l…","labels":["def:one-div-neg-log-cdf"],"detail_key":"p36"},{"id":"n32191","layer":"informal","project":"p36","title":"Continuity points of one over negative logarithm c.d.f.","kind":"lemma","summary":"[Continuity points of one over negative logarithm c.d.f.] The transform \\frac1\\widetilde\\log \\b…","labels":["lem:one-div-neg-log-cdf-continuity-pts"],"detail_key":"p36"},{"id":"n32192","layer":"informal","project":"p36","title":"Since \\lim_x \\to +\\infty \\widetildeF(x) = \\widetildeF(+\\infty) = 1 by Lemma~\\reflem:cdf-e…","kind":"proof","summary":"Since \\lim_x \\to +\\infty \\widetildeF(x) = \\widetildeF(+\\infty) = 1 by Lemma~\\reflem:cdf-extend-…","labels":[],"detail_key":"p36"},{"id":"n32193","layer":"informal","project":"p36","title":"Functional equation in one parameter subgroups of affine isomorphisms","kind":"lemma","summary":"[Functional equation in one parameter subgroups of affine isomorphisms] Suppose that t \\mapsto…","labels":["lem:affine-one-parameter-subgroup-functional-eqn"],"detail_key":"p36"},{"id":"n32194","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32195","layer":"informal","project":"p36","title":"Functional equation scaling coefficient solution","kind":"lemma","summary":"[Functional equation scaling coefficient solution] Suppose that a \\colon (0,+\\infty) \\to (0,+\\i…","labels":["lem:solution-functional-eqn-scaling"],"detail_key":"p36"},{"id":"n32196","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32197","layer":"informal","project":"p36","title":"Functional equation translation coefficient solution with \\rho = 0","kind":"lemma","summary":"[Functional equation translation coefficient solution with \\rho = 0] Suppose that b \\colon (0,+…","labels":["lem:solution-functional-eqn-translation-no-scaling"],"detail_key":"p36"},{"id":"n32198","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32199","layer":"informal","project":"p36","title":"Functional equation translation coefficient solution with \\rho \\ne 0","kind":"lemma","summary":"[Functional equation translation coefficient solution with \\rho \\ne 0] Suppose that \\rho \\in R\\…","labels":["lem:solution-functional-eqn-translation-with-scaling"],"detail_key":"p36"},{"id":"n32200","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32201","layer":"informal","project":"p36","title":"One-parameter subgroups of affine isomorphisms of R","kind":"theorem","summary":"[One-parameter subgroups of affine isomorphisms of R] \\emph[TODO: Switch to additive notation a…","labels":["thm:one-parameter-subgroups-of-affine-isomorphisms"],"detail_key":"p36"},{"id":"n32202","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32203","layer":"informal","project":"p36","title":"Type (of distribution on R)","kind":"definition","summary":"[Type (of distribution on R)] Two c.d.f.s F, G are said to be of the same type, if there exists…","labels":["def:cdf-type"],"detail_key":"p36"},{"id":"n32204","layer":"informal","project":"p36","title":"Unique affine relation among two nondegenerate c.d.f.s","kind":"lemma","summary":"[Unique affine relation among two nondegenerate c.d.f.s] Let F, G be two c.d.f.s of the same ty…","labels":["lem:unique-affine-relation-cdf"],"detail_key":"p36"},{"id":"n32205","layer":"informal","project":"p36","title":"Since F is nondegenerate, we can find two different points x_1 < x_2 such that 0 < F(x_1)…","kind":"proof","summary":"Since F is nondegenerate, we can find two different points x_1 < x_2 such that 0 < F(x_1) < F(x…","labels":[],"detail_key":"p36"},{"id":"n32206","layer":"informal","project":"p36","title":"Degeneration by shrinking affine transformations","kind":"lemma","summary":"[Degeneration by shrinking affine transformations] Let (F_n)_n \\in N be a sequence of c.d.f.s w…","labels":["lem:degenerate-shrinking-limit"],"detail_key":"p36"},{"id":"n32207","layer":"informal","project":"p36","title":"It suffices to prove that for any x < \\beta we have \\widetildeG(x) = 0 and for any x > \\b…","kind":"proof","summary":"It suffices to prove that for any x < \\beta we have \\widetildeG(x) = 0 and for any x > \\beta we…","labels":[],"detail_key":"p36"},{"id":"n32208","layer":"informal","project":"p36","title":"Impossibility of expanding affine transformations","kind":"lemma","summary":"[Impossibility of expanding affine transformations] Let (F_n)_n \\in N be a sequence of c.d.f.s…","labels":["lem:impossible-expanding-limit"],"detail_key":"p36"},{"id":"n32209","layer":"informal","project":"p36","title":"Since G is assumed nondegenerate, by Lemma~\\reflem:exists-two-nontrivial-continuity-pts-c…","kind":"proof","summary":"Since G is assumed nondegenerate, by Lemma~\\reflem:exists-two-nontrivial-continuity-pts-cdf we…","labels":[],"detail_key":"p36"},{"id":"n32210","layer":"informal","project":"p36","title":"Convergence to types","kind":"theorem","summary":"[Convergence to types] Suppose that (F_n)_n \\in N is a sequence of c.d.f.s which converges to a…","labels":["thm:convergence-to-types"],"detail_key":"p36"},{"id":"n32211","layer":"informal","project":"p36","title":"Let us first argue that (a_n)_n \\in N are bounded. If not, then by passing to a subsequen…","kind":"proof","summary":"Let us first argue that (a_n)_n \\in N are bounded. If not, then by passing to a subsequence, we…","labels":[],"detail_key":"p36"},{"id":"n32212","layer":"informal","project":"p36","title":"Convergence to types again","kind":"theorem","summary":"[Convergence to types again] Let (A_n)_n \\in N and (\\widetildeA_n)_n \\in N be two sequences of…","labels":["thm:convergence-to-types-different-affine"],"detail_key":"p36"},{"id":"n32213","layer":"informal","project":"p36","title":"\\emph(This is actually just a special case of what is stated as Corollary~\\refcor:converg…","kind":"proof","summary":"\\emph(This is actually just a special case of what is stated as Corollary~\\refcor:convergence-t…","labels":[],"detail_key":"p36"},{"id":"n32214","layer":"informal","project":"p36","title":"Convergence to types with different limits","kind":"corollary","summary":"[Convergence to types with different limits] Let (A_n)_n \\in N and (\\widetildeA_n)_n \\in N be t…","labels":["cor:convergence-to-types-different-limit"],"detail_key":"p36"},{"id":"n32215","layer":"informal","project":"p36","title":"We will apply the convergence to types with the reference sequence (A_n.F_n)_n \\in N, whi…","kind":"proof","summary":"We will apply the convergence to types with the reference sequence (A_n.F_n)_n \\in N, which by…","labels":[],"detail_key":"p36"},{"id":"n32216","layer":"informal","project":"p36","title":"A choice of normalizing constants for convergence to types","kind":"lemma","summary":"[A choice of normalizing constants for convergence to types] (It is possible to choose normaliz…","labels":["lemma:convergence-to-types-normalization"],"detail_key":"p36"},{"id":"n32217","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32218","layer":"informal","project":"p36","title":"Subgroup of translations","kind":"definition","summary":"[Subgroup of translations] The mapping s \\mapsto A_s with A_s(x) = x + s is a homomorphism R\\to…","labels":["def:translation-subgroup"],"detail_key":"p36"},{"id":"n32219","layer":"informal","project":"p36","title":"Only translations have no fixed points","kind":"lemma","summary":"[Only translations have no fixed points] If A \\in Aff^+_R has no fixed points (no x \\in R such…","labels":["lem:no-fixed-point-implies-translation"],"detail_key":"p36"},{"id":"n32220","layer":"informal","project":"p36","title":"Let us prove this by contrapositive: that any element A which is not a translation must h…","kind":"proof","summary":"Let us prove this by contrapositive: that any element A which is not a translation must have a…","labels":[],"detail_key":"p36"},{"id":"n32221","layer":"informal","project":"p36","title":"Conjugate of translation is translation","kind":"lemma","summary":"[Conjugate of translation is translation] Let A^(\\beta)_s = x + \\beta s for s, \\beta \\in R as i…","labels":["lem:conjugate-translation"],"detail_key":"p36"},{"id":"n32222","layer":"informal","project":"p36","title":"Calculate, for x \\in R (B A^(\\beta)_s B^-1)(x) \\; = \\; & (B A^(\\beta)_s)\\big( \\fracx-ba \\…","kind":"proof","summary":"Calculate, for x \\in R (B A^(\\beta)_s B^-1)(x) \\; = \\; & (B A^(\\beta)_s)\\big( \\fracx-ba \\big) \\…","labels":[],"detail_key":"p36"},{"id":"n32223","layer":"informal","project":"p36","title":"Subgroup fixing a point","kind":"definition","summary":"[Subgroup fixing a point] The mapping s \\mapsto A_s with A_s(x) = e^s (x - c) + c is a homomorp…","labels":["def:fixing-subgroup"],"detail_key":"p36"},{"id":"n32224","layer":"informal","project":"p36","title":"Characterization of the subgroup fixing a point","kind":"lemma","summary":"[Characterization of the subgroup fixing a point] An orientation-preserving affine transformati…","labels":["lem:fixing-subgroup-characterization"],"detail_key":"p36"},{"id":"n32225","layer":"informal","project":"p36","title":"Suppose first that A is an element of the said subgroup, i.e., A(x) = e^s (x - c) + c for…","kind":"proof","summary":"Suppose first that A is an element of the said subgroup, i.e., A(x) = e^s (x - c) + c for some…","labels":[],"detail_key":"p36"},{"id":"n32226","layer":"informal","project":"p36","title":"Conjugate of fixing is fixing image","kind":"lemma","summary":"[Conjugate of fixing is fixing image] Let A^(\\alpha;c)_s = e^\\alpha s (x - c) + c for \\alpha, c…","labels":["lem:conjugate-fixing"],"detail_key":"p36"},{"id":"n32227","layer":"informal","project":"p36","title":"Calculate, for x \\in R (B A^(\\alpha;c)_s B^-1)(x) \\; = \\; & (B A^(\\alpha;c)_s)\\big( \\frac…","kind":"proof","summary":"Calculate, for x \\in R (B A^(\\alpha;c)_s B^-1)(x) \\; = \\; & (B A^(\\alpha;c)_s)\\big( \\fracx-ba \\…","labels":[],"detail_key":"p36"},{"id":"n32228","layer":"informal","project":"p36","title":"Continuous parameter extreme value limit relation","kind":"lemma","summary":"[Continuous parameter extreme value limit relation] Let F be a c.d.f. \\emph(Note that below we…","labels":["lem:continuous-parameter-ev-limit"],"detail_key":"p36"},{"id":"n32229","layer":"informal","project":"p36","title":"Let t > 0 and let x \\in R be a continuity point of G. For n \\in N, calculate \\big((A_n .…","kind":"proof","summary":"Let t > 0 and let x \\in R be a continuity point of G. For n \\in N, calculate \\big((A_n . F) (x)…","labels":[],"detail_key":"p36"},{"id":"n32230","layer":"informal","project":"p36","title":"Self-similarity of extreme value distributions","kind":"lemma","summary":"[Self-similarity of extreme value distributions] Suppose that G is an extreme-value distributio…","labels":["lem:self-similarity-of-extreme-value-distributions"],"detail_key":"p36"},{"id":"n32231","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32232","layer":"informal","project":"p36","title":"Self-similar continuous c.d.f. family characterization \\gamma = 0","kind":"lemma","summary":"[Self-similar continuous c.d.f. family characterization \\gamma = 0] Suppose that G is a nondege…","labels":["lem:characterization-self-similar-family-zero-index"],"detail_key":"p36"},{"id":"n32233","layer":"informal","project":"p36","title":"Since G is nondegenerate, there exists an x_0 \\in R with 0 < G(x_0) < 1. Write q = -\\log…","kind":"proof","summary":"Since G is nondegenerate, there exists an x_0 \\in R with 0 < G(x_0) < 1. Write q = -\\log G(x_0)…","labels":[],"detail_key":"p36"},{"id":"n32234","layer":"informal","project":"p36","title":"Self-similar continuous c.d.f. family characterization \\gamma > 0","kind":"lemma","summary":"[Self-similar continuous c.d.f. family characterization \\gamma > 0] Suppose that G is a nondege…","labels":["lem:characterization-self-similar-family-pos-index"],"detail_key":"p36"},{"id":"n32235","layer":"informal","project":"p36","title":"Since G is nondegenerate, there exists an x_0 \\in R with 0 < G(x_0) < 1. Write q = -\\log…","kind":"proof","summary":"Since G is nondegenerate, there exists an x_0 \\in R with 0 < G(x_0) < 1. Write q = -\\log G(x_0)…","labels":[],"detail_key":"p36"},{"id":"n32236","layer":"informal","project":"p36","title":"Self-similar continuous c.d.f. family characterization \\gamma < 0","kind":"lemma","summary":"[Self-similar continuous c.d.f. family characterization \\gamma < 0] Suppose that G is a nondege…","labels":["lem:characterization-self-similar-family-neg-index"],"detail_key":"p36"},{"id":"n32237","layer":"informal","project":"p36","title":"\\emph(Note: The Lean formalized statement uses the opposite sign of \\alpha: it is assumed…","kind":"proof","summary":"\\emph(Note: The Lean formalized statement uses the opposite sign of \\alpha: it is assumed that…","labels":[],"detail_key":"p36"},{"id":"n32238","layer":"informal","project":"p36","title":"Three types of extreme value distributions [Fisher-Tippett-Gnedenko]","kind":"theorem","summary":"[Three types of extreme value distributions [Fisher-Tippett-Gnedenko]] For any extreme value di…","labels":["thm:three-types-of-extr-val-distr"],"detail_key":"p36"},{"id":"n32239","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32240","layer":"informal","project":"p36","title":"def:lc-inverse","kind":"definition","summary":"Let f \\colon R \\to S be a function (usually assumed nondecreasing). The left-continuous inverse…","labels":["def:lc-inverse"],"detail_key":"p36"},{"id":"n32241","layer":"informal","project":"p36","title":"Countably many connected components for an open set","kind":"lemma","summary":"[Countably many connected components for an open set] Let X be a locally connected separable sp…","labels":["lem:countably-many-connected-components-of-open"],"detail_key":"p36"},{"id":"n32242","layer":"informal","project":"p36","title":"(The proof is already formalized, see: \\textttIsOpen.countable-setOf-connectedComponentIn…","kind":"proof","summary":"(The proof is already formalized, see: \\textttIsOpen.countable-setOf-connectedComponentIn.)","labels":[],"detail_key":"p36"},{"id":"n32243","layer":"informal","project":"p36","title":"Finding an interval with high overlap","kind":"lemma","summary":"[Finding an interval with high overlap] Let A \\subset R be a measurable set such that 0 < \\Lamb…","labels":["lem:exists-high-overlap-interval"],"detail_key":"p36"},{"id":"n32244","layer":"informal","project":"p36","title":"Assume, without loss of generality, 0 < r < 1. Since the Lebesgue measure is outer regula…","kind":"proof","summary":"Assume, without loss of generality, 0 < r < 1. Since the Lebesgue measure is outer regular, we…","labels":[],"detail_key":"p36"},{"id":"n32245","layer":"informal","project":"p36","title":"Shifts of a smaller interval contained in a larger interval","kind":"lemma","summary":"[Shifts of a smaller interval contained in a larger interval] Let I, J be nontrivial intervals,…","labels":["lem:shift-interval-contained-in-larger"],"detail_key":"p36"},{"id":"n32246","layer":"informal","project":"p36","title":"\\ldots","kind":"proof","summary":"\\ldots","labels":[],"detail_key":"p36"},{"id":"n32247","layer":"informal","project":"p36","title":"Overlapping union of copies of an interval","kind":"lemma","summary":"[Overlapping union of copies of an interval] Let J be a nontrivial interval of finite length (0…","labels":["lem:overlapping-union-interval-copies"],"detail_key":"p36"},{"id":"n32248","layer":"informal","project":"p36","title":"Denote a = \\inf J and b = \\sup J. We have -\\infty < a < b < +\\infty and (a,b) \\; \\subsete…","kind":"proof","summary":"Denote a = \\inf J and b = \\sup J. We have -\\infty < a < b < +\\infty and (a,b) \\; \\subseteq \\; J…","labels":[],"detail_key":"p36"},{"id":"n32249","layer":"informal","project":"p36","title":"Difference set of positive measure set contains an interval","kind":"lemma","summary":"[Difference set of positive measure set contains an interval] Let A \\subset R be a measurable s…","labels":["lem:difference-set-contains-interval"],"detail_key":"p36"},{"id":"n32250","layer":"informal","project":"p36","title":"Pick a measurable subset A_0 \\subseteq A such that 0 < \\Lambda[A_0] < +\\infty. By Lemma~\\…","kind":"proof","summary":"Pick a measurable subset A_0 \\subseteq A such that 0 < \\Lambda[A_0] < +\\infty. By Lemma~\\reflem…","labels":[],"detail_key":"p36"},{"id":"n32251","layer":"informal","project":"p36","title":"Dividing a high overlap interval","kind":"lemma","summary":"[Dividing a high overlap interval] Let A be a measurable. Suppose that for some a < b, the inte…","labels":["lem:dividing-high-operlap-interval"],"detail_key":"p36"},{"id":"n32252","layer":"informal","project":"p36","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p36"},{"id":"n32253","layer":"informal","project":"p36","title":"Difference of two positive measure sets contains an interval","kind":"lemma","summary":"[Difference of two positive measure sets contains an interval] Let A, B \\subset R be two measur…","labels":["lem:different-difference-set-contains-interval"],"detail_key":"p36"},{"id":"n32254","layer":"informal","project":"p36","title":"Pick measurable subsets A_0 \\subseteq A and B_0 \\subseteq B such that 0 < \\Lambda[A_0] <…","kind":"proof","summary":"Pick measurable subsets A_0 \\subseteq A and B_0 \\subseteq B such that 0 < \\Lambda[A_0] < +\\inft…","labels":[],"detail_key":"p36"},{"id":"n32255","layer":"formal","project":"p36","title":"AffineIncrEquiv","kind":"def","summary":"Type","labels":[],"detail_key":"p36","name":"AffineIncrEquiv","module":"ExtremeValueProject.AffineTransformation"},{"id":"n32256","layer":"formal","project":"p36","title":"AffineIncrEquiv.continuous_coefs_fst","kind":"theorem","summary":"Continuous fun A => A.coefs.1","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.continuous_coefs_fst","module":"ExtremeValueProject.AffineTransformation"},{"id":"n32257","layer":"formal","project":"p36","title":"AffineIncrEquiv.continuous_coefs_snd","kind":"theorem","summary":"Continuous fun A => A.coefs.2","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.continuous_coefs_snd","module":"ExtremeValueProject.AffineTransformation"},{"id":"n32258","layer":"formal","project":"p36","title":"AffineIncrEquiv.continuous_inv","kind":"theorem","summary":"Continuous fun A => Inv.inv 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:…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.affine_continuousAt_of_continuousAt","module":"ExtremeValueProject.AffineTransformation"},{"id":"n32264","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.affine_isDegenerate_iff","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) (A : AffineIncrEquiv), Iff (HSMul.hSMul A F).IsDegenerat…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.affine_isDegenerate_iff","module":"ExtremeValueProject.AffineTransformation"},{"id":"n32265","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.instMulActionAffineIncrEquiv","kind":"def","summary":"MulAction AffineIncrEquiv 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1), Eq γ (deriv (deriv fun s => E (Real.exp s…","labels":[],"detail_key":"p36","name":"ExtremeValueProject.solve_E","module":"ExtremeValueProject.ClassificationCalculation"},{"id":"n32269","layer":"formal","project":"p36","title":"ExtremeValueProject.solve_Q","kind":"theorem","summary":"∀ Q : Real → Real γ : Real, ContDiff Real 2 Q → Eq (Q 0) 0 → Eq (deriv Q 0) 1 → (∀ (s : Real),…","labels":[],"detail_key":"p36","name":"ExtremeValueProject.solve_Q","module":"ExtremeValueProject.ClassificationCalculation"},{"id":"n32270","layer":"formal","project":"p36","title":"CumulativeDistributionFunction","kind":"inductive","summary":"Type","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction","module":"ExtremeValueProject.CumulativeDistributionFunction"},{"id":"n32271","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.continuousAt_iff","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) (x : Real), Iff (ContinuousAt (↑F.toStieltjesFunction) x…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.continuousAt_iff","module":"ExtremeValueProject.CumulativeDistributionFunction"},{"id":"n32272","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.eq_of_forall_dense_eq","kind":"theorem","summary":"∀ S : Set Real, Dense S → ∀ (F G : CumulativeDistributionFunction), (∀ (x : Real), Membership.m…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.eq_of_forall_dense_eq","module":"ExtremeValueProject.CumulativeDistributionFunction"},{"id":"n32273","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.tendsto_apply_of_tendsto_of_continuousAt","kind":"theorem","summary":"∀ ι : Type u_1 L : Filter ι μs : ι → MeasureTheory.ProbabilityMeasure Real μ : MeasureTheory.Pr…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.tendsto_apply_of_tendsto_of_continuousAt","module":"ExtremeValueProject.CumulativeDistributionFunction"},{"id":"n32274","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.IsDegenerate","kind":"def","summary":"CumulativeDistributionFunction → Prop","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.IsDegenerate","module":"ExtremeValueProject.DegenerateCDF"},{"id":"n32275","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.diracProba_is_degenerate","kind":"theorem","summary":"∀ (x₀ : Real), (MeasureTheory.diracProba x₀).cdf.IsDegenerate","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.diracProba_is_degenerate","module":"ExtremeValueProject.DegenerateCDF"},{"id":"n32276","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.eq_diracProba_of_isDegenerate","kind":"theorem","summary":"∀ (μ : MeasureTheory.ProbabilityMeasure Real), μ.cdf.IsDegenerate → Exists fun x₀ => Eq μ (Meas…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.eq_diracProba_of_isDegenerate","module":"ExtremeValueProject.DegenerateCDF"},{"id":"n32277","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.isDegenerate_iff","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), Iff F.IsDegenerate (Exists fun x₀ => Eq (↑F.toStieltjes…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.isDegenerate_iff","module":"ExtremeValueProject.DegenerateCDF"},{"id":"n32278","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.IsExtremeValueDistr","kind":"def","summary":"CumulativeDistributionFunction → 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ξ_pos).IsExtremeValueDistr","labels":[],"detail_key":"p36","name":"isExtremeValueDistr_standardFrechetCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32281","layer":"formal","project":"p36","title":"isExtremeValueDistr_standardGumbelCDF","kind":"theorem","summary":"standardGumbelCDF.IsExtremeValueDistr","labels":[],"detail_key":"p36","name":"isExtremeValueDistr_standardGumbelCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32282","layer":"formal","project":"p36","title":"isExtremeValueDistr_standardWeibullCDF","kind":"theorem","summary":"∀ ξ : Real (ξ_pos : LT.lt 0 ξ), (standardWeibullCDF ξ_pos).IsExtremeValueDistr","labels":[],"detail_key":"p36","name":"isExtremeValueDistr_standardWeibullCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32283","layer":"formal","project":"p36","title":"standardFrechetCDF","kind":"def","summary":"α : Real → LT.lt 0 α → CumulativeDistributionFunction","labels":[],"detail_key":"p36","name":"standardFrechetCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32284","layer":"formal","project":"p36","title":"standardGumbelCDF","kind":"def","summary":"CumulativeDistributionFunction","labels":[],"detail_key":"p36","name":"standardGumbelCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32285","layer":"formal","project":"p36","title":"standardWeibullCDF","kind":"def","summary":"α : Real → LT.lt 0 α → CumulativeDistributionFunction","labels":[],"detail_key":"p36","name":"standardWeibullCDF","module":"ExtremeValueProject.ExtremeValueDistribution"},{"id":"n32286","layer":"formal","project":"p36","title":"ev_limit_iff_log_ev_limit","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction As : Nat → AffineIncrEquiv x : Real, Membership.mem (Set…","labels":[],"detail_key":"p36","name":"ev_limit_iff_log_ev_limit","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32287","layer":"formal","project":"p36","title":"log_ev_limit_iff_taylored_ev_limit","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction As : Nat → AffineIncrEquiv x : Real, Membership.mem (Set…","labels":[],"detail_key":"p36","name":"log_ev_limit_iff_taylored_ev_limit","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32288","layer":"formal","project":"p36","title":"taylored_ev_limit_iff_oneDivOneSub_limit","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction As : Nat → AffineIncrEquiv x : Real, Membership.mem (Set…","labels":[],"detail_key":"p36","name":"taylored_ev_limit_iff_oneDivOneSub_limit","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32289","layer":"formal","project":"p36","title":"tendsto_one_of_ev_limit","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction As : Nat → AffineIncrEquiv x : Real, Membership.mem (Set…","labels":[],"detail_key":"p36","name":"tendsto_one_of_ev_limit","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32290","layer":"formal","project":"p36","title":"tendsto_smul_apply_smul_deriv_of_tendsto_atTop_of_tendsto_smul_apply_smul_deriv","kind":"theorem","summary":"∀ ι : Type u_1 L : Filter ι [L.NeBot] E : Type u_2 [inst : NormedAddCommGroup E] [inst_1 : Norm…","labels":[],"detail_key":"p36","name":"tendsto_smul_apply_smul_deriv_of_tendsto_atTop_of_tendsto_smul_apply_smul_deriv","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32291","layer":"formal","project":"p36","title":"tendsto_zero_of_tendsto_atTop_of_tendsto_smul","kind":"theorem","summary":"∀ ι : Type u_1 L : Filter ι E : Type u_2 [inst : SeminormedAddCommGroup E] [inst_1 : NormedSpac…","labels":[],"detail_key":"p36","name":"tendsto_zero_of_tendsto_atTop_of_tendsto_smul","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32292","layer":"formal","project":"p36","title":"tfae_ev_limit_relation","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction (As : Nat → AffineIncrEquiv) x : Real, Membership.mem (S…","labels":[],"detail_key":"p36","name":"tfae_ev_limit_relation","module":"ExtremeValueProject.LimitRelationManipulation"},{"id":"n32293","layer":"formal","project":"p36","title":"AffineIncrEquiv.conjugate_homOfIndex","kind":"theorem","summary":"∀ (A : AffineIncrEquiv) (α c s : Real), Eq (HMul.hMul (HMul.hMul A ((AffineIncrEquiv.homOfIndex…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.conjugate_homOfIndex","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32294","layer":"formal","project":"p36","title":"AffineIncrEquiv.conjugate_homOfIndex₀","kind":"theorem","summary":"∀ (A : AffineIncrEquiv) (β s : Real), Eq (HMul.hMul (HMul.hMul A ((AffineIncrEquiv.homOfIndex₀…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.conjugate_homOfIndex₀","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32295","layer":"formal","project":"p36","title":"AffineIncrEquiv.homOfIndex","kind":"def","summary":"Real → Real → MonoidHom (Multiplicative Real) AffineIncrEquiv","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.homOfIndex","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32296","layer":"formal","project":"p36","title":"AffineIncrEquiv.homOfIndex₀","kind":"def","summary":"Real → MonoidHom (Multiplicative Real) AffineIncrEquiv","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.homOfIndex₀","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32297","layer":"formal","project":"p36","title":"AffineIncrEquiv.homomorphism_from_Real_characterization","kind":"theorem","summary":"∀ (f : MonoidHom (Multiplicative Real) AffineIncrEquiv), Measurable ⇑f → Or (Exists fun β => Eq…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.homomorphism_from_Real_characterization","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32298","layer":"formal","project":"p36","title":"AffineIncrEquiv.homomorphism_from_Real_characterization_of_nontrivial","kind":"theorem","summary":"∀ f : MonoidHom (Multiplicative Real) AffineIncrEquiv, Not (Eq f 1) → Measurable ⇑f → Or (Exist…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.homomorphism_from_Real_characterization_of_nontrivial","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32299","layer":"formal","project":"p36","title":"AffineIncrEquiv.mem_subGroupOfIndex_iff_fixed_point","kind":"theorem","summary":"∀ (A : AffineIncrEquiv) α : Real, Ne α 0 → ∀ (c : Real), Iff (Membership.mem (AffineIncrEquiv.s…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.mem_subGroupOfIndex_iff_fixed_point","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32300","layer":"formal","project":"p36","title":"AffineIncrEquiv.mem_subGroupOfIndex₀_of_no_fixed_point","kind":"theorem","summary":"∀ (A : AffineIncrEquiv) α : Real, Ne α 0 → ∀ (c : Real), (∀ (x : Real), Ne (A x) x) → Membershi…","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.mem_subGroupOfIndex₀_of_no_fixed_point","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32301","layer":"formal","project":"p36","title":"AffineIncrEquiv.subGroupOfIndex","kind":"def","summary":"Real → Real → Subgroup AffineIncrEquiv","labels":[],"detail_key":"p36","name":"AffineIncrEquiv.subGroupOfIndex","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32302","layer":"formal","project":"p36","title":"AffineIncrEquiv.subGroupOfIndex₀","kind":"def","summary":"Subgroup 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Mea…","labels":[],"detail_key":"p36","name":"eq_const_mul_of_additive_of_measurable","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32305","layer":"formal","project":"p36","title":"eq_const_mul_one_sub_exp_of_multiplicative_with_scaling_of_measurable","kind":"theorem","summary":"∀ α : Real, Ne α 0 → ∀ b : Real → Real, (∀ (s t : Real), Eq (b (HAdd.hAdd s t)) (HAdd.hAdd (HMu…","labels":[],"detail_key":"p36","name":"eq_const_mul_one_sub_exp_of_multiplicative_with_scaling_of_measurable","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32306","layer":"formal","project":"p36","title":"eq_exp_const_mul_of_multiplicative_of_measurable","kind":"theorem","summary":"∀ f : Real → Real, (∀ (s : Real), LT.lt 0 (f s)) → (∀ (s₁ s₂ : Real), Eq (f (HAdd.hAdd s₁ s₂))…","labels":[],"detail_key":"p36","name":"eq_exp_const_mul_of_multiplicative_of_measurable","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32307","layer":"formal","project":"p36","title":"exists_Ioo_subset_diff_self_of_measure_pos","kind":"theorem","summary":"∀ A : Set Real, MeasurableSet A → LT.lt 0 (MeasureTheory.volume A) → Exists fun δ => And (GT.gt…","labels":[],"detail_key":"p36","name":"exists_Ioo_subset_diff_self_of_measure_pos","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32308","layer":"formal","project":"p36","title":"exists_interval_measure_inter_gt_mul_measure","kind":"theorem","summary":"∀ A : Set Real, MeasurableSet A → LT.lt 0 (MeasureTheory.volume A) → LT.lt (MeasureTheory.volum…","labels":[],"detail_key":"p36","name":"exists_interval_measure_inter_gt_mul_measure","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32309","layer":"formal","project":"p36","title":"isConnected_of_Ioo_subset_of_subset_Icc","kind":"theorem","summary":"∀ J : Set Real a b : Real, LT.lt a b → Subset (Set.Ioo a b) J → Subset J (Set.Icc a b) → IsConn…","labels":[],"detail_key":"p36","name":"isConnected_of_Ioo_subset_of_subset_Icc","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32310","layer":"formal","project":"p36","title":"volume_union_add_self_ge_of_Ioo_subset","kind":"theorem","summary":"∀ J : Set Real a b : Real, LE.le a b → Subset (Set.Ioo a b) J → ∀ (t : Real), LT.lt (abs t) (HS…","labels":[],"detail_key":"p36","name":"volume_union_add_self_ge_of_Ioo_subset","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32311","layer":"formal","project":"p36","title":"volume_union_add_self_le_of_subset_Icc","kind":"theorem","summary":"∀ J : Set Real a b : Real, LE.le a b → Subset J (Set.Icc a b) → ∀ (t : Real), LE.le (MeasureThe…","labels":[],"detail_key":"p36","name":"volume_union_add_self_le_of_subset_Icc","module":"ExtremeValueProject.OneParameterAffine"},{"id":"n32312","layer":"formal","project":"p36","title":"lcInv","kind":"def","summary":"R : Type u_1 → S : Type u_2 → [CompleteLinearOrder R] → [CompleteLinearOrder S] → (R → S) → S →…","labels":[],"detail_key":"p36","name":"lcInv","module":"ExtremeValueProject.PseudoInverses"},{"id":"n32313","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.IsExtremeValueDistr.classification","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, G.IsExtremeValueDistr → Or (Exists fun A => Eq (HSMul.hSM…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.IsExtremeValueDistr.classification","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32314","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.IsExtremeValueDistr.self_similar","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, G.IsExtremeValueDistr → Exists fun f => And (Ne f 1) (And…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.IsExtremeValueDistr.self_similar","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32315","layer":"formal","project":"p36","title":"continuous_parameter_ev_limit_relation","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction As : Nat → AffineIncrEquiv x : Real, Filter.Tendsto (fun…","labels":[],"detail_key":"p36","name":"continuous_parameter_ev_limit_relation","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32316","layer":"formal","project":"p36","title":"frechet_type_of_selfSimilar_index_pos","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction (G_nondeg : Not G.IsDegenerate) α c : Real (α_pos : LT.lt…","labels":[],"detail_key":"p36","name":"frechet_type_of_selfSimilar_index_pos","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32317","layer":"formal","project":"p36","title":"frechet_type_of_selfSimilar_index_pos'","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, Not G.IsDegenerate → ∀ α c : Real, LT.lt 0 α → (∀ (s : Mu…","labels":[],"detail_key":"p36","name":"frechet_type_of_selfSimilar_index_pos'","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32318","layer":"formal","project":"p36","title":"gumbel_type_of_selfSimilar_index_zero","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, Not G.IsDegenerate → ∀ β : Real (β_pos : LT.lt 0 β), (∀ (…","labels":[],"detail_key":"p36","name":"gumbel_type_of_selfSimilar_index_zero","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32319","layer":"formal","project":"p36","title":"gumbel_type_of_selfSimilar_index_zero'","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, Not G.IsDegenerate → ∀ β : Real, LT.lt 0 β → (∀ (s : Mult…","labels":[],"detail_key":"p36","name":"gumbel_type_of_selfSimilar_index_zero'","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32320","layer":"formal","project":"p36","title":"weibull_type_of_selfSimilar_index_neg","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction (G_nondeg : Not G.IsDegenerate) α c : Real (α_neg : LT.lt…","labels":[],"detail_key":"p36","name":"weibull_type_of_selfSimilar_index_neg","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32321","layer":"formal","project":"p36","title":"weibull_type_of_selfSimilar_index_neg'","kind":"theorem","summary":"∀ G : CumulativeDistributionFunction, Not G.IsDegenerate → ∀ α c : Real, LT.lt α 0 → (∀ (s : Mu…","labels":[],"detail_key":"p36","name":"weibull_type_of_selfSimilar_index_neg'","module":"ExtremeValueProject.SelfSimilarCDF"},{"id":"n32322","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.extend","kind":"def","summary":"CumulativeDistributionFunction → EReal → ENNReal","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.extend","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32323","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.extend_continuousAt","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) x : Real, ContinuousAt (↑F.toStieltjesFunction) x → Cont…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.extend_continuousAt","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32324","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.extend_continuousAt_bot","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.extend Bot.bot","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.extend_continuousAt_bot","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32325","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.extend_continuousAt_top","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.extend Top.top","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.extend_continuousAt_top","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32326","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivNegLog","kind":"def","summary":"CumulativeDistributionFunction → EReal → ENNReal","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivNegLog","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32327","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivNegLog_continuousAt","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) x : Real, ContinuousAt (↑F.toStieltjesFunction) x → Cont…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivNegLog_continuousAt","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32328","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivNegLog_continuousAt_bot","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.oneDivNegLog Bot.bot","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivNegLog_continuousAt_bot","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32329","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivNegLog_continuousAt_top","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.oneDivNegLog Top.top","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivNegLog_continuousAt_top","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32330","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivOneSub","kind":"def","summary":"CumulativeDistributionFunction → EReal → ENNReal","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivOneSub","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32331","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivOneSub_continuousAt","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) x : Real, ContinuousAt (↑F.toStieltjesFunction) x → Cont…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivOneSub_continuousAt","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32332","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivOneSub_continuousAt_bot","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.oneDivOneSub Bot.bot","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivOneSub_continuousAt_bot","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32333","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.oneDivOneSub_continuousAt_top","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), ContinuousAt F.oneDivOneSub Top.top","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.oneDivOneSub_continuousAt_top","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32334","layer":"formal","project":"p36","title":"oneDivSub_limit_iff","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction (As : Nat → AffineIncrEquiv) x : Real, Membership.mem (S…","labels":[],"detail_key":"p36","name":"oneDivSub_limit_iff","module":"ExtremeValueProject.TransformedCDF"},{"id":"n32335","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.exists₂_continuousAt_of_not_isDegenerate","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction), Not F.IsDegenerate → Exists fun x₁ => Exists fun x₂ =>…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.exists₂_continuousAt_of_not_isDegenerate","module":"ExtremeValueProject.TypeOfCDF"},{"id":"n32336","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.isDegenerate_of_tendsto_shrinking","kind":"theorem","summary":"∀ F : Nat → CumulativeDistributionFunction G G' : CumulativeDistributionFunction a : Nat → Real…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.isDegenerate_of_tendsto_shrinking","module":"ExtremeValueProject.TypeOfCDF"},{"id":"n32337","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.not_tendsto_cdf_of_expanding_of_tendsto_not_isDegenerate","kind":"theorem","summary":"∀ F : Nat → CumulativeDistributionFunction G G' : CumulativeDistributionFunction, (∀ (x : Real)…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.not_tendsto_cdf_of_expanding_of_tendsto_not_isDegenerate","module":"ExtremeValueProject.TypeOfCDF"},{"id":"n32338","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.unique_orientationPreservingAffineEquiv_smul_eq_not_isDege…","kind":"theorem","summary":"∀ F G : CumulativeDistributionFunction A₁ A₂ : AffineIncrEquiv, Not G.IsDegenerate → Eq (HSMul.…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.unique_orientationPreservingAffineEquiv_smul_eq_not_isDegenerate","module":"ExtremeValueProject.TypeOfCDF"},{"id":"n32339","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.continuous_mulAction","kind":"theorem","summary":"Continuous fun x => CumulativeDistributionFunction.continuous_mulAction.match_1 (fun x => Cumul…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.continuous_mulAction","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32340","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.forall_pos_exists_lt_gt_continuousAt","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) ε : Real, LT.lt 0 ε → Exists fun a => Exists fun b => An…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.forall_pos_exists_lt_gt_continuousAt","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32341","layer":"formal","project":"p36","title":"CumulativeDistributionFunction.integral_sum_indicator_eq","kind":"theorem","summary":"∀ (F : CumulativeDistributionFunction) E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : Nor…","labels":[],"detail_key":"p36","name":"CumulativeDistributionFunction.integral_sum_indicator_eq","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32342","layer":"formal","project":"p36","title":"forall_exists_subdivision_diff_lt_of_dense","kind":"theorem","summary":"∀ D : Set Real, Dense D → ∀ a b : Real, Membership.mem D a → Membership.mem D b → LT.lt a b → ∀…","labels":[],"detail_key":"p36","name":"forall_exists_subdivision_diff_lt_of_dense","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32343","layer":"formal","project":"p36","title":"forall_exists_subdivision_dist_apply_lt_of_dense_of_continuous","kind":"theorem","summary":"∀ D : Set Real, Dense D → ∀ f : Real → Real, Continuous f → ∀ a b : Real, Membership.mem D a →…","labels":[],"detail_key":"p36","name":"forall_exists_subdivision_dist_apply_lt_of_dense_of_continuous","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32344","layer":"formal","project":"p36","title":"tendsto_of_forall_continuousAt_tendsto_cdf","kind":"theorem","summary":"∀ (μs : Nat → MeasureTheory.ProbabilityMeasure Real) (μ : MeasureTheory.ProbabilityMeasure Real…","labels":[],"detail_key":"p36","name":"tendsto_of_forall_continuousAt_tendsto_cdf","module":"ExtremeValueProject.WeakConvergenceCDF"},{"id":"n32345","layer":"informal","project":"p37","title":"model parameters","kind":"definition","summary":"[model parameters] A collection of \\emphmodel parameters is a tuple \\( (\\lambda, <_\\lambda, \\ka…","labels":["def:Params"],"detail_key":"p37"},{"id":"n32346","layer":"informal","project":"p37","title":"prop:Params.minimal","kind":"proposition","summary":"The tuple \\( ( N, <_ N, \\aleph_1, \\beth_\\omega_1) \\) is a collection of model parameters, where…","labels":["prop:Params.minimal"],"detail_key":"p37"},{"id":"n32347","layer":"informal","project":"p37","title":"Direct.","kind":"proof","summary":"Direct.","labels":[],"detail_key":"p37"},{"id":"n32348","layer":"informal","project":"p37","title":"type index","kind":"definition","summary":"[type index] The type of \\emphtype indices is \\( \\lambda^\\bot \\coloneq \\textttWithBot(\\lambda)…","labels":["def:TypeIndex"],"detail_key":"p37"},{"id":"n32349","layer":"informal","project":"p37","title":"small","kind":"definition","summary":"[small] A set \\( s : \\mathsfSet(\\tau) \\) is called \\emphsmall if \\( \\#s < \\#\\kappa \\). Smallnes…","labels":["def:Small"],"detail_key":"p37"},{"id":"n32350","layer":"informal","project":"p37","title":"litter","kind":"definition","summary":"[litter] A \\emphlitter is a triple \\( L = (\\nu, \\beta, \\gamma) : \\mu \\times \\lambda^\\bot \\times…","labels":["def:Litter"],"detail_key":"p37"},{"id":"n32351","layer":"informal","project":"p37","title":"atom","kind":"definition","summary":"[atom] An \\emphatom is a pair \\( a = (L, i) : L\\times \\kappa \\).\\footnoteThis should be formali…","labels":["def:Atom"],"detail_key":"p37"},{"id":"n32352","layer":"informal","project":"p37","title":"near-litter","kind":"definition","summary":"[near-litter] A \\emphnear-litter is a pair \\( N = (L, s) : L\\times \\mathsfSet A\\) such that \\(…","labels":["def:NearLitter"],"detail_key":"p37"},{"id":"n32353","layer":"informal","project":"p37","title":"base permutation","kind":"definition","summary":"[base permutation] A \\emphbase permutation is a pair \\( \\pi = (\\pi^ A, \\pi^ L) \\), where \\( \\pi…","labels":["def:BasePerm"],"detail_key":"p37"},{"id":"n32354","layer":"informal","project":"p37","title":"path","kind":"definition","summary":"[path] If \\( \\alpha, \\beta \\) are type indices, then a \\emphpath \\( \\alpha \\rightsquigarrow\\bet…","labels":["def:Path"],"detail_key":"p37"},{"id":"n32355","layer":"informal","project":"p37","title":"tree","kind":"definition","summary":"[tree] Let \\( \\tau \\) be any type, and let \\( \\alpha \\) be a type index. An \\emph\\( \\alpha \\)-t…","labels":["def:Tree"],"detail_key":"p37"},{"id":"n32356","layer":"informal","project":"p37","title":"structural permutation","kind":"definition","summary":"[structural permutation] Let \\( \\alpha \\) be a type index. Then an \\emph\\( \\alpha \\)-structural…","labels":["def:StrPerm"],"detail_key":"p37"},{"id":"n32357","layer":"informal","project":"p37","title":"enumeration","kind":"definition","summary":"[enumeration] Let \\( \\tau \\) be a type. An \\emphenumeration of \\( \\tau \\) is a pair \\( E = (i,…","labels":["def:Enumeration"],"detail_key":"p37"},{"id":"n32358","layer":"informal","project":"p37","title":"base support","kind":"definition","summary":"[base support] A \\emphbase support is a pair \\( S = (S^ A, S^ N) \\) where \\( S^ A\\) is an enume…","labels":["def:BaseSupport"],"detail_key":"p37"},{"id":"n32359","layer":"informal","project":"p37","title":"structural support","kind":"definition","summary":"[structural support] A \\emph\\( \\beta \\)-structural support (or just \\emph\\( \\beta \\)-support) i…","labels":["def:StrSupport"],"detail_key":"p37"},{"id":"n32360","layer":"informal","project":"p37","title":"structural set","kind":"definition","summary":"[structural set] The type of \\emph\\( \\alpha \\)-structural sets, denoted \\( \\mathsfStrSet_\\alpha…","labels":["def:StrSet"],"detail_key":"p37"},{"id":"n32361","layer":"informal","project":"p37","title":"position function","kind":"definition","summary":"[position function] Let \\( \\tau \\) be a type. A \\emphposition function for \\( \\tau \\) is an inj…","labels":["def:Position"],"detail_key":"p37"},{"id":"n32362","layer":"informal","project":"p37","title":"injective functions from denied sets","kind":"proposition","summary":"[injective functions from denied sets] Let \\( \\tau \\) be a type such that \\( \\#\\tau \\leq \\#\\mu…","labels":["prop:funOfDeny"],"detail_key":"p37"},{"id":"n32363","layer":"informal","project":"p37","title":"Pick a well-ordering \\( \\prec \\) of \\( \\tau \\) of length at most \\( \\mathsford(\\#\\mu) \\).…","kind":"proof","summary":"Pick a well-ordering \\( \\prec \\) of \\( \\tau \\) of length at most \\( \\mathsford(\\#\\mu) \\). Defin…","labels":[],"detail_key":"p37"},{"id":"n32364","layer":"informal","project":"p37","title":"base positions","kind":"proposition","summary":"[base positions] There are position functions on \\( A, N\\) that are jointly injective and satis…","labels":["prop:BasePositions"],"detail_key":"p37"},{"id":"n32365","layer":"informal","project":"p37","title":"First, establish an equivalence \\( f_ L: L\\simeq \\mu \\). Use \\crefprop:funOfDeny to obtai…","kind":"proof","summary":"First, establish an equivalence \\( f_ L: L\\simeq \\mu \\). Use \\crefprop:funOfDeny to obtain an i…","labels":[],"detail_key":"p37"},{"id":"n32366","layer":"informal","project":"p37","title":"model data","kind":"definition","summary":"[model data] Let \\( \\alpha \\) be a type index. \\emphModel data at type \\( \\alpha \\) consists of…","labels":["def:ModelData"],"detail_key":"p37"},{"id":"n32367","layer":"informal","project":"p37","title":"tangle","kind":"definition","summary":"[tangle] Let \\( \\alpha \\) be a type index with model data. An \\emph\\( \\alpha \\)-tangle is a pai…","labels":["def:Tangle"],"detail_key":"p37"},{"id":"n32368","layer":"informal","project":"p37","title":"fuzz maps","kind":"proposition","summary":"[fuzz maps] Let \\( \\beta \\) be a type index with model data, and suppose that \\( \\mathsfTang_\\b…","labels":["prop:fuzz"],"detail_key":"p37"},{"id":"n32369","layer":"informal","project":"p37","title":"We define \\( g : \\mathsfTang_\\beta \\to \\mu \\) by \\crefprop:funOfDeny, where the denied se…","kind":"proof","summary":"We define \\( g : \\mathsfTang_\\beta \\to \\mu \\) by \\crefprop:funOfDeny, where the denied sets are…","labels":[],"detail_key":"p37"},{"id":"n32370","layer":"informal","project":"p37","title":"inflexible path","kind":"definition","summary":"[inflexible path] Let \\( \\alpha \\) be a proper type index. Suppose that we have model data for…","labels":["def:InflexiblePath"],"detail_key":"p37"},{"id":"n32371","layer":"informal","project":"p37","title":"typed near-litter","kind":"definition","summary":"[typed near-litter] Let \\( \\alpha \\) be a proper type index with model data, and suppose that \\…","labels":["def:TypedNearLitter"],"detail_key":"p37"},{"id":"n32372","layer":"informal","project":"p37","title":"coherent data","kind":"definition","summary":"[coherent data] Let \\( \\alpha \\) be a proper type index. Suppose that we have model data for al…","labels":["def:CoherentData"],"detail_key":"p37"},{"id":"n32373","layer":"informal","project":"p37","title":"code","kind":"definition","summary":"[code] A \\emphcode is a pair \\( c = (\\beta, s) \\) where \\( \\beta < \\alpha \\) is a type index an…","labels":["def:Code"],"detail_key":"p37"},{"id":"n32374","layer":"informal","project":"p37","title":"cloud","kind":"definition","summary":"[cloud] The \\emphcloud relation \\( \\prec \\) on codes is given by the constructor \\[ (\\beta, s)…","labels":["def:cloud"],"detail_key":"p37"},{"id":"n32375","layer":"informal","project":"p37","title":"prop:eq_of_cloud","kind":"proposition","summary":"If \\( c \\prec (\\gamma, s_1) \\) and \\( c \\prec (\\gamma, s_2) \\), then \\( s_1 = s_2 \\).","labels":["prop:eq_of_cloud"],"detail_key":"p37"},{"id":"n32376","layer":"informal","project":"p37","title":"Let \\( c = (\\beta, s) \\). We obtain \\[ s_1 = \\ \\mathsftyped_\\gamma(N) \\mid \\exists t : \\m…","kind":"proof","summary":"Let \\( c = (\\beta, s) \\). We obtain \\[ s_1 = \\ \\mathsftyped_\\gamma(N) \\mid \\exists t : \\mathsfT…","labels":[],"detail_key":"p37"},{"id":"n32377","layer":"informal","project":"p37","title":"prop:cloud_injective","kind":"proposition","summary":"The cloud relation is injective (\\crefdef:relation_props). That is, if \\( c_1, c_2 \\prec d \\),…","labels":["prop:cloud_injective"],"detail_key":"p37"},{"id":"n32378","layer":"informal","project":"p37","title":"Let \\( c_i = (\\beta_i, s_i) \\) for \\( i = 1, 2 \\), and let \\( d = (\\gamma, s') \\). We fir…","kind":"proof","summary":"Let \\( c_i = (\\beta_i, s_i) \\) for \\( i = 1, 2 \\), and let \\( d = (\\gamma, s') \\). We first sho…","labels":[],"detail_key":"p37"},{"id":"n32379","layer":"informal","project":"p37","title":"prop:cloud_wf","kind":"proposition","summary":"The cloud relation is well-founded.","labels":["prop:cloud_wf"],"detail_key":"p37"},{"id":"n32380","layer":"informal","project":"p37","title":"Define a function \\( F \\) that maps a code \\( c = (\\beta, s) \\) to the set \\[ \\ \\iota(t)…","kind":"proof","summary":"Define a function \\( F \\) that maps a code \\( c = (\\beta, s) \\) to the set \\[ \\ \\iota(t) \\mid \\…","labels":[],"detail_key":"p37"},{"id":"n32381","layer":"informal","project":"p37","title":"prop:odd_iff_not_even","kind":"proposition","summary":"Let \\( \\prec \\) be a relation on a type \\( \\tau \\). We say that an object \\( x : \\tau \\) is \\em…","labels":["prop:odd_iff_not_even"],"detail_key":"p37"},{"id":"n32382","layer":"informal","project":"p37","title":"\\emphPart 1. Direct from the definition. \\emphPart 2. We show this by induction along \\(…","kind":"proof","summary":"\\emphPart 1. Direct from the definition. \\emphPart 2. We show this by induction along \\( \\prec…","labels":[],"detail_key":"p37"},{"id":"n32383","layer":"informal","project":"p37","title":"def:Code.Represents","kind":"definition","summary":"We define the relation \\( \\looparrowright \\) between codes by the following two constructors. \\…","labels":["def:Code.Represents"],"detail_key":"p37"},{"id":"n32384","layer":"informal","project":"p37","title":"Proof of claim","kind":"proof","summary":"[Proof of claim] If \\( d \\) is even, then \\( d \\looparrowright d \\). If \\( c \\) is any other ev…","labels":[],"detail_key":"p37"},{"id":"n32385","layer":"informal","project":"p37","title":"extensionality","kind":"proposition","summary":"[extensionality] Let \\( x : \\mathsfTSet_\\beta \\) for some type index \\( \\beta < \\alpha \\), and…","labels":["prop:Code.ext"],"detail_key":"p37"},{"id":"n32386","layer":"informal","project":"p37","title":"Suppose that there is no \\( x : \\mathsfTSet_\\beta \\) such that \\( x \\in_\\beta c_1 \\). The…","kind":"proof","summary":"Suppose that there is no \\( x : \\mathsfTSet_\\beta \\) such that \\( x \\in_\\beta c_1 \\). Then it i…","labels":[],"detail_key":"p37"},{"id":"n32387","layer":"informal","project":"p37","title":"new allowable permutation","kind":"definition","summary":"[new allowable permutation] A \\emphnew allowable permutation is a dependent function \\( \\rho \\)…","labels":["def:NewAllPerm"],"detail_key":"p37"},{"id":"n32388","layer":"informal","project":"p37","title":"prop:AllPerm.smul_cloud_smul","kind":"proposition","summary":"Define an action of allowable permutations on codes by \\[ \\rho(\\beta, s) = (\\beta, \\rho(\\beta)[…","labels":["prop:AllPerm.smul_cloud_smul"],"detail_key":"p37"},{"id":"n32389","layer":"informal","project":"p37","title":"\\emphPart 1. Suppose that \\( c \\prec d \\). Then, writing \\( c = (\\beta, s) \\) and \\( d =…","kind":"proof","summary":"\\emphPart 1. Suppose that \\( c \\prec d \\). Then, writing \\( c = (\\beta, s) \\) and \\( d = (\\gamm…","labels":[],"detail_key":"p37"},{"id":"n32390","layer":"informal","project":"p37","title":"new t-set","kind":"definition","summary":"[new t-set] A \\emphnew t-set is an even code \\( c \\) such that there is an \\( \\alpha \\)-support…","labels":["def:NewTSet"],"detail_key":"p37"},{"id":"n32391","layer":"informal","project":"p37","title":"new model data","kind":"definition","summary":"[new model data] Given model data, position functions, and typed near-litters for all types \\(…","labels":["def:NewModelData"],"detail_key":"p37"},{"id":"n32392","layer":"informal","project":"p37","title":"typed near-litters","kind":"definition","summary":"[typed near-litters] We define a function \\( \\mathsftyped_\\alpha \\) from the type of near-litte…","labels":["def:newTypedNearLitter"],"detail_key":"p37"},{"id":"n32393","layer":"informal","project":"p37","title":"singletons","kind":"definition","summary":"[singletons] We define a function \\( \\mathsfsingleton_\\alpha \\) for each lower type index \\( \\b…","labels":["def:newSingleton"],"detail_key":"p37"},{"id":"n32394","layer":"informal","project":"p37","title":"position function","kind":"proposition","summary":"[position function] Using the model data from \\crefdef:NewModelData, if \\( \\#\\mathsfTang_\\alpha…","labels":["prop:newPos"],"detail_key":"p37"},{"id":"n32395","layer":"informal","project":"p37","title":"We use \\crefprop:funOfDeny to construct the position function, using denied set \\[ D(t) =…","kind":"proof","summary":"We use \\crefprop:funOfDeny to construct the position function, using denied set \\[ D(t) = \\ \\io…","labels":[],"detail_key":"p37"},{"id":"n32396","layer":"informal","project":"p37","title":"base approximation","kind":"definition","summary":"[base approximation] A \\emphbase approximation is a pair \\( \\psi = (\\psi^E A, \\psi^ L) \\) such…","labels":["def:BaseApprox"],"detail_key":"p37"},{"id":"n32397","layer":"informal","project":"p37","title":"atom graph of an approximation","kind":"definition","summary":"[atom graph of an approximation] The \\emphtypical atom graph of \\( \\psi \\) is the relation \\( \\…","labels":["def:atomGraph"],"detail_key":"p37"},{"id":"n32398","layer":"informal","project":"p37","title":"prop:atomGraph_inv","kind":"proposition","summary":"\\( (\\psi^T A)^-1 = (\\psi^-1)^T A \\) and hence \\( (\\psi^ A)^-1 = (\\psi^-1)^ A\\).","labels":["prop:atomGraph_inv"],"detail_key":"p37"},{"id":"n32399","layer":"informal","project":"p37","title":"This follows directly from the fact that \\( L_\\psi = L_\\psi^-1 \\) for any litter \\( L \\).","kind":"proof","summary":"This follows directly from the fact that \\( L_\\psi = L_\\psi^-1 \\) for any litter \\( L \\).","labels":[],"detail_key":"p37"},{"id":"n32400","layer":"informal","project":"p37","title":"prop:atomGraph_permutative","kind":"proposition","summary":"The graphs \\( \\psi^T A \\) and \\( \\psi^ A\\) are permutative.","labels":["prop:atomGraph_permutative"],"detail_key":"p37"},{"id":"n32401","layer":"informal","project":"p37","title":"The typical atom graph is injective, because the equation \\( h_L_\\psi(i)^\\circ = L \\) can…","kind":"proof","summary":"The typical atom graph is injective, because the equation \\( h_L_\\psi(i)^\\circ = L \\) can be us…","labels":[],"detail_key":"p37"},{"id":"n32402","layer":"informal","project":"p37","title":"prop:comp_atomGraph","kind":"proposition","summary":"If \\( \\psi, \\chi \\) have equal exceptional atom and litter coimages, then \\( (\\psi \\circ \\chi)^…","labels":["prop:comp_atomGraph"],"detail_key":"p37"},{"id":"n32403","layer":"informal","project":"p37","title":"Suppose that \\( (a_1, a_3) \\in (\\psi \\circ \\chi)^T A \\), so \\[ a_1 = h_(L_1)_\\psi \\circ \\…","kind":"proof","summary":"Suppose that \\( (a_1, a_3) \\in (\\psi \\circ \\chi)^T A \\), so \\[ a_1 = h_(L_1)_\\psi \\circ \\chi(i)…","labels":[],"detail_key":"p37"},{"id":"n32404","layer":"informal","project":"p37","title":"near-litter graph of an approximation","kind":"definition","summary":"[near-litter graph of an approximation] The \\emphnear-litter graph","labels":["def:nearLitterGraph"],"detail_key":"p37"},{"id":"n32405","layer":"informal","project":"p37","title":"prop:approx_near","kind":"proposition","summary":"Let \\( s \\) be a set of atoms near \\( \\mathsfLS(L) \\) for some litter \\( L \\). If \\( (L, L') \\i…","labels":["prop:approx_near"],"detail_key":"p37"},{"id":"n32406","layer":"informal","project":"p37","title":"We calculate \\mathsfim\\psi^ A|_s &= \\mathsfim\\psi^ A|_\\mathsfLS(L) \\mathrel\\raisebox1pt\\(…","kind":"proof","summary":"We calculate \\mathsfim\\psi^ A|_s &= \\mathsfim\\psi^ A|_\\mathsfLS(L) \\mathrel\\raisebox1pt\\( \\math…","labels":[],"detail_key":"p37"},{"id":"n32407","layer":"informal","project":"p37","title":"prop:nearLitterGraph_permutative","kind":"proposition","summary":"\\( (\\psi^-1)^ N= (\\psi^ N)^-1 \\), and \\( \\psi^ N\\) is permutative.","labels":["prop:nearLitterGraph_permutative"],"detail_key":"p37"},{"id":"n32408","layer":"informal","project":"p37","title":"The first part follows from \\crefprop:atomGraph_inv. To show \\( \\psi^ N\\) is permutative,…","kind":"proof","summary":"The first part follows from \\crefprop:atomGraph_inv. To show \\( \\psi^ N\\) is permutative, it su…","labels":[],"detail_key":"p37"},{"id":"n32409","layer":"informal","project":"p37","title":"def:smulApproxSupport","kind":"definition","summary":"Base approximations act on base supports in the following way. If \\( S^ A= (i, f) \\), then \\( \\…","labels":["def:smulApproxSupport"],"detail_key":"p37"},{"id":"n32410","layer":"informal","project":"p37","title":"def:BaseApprox.LE","kind":"definition","summary":"We define a partial order on base approximations by setting \\( \\psi \\leq \\chi \\) when \\( \\psi^E…","labels":["def:BaseApprox.LE"],"detail_key":"p37"},{"id":"n32411","layer":"informal","project":"p37","title":"adding orbits","kind":"proposition","summary":"[adding orbits] Let \\( \\psi \\) be a base approximation, and let \\( L : N \\to L\\) be a function…","labels":["prop:BaseApprox.addOrbit"],"detail_key":"p37"},{"id":"n32412","layer":"informal","project":"p37","title":"Define the relation \\[ R = \\ (L(n), L(n+1)) \\mid n : Z \\ \\] This clearly has equal image…","kind":"proof","summary":"Define the relation \\[ R = \\ (L(n), L(n+1)) \\mid n : Z \\ \\] This clearly has equal image and co…","labels":[],"detail_key":"p37"},{"id":"n32413","layer":"informal","project":"p37","title":"def:StrApprox","kind":"definition","summary":"For a type index \\( \\beta \\), a \\emph\\( \\beta \\)-approximation is a \\( \\beta \\)-tree of base ap…","labels":["def:StrApprox"],"detail_key":"p37"},{"id":"n32414","layer":"informal","project":"p37","title":"def:Inflexible","kind":"definition","summary":"Let \\( A \\) be a \\( \\beta \\)-extended type index. A litter \\( L \\) is \\emph\\( A \\)-inflexible i…","labels":["def:Inflexible"],"detail_key":"p37"},{"id":"n32415","layer":"informal","project":"p37","title":"def:StrApprox.Coherent","kind":"definition","summary":"A \\( \\beta \\)-approximation \\( \\psi \\) is \\emphcoherent at \\( (A, L_1, L_2) \\) if: \\item If \\(…","labels":["def:StrApprox.Coherent"],"detail_key":"p37"},{"id":"n32416","layer":"informal","project":"p37","title":"adding orbits coherently","kind":"proposition","summary":"[adding orbits coherently] Suppose that \\( \\psi \\) is an approximation and \\( L : Z \\to L\\) is…","labels":["prop:StrApprox.addOrbit"],"detail_key":"p37"},{"id":"n32417","layer":"informal","project":"p37","title":"This proof just relies on the fact that if \\( (\\psi_B)_\\delta(\\mathsfsupp(t)) = \\rho(\\mat…","kind":"proof","summary":"This proof just relies on the fact that if \\( (\\psi_B)_\\delta(\\mathsfsupp(t)) = \\rho(\\mathsfsup…","labels":[],"detail_key":"p37"},{"id":"n32418","layer":"informal","project":"p37","title":"prop:StrApprox.Coherent.inv","kind":"proposition","summary":"If \\( \\psi \\) is coherent, then \\( \\psi^-1 \\) is coherent.","labels":["prop:StrApprox.Coherent.inv"],"detail_key":"p37"},{"id":"n32419","layer":"informal","project":"p37","title":"Suppose that \\( (L_1, L_2) \\in (\\psi^-1_A)^ L\\), so \\( (L_2, L_1) \\in \\psi_A^ L\\). Suppos…","kind":"proof","summary":"Suppose that \\( (L_1, L_2) \\in (\\psi^-1_A)^ L\\), so \\( (L_2, L_1) \\in \\psi_A^ L\\). Suppose firs…","labels":[],"detail_key":"p37"},{"id":"n32420","layer":"informal","project":"p37","title":"prop:StrApprox.Coherent.comp","kind":"proposition","summary":"If \\( \\psi \\) and \\( \\chi \\) are coherent and have equal coimages along all paths, then \\( \\psi…","labels":["prop:StrApprox.Coherent.comp"],"detail_key":"p37"},{"id":"n32421","layer":"informal","project":"p37","title":"Suppose that \\( (L_1, L_3) \\in ((\\psi \\circ \\chi)_A)^ L\\), so \\( (L_1, L_2) \\in \\psi_A^ L…","kind":"proof","summary":"Suppose that \\( (L_1, L_3) \\in ((\\psi \\circ \\chi)_A)^ L\\), so \\( (L_1, L_2) \\in \\psi_A^ L\\) and…","labels":[],"detail_key":"p37"},{"id":"n32422","layer":"informal","project":"p37","title":"prop:StrApprox.Coherent.deriv","kind":"proposition","summary":"If \\( \\psi \\) is a coherent \\( \\beta \\)-approximation and \\( A \\) is a path \\( \\beta \\rightsqui…","labels":["prop:StrApprox.Coherent.deriv"],"detail_key":"p37"},{"id":"n32423","layer":"informal","project":"p37","title":"Let \\( (L_1, L_2) \\in (\\psi_A)_B^ L\\). Suppose that \\( L_1 \\) is \\( B \\)-inflexible with…","kind":"proof","summary":"Let \\( (L_1, L_2) \\in (\\psi_A)_B^ L\\). Suppose that \\( L_1 \\) is \\( B \\)-inflexible with path \\…","labels":[],"detail_key":"p37"},{"id":"n32424","layer":"informal","project":"p37","title":"approximates","kind":"definition","summary":"[approximates] We say that a \\( \\beta \\)-approximation \\( \\psi \\) \\emphapproximates a \\( \\beta…","labels":["def:StrApprox.Approximates"],"detail_key":"p37"},{"id":"n32425","layer":"informal","project":"p37","title":"freedom of action","kind":"definition","summary":"[freedom of action] We say that \\emphfreedom of action holds at a type index \\( \\delta \\) if ev…","labels":["def:FreedomOfAction"],"detail_key":"p37"},{"id":"n32426","layer":"informal","project":"p37","title":"adding flexible litters","kind":"proposition","summary":"[adding flexible litters] Let \\( \\psi \\) be a coherent \\( \\beta \\)-approximation, and let \\( L…","labels":["prop:StrApprox.addFlexible"],"detail_key":"p37"},{"id":"n32427","layer":"informal","project":"p37","title":"Define \\( L' : Z \\to L\\) by \\( L'(n) = L \\), then appeal to \\crefprop:StrApprox.addOrbit…","kind":"proof","summary":"Define \\( L' : Z \\to L\\) by \\( L'(n) = L \\), then appeal to \\crefprop:StrApprox.addOrbit to obt…","labels":[],"detail_key":"p37"},{"id":"n32428","layer":"informal","project":"p37","title":"adding inflexible litters","kind":"proposition","summary":"[adding inflexible litters] Let \\( \\psi \\) be a coherent \\( \\beta \\)-approximation, and let \\(…","labels":["prop:StrApprox.addInflexible"],"detail_key":"p37"},{"id":"n32429","layer":"informal","project":"p37","title":"Let \\( \\rho \\) be a \\( \\delta \\)-allowable permutation that \\( (\\psi_B)_\\delta \\) approxi…","kind":"proof","summary":"Let \\( \\rho \\) be a \\( \\delta \\)-allowable permutation that \\( (\\psi_B)_\\delta \\) approximates.…","labels":[],"detail_key":"p37"},{"id":"n32430","layer":"informal","project":"p37","title":"prop:StrApprox.chain","kind":"proposition","summary":"If \\( (\\psi_i)_i : I \\) is a chain of coherent approximations where \\( I \\) is a linear order,…","labels":["prop:StrApprox.chain"],"detail_key":"p37"},{"id":"n32431","layer":"informal","project":"p37","title":"Direct, using the same idea as the proof of \\crefprop:StrApprox.addOrbit.","kind":"proof","summary":"Direct, using the same idea as the proof of \\crefprop:StrApprox.addOrbit.","labels":[],"detail_key":"p37"},{"id":"n32432","layer":"informal","project":"p37","title":"freedom of action","kind":"theorem","summary":"[freedom of action] Freedom of action holds at all type indices \\( \\beta \\leq \\alpha \\).","labels":["thm:StrApprox.foa"],"detail_key":"p37"},{"id":"n32433","layer":"informal","project":"p37","title":"By induction, we may assume freedom of action holds at all \\( \\delta < \\beta \\). Let \\( \\…","kind":"proof","summary":"By induction, we may assume freedom of action holds at all \\( \\delta < \\beta \\). Let \\( \\psi \\)…","labels":[],"detail_key":"p37"},{"id":"n32434","layer":"informal","project":"p37","title":"def:Interference","kind":"definition","summary":"The \\emphinterference of near-litters \\( N_1, N_2 \\) is \\[ \\mathsfinterf(N_1, N_2) = N_1 \\mathr…","labels":["def:Interference"],"detail_key":"p37"},{"id":"n32435","layer":"informal","project":"p37","title":"def:BaseAction","kind":"definition","summary":"A \\emphbase action is a pair \\( \\xi = (\\xi^ A, \\xi^ N) \\) such that \\( \\xi^ A\\) and \\( \\xi^ N\\)…","labels":["def:BaseAction"],"detail_key":"p37"},{"id":"n32436","layer":"informal","project":"p37","title":"def:BaseAction.Nice","kind":"definition","summary":"A base action \\( \\xi \\) is \\emphnice if whenever \\( (N_1, N_2) \\in \\xi^ N\\), \\[ N_1 \\mathrel\\ra…","labels":["def:BaseAction.Nice"],"detail_key":"p37"},{"id":"n32437","layer":"informal","project":"p37","title":"extending orbits inside near-litters","kind":"proposition","summary":"[extending orbits inside near-litters] Every base action \\( \\xi \\) admits an extension \\( \\zeta…","labels":["prop:BaseAction.exists_inside"],"detail_key":"p37"},{"id":"n32438","layer":"informal","project":"p37","title":"For each litter \\( L \\), there is an injection \\[ i_L : \\bigcup_N \\in \\mathsfcoim\\xi^ N (…","kind":"proof","summary":"For each litter \\( L \\), there is an injection \\[ i_L : \\bigcup_N \\in \\mathsfcoim\\xi^ N (N \\set…","labels":[],"detail_key":"p37"},{"id":"n32439","layer":"informal","project":"p37","title":"extending orbits outside near-litters","kind":"proposition","summary":"[extending orbits outside near-litters] Every base action \\( \\xi \\) admits an extension \\( \\zet…","labels":["prop:BaseAction.exists_outside"],"detail_key":"p37"},{"id":"n32440","layer":"informal","project":"p37","title":"Without loss of generality (as extensions are transitive), let \\( \\xi \\) satisfy the conc…","kind":"proof","summary":"Without loss of generality (as extensions are transitive), let \\( \\xi \\) satisfy the conclusion…","labels":[],"detail_key":"p37"},{"id":"n32441","layer":"informal","project":"p37","title":"prop:BaseAction.exists_nice","kind":"proposition","summary":"Every base action has a nice extension.","labels":["prop:BaseAction.exists_nice"],"detail_key":"p37"},{"id":"n32442","layer":"informal","project":"p37","title":"Apply \\crefprop:BaseAction.exists_inside to \\( \\xi \\) to obtain \\( \\xi_1 \\); apply \\crefp…","kind":"proof","summary":"Apply \\crefprop:BaseAction.exists_inside to \\( \\xi \\) to obtain \\( \\xi_1 \\); apply \\crefprop:Ba…","labels":[],"detail_key":"p37"},{"id":"n32443","layer":"informal","project":"p37","title":"def:StrAction","kind":"definition","summary":"For a type index \\( \\beta \\), a \\emph\\( \\beta \\)-action is a \\( \\beta \\)-tree of base actions.…","labels":["def:StrAction"],"detail_key":"p37"},{"id":"n32444","layer":"informal","project":"p37","title":"def:StrAction.Coherent","kind":"definition","summary":"A \\( \\beta \\)-action \\( \\xi \\) is \\emphcoherent at \\( (A, L_1, L_2) \\) if: \\item If \\( L_1 \\) i…","labels":["def:StrAction.Coherent"],"detail_key":"p37"},{"id":"n32445","layer":"informal","project":"p37","title":"def:FlexApprox","kind":"definition","summary":"Let \\( A : \\beta \\rightsquigarrow\\bot \\). An \\emph\\( A \\)-flexible approximation of a base acti…","labels":["def:FlexApprox"],"detail_key":"p37"},{"id":"n32446","layer":"informal","project":"p37","title":"prop:exists_flexApprox","kind":"proposition","summary":"Every base action has an \\( A \\)-flexible approximation. Hence, every \\( \\beta \\)-action has a…","labels":["prop:exists_flexApprox"],"detail_key":"p37"},{"id":"n32447","layer":"informal","project":"p37","title":"If \\( \\xi \\leq \\zeta \\) and \\( \\psi \\) is an \\( A \\)-flexible approximation for \\( \\zeta…","kind":"proof","summary":"If \\( \\xi \\leq \\zeta \\) and \\( \\psi \\) is an \\( A \\)-flexible approximation for \\( \\zeta \\), th…","labels":[],"detail_key":"p37"},{"id":"n32448","layer":"informal","project":"p37","title":"approximates","kind":"definition","summary":"[approximates] We say that a \\( \\beta \\)-action \\( \\xi \\) \\emphapproximates a \\( \\beta \\)-allow…","labels":["def:StrAction.Approximates"],"detail_key":"p37"},{"id":"n32449","layer":"informal","project":"p37","title":"prop:FlexApprox.smul_nearLitter_eq","kind":"proposition","summary":"Let \\( \\xi \\) be a base action, and let \\( \\psi \\) be an \\( A \\)-flexible approximation of it.…","labels":["prop:FlexApprox.smul_nearLitter_eq"],"detail_key":"p37"},{"id":"n32450","layer":"informal","project":"p37","title":"First, note that \\pi[N_1] &= \\pi[\\mathsfLS(N_1^\\circ)] \\mathrel\\raisebox1pt\\( \\mathsmalle…","kind":"proof","summary":"First, note that \\pi[N_1] &= \\pi[\\mathsfLS(N_1^\\circ)] \\mathrel\\raisebox1pt\\( \\mathsmaller\\tria…","labels":[],"detail_key":"p37"},{"id":"n32451","layer":"informal","project":"p37","title":"prop:approximates_of_flexApprox","kind":"proposition","summary":"Let \\( \\xi \\) be a coherent \\( \\beta \\)-action, and let \\( \\psi \\) be a flexible approximation…","labels":["prop:approximates_of_flexApprox"],"detail_key":"p37"},{"id":"n32452","layer":"informal","project":"p37","title":"First, note that \\( \\xi_A^ A\\leq \\psi_A^E A \\) and \\( \\psi_A^ A\\leq \\rho_A^ A\\) give the…","kind":"proof","summary":"First, note that \\( \\xi_A^ A\\leq \\psi_A^E A \\) and \\( \\psi_A^ A\\leq \\rho_A^ A\\) give the requir…","labels":[],"detail_key":"p37"},{"id":"n32453","layer":"informal","project":"p37","title":"freedom of action for actions","kind":"theorem","summary":"[freedom of action for actions] Every coherent action approximates some allowable permutation.","labels":["thm:StrAction.foa"],"detail_key":"p37"},{"id":"n32454","layer":"informal","project":"p37","title":"Let \\( \\xi \\) be a coherent \\( \\beta \\)-action, and let \\( \\psi \\) be a flexible approxim…","kind":"proof","summary":"Let \\( \\xi \\) be a coherent \\( \\beta \\)-action, and let \\( \\psi \\) be a flexible approximation…","labels":[],"detail_key":"p37"},{"id":"n32455","layer":"informal","project":"p37","title":"def:StrSupport.Occurs","kind":"definition","summary":"We define a preorder \\( \\preceq \\) on base supports by \\( S \\preceq T \\) if and only if \\( \\mat…","labels":["def:StrSupport.Occurs"],"detail_key":"p37"},{"id":"n32456","layer":"informal","project":"p37","title":"def:Strong","kind":"definition","summary":"A \\( \\beta \\)-support \\( S \\) is \\emphstrong if: \\item for every pair of near-litters \\( N_1, N…","labels":["def:Strong"],"detail_key":"p37"},{"id":"n32457","layer":"informal","project":"p37","title":"prop:Strong.smul","kind":"proposition","summary":"If \\( S \\) is a strong \\( \\beta \\)-support and \\( \\rho \\) is \\( \\beta \\)-allowable, then \\( \\rh…","labels":["prop:Strong.smul"],"detail_key":"p37"},{"id":"n32458","layer":"informal","project":"p37","title":"Interference is stable under application of allowable permutations, and the required supp…","kind":"proof","summary":"Interference is stable under application of allowable permutations, and the required supports a…","labels":[],"detail_key":"p37"},{"id":"n32459","layer":"informal","project":"p37","title":"prop:exists_strong","kind":"proposition","summary":"For every support \\( S \\), there is a strong support \\( T \\succeq S \\).","labels":["prop:exists_strong"],"detail_key":"p37"},{"id":"n32460","layer":"informal","project":"p37","title":"We define a relation \\( R \\) on pairs \\( (A, N) \\) where \\( A : \\beta \\rightsquigarrow\\bo…","kind":"proof","summary":"We define a relation \\( R \\) on pairs \\( (A, N) \\) where \\( A : \\beta \\rightsquigarrow\\bot \\) a…","labels":[],"detail_key":"p37"},{"id":"n32461","layer":"informal","project":"p37","title":"def:SupportOrbit","kind":"definition","summary":"For a type index \\( \\beta \\leq \\alpha \\), a \\emph\\( \\beta \\)-support orbit is the quotient of \\…","labels":["def:SupportOrbit"],"detail_key":"p37"},{"id":"n32462","layer":"informal","project":"p37","title":"def:CodingFunction","kind":"definition","summary":"For any type index \\( \\beta \\leq \\alpha \\), a \\emph\\( \\beta \\)-coding function is a relation \\(…","labels":["def:CodingFunction"],"detail_key":"p37"},{"id":"n32463","layer":"informal","project":"p37","title":"extensionality for coding functions","kind":"proposition","summary":"[extensionality for coding functions] Let \\( \\chi_1, \\chi_2 \\) be \\( \\beta \\)-coding functions.…","labels":["prop:CodingFunction.ext"],"detail_key":"p37"},{"id":"n32464","layer":"informal","project":"p37","title":"We show \\( \\chi_1 \\subseteq \\chi_2 \\); the result then follows by antisymmetry. Suppose \\…","kind":"proof","summary":"We show \\( \\chi_1 \\subseteq \\chi_2 \\); the result then follows by antisymmetry. Suppose \\( (T,…","labels":[],"detail_key":"p37"},{"id":"n32465","layer":"informal","project":"p37","title":"def:code","kind":"definition","summary":"Let \\( t : \\mathsfTang_\\beta \\). Then we define the coding function \\( \\chi_t \\) by the constru…","labels":["def:code"],"detail_key":"p37"},{"id":"n32466","layer":"informal","project":"p37","title":"def:code_eq_code_iff","kind":"proposition","summary":"Let \\( t, u : \\mathsfTang_\\beta \\). Then \\( \\chi_t = \\chi_u \\) if and only if there is a \\( \\be…","labels":["def:code_eq_code_iff"],"detail_key":"p37"},{"id":"n32467","layer":"informal","project":"p37","title":"If \\( \\rho(t) = u \\), then \\( (\\mathsfsupp(t), \\mathsfset(t)) \\in \\chi_t \\) implies \\( (\\…","kind":"proof","summary":"If \\( \\rho(t) = u \\), then \\( (\\mathsfsupp(t), \\mathsfset(t)) \\in \\chi_t \\) implies \\( (\\mathsf…","labels":[],"detail_key":"p37"},{"id":"n32468","layer":"informal","project":"p37","title":"def:Spec","kind":"definition","summary":"An \\emphatom condition is a pair \\( (s, t) \\) where \\( s, t : \\mathsfSet\\kappa \\). A \\emph\\( \\b…","labels":["def:Spec"],"detail_key":"p37"},{"id":"n32469","layer":"informal","project":"p37","title":"def:StrSupport.spec","kind":"definition","summary":"Let \\( S \\) be a \\( \\beta \\)-support. Then \\emphits specification is the \\( \\beta \\)-specificat…","labels":["def:StrSupport.spec"],"detail_key":"p37"},{"id":"n32470","layer":"informal","project":"p37","title":"prop:spec_eq_spec_iff","kind":"proposition","summary":"Let \\( S, T \\) be \\( \\beta \\)-supports. Then \\( \\mathsfspec(S) = \\mathsfspec(T) \\) if and only…","labels":["prop:spec_eq_spec_iff"],"detail_key":"p37"},{"id":"n32471","layer":"informal","project":"p37","title":"We will only sketch the fourth bullet point of this proof; the remainder is direct (but q…","kind":"proof","summary":"We will only sketch the fourth bullet point of this proof; the remainder is direct (but quite l…","labels":[],"detail_key":"p37"},{"id":"n32472","layer":"informal","project":"p37","title":"prop:spec_smul","kind":"proposition","summary":"Let \\( \\rho \\) be \\( \\beta \\)-allowable, and let \\( S \\) be a \\( \\beta \\)-support. Then \\( \\mat…","labels":["prop:spec_smul"],"detail_key":"p37"},{"id":"n32473","layer":"informal","project":"p37","title":"We appeal to \\crefprop:spec_eq_spec_iff. Clearly the coimage condition holds. For the ato…","kind":"proof","summary":"We appeal to \\crefprop:spec_eq_spec_iff. Clearly the coimage condition holds. For the atom cond…","labels":[],"detail_key":"p37"},{"id":"n32474","layer":"informal","project":"p37","title":"def:conv_base","kind":"definition","summary":"Let \\( S \\) and \\( T \\) be base supports. We define the relations \\( \\mathsfconv_S,T^ A, \\maths…","labels":["def:conv_base"],"detail_key":"p37"},{"id":"n32475","layer":"informal","project":"p37","title":"prop:conv_one_to_one","kind":"proposition","summary":"Let \\( S, T \\) be supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\). Then \\( \\mathsfconv…","labels":["prop:conv_one_to_one"],"detail_key":"p37"},{"id":"n32476","layer":"informal","project":"p37","title":"If \\( (a_1, a_2), (a_1, a_3) \\in \\mathsfconv_S_A, T_A^ A\\), then there are \\( i, j \\) suc…","kind":"proof","summary":"If \\( (a_1, a_2), (a_1, a_3) \\in \\mathsfconv_S_A, T_A^ A\\), then there are \\( i, j \\) such that…","labels":[],"detail_key":"p37"},{"id":"n32477","layer":"informal","project":"p37","title":"prop:conv_mem_nearLitter_iff","kind":"proposition","summary":"Let \\( S, T \\) be supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\). If \\( (a_1, a_2) \\i…","labels":["prop:conv_mem_nearLitter_iff"],"detail_key":"p37"},{"id":"n32478","layer":"informal","project":"p37","title":"As \\( (a_1, a_2) \\in \\mathsfconv_S_A, T_A^ A\\), there is \\( i \\) such that \\( (i, a_1) \\i…","kind":"proof","summary":"As \\( (a_1, a_2) \\in \\mathsfconv_S_A, T_A^ A\\), there is \\( i \\) such that \\( (i, a_1) \\in S_A^…","labels":[],"detail_key":"p37"},{"id":"n32479","layer":"informal","project":"p37","title":"prop:conv_circ_eq_circ_iff","kind":"proposition","summary":"Let \\( S, T \\) be supports such that \\( T \\) is strong and \\( \\mathsfspec(S) = \\mathsfspec(T) \\…","labels":["prop:conv_circ_eq_circ_iff"],"detail_key":"p37"},{"id":"n32480","layer":"informal","project":"p37","title":"There are \\( i, j \\) such that \\( (i, N_1), (j, N_2) \\in S_A^ N\\) and \\( (i, N_3), (j, N_…","kind":"proof","summary":"There are \\( i, j \\) such that \\( (i, N_1), (j, N_2) \\in S_A^ N\\) and \\( (i, N_3), (j, N_4) \\in…","labels":[],"detail_key":"p37"},{"id":"n32481","layer":"informal","project":"p37","title":"prop:conv_interf","kind":"proposition","summary":"Let \\( S, T \\) be strong supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\). Then for eac…","labels":["prop:conv_interf"],"detail_key":"p37"},{"id":"n32482","layer":"informal","project":"p37","title":"As \\( S \\) is strong, we have \\( \\mathsfinterf(N_1, N_2) \\subseteq \\mathsfimS_A^ A\\). But…","kind":"proof","summary":"As \\( S \\) is strong, we have \\( \\mathsfinterf(N_1, N_2) \\subseteq \\mathsfimS_A^ A\\). But \\( \\m…","labels":[],"detail_key":"p37"},{"id":"n32483","layer":"informal","project":"p37","title":"def:conv","kind":"definition","summary":"Let \\( S, T \\) be strong \\( \\beta \\)-supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\).…","labels":["def:conv"],"detail_key":"p37"},{"id":"n32484","layer":"informal","project":"p37","title":"prop:conv_coherent","kind":"proposition","summary":"Let \\( S, T \\) be strong supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\). Then \\( \\mat…","labels":["prop:conv_coherent"],"detail_key":"p37"},{"id":"n32485","layer":"informal","project":"p37","title":"Suppose that \\( (N_1, N_2) \\in \\mathsfconv_S_A,T_A^ N\\), so there is \\( i \\) such that \\(…","kind":"proof","summary":"Suppose that \\( (N_1, N_2) \\in \\mathsfconv_S_A,T_A^ N\\), so there is \\( i \\) such that \\( (i, N…","labels":[],"detail_key":"p37"},{"id":"n32486","layer":"informal","project":"p37","title":"prop:exists_allowable_of_spec_eq_spec","kind":"proposition","summary":"Let \\( S, T \\) be strong supports such that \\( \\mathsfspec(S) = \\mathsfspec(T) \\). Then there i…","labels":["prop:exists_allowable_of_spec_eq_spec"],"detail_key":"p37"},{"id":"n32487","layer":"informal","project":"p37","title":"By \\crefprop:conv_coherent, we may apply \\crefthm:StrAction.foa to \\( \\mathsfconv_S,T \\)…","kind":"proof","summary":"By \\crefprop:conv_coherent, we may apply \\crefthm:StrAction.foa to \\( \\mathsfconv_S,T \\) to obt…","labels":[],"detail_key":"p37"},{"id":"n32488","layer":"informal","project":"p37","title":"def:Combination","kind":"definition","summary":"Let \\( \\gamma < \\beta \\) be proper type indices at most \\( \\alpha \\). An object \\( x : \\mathsfT…","labels":["def:Combination"],"detail_key":"p37"},{"id":"n32489","layer":"informal","project":"p37","title":"prop:Combination.smul","kind":"proposition","summary":"If \\( x \\) is a \\( \\gamma \\)-combination of \\( s \\) with respect to \\( S \\) then \\( \\rho(x) \\)…","labels":["prop:Combination.smul"],"detail_key":"p37"},{"id":"n32490","layer":"informal","project":"p37","title":"We can calculate U_\\beta(\\rho(x))(\\gamma) &= \\rho_\\gamma[U_\\beta(x)(\\gamma)] \\\\ &= \\rho_\\…","kind":"proof","summary":"We can calculate U_\\beta(\\rho(x))(\\gamma) &= \\rho_\\gamma[U_\\beta(x)(\\gamma)] \\\\ &= \\rho_\\gamma\\…","labels":[],"detail_key":"p37"},{"id":"n32491","layer":"informal","project":"p37","title":"def:raisedCodingFunction","kind":"definition","summary":"Let \\( s \\) be a set of \\( \\beta \\)-coding functions, and let \\( o \\) be a \\( \\beta \\)-support…","labels":["def:raisedCodingFunction"],"detail_key":"p37"},{"id":"n32492","layer":"informal","project":"p37","title":"prop:raisedCodingFunction_spec","kind":"proposition","summary":"The \\( (\\gamma,\\beta) \\)-raised coding function for \\( (s, o) \\) is a coding function.","labels":["prop:raisedCodingFunction_spec"],"detail_key":"p37"},{"id":"n32493","layer":"informal","project":"p37","title":"Coinjectivity follows from uniqueness of combinations. The nonemptiness and support orbit…","kind":"proof","summary":"Coinjectivity follows from uniqueness of combinations. The nonemptiness and support orbit condi…","labels":[],"detail_key":"p37"},{"id":"n32494","layer":"informal","project":"p37","title":"designated support","kind":"definition","summary":"[designated support] For a type index \\( \\beta \\leq \\alpha \\), a \\emph\\( \\beta \\)-set orbit is…","labels":["def:designatedSupport"],"detail_key":"p37"},{"id":"n32495","layer":"informal","project":"p37","title":"def:raisedSingleton","kind":"definition","summary":"Let \\( \\gamma < \\beta \\) be proper type indices. Let \\( S \\) be a \\( \\beta \\)-support and let \\…","labels":["def:raisedSingleton"],"detail_key":"p37"},{"id":"n32496","layer":"informal","project":"p37","title":"prop:raise_combination","kind":"proposition","summary":"Let \\( \\gamma < \\beta \\) be proper type indices, and let \\( x : \\mathsfTSet_\\beta \\). Then for…","labels":["prop:raise_combination"],"detail_key":"p37"},{"id":"n32497","layer":"informal","project":"p37","title":"We must show that \\[ U_\\beta(x)(\\gamma) = \\bigcup_u \\in U_\\beta(x)(\\gamma),\\,(V,v) \\in \\m…","kind":"proof","summary":"We must show that \\[ U_\\beta(x)(\\gamma) = \\bigcup_u \\in U_\\beta(x)(\\gamma),\\,(V,v) \\in \\mathsfr…","labels":[],"detail_key":"p37"},{"id":"n32498","layer":"informal","project":"p37","title":"prop:recode","kind":"proposition","summary":"Let \\( \\gamma < \\beta \\) be proper type indices, and let \\( \\chi \\) be a \\( \\beta \\)-coding fun…","labels":["prop:recode"],"detail_key":"p37"},{"id":"n32499","layer":"informal","project":"p37","title":"Let \\( \\chi \\) be a \\( \\beta \\)-coding function, and let \\( (S, x) \\in \\chi \\). Let \\[ s…","kind":"proof","summary":"Let \\( \\chi \\) be a \\( \\beta \\)-coding function, and let \\( (S, x) \\in \\chi \\). Let \\[ s = \\ \\m…","labels":[],"detail_key":"p37"},{"id":"n32500","layer":"informal","project":"p37","title":"the swap permutation","kind":"proposition","summary":"[the swap permutation] Let \\( S \\) be a base support that is closed under interference of near-…","labels":["prop:exists_swap"],"detail_key":"p37"},{"id":"n32501","layer":"informal","project":"p37","title":"Let \\( i \\) be an index that does not occur in \\( \\mathsfcoimS^ A\\), and define \\( T_1, T…","kind":"proof","summary":"Let \\( i \\) be an index that does not occur in \\( \\mathsfcoimS^ A\\), and define \\( T_1, T_2 \\)…","labels":[],"detail_key":"p37"},{"id":"n32502","layer":"informal","project":"p37","title":"prop:supports_atoms_iff","kind":"proposition","summary":"Let \\( S \\) be a base support that is closed under interference of near-litters. Suppose that \\…","labels":["prop:supports_atoms_iff"],"detail_key":"p37"},{"id":"n32503","layer":"informal","project":"p37","title":"Let \\( a_1, a_2 \\) be atoms that satisfy the two statements. By \\crefprop:exists_swap, th…","kind":"proof","summary":"Let \\( a_1, a_2 \\) be atoms that satisfy the two statements. By \\crefprop:exists_swap, there is…","labels":[],"detail_key":"p37"},{"id":"n32504","layer":"informal","project":"p37","title":"prop:card_supports_atoms","kind":"proposition","summary":"Let \\( S \\) be a base support. Then \\( S \\) supports at most \\( 2^\\#\\kappa \\)-many sets \\( s :…","labels":["prop:card_supports_atoms"],"detail_key":"p37"},{"id":"n32505","layer":"informal","project":"p37","title":"Without loss of generality, we may assume \\( S \\) is closed under interference of near-li…","kind":"proof","summary":"Without loss of generality, we may assume \\( S \\) is closed under interference of near-litters,…","labels":[],"detail_key":"p37"},{"id":"n32506","layer":"informal","project":"p37","title":"prop:card_spec","kind":"proposition","summary":"Suppose that for all type indices \\( \\delta < \\beta \\), there are strictly less than \\( \\#\\mu \\…","labels":["prop:card_spec"],"detail_key":"p37"},{"id":"n32507","layer":"informal","project":"p37","title":"There are less than \\( \\#\\mu \\) atom conditions as \\( \\#\\kappa < \\#\\mu \\) and \\( \\#\\mu \\)…","kind":"proof","summary":"There are less than \\( \\#\\mu \\) atom conditions as \\( \\#\\kappa < \\#\\mu \\) and \\( \\#\\mu \\) is a…","labels":[],"detail_key":"p37"},{"id":"n32508","layer":"informal","project":"p37","title":"def:WeakSpec","kind":"definition","summary":"A \\emphweak \\( \\beta \\)-specification is a triple \\( W = (R^ A, R^ N, \\sigma) \\) where \\( R^ A,…","labels":["def:WeakSpec"],"detail_key":"p37"},{"id":"n32509","layer":"informal","project":"p37","title":"prop:exists_weakSpec","kind":"proposition","summary":"Every support has a weak specification that specifies it.","labels":["prop:exists_weakSpec"],"detail_key":"p37"},{"id":"n32510","layer":"informal","project":"p37","title":"Let \\( S \\) be a support, and let \\( T \\) be a strong support such that \\( S \\preceq T \\)…","kind":"proof","summary":"Let \\( S \\) be a support, and let \\( T \\) be a strong support such that \\( S \\preceq T \\), whic…","labels":[],"detail_key":"p37"},{"id":"n32511","layer":"informal","project":"p37","title":"prop:exists_allowable_of_weakSpec","kind":"proposition","summary":"If \\( W \\) is a weak specification that specifies supports \\( S \\) and \\( T \\), then there is a…","labels":["prop:exists_allowable_of_weakSpec"],"detail_key":"p37"},{"id":"n32512","layer":"informal","project":"p37","title":"Let \\( W = (R^ A, R^ N, \\sigma) \\), and let \\( U, V \\) be strong supports such that \\( \\m…","kind":"proof","summary":"Let \\( W = (R^ A, R^ N, \\sigma) \\), and let \\( U, V \\) be strong supports such that \\( \\mathsfs…","labels":[],"detail_key":"p37"},{"id":"n32513","layer":"informal","project":"p37","title":"prop:card_weakSpec","kind":"proposition","summary":"Suppose that for all type indices \\( \\delta < \\beta \\), there are strictly less than \\( \\#\\mu \\…","labels":["prop:card_weakSpec"],"detail_key":"p37"},{"id":"n32514","layer":"informal","project":"p37","title":"Follows directly from \\crefprop:card_spec and the remark that there are less than \\( \\#\\m…","kind":"proof","summary":"Follows directly from \\crefprop:card_spec and the remark that there are less than \\( \\#\\mu \\) \\…","labels":[],"detail_key":"p37"},{"id":"n32515","layer":"informal","project":"p37","title":"prop:card_supportOrbit","kind":"proposition","summary":"Suppose that for all type indices \\( \\delta < \\beta \\), there are strictly less than \\( \\#\\mu \\…","labels":["prop:card_supportOrbit"],"detail_key":"p37"},{"id":"n32516","layer":"informal","project":"p37","title":"Define a function from the type of \\( \\beta \\)-support orbits into the type of weak \\( \\b…","kind":"proof","summary":"Define a function from the type of \\( \\beta \\)-support orbits into the type of weak \\( \\beta \\)…","labels":[],"detail_key":"p37"},{"id":"n32517","layer":"informal","project":"p37","title":"prop:card_codingFunction_of_card_supports","kind":"proposition","summary":"Let \\( \\beta \\) be a type index (which in practice will be \\( \\bot \\) or the lowest proper type…","labels":["prop:card_codingFunction_of_card_supports"],"detail_key":"p37"},{"id":"n32518","layer":"informal","project":"p37","title":"Every \\( \\beta \\)-coding function is of the form \\( \\chi_(x, S) \\) where \\( S \\) is a rep…","kind":"proof","summary":"Every \\( \\beta \\)-coding function is of the form \\( \\chi_(x, S) \\) where \\( S \\) is a represent…","labels":[],"detail_key":"p37"},{"id":"n32519","layer":"informal","project":"p37","title":"prop:card_codingFunction_bot","kind":"proposition","summary":"There are less than \\( \\#\\mu \\) \\( \\bot \\)-coding functions.","labels":["prop:card_codingFunction_bot"],"detail_key":"p37"},{"id":"n32520","layer":"informal","project":"p37","title":"By \\crefprop:card_codingFunction_of_card_supports, it suffices to show that there are les…","kind":"proof","summary":"By \\crefprop:card_codingFunction_of_card_supports, it suffices to show that there are less than…","labels":[],"detail_key":"p37"},{"id":"n32521","layer":"informal","project":"p37","title":"prop:card_codingFunction_min","kind":"proposition","summary":"There are less than \\( \\#\\mu \\) \\( \\beta \\)-coding functions if \\( \\beta \\) is the minimal inha…","labels":["prop:card_codingFunction_min"],"detail_key":"p37"},{"id":"n32522","layer":"informal","project":"p37","title":"Again, we apply \\crefprop:card_codingFunction_of_card_supports. The first claim follows f…","kind":"proof","summary":"Again, we apply \\crefprop:card_codingFunction_of_card_supports. The first claim follows from \\c…","labels":[],"detail_key":"p37"},{"id":"n32523","layer":"informal","project":"p37","title":"prop:card_raisedSingleton","kind":"proposition","summary":"Suppose that for all type indices \\( \\delta < \\beta \\), there are strictly less than \\( \\#\\mu \\…","labels":["prop:card_raisedSingleton"],"detail_key":"p37"},{"id":"n32524","layer":"informal","project":"p37","title":"A \\( (\\gamma,\\beta) \\)-raised singleton \\( \\mathsfraise(S,u) = \\chi_(\\mathsfsingleton_\\be…","kind":"proof","summary":"A \\( (\\gamma,\\beta) \\)-raised singleton \\( \\mathsfraise(S,u) = \\chi_(\\mathsfsingleton_\\beta(u),…","labels":[],"detail_key":"p37"},{"id":"n32525","layer":"informal","project":"p37","title":"prop:card_codingFunction","kind":"proposition","summary":"There are less than \\( \\#\\mu \\)-many \\( \\beta \\)-coding functions for all type indices \\( \\beta…","labels":["prop:card_codingFunction"],"detail_key":"p37"},{"id":"n32526","layer":"informal","project":"p37","title":"By induction we may assume that for all type indices \\( \\delta < \\beta \\), there are stri…","kind":"proof","summary":"By induction we may assume that for all type indices \\( \\delta < \\beta \\), there are strictly l…","labels":[],"detail_key":"p37"},{"id":"n32527","layer":"informal","project":"p37","title":"prop:card_tSet","kind":"proposition","summary":"For each type index \\( \\beta \\leq \\alpha \\), \\( \\#\\mathsfTSet_\\beta = \\#\\mu \\).","labels":["prop:card_tSet"],"detail_key":"p37"},{"id":"n32528","layer":"informal","project":"p37","title":"If \\( \\beta \\) is \\( \\bot \\), we already know \\( \\#\\mathsfTSet_\\bot = \\# A= \\#\\mu \\), so…","kind":"proof","summary":"If \\( \\beta \\) is \\( \\bot \\), we already know \\( \\#\\mathsfTSet_\\bot = \\# A= \\#\\mu \\), so suppos…","labels":[],"detail_key":"p37"},{"id":"n32529","layer":"informal","project":"p37","title":"prop:card_tangle","kind":"proposition","summary":"For each type index \\( \\beta \\leq \\alpha \\), \\( \\#\\mathsfTang_\\beta = \\#\\mu \\).","labels":["prop:card_tangle"],"detail_key":"p37"},{"id":"n32530","layer":"informal","project":"p37","title":"Use \\crefprop:card_tSet and the fact that there are precisely \\( \\#\\mu \\)-many \\( \\beta \\…","kind":"proof","summary":"Use \\crefprop:card_tSet and the fact that there are precisely \\( \\#\\mu \\)-many \\( \\beta \\)-supp…","labels":[],"detail_key":"p37"},{"id":"n32531","layer":"informal","project":"p37","title":"def:IC","kind":"definition","summary":"Let \\( I : \\mathsfType_u \\) be a type with a well-founded transitive relation \\( \\prec \\). Let…","labels":["def:IC"],"detail_key":"p37"},{"id":"n32532","layer":"informal","project":"p37","title":"inductive construction theorem for propositions","kind":"proposition","summary":"[inductive construction theorem for propositions] Let \\( (F_A, F_B) \\) be an inductive construc…","labels":["prop:IC.fix_prop"],"detail_key":"p37"},{"id":"n32533","layer":"informal","project":"p37","title":"Recall that \\( \\mathsfPart\\alpha \\) denotes the type \\( \\sum_p: \\mathsfProp (p \\to \\alpha…","kind":"proof","summary":"Recall that \\( \\mathsfPart\\alpha \\) denotes the type \\( \\sum_p: \\mathsfProp (p \\to \\alpha) \\).…","labels":[],"detail_key":"p37"},{"id":"n32534","layer":"informal","project":"p37","title":"inductive construction theorem","kind":"theorem","summary":"[inductive construction theorem] Let \\( (F_A, F_B) \\) be an inductive construction for \\( (I, A…","labels":["prop:IC.fix"],"detail_key":"p37"},{"id":"n32535","layer":"informal","project":"p37","title":"Define \\[ C : \\prod_i : I A_i \\to \\left(\\prod_j : I j \\prec i \\to A_j\\right) \\to \\mathsfP…","kind":"proof","summary":"Define \\[ C : \\prod_i : I A_i \\to \\left(\\prod_j : I j \\prec i \\to A_j\\right) \\to \\mathsfProp\\]…","labels":[],"detail_key":"p37"},{"id":"n32536","layer":"informal","project":"p37","title":"def:MainMotive","kind":"definition","summary":"For a proper t","labels":["def:MainMotive"],"detail_key":"p37"},{"id":"n32537","layer":"informal","project":"p37","title":"def:MainHypothesis","kind":"definition","summary":"We define the \\emphmain hypothesis \\[ \\mathsfHypothesis: \\prod_\\alpha : \\lambda \\mathsfMotive_\\…","labels":["def:MainHypothesis"],"detail_key":"p37"},{"id":"n32538","layer":"informal","project":"p37","title":"def:motiveStep","kind":"definition","summary":"The \\emphinductive step for the main motive is the function &\\mathsfStep_M : \\prod_\\alpha : \\la…","labels":["def:motiveStep"],"detail_key":"p37"},{"id":"n32539","layer":"informal","project":"p37","title":"def:hypothesisStep","kind":"definition","summary":"The \\emphinductive step for the main hypothesis is the function &\\mathsfStep_H : \\prod_\\alpha :…","labels":["def:hypothesisStep"],"detail_key":"p37"},{"id":"n32540","layer":"informal","project":"p37","title":"model construction","kind":"theorem","summary":"[model construction] There are noncomputable functions \\[ \\mathsfComputeMotive : \\prod_\\alpha :…","labels":["thm:model_construction"],"detail_key":"p37"},{"id":"n32541","layer":"informal","project":"p37","title":"Direct from \\crefprop:IC.fix.","kind":"proof","summary":"Direct from \\crefprop:IC.fix.","labels":[],"detail_key":"p37"},{"id":"n32542","layer":"informal","project":"p37","title":"prop:raiseStrong_length","kind":"proposition","summary":"Let \\( T \\) be a \\( \\gamma \\)-support, and let \\( U \\) be the strong support generated by \\( T^…","labels":["prop:raiseStrong_length"],"detail_key":"p37"},{"id":"n32543","layer":"informal","project":"p37","title":"Let \\( N \\in \\mathsfimU_A^ N\\). Either \\( A \\) is of the form \\( B^\\beta \\) for \\( B : \\g…","kind":"proof","summary":"Let \\( N \\in \\mathsfimU_A^ N\\). Either \\( A \\) is of the form \\( B^\\beta \\) for \\( B : \\gamma \\…","labels":[],"detail_key":"p37"},{"id":"n32544","layer":"informal","project":"p37","title":"prop:raiseStrong","kind":"proposition","summary":"Let \\( T \\) be a \\( \\gamma \\)-support, and let \\( U \\) be the strong support generated by \\( T^…","labels":["prop:raiseStrong"],"detail_key":"p37"},{"id":"n32545","layer":"informal","project":"p37","title":"The interference condition is clear. First we show that proofs that \\( A^\\alpha \\)-inflex…","kind":"proof","summary":"The interference condition is clear. First we show that proofs that \\( A^\\alpha \\)-inflexibilit…","labels":[],"detail_key":"p37"},{"id":"n32546","layer":"informal","project":"p37","title":"prop:raiseRaise_strong","kind":"proposition","summary":"Let \\( S \\) be a strong \\( \\alpha \\)-support and let \\( T \\) be a \\( \\gamma \\)-support. Let \\(…","labels":["prop:raiseRaise_strong"],"detail_key":"p37"},{"id":"n32547","layer":"informal","project":"p37","title":"Follows directly from \\crefprop:Strong.smul,prop:raiseStrong.","kind":"proof","summary":"Follows directly from \\crefprop:Strong.smul,prop:raiseStrong.","labels":[],"detail_key":"p37"},{"id":"n32548","layer":"informal","project":"p37","title":"prop:combineStrong","kind":"proposition","summary":"Let \\( S \\) be a strong \\( \\alpha \\)-support. Let \\( U \\) be a strong \\( \\beta \\)-support with…","labels":["prop:combineStrong"],"detail_key":"p37"},{"id":"n32549","layer":"informal","project":"p37","title":"Appeal to \\crefprop:spec_eq_spec_iff. First, note that \\( (i, a) \\in (S + (\\rho(U))^\\alph…","kind":"proof","summary":"Appeal to \\crefprop:spec_eq_spec_iff. First, note that \\( (i, a) \\in (S + (\\rho(U))^\\alpha)_A^…","labels":[],"detail_key":"p37"},{"id":"n32550","layer":"informal","project":"p37","title":"prop:exists_allowable_of_fixes","kind":"proposition","summary":"Let \\( S \\) be a strong \\( \\alpha \\)-support and let \\( T \\) be a \\( \\gamma \\)-support. Let \\(…","labels":["prop:exists_allowable_of_fixes"],"detail_key":"p37"},{"id":"n32551","layer":"informal","project":"p37","title":"Let \\( U \\) be the strong support generated by \\( T^\\beta \\), and let \\( V \\) be the supp…","kind":"proof","summary":"Let \\( U \\) be the strong support generated by \\( T^\\beta \\), and let \\( V \\) be the support wh…","labels":[],"detail_key":"p37"},{"id":"n32552","layer":"informal","project":"p37","title":"tangled membership","kind":"definition","summary":"[tangled membership] We define the membership relation \\( \\in^\\alpha_\\beta : \\mathsfTSet_\\beta…","labels":["def:TSet.mem"],"detail_key":"p37"},{"id":"n32553","layer":"informal","project":"p37","title":"symmetric","kind":"definition","summary":"[symmetric] Let \\( \\beta < \\alpha \\) be proper type indices. A set \\( s : \\mathsfTSet_\\beta \\)…","labels":["def:Symmetric"],"detail_key":"p37"},{"id":"n32554","layer":"informal","project":"p37","title":"prop:TSet.of_symmetric","kind":"proposition","summary":"Let \\( \\beta < \\alpha \\) be proper type indices. Let \\( s : \\mathsfTSet_\\beta \\) be \\( \\alpha \\…","labels":["prop:TSet.of_symmetric"],"detail_key":"p37"},{"id":"n32555","layer":"informal","project":"p37","title":"If \\( s \\) is empty, the result follows directly from the definition of \\( \\mathsfTSet_\\a…","kind":"proof","summary":"If \\( s \\) is empty, the result follows directly from the definition of \\( \\mathsfTSet_\\alpha \\…","labels":[],"detail_key":"p37"},{"id":"n32556","layer":"informal","project":"p37","title":"unions of singletons","kind":"proposition","summary":"[unions of singletons] Let \\( \\gamma < \\beta < \\alpha \\) be proper type indices. Let \\( s : \\ma…","labels":["prop:singleton_union"],"detail_key":"p37"},{"id":"n32557","layer":"informal","project":"p37","title":"Let \\( S \\) be an \\( \\alpha \\)-support for \\( \\mathsfsingleton_\\beta[s] \\), which without…","kind":"proof","summary":"Let \\( S \\) be an \\( \\alpha \\)-support for \\( \\mathsfsingleton_\\beta[s] \\), which without loss…","labels":[],"detail_key":"p37"},{"id":"n32558","layer":"informal","project":"p37","title":"consistency of tangled type theory","kind":"theorem","summary":"[consistency of tangled type theory] Let \\( \\ x \\_\\beta \\) be an abbreviation for \\( \\mathsfsin…","labels":["thm:ttt_consistent"],"detail_key":"p37"},{"id":"n32559","layer":"informal","project":"p37","title":"The axiom of extensionality was proven in \\crefprop:Code.ext. All axioms except for the t…","kind":"proof","summary":"The axiom of extensionality was proven in \\crefprop:Code.ext. All axioms except for the type lo…","labels":[],"detail_key":"p37"},{"id":"n32560","layer":"informal","project":"p37","title":"A \\emph\\( \\Sigma \\)-language consists of a map \\( \\mathsfFunctions : \\prod_n : N (\\mathsf…","kind":"definition","summary":"A \\emph\\( \\Sigma \\)-language consists of a map \\( \\mathsfFunctions : \\prod_n : N (\\mathsfFinn \\…","labels":[],"detail_key":"p37"},{"id":"n32561","layer":"informal","project":"p37","title":"Let \\( \\Phi : \\Sigma \\to \\Sigma' \\). Let \\( L \\) be a \\( \\Sigma \\)-language and let \\( L'…","kind":"definition","summary":"Let \\( \\Phi : \\Sigma \\to \\Sigma' \\). Let \\( L \\) be a \\( \\Sigma \\)-language and let \\( L' \\) be…","labels":[],"detail_key":"p37"},{"id":"n32562","layer":"informal","project":"p37","title":"Let \\( L, L' \\) be \\( \\Sigma \\)-languages. We define \\( L \\oplus L' \\) to be the \\( \\Sigm…","kind":"definition","summary":"Let \\( L, L' \\) be \\( \\Sigma \\)-languages. We define \\( L \\oplus L' \\) to be the \\( \\Sigma \\)-l…","labels":[],"detail_key":"p37"},{"id":"n32563","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( M : \\Sigma \\to \\mathsfType_w \\). An \\e…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( M : \\Sigma \\to \\mathsfType_w \\). An \\emph\\(…","labels":[],"detail_key":"p37"},{"id":"n32564","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language. A \\emphmorphism of \\( L \\)-structures \\( M \\to N…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language. A \\emphmorphism of \\( L \\)-structures \\( M \\to N \\) con…","labels":[],"detail_key":"p37"},{"id":"n32565","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( \\alpha : \\mathsfType_u' \\) be a sort o…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( \\alpha : \\mathsfType_u' \\) be a sort of vari…","labels":[],"detail_key":"p37"},{"id":"n32566","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( \\alpha : \\mathsfType_u' \\) and \\( S :…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language, and let \\( \\alpha : \\mathsfType_u' \\) and \\( S : \\alpha…","labels":[],"detail_key":"p37"},{"id":"n32567","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language. A \\emph1-\\( L \\)-formula of sort \\( A \\) is an \\(…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language. A \\emph1-\\( L \\)-formula of sort \\( A \\) is an \\( L \\)-…","labels":[],"detail_key":"p37"},{"id":"n32568","layer":"informal","project":"p37","title":"Let \\( L \\) be a \\( \\Sigma \\)-language. The \\emphwitness symbols for \\( L \\) is the langu…","kind":"definition","summary":"Let \\( L \\) be a \\( \\Sigma \\)-language. The \\emphwitness symbols for \\( L \\) is the language \\(…","labels":[],"detail_key":"p37"},{"id":"n32569","layer":"informal","project":"p37","title":"Let \\( M \\) be a nonempty \\( L \\)-structure. Then \\( M \\) has an \\( L_W \\)-structure such…","kind":"proposition","summary":"Let \\( M \\) be a nonempty \\( L \\)-structure. Then \\( M \\) has an \\( L_W \\)-structure such that…","labels":[],"detail_key":"p37"},{"id":"n32570","layer":"informal","project":"p37","title":"We define the interpretation of the constant for \\( \\phi \\) to be some \\( x : M \\) such t…","kind":"proof","summary":"We define the interpretation of the constant for \\( \\phi \\) to be some \\( x : M \\) such that \\(…","labels":[],"detail_key":"p37"},{"id":"n32571","layer":"informal","project":"p37","title":"For each \\( n : N \\), we define \\[ L^(0) = L;\\quad L^(n+1) = L^(n) \\oplus (L^(n))_W \\] Th…","kind":"definition","summary":"For each \\( n : N \\), we define \\[ L^(0) = L;\\quad L^(n+1) = L^(n) \\oplus (L^(n))_W \\] This for…","labels":[],"detail_key":"p37"},{"id":"n32572","layer":"informal","project":"p37","title":"Let \\( M \\) be a nonempty \\( L \\)-structure. Then \\( M \\) has an \\( L^(\\omega) \\)-structu…","kind":"proposition","summary":"Let \\( M \\) be a nonempty \\( L \\)-structure. Then \\( M \\) has an \\( L^(\\omega) \\)-structure suc…","labels":[],"detail_key":"p37"},{"id":"n32573","layer":"informal","project":"p37","title":"Let \\( L \\) be an \\( N \\)-language. A \\emphtype raising morphism is a map of languages \\(…","kind":"definition","summary":"Let \\( L \\) be an \\( N \\)-language. A \\emphtype raising morphism is a map of languages \\( L \\xr…","labels":[],"detail_key":"p37"},{"id":"n32574","layer":"informal","project":"p37","title":"def:relation_props","kind":"definition","summary":"Let \\( R : \\sigma \\to \\tau \\to \\mathsfProp\\). We define \\item the \\emphimage of \\( R \\) to be t…","labels":["def:relation_props"],"detail_key":"p37"},{"id":"n32575","layer":"informal","project":"p37","title":"prop:relation_results","kind":"proposition","summary":"\\mbox\\negthinspace \\item \\( R : \\tau \\to \\tau \\to \\mathsfProp\\) is permutative if and only if i…","labels":["prop:relation_results"],"detail_key":"p37"},{"id":"n32576","layer":"informal","project":"p37","title":"def:OrbitRestriction","kind":"definition","summary":"Let \\( s : \\mathsfSet\\tau \\). An \\emphorbit restriction for \\( s \\) (over some type \\( \\sigma \\…","labels":["def:OrbitRestriction"],"detail_key":"p37"},{"id":"n32577","layer":"informal","project":"p37","title":"completing restricted orbits","kind":"proposition","summary":"[completing restricted orbits] Let \\( R : \\tau \\to \\tau \\to \\mathsfProp\\) be a one-to-one relat…","labels":["prop:completing_restricted_orbits"],"detail_key":"p37"},{"id":"n32578","layer":"informal","project":"p37","title":"For each \\( u : \\sigma \\), define an injection \\( i_u : \\mathsffieldR \\times N \\to \\tau \\…","kind":"proof","summary":"For each \\( u : \\sigma \\), define an injection \\( i_u : \\mathsffieldR \\times N \\to \\tau \\) wher…","labels":[],"detail_key":"p37"},{"id":"n32579","layer":"informal","project":"p37","title":"completing orbits","kind":"proposition","summary":"[completing orbits] Let \\( R : \\tau \\to \\tau \\to \\mathsfProp\\) be a one-to-one relation. Let \\(…","labels":["prop:completing_orbits"],"detail_key":"p37"},{"id":"n32580","layer":"informal","project":"p37","title":"Define the orbit restriction \\( (s, f, \\pi) \\) for \\( \\mathsffieldR \\) over \\( \\mathsfUni…","kind":"proof","summary":"Define the orbit restriction \\( (s, f, \\pi) \\) for \\( \\mathsffieldR \\) over \\( \\mathsfUnit\\). N…","labels":[],"detail_key":"p37"},{"id":"n32581","layer":"informal","project":"p37","title":"mathlib","kind":"lemma","summary":"[mathlib] Let \\( \\#\\mu \\) be a strong limit cardinal. Then there are precisely \\( \\#\\mu \\)-many…","labels":["prop:card_subset_card_lt_cof"],"detail_key":"p37"},{"id":"n32582","layer":"informal","project":"p37","title":"Endow \\( \\mu \\) with its initial well-ordering. Each such subset is bounded in \\( \\mu \\)…","kind":"proof","summary":"Endow \\( \\mu \\) with its initial well-ordering. Each such subset is bounded in \\( \\mu \\) with r…","labels":[],"detail_key":"p37"},{"id":"n32583","layer":"formal","project":"p37","title":"Rel.OrbitRestriction","kind":"inductive","summary":"α : Type u_1 → Set α → Type u_2 → Type (max u_1 u_2)","labels":[],"detail_key":"p37","name":"Rel.OrbitRestriction","module":"ConNF.Background.PermutativeExtension"},{"id":"n32584","layer":"formal","project":"p37","title":"Rel.permutativeExtension","kind":"def","summary":"α : Type u_1 → β : Type u_2 → (r : Rel α α) → Rel.OrbitRestriction (Union.union r.dom r.codom)…","labels":[],"detail_key":"p37","name":"Rel.permutativeExtension","module":"ConNF.Background.PermutativeExtension"},{"id":"n32585","layer":"formal","project":"p37","title":"Rel.permutativeExtension'","kind":"def","summary":"α : Type u_1 → (r : Rel α α) → r.OneOne → (s : Set α) → s.Infinite → LE.le (Cardinal.mk ↑r.dom)…","labels":[],"detail_key":"p37","name":"Rel.permutativeExtension'","module":"ConNF.Background.PermutativeExtension"},{"id":"n32586","layer":"formal","project":"p37","title":"ConNF.TypedNearLitters","kind":"inductive","summary":"[inst : ConNF.Params] → (α : ConNF.Λ) → [inst_1 : ConNF.ModelData ↑α] → [ConNF.Position (ConNF.…","labels":[],"detail_key":"p37","name":"ConNF.TypedNearLitters","module":"ConNF.Construction.Code"},{"id":"n32587","layer":"formal","project":"p37","title":"ConNF.BaseAction","kind":"inductive","summary":"[ConNF.Params] → Type u","labels":[],"detail_key":"p37","name":"ConNF.BaseAction","module":"ConNF.FOA.BaseAction"},{"id":"n32588","layer":"formal","project":"p37","title":"ConNF.BaseAction.Nice","kind":"inductive","summary":"[inst : ConNF.Params] → ConNF.BaseAction → Prop","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.Nice","module":"ConNF.FOA.BaseAction"},{"id":"n32589","layer":"formal","project":"p37","title":"ConNF.BaseAction.insideExtension","kind":"def","summary":"[inst : ConNF.Params] → ConNF.BaseAction → ConNF.BaseAction","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.insideExtension","module":"ConNF.FOA.BaseAction"},{"id":"n32590","layer":"formal","project":"p37","title":"ConNF.BaseAction.niceExtension","kind":"def","summary":"[inst : ConNF.Params] → ConNF.BaseAction → ConNF.BaseAction","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.niceExtension","module":"ConNF.FOA.BaseAction"},{"id":"n32591","layer":"formal","project":"p37","title":"ConNF.BaseAction.outsideExtension","kind":"def","summary":"[inst : ConNF.Params] → (ξ : ConNF.BaseAction) → (∀ (N : ConNF.NearLitter), Membership.mem (Con…","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.outsideExtension","module":"ConNF.FOA.BaseAction"},{"id":"n32592","layer":"formal","project":"p37","title":"ConNF.BaseApprox.addOrbit","kind":"def","summary":"[inst : ConNF.Params] → (ψ : ConNF.BaseApprox) → (f : Int → ConNF.Litter) → (∀ (m n k : Int), E…","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.addOrbit","module":"ConNF.FOA.BaseApprox"},{"id":"n32593","layer":"formal","project":"p37","title":"ConNF.BaseApprox.atoms","kind":"def","summary":"[inst : ConNF.Params] → ConNF.BaseApprox → Rel ConNF.Atom ConNF.Atom","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.atoms","module":"ConNF.FOA.BaseApprox"},{"id":"n32594","layer":"formal","project":"p37","title":"ConNF.BaseApprox.atoms_permutative","kind":"theorem","summary":"∀ [inst : ConNF.Params] (ψ : ConNF.BaseApprox), (ConNF.SuperA.superA ψ).Permutative","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.atoms_permutative","module":"ConNF.FOA.BaseApprox"},{"id":"n32595","layer":"formal","project":"p37","title":"ConNF.BaseApprox.image_near_of_near","kind":"theorem","summary":"∀ [inst : ConNF.Params] (ψ : ConNF.BaseApprox) (s : Set ConNF.Atom) L₁ L₂ : ConNF.Litter, ConNF…","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.image_near_of_near","module":"ConNF.FOA.BaseApprox"},{"id":"n32596","layer":"formal","project":"p37","title":"ConNF.BaseApprox.instInv","kind":"def","summary":"[inst : ConNF.Params] → Inv ConNF.BaseApprox","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.instInv","module":"ConNF.FOA.BaseApprox"},{"id":"n32597","layer":"formal","project":"p37","title":"ConNF.BaseApprox.instPartialOrder","kind":"def","summary":"[inst : ConNF.Params] → PartialOrder ConNF.BaseApprox","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.instPartialOrder","module":"ConNF.FOA.BaseApprox"},{"id":"n32598","layer":"formal","project":"p37","title":"ConNF.BaseApprox.inv_nearLitters","kind":"theorem","summary":"∀ [inst : ConNF.Params] (ψ : ConNF.BaseApprox), Eq (ConNF.SuperN.superN (Inv.inv ψ)) (ConNF.Sup…","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.inv_nearLitters","module":"ConNF.FOA.BaseApprox"},{"id":"n32599","layer":"formal","project":"p37","title":"ConNF.BaseApprox.nearLitters","kind":"def","summary":"[inst : ConNF.Params] → ConNF.BaseApprox → Rel ConNF.NearLitter ConNF.NearLitter","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.nearLitters","module":"ConNF.FOA.BaseApprox"},{"id":"n32600","layer":"formal","project":"p37","title":"ConNF.BaseApprox.nearLitters_permutative","kind":"theorem","summary":"∀ [inst : ConNF.Params] (ψ : ConNF.BaseApprox), (ConNF.SuperN.superN ψ).Permutative","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.nearLitters_permutative","module":"ConNF.FOA.BaseApprox"},{"id":"n32601","layer":"formal","project":"p37","title":"ConNF.BaseApprox.typical_permutative","kind":"theorem","summary":"∀ [inst : ConNF.Params] (ψ : ConNF.BaseApprox), ψ.typical.Permutative","labels":[],"detail_key":"p37","name":"ConNF.BaseApprox.typical_permutative","module":"ConNF.FOA.BaseApprox"},{"id":"n32602","layer":"formal","project":"p37","title":"ConNF.BaseAction.FlexApprox","kind":"inductive","summary":"[inst : ConNF.Params] → [inst_1 : ConNF.Level] → [ConNF.CoherentData] → β : ConNF.TypeIndex → […","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.FlexApprox","module":"ConNF.FOA.FlexApprox"},{"id":"n32603","layer":"formal","project":"p37","title":"ConNF.BaseAction.flexApprox_flexApprox","kind":"theorem","summary":"∀ [inst : ConNF.Params] [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData] β : ConNF.TypeInde…","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.flexApprox_flexApprox","module":"ConNF.FOA.FlexApprox"},{"id":"n32604","layer":"formal","project":"p37","title":"ConNF.BaseAction.smul_nearLitter_of_smul_litter","kind":"theorem","summary":"∀ [inst : ConNF.Params] [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData] β : ConNF.TypeInde…","labels":[],"detail_key":"p37","name":"ConNF.BaseAction.smul_nearLitter_of_smul_litter","module":"ConNF.FOA.FlexApprox"},{"id":"n32605","layer":"formal","project":"p37","title":"ConNF.InflexiblePath","kind":"inductive","summary":"[inst : ConNF.Params] → ConNF.TypeIndex → Type u","labels":[],"detail_key":"p37","name":"ConNF.InflexiblePath","module":"ConNF.FOA.Inflexible"},{"id":"n32606","layer":"formal","project":"p37","title":"ConNF.StrAction.approximates_of_flexApprox_exactlyApproximates","kind":"theorem","summary":"∀ [inst : ConNF.Params] [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData] β : ConNF.TypeInde…","labels":[],"detail_key":"p37","name":"ConNF.StrAction.approximates_of_flexApprox_exactlyApproximates","module":"ConNF.FOA.StrActionFOA"},{"id":"n32607","layer":"formal","project":"p37","title":"ConNF.StrAction.freedomOfAction","kind":"theorem","summary":"∀ [inst : ConNF.Params] [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData] β : ConNF.TypeInde…","labels":[],"detail_key":"p37","name":"ConNF.StrAction.freedomOfAction","module":"ConNF.FOA.StrActionFOA"},{"id":"n32608","layer":"formal","project":"p37","title":"ConNF.StrApprox","kind":"def","summary":"[inst : ConNF.Params] → ConNF.TypeIndex → Type u","labels":[],"detail_key":"p37","name":"ConNF.StrApprox","module":"ConNF.FOA.StrApprox"},{"id":"n32609","layer":"formal","project":"p37","title":"ConNF.StrApprox.Coherent","kind":"def","summary":"[inst : ConNF.Params] → β : ConNF.TypeIndex → [inst_1 : ConNF.Level] → [ConNF.CoherentData] → […","labels":[],"detail_key":"p37","name":"ConNF.StrApprox.Coherent","module":"ConNF.FOA.StrApprox"},{"id":"n32610","layer":"formal","project":"p37","title":"ConNF.StrApprox.addOrbit_coherent","kind":"theorem","summary":"∀ [inst : ConNF.Params] β : ConNF.TypeIndex [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData…","labels":[],"detail_key":"p37","name":"ConNF.StrApprox.addOrbit_coherent","module":"ConNF.FOA.StrApprox"},{"id":"n32611","layer":"formal","project":"p37","title":"ConNF.StrApprox.Approximates","kind":"def","summary":"[inst : ConNF.Params] → [inst_1 : ConNF.Level] → [inst_2 : ConNF.CoherentData] → β : ConNF.Type…","labels":[],"detail_key":"p37","name":"ConNF.StrApprox.Approximates","module":"ConNF.FOA.StrApproxFOA"},{"id":"n32612","layer":"formal","project":"p37","title":"ConNF.StrApprox.FreedomOfAction","kind":"def","summary":"[inst : ConNF.Params] → [inst_1 : ConNF.Level] → [ConNF.CoherentData] → (β : ConNF.TypeIndex) →…","labels":[],"detail_key":"p37","name":"ConNF.StrApprox.FreedomOfAction","module":"ConNF.FOA.StrApproxFOA"},{"id":"n32613","layer":"formal","project":"p37","title":"ConNF.StrApprox.addFlexible","kind":"def","summary":"[inst : ConNF.Params] → β : ConNF.TypeIndex → (ψ : ConNF.StrApprox β) → (A : ConNF.Path β Bot.b…","labels":[],"detail_key":"p37","name":"ConNF.StrApprox.addFlexible","module":"ConNF.FOA.StrApproxFOA"},{"id":"n32614","layer":"formal","project":"p37","title":"ConNF.Path","kind":"inductive","summary":"[inst : ConNF.Params] → ConNF.TypeIndex → ConNF.TypeIndex → Type u","labels":[],"detail_key":"p37","name":"ConNF.Path","module":"ConNF.Levels.Path"},{"id":"n32615","layer":"formal","project":"p37","title":"ConNF.StrPerm","kind":"def","summary":"[inst : ConNF.Params] → ConNF.TypeIndex → Type u","labels":[],"detail_key":"p37","name":"ConNF.StrPerm","module":"ConNF.Levels.StrPerm"},{"id":"n32616","layer":"formal","project":"p37","title":"ConNF.StrSet","kind":"def","summary":"[inst : ConNF.Params] → ConNF.TypeIndex → Type u","labels":[],"detail_key":"p37","name":"ConNF.StrSet","module":"ConNF.Levels.StrSet"},{"id":"n32617","layer":"formal","project":"p37","title":"ConNF.CoherentData","kind":"inductive","summary":"[inst : ConNF.Params] → [ConNF.Level] → Type (u + 1)","labels":[],"detail_key":"p37","name":"ConNF.CoherentData","module":"ConNF.ModelData.CoherentData"},{"id":"n32618","layer":"formal","project":"p37","title":"ConNF.BaseSupport","kind":"inductive","summary":"[ConNF.Params] → Type u","labels":[],"detail_key":"p37","name":"ConNF.BaseSupport","module":"ConNF.ModelData.Support"},{"id":"n32619","layer":"formal","project":"p37","title":"ConNF.instPreorderSupport","kind":"def","summary":"[inst : ConNF.Params] → α : ConNF.TypeIndex → Preorder (ConNF.Support α)","labels":[],"detail_key":"p37","name":"ConNF.instPreorderSupport","module":"ConNF.ModelData.Support"},{"id":"n32620","layer":"formal","project":"p37","title":"ConNF.funOfDeny","kind":"def","summary":"[inst : ConNF.Params] → X : Type u → LE.le (Cardinal.mk X) (Cardinal.mk ConNF.μ) → (deny : X →…","labels":[],"detail_key":"p37","name":"ConNF.funOfDeny","module":"ConNF.Position.Deny"},{"id":"n32621","layer":"formal","project":"p37","title":"ConNF.Support.strong_strong","kind":"theorem","summary":"∀ [inst : ConNF.Params] β : ConNF.TypeIndex [inst_1 : ConNF.Level] [inst_2 : ConNF.CoherentData…","labels":[],"detail_key":"p37","name":"ConNF.Support.strong_strong","module":"ConNF.Strong.Strong"},{"id":"n32622","layer":"formal","project":"p37","title":"ConNF.minimalParams","kind":"def","summary":"","labels":[],"detail_key":"p37","name":"ConNF.minimalParams","module":"Old.ConNF.BaseType.Params"},{"id":"n32623","layer":"informal","project":"p38","title":"Proof of the sphere eversion corollary","kind":"proof","summary":"[Proof of the sphere eversion corollary] We denote by ι the inclusion of 𝕊^2 into ℝ^3. We set j…","labels":[],"detail_key":"p38"},{"id":"n32624","layer":"informal","project":"p38","title":"def:loop","kind":"definition","summary":"A loop is a map defined on the circle 𝕊^1 = ℝ/ℤ with values in a finite-dimensional vector spac…","labels":["def:loop"],"detail_key":"p38"},{"id":"n32625","layer":"informal","project":"p38","title":"prop:∃_loops","kind":"proposition","summary":"Let K a compact set in E. Let Ω be an open set in E × F. Let β and g be smooth maps from E to F…","labels":["prop:∃_loops"],"detail_key":"p38"},{"id":"n32626","layer":"informal","project":"p38","title":"def:surrounds_points","kind":"definition","summary":"A point x in E is surrounded by points p_0, \\dots, p_d if those points are affinely independent…","labels":["def:surrounds_points"],"detail_key":"p38"},{"id":"n32627","layer":"informal","project":"p38","title":"lem:smooth_barycentric_coord","kind":"lemma","summary":"For every x in E and every collection of points p ∈ E^d+1 surrounding x, there is a function w…","labels":["lem:smooth_barycentric_coord"],"detail_key":"p38"},{"id":"n32628","layer":"informal","project":"p38","title":"Let: A = E \\times \\ q \\in E^d+1 ~|~ \\mboxq is an affine basis for E \\, and define: w \\co…","kind":"proof","summary":"Let: A = E \\times \\ q \\in E^d+1 ~|~ \\mboxq is an affine basis for E \\, and define: w \\co A &\\to…","labels":[],"detail_key":"p38"},{"id":"n32629","layer":"informal","project":"p38","title":"prop:surrounded_by_open","kind":"proposition","summary":"If a point x of E lies in the convex hull of an open set P, then it is surrounded by some colle…","labels":["prop:surrounded_by_open"],"detail_key":"p38"},{"id":"n32630","layer":"informal","project":"p38","title":"Carathéodory's lemma","kind":"lemma","summary":"[Carathéodory's lemma] If a point x of E lies in the convex hull of a set P, then x belongs to…","labels":["lem:caratheodory"],"detail_key":"p38"},{"id":"n32631","layer":"informal","project":"p38","title":"By assumption, there is a finite set of points t_i in P and weights f_i such that x = \\su…","kind":"proof","summary":"By assumption, there is a finite set of points t_i in P and weights f_i such that x = \\sum f_i…","labels":[],"detail_key":"p38"},{"id":"n32632","layer":"informal","project":"p38","title":"lem:interior_chab","kind":"lemma","summary":"Given an affine basis b of E, the interior of the convex hull of b is the set of points with st…","labels":["lem:interior_chab"],"detail_key":"p38"},{"id":"n32633","layer":"informal","project":"p38","title":"For each i, let: \\[ w_i \\co E \\to ℝ \\] be the i^\\rm th barycentric coordinate with respec…","kind":"proof","summary":"For each i, let: \\[ w_i \\co E \\to ℝ \\] be the i^\\rm th barycentric coordinate with respect to t…","labels":[],"detail_key":"p38"},{"id":"n32634","layer":"informal","project":"p38","title":"lem:int_homothety_cvx","kind":"lemma","summary":"Given a point c of E and a real number t, let: \\[ h^c_t \\co E \\to E \\] be the homothety which d…","labels":["lem:int_homothety_cvx"],"detail_key":"p38"},{"id":"n32635","layer":"informal","project":"p38","title":"Since h^c_t is a homeomorphism with inverse h^c_t^-1, taking s = t^-1, the required resul…","kind":"proof","summary":"Since h^c_t is a homeomorphism with inverse h^c_t^-1, taking s = t^-1, the required result is e…","labels":[],"detail_key":"p38"},{"id":"n32636","layer":"informal","project":"p38","title":"Proof of \\Crefprop:surrounded_by_open","kind":"proof","summary":"[Proof of \\Crefprop:surrounded_by_open] It follows from \\Creflem:interior_chab that we need onl…","labels":[],"detail_key":"p38"},{"id":"n32637","layer":"informal","project":"p38","title":"def:surrounds","kind":"definition","summary":"We say a loop γ surrounds a vector v if v is surrounded by a collection of points belonging to…","labels":["def:surrounds"],"detail_key":"p38"},{"id":"n32638","layer":"informal","project":"p38","title":"lem:loop_of_hull","kind":"lemma","summary":"If a vector v is in the convex hull of a connected open subset O then, for every base point b ∈…","labels":["lem:loop_of_hull"],"detail_key":"p38"},{"id":"n32639","layer":"informal","project":"p38","title":"Since O is open, \\Crefprop:surrounded_by_open gives points p_i in O surrounding x. Since…","kind":"proof","summary":"Since O is open, \\Crefprop:surrounded_by_open gives points p_i in O surrounding x. Since O is o…","labels":[],"detail_key":"p38"},{"id":"n32640","layer":"informal","project":"p38","title":"def:family_surrounds","kind":"definition","summary":"A continuous family of loops γ \\co E × [0, 1] × 𝕊^1 → F, (x, t, s) ↦ γ^t_x(s) surrounds a map g…","labels":["def:family_surrounds"],"detail_key":"p38"},{"id":"n32641","layer":"informal","project":"p38","title":"lem:local_loops","kind":"lemma","summary":"Assume Ω is open over some neighborhood of x_0. If g(x_0) is in the convex hull of the connecte…","labels":["lem:local_loops"],"detail_key":"p38"},{"id":"n32642","layer":"informal","project":"p38","title":"In this proof we don't mention the t parameter since it plays no role, but it is still th…","kind":"proof","summary":"In this proof we don't mention the t parameter since it plays no role, but it is still there. \\…","labels":[],"detail_key":"p38"},{"id":"n32643","layer":"informal","project":"p38","title":"lem:satisfied_or_refund","kind":"lemma","summary":"For every set U ⊂ E, \\Loop(g, β, U, Ω) is ``path connected'': for every γ_0 and γ_1 in \\Loop(g,…","labels":["lem:satisfied_or_refund"],"detail_key":"p38"},{"id":"n32644","layer":"informal","project":"p38","title":"Let ρ be the piecewise affine map from ℝ to ℝ such that ρ(τ) = 1 if τ ≤ 1/2, ρ is affine…","kind":"proof","summary":"Let ρ be the piecewise affine map from ℝ to ℝ such that ρ(τ) = 1 if τ ≤ 1/2, ρ is affine on [1/…","labels":[],"detail_key":"p38"},{"id":"n32645","layer":"informal","project":"p38","title":"cor:extend_loops","kind":"corollary","summary":"Let U_0 and U_1 be open sets in E. Let K_0 ⊂ U_0 and K_1 ⊂ U_1 be compact subsets. For any γ_0…","labels":["cor:extend_loops"],"detail_key":"p38"},{"id":"n32646","layer":"informal","project":"p38","title":"Let C_0 = K_0\\cup U_1^c and C_1 := K_1 ∖ U_0. Since C_0 and C_1 are disjoint closed sets,…","kind":"proof","summary":"Let C_0 = K_0\\cup U_1^c and C_1 := K_1 ∖ U_0. Since C_0 and C_1 are disjoint closed sets, there…","labels":[],"detail_key":"p38"},{"id":"n32647","layer":"informal","project":"p38","title":"lem:∃_surrounding_loops","kind":"lemma","summary":"In the setup of \\Crefprop:∃_loops, assume we have a continuous family γ of loops defined near K…","labels":["lem:∃_surrounding_loops"],"detail_key":"p38"},{"id":"n32648","layer":"informal","project":"p38","title":"\\Creflem:local_loops proves the existence of local families of surrounding loops and \\Cre…","kind":"proof","summary":"\\Creflem:local_loops proves the existence of local families of surrounding loops and \\Crefcor:e…","labels":[],"detail_key":"p38"},{"id":"n32649","layer":"informal","project":"p38","title":"lem:exists_cont_diff_of_convex","kind":"lemma","summary":"Let E and F be real normed vector spaces. Assume that E is finite dimensional. Let P be a predi…","labels":["lem:exists_cont_diff_of_convex"],"detail_key":"p38"},{"id":"n32650","layer":"informal","project":"p38","title":"The assumption give us an open cover (U_i)_i ∈ I of E and functions f_i \\co E → F that ar…","kind":"proof","summary":"The assumption give us an open cover (U_i)_i ∈ I of E and functions f_i \\co E → F that are smoo…","labels":[],"detail_key":"p38"},{"id":"n32651","layer":"informal","project":"p38","title":"lem:exists_cont_diff_of_convex₂","kind":"lemma","summary":"Let E₁, E₂ and F be real vector spaces. Assume E₁ and E₂ are finite dimensional. Let n be a nat…","labels":["lem:exists_cont_diff_of_convex₂"],"detail_key":"p38"},{"id":"n32652","layer":"informal","project":"p38","title":"This is completely analogous to the previous proof.","kind":"proof","summary":"This is completely analogous to the previous proof.","labels":[],"detail_key":"p38"},{"id":"n32653","layer":"informal","project":"p38","title":"lem:reparametrization","kind":"lemma","summary":"Let γ \\co E × 𝕊^1 → F be a smooth family of loops surrounding a map g. There is a smooth family…","labels":["lem:reparametrization"],"detail_key":"p38"},{"id":"n32654","layer":"informal","project":"p38","title":"Gromov's main idea in order to prove this result is to translate the problem of construct…","kind":"proof","summary":"Gromov's main idea in order to prove this result is to translate the problem of constructing a…","labels":[],"detail_key":"p38"},{"id":"n32655","layer":"informal","project":"p38","title":"Proof of \\Crefprop:∃_loops","kind":"proof","summary":"[Proof of \\Crefprop:∃_loops] Let γ^* be a family of loops surrounding the origin in B_F(0,1) th…","labels":[],"detail_key":"p38"},{"id":"n32656","layer":"informal","project":"p38","title":"def:dual_pair","kind":"definition","summary":"A dual pair on a vector space E is a pair (π, v) where π is a linear form on E and v a vector i…","labels":["def:dual_pair"],"detail_key":"p38"},{"id":"n32657","layer":"informal","project":"p38","title":"Theillière 2018","kind":"definition","summary":"[Theillière 2018] The map obtained by corrugation of f in direction (π, v) using γ with oscilla…","labels":["def:corrugation"],"detail_key":"p38"},{"id":"n32658","layer":"informal","project":"p38","title":"Theillière 2018","kind":"proposition","summary":"[Theillière 2018] Let f be a C^1 function from E to F. Let (π, v) be a dual pair on E. Let γ \\c…","labels":["prop:theilliere","CP:C0","CP:kerpi","CP:v"],"detail_key":"p38"},{"id":"n32659","layer":"informal","project":"p38","title":"We set Γ_x(t) = ∫_0^t \\left(γ_x(s) - \\overlineγ_x\\right)ds, so that f'(x) = f(x) + Γ_x(Nπ…","kind":"proof","summary":"We set Γ_x(t) = ∫_0^t \\left(γ_x(s) - \\overlineγ_x\\right)ds, so that f'(x) = f(x) + Γ_x(Nπ(x))/N…","labels":[],"detail_key":"p38"},{"id":"n32660","layer":"informal","project":"p38","title":"def:hol_partial","kind":"definition","summary":"Let E' be a linear subspace of E. A map \\F = (f, φ) : E → F × \\Hom(E, F) is E'--holonomic if, f…","labels":["def:hol_partial"],"detail_key":"p38"},{"id":"n32661","layer":"informal","project":"p38","title":"def:rel_loc","kind":"definition","summary":"A first order differential relation for maps from E to F is a subset \\Rel of E × F × \\Hom(E, F).","labels":["def:rel_loc"],"detail_key":"p38"},{"id":"n32662","layer":"informal","project":"p38","title":"def:formal_sol_loc","kind":"definition","summary":"A formal solution of a differential relation \\Rel is a map \\F = (f, φ) \\co E → F × \\Hom(E, F) s…","labels":["def:formal_sol_loc"],"detail_key":"p38"},{"id":"n32663","layer":"informal","project":"p38","title":"def:htpy_jet_sec_loc","kind":"definition","summary":"A 1-jet section from E to F is a function from E to F × \\Hom(E, F). A homotopy of 1-jet section…","labels":["def:htpy_jet_sec_loc"],"detail_key":"p38"},{"id":"n32664","layer":"informal","project":"p38","title":"def:rel_slice","kind":"definition","summary":"For every σ = (x, y, φ), the slice of \\Rel at σ with respect to (π, v) is: \\[ \\Rel(σ, π, v) = \\…","labels":["def:rel_slice"],"detail_key":"p38"},{"id":"n32665","layer":"informal","project":"p38","title":"lem:update_lin_map","kind":"lemma","summary":"The linear map φ + (w - φ(v)) ⊗ π) coincides with φ on \\ker π and sends v to w. If \\sigma belon…","labels":["lem:update_lin_map"],"detail_key":"p38"},{"id":"n32666","layer":"informal","project":"p38","title":"These are direct checks.","kind":"proof","summary":"These are direct checks.","labels":[],"detail_key":"p38"},{"id":"n32667","layer":"informal","project":"p38","title":"def:short_formal_sol","kind":"definition","summary":"A formal solution \\F of \\Rel is (π, v)--short if, for every x, Df(x)v belongs to the interior o…","labels":["def:short_formal_sol"],"detail_key":"p38"},{"id":"n32668","layer":"informal","project":"p38","title":"lem:integration_step","kind":"lemma","summary":"Let \\F be a formal solution of \\Rel. Let K_1 ⊂ E be a compact subset, and let K_0 be a compact…","labels":["lem:integration_step"],"detail_key":"p38"},{"id":"n32669","layer":"informal","project":"p38","title":"We denote the components of \\F by f and φ. Since \\F is short, \\crefprop:∃_loops applied t…","kind":"proof","summary":"We denote the components of \\F by f and φ. Since \\F is short, \\crefprop:∃_loops applied to g \\c…","labels":[],"detail_key":"p38"},{"id":"n32670","layer":"informal","project":"p38","title":"def:ample_subset","kind":"definition","summary":"A subset Ω of a real vector space E is ample if the convex hull of each connected component of…","labels":["def:ample_subset"],"detail_key":"p38"},{"id":"n32671","layer":"informal","project":"p38","title":"lem:ample_codim_two","kind":"lemma","summary":"The complement of a linear subspace of codimension at least 2 is ample.","labels":["lem:ample_codim_two"],"detail_key":"p38"},{"id":"n32672","layer":"informal","project":"p38","title":"Let F be subspace of E with codimension at least 2. Let F' be a complement subspace. Its…","kind":"proof","summary":"Let F be subspace of E with codimension at least 2. Let F' be a complement subspace. Its dimens…","labels":[],"detail_key":"p38"},{"id":"n32673","layer":"informal","project":"p38","title":"def:ample_relation_loc","kind":"definition","summary":"A first order differential relation \\Rel is ample if all its slices are ample.","labels":["def:ample_relation_loc"],"detail_key":"p38"},{"id":"n32674","layer":"informal","project":"p38","title":"lem:h_principle_open_ample_loc","kind":"lemma","summary":"Let \\F be a formal solution of \\Rel. Let K_1 ⊂ E be a compact subset, and let K_0 be a compact…","labels":["lem:h_principle_open_ample_loc"],"detail_key":"p38"},{"id":"n32675","layer":"informal","project":"p38","title":"This is a straightforward induction using \\creflem:integration_step. Let (e_1, \\dots, e_n…","kind":"proof","summary":"This is a straightforward induction using \\creflem:integration_step. Let (e_1, \\dots, e_n) be a…","labels":[],"detail_key":"p38"},{"id":"n32676","layer":"informal","project":"p38","title":"def:update","kind":"definition","summary":"Given smooth open embeddings φ : X → M and ψ : Y → N, the update of a map f : M → N, using a ma…","labels":["def:update"],"detail_key":"p38"},{"id":"n32677","layer":"informal","project":"p38","title":"lem:smooth_updating","kind":"lemma","summary":"Let φ : P × X → M and ψ : P × Y → N be families of smooth open embeddings. Let K be a set in X…","labels":["lem:smooth_updating"],"detail_key":"p38"},{"id":"n32678","layer":"informal","project":"p38","title":"Note that P × M = (P × φ(X)) ∪ (P × φ(K)^c). Both those sets are open and the updated map…","kind":"proof","summary":"Note that P × M = (P × φ(X)) ∪ (P × φ(K)^c). Both those sets are open and the updated maps coin…","labels":[],"detail_key":"p38"},{"id":"n32679","layer":"informal","project":"p38","title":"lem:dist_updating","kind":"lemma","summary":"Let φ : X → M and ψ : Y → N be smooth open embeddings. Let K_X and K_P be compact sets in X and…","labels":["lem:dist_updating"],"detail_key":"p38"},{"id":"n32680","layer":"informal","project":"p38","title":"Let ε be a positive continuous function on M. Since K_X is compact, we get a positive num…","kind":"proof","summary":"Let ε be a positive continuous function on M. Since K_X is compact, we get a positive number ε₀…","labels":[],"detail_key":"p38"},{"id":"n32681","layer":"informal","project":"p38","title":"lem:nice_atlas","kind":"lemma","summary":"Let M be a manifold modelled on the normed space E and (V_j)_j ∈ J a cover of M by open sets. T…","labels":["lem:nice_atlas"],"detail_key":"p38"},{"id":"n32682","layer":"informal","project":"p38","title":"The proof is a standard compact-exhaustion argument. Let K_0, K_1, K_2, \\ldots be a compa…","kind":"proof","summary":"The proof is a standard compact-exhaustion argument. Let K_0, K_1, K_2, \\ldots be a compact exh…","labels":[],"detail_key":"p38"},{"id":"n32683","layer":"informal","project":"p38","title":"def:localisation_data","kind":"definition","summary":"Let f : M → N be a continuous map between manifolds. A localisation data for f is a tuple (E, F…","labels":["def:localisation_data"],"detail_key":"p38"},{"id":"n32684","layer":"informal","project":"p38","title":"lem:ex_localisation","kind":"lemma","summary":"Any continuous map between manifolds has some localisation data.","labels":["lem:ex_localisation"],"detail_key":"p38"},{"id":"n32685","layer":"informal","project":"p38","title":"The preceding lemma (applied to the trivial cover of N by itself) gives a family of ψ : ι…","kind":"proof","summary":"The preceding lemma (applied to the trivial cover of N by itself) gives a family of ψ : ι' × F…","labels":[],"detail_key":"p38"},{"id":"n32686","layer":"informal","project":"p38","title":"lem:stability_cover","kind":"lemma","summary":"In a metric space X, let U : ι → \\setX be a family of open subsets of X and let K : ι → \\setX b…","labels":["lem:stability_cover"],"detail_key":"p38"},{"id":"n32687","layer":"informal","project":"p38","title":"We first note that, for any given i, compactness of K and openness of V_i give a positive…","kind":"proof","summary":"We first note that, for any given i, compactness of K and openness of V_i give a positive numbe…","labels":[],"detail_key":"p38"},{"id":"n32688","layer":"informal","project":"p38","title":"lem:localisation_stability","kind":"lemma","summary":"Let f : M → N be a continuous map between manifolds, and let (φ, ψ, i) be some localisation dat…","labels":["lem:localisation_stability"],"detail_key":"p38"},{"id":"n32689","layer":"informal","project":"p38","title":"The preceding lemma applied to the family of open sets ψ_j(F) and the family of compact s…","kind":"proof","summary":"The preceding lemma applied to the family of open sets ψ_j(F) and the family of compact sets ψ_…","labels":[],"detail_key":"p38"},{"id":"n32690","layer":"informal","project":"p38","title":"def:pull_back_bundle","kind":"definition","summary":"For every bundle p : E → B and every map f \\co B' → B, the pull-back bundle f^*E → B' is define…","labels":["def:pull_back_bundle"],"detail_key":"p38"},{"id":"n32691","layer":"informal","project":"p38","title":"def:hom_bundle","kind":"definition","summary":"Let E → B and F → B be two vector bundles over some smooth manifold B. The bundle \\Hom(E, F) →…","labels":["def:hom_bundle"],"detail_key":"p38"},{"id":"n32692","layer":"informal","project":"p38","title":"def:one_jet_space","kind":"definition","summary":"Let M and N be smooth manifolds. Denote by p_1 and p_2 the projections of M × N to M and N resp…","labels":["def:one_jet_space"],"detail_key":"p38"},{"id":"n32693","layer":"informal","project":"p38","title":"def:one_jet_extension","kind":"definition","summary":"The 1-jet of a smooth map f \\co M → N is the map from m to J^1(M, N) defined by j^1f(m) = (m, f…","labels":["def:one_jet_extension"],"detail_key":"p38"},{"id":"n32694","layer":"informal","project":"p38","title":"lem:one_jet_extension_prop","kind":"lemma","summary":"For every smooth map f \\co M → N, \\item j^1f is smooth \\item j^1f is a section of J^1(M, N) → M","labels":["lem:one_jet_extension_prop","lem:one_jet_smooth","lem:one_jet_section"],"detail_key":"p38"},{"id":"n32695","layer":"informal","project":"p38","title":"Points 2 and 3 are obvious by construction. To show that j^1f is smooth, suppose that M i…","kind":"proof","summary":"Points 2 and 3 are obvious by construction. To show that j^1f is smooth, suppose that M is mode…","labels":[],"detail_key":"p38"},{"id":"n32696","layer":"informal","project":"p38","title":"def:holonomic_section","kind":"definition","summary":"A section \\F of J^1(M, N) → M is called holonomic if it is the 1--jet of its base map. Equivale…","labels":["def:holonomic_section"],"detail_key":"p38"},{"id":"n32697","layer":"informal","project":"p38","title":"def:rel","kind":"definition","summary":"A first order differential relation for maps from M to N is a subset \\Rel of J^1(M, N).","labels":["def:rel"],"detail_key":"p38"},{"id":"n32698","layer":"informal","project":"p38","title":"def:formal_sol","kind":"definition","summary":"A formal solution of a differential relation \\Rel ⊂ J^1(M, N) is a section of J^1(M, N) → M tak…","labels":["def:formal_sol"],"detail_key":"p38"},{"id":"n32699","layer":"informal","project":"p38","title":"def:htpy_formal_sol","kind":"definition","summary":"A homotopy of formal solutions of \\Rel is a smooth family of sections \\F : ℝ × M → J^1(M, N) su…","labels":["def:htpy_formal_sol"],"detail_key":"p38"},{"id":"n32700","layer":"informal","project":"p38","title":"def:transfer_map","kind":"definition","summary":"Given manifolds M, X, N and Y and smooth open embeddings g : Y → N and h : X → M we get a trans…","labels":["def:transfer_map"],"detail_key":"p38"},{"id":"n32701","layer":"informal","project":"p38","title":"lem:transfer","kind":"lemma","summary":"In the situation of the previous definition, given a section \\F : M → J^1(M, N): \\item Ψ_g, h(\\…","labels":["lem:transfer"],"detail_key":"p38"},{"id":"n32702","layer":"informal","project":"p38","title":"The first point is clear by composition. In order to prove the second point while keeping…","kind":"proof","summary":"The first point is clear by composition. In order to prove the second point while keeping notat…","labels":[],"detail_key":"p38"},{"id":"n32703","layer":"informal","project":"p38","title":"def:h-princ","kind":"definition","summary":"A first order differential relation \\Rel ⊂ J^1(M, N) satisfies the h-principle if every formal…","labels":["def:h-princ"],"detail_key":"p38"},{"id":"n32704","layer":"informal","project":"p38","title":"lem:param_trick","kind":"lemma","summary":"In the above setup, we have: \\item \\bar F is holonomic at (x, p) if and only if F_p is holonomi…","labels":["lem:param_trick"],"detail_key":"p38"},{"id":"n32705","layer":"informal","project":"p38","title":"For the first part, the derivative of \\bar F is ∂f/∂x(x, p) ⊕ ∂f/∂p(x, p), which is equal…","kind":"proof","summary":"For the first part, the derivative of \\bar F is ∂f/∂x(x, p) ⊕ ∂f/∂p(x, p), which is equal to \\b…","labels":[],"detail_key":"p38"},{"id":"n32706","layer":"informal","project":"p38","title":"lem:param_for_free","kind":"lemma","summary":"Let \\Rel be a first order differential relation for maps from M to N. If, for every manifold wi…","labels":["lem:param_for_free"],"detail_key":"p38"},{"id":"n32707","layer":"informal","project":"p38","title":"By \\creflem:param_trick we can turn a formal solution of \\Rel into a formal solution of \\…","kind":"proof","summary":"By \\creflem:param_trick we can turn a formal solution of \\Rel into a formal solution of \\Rel^P,…","labels":[],"detail_key":"p38"},{"id":"n32708","layer":"informal","project":"p38","title":"def:ample_relation","kind":"definition","summary":"A relation \\Rel is ample if, for every σ = (x, y, φ) in \\Rel and every (λ, v), the slice \\Rel_σ…","labels":["def:ample_relation"],"detail_key":"p38"},{"id":"n32709","layer":"informal","project":"p38","title":"lem:ample_iff_loc","kind":"lemma","summary":"Given manifolds W, X, Y and Z and smooth open embeddings g : Z → Y and h : W → X, the relation…","labels":["lem:ample_iff_loc"],"detail_key":"p38"},{"id":"n32710","layer":"informal","project":"p38","title":"By definition, the relation induced by \\Rel is ψ_g, h⁻¹\\Rel where ψ_g, h(w, z, φ) = (h(w)…","kind":"proof","summary":"By definition, the relation induced by \\Rel is ψ_g, h⁻¹\\Rel where ψ_g, h(w, z, φ) = (h(w), g(z)…","labels":[],"detail_key":"p38"},{"id":"n32711","layer":"informal","project":"p38","title":"lem:open_ample_immersion","kind":"lemma","summary":"The relation of immersions of M into N in positive codimension is open and ample.","labels":["lem:open_ample_immersion"],"detail_key":"p38"},{"id":"n32712","layer":"informal","project":"p38","title":"immersionRel_open_ample","kind":"proof","summary":"For every σ = (x, y, φ) in the immersion relation \\Rel, and for every dual pair (π, v), the sli…","labels":[],"detail_key":"p38"},{"id":"n32713","layer":"informal","project":"p38","title":"Gromov","kind":"theorem","summary":"[Gromov] For any manifolds X and Y, any relation \\Rel ⊂ J^1(X, Y) that is open and ample satisf…","labels":["thm:open_ample"],"detail_key":"p38"},{"id":"n32714","layer":"informal","project":"p38","title":"lem:ample_parameter","kind":"lemma","summary":"If \\Rel is ample then, for any parameter space P, \\Rel^P is also ample.","labels":["lem:ample_parameter"],"detail_key":"p38"},{"id":"n32715","layer":"informal","project":"p38","title":"We fix σ = (x, y, ψ) in \\Rel^P. For any λ = (λ_X, λ_P) ∈ T^*_xX × T^*_pP and v = (v_X, v_…","kind":"proof","summary":"We fix σ = (x, y, ψ) in \\Rel^P. For any λ = (λ_X, λ_P) ∈ T^*_xX × T^*_pP and v = (v_X, v_P) ∈ T…","labels":[],"detail_key":"p38"},{"id":"n32716","layer":"informal","project":"p38","title":"Proof of Theorem~\\refthm:open_ample","kind":"proof","summary":"[Proof of Theorem~\\refthm:open_ample] Lemmas~\\reflem:param_for_free and~\\reflem:ample_parameter…","labels":[],"detail_key":"p38"},{"id":"n32717","layer":"informal","project":"p38","title":"Smale 1958","kind":"theorem","summary":"[Smale 1958] There is a homotopy of immersions of 𝕊^2 into ℝ^3 from the inclusion map to the an…","labels":["thm:sphere_eversion"],"detail_key":"p38"},{"id":"n32718","layer":"informal","project":"p38","title":"We denote by ι the inclusion of 𝕊^2 into ℝ^3. We set j_t = (1-t)ι + ta. This is a homotop…","kind":"proof","summary":"We denote by ι the inclusion of 𝕊^2 into ℝ^3. We set j_t = (1-t)ι + ta. This is a homotopy from…","labels":[],"detail_key":"p38"},{"id":"n32719","layer":"informal","project":"p38","title":"lem:loc_immersion_rel_open","kind":"lemma","summary":"The relation \\Rel above is open.","labels":["lem:loc_immersion_rel_open"],"detail_key":"p38"},{"id":"n32720","layer":"informal","project":"p38","title":"The main task is to fix x_0 \\notin B and \\varphi_0 \\in L(E, E) which is injective on x_0^…","kind":"proof","summary":"The main task is to fix x_0 \\notin B and \\varphi_0 \\in L(E, E) which is injective on x_0^\\perp…","labels":[],"detail_key":"p38"},{"id":"n32721","layer":"informal","project":"p38","title":"lem:loc_immersion_rel_ample","kind":"lemma","summary":"The relation \\Rel above is ample.","labels":["lem:loc_immersion_rel_ample"],"detail_key":"p38"},{"id":"n32722","layer":"informal","project":"p38","title":"The core fact here is that if one fixes vector spaces F and F', a dual pair (\\pi, v) on F…","kind":"proof","summary":"The core fact here is that if one fixes vector spaces F and F', a dual pair (\\pi, v) on F and a…","labels":[],"detail_key":"p38"},{"id":"n32723","layer":"informal","project":"p38","title":"Smale 1958","kind":"theorem","summary":"[Smale 1958] There is a homotopy of immersion of 𝕊^2 into ℝ^3 from the inclusion map to the ant…","labels":["sphere_eversion_of_loc"],"detail_key":"p38"},{"id":"n32724","layer":"informal","project":"p38","title":"We denote by ι the inclusion of 𝕊^2 into ℝ^3. We set j_t = (1-t)ι + ta. This is a homotop…","kind":"proof","summary":"We denote by ι the inclusion of 𝕊^2 into ℝ^3. We set j_t = (1-t)ι + ta. This is a homotopy from…","labels":[],"detail_key":"p38"},{"id":"n32725","layer":"informal","project":"p38","title":"def:index_type","kind":"definition","summary":"For every natural number N we set \\[ \\ITN = ℕ \\text if N = 0\\\\ \\0, \\dots, N - 1\\ \\textotherwise…","labels":["def:index_type"],"detail_key":"p38"},{"id":"n32726","layer":"informal","project":"p38","title":"lem:exists_forall_eventually_of_index_type","kind":"lemma","summary":"Let X be a topological space and let Y be any set. Let f be a sequence of functions from X to Y…","labels":["lem:exists_forall_eventually_of_index_type"],"detail_key":"p38"},{"id":"n32727","layer":"informal","project":"p38","title":"The assumption that V is locally finite gives, for every x in X, a subset U_x of X such t…","kind":"proof","summary":"The assumption that V is locally finite gives, for every x in X, a subset U_x of X such that U_…","labels":[],"detail_key":"p38"},{"id":"n32728","layer":"informal","project":"p38","title":"def:germ","kind":"definition","summary":"Let X be a topological space, x a point in X and Y a set. A germ of function from X to Y at x i…","labels":["def:germ"],"detail_key":"p38"},{"id":"n32729","layer":"informal","project":"p38","title":"def:restrict_germ_predicate","kind":"definition","summary":"Let X be a topological space, A a subset of X, Y a set and P a local predicate on functions fro…","labels":["def:restrict_germ_predicate"],"detail_key":"p38"},{"id":"n32730","layer":"informal","project":"p38","title":"lem:inductive_construction","kind":"lemma","summary":"Let X be a topological space and Y be any set. Let U be a locally finite family of subsets of X…","labels":["lem:inductive_construction"],"detail_key":"p38"},{"id":"n32731","layer":"informal","project":"p38","title":"The main assumption from the lemma allows to build by induction a sequence f of functions…","kind":"proof","summary":"The main assumption from the lemma allows to build by induction a sequence f of functions from…","labels":[],"detail_key":"p38"},{"id":"n32732","layer":"informal","project":"p38","title":"lem:inductive_htpy_construction","kind":"lemma","summary":"Let X be a topological space and Y be any set. Let P₀ and P₁ be local predicates on maps from X…","labels":["lem:inductive_htpy_construction"],"detail_key":"p38"},{"id":"n32733","layer":"informal","project":"p38","title":"Carefully checking all details is a bit technical but the strategy is as follows. We fix…","kind":"proof","summary":"Carefully checking all details is a bit technical but the strategy is as follows. We fix an inc…","labels":[],"detail_key":"p38"},{"id":"n32734","layer":"informal","project":"p38","title":"lem:exists_locally_finite_subcover_of_locally","kind":"lemma","summary":"Let X be a metrizable locally compact second countable topological space. Let C be a closed sub…","labels":["lem:exists_locally_finite_subcover_of_locally"],"detail_key":"p38"},{"id":"n32735","layer":"informal","project":"p38","title":"This is a classical result.","kind":"proof","summary":"This is a classical result.","labels":[],"detail_key":"p38"},{"id":"n32736","layer":"informal","project":"p38","title":"lem:inductive_construction_of_loc","kind":"lemma","summary":"Let X a second countable locally compact metrizable topological space. Let P₀, P₀' and P₁ be lo…","labels":["lem:inductive_construction_of_loc"],"detail_key":"p38"},{"id":"n32737","layer":"informal","project":"p38","title":"The assumptions on the topology of X and local existence of solutions allow to apply \\Cre…","kind":"proof","summary":"The assumptions on the topology of X and local existence of solutions allow to apply \\Creflem:e…","labels":[],"detail_key":"p38"},{"id":"n32738","layer":"informal","project":"p38","title":"lem:relative_inductive_construction_of_loc","kind":"lemma","summary":"Let X a second countable locally compact metrizable topological space. Let P₀ and P₁ be local p…","labels":["lem:relative_inductive_construction_of_loc"],"detail_key":"p38"},{"id":"n32739","layer":"informal","project":"p38","title":"We reduce this to \\Creflem:inductive_construction_of_loc using as auxilliary local predic…","kind":"proof","summary":"We reduce this to \\Creflem:inductive_construction_of_loc using as auxilliary local predicate P₀…","labels":[],"detail_key":"p38"},{"id":"n32740","layer":"formal","project":"p38","title":"RelMfld.Ample.satisfiesHPrincipleWith","kind":"theorem","summary":"∀ EM : Type u_1 [inst : NormedAddCommGroup EM] [inst_1 : NormedSpace Real EM] [FiniteDimensiona…","labels":[],"detail_key":"p38","name":"RelMfld.Ample.satisfiesHPrincipleWith","module":"SphereEversion.Global.Gromov"},{"id":"n32741","layer":"formal","project":"p38","title":"immersionRel_open_ample","kind":"theorem","summary":"∀ E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : NormedSpace Real E] H : Type u_2 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NormedAddCommGroup…","labels":[],"detail_key":"p38","name":"OneJetBundle","module":"SphereEversion.Global.OneJetBundle"},{"id":"n32748","layer":"formal","project":"p38","title":"FamilyOneJetSec.isHolonomicAt_uncurry","kind":"theorem","summary":"∀ E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : NormedSpace Real E] H : Type u_2 [inst_2…","labels":[],"detail_key":"p38","name":"FamilyOneJetSec.isHolonomicAt_uncurry","module":"SphereEversion.Global.OneJetSec"},{"id":"n32749","layer":"formal","project":"p38","title":"OneJetSec.IsHolonomicAt","kind":"def","summary":"𝕜 : Type u_1 → [inst : NontriviallyNormedField 𝕜] → E : Type u_2 → [inst_1 : NormedAddCommGroup…","labels":[],"detail_key":"p38","name":"OneJetSec.IsHolonomicAt","module":"SphereEversion.Global.OneJetSec"},{"id":"n32750","layer":"formal","project":"p38","title":"OneJetSec.isHolonomicAt_iff","kind":"theorem","summary":"∀ 𝕜 : Type u_1 [inst : NontriviallyNormedField 𝕜] E : Type u_2 [inst_1 : NormedAddCommGroup E]…","labels":[],"detail_key":"p38","name":"OneJetSec.isHolonomicAt_iff","module":"SphereEversion.Global.OneJetSec"},{"id":"n32751","layer":"formal","project":"p38","title":"FamilyOneJetSec.uncurry_mem_relativize","kind":"theorem","summary":"∀ E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : NormedSpace Real E] H : Type u_2 [inst_2…","labels":[],"detail_key":"p38","name":"FamilyOneJetSec.uncurry_mem_relativize","module":"SphereEversion.Global.ParametricityForFree"},{"id":"n32752","layer":"formal","project":"p38","title":"RelMfld.Ample.relativize","kind":"theorem","summary":"∀ E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : NormedSpace Real E] H : Type u_2 [inst_2…","labels":[],"detail_key":"p38","name":"RelMfld.Ample.relativize","module":"SphereEversion.Global.ParametricityForFree"},{"id":"n32753","layer":"formal","project":"p38","title":"RelMfld.SatisfiesHPrinciple.satisfiesHPrincipleWith","kind":"theorem","summary":"∀ E : Type u_1 [inst : NormedAddCommGroup E] [inst_1 : 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u_2…","labels":[],"detail_key":"p38","name":"RelMfld.localize","module":"SphereEversion.Global.Relation"},{"id":"n32763","layer":"formal","project":"p38","title":"RelMfld.slice","kind":"def","summary":"E : Type u_1 → [inst : NormedAddCommGroup E] → [inst_1 : NormedSpace Real E] → H : Type u_2 → […","labels":[],"detail_key":"p38","name":"RelMfld.slice","module":"SphereEversion.Global.Relation"},{"id":"n32764","layer":"formal","project":"p38","title":"isHolonomicAt_localize_iff","kind":"theorem","summary":"∀ EX : Type u_1 [inst : NormedAddCommGroup EX] [inst_1 : NormedSpace Real EX] HX : Type u_2 [in…","labels":[],"detail_key":"p38","name":"isHolonomicAt_localize_iff","module":"SphereEversion.Global.Relation"},{"id":"n32765","layer":"formal","project":"p38","title":"OpenSmoothEmbedding.contMDiff_update","kind":"theorem","summary":"∀ 𝕜 : Type u_1 EX : Type u_2 EM : Type u_3 EY : Type u_4 EN : Type u_5 EM' : Type u_6 X : Type…","labels":[],"detail_key":"p38","name":"OpenSmoothEmbedding.contMDiff_update","module":"SphereEversion.Global.SmoothEmbedding"},{"id":"n32766","layer":"formal","project":"p38","title":"OpenSmoothEmbedding.dist_update","kind":"theorem","summary":"∀ 𝕜 : Type u_1 EX : Type u_2 EM : Type u_3 EY : Type u_4 EN : Type u_5 X : Type u_7 M : Type u_…","labels":[],"detail_key":"p38","name":"OpenSmoothEmbedding.dist_update","module":"SphereEversion.Global.SmoothEmbedding"},{"id":"n32767","layer":"formal","project":"p38","title":"OpenSmoothEmbedding.update","kind":"def","summary":"𝕜 : Type u_1 → EX : Type u_2 → EM : Type u_3 → EY : Type u_4 → EN : Type u_5 → X : Type u_7 → M…","labels":[],"detail_key":"p38","name":"OpenSmoothEmbedding.update","module":"SphereEversion.Global.SmoothEmbedding"},{"id":"n32768","layer":"formal","project":"p38","title":"nice_atlas","kind":"theorem","summary":"∀ (F : Type u_1) H : Type u_2 M : Type u [inst : NormedAddCommGroup F] [inst_1 : NormedSpace Re…","labels":[],"detail_key":"p38","name":"nice_atlas","module":"SphereEversion.Global.SmoothEmbedding"},{"id":"n32769","layer":"formal","project":"p38","title":"IndexType","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p38","name":"IndexType","module":"SphereEversion.Indexing"},{"id":"n32770","layer":"formal","project":"p38","title":"LocallyFinite.exists_forall_eventually_of_indexType","kind":"theorem","summary":"∀ α : Type u_1 X : Type u_2 [inst : TopologicalSpace X] N : Nat f : IndexType N → X → α V : Ind…","labels":[],"detail_key":"p38","name":"LocallyFinite.exists_forall_eventually_of_indexType","module":"SphereEversion.InductiveConstructions"},{"id":"n32771","layer":"formal","project":"p38","title":"inductive_construction","kind":"theorem","summary":"∀ X : Type u_1 Y : Type u_2 [inst : TopologicalSpace X] N : Nat U : IndexType N → Set X (P₀ : (…","labels":[],"detail_key":"p38","name":"inductive_construction","module":"SphereEversion.InductiveConstructions"},{"id":"n32772","layer":"formal","project":"p38","title":"inductive_construction_of_loc","kind":"theorem","summary":"∀ X : Type u_1 Y : Type u_2 [inst : EMetricSpace X] [LocallyCompactSpace X] [SecondCountableTop…","labels":[],"detail_key":"p38","name":"inductive_construction_of_loc","module":"SphereEversion.InductiveConstructions"},{"id":"n32773","layer":"formal","project":"p38","title":"inductive_htpy_construction","kind":"theorem","summary":"∀ X : Type u_1 Y : Type u_2 [inst : EMetricSpace X] [LocallyCompactSpace X] [SecondCountableTop…","labels":[],"detail_key":"p38","name":"inductive_htpy_construction","module":"SphereEversion.InductiveConstructions"},{"id":"n32774","layer":"formal","project":"p38","title":"relative_inductive_construction_of_loc","kind":"theorem","summary":"∀ X : Type u_1 Y : Type u_2 [inst : EMetricSpace X] [LocallyCompactSpace X] [SecondCountableTop…","labels":[],"detail_key":"p38","name":"relative_inductive_construction_of_loc","module":"SphereEversion.InductiveConstructions"},{"id":"n32775","layer":"formal","project":"p38","title":"RelLoc.FormalSol.IsShortAt","kind":"def","summary":"E : Type u_1 → [inst : NormedAddCommGroup E] → [inst_1 : NormedSpace Real E] → F : Type u_2 → […","labels":[],"detail_key":"p38","name":"RelLoc.FormalSol.IsShortAt","module":"SphereEversion.Local.AmpleRelation"},{"id":"n32776","layer":"formal","project":"p38","title":"RelLoc.IsAmple","kind":"def","summary":"E : Type u_1 → [inst : NormedAddCommGroup E] → [inst_1 : NormedSpace Real E] → F : Type u_2 → […","labels":[],"detail_key":"p38","name":"RelLoc.IsAmple","module":"SphereEversion.Local.AmpleRelation"},{"id":"n32777","layer":"formal","project":"p38","title":"corrugation","kind":"def","summary":"E : Type u_1 → [inst : NormedAddCommGroup E] → [inst_1 : NormedSpace Real E] → F : Type u_2 → 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It is impossible to construct two distinct points…","labels":["proposition_7"],"detail_key":"p39"},{"id":"n32869","layer":"informal","project":"p39","title":"Euclid's proof only works with the last two conditions, though he probably intended to pr…","kind":"proof","summary":"Euclid's proof only works with the last two conditions, though he probably intended to prove a…","labels":[],"detail_key":"p39"},{"id":"n32870","layer":"informal","project":"p39","title":"proposition_8","kind":"proposition","summary":"\\triangle~ABC and \\triangle~DEF are two triangles with AB = DE, AC = DF, and BC = EF. Then, \\an…","labels":["proposition_8"],"detail_key":"p39"},{"id":"n32871","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32872","layer":"informal","project":"p39","title":"proposition_9","kind":"proposition","summary":"Given an \\angle~BAC, there must exist a point F, s.t., F \\neq A and \\angle~BAF = \\angle~CAF.","labels":["proposition_9"],"detail_key":"p39"},{"id":"n32873","layer":"informal","project":"p39","title":"Euclid's proof has two problems. First, when constructing F, it fails to state the requir…","kind":"proof","summary":"Euclid's proof has two problems. First, when constructing F, it fails to state the requirement…","labels":[],"detail_key":"p39"},{"id":"n32874","layer":"informal","project":"p39","title":"proposition_9'","kind":"proposition","summary":"Given an \\angle~BAC, there must exist a point F, s.t., F \\neq A, \\angle~BAF = \\angle~CAF; F, C…","labels":["proposition_9'"],"detail_key":"p39"},{"id":"n32875","layer":"informal","project":"p39","title":"Same as the proof of Prop.~\\refproposition_9.","kind":"proof","summary":"Same as the proof of Prop.~\\refproposition_9.","labels":[],"detail_key":"p39"},{"id":"n32876","layer":"informal","project":"p39","title":"proposition_10","kind":"proposition","summary":"A and B are two distinct points on a line AB. There must exist a point D between them, s.t., |A…","labels":["proposition_10"],"detail_key":"p39"},{"id":"n32877","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32878","layer":"informal","project":"p39","title":"proposition_11","kind":"proposition","summary":"A, B are two distinct points on a line AB. C is a point between them. Then, there must exist a…","labels":["proposition_11"],"detail_key":"p39"},{"id":"n32879","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32880","layer":"informal","project":"p39","title":"proposition_11'","kind":"proposition","summary":"A, B are two distinct points on a line AB. C is a point between them, and X is a point not on A…","labels":["proposition_11'"],"detail_key":"p39"},{"id":"n32881","layer":"informal","project":"p39","title":"Similar to the original proof by Euclid.","kind":"proof","summary":"Similar to the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32882","layer":"informal","project":"p39","title":"proposition_11''","kind":"proposition","summary":"For two distinct points A and B on a line AB, there must exist a point F not on AB, s.t., \\angl…","labels":["proposition_11''"],"detail_key":"p39"},{"id":"n32883","layer":"informal","project":"p39","title":"Let C be a point on AB, s.t., A is between B and C. Apply Prop.~\\refproposition_11.","kind":"proof","summary":"Let C be a point on AB, s.t., A is between B and C. Apply Prop.~\\refproposition_11.","labels":[],"detail_key":"p39"},{"id":"n32884","layer":"informal","project":"p39","title":"proposition_11'''","kind":"proposition","summary":"A, B are two distinct points on a line AB. X is a point not on AB. Then, there must exist a poi…","labels":["proposition_11'''"],"detail_key":"p39"},{"id":"n32885","layer":"informal","project":"p39","title":"Let C be a point on AB, s.t., A is between B and C. Let Y be any point on the different s…","kind":"proof","summary":"Let C be a point on AB, s.t., A is between B and C. Let Y be any point on the different side of…","labels":[],"detail_key":"p39"},{"id":"n32886","layer":"informal","project":"p39","title":"proposition_12","kind":"proposition","summary":"A and B are two distinct points on a line AB. C is a point not on AB. Then, there must exist a…","labels":["proposition_12"],"detail_key":"p39"},{"id":"n32887","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32888","layer":"informal","project":"p39","title":"proposition_13","kind":"proposition","summary":"A, B are two distinct points on a line AB. C, D are two distinct points on a different line CD.…","labels":["proposition_13"],"detail_key":"p39"},{"id":"n32889","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32890","layer":"informal","project":"p39","title":"proposition_14","kind":"proposition","summary":"A, B are two distinct points on a line AB. C and D are two points on different sides of AB. B,…","labels":["proposition_14"],"detail_key":"p39"},{"id":"n32891","layer":"informal","project":"p39","title":"Euclid only discussed the case where E and A are on the same side of BD, though the proof…","kind":"proof","summary":"Euclid only discussed the case where E and A are on the same side of BD, though the proof for o…","labels":[],"detail_key":"p39"},{"id":"n32892","layer":"informal","project":"p39","title":"proposition_15","kind":"proposition","summary":"AB and CD are two different lines intersecting at E. A and B are two distinct points on AB. C a…","labels":["proposition_15"],"detail_key":"p39"},{"id":"n32893","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32894","layer":"informal","project":"p39","title":"proposition_16","kind":"proposition","summary":"BC is an edge of \\triangle~ABC and is extended to D. Then, we have \\angle~ACD~>~\\angle~CBA and…","labels":["proposition_16"],"detail_key":"p39"},{"id":"n32895","layer":"informal","project":"p39","title":"Euclid only proved \\angle~ACD~>~\\angle~CBA, though the proof of \\angle~ACD~>~\\angle~BAC i…","kind":"proof","summary":"Euclid only proved \\angle~ACD~>~\\angle~CBA, though the proof of \\angle~ACD~>~\\angle~BAC is almo…","labels":[],"detail_key":"p39"},{"id":"n32896","layer":"informal","project":"p39","title":"proposition_17","kind":"proposition","summary":"In \\triangle~ABC, we have \\angle~ABC + \\angle~BCA is less than 180 degrees.","labels":["proposition_17"],"detail_key":"p39"},{"id":"n32897","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32898","layer":"informal","project":"p39","title":"proposition_18","kind":"proposition","summary":"In \\triangle~ABC, if |AC|~>~|AB|, then \\angle~ABC~>~\\angle~BCA","labels":["proposition_18"],"detail_key":"p39"},{"id":"n32899","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32900","layer":"informal","project":"p39","title":"proposition_19","kind":"proposition","summary":"In \\triangle~ABC, if \\angle~ABC~>~\\angle~BCA, then |AC|~>~|AB|.","labels":["proposition_19"],"detail_key":"p39"},{"id":"n32901","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32902","layer":"informal","project":"p39","title":"proposition_20","kind":"proposition","summary":"In \\triangle~ABC, we have |BA| + |AC|~>~|BC|.","labels":["proposition_20"],"detail_key":"p39"},{"id":"n32903","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32904","layer":"informal","project":"p39","title":"proposition_21","kind":"proposition","summary":"D is a point inside \\triangle~ABC. We have |BD| + |DC|~<~|BA| + |AC|, and \\angle~BDC~>~\\angle~B…","labels":["proposition_21"],"detail_key":"p39"},{"id":"n32905","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32906","layer":"informal","project":"p39","title":"proposition_22","kind":"proposition","summary":"A and A' are two disctinct points on a line AA'. B and B' are two distinct points on a line BB'…","labels":["proposition_22"],"detail_key":"p39"},{"id":"n32907","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32908","layer":"informal","project":"p39","title":"proposition_22'","kind":"proposition","summary":"A and A'","labels":["proposition_22'"],"detail_key":"p39"},{"id":"n32909","layer":"informal","project":"p39","title":"The same proof as Euclid's.","kind":"proof","summary":"The same proof as Euclid's.","labels":[],"detail_key":"p39"},{"id":"n32910","layer":"informal","project":"p39","title":"proposition_22''","kind":"proposition","summary":"A and A'","labels":["proposition_22''"],"detail_key":"p39"},{"id":"n32911","layer":"informal","project":"p39","title":"The same proof as Euclid's.","kind":"proof","summary":"The same proof as Euclid's.","labels":[],"detail_key":"p39"},{"id":"n32912","layer":"informal","project":"p39","title":"proposition_23","kind":"proposition","summary":"\\angle~DCE is an angle. A and B are two distinct points on a line AB. Then, there must exists a…","labels":["proposition_23"],"detail_key":"p39"},{"id":"n32913","layer":"informal","project":"p39","title":"Euclid's proof omitted the degenerated cases that D is on CE, i.e., \\angle~DCE is either…","kind":"proof","summary":"Euclid's proof omitted the degenerated cases that D is on CE, i.e., \\angle~DCE is either 0 or \\…","labels":[],"detail_key":"p39"},{"id":"n32914","layer":"informal","project":"p39","title":"proposition_23'","kind":"proposition","summary":"\\angle~DCE is an angle. A and B are two distinct points on a line AB. X is a point not on AB. T…","labels":["proposition_23'"],"detail_key":"p39"},{"id":"n32915","layer":"informal","project":"p39","title":"Similar to the previous proof.","kind":"proof","summary":"Similar to the previous proof.","labels":[],"detail_key":"p39"},{"id":"n32916","layer":"informal","project":"p39","title":"proposition_24","kind":"proposition","summary":"\\triangle~ABC and \\triangle~DEF are two triangles s.t., |AB|=|DE|, |AC|=|DF|, and \\angle~BAC~>~…","labels":["proposition_24"],"detail_key":"p39"},{"id":"n32917","layer":"informal","project":"p39","title":"We only prove the case missed by Euclid. Let's construct \\triangle~EDG s.t., \\angle~EDG =…","kind":"proof","summary":"We only prove the case missed by Euclid. Let's construct \\triangle~EDG s.t., \\angle~EDG = \\angl…","labels":[],"detail_key":"p39"},{"id":"n32918","layer":"informal","project":"p39","title":"proposition_25","kind":"proposition","summary":"\\triangle~ABC and \\triangle~DEF are two triangles s.t. |AB|=|DE|, |AC|=|DF|, and |BC|~>~|EF|. T…","labels":["proposition_25"],"detail_key":"p39"},{"id":"n32919","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32920","layer":"informal","project":"p39","title":"proposition_26","kind":"proposition","summary":"\\triangle~ABC and \\triangle~DEF are two triangles, s.t., \\angle~ABC=\\angle~DEF, \\angle~BCA=\\ang…","labels":["proposition_26"],"detail_key":"p39"},{"id":"n32921","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32922","layer":"informal","project":"p39","title":"proposition_27","kind":"proposition","summary":"A and E are two distcint points on the line AE. F and D are two distinct points on FD. E and F…","labels":["proposition_27"],"detail_key":"p39"},{"id":"n32923","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32924","layer":"informal","project":"p39","title":"proposition_28","kind":"proposition","summary":"A and B are two distcint points on AB. C and D are two distinct points on CD. E and F are two d…","labels":["proposition_28"],"detail_key":"p39"},{"id":"n32925","layer":"informal","project":"p39","title":"See the original proof by Euclid.","kind":"proof","summary":"See the original proof by Euclid.","labels":[],"detail_key":"p39"},{"id":"n32926","layer":"informal","project":"p40","title":"The entrywise measurable structure on matrices","kind":"definition","summary":"[The entrywise measurable structure on matrices] A matrix is a function \\(m \\to n \\to \\alpha\\),…","labels":["instMeasurableSpaceMatrix"],"detail_key":"p40"},{"id":"n32927","layer":"informal","project":"p40","title":"The iterated linear cocycle","kind":"definition","summary":"[The iterated linear cocycle] Given \\(A : X \\to Matrix(Find)(Find)\\,R\\) and \\(T : X \\to X\\), de…","labels":["cocycle"],"detail_key":"p40"},{"id":"n32928","layer":"informal","project":"p40","title":"The cocycle identity","kind":"theorem","summary":"[The cocycle identity] For all \\(m, n \\in N\\) and \\(x \\in X\\), \\[ A^(m+n)(x) = A^(m)(T^nx)\\cdot…","labels":["cocycle_add"],"detail_key":"p40"},{"id":"n32929","layer":"informal","project":"p40","title":"Induction on \\(n\\) with \\(x\\) generalized. For \\(n = 0\\) both sides equal \\(A^(m)(x)\\). F…","kind":"proof","summary":"Induction on \\(n\\) with \\(x\\) generalized. For \\(n = 0\\) both sides equal \\(A^(m)(x)\\). For the…","labels":[],"detail_key":"p40"},{"id":"n32930","layer":"informal","project":"p40","title":"One-sided log-integrability of the generator","kind":"definition","summary":"[One-sided log-integrability of the generator] The hypothesis \\(\\textttIntegrableLogNorm\\,A\\,\\m…","labels":["IntegrableLogNorm"],"detail_key":"p40"},{"id":"n32931","layer":"informal","project":"p40","title":"Measurability of the cocycle iterates","kind":"theorem","summary":"[Measurability of the cocycle iterates] If \\(A\\) and \\(T\\) are measurable then for each \\(n\\) t…","labels":["measurable_cocycle"],"detail_key":"p40"},{"id":"n32932","layer":"informal","project":"p40","title":"Induction on \\(n\\). The base case is a constant map. For the step, the recursion writes \\…","kind":"proof","summary":"Induction on \\(n\\). The base case is a constant map. For the step, the recursion writes \\(A^(n+…","labels":[],"detail_key":"p40"},{"id":"n32933","layer":"informal","project":"p40","title":"The Pi structure is an opens-measurable space","kind":"lemma","summary":"[The Pi structure is an opens-measurable space] The entrywise (Pi) measurable structure on \\(Ma…","labels":["instOpensMeasurableSpaceMatrix"],"detail_key":"p40"},{"id":"n32934","layer":"informal","project":"p40","title":"Measurability of the L2 operator norm","kind":"theorem","summary":"[Measurability of the L2 operator norm] The map \\(M \\mapsto \\left\\lVert M \\right\\rVert\\) on \\(M…","labels":["measurable_l2_opNorm"],"detail_key":"p40"},{"id":"n32935","layer":"informal","project":"p40","title":"The norm is continuous for the operator-norm topology, and by \\refinstOpensMeasurableSpac…","kind":"proof","summary":"The norm is continuous for the operator-norm topology, and by \\refinstOpensMeasurableSpaceMatri…","labels":[],"detail_key":"p40"},{"id":"n32936","layer":"informal","project":"p40","title":"Measurability of the determinant","kind":"theorem","summary":"[Measurability of the determinant] The determinant \\(M \\mapsto \\det M\\) is measurable.","labels":["measurable_det"],"detail_key":"p40"},{"id":"n32937","layer":"informal","project":"p40","title":"By the Leibniz formula \\(\\det M = \\sum_\\sigmasgn(\\sigma)\\prod_i M_i,\\sigma(i)\\), the dete…","kind":"proof","summary":"By the Leibniz formula \\(\\det M = \\sum_\\sigmasgn(\\sigma)\\prod_i M_i,\\sigma(i)\\), the determinan…","labels":[],"detail_key":"p40"},{"id":"n32938","layer":"informal","project":"p40","title":"Measurability of the matrix inverse","kind":"theorem","summary":"[Measurability of the matrix inverse] The inverse \\(M \\mapsto M^-1\\) is measurable on the entry…","labels":["measurable_inv_matrix"],"detail_key":"p40"},{"id":"n32939","layer":"informal","project":"p40","title":"Writing \\(M^-1 = (\\det M)^-1\\cdot adj(M)\\), each entry is a ratio of polynomials in the e…","kind":"proof","summary":"Writing \\(M^-1 = (\\det M)^-1\\cdot adj(M)\\), each entry is a ratio of polynomials in the entries…","labels":[],"detail_key":"p40"},{"id":"n32940","layer":"informal","project":"p40","title":"Invertibility of the iterates","kind":"lemma","summary":"[Invertibility of the iterates] If \\(\\det A(x) \\neq 0\\) for every \\(x\\), then \\(\\det A^(n)(x) \\…","labels":["det_cocycle_ne_zero"],"detail_key":"p40"},{"id":"n32941","layer":"informal","project":"p40","title":"Induction on \\(n\\). The base case is \\(\\det 1 = 1\\). For the step, \\(\\det(A^(n)(Tx)\\cdot…","kind":"proof","summary":"Induction on \\(n\\). The base case is \\(\\det 1 = 1\\). For the step, \\(\\det(A^(n)(Tx)\\cdot A(x))…","labels":[],"detail_key":"p40"},{"id":"n32942","layer":"informal","project":"p40","title":"The unit matrix has norm one","kind":"lemma","summary":"[The unit matrix has norm one] When \\(d \\neq 0\\), \\(\\left\\lVert (1 : Matrix(Find)(Find)\\,R) \\ri…","labels":["norm_one_matrix"],"detail_key":"p40"},{"id":"n32943","layer":"informal","project":"p40","title":"For \\(d \\neq 0\\) the space \\(EuclideanSpace\\;R\\,(Find)\\) is nontrivial. The star-algebra…","kind":"proof","summary":"For \\(d \\neq 0\\) the space \\(EuclideanSpace\\;R\\,(Find)\\) is nontrivial. The star-algebra equiva…","labels":[],"detail_key":"p40"},{"id":"n32944","layer":"informal","project":"p40","title":"Positivity of the iterate norms","kind":"lemma","summary":"[Positivity of the iterate norms] Assume \\(\\det A(x) \\neq 0\\) for all \\(x\\) and \\(d \\neq 0\\). T…","labels":["norm_cocycle_pos"],"detail_key":"p40"},{"id":"n32945","layer":"informal","project":"p40","title":"If \\(\\left\\lVert A^(n)(x) \\right\\rVert = 0\\) then \\(A^(n)(x)\\) is the zero matrix, whose…","kind":"proof","summary":"If \\(\\left\\lVert A^(n)(x) \\right\\rVert = 0\\) then \\(A^(n)(x)\\) is the zero matrix, whose determ…","labels":[],"detail_key":"p40"},{"id":"n32946","layer":"informal","project":"p40","title":"Subadditivity of the log-norm cocycle","kind":"theorem","summary":"[Subadditivity of the log-norm cocycle] Assume \\(\\det A \\neq 0\\) everywhere and \\(d \\neq 0\\). T…","labels":["isSubadditiveCocycle_logNorm"],"detail_key":"p40"},{"id":"n32947","layer":"informal","project":"p40","title":"Rewrite \\(m+n\\) as \\(n+m\\) and apply the cocycle identity \\refcocycle_add to get \\(A^(m+n…","kind":"proof","summary":"Rewrite \\(m+n\\) as \\(n+m\\) and apply the cocycle identity \\refcocycle_add to get \\(A^(m+n)(x) =…","labels":[],"detail_key":"p40"},{"id":"n32948","layer":"informal","project":"p40","title":"Subadditivity of the inverse log-norm cocycle","kind":"theorem","summary":"[Subadditivity of the inverse log-norm cocycle] Under the same hypotheses, \\(g_n(x) = \\log\\left…","labels":["isSubadditiveCocycle_logNorm_inv"],"detail_key":"p40"},{"id":"n32949","layer":"informal","project":"p40","title":"As above, after \\(\\refcocycle_add\\) the inverse of the product reverses the order: \\((A^(…","kind":"proof","summary":"As above, after \\(\\refcocycle_add\\) the inverse of the product reverses the order: \\((A^(m+n)(x…","labels":[],"detail_key":"p40"},{"id":"n32950","layer":"informal","project":"p40","title":"Upper Fekete bound","kind":"lemma","summary":"[Upper Fekete bound] For \\(\\det A \\neq 0\\) everywhere and \\(d \\neq 0\\), \\[ \\log\\left\\lVert A^(n…","labels":["logNorm_cocycle_le_birkhoffSum"],"detail_key":"p40"},{"id":"n32951","layer":"informal","project":"p40","title":"Induction on \\(n\\). The base case is \\(0 \\le 0\\). For the step, the recursion gives \\(A^(…","kind":"proof","summary":"Induction on \\(n\\). The base case is \\(0 \\le 0\\). For the step, the recursion gives \\(A^(n+1)(x…","labels":[],"detail_key":"p40"},{"id":"n32952","layer":"informal","project":"p40","title":"Lower bound via the inverse Birkhoff sum","kind":"lemma","summary":"[Lower bound via the inverse Birkhoff sum] Under the same hypotheses, \\[ -\\sum_k<n\\log^+\\left\\l…","labels":["neg_birkhoffSum_le_logNorm_cocycle"],"detail_key":"p40"},{"id":"n32953","layer":"informal","project":"p40","title":"From \\(A^(n)(x)\\cdot (A^(n)(x))^-1 = 1\\) and submultiplicativity, \\(1 = \\left\\lVert 1 \\ri…","kind":"proof","summary":"From \\(A^(n)(x)\\cdot (A^(n)(x))^-1 = 1\\) and submultiplicativity, \\(1 = \\left\\lVert 1 \\right\\rV…","labels":[],"detail_key":"p40"},{"id":"n32954","layer":"informal","project":"p40","title":"Upper Fekete bound for the inverse cocycle","kind":"lemma","summary":"[Upper Fekete bound for the inverse cocycle] Under the same hypotheses, \\[ \\log\\left\\lVert (A^(…","labels":["logNorm_inv_cocycle_le_birkhoffSum"],"detail_key":"p40"},{"id":"n32955","layer":"informal","project":"p40","title":"Induction on \\(n\\), exactly as for the forward bound but with the inverse generator: the…","kind":"proof","summary":"Induction on \\(n\\), exactly as for the forward bound but with the inverse generator: the recurs…","labels":[],"detail_key":"p40"},{"id":"n32956","layer":"informal","project":"p40","title":"Integral of a Birkhoff sum","kind":"lemma","summary":"[Integral of a Birkhoff sum] For measure-preserving \\(T\\) and integrable \\(f\\), \\(\\int \\sum_k<n…","labels":["integral_birkhoffSum"],"detail_key":"p40"},{"id":"n32957","layer":"informal","project":"p40","title":"Each composition \\(f\\circ T^k\\) is integrable and integral-preserving because \\(T^k\\) is…","kind":"proof","summary":"Each composition \\(f\\circ T^k\\) is integrable and integral-preserving because \\(T^k\\) is measur…","labels":[],"detail_key":"p40"},{"id":"n32958","layer":"informal","project":"p40","title":"Integrability of the log-norm levels","kind":"theorem","summary":"[Integrability of the log-norm levels] Let \\(T\\) be measure-preserving for a finite measure \\(\\…","labels":["integrable_logNorm_cocycle"],"detail_key":"p40"},{"id":"n32959","layer":"informal","project":"p40","title":"The level \\(g_n\\) is sandwiched between \\(-B_n^-\\) and \\(B_n^+\\), where \\(B_n^\\pm\\) are t…","kind":"proof","summary":"The level \\(g_n\\) is sandwiched between \\(-B_n^-\\) and \\(B_n^+\\), where \\(B_n^\\pm\\) are the Bir…","labels":[],"detail_key":"p40"},{"id":"n32960","layer":"informal","project":"p40","title":"Furstenberg--Kesten, top exponent","kind":"theorem","summary":"[Furstenberg--Kesten, top exponent] Let \\(T\\) be ergodic for a probability measure \\(\\mu\\), let…","labels":["furstenbergKesten_norm"],"detail_key":"p40"},{"id":"n32961","layer":"informal","project":"p40","title":"If \\(d = 0\\) the matrix algebra is trivial, every norm is \\(0\\), and the limit is the con…","kind":"proof","summary":"If \\(d = 0\\) the matrix algebra is trivial, every norm is \\(0\\), and the limit is the constant…","labels":[],"detail_key":"p40"},{"id":"n32962","layer":"informal","project":"p40","title":"Furstenberg--Kesten, bottom exponent","kind":"theorem","summary":"[Furstenberg--Kesten, bottom exponent] Under the same hypotheses there is a constant \\(\\lambda…","labels":["furstenbergKesten_norm_inv"],"detail_key":"p40"},{"id":"n32963","layer":"informal","project":"p40","title":"Identical to the top case with the inverse subadditive cocycle \\(g_n = \\log\\left\\lVert (A…","kind":"proof","summary":"Identical to the top case with the inverse subadditive cocycle \\(g_n = \\log\\left\\lVert (A^(n))^…","labels":[],"detail_key":"p40"},{"id":"n32964","layer":"informal","project":"p40","title":"Garsia's maximal function","kind":"definition","summary":"[Garsia's maximal function] For a measurable \\(T\\), a function \\(g : X \\to R\\), \\(N : N\\) and \\…","labels":["maxBirkhoff"],"detail_key":"p40"},{"id":"n32965","layer":"informal","project":"p40","title":"Nonnegativity of the maximal function","kind":"lemma","summary":"[Nonnegativity of the maximal function] For all \\(g, N, x\\) one has \\(0 \\le \\textttmaxBirkhoff\\…","labels":["maxBirkhoff_nonneg"],"detail_key":"p40"},{"id":"n32966","layer":"informal","project":"p40","title":"The index \\(k = 0\\) lies in \\(\\textttrange\\,(N+1)\\), and the corresponding term is \\(\\tex…","kind":"proof","summary":"The index \\(k = 0\\) lies in \\(\\textttrange\\,(N+1)\\), and the corresponding term is \\(\\textttbir…","labels":[],"detail_key":"p40"},{"id":"n32967","layer":"informal","project":"p40","title":"Recursion for the maximal function","kind":"lemma","summary":"[Recursion for the maximal function] For all \\(g, N, x\\), \\[ \\textttmaxBirkhoff\\,T\\,g\\,(N+1)\\,x…","labels":["maxBirkhoff_succ"],"detail_key":"p40"},{"id":"n32968","layer":"informal","project":"p40","title":"This is the identity \\(\\textttrange\\,(N+2) = \\textttinsert\\,(N+1)\\, (\\textttrange\\,(N+1))…","kind":"proof","summary":"This is the identity \\(\\textttrange\\,(N+2) = \\textttinsert\\,(N+1)\\, (\\textttrange\\,(N+1))\\) app…","labels":[],"detail_key":"p40"},{"id":"n32969","layer":"informal","project":"p40","title":"Garsia's pointwise inequality","kind":"lemma","summary":"[Garsia's pointwise inequality] On the set where the maximal function is positive, for every \\(…","labels":["maxBirkhoff_le_add"],"detail_key":"p40"},{"id":"n32970","layer":"informal","project":"p40","title":"Pull the constant \\(g\\,x\\) through the supremum: by the additive shift of a nonempty \\tex…","kind":"proof","summary":"Pull the constant \\(g\\,x\\) through the supremum: by the additive shift of a nonempty \\textttsup…","labels":[],"detail_key":"p40"},{"id":"n32971","layer":"informal","project":"p40","title":"Measurability and integrability of the maximal function","kind":"lemma","summary":"[Measurability and integrability of the maximal function] If \\(T\\) is measure-preserving and \\(…","labels":["integrable_maxBirkhoff"],"detail_key":"p40"},{"id":"n32972","layer":"informal","project":"p40","title":"Each summand \\(g \\circ T^[j]\\) is integrable because \\(T^[j]\\) is measure-preserving, so…","kind":"proof","summary":"Each summand \\(g \\circ T^[j]\\) is integrable because \\(T^[j]\\) is measure-preserving, so the fi…","labels":[],"detail_key":"p40"},{"id":"n32973","layer":"informal","project":"p40","title":"Garsia's inequality at a fixed level","kind":"proposition","summary":"[Garsia's inequality at a fixed level] For a measure-preserving \\(T\\) and a measurable integrab…","labels":["setIntegral_maxBirkhoff_pos_nonneg"],"detail_key":"p40"},{"id":"n32974","layer":"informal","project":"p40","title":"Write \\(E\\) for the level set and \\(M = \\textttmaxBirkhoff\\,T\\,g\\,N\\). Integrating the po…","kind":"proof","summary":"Write \\(E\\) for the level set and \\(M = \\textttmaxBirkhoff\\,T\\,g\\,N\\). Integrating the pointwis…","labels":[],"detail_key":"p40"},{"id":"n32975","layer":"informal","project":"p40","title":"Level sets exhaust the target set","kind":"lemma","summary":"[Level sets exhaust the target set] The level sets \\(\\x : 0 < \\textttmaxBirkhoff\\,T\\,g\\,N\\,x\\\\)…","labels":["iUnion_setOf_maxBirkhoff_pos"],"detail_key":"p40"},{"id":"n32976","layer":"informal","project":"p40","title":"Monotonicity is \\textttFinset.sup' monotonicity over the nested ranges. For the union: \\(…","kind":"proof","summary":"Monotonicity is \\textttFinset.sup' monotonicity over the nested ranges. For the union: \\(0 < \\t…","labels":[],"detail_key":"p40"},{"id":"n32977","layer":"informal","project":"p40","title":"Maximal ergodic inequality (Hopf--Garsia)","kind":"theorem","summary":"[Maximal ergodic inequality (Hopf--Garsia)] For a measure-preserving \\(T\\) and an integrable \\(…","labels":["setIntegral_birkhoffSum_pos_nonneg"],"detail_key":"p40"},{"id":"n32978","layer":"informal","project":"p40","title":"First treat a measurable integrable \\(g\\): by \\crefsetIntegral_maxBirkhoff_pos_nonneg eac…","kind":"proof","summary":"First treat a measurable integrable \\(g\\): by \\crefsetIntegral_maxBirkhoff_pos_nonneg each leve…","labels":[],"detail_key":"p40"},{"id":"n32979","layer":"informal","project":"p40","title":"Conditional expectation commutes with the dynamics","kind":"theorem","summary":"[Conditional expectation commutes with the dynamics] For a finite measure, a measure-preserving…","labels":["condExp_invariants_comp_self"],"detail_key":"p40"},{"id":"n32980","layer":"informal","project":"p40","title":"Set-integral invariance of \\(h \\circ T\\) over a measurable invariant set reduces, via \\(\\…","kind":"proof","summary":"Set-integral invariance of \\(h \\circ T\\) over a measurable invariant set reduces, via \\(\\texttt…","labels":[],"detail_key":"p40"},{"id":"n32981","layer":"informal","project":"p40","title":"Subexponential orbital tail","kind":"lemma","summary":"[Subexponential orbital tail] For a measure-preserving \\(T\\) and integrable \\(g\\), the orbital…","labels":["ae_tendsto_orbit_div_atTop_zero"],"detail_key":"p40"},{"id":"n32982","layer":"informal","project":"p40","title":"A Borel--Cantelli argument. For each threshold \\(\\delta = 1/(k+1)\\) the series \\(\\sum_n \\…","kind":"proof","summary":"A Borel--Cantelli argument. For each threshold \\(\\delta = 1/(k+1)\\) the series \\(\\sum_n \\mu\\x :…","labels":[],"detail_key":"p40"},{"id":"n32983","layer":"informal","project":"p40","title":"A.e.\\ boundedness of Birkhoff averages","kind":"lemma","summary":"[A.e.\\ boundedness of Birkhoff averages] For a finite measure, a measure-preserving \\(T\\) and i…","labels":["ae_bddAbove_birkhoffAverage"],"detail_key":"p40"},{"id":"n32984","layer":"informal","project":"p40","title":"The maximal ergodic inequality applied to \\(g - c\\) yields, for the maximal set \\(B_c = \\…","kind":"proof","summary":"The maximal ergodic inequality applied to \\(g - c\\) yields, for the maximal set \\(B_c = \\x : \\e…","labels":[],"detail_key":"p40"},{"id":"n32985","layer":"informal","project":"p40","title":"Limsup invariance and vanishing perturbations","kind":"lemma","summary":"[Limsup invariance and vanishing perturbations] The pointwise limsup \\(x \\mapsto \\limsup_n \\tex…","labels":["limsup_birkhoffAverage_comp_ae"],"detail_key":"p40"},{"id":"n32986","layer":"informal","project":"p40","title":"The difference \\(A_n(g)(Tx) - A_n(g)(x) = n^-1(g(T^[n]x) - g\\,x)\\) tends to \\(0\\) a.e.\\ b…","kind":"proof","summary":"The difference \\(A_n(g)(Tx) - A_n(g)(x) = n^-1(g(T^[n]x) - g\\,x)\\) tends to \\(0\\) a.e.\\ by the…","labels":[],"detail_key":"p40"},{"id":"n32987","layer":"informal","project":"p40","title":"The core maximal-inequality step","kind":"proposition","summary":"[The core maximal-inequality step] For a finite measure, measure-preserving \\(T\\), integrable \\…","labels":["measure_setOf_lt_limsup_eq_zero"],"detail_key":"p40"},{"id":"n32988","layer":"informal","project":"p40","title":"Write \\(L = \\mu[g\\mid I]\\) and \\(Ls = \\limsup A_\\bullet(g)\\). The set \\(E = \\L + \\varepsi…","kind":"proof","summary":"Write \\(L = \\mu[g\\mid I]\\) and \\(Ls = \\limsup A_\\bullet(g)\\). The set \\(E = \\L + \\varepsilon <…","labels":[],"detail_key":"p40"},{"id":"n32989","layer":"informal","project":"p40","title":"Pointwise (Birkhoff) ergodic theorem","kind":"theorem","summary":"[Pointwise (Birkhoff) ergodic theorem] For a finite measure, a measure-preserving \\(T\\) and an…","labels":["tendsto_birkhoffAverage_ae"],"detail_key":"p40"},{"id":"n32990","layer":"informal","project":"p40","title":"Unioning the null superlevel sets of \\crefmeasure_setOf_lt_limsup_eq_zero over \\(\\varepsi…","kind":"proof","summary":"Unioning the null superlevel sets of \\crefmeasure_setOf_lt_limsup_eq_zero over \\(\\varepsilon =…","labels":[],"detail_key":"p40"},{"id":"n32991","layer":"informal","project":"p40","title":"Ergodic case of Birkhoff","kind":"corollary","summary":"[Ergodic case of Birkhoff] For an ergodic \\(T\\) on a probability space and integrable \\(g\\), th…","labels":["tendsto_birkhoffAverage_ae_integral"],"detail_key":"p40"},{"id":"n32992","layer":"informal","project":"p40","title":"By \\creftendsto_birkhoffAverage_ae the limit is \\(\\mu[g\\mid I]\\), which is a.e.\\ \\(T\\)-in…","kind":"proof","summary":"By \\creftendsto_birkhoffAverage_ae the limit is \\(\\mu[g\\mid I]\\), which is a.e.\\ \\(T\\)-invarian…","labels":[],"detail_key":"p40"},{"id":"n32993","layer":"informal","project":"p40","title":"Subadditive cocycle","kind":"definition","summary":"[Subadditive cocycle] A sequence \\(g : N\\to X \\to R\\) is a \\emphsubadditive cocycle over \\(T\\)…","labels":["IsSubadditiveCocycle"],"detail_key":"p40"},{"id":"n32994","layer":"informal","project":"p40","title":"Singleton and block subadditivity","kind":"lemma","summary":"[Singleton and block subadditivity] For a subadditive cocycle and \\(n : N\\), one has \\(g\\,(n+1)…","labels":["le_birkhoffSum_one"],"detail_key":"p40"},{"id":"n32995","layer":"informal","project":"p40","title":"Both are inductions that peel off the last block and apply the defining subadditivity at…","kind":"proof","summary":"Both are inductions that peel off the last block and apply the defining subadditivity at the sp…","labels":[],"detail_key":"p40"},{"id":"n32996","layer":"informal","project":"p40","title":"Normalized cocycle","kind":"definition","summary":"[Normalized cocycle] The \\emphnormalized cocycle is \\(\\textttcdiv\\,g\\,n\\,x = g\\,(n+1)\\,x / (n+1…","labels":["cdiv"],"detail_key":"p40"},{"id":"n32997","layer":"informal","project":"p40","title":"Fekete limit of the normalized integrals","kind":"lemma","summary":"[Fekete limit of the normalized integrals] For a measure-preserving \\(T\\), an integrable subadd…","labels":["exists_fekete"],"detail_key":"p40"},{"id":"n32998","layer":"informal","project":"p40","title":"Integrating the cocycle inequality and using measure-preservation (\\(\\int g\\,n\\,(T^[m]\\cd…","kind":"proof","summary":"Integrating the cocycle inequality and using measure-preservation (\\(\\int g\\,n\\,(T^[m]\\cdot) =…","labels":[],"detail_key":"p40"},{"id":"n32999","layer":"informal","project":"p40","title":"Invariance from a one-sided orbital bound","kind":"lemma","summary":"[Invariance from a one-sided orbital bound] For a finite measure, measure-preserving \\(T\\) and…","labels":["ae_eq_comp_of_le_comp"],"detail_key":"p40"},{"id":"n33000","layer":"informal","project":"p40","title":"For each rational \\(c\\) the upper level set \\(\\c \\le F\\\\) is null-measurable (via a measu…","kind":"proof","summary":"For each rational \\(c\\) the upper level set \\(\\c \\le F\\\\) is null-measurable (via a measurable…","labels":[],"detail_key":"p40"},{"id":"n33001","layer":"informal","project":"p40","title":"Envelopes are a.e.\\ measurable and \\(T\\)-invariant","kind":"lemma","summary":"[Envelopes are a.e.\\ measurable and \\(T\\)-invariant] For a finite measure, measure-preserving \\…","labels":["limsup_div_comp_ae"],"detail_key":"p40"},{"id":"n33002","layer":"informal","project":"p40","title":"The subadditive bound \\(\\textttcdiv\\,g\\,n\\,x \\le g\\,1\\,x/(n+1) + g\\,n\\,(Tx)/(n+1)\\) diffe…","kind":"proof","summary":"The subadditive bound \\(\\textttcdiv\\,g\\,n\\,x \\le g\\,1\\,x/(n+1) + g\\,n\\,(Tx)/(n+1)\\) differs fro…","labels":[],"detail_key":"p40"},{"id":"n33003","layer":"informal","project":"p40","title":"Integrability of the limsup envelope","kind":"lemma","summary":"[Integrability of the limsup envelope] Under the same hypotheses, the limsup envelope \\(f_+\\) i…","labels":["int_limsup_div_integrable"],"detail_key":"p40"},{"id":"n33004","layer":"informal","project":"p40","title":"The nonnegative Fatou defect \\(d_n(x) = \\textttbirkhoffAverage\\,R\\,T\\,(g\\,1)\\, (n+1)\\,x -…","kind":"proof","summary":"The nonnegative Fatou defect \\(d_n(x) = \\textttbirkhoffAverage\\,R\\,T\\,(g\\,1)\\, (n+1)\\,x - \\text…","labels":[],"detail_key":"p40"},{"id":"n33005","layer":"informal","project":"p40","title":"Hard direction: \\(\\limsup \\le \\liminf\\) a.e.","kind":"proposition","summary":"[Hard direction: \\(\\limsup \\le \\liminf\\) a.e.] Under the same hypotheses, for a.e.\\ \\(x\\) the \\…","labels":["ae_ereal_limsup_le_liminf"],"detail_key":"p40"},{"id":"n33006","layer":"informal","project":"p40","title":"This is the stopping-time / greedy block argument of Katznelson--Weiss and Karlsson. Afte…","kind":"proof","summary":"This is the stopping-time / greedy block argument of Katznelson--Weiss and Karlsson. After the…","labels":[],"detail_key":"p40"},{"id":"n33007","layer":"informal","project":"p40","title":"Kingman core: a.e.\\ existence of an integrable limit","kind":"theorem","summary":"[Kingman core: a.e.\\ existence of an integrable limit] For a finite measure, measure-preserving…","labels":["ae_tendsto_cdiv"],"detail_key":"p40"},{"id":"n33008","layer":"informal","project":"p40","title":"Take \\(G = f_+\\), integrable by \\crefint_limsup_div_integrable. On the a.e.\\ good set the…","kind":"proof","summary":"Take \\(G = f_+\\), integrable by \\crefint_limsup_div_integrable. On the a.e.\\ good set the \\text…","labels":[],"detail_key":"p40"},{"id":"n33009","layer":"informal","project":"p40","title":"Kingman's subadditive ergodic theorem","kind":"theorem","summary":"[Kingman's subadditive ergodic theorem] For a finite measure, a measure-preserving \\(T\\), an in…","labels":["tendsto_kingman"],"detail_key":"p40"},{"id":"n33010","layer":"informal","project":"p40","title":"Take \\(G = f_-\\). On the a.e.\\ set where the normalized cocycle is bounded, \\(\\liminf \\le…","kind":"proof","summary":"Take \\(G = f_-\\). On the a.e.\\ set where the normalized cocycle is bounded, \\(\\liminf \\le \\lims…","labels":[],"detail_key":"p40"},{"id":"n33011","layer":"informal","project":"p40","title":"Kingman, ergodic case","kind":"corollary","summary":"[Kingman, ergodic case] For an ergodic \\(T\\) on a probability space, an integrable subadditive…","labels":["tendsto_kingman_ergodic"],"detail_key":"p40"},{"id":"n33012","layer":"informal","project":"p40","title":"Kingman's theorem \\creftendsto_kingman gives a \\(T\\)-invariant integrable limit \\(G\\); er…","kind":"proof","summary":"Kingman's theorem \\creftendsto_kingman gives a \\(T\\)-invariant integrable limit \\(G\\); ergodici…","labels":[],"detail_key":"p40"},{"id":"n33013","layer":"informal","project":"p40","title":"Ultrametric growth function","kind":"definition","summary":"[Ultrametric growth function] Let E be a real vector space. A function g\\colon E\\toR is an \\emp…","labels":["IsUltrametricGrowth"],"detail_key":"p40"},{"id":"n33014","layer":"informal","project":"p40","title":"Strict ultrametric equality","kind":"lemma","summary":"[Strict ultrametric equality] If g is an ultrametric growth function and g(v)\\neq g(w) (with v,…","labels":["add_eq_max_of_ne"],"detail_key":"p40"},{"id":"n33015","layer":"informal","project":"p40","title":"By symmetry (v+w=w+v) assume g(v)<g(w), so \\max=g(w). The non-Archimedean inequality give…","kind":"proof","summary":"By symmetry (v+w=w+v) assume g(v)<g(w), so \\max=g(w). The non-Archimedean inequality gives g(v+…","labels":[],"detail_key":"p40"},{"id":"n33016","layer":"informal","project":"p40","title":"Sum of distinct-value vectors","kind":"lemma","summary":"[Sum of distinct-value vectors] Let g be an ultrametric growth function, s a nonempty finite in…","labels":["sum_ne_zero_and_g_eq_sup'"],"detail_key":"p40"},{"id":"n33017","layer":"informal","project":"p40","title":"Strong induction on s, peeling off one element a. For the inductive step, the tail sum is…","kind":"proof","summary":"Strong induction on s, peeling off one element a. For the inductive step, the tail sum is nonze…","labels":[],"detail_key":"p40"},{"id":"n33018","layer":"informal","project":"p40","title":"Distinct values are independent","kind":"lemma","summary":"[Distinct values are independent] If g is an ultrametric growth function and v\\colon \\iota\\to E…","labels":["linearIndependent_of_injOn"],"detail_key":"p40"},{"id":"n33019","layer":"informal","project":"p40","title":"Suppose \\sum_j c_j v_j = 0 with some c_i\\neq 0. Restrict to the support t=\\j : c_j\\neq 0\\…","kind":"proof","summary":"Suppose \\sum_j c_j v_j = 0 with some c_i\\neq 0. Restrict to the support t=\\j : c_j\\neq 0\\\\ni i,…","labels":[],"detail_key":"p40"},{"id":"n33020","layer":"informal","project":"p40","title":"Finiteness of the value set","kind":"lemma","summary":"[Finiteness of the value set] If E is finite-dimensional and g is an ultrametric growth functio…","labels":["finite_range"],"detail_key":"p40"},{"id":"n33021","layer":"informal","project":"p40","title":"If the value set were infinite, pick \\dim_R E + 1 distinct values and witnessing nonzero…","kind":"proof","summary":"If the value set were infinite, pick \\dim_R E + 1 distinct values and witnessing nonzero vector…","labels":[],"detail_key":"p40"},{"id":"n33022","layer":"informal","project":"p40","title":"Sublevel submodule","kind":"definition","summary":"[Sublevel submodule] For an ultrametric growth function g and threshold t\\inR, the \\emphsubleve…","labels":["sublevel"],"detail_key":"p40"},{"id":"n33023","layer":"informal","project":"p40","title":"Defining sequence","kind":"definition","summary":"[Defining sequence] For x\\in X and v\\inR^d the \\emphgrowth sequence is \\[ growthSeq(A,T,x,v)(n)…","labels":["growthSeq"],"detail_key":"p40"},{"id":"n33024","layer":"informal","project":"p40","title":"Upper Lyapunov growth function","kind":"definition","summary":"[Upper Lyapunov growth function] The \\emphupper Lyapunov growth function is the \\limsup of the…","labels":["lambdaBar"],"detail_key":"p40"},{"id":"n33025","layer":"informal","project":"p40","title":"Scaling invariance","kind":"lemma","summary":"[Scaling invariance] For c\\neq 0 and v\\neq 0, \\bar\\lambda(c\\cdot v)=\\bar\\lambda(v). By linearit…","labels":["lambdaBar_smul"],"detail_key":"p40"},{"id":"n33026","layer":"informal","project":"p40","title":"By linearity \\left\\lVert A^(n)(x)(c\\cdot v) \\right\\rVert=\\left\\lvert c \\right\\rvert\\,\\lef…","kind":"proof","summary":"By linearity \\left\\lVert A^(n)(x)(c\\cdot v) \\right\\rVert=\\left\\lvert c \\right\\rvert\\,\\left\\lVer…","labels":[],"detail_key":"p40"},{"id":"n33027","layer":"informal","project":"p40","title":"Finiteness sandwich","kind":"lemma","summary":"[Finiteness sandwich] Assume T ergodic on a probability space, A measurable and invertible, wit…","labels":["lambdaBar_mem_Icc"],"detail_key":"p40"},{"id":"n33028","layer":"informal","project":"p40","title":"Take \\lambda_top and \\lambda_k' from the Furstenberg--Kesten limits of \\tfrac1n\\log\\left\\…","kind":"proof","summary":"Take \\lambda_top and \\lambda_k' from the Furstenberg--Kesten limits of \\tfrac1n\\log\\left\\lVert…","labels":[],"detail_key":"p40"},{"id":"n33029","layer":"informal","project":"p40","title":"Non-Archimedean inequality","kind":"lemma","summary":"[Non-Archimedean inequality] For nonzero v,w,v+w, if the three growth sequences are bounded, th…","labels":["lambdaBar_add_le"],"detail_key":"p40"},{"id":"n33030","layer":"informal","project":"p40","title":"From the triangle inequality \\left\\lVert A^(n)(v+w) \\right\\rVert\\le\\left\\lVert A^(n)v \\ri…","kind":"proof","summary":"From the triangle inequality \\left\\lVert A^(n)(v+w) \\right\\rVert\\le\\left\\lVert A^(n)v \\right\\rV…","labels":[],"detail_key":"p40"},{"id":"n33031","layer":"informal","project":"p40","title":"\\bar\\lambda is an ultrametric growth function, a.e.","kind":"theorem","summary":"[\\bar\\lambda is an ultrametric growth function, a.e.] Under the hypotheses of \\reflambdaBar_mem…","labels":["isUltrametricGrowth_lambdaBar"],"detail_key":"p40"},{"id":"n33032","layer":"informal","project":"p40","title":"Scaling-invariance is \\reflambdaBar_smul (trivial on v=0). The non-Archimedean axiom is \\…","kind":"proof","summary":"Scaling-invariance is \\reflambdaBar_smul (trivial on v=0). The non-Archimedean axiom is \\reflam…","labels":[],"detail_key":"p40"},{"id":"n33033","layer":"informal","project":"p40","title":"A-equivariance, a.e.","kind":"theorem","summary":"[A-equivariance, a.e.] Under the same hypotheses, for a.e.\\ x and every v\\neq 0, \\[ \\bar\\lambda…","labels":["lambdaBar_equivariant_ae"],"detail_key":"p40"},{"id":"n33034","layer":"informal","project":"p40","title":"The cocycle identity A^(n+1)(x)=A^(n)(Tx)\\,A(x) gives growthSeq_x(v)(n+1)=\\tfrac1n+1\\log\\…","kind":"proof","summary":"The cocycle identity A^(n+1)(x)=A^(n)(Tx)\\,A(x) gives growthSeq_x(v)(n+1)=\\tfrac1n+1\\log\\left\\l…","labels":[],"detail_key":"p40"},{"id":"n33035","layer":"informal","project":"p40","title":"Lyapunov spectrum","kind":"definition","summary":"[Lyapunov spectrum] The \\emphLyapunov spectrum at x is the finite set of realized values \\[ lya…","labels":["lyapunovSpectrum"],"detail_key":"p40"},{"id":"n33036","layer":"informal","project":"p40","title":"Multiplicity count and descending list","kind":"definition","summary":"[Multiplicity count and descending list] Write k=specCard(A,T,x) for the number of distinct exp…","labels":["specList"],"detail_key":"p40"},{"id":"n33037","layer":"informal","project":"p40","title":"Sublevel subspace","kind":"definition","summary":"[Sublevel subspace] The \\emphsublevel subspace at threshold t, \\[ lambdaSublevel(A,T,x,t) = \\\\,…","labels":["lambdaSublevel"],"detail_key":"p40"},{"id":"n33038","layer":"informal","project":"p40","title":"Oseledets filtration / limsup flag","kind":"definition","summary":"[Oseledets filtration / limsup flag] The \\emphlimsup flag at x is the family vflag(A,T,x)\\colon…","labels":["vflag"],"detail_key":"p40"},{"id":"n33039","layer":"informal","project":"p40","title":"Extremal levels","kind":"lemma","summary":"[Extremal levels] On the good set, vflag(A,T,x)(0)=\\top; and unconditionally vflag(A,T,x)(last)…","labels":["vflag_zero"],"detail_key":"p40"},{"id":"n33040","layer":"informal","project":"p40","title":"For any v\\neq 0, \\bar\\lambda_x(v) lies in the spectrum, so k>0 and specList(0) is the max…","kind":"proof","summary":"For any v\\neq 0, \\bar\\lambda_x(v) lies in the spectrum, so k>0 and specList(0) is the maximum o…","labels":[],"detail_key":"p40"},{"id":"n33041","layer":"informal","project":"p40","title":"Strict decrease","kind":"theorem","summary":"[Strict decrease] On the good set, vflag(A,T,x)(i+1) \\subsetneq vflag(A,T,x)(i) for each interi…","labels":["vflag_strictAnti"],"detail_key":"p40"},{"id":"n33042","layer":"informal","project":"p40","title":"Inclusion: since specList is strictly antitone, specList(i+1)<specList(i), so the subleve…","kind":"proof","summary":"Inclusion: since specList is strictly antitone, specList(i+1)<specList(i), so the sublevel at t…","labels":[],"detail_key":"p40"},{"id":"n33043","layer":"informal","project":"p40","title":"Stratum exactness","kind":"lemma","summary":"[Stratum exactness] On the good set, if v\\invflag(A,T,x)(i) but v\\notinvflag(A,T,x)(i+1), then…","labels":["lambdaBar_eq_on_stratum"],"detail_key":"p40"},{"id":"n33044","layer":"informal","project":"p40","title":"Membership in level i gives \\bar\\lambda_x(v)\\le\\lambda_i. Since \\bar\\lambda_x(v) is a spe…","kind":"proof","summary":"Membership in level i gives \\bar\\lambda_x(v)\\le\\lambda_i. Since \\bar\\lambda_x(v) is a spectrum…","labels":[],"detail_key":"p40"},{"id":"n33045","layer":"informal","project":"p40","title":"A-equivariance of spectrum and flag, a.e.","kind":"theorem","summary":"[A-equivariance of spectrum and flag, a.e.] Under the standing hypotheses, for a.e.\\ x the spec…","labels":["vflag_equivariant"],"detail_key":"p40"},{"id":"n33046","layer":"informal","project":"p40","title":"The bijection v\\mapsto A x\\cdot v preserves \\bar\\lambda by \\reflambdaBar_equivariant_ae (…","kind":"proof","summary":"The bijection v\\mapsto A x\\cdot v preserves \\bar\\lambda by \\reflambdaBar_equivariant_ae (a.e.),…","labels":[],"detail_key":"p40"},{"id":"n33047","layer":"informal","project":"p40","title":"Projection matrix encoding","kind":"definition","summary":"[Projection matrix encoding] For K\\leR^d, orthProjMatrix(K) is the matrix of the orthogonal pro…","labels":["orthProjMatrix"],"detail_key":"p40"},{"id":"n33048","layer":"informal","project":"p40","title":"Measurable family of subspaces","kind":"definition","summary":"[Measurable family of subspaces] A subspace-valued map V\\colon X\\toSubmodule\\,R\\,R^d is a \\emph…","labels":["MeasurableSubspace"],"detail_key":"p40"},{"id":"n33049","layer":"informal","project":"p40","title":"Scalar growth is measurable","kind":"lemma","summary":"[Scalar growth is measurable] For fixed v, the map x\\mapsto\\bar\\lambda_x(v) is measurable. It i…","labels":["measurable_lambdaBar_apply"],"detail_key":"p40"},{"id":"n33050","layer":"informal","project":"p40","title":"It is the \\limsup of the sequence x\\mapsto\\tfrac1n\\log\\left\\lVert A^(n)(x)\\cdot v \\right\\…","kind":"proof","summary":"It is the \\limsup of the sequence x\\mapsto\\tfrac1n\\log\\left\\lVert A^(n)(x)\\cdot v \\right\\rVert.…","labels":[],"detail_key":"p40"},{"id":"n33051","layer":"informal","project":"p40","title":"Polynomial in a measurable matrix","kind":"lemma","summary":"[Polynomial in a measurable matrix] For a fixed real polynomial q, the map a\\mapsto q(a) on Mat…","labels":["measurable_aeval_matrix"],"detail_key":"p40"},{"id":"n33052","layer":"informal","project":"p40","title":"Induction on q over the constant/sum/monomial generators, using that matrix addition and…","kind":"proof","summary":"Induction on q over the constant/sum/monomial generators, using that matrix addition and multip…","labels":[],"detail_key":"p40"},{"id":"n33053","layer":"informal","project":"p40","title":"CFC measurability via interpolating polynomial","kind":"theorem","summary":"[CFC measurability via interpolating polynomial] Let M\\colon X\\toMat_d\\times d(R) be measurable…","labels":["measurable_cfc_eqOn_polynomial"],"detail_key":"p40"},{"id":"n33054","layer":"informal","project":"p40","title":"On the spectrum of M x the continuous functional calculus of g coincides with that of q,…","kind":"proof","summary":"On the spectrum of M x the continuous functional calculus of g coincides with that of q, and fo…","labels":[],"detail_key":"p40"},{"id":"n33055","layer":"informal","project":"p40","title":"CFC measurability for continuous functions","kind":"theorem","summary":"[CFC measurability for continuous functions] Let M be measurable with each M x self-adjoint, an…","labels":["measurable_cfc_continuous"],"detail_key":"p40"},{"id":"n33056","layer":"informal","project":"p40","title":"A single polynomial need not agree with f on the unbounded family of spectra, so approxim…","kind":"proof","summary":"A single polynomial need not agree with f on the unbounded family of spectra, so approximate pe…","labels":[],"detail_key":"p40"},{"id":"n33057","layer":"informal","project":"p40","title":"Deterministic singular-value exponents","kind":"theorem","summary":"[Deterministic singular-value exponents] There is an antitone sequence \\(\\lambda^0 : N\\toR\\) (a…","labels":["exists_lam_tendsto_singularValue"],"detail_key":"p40"},{"id":"n33058","layer":"informal","project":"p40","title":"Package the ergodic limits \\(\\Gamma_k = \\lim \\tfrac1n \\log s_k(A^(n)(x))\\) of the product…","kind":"proof","summary":"Package the ergodic limits \\(\\Gamma_k = \\lim \\tfrac1n \\log s_k(A^(n)(x))\\) of the products of t…","labels":[],"detail_key":"p40"},{"id":"n33059","layer":"informal","project":"p40","title":"Block-value step function reproduces the spectrum","kind":"lemma","summary":"[Block-value step function reproduces the spectrum] Let \\(stepVal\\,\\lambda^0\\,D\\) be the step f…","labels":["stepVal_exp_lam"],"detail_key":"p40"},{"id":"n33060","layer":"informal","project":"p40","title":"At the argument \\(e^\\lambda^0_j\\) the threshold indicator \\( 1_(c_k,\\infty)\\) is \\(1\\) ex…","kind":"proof","summary":"At the argument \\(e^\\lambda^0_j\\) the threshold indicator \\( 1_(c_k,\\infty)\\) is \\(1\\) exactly…","labels":[],"detail_key":"p40"},{"id":"n33061","layer":"informal","project":"p40","title":"Spectral deviation bound","kind":"lemma","summary":"[Spectral deviation bound] For a self-adjoint matrix \\(M\\) and any function \\(g\\), \\[ \\left\\lVe…","labels":["norm_sub_cfc_le_sum_eigenvalue_dev"],"detail_key":"p40"},{"id":"n33062","layer":"informal","project":"p40","title":"Writing \\(M = id(M)\\) gives \\(M - g(M) = (id-g)(M)\\) by linearity of the continuous funct…","kind":"proof","summary":"Writing \\(M = id(M)\\) gives \\(M - g(M) = (id-g)(M)\\) by linearity of the continuous functional…","labels":[],"detail_key":"p40"},{"id":"n33063","layer":"informal","project":"p40","title":"Per-term band-projector convergence","kind":"lemma","summary":"[Per-term band-projector convergence] For \\(\\mu\\)-a.e.\\ \\(x\\) and every threshold index \\(k\\in[…","labels":["ae_forall_tendsto_block_term"],"detail_key":"p40"},{"id":"n33064","layer":"informal","project":"p40","title":"At a genuine gap \\(\\lambda^0_k < \\lambda^0_k-1\\) the threshold \\(c_k\\) is strictly separa…","kind":"proof","summary":"At a genuine gap \\(\\lambda^0_k < \\lambda^0_k-1\\) the threshold \\(c_k\\) is strictly separated fr…","labels":[],"detail_key":"p40"},{"id":"n33065","layer":"informal","project":"p40","title":"Existence of the Oseledets limit","kind":"theorem","summary":"[Existence of the Oseledets limit] For \\(\\mu\\)-a.e.\\ \\(x\\) the approximants \\(q_n(x)\\) converge…","labels":["tendsto_qpow"],"detail_key":"p40"},{"id":"n33066","layer":"informal","project":"p40","title":"The eigenvalues \\(\\mu_j,n=\\sigma_j^1/n\\) of \\(q_n(x)\\) converge a.e.\\ to the exponentials…","kind":"proof","summary":"The eigenvalues \\(\\mu_j,n=\\sigma_j^1/n\\) of \\(q_n(x)\\) converge a.e.\\ to the exponentials \\(e^\\…","labels":[],"detail_key":"p40"},{"id":"n33067","layer":"informal","project":"p40","title":"The named Oseledets limit","kind":"definition","summary":"[The named Oseledets limit] \\(\\Lambda(x) := (\\,\\lim_n (q_n(x))_ij\\,)_ij\\) is the entrywise real…","labels":["oseledetsLimit"],"detail_key":"p40"},{"id":"n33068","layer":"informal","project":"p40","title":"The limit is the a.e.\\ limit of the approximants","kind":"theorem","summary":"[The limit is the a.e.\\ limit of the approximants] For \\(\\mu\\)-a.e.\\ \\(x\\), \\(q_n(x)\\to \\Lambda…","labels":["tendsto_oseledetsLimit"],"detail_key":"p40"},{"id":"n33069","layer":"informal","project":"p40","title":"On the a.e.\\ full convergence set of Theorem~\\reftendsto_qpow the entrywise limit recover…","kind":"proof","summary":"On the a.e.\\ full convergence set of Theorem~\\reftendsto_qpow the entrywise limit recovers the…","labels":[],"detail_key":"p40"},{"id":"n33070","layer":"informal","project":"p40","title":"Structure of the limit","kind":"proposition","summary":"[Structure of the limit] For \\(\\mu\\)-a.e.\\ \\(x\\), \\(\\Lambda(x)\\) is self-adjoint and positive s…","labels":["oseledetsLimit_posSemidef"],"detail_key":"p40"},{"id":"n33071","layer":"informal","project":"p40","title":"Self-adjointness \\(M^\\top=M\\) is an entrywise closed condition preserved under the matrix…","kind":"proof","summary":"Self-adjointness \\(M^\\top=M\\) is an entrywise closed condition preserved under the matrix limit…","labels":[],"detail_key":"p40"},{"id":"n33072","layer":"informal","project":"p40","title":"Gram quadratic-form band bound","kind":"lemma","summary":"[Gram quadratic-form band bound] For self-adjoint \\(Q\\), a band indicator \\(\\chi= 1_(c,\\infty)\\…","labels":["inner_cfc_ge_band"],"detail_key":"p40"},{"id":"n33073","layer":"informal","project":"p40","title":"The band projector \\(\\chi(Q)\\) is a self-adjoint idempotent, so \\(\\left\\lVert \\chi(Q)v \\r…","kind":"proof","summary":"The band projector \\(\\chi(Q)\\) is a self-adjoint idempotent, so \\(\\left\\lVert \\chi(Q)v \\right\\r…","labels":[],"detail_key":"p40"},{"id":"n33074","layer":"informal","project":"p40","title":"Band lower bound for the cocycle","kind":"lemma","summary":"[Band lower bound for the cocycle] For \\(c\\ge0\\) and \\(n\\ge1\\), \\[ c^2n\\,\\left\\lVert P^c_n(x)\\,…","labels":["cocycle_apply_sq_ge_band"],"detail_key":"p40"},{"id":"n33075","layer":"informal","project":"p40","title":"Raising \\(q_n(x)=(gram_n)^1/2n\\) to the \\(2n\\)-th power via the functional calculus recov…","kind":"proof","summary":"Raising \\(q_n(x)=(gram_n)^1/2n\\) to the \\(2n\\)-th power via the functional calculus recovers th…","labels":[],"detail_key":"p40"},{"id":"n33076","layer":"informal","project":"p40","title":"The band correction vanishes","kind":"lemma","summary":"[The band correction vanishes] If \\(P^c_n(x)\\to P\\) with \\(Pv\\neq0\\), then \\(\\tfrac1n\\log\\left\\…","labels":["tendsto_inv_mul_log_norm_bandProjector_apply"],"detail_key":"p40"},{"id":"n33077","layer":"informal","project":"p40","title":"The evaluation \\(M\\mapsto Mv\\) is continuous in finite dimensions, so \\(P^c_n(x)v\\to Pv\\n…","kind":"proof","summary":"The evaluation \\(M\\mapsto Mv\\) is continuous in finite dimensions, so \\(P^c_n(x)v\\to Pv\\neq0\\)…","labels":[],"detail_key":"p40"},{"id":"n33078","layer":"informal","project":"p40","title":"Per-vector liminf lower bound","kind":"proposition","summary":"[Per-vector liminf lower bound] If \\(P^c_n(x)\\to P\\) with \\(c>0\\) and \\(Pv\\neq0\\), and the cocy…","labels":["log_le_liminf_log_cocycle_apply"],"detail_key":"p40"},{"id":"n33079","layer":"informal","project":"p40","title":"Taking logs in Lemma~\\refcocycle_apply_sq_ge_band and dividing by \\(2n\\) gives, eventuall…","kind":"proof","summary":"Taking logs in Lemma~\\refcocycle_apply_sq_ge_band and dividing by \\(2n\\) gives, eventually, \\[…","labels":[],"detail_key":"p40"},{"id":"n33080","layer":"informal","project":"p40","title":"Band-projector nesting, kernel propagation","kind":"lemma","summary":"[Band-projector nesting, kernel propagation] For thresholds \\(c\\le c'\\) with limit band project…","labels":["limitBandProjector_apply_eq_zero_of_le"],"detail_key":"p40"},{"id":"n33081","layer":"informal","project":"p40","title":"The finite-\\(n\\) bands are nested: \\( 1_(c,\\infty)\\cdot 1_(c',\\infty)= 1_(c',\\infty)\\) on…","kind":"proof","summary":"The finite-\\(n\\) bands are nested: \\( 1_(c,\\infty)\\cdot 1_(c',\\infty)= 1_(c',\\infty)\\) on the s…","labels":[],"detail_key":"p40"},{"id":"n33082","layer":"informal","project":"p40","title":"Tempering","kind":"lemma","summary":"[Tempering] If \\(T\\) is measure-preserving and \\(g\\in L^1(\\mu)\\), then for \\(\\mu\\)-a.e.\\ \\(x\\),…","labels":["tempering_posLog"],"detail_key":"p40"},{"id":"n33083","layer":"informal","project":"p40","title":"The series \\(\\sum_n g(T^n x)/n^2\\) is a.e.\\ finite by integrability and invariance of \\(\\…","kind":"proof","summary":"The series \\(\\sum_n g(T^n x)/n^2\\) is a.e.\\ finite by integrability and invariance of \\(\\mu\\),…","labels":[],"detail_key":"p40"},{"id":"n33084","layer":"informal","project":"p40","title":"Slow-volume exponent squeeze (superseded route)","kind":"lemma","summary":"[Slow-volume exponent squeeze (superseded route)] \\emphThis lemma belonged to a superseded dete…","labels":["limsup_topSlow_le_of_squeeze"],"detail_key":"p40"},{"id":"n33085","layer":"informal","project":"p40","title":"The total volume exponent is the Furstenberg--Kesten determinant limit \\(\\sum_j\\lambda^0_…","kind":"proof","summary":"The total volume exponent is the Furstenberg--Kesten determinant limit \\(\\sum_j\\lambda^0_j\\); t…","labels":[],"detail_key":"p40"},{"id":"n33086","layer":"informal","project":"p40","title":"Spectral upper bound on each stratum","kind":"theorem","summary":"[Spectral upper bound on each stratum] For \\(\\mu\\)-a.e.\\ \\(x\\), every index \\(i\\) and every vec…","labels":["spectral_upper_bound_of_squeeze"],"detail_key":"p40"},{"id":"n33087","layer":"informal","project":"p40","title":"On the \\(IsUltrametricGrowth\\) good set the per-vector upper growth function \\(\\overline\\…","kind":"proof","summary":"On the \\(IsUltrametricGrowth\\) good set the per-vector upper growth function \\(\\overline\\lambda…","labels":[],"detail_key":"p40"},{"id":"n33088","layer":"informal","project":"p40","title":"Upper bound on the slow space","kind":"theorem","summary":"[Upper bound on the slow space] On the ultrametric-growth good set, every vector \\(v\\) of the \\…","labels":["vslow_subset_lambdaSublevel_of_upper"],"detail_key":"p40"},{"id":"n33089","layer":"informal","project":"p40","title":"The \\(\\limsup\\) of \\(\\tfrac1n\\log\\left\\lVert A^(n)v \\right\\rVert\\) is by definition the u…","kind":"proof","summary":"The \\(\\limsup\\) of \\(\\tfrac1n\\log\\left\\lVert A^(n)v \\right\\rVert\\) is by definition the upper g…","labels":[],"detail_key":"p40"},{"id":"n33090","layer":"informal","project":"p40","title":"Band projectors converge to the CFC indicator","kind":"theorem","summary":"[Band projectors converge to the CFC indicator] For \\(\\mu\\)-a.e.\\ \\(x\\), every \\(c>0\\) that is…","labels":["ae_tendsto_bandProjector_cfc_indicator"],"detail_key":"p40"},{"id":"n33091","layer":"informal","project":"p40","title":"Since \\(c\\) avoids the spectrum of \\(\\Lambda(x)\\), there is a gap \\(\\delta>0\\) between \\(…","kind":"proof","summary":"Since \\(c\\) avoids the spectrum of \\(\\Lambda(x)\\), there is a gap \\(\\delta>0\\) between \\(c\\) an…","labels":[],"detail_key":"p40"},{"id":"n33092","layer":"informal","project":"p40","title":"Reverse slow-flag inclusion","kind":"theorem","summary":"[Reverse slow-flag inclusion] For \\(\\mu\\)-a.e.\\ \\(x\\) and every \\(t\\), \\(lambdaSublevel(x,t) \\l…","labels":["ae_lambdaSublevel_le_vslow"],"detail_key":"p40"},{"id":"n33093","layer":"informal","project":"p40","title":"Contrapositively, a vector \\(v\\notinvslow(e^t)\\) has nonzero component in the band of \\(\\…","kind":"proof","summary":"Contrapositively, a vector \\(v\\notinvslow(e^t)\\) has nonzero component in the band of \\(\\Lambda…","labels":[],"detail_key":"p40"},{"id":"n33094","layer":"informal","project":"p40","title":"The slow flag equals the limsup sublevel","kind":"theorem","summary":"[The slow flag equals the limsup sublevel] Under the spectral upper bound and the reverse inclu…","labels":["vslow_eq_lambdaSublevel_of_upper"],"detail_key":"p40"},{"id":"n33095","layer":"informal","project":"p40","title":"Two inclusions: the forward one (Theorem~\\refvslow_subset_lambdaSublevel_of_upper, from t…","kind":"proof","summary":"Two inclusions: the forward one (Theorem~\\refvslow_subset_lambdaSublevel_of_upper, from the upp…","labels":[],"detail_key":"p40"},{"id":"n33096","layer":"informal","project":"p40","title":"Ruelle reverse cofactor bound","kind":"lemma","summary":"[Ruelle reverse cofactor bound] Let \\(S\\) be orthogonal (\\(S S^\\top=1\\)) with the graded forwar…","labels":["entry_reverse_bound_of_orthogonal"],"detail_key":"p40"},{"id":"n33097","layer":"informal","project":"p40","title":"Since \\(S^-1=S^\\top\\), the entry \\(S_ij\\) is \\((\\det S)^-1\\) times the cofactor \\(adj(S)_…","kind":"proof","summary":"Since \\(S^-1=S^\\top\\), the entry \\(S_ij\\) is \\((\\det S)^-1\\) times the cofactor \\(adj(S)_ji\\),…","labels":[],"detail_key":"p40"},{"id":"n33098","layer":"informal","project":"p40","title":"Top-gap fast-band-mass envelope","kind":"definition","summary":"[Top-gap fast-band-mass envelope] \\(TopGapMassEnvelope\\,A\\,T\\,\\lambda^0\\,x\\) asserts the unifor…","labels":["TopGapMassEnvelope"],"detail_key":"p40"},{"id":"n33099","layer":"informal","project":"p40","title":"Multi-source geometric envelope","kind":"lemma","summary":"[Multi-source geometric envelope] If a nonnegative sequence \\(a\\) obeys a one-step recursion fe…","labels":["multi_source_envelope"],"detail_key":"p40"},{"id":"n33100","layer":"informal","project":"p40","title":"Each single source contributes a geometric partial sum bounded by \\(K/(1-\\rho)\\); summing…","kind":"proof","summary":"Each single source contributes a geometric partial sum bounded by \\(K/(1-\\rho)\\); summing the f…","labels":[],"detail_key":"p40"},{"id":"n33101","layer":"informal","project":"p40","title":"Per-stratum envelope step","kind":"lemma","summary":"[Per-stratum envelope step] At a fixed cut strictly inside a gap of width \\(\\ge G\\), the one-st…","labels":["perStratumEnvelope_step"],"detail_key":"p40"},{"id":"n33102","layer":"informal","project":"p40","title":"In the sorted-Gram-eigenbasis block decomposition, the band mass above the cut at step \\(…","kind":"proof","summary":"In the sorted-Gram-eigenbasis block decomposition, the band mass above the cut at step \\(n+1\\)…","labels":[],"detail_key":"p40"},{"id":"n33103","layer":"informal","project":"p40","title":"The top-gap envelope, a.e.","kind":"theorem","summary":"[The top-gap envelope, a.e.] For \\(\\mu\\)-a.e.\\ \\(x\\), the top-gap fast-band-mass envelope \\(Top…","labels":["topGapMassEnvelope_ae"],"detail_key":"p40"},{"id":"n33104","layer":"informal","project":"p40","title":"Fix the deterministic distinct gap \\(G>0\\) separating distinct exponents. On the a.e.\\ se…","kind":"proof","summary":"Fix the deterministic distinct gap \\(G>0\\) separating distinct exponents. On the a.e.\\ set wher…","labels":[],"detail_key":"p40"},{"id":"n33105","layer":"informal","project":"p40","title":"Spectrum identity from two inclusions","kind":"lemma","summary":"[Spectrum identity from two inclusions] If at \\(x\\) every realized exponent is a deterministic…","labels":["lyapunovSpectrum_eq_of_subsets"],"detail_key":"p40"},{"id":"n33106","layer":"informal","project":"p40","title":"Both directions are finite-set inclusions; antisymmetry of \\(\\subseteq\\) gives the equali…","kind":"proof","summary":"Both directions are finite-set inclusions; antisymmetry of \\(\\subseteq\\) gives the equality of…","labels":[],"detail_key":"p40"},{"id":"n33107","layer":"informal","project":"p40","title":"Ergodic constancy of the spectrum","kind":"theorem","summary":"[Ergodic constancy of the spectrum] Given a.e.\\ that every realized value of the upper growth f…","labels":["lyapunovSpectrum_eq_distinctExp_of_lambdaBar"],"detail_key":"p40"},{"id":"n33108","layer":"informal","project":"p40","title":"On the ultrametric-growth good set the two Finset inclusions are equivalent to native sta…","kind":"proof","summary":"On the ultrametric-growth good set the two Finset inclusions are equivalent to native statement…","labels":[],"detail_key":"p40"},{"id":"n33109","layer":"informal","project":"p40","title":"Filtration from the spectral upper bound","kind":"theorem","summary":"[Filtration from the spectral upper bound] Assume the per-vector spectral upper bound on the sl…","labels":["oseledets_filtration_of_upper"],"detail_key":"p40"},{"id":"n33110","layer":"informal","project":"p40","title":"The deterministic exponents \\(\\lambda^0\\) come from Theorem~\\refexists_lam_tendsto_singul…","kind":"proof","summary":"The deterministic exponents \\(\\lambda^0\\) come from Theorem~\\refexists_lam_tendsto_singularValu…","labels":[],"detail_key":"p40"},{"id":"n33111","layer":"informal","project":"p40","title":"Filtration from the top-gap envelope","kind":"theorem","summary":"[Filtration from the top-gap envelope] Under the standing ergodic, invertible, log-integrable h…","labels":["oseledets_filtration_of_topgap"],"detail_key":"p40"},{"id":"n33112","layer":"informal","project":"p40","title":"Diagonalize \\(\\Lambda(x)\\) by its limit eigenbasis with eigenvalues \\(e^\\lambda_sing\\) an…","kind":"proof","summary":"Diagonalize \\(\\Lambda(x)\\) by its limit eigenbasis with eigenvalues \\(e^\\lambda_sing\\) and slow…","labels":[],"detail_key":"p40"},{"id":"n33113","layer":"informal","project":"p40","title":"One-sided Oseledets multiplicative ergodic theorem","kind":"theorem","summary":"[One-sided Oseledets multiplicative ergodic theorem] Let \\(\\mu\\) be a probability measure, \\(T:…","labels":["oseledets_filtration"],"detail_key":"p40"},{"id":"n33114","layer":"informal","project":"p40","title":"If \\(d=0\\) the trivial flag \\(\\top=\\bot\\) with no exponents discharges the statement. For…","kind":"proof","summary":"If \\(d=0\\) the trivial flag \\(\\top=\\bot\\) with no exponents discharges the statement. For \\(d>0…","labels":[],"detail_key":"p40"},{"id":"n33115","layer":"informal","project":"p40","title":"Bundled Oseledets filtration","kind":"definition","summary":"[Bundled Oseledets filtration] For a measure","labels":["IsOseledetsFiltration"],"detail_key":"p40"},{"id":"n33116","layer":"informal","project":"p40","title":"Repackaged existence","kind":"theorem","summary":"[Repackaged existence] Under the standing hypotheses (ergodic T, invertible measurable A with \\…","labels":["oseledets_filtration'"],"detail_key":"p40"},{"id":"n33117","layer":"informal","project":"p40","title":"Destructure the conclusion of \\textttoseledets\\_filtration and repackage its three conjun…","kind":"proof","summary":"Destructure the conclusion of \\textttoseledets\\_filtration and repackage its three conjuncts as…","labels":[],"detail_key":"p40"},{"id":"n33118","layer":"informal","project":"p40","title":"Canonical sublevel characterization","kind":"theorem","summary":"[Canonical sublevel characterization] If IsOseledetsFiltration\\ \\mu\\ T\\ A\\ k\\ \\lambda\\ V holds,…","labels":["ae_mem_iff_limsup_le"],"detail_key":"p40"},{"id":"n33119","layer":"informal","project":"p40","title":"At a good point pick the stratum index j of v \\ne 0; the per-stratum convergence gives \\l…","kind":"proof","summary":"At a good point pick the stratum index j of v \\ne 0; the per-stratum convergence gives \\limsup…","labels":[],"detail_key":"p40"},{"id":"n33120","layer":"informal","project":"p40","title":"Uniqueness of the spectrum and filtration","kind":"theorem","summary":"[Uniqueness of the spectrum and filtration] On a probability space, any two Oseledets filtratio…","labels":["unique"],"detail_key":"p40"},{"id":"n33121","layer":"informal","project":"p40","title":"At a single good point the set of realized growth limits equals range\\,\\lambda and range\\…","kind":"proof","summary":"At a single good point the set of realized growth limits equals range\\,\\lambda and range\\,\\lamb…","labels":[],"detail_key":"p40"},{"id":"n33122","layer":"informal","project":"p40","title":"Nontriviality","kind":"lemma","summary":"[Nontriviality] On a probability space with 0 < d, any Oseledets filtration has 0 < k.","labels":["k_pos"],"detail_key":"p40"},{"id":"n33123","layer":"informal","project":"p40","title":"If k = 0 then at a good point R^d = V_0(x) = V_last(x) = 0, forcing finrank = 0, contradi…","kind":"proof","summary":"If k = 0 then at a good point R^d = V_0(x) = V_last(x) = 0, forcing finrank = 0, contradicting…","labels":[],"detail_key":"p40"},{"id":"n33124","layer":"informal","project":"p40","title":"Top exponent = norm growth","kind":"theorem","summary":"[Top exponent = norm growth] On a probability space, with A invertible and 0 < k, \\mu-a.e.\\ the…","labels":["tendsto_log_opNorm_cocycle"],"detail_key":"p40"},{"id":"n33125","layer":"informal","project":"p40","title":"Two-sided","kind":"proof","summary":"Two-sided","labels":[],"detail_key":"p40"},{"id":"n33126","layer":"informal","project":"p40","title":"Identification of the Furstenberg--Kesten constant","kind":"corollary","summary":"[Identification of the Furstenberg--Kesten constant] Any constant c to which \\tfrac1n\\log\\left\\…","labels":["oseledets_top_exponent_eq_furstenbergKesten"],"detail_key":"p40"},{"id":"n33127","layer":"informal","project":"p40","title":"Both \\reftendsto_log_opNorm_cocycle and the hypothesis hold at a common good point; uniqu…","kind":"proof","summary":"Both \\reftendsto_log_opNorm_cocycle and the hypothesis hold at a common good point; uniqueness…","labels":[],"detail_key":"p40"},{"id":"n33128","layer":"informal","project":"p40","title":"Deterministic dimension profile","kind":"theorem","summary":"[Deterministic dimension profile] For ergodic T and invertible A, every Oseledets filtration ha…","labels":["exists_finrank_ae_eq"],"detail_key":"p40"},{"id":"n33129","layer":"informal","project":"p40","title":"The dimension x \\mapsto finrank\\,V_i(x) is measurable via the trace of the orthogonal pro…","kind":"proof","summary":"The dimension x \\mapsto finrank\\,V_i(x) is measurable via the trace of the orthogonal projector…","labels":[],"detail_key":"p40"},{"id":"n33130","layer":"informal","project":"p40","title":"Per-exponent multiplicities","kind":"corollary","summary":"[Per-exponent multiplicities] For ergodic T and invertible A, each exponent \\lambda_i carries a…","labels":["exists_multiplicity"],"detail_key":"p40"},{"id":"n33131","layer":"informal","project":"p40","title":"Set m_i to the consecutive dimension drops of \\refexists_finrank_ae_eq; positivity is str…","kind":"proof","summary":"Set m_i to the consecutive dimension drops of \\refexists_finrank_ae_eq; positivity is strict an…","labels":[],"detail_key":"p40"},{"id":"n33132","layer":"informal","project":"p40","title":"MET with multiplicities","kind":"theorem","summary":"[MET with multiplicities] Under the standing hypotheses there exist k, \\lambda, V and a strictl…","labels":["oseledets_filtration_with_multiplicities"],"detail_key":"p40"},{"id":"n33133","layer":"informal","project":"p40","title":"Obtain a witness from \\refoseledets_filtration' and apply \\refexists_finrank_ae_eq to it.","kind":"proof","summary":"Obtain a witness from \\refoseledets_filtration' and apply \\refexists_finrank_ae_eq to it.","labels":[],"detail_key":"p40"},{"id":"n33134","layer":"informal","project":"p40","title":"Sorted spectrum","kind":"definition","summary":"[Sorted spectrum] The full Lyapunov spectrum with multiplicity is the total function exponents…","labels":["exponents"],"detail_key":"p40"},{"id":"n33135","layer":"informal","project":"p40","title":"Defining \\sigma-limit and order","kind":"theorem","summary":"[Defining \\sigma-limit and order] For each sorted index i and \\mu-a.e.\\ x, \\tfrac1n\\log\\sigma_i…","labels":["exponents_tendsto_log_singularValue"],"detail_key":"p40"},{"id":"n33136","layer":"informal","project":"p40","title":"The deterministic exponent sequence is extracted by Classical.choose from the singular-va…","kind":"proof","summary":"The deterministic exponent sequence is extracted by Classical.choose from the singular-value co…","labels":[],"detail_key":"p40"},{"id":"n33137","layer":"informal","project":"p40","title":"Eigenvalue tie","kind":"theorem","summary":"[Eigenvalue tie] \\mu-a.e., \\exp(exponents_i) is the i-th sorted eigenvalue of the Oseledets lim…","labels":["exp_exponents_eq_eigenvalues₀_oseledetsLimit"],"detail_key":"p40"},{"id":"n33138","layer":"informal","project":"p40","title":"The \\sigma-limit identifies lamSing(x,i) = exponents_i a.e.; combine with the eigenvalues…","kind":"proof","summary":"The \\sigma-limit identifies lamSing(x,i) = exponents_i a.e.; combine with the eigenvalues of \\L…","labels":[],"detail_key":"p40"},{"id":"n33139","layer":"informal","project":"p40","title":"Sign characterizations of exponent sums","kind":"theorem","summary":"[Sign characterizations of exponent sums] The sum sumPosExp of the strictly positive exponents…","labels":["sumPosExp_pos_iff"],"detail_key":"p40"},{"id":"n33140","layer":"informal","project":"p40","title":"Each summand of sumPosExp is strictly positive on the filter, so the sum is \\ge 0; a sum…","kind":"proof","summary":"Each summand of sumPosExp is strictly positive on the filter, so the sum is \\ge 0; a sum of non…","labels":[],"detail_key":"p40"},{"id":"n33141","layer":"informal","project":"p40","title":"Telescoping identity for partial sums","kind":"theorem","summary":"[Telescoping identity for partial sums] For k \\le d, the ergodic growth rate \\Gamma_k of the pr…","labels":["gammaK_eq_sum_top_exponents"],"detail_key":"p40"},{"id":"n33142","layer":"informal","project":"p40","title":"Since sprod_k = \\prod_i<k\\sigma_i, the normalized \\logsprod_k is the finite sum of the pe…","kind":"proof","summary":"Since sprod_k = \\prod_i<k\\sigma_i, the normalized \\logsprod_k is the finite sum of the per-inde…","labels":[],"detail_key":"p40"},{"id":"n33143","layer":"informal","project":"p40","title":"Exterior cocycle generator","kind":"definition","summary":"[Exterior cocycle generator] The k-th exterior generator extGen\\,k\\,A sends x to the k-th compo…","labels":["extGen"],"detail_key":"p40"},{"id":"n33144","layer":"informal","project":"p40","title":"k-volume growth rate","kind":"theorem","summary":"[k-volume growth rate] For k \\le d and \\mu-a.e.\\ x, the operator-norm growth of the compound co…","labels":["tendsto_log_opNorm_compound_cocycle"],"detail_key":"p40"},{"id":"n33145","layer":"informal","project":"p40","title":"The operator norm of the compound matrix is sprod_k, the product of the top-k singular va…","kind":"proof","summary":"The operator norm of the compound matrix is sprod_k, the product of the top-k singular values;…","labels":[],"detail_key":"p40"},{"id":"n33146","layer":"informal","project":"p40","title":"Positive sum as a maximal partial sum","kind":"corollary","summary":"[Positive sum as a maximal partial sum] Writing k_+ = \\#\\i : 0 < exponents_i\\, the positive-exp…","labels":["sumPosExp_eq_gammaK_card_pos"],"detail_key":"p40"},{"id":"n33147","layer":"informal","project":"p40","title":"By antitonicity the strictly positive entries are exactly the top k_+ indices, so the fil…","kind":"proof","summary":"By antitonicity the strictly positive entries are exactly the top k_+ indices, so the filtered…","labels":[],"detail_key":"p40"},{"id":"n33148","layer":"informal","project":"p40","title":"Product of singular values is the absolute determinant","kind":"lemma","summary":"[Product of singular values is the absolute determinant] For every n, x: sprod\\,A\\,T\\,d\\,n\\,x =…","labels":["sprod_d_eq_abs_det"],"detail_key":"p40"},{"id":"n33149","layer":"informal","project":"p40","title":"Squaring, sprod_d^2 = \\prod_i \\sigma_i^2 = \\det(M^\\top M) = (\\det M)^2 for the symmetric…","kind":"proof","summary":"Squaring, sprod_d^2 = \\prod_i \\sigma_i^2 = \\det(M^\\top M) = (\\det M)^2 for the symmetric Gram o…","labels":[],"detail_key":"p40"},{"id":"n33150","layer":"informal","project":"p40","title":"Determinant identity","kind":"theorem","summary":"[Determinant identity] The sum of all Lyapunov exponents equals the integral of \\log\\left\\lvert…","labels":["sumAllExp_eq_integral_log_abs_det"],"detail_key":"p40"},{"id":"n33151","layer":"informal","project":"p40","title":"Two a.e.\\","kind":"proof","summary":"Two a.e.\\","labels":[],"detail_key":"p40"},{"id":"n33152","layer":"informal","project":"p40","title":"Volume contraction","kind":"corollary","summary":"[Volume contraction] If \\sum_i exponents_i < 0 then \\mu-a.e.\\ \\left\\lvert \\det A^(n)(x) \\right\\…","labels":["tendsto_abs_det_cocycle_atTop_zero"],"detail_key":"p40"},{"id":"n33153","layer":"informal","project":"p40","title":"Since \\tfrac1n\\log\\left\\lvert \\det A^(n) \\right\\rvert tends to a negative constant, \\log\\…","kind":"proof","summary":"Since \\tfrac1n\\log\\left\\lvert \\det A^(n) \\right\\rvert tends to a negative constant, \\log\\left\\l…","labels":[],"detail_key":"p40"},{"id":"n33154","layer":"informal","project":"p40","title":"Inverse cocycle exponents","kind":"theorem","summary":"[Inverse cocycle exponents] For each sorted index i and \\mu-a.e.\\ x, the singular-value exponen…","labels":["tendsto_log_singularValue_inv_cocycle"],"detail_key":"p40"},{"id":"n33155","layer":"informal","project":"p40","title":"Singular-value reciprocity \\sigma_i(M^-1) = \\sigma_rev\\,i(M)^-1 for invertible M, applied…","kind":"proof","summary":"Singular-value reciprocity \\sigma_i(M^-1) = \\sigma_rev\\,i(M)^-1 for invertible M, applied to th…","labels":[],"detail_key":"p40"},{"id":"n33156","layer":"informal","project":"p40","title":"Top of the reversed spectrum is minus the bottom","kind":"corollary","summary":"[Top of the reversed spectrum is minus the bottom] \\mu-a.e.\\ the largest exponent of the invers…","labels":["topExponent_inv_eq_neg_bot"],"detail_key":"p40"},{"id":"n33157","layer":"informal","project":"p40","title":"Specialize \\reftendsto_log_singularValue_inv_cocycle at i = 0, where rev\\,0 = d-1.","kind":"proof","summary":"Specialize \\reftendsto_log_singularValue_inv_cocycle at i = 0, where rev\\,0 = d-1.","labels":[],"detail_key":"p40"},{"id":"n33158","layer":"informal","project":"p40","title":"Invariant subbundle","kind":"definition","summary":"[Invariant subbundle] An invariant subbundle is a measurable family of fibre subspaces W(x) \\le…","labels":["InvariantSubbundle"],"detail_key":"p40"},{"id":"n33159","layer":"informal","project":"p40","title":"Dimension interlacing","kind":"lemma","summary":"[Dimension interlacing] At each ambient flag level, the dimension captured by the subbundle is…","labels":["restricted_finrank_le"],"detail_key":"p40"},{"id":"n33160","layer":"informal","project":"p40","title":"Monotonicity of finrank under W \\cap V_i \\le V_i.","kind":"proof","summary":"Monotonicity of finrank under W \\cap V_i \\le V_i.","labels":[],"detail_key":"p40"},{"id":"n33161","layer":"informal","project":"p40","title":"Restricted strict Oseledets filtration","kind":"theorem","summary":"[Restricted strict Oseledets filtration] For ergodic","labels":["restricted_strict_filtration"],"detail_key":"p40"},{"id":"n33162","layer":"informal","project":"p40","title":"Obtain a","kind":"proof","summary":"Obtain a","labels":[],"detail_key":"p40"},{"id":"n33163","layer":"informal","project":"p40","title":"Non-ergodic exponents","kind":"theorem","summary":"[Non-ergodic exponents] For merely measure-preserving T (no ergodicity) and invertible measurab…","labels":["exists_exponents_nonergodic"],"detail_key":"p40"},{"id":"n33164","layer":"informal","project":"p40","title":"The non-ergodic Kingman theorem applied to the subadditive cocycle \\logsprod_k produces i…","kind":"proof","summary":"The non-ergodic Kingman theorem applied to the subadditive cocycle \\logsprod_k produces invaria…","labels":[],"detail_key":"p40"},{"id":"n33165","layer":"informal","project":"p40","title":"Non-ergodic positive-exponent sum","kind":"corollary","summary":"[Non-ergodic positive-exponent sum] Summing the positive parts \\max(\\lambda_i(x),0) over i < d…","labels":["exists_sumPosExp_nonergodic"],"detail_key":"p40"},{"id":"n33166","layer":"informal","project":"p40","title":"A finite sum of positive parts of the invariant integrable functions of \\refexists_expone…","kind":"proof","summary":"A finite sum of positive parts of the invariant integrable functions of \\refexists_exponents_no…","labels":[],"detail_key":"p40"},{"id":"n33167","layer":"informal","project":"p40","title":"Fekete infimum representation","kind":"theorem","summary":"[Fekete infimum representation] The partial-sum growth rate is the infimum over n of the normal…","labels":["gammaK_eq_iInf"],"detail_key":"p40"},{"id":"n33168","layer":"informal","project":"p40","title":"The integral sequence is subadditive (Fekete), so it converges to its infimum; a Fatou es…","kind":"proof","summary":"The integral sequence is subadditive (Fekete), so it converges to its infimum; a Fatou estimate…","labels":[],"detail_key":"p40"},{"id":"n33169","layer":"informal","project":"p40","title":"Upper semicontinuity of partial sums and top exponent","kind":"theorem","summary":"[Upper semicontinuity of partial sums and top exponent] Along a filter of generators B_i \\to A…","labels":["gammaK_upperSemicontinuous"],"detail_key":"p40"},{"id":"n33170","layer":"informal","project":"p40","title":"\\Gamma_k is an infimum of the per-n continuous normalized integrals (\\refgammaK_eq_iInf);…","kind":"proof","summary":"\\Gamma_k is an infimum of the per-n continuous normalized integrals (\\refgammaK_eq_iInf); for e…","labels":[],"detail_key":"p40"},{"id":"n33171","layer":"informal","project":"p40","title":"Lower semicontinuity of the bottom exponent","kind":"theorem","summary":"[Lower semicontinuity of the bottom exponent] The bottom exponent \\lambda_d = \\Gamma_d - \\Gamma…","labels":["botExp_lowerSemicontinuous"],"detail_key":"p40"},{"id":"n33172","layer":"informal","project":"p40","title":"Writing \\Gamma_d = \\int\\log\\left\\lvert \\det \\right\\rvert (the determinant identity \\refsu…","kind":"proof","summary":"Writing \\Gamma_d = \\int\\log\\left\\lvert \\det \\right\\rvert (the determinant identity \\refsumAllEx…","labels":[],"detail_key":"p40"},{"id":"n33173","layer":"informal","project":"p40","title":"Forward top value","kind":"theorem","summary":"[Forward top value] For ergodic T and a possibly-singular measurable generator with only \\log^+…","labels":["tendsto_top_posLogNorm"],"detail_key":"p40"},{"id":"n33174","layer":"informal","project":"p40","title":"Apply the ergodic Kingman theorem to the subadditive, bounded-below, integrable cocycle \\…","kind":"proof","summary":"Apply the ergodic Kingman theorem to the subadditive, bounded-below, integrable cocycle \\log^+\\…","labels":[],"detail_key":"p40"},{"id":"n33175","layer":"informal","project":"p40","title":"Upper bound on the singular top exponent","kind":"theorem","summary":"[Upper bound on the singular top exponent] Under the same singular hypotheses there is \\lambda_…","labels":["limsup_logNorm_le_top"],"detail_key":"p40"},{"id":"n33176","layer":"informal","project":"p40","title":"Termwise \\log \\le \\log^+, then pass to the EReal \\limsup; since the \\log^+ sequence conve…","kind":"proof","summary":"Termwise \\log \\le \\log^+, then pass to the EReal \\limsup; since the \\log^+ sequence converges t…","labels":[],"detail_key":"p40"},{"id":"n33177","layer":"informal","project":"p40","title":"Sharp \\limsup in the expanding case","kind":"theorem","summary":"[Sharp \\limsup in the expanding case] There is a forward top value \\lambda_1^+ (the a.e.\\ limit…","labels":["limsup_logNorm_eq_top_of_pos"],"detail_key":"p40"},{"id":"n33178","layer":"informal","project":"p40","title":"The \\le half is \\reflimsup_logNorm_le_top. For \\ge: where \\tfrac1n\\log^+\\left\\lVert A^(n)…","kind":"proof","summary":"The \\le half is \\reflimsup_logNorm_le_top. For \\ge: where \\tfrac1n\\log^+\\left\\lVert A^(n) \\righ…","labels":[],"detail_key":"p40"},{"id":"n33179","layer":"informal","project":"p40","title":"Singular top-k volume upper bound","kind":"theorem","summary":"[Singular top-k volume upper bound] For ergodic T and a possibly-singular generator with \\log^+…","labels":["limsup_logSprod_le_top"],"detail_key":"p40"},{"id":"n33180","layer":"informal","project":"p40","title":"Since sprod_k \\ge 0 is submultiplicative with no invertibility, the same \\log^+-of-a-nonn…","kind":"proof","summary":"Since sprod_k \\ge 0 is submultiplicative with no invertibility, the same \\log^+-of-a-nonnegativ…","labels":[],"detail_key":"p40"},{"id":"n33181","layer":"informal","project":"p40","title":"Backward generator","kind":"definition","summary":"[Backward generator] For \\(A : X \\to Matrix\\,(Fin\\,d)\\,(Fin\\,d)\\,R\\) and \\(T : X \\simeq_m X\\),…","labels":["backwardGen"],"detail_key":"p40"},{"id":"n33182","layer":"informal","project":"p40","title":"Cocycle recursion, newest factor on the right","kind":"lemma","summary":"[Cocycle recursion, newest factor on the right] For any generator \\(A\\) and map \\(T\\), \\(A^(n+1…","labels":["cocycle_succ'"],"detail_key":"p40"},{"id":"n33183","layer":"informal","project":"p40","title":"This is the companion of the standard recursion \\(A^(n+1)(x) = A^(n)(Tx)\\cdot A(x)\\): app…","kind":"proof","summary":"This is the companion of the standard recursion \\(A^(n+1)(x) = A^(n)(Tx)\\cdot A(x)\\): apply the…","labels":[],"detail_key":"p40"},{"id":"n33184","layer":"informal","project":"p40","title":"Backward cocycle identity","kind":"lemma","summary":"[Backward cocycle identity] Writing \\(B = backwardGen\\,A\\,T\\), the cocycle of \\(B\\) over \\(T^-1…","labels":["cocycle_backwardGen"],"detail_key":"p40"},{"id":"n33185","layer":"informal","project":"p40","title":"Induct on","kind":"proof","summary":"Induct on","labels":[],"detail_key":"p40"},{"id":"n33186","layer":"informal","project":"p40","title":"Backward standing hypotheses","kind":"proposition","summary":"[Backward standing hypotheses] If \\(\\det A(x) \\neq 0\\) for all \\(x\\), \\(A\\) is measurable, \\(T\\…","labels":["backwardData_of"],"detail_key":"p40"},{"id":"n33187","layer":"informal","project":"p40","title":"Invertibi","kind":"proof","summary":"Invertibi","labels":[],"detail_key":"p40"},{"id":"n33188","layer":"informal","project":"p40","title":"Biinvariant conull set","kind":"lemma","summary":"[Biinvariant conull set] For \\(T\\) measure-preserving on a probability space and a conull measu…","labels":["exists_conull_biinvariant"],"detail_key":"p40"},{"id":"n33189","layer":"informal","project":"p40","title":"Take \\(S' = \\bigl(\\bigcap_n (T^[n])^-1S\\bigr) \\cap \\bigl(\\bigcap_n (T^-[n])^-1S\\bigr)\\).…","kind":"proof","summary":"Take \\(S' = \\bigl(\\bigcap_n (T^[n])^-1S\\bigr) \\cap \\bigl(\\bigcap_n (T^-[n])^-1S\\bigr)\\). Each i…","labels":[],"detail_key":"p40"},{"id":"n33190","layer":"informal","project":"p40","title":"Strong one-sided export","kind":"proposition","summary":"[Strong one-sided export] Under the one-sided hypotheses (with \\([NeZero\\,d]\\)) there exist a d…","labels":["oseledets_filtration_dims"],"detail_key":"p40"},{"id":"n33191","layer":"informal","project":"p40","title":"This is t","kind":"proof","summary":"This is t","labels":[],"detail_key":"p40"},{"id":"n33192","layer":"informal","project":"p40","title":"Rank of the slow space","kind":"proposition","summary":"[Rank of the slow space] For a.e.\\ \\(x\\) and all \\(t \\in R\\), \\[ finrank\\,\\bigl(vslow\\,A\\,T\\,(\\…","labels":["ae_finrank_vslow"],"detail_key":"p40"},{"id":"n33193","layer":"informal","project":"p40","title":"The sanit","kind":"proof","summary":"The sanit","labels":[],"detail_key":"p40"},{"id":"n33194","layer":"informal","project":"p40","title":"Functional calculus on an eigenvector","kind":"lemma","summary":"[Functional calculus on an eigenvector] If \\(M\\) is self-adjoint and \\(M v = c\\,v\\) for \\(v \\ne…","labels":["cfc_apply_of_eigenvector"],"detail_key":"p40"},{"id":"n33195","layer":"informal","project":"p40","title":"The eigenvalue \\(c\\) lies in the (finite) spectrum of \\(M\\). Pick a Lagrange interpolatin…","kind":"proof","summary":"The eigenvalue \\(c\\) lies in the (finite) spectrum of \\(M\\). Pick a Lagrange interpolating poly…","labels":[],"detail_key":"p40"},{"id":"n33196","layer":"informal","project":"p40","title":"Kingman constant as the limit of integral means","kind":"proposition","summary":"[Kingman constant as the limit of integral means] Under the hypotheses of the ergodic Kingman t…","labels":["tendsto_kingman_ergodic_means"],"detail_key":"p40"},{"id":"n33197","layer":"informal","project":"p40","title":"Let \\(L\\)","kind":"proof","summary":"Let \\(L\\)","labels":[],"detail_key":"p40"},{"id":"n33198","layer":"informal","project":"p40","title":"Floored restricted log-cocycle","kind":"definition","summary":"[Floored restricted log-cocycle] For a measurable family \\(V : X \\to Submodule\\,R\\,(R^d)\\), set…","labels":["restLog"],"detail_key":"p40"},{"id":"n33199","layer":"informal","project":"p40","title":"Everywhere subadditivity","kind":"lemma","summary":"[Everywhere subadditivity] \\(restLog\\,A\\,V\\,T\\) is an everywhere subadditive cocycle: for all \\…","labels":["isSubadditiveCocycle_restLog"],"detail_key":"p40"},{"id":"n33200","layer":"informal","project":"p40","title":"The floor","kind":"proof","summary":"The floor","labels":[],"detail_key":"p40"},{"id":"n33201","layer":"informal","project":"p40","title":"Restricted Kingman, both directions of time","kind":"lemma","summary":"[Restricted Kingman, both directions of time] Kingman applied to \\(restLog\\) over \\(T\\) yields…","labels":["restLog_backward_kingman"],"detail_key":"p40"},{"id":"n33202","layer":"informal","project":"p40","title":"Over \\(T\\","kind":"proof","summary":"Over \\(T\\","labels":[],"detail_key":"p40"},{"id":"n33203","layer":"informal","project":"p40","title":"Restricted exponent equals \\(\\lambda_i\\)","kind":"proposition","summary":"[Restricted exponent equals \\(\\lambda_i\\)] For the forward level \\(V_i\\), the restricted Kingma…","labels":["restricted_const_eq"],"detail_key":"p40"},{"id":"n33204","layer":"informal","project":"p40","title":"Lower bou","kind":"proof","summary":"Lower bou","labels":[],"detail_key":"p40"},{"id":"n33205","layer":"informal","project":"p40","title":"Backward-orbit growth envelope","kind":"proposition","summary":"[Backward-orbit growth envelope] For a.e.\\ \\(x\\), \\[ \\limsup_n \\tfrac1n\\log\\bigl\\lVert A^(n)(T^…","labels":["ae_limsup_restricted_backward_le"],"detail_key":"p40"},{"id":"n33206","layer":"informal","project":"p40","title":"This is the analytic heart. By \\crefrestLog_backward_kingman the reversed restricted log…","kind":"proof","summary":"This is the analytic heart. By \\crefrestLog_backward_kingman the reversed restricted log conver…","labels":[],"detail_key":"p40"},{"id":"n33207","layer":"informal","project":"p40","title":"Sublevels of opposite sign are transverse","kind":"proposition","summary":"[Sublevels of opposite sign are transverse] Fix \\(x\\). If the forward envelope for \\(V_x\\) hold…","labels":["inf_eq_bot_of_neg_sum"],"detail_key":"p40"},{"id":"n33208","layer":"informal","project":"p40","title":"Suppose \\","kind":"proof","summary":"Suppose \\","labels":[],"detail_key":"p40"},{"id":"n33209","layer":"informal","project":"p40","title":"The a.e.\\ crux","kind":"proposition","summary":"[The a.e.\\ crux] For a.e.\\ \\(x\\), for every forward level \\(i\\) and backward level \\(s\\) with \\…","labels":["ae_crux"],"detail_key":"p40"},{"id":"n33210","layer":"informal","project":"p40","title":"Bundle al","kind":"proof","summary":"Bundle al","labels":[],"detail_key":"p40"},{"id":"n33211","layer":"informal","project":"p40","title":"Counting bound","kind":"corollary","summary":"[Counting bound] For all \\(a, b \\in R\\) with \\(a + b < 0\\), \\[ \\#\\\\, j < d : lam0\\,j \\le a \\,\\…","labels":["ae_counting"],"detail_key":"p40"},{"id":"n33212","layer":"informal","project":"p40","title":"Convert thresholds to filtration levels and apply \\crefae_crux at one good point: the two…","kind":"proof","summary":"Convert thresholds to filtration levels and apply \\crefae_crux at one good point: the two suble…","labels":[],"detail_key":"p40"},{"id":"n33213","layer":"informal","project":"p40","title":"Forward determinant sum","kind":"proposition","summary":"[Forward determinant sum] Any exponent sequence \\(lam0\\) realizing the a.e.\\ singular-value lim…","labels":["sum_lam0_eq_integral_log_abs_det"],"detail_key":"p40"},{"id":"n33214","layer":"informal","project":"p40","title":"By uniqueness of a.e.\\ limits at a common conull point, \\(lam0\\,j\\) equals the chosen spe…","kind":"proof","summary":"By uniqueness of a.e.\\ limits at a common conull point, \\(lam0\\,j\\) equals the chosen spectrum…","labels":[],"detail_key":"p40"},{"id":"n33215","layer":"informal","project":"p40","title":"Backward sum is the negated forward sum","kind":"proposition","summary":"[Backward sum is the negated forward sum] For a backward exponent sequence \\(mu0\\), \\(\\sum_j<dm…","labels":["sum_mu0_eq_neg_sum_lam0"],"detail_key":"p40"},{"id":"n33216","layer":"informal","project":"p40","title":"Apply \\cr","kind":"proof","summary":"Apply \\cr","labels":[],"detail_key":"p40"},{"id":"n33217","layer":"informal","project":"p40","title":"Reflection lemma","kind":"proposition","summary":"[Reflection lemma] Let \\(p, q : N\\to R\\) be antitone on \\([0,d)\\). If \\(\\#\\p \\le a\\ + \\#\\q \\le…","labels":["reflect_of_counting_and_sum"],"detail_key":"p40"},{"id":"n33218","layer":"informal","project":"p40","title":"Apply the counting bound at \\(b\\) and \\(a = -q(j) - \\varepsilon\\): since antitone tuples…","kind":"proof","summary":"Apply the counting bound at \\(b\\) and \\(a = -q(j) - \\varepsilon\\): since antitone tuples are th…","labels":[],"detail_key":"p40"},{"id":"n33219","layer":"informal","project":"p40","title":"Aligned backward index","kind":"corollary","summary":"[Aligned backward index] Under the reflection \\(mu0\\,j = -lam0\\,(d-1-j)\\), the backward count o…","labels":["sidx_exp"],"detail_key":"p40"},{"id":"n33220","layer":"informal","project":"p40","title":"From \\cre","kind":"proof","summary":"From \\cre","labels":[],"detail_key":"p40"},{"id":"n33221","layer":"informal","project":"p40","title":"Powers of the projector triple converge to the intersection projector","kind":"proposition","summary":"[Powers of the projector triple converge to the intersection projector] For subspaces \\(K, L\\)…","labels":["tendsto_pow_orthProj_inf"],"detail_key":"p40"},{"id":"n33222","layer":"informal","project":"p40","title":"The matrix \\(S = P_K P_L P_K\\) is self-adjoint, PSD, and a contraction, so its eigenvalue…","kind":"proof","summary":"The matrix \\(S = P_K P_L P_K\\) is self-adjoint, PSD, and a contraction, so its eigenvalues lie…","labels":[],"detail_key":"p40"},{"id":"n33223","layer":"informal","project":"p40","title":"The intersection fixes exactly \\(K \\sqcap L\\)","kind":"lemma","summary":"[The intersection fixes exactly \\(K \\sqcap L\\)] \\((P_K\\,P_L\\,P_K)\\,v = v\\) if and only if \\(v \\…","labels":["one_eigenspace_projComp"],"detail_key":"p40"},{"id":"n33224","layer":"informal","project":"p40","title":"The \"if\" direction is immediate since both projectors fix vectors in \\(K \\sqcap L\\). For…","kind":"proof","summary":"The \"if\" direction is immediate since both projectors fix vectors in \\(K \\sqcap L\\). For \"only…","labels":[],"detail_key":"p40"},{"id":"n33225","layer":"informal","project":"p40","title":"Measurable intersection of measurable subspaces","kind":"proposition","summary":"[Measurable intersection of measurable subspaces] If \\(x \\mapsto V(x)\\) and \\(x \\mapsto W(x)\\)…","labels":["inf"],"detail_key":"p40"},{"id":"n33226","layer":"informal","project":"p40","title":"Each \\(x \\mapsto (P_V(x) P_W(x) P_V(x))^n\\) is measurable (matrix powers of measurable ma…","kind":"proof","summary":"Each \\(x \\mapsto (P_V(x) P_W(x) P_V(x))^n\\) is measurable (matrix powers of measurable matrices…","labels":[],"detail_key":"p40"},{"id":"n33227","layer":"informal","project":"p40","title":"Telescoping-flag lattice lemma","kind":"lemma","summary":"[Telescoping-flag lattice lemma] A descending flag \\(V_0 \\supseteq \\cdots \\supseteq V_k = \\bot\\…","labels":["flag_iSupIndep_and_iSup"],"detail_key":"p40"},{"id":"n33228","layer":"informal","project":"p40","title":"Induct on \\(k\\). The cons step shows that prepending a head \\(E_0\\) disjoint from the sup…","kind":"proof","summary":"Induct on \\(k\\). The cons step shows that prepending a head \\(E_0\\) disjoint from the supremum…","labels":[],"detail_key":"p40"},{"id":"n33229","layer":"informal","project":"p40","title":"Splitting at a point","kind":"proposition","summary":"[Splitting at a point] Define \\(E_i = V_i \\sqcap W_sidx\\,i\\). Then \\(finrank\\,E_i \\ge 1\\), the…","labels":["esplitAt"],"detail_key":"p40"},{"id":"n33230","layer":"informal","project":"p40","title":"The crux","kind":"proof","summary":"The crux","labels":[],"detail_key":"p40"},{"id":"n33231","layer":"informal","project":"p40","title":"The two-sided Oseledets splitting","kind":"theorem","summary":"[The two-sided Oseledets splitting] Let \\(T : X \\simeq_m X\\) be an invertible ergodic measure-p…","labels":["oseledets_splitting"],"detail_key":"p40"},{"id":"n33232","layer":"informal","project":"p40","title":"For \\(d =","kind":"proof","summary":"For \\(d =","labels":[],"detail_key":"p40"},{"id":"n33233","layer":"informal","project":"p40","title":"Measure-preserving one-parameter flow","kind":"definition","summary":"[Measure-preserving one-parameter flow] A \\emphmeasure-preserving one-parameter flow on a measu…","labels":["MeasurePreservingFlow"],"detail_key":"p40"},{"id":"n33234","layer":"informal","project":"p40","title":"Integer times are iterates of the time-one map","kind":"lemma","summary":"[Integer times are iterates of the time-one map] For a measure-preserving flow \\( \\varphi \\) an…","labels":["natCast_eq_iterate"],"detail_key":"p40"},{"id":"n33235","layer":"informal","project":"p40","title":"Induction on \\( n \\). The base case \\( \\varphi(0) = id = (\\varphi(1))^[0] \\) is the time-…","kind":"proof","summary":"Induction on \\( n \\). The base case \\( \\varphi(0) = id = (\\varphi(1))^[0] \\) is the time-zero l…","labels":[],"detail_key":"p40"},{"id":"n33236","layer":"informal","project":"p40","title":"Continuous-time linear cocycle over a flow","kind":"definition","summary":"[Continuous-time linear cocycle over a flow] A \\emphcontinuous-time linear cocycle over a measu…","labels":["FlowCocycle"],"detail_key":"p40"},{"id":"n33237","layer":"informal","project":"p40","title":"Reduction identity at integer times","kind":"proposition","summary":"[Reduction identity at integer times] For a flow cocycle \\( A \\) over \\( \\varphi \\), every \\( n…","labels":["toCocycle_eq"],"detail_key":"p40"},{"id":"n33238","layer":"informal","project":"p40","title":"Induction on \\( n \\). At \\( n = 0 \\) both sides are the identity. For the step, split \\(…","kind":"proof","summary":"Induction on \\( n \\). At \\( n = 0 \\) both sides are the identity. For the step, split \\( A((n+1…","labels":[],"detail_key":"p40"},{"id":"n33239","layer":"informal","project":"p40","title":"Integrability of the time-one log-norm","kind":"lemma","summary":"[Integrability of the time-one log-norm] If \\( g \\in L^1(\\mu) \\) dominates \\( \\log^+\\left\\lVert…","labels":["integrableLogNorm_timeOne"],"detail_key":"p40"},{"id":"n33240","layer":"informal","project":"p40","title":"The map \\( x \\mapsto \\log^+\\left\\lVert A(1,x) \\right\\rVert \\) is measurable (composition…","kind":"proof","summary":"The map \\( x \\mapsto \\log^+\\left\\lVert A(1,x) \\right\\rVert \\) is measurable (composition of the…","labels":[],"detail_key":"p40"},{"id":"n33241","layer":"informal","project":"p40","title":"Integrability of the inverse time-one log-norm","kind":"lemma","summary":"[Integrability of the inverse time-one log-norm] If \\( g' \\in L^1(\\mu) \\) dominates \\( \\log^+\\l…","labels":["integrableLogNorm_timeOne_inv"],"detail_key":"p40"},{"id":"n33242","layer":"informal","project":"p40","title":"Identical to \\refintegrableLogNorm_timeOne, inserting the measurable matrix-inversion map…","kind":"proof","summary":"Identical to \\refintegrableLogNorm_timeOne, inserting the measurable matrix-inversion map and e…","labels":[],"detail_key":"p40"},{"id":"n33243","layer":"informal","project":"p40","title":"Discrete filtration for the time-one data","kind":"proposition","summary":"[Discrete filtration for the time-one data] Let \\( \\mu \\) be a probability measure, \\( \\varphi…","labels":["exists_isOseledetsFiltration_timeOne"],"detail_key":"p40"},{"id":"n33244","layer":"informal","project":"p40","title":"Apply the discrete theorem \\refoseledets_filtration to the ergodic map \\( \\varphi(1) \\) a…","kind":"proof","summary":"Apply the discrete theorem \\refoseledets_filtration to the ergodic map \\( \\varphi(1) \\) and gen…","labels":[],"detail_key":"p40"},{"id":"n33245","layer":"informal","project":"p40","title":"Error sublinearity along the integer orbit","kind":"lemma","summary":"[Error sublinearity along the integer orbit] For integrable \\( g, g' \\) and a measure-preservin…","labels":["ae_tendsto_flowError_zero"],"detail_key":"p40"},{"id":"n33246","layer":"informal","project":"p40","title":"Apply the Birkhoff orbital-tail estimate (a.e.\\ \\( n^-1 h(T^[n]x) \\to 0 \\) for integrable…","kind":"proof","summary":"Apply the Birkhoff orbital-tail estimate (a.e.\\ \\( n^-1 h(T^[n]x) \\to 0 \\) for integrable \\( h…","labels":[],"detail_key":"p40"},{"id":"n33247","layer":"informal","project":"p40","title":"Between-times sandwich: continuous growth equals integer-time growth","kind":"theorem","summary":"[Between-times sandwich: continuous growth equals integer-time growth] Fix a flow \\( \\varphi \\)…","labels":["tendsto_log_norm_atTop_of_discrete"],"detail_key":"p40"},{"id":"n33248","layer":"informal","project":"p40","title":"Write \\( t = r + n \\) with \\( n = \\lfloor t \\rfloor \\ge 1 \\) and \\( r \\in [0,1) \\). The c…","kind":"proof","summary":"Write \\( t = r + n \\) with \\( n = \\lfloor t \\rfloor \\ge 1 \\) and \\( r \\in [0,1) \\). The cocycle…","labels":[],"detail_key":"p40"},{"id":"n33249","layer":"informal","project":"p40","title":"Fixed-time log-norm is sublinear","kind":"lemma","summary":"[Fixed-time log-norm is sublinear] Fix a real time \\( t_0 \\). For almost every \\( x \\), both \\(…","labels":["ae_tendsto_logNorm_fixedTime_zero"],"detail_key":"p40"},{"id":"n33250","layer":"informal","project":"p40","title":"One builds an integrable function \\( H \\) dominating both \\( \\log^+\\left\\lVert A(t_0,\\cdo…","kind":"proof","summary":"One builds an integrable function \\( H \\) dominating both \\( \\log^+\\left\\lVert A(t_0,\\cdot) \\ri…","labels":[],"detail_key":"p40"},{"id":"n33251","layer":"informal","project":"p40","title":"Shift-invariance of the growth limsup","kind":"theorem","summary":"[Shift-invariance of the growth limsup] Fix a real time \\( t_0 \\). For almost every \\( x \\) and…","labels":["glim_shift"],"detail_key":"p40"},{"id":"n33252","layer":"informal","project":"p40","title":"First, a.e.\\ the discrete growth average \\( n^-1\\log\\left\\lVert cocycle\\,n\\,x\\,u \\right\\r…","kind":"proof","summary":"First, a.e.\\ the discrete growth average \\( n^-1\\log\\left\\lVert cocycle\\,n\\,x\\,u \\right\\rVert \\…","labels":[],"detail_key":"p40"},{"id":"n33253","layer":"informal","project":"p40","title":"Flow-equivariance of the filtration at every real time","kind":"theorem","summary":"[Flow-equivariance of the filtration at every real time] Let \\( V \\) be the Oseledets filtratio…","labels":["ae_flow_equivariant"],"detail_key":"p40"},{"id":"n33254","layer":"informal","project":"p40","title":"Use the growth characterization \\( v \\in V_i\\,x \\iff v = 0 \\lor \\limsup \\le \\lambda_i \\)…","kind":"proof","summary":"Use the growth characterization \\( v \\in V_i\\,x \\iff v = 0 \\lor \\limsup \\le \\lambda_i \\) at \\(…","labels":[],"detail_key":"p40"},{"id":"n33255","layer":"informal","project":"p40","title":"Continuous-flow multiplicative ergodic theorem","kind":"theorem","summary":"[Continuous-flow multiplicative ergodic theorem] Let \\( \\mu \\) be a probability measure on \\( X…","labels":["oseledets_flow"],"detail_key":"p40"},{"id":"n33256","layer":"informal","project":"p40","title":"Take \\( (k, \\lambda, V) \\) from the reduction \\refexists_isOseledetsFiltration_timeOne; t…","kind":"proof","summary":"Take \\( (k, \\lambda, V) \\) from the reduction \\refexists_isOseledetsFiltration_timeOne; this su…","labels":[],"detail_key":"p40"},{"id":"n33257","layer":"informal","project":"p40","title":"The suspension (mapping-torus) space","kind":"definition","summary":"[The suspension (mapping-torus) space] The \\emphsuspension space \\Sigma = SuspensionSpace T\\,\\t…","labels":["thm:susp-space"],"detail_key":"p40"},{"id":"n33258","layer":"informal","project":"p40","title":"The invariant suspension probability measure","kind":"definition","summary":"[The invariant suspension probability measure] The \\emphsuspension measure \\hat\\mu = suspension…","labels":["thm:susp-measure"],"detail_key":"p40"},{"id":"n33259","layer":"informal","project":"p40","title":"\\hat\\mu is a probability measure","kind":"theorem","summary":"[\\hat\\mu is a probability measure] For a nonnegative integrable roof with 0 < \\int\\tau, the nor…","labels":["thm:susp-prob"],"detail_key":"p40"},{"id":"n33260","layer":"informal","project":"p40","title":"The raw box push-forward has total mass \\int\\tau (Fubini: the x-fibre of the box is [0, \\…","kind":"proof","summary":"The raw box push-forward has total mass \\int\\tau (Fubini: the x-fibre of the box is [0, \\tau x)…","labels":[],"detail_key":"p40"},{"id":"n33261","layer":"informal","project":"p40","title":"The cover flow cocycle","kind":"definition","summary":"[The cover flow cocycle] For a cover point p = (x, h) the \\emphcover cocycle coverCocycle A\\,T\\…","labels":["thm:susp-coverCocycle"],"detail_key":"p40"},{"id":"n33262","layer":"informal","project":"p40","title":"Agreement on the base section","kind":"lemma","summary":"[Agreement on the base section] At height 0 the cover cocycle is the cross-section flow cocycle…","labels":["thm:susp-coverBase"],"detail_key":"p40"},{"id":"n33263","layer":"informal","project":"p40","title":"Immediate from the definition on rewriting 0 + t = t in the fibre coordinate.","kind":"proof","summary":"Immediate from the definition on rewriting 0 + t = t in the fibre coordinate.","labels":[],"detail_key":"p40"},{"id":"n33264","layer":"informal","project":"p40","title":"Section multiplicativity at a return boundary","kind":"theorem","summary":"[Section multiplicativity at a return boundary] Starting on the base section at x, the cover co…","labels":["thm:susp-coverReturn"],"detail_key":"p40"},{"id":"n33265","layer":"informal","project":"p40","title":"The accumulated matrix splits at the return boundary because the lap counter does; the re…","kind":"proof","summary":"The accumulated matrix splits at the return boundary because the lap counter does; the return c…","labels":[],"detail_key":"p40"},{"id":"n33266","layer":"informal","project":"p40","title":"Additive \\log-discrepancy across an orbit step","kind":"theorem","summary":"[Additive \\log-discrepancy across an orbit step] Under invertibility of the base cocycle and st…","labels":["thm:susp-logDisc"],"detail_key":"p40"},{"id":"n33267","layer":"informal","project":"p40","title":"Take \\log of the two operator-norm brackets supplied by \\Crefthm:susp-coverReturn (which…","kind":"proof","summary":"Take \\log of the two operator-norm brackets supplied by \\Crefthm:susp-coverReturn (which bound…","labels":[],"detail_key":"p40"},{"id":"n33268","layer":"informal","project":"p40","title":"The Lyapunov-exponent limit transfer","kind":"theorem","summary":"[The Lyapunov-exponent limit transfer] If the cover-cocycle growth rate t^-1\\log\\left\\lVert cov…","labels":["thm:susp-limitTransfer"],"detail_key":"p40"},{"id":"n33269","layer":"informal","project":"p40","title":"The per-t average at the re-based point lies within C/t of the average at (x, s), where C…","kind":"proof","summary":"The per-t average at the re-based point lies within C/t of the average at (x, s), where C is th…","labels":[],"detail_key":"p40"},{"id":"n33270","layer":"informal","project":"p40","title":"The flow Lyapunov exponent of an orbit class","kind":"definition","summary":"[The flow Lyapunov exponent of an orbit class] HasFlowExponent q\\,L holds when \\emphsome repres…","labels":["thm:susp-hasFlow"],"detail_key":"p40"},{"id":"n33271","layer":"informal","project":"p40","title":"Two-sided well-definedness across a forward step","kind":"theorem","summary":"[Two-sided well-definedness across a forward step] If (x_2, s_2) = suspensionAct(n)\\,(x, s) for…","labels":["thm:susp-iffForward"],"detail_key":"p40"},{"id":"n33272","layer":"informal","project":"p40","title":"The \\to direction is \\Crefthm:susp-limitTransfer. The \\leftarrow direction is its mirror:…","kind":"proof","summary":"The \\to direction is \\Crefthm:susp-limitTransfer. The \\leftarrow direction is its mirror: the a…","labels":[],"detail_key":"p40"},{"id":"n33273","layer":"informal","project":"p40","title":"Orbit-class invariance of the flow exponent","kind":"theorem","summary":"[Orbit-class invariance of the flow exponent] If two cover points are connected by a forward or…","labels":["thm:susp-classInv"],"detail_key":"p40"},{"id":"n33274","layer":"informal","project":"p40","title":"Orbit-equivalent points have equal \\pi-images, so (x, s) itself serves as the witness for…","kind":"proof","summary":"Orbit-equivalent points have equal \\pi-images, so (x, s) itself serves as the witness for the c…","labels":[],"detail_key":"p40"},{"id":"n33275","layer":"informal","project":"p40","title":"The special-flow Lyapunov exponent, \\lambda_flow = \\lambda_base/\\int\\tau","kind":"theorem","summary":"[The special-flow Lyapunov exponent, \\lambda_flow = \\lambda_base/\\int\\tau] Under a bounded roof…","labels":["thm:susp-abramovExp"],"detail_key":"p40"},{"id":"n33276","layer":"informal","project":"p40","title":"The base exponent-set measurability is supplied internally: the full-time cover-cocycle e…","kind":"proof","summary":"The base exponent-set measurability is supplied internally: the full-time cover-cocycle exponen…","labels":[],"detail_key":"p40"},{"id":"n33277","layer":"informal","project":"p40","title":"The exponent along genuine flow orbits","kind":"theorem","summary":"[The exponent along genuine flow orbits] Under the same hypotheses (and T measure-preserving),…","labels":["thm:susp-abramovExpFlow"],"detail_key":"p40"},{"id":"n33278","layer":"informal","project":"p40","title":"The disintegration already identifies each a.e.\\ class with a flow-orbit point of the bas…","kind":"proof","summary":"The disintegration already identifies each a.e.\\ class with a flow-orbit point of the base sect…","labels":[],"detail_key":"p40"},{"id":"n33279","layer":"informal","project":"p40","title":"Strict positivity under global invertibility","kind":"lemma","summary":"[Strict positivity under global invertibility] If the base generator A is everywhere invertible…","labels":["thm:susp-normPos"],"detail_key":"p40"},{"id":"n33280","layer":"informal","project":"p40","title":"Unfold the cover cocycle to cocycle A\\,T\\,(\\textlap count)\\,x, a product of invertible ma…","kind":"proof","summary":"Unfold the cover cocycle to cocycle A\\,T\\,(\\textlap count)\\,x, a product of invertible matrices…","labels":[],"detail_key":"p40"},{"id":"n33281","layer":"informal","project":"p40","title":"Signed-step cross-representative uniqueness","kind":"theorem","summary":"[Signed-step cross-representative uniqueness] If two cover points are connected by a \\emphsigne…","labels":["thm:susp-iffSigned"],"detail_key":"p40"},{"id":"n33282","layer":"informal","project":"p40","title":"Both signs reduce to the forward iff of \\Crefthm:susp-iffForward at a different base poin…","kind":"proof","summary":"Both signs reduce to the forward iff of \\Crefthm:susp-iffForward at a different base point: for…","labels":[],"detail_key":"p40"},{"id":"n33283","layer":"informal","project":"p40","title":"The representative-level and descended exponents","kind":"definition","summary":"[The representative-level and descended exponents] repExponent p ( ) is the growth-rate limit \\…","labels":["thm:susp-flowExpAt"],"detail_key":"p40"},{"id":"n33284","layer":"informal","project":"p40","title":"flowExponentAt reads off the exponent","kind":"theorem","summary":"[flowExponentAt reads off the exponent] If q carries the flow exponent L (some representative h…","labels":["thm:susp-flowExpEq"],"detail_key":"p40"},{"id":"n33285","layer":"informal","project":"p40","title":"Well-definedness of the lift (\\Crefthm:susp-iffSigned) transfers existence of the limit a…","kind":"proof","summary":"Well-definedness of the lift (\\Crefthm:susp-iffSigned) transfers existence of the limit across…","labels":[],"detail_key":"p40"},{"id":"n33286","layer":"informal","project":"p40","title":"The representative-free Abramov exponent","kind":"theorem","summary":"[The representative-free Abramov exponent] Under a bounded roof, positive \\int\\tau, measurable…","labels":["thm:susp-flowExpAe"],"detail_key":"p40"},{"id":"n33287","layer":"informal","project":"p40","title":"Combine the existential a.e.\\ exponent of \\Crefthm:susp-abramovExp with the read-off equa…","kind":"proof","summary":"Combine the existential a.e.\\ exponent of \\Crefthm:susp-abramovExp with the read-off equality \\…","labels":[],"detail_key":"p40"},{"id":"n33288","layer":"informal","project":"p40","title":"The cat suspension realises the base derivative-cocycle exponent","kind":"theorem","summary":"[The cat suspension realises the base derivative-cocycle exponent] For \\hat\\mu-a.e.\\ orbit clas…","labels":["thm:susp-catHasFlow"],"detail_key":"p40"},{"id":"n33289","layer":"informal","project":"p40","title":"The generator is constant in the base point, so its discrete growth rate is the determini…","kind":"proof","summary":"The generator is constant in the base point, so its discrete growth rate is the deterministic G…","labels":[],"detail_key":"p40"},{"id":"n33290","layer":"informal","project":"p40","title":"Positivity of the cat suspension exponent (issue \\#30)","kind":"theorem","summary":"[Positivity of the cat suspension exponent (issue \\#30)] For \\hat\\mu-a.e.\\ orbit class q there…","labels":["thm:susp-catPos"],"detail_key":"p40"},{"id":"n33291","layer":"informal","project":"p40","title":"Take L = \\log((3+\\sqrt5)/2) from \\Crefthm:susp-catHasFlow; positivity is \\log of a number…","kind":"proof","summary":"Take L = \\log((3+\\sqrt5)/2) from \\Crefthm:susp-catHasFlow; positivity is \\log of a number excee…","labels":[],"detail_key":"p40"},{"id":"n33292","layer":"informal","project":"p40","title":"Abramov quotient reading, existential form","kind":"theorem","summary":"[Abramov quotient reading, existential form] The flow exponent equals the base top Lyapunov exp…","labels":["thm:susp-catBaseDiv"],"detail_key":"p40"},{"id":"n33293","layer":"informal","project":"p40","title":"Rewrite the base top exponent as the Grade-1 spectral value \\log((3+\\sqrt5)/2) and divide…","kind":"proof","summary":"Rewrite the base top exponent as the Grade-1 spectral value \\log((3+\\sqrt5)/2) and divide by \\i…","labels":[],"detail_key":"p40"},{"id":"n33294","layer":"informal","project":"p40","title":"Representative-free cat exponent","kind":"theorem","summary":"[Representative-free cat exponent] For \\hat\\mu-a.e.\\ q, the descended flow exponent is flowExpo…","labels":["thm:susp-catFlowExpEq"],"detail_key":"p40"},{"id":"n33295","layer":"informal","project":"p40","title":"Apply \\Crefthm:susp-flowExpEq to the existential exponent \\Crefthm:susp-catHasFlow; globa…","kind":"proof","summary":"Apply \\Crefthm:susp-flowExpEq to the existential exponent \\Crefthm:susp-catHasFlow; global inve…","labels":[],"detail_key":"p40"},{"id":"n33296","layer":"informal","project":"p40","title":"Representative-free Abramov reading","kind":"theorem","summary":"[Representative-free Abramov reading] The descended flow exponent equals the base top exponent…","labels":["thm:susp-catFlowExpBaseDiv"],"detail_key":"p40"},{"id":"n33297","layer":"informal","project":"p40","title":"Rewrite the base exponent as \\log((3+\\sqrt5)/2) and \\int\\tau = 1 in \\Crefthm:susp-catFlow…","kind":"proof","summary":"Rewrite the base exponent as \\log((3+\\sqrt5)/2) and \\int\\tau = 1 in \\Crefthm:susp-catFlowExpEq.","labels":[],"detail_key":"p40"},{"id":"n33298","layer":"informal","project":"p40","title":"Representative-free positivity","kind":"theorem","summary":"[Representative-free positivity] For \\hat\\mu-a.e.\\ q, the descended flow exponent is strictly p…","labels":["thm:susp-catFlowExpPos"],"detail_key":"p40"},{"id":"n33299","layer":"informal","project":"p40","title":"Substitute the value \\log((3+\\sqrt5)/2) from \\Crefthm:susp-catFlowExpEq; it is positive b…","kind":"proof","summary":"Substitute the value \\log((3+\\sqrt5)/2) from \\Crefthm:susp-catFlowExpEq; it is positive because…","labels":[],"detail_key":"p40"},{"id":"n33300","layer":"informal","project":"p40","title":"Strong mixing of the two-sided Bernoulli shift","kind":"theorem","summary":"[Strong mixing of the two-sided Bernoulli shift] For the invertible two-sided Bernoulli shift w…","labels":["thm:susp-bernMixing"],"detail_key":"p40"},{"id":"n33301","layer":"informal","project":"p40","title":"Approximate A, B by finite-block cylinder sets in symmetric difference; on cylinders the…","kind":"proof","summary":"Approximate A, B by finite-block cylinder sets in symmetric difference; on cylinders the correl…","labels":[],"detail_key":"p40"},{"id":"n33302","layer":"informal","project":"p40","title":"Mixing kills eigenvalues","kind":"theorem","summary":"[Mixing kills eigenvalues] A measurable eigenfunction g (with g \\circ f = l\\,g) of a strongly-m…","labels":["thm:susp-eigZero"],"detail_key":"p40"},{"id":"n33303","layer":"informal","project":"p40","title":"ErgodicTheory.frequently_pow_far_from_one","kind":"proof","summary":"Purely set-theoretic. The powers l^n stay a fixed distance \\delta = \\left\\lVert l - 1 \\right\\rV…","labels":[],"detail_key":"p40"},{"id":"n33304","layer":"informal","project":"p40","title":"The fibre Fourier coefficient","kind":"definition","summary":"[The fibre Fourier coefficient] For F : X \\times R\\to C the n-th \\emphfibre Fourier coefficient…","labels":["thm:susp-coeffFn"],"detail_key":"p40"},{"id":"n33305","layer":"informal","project":"p40","title":"The twisted eigenfunction relation","kind":"theorem","summary":"[The twisted eigenfunction relation] If F is 1-periodic in the fibre and satisfies the deck ide…","labels":["thm:susp-twist"],"detail_key":"p40"},{"id":"n33306","layer":"informal","project":"p40","title":"A three-step change of variables over the unit window: character algebra factoring out e^…","kind":"proof","summary":"A three-step change of variables over the unit window: character algebra factoring out e^2\\pi i…","labels":[],"detail_key":"p40"},{"id":"n33307","layer":"informal","project":"p40","title":"Per-fibre Parseval bridge","kind":"theorem","summary":"[Per-fibre Parseval bridge] For a bounded measurable f : R\\to C, the squared fibre-coefficient…","labels":["thm:susp-parseval"],"detail_key":"p40"},{"id":"n33308","layer":"informal","project":"p40","title":"Lift f to the length-1 circle AddCircle(1); the interval coefficients are the circle's Fo…","kind":"proof","summary":"Lift f to the length-1 circle AddCircle(1); the interval coefficients are the circle's Fourier…","labels":[],"detail_key":"p40"},{"id":"n33309","layer":"informal","project":"p40","title":"Indicator dichotomy","kind":"theorem","summary":"[Indicator dichotomy] Let g : R\\to R be measurable, 1-periodic, \\0, 1\\-valued, with all nonzero…","labels":["thm:susp-dichotomy"],"detail_key":"p40"},{"id":"n33310","layer":"informal","project":"p40","title":"ErgodicTheory.measure_periodic_ae_zero_spread","kind":"proof","summary":"Parseval collapses to the zero mode: \\bigl(\\int_0^1 g\\bigr)^2 = \\int_0^1\\left\\lVert g \\right\\rV…","labels":[],"detail_key":"p40"},{"id":"n33311","layer":"informal","project":"p40","title":"Time-one ergodicity, abstract base-generic form (issue \\#35)","kind":"theorem","summary":"[Time-one ergodicity, abstract base-generic form (issue \\#35)] Let T be ergodic and measure-pre…","labels":["thm:susp-timeOneErgodic"],"detail_key":"p40"},{"id":"n33312","layer":"informal","project":"p40","title":"Fix a \\zeta_1-invariant A with lifted indicator F. For n \\ne 0 the twist eigenvalue e^2\\p…","kind":"proof","summary":"Fix a \\zeta_1-invariant A with lifted indicator F. For n \\ne 0 the twist eigenvalue e^2\\pi i n…","labels":[],"detail_key":"p40"},{"id":"n33313","layer":"informal","project":"p40","title":"Time-one ergodicity of the irrational-roof Bernoulli suspension","kind":"theorem","summary":"[Time-one ergodicity of the irrational-roof Bernoulli suspension] For the two-sided Bernoulli s…","labels":["thm:susp-bernTimeOne"],"detail_key":"p40"},{"id":"n33314","layer":"informal","project":"p40","title":"Discharge the abstract \\Crefthm:susp-timeOneErgodic with base ergodicity of the shift and…","kind":"proof","summary":"Discharge the abstract \\Crefthm:susp-timeOneErgodic with base ergodicity of the shift and the n…","labels":[],"detail_key":"p40"},{"id":"n33315","layer":"informal","project":"p40","title":"Concrete non-vacuity witness r = \\sqrt2","kind":"theorem","summary":"[Concrete non-vacuity witness r = \\sqrt2] The irrational roof r := \\sqrt2 yields an ergodic tim…","labels":["thm:susp-sqrt2"],"detail_key":"p40"},{"id":"n33316","layer":"informal","project":"p40","title":"Instantiate \\Crefthm:susp-bernTimeOne at r = \\sqrt2, positive and irrational.","kind":"proof","summary":"Instantiate \\Crefthm:susp-bernTimeOne at r = \\sqrt2, positive and irrational.","labels":[],"detail_key":"p40"},{"id":"n33317","layer":"informal","project":"p40","title":"Non-ergodicity at the unit roof","kind":"theorem","summary":"[Non-ergodicity at the unit roof] The time-1 map of the \\emphunit-roof Bernoulli suspension flo…","labels":["thm:susp-notErgodic"],"detail_key":"p40"},{"id":"n33318","layer":"informal","project":"p40","title":"With r = 1 the deck translation advances the fibre by exactly the invariance period, so t…","kind":"proof","summary":"With r = 1 the deck translation advances the fibre by exactly the invariance period, so the sat…","labels":[],"detail_key":"p40"},{"id":"n33319","layer":"informal","project":"p40","title":"Discrete entropy power rule","kind":"theorem","summary":"[Discrete entropy power rule] For a measure-preserving T on a probability space and n \\in N, h(…","labels":["thm:susp-ksPow"],"detail_key":"p40"},{"id":"n33320","layer":"informal","project":"p40","title":"Walters, \\emphAn Introduction to Ergodic Theory, Theorem 4.13. The n-fold refinement of a…","kind":"proof","summary":"Walters, \\emphAn Introduction to Ergodic Theory, Theorem 4.13. The n-fold refinement of a parti…","labels":[],"detail_key":"p40"},{"id":"n33321","layer":"informal","project":"p40","title":"Flow iterate identity","kind":"theorem","summary":"[Flow iterate identity] For a measure-preserving flow \\varphi, the n-th iterate of the time-t m…","labels":["thm:susp-flowIter"],"detail_key":"p40"},{"id":"n33322","layer":"informal","project":"p40","title":"Induction on n using \\varphi_s+t = \\varphi_s \\circ \\varphi_t and the iterate recursion.","kind":"proof","summary":"Induction on n using \\varphi_s+t = \\varphi_s \\circ \\varphi_t and the iterate recursion.","labels":[],"detail_key":"p40"},{"id":"n33323","layer":"informal","project":"p40","title":"Flow entropy homogeneity along N-multiples","kind":"theorem","summary":"[Flow entropy homogeneity along N-multiples] For a measure-preserving flow \\varphi on a probabi…","labels":["thm:susp-nsmulKs"],"detail_key":"p40"},{"id":"n33324","layer":"informal","project":"p40","title":"Apply the power rule \\Crefthm:susp-ksPow to T = \\varphi_t, whose n-th iterate is \\varphi_…","kind":"proof","summary":"Apply the power rule \\Crefthm:susp-ksPow to T = \\varphi_t, whose n-th iterate is \\varphi_nt by…","labels":[],"detail_key":"p40"},{"id":"n33325","layer":"informal","project":"p40","title":"The fibre-rescaling equivalence","kind":"definition","summary":"[The fibre-rescaling equivalence] The fibre map (x, s) \\mapsto (x, s/r) descends to a measurabl…","labels":["thm:susp-rescale"],"detail_key":"p40"},{"id":"n33326","layer":"informal","project":"p40","title":"The time-r entropy conjugacy","kind":"theorem","summary":"[The time-r entropy conjugacy] The time-t map of the constant-r suspension flow has the same Ko…","labels":["thm:susp-rescaleKs"],"detail_key":"p40"},{"id":"n33327","layer":"informal","project":"p40","title":"Measurable-conjugacy invariance of entropy applied to suspensionRescale: it intertwines \\…","kind":"proof","summary":"Measurable-conjugacy invariance of entropy applied to suspensionRescale: it intertwines \\zeta^(…","labels":[],"detail_key":"p40"},{"id":"n33328","layer":"informal","project":"p40","title":"Time-r entropy of the Bernoulli suspension","kind":"theorem","summary":"[Time-r entropy of the Bernoulli suspension] For the constant-roof (\\tau \\equiv r) suspension o…","labels":["thm:susp-timeRBern"],"detail_key":"p40"},{"id":"n33329","layer":"informal","project":"p40","title":"\\Crefthm:susp-rescaleKs at t = r (so t/r = 1) reduces to the unit-roof time-1 value h(\\ze…","kind":"proof","summary":"\\Crefthm:susp-rescaleKs at t = r (so t/r = 1) reduces to the unit-roof time-1 value h(\\zeta^(1)…","labels":[],"detail_key":"p40"},{"id":"n33330","layer":"informal","project":"p40","title":"Constant-roof time-one entropy, rational roof (issue \\#38)","kind":"theorem","summary":"[Constant-roof time-one entropy, rational roof (issue \\#38)] For a \\emphrational roof r = a/b (…","labels":["thm:susp-timeOneEntropy"],"detail_key":"p40"},{"id":"n33331","layer":"informal","project":"p40","title":"Fibre time-rescaling reduces h(\\zeta^(r)_1) to the unit-roof h(\\zeta^(1)_1/r); writing 1/…","kind":"proof","summary":"Fibre time-rescaling reduces h(\\zeta^(r)_1) to the unit-roof h(\\zeta^(1)_1/r); writing 1/r = b/…","labels":[],"detail_key":"p40"},{"id":"n33332","layer":"informal","project":"p40","title":"Multiplicative form","kind":"theorem","summary":"[Multiplicative form] For a rational roof r = a/b, \\;h(\\zeta^(r)_1) \\cdot r = H_\\nu.","labels":["thm:susp-timeOneEntropyMul"],"detail_key":"p40"},{"id":"n33333","layer":"informal","project":"p40","title":"Multiply the value form \\Crefthm:susp-timeOneEntropy by r; the product (H_\\nu/r) \\cdot r…","kind":"proof","summary":"Multiply the value form \\Crefthm:susp-timeOneEntropy by r; the product (H_\\nu/r) \\cdot r = H_\\n…","labels":[],"detail_key":"p40"},{"id":"n33334","layer":"informal","project":"p40","title":"Measure-continuity of a flow","kind":"definition","summary":"[Measure-continuity of a flow] A measure-preserving flow \\varphi is \\emphmeasure-continuous if…","labels":["def:flow-measCont"],"detail_key":"p40"},{"id":"n33335","layer":"informal","project":"p40","title":"Measure-continuity of the suspension flow","kind":"theorem","summary":"[Measure-continuity of the suspension flow] The unit-roof suspension flow of an \\empharbitrary…","labels":["thm:flow-suspMeasCont"],"detail_key":"p40"},{"id":"n33336","layer":"informal","project":"p40","title":"Fibre-translation continuity of the roof-1 fundamental box, combined with dominated conve…","kind":"proof","summary":"Fibre-translation continuity of the roof-1 fundamental box, combined with dominated convergence…","labels":[],"detail_key":"p40"},{"id":"n33337","layer":"informal","project":"p40","title":"Ito's L1: the partition moves little under a small shift","kind":"theorem","summary":"[Ito's L1: the partition moves little under a small shift] For a measure-continuous flow and a…","labels":["thm:flow-L1"],"detail_key":"p40"},{"id":"n33338","layer":"informal","project":"p40","title":"The self-conditioning H(P \\mid P) = 0, and the cell measures \\mu(\\varphi_t^-1P_i \\cap P_j…","kind":"proof","summary":"The self-conditioning H(P \\mid P) = 0, and the cell measures \\mu(\\varphi_t^-1P_i \\cap P_j) are…","labels":[],"detail_key":"p40"},{"id":"n33339","layer":"informal","project":"p40","title":"Two-family Shannon comparison","kind":"theorem","summary":"[Two-family Shannon comparison] For two finite families \\beta = (\\beta_k), \\gamma = (\\gamma_k)…","labels":["thm:flow-finJoin"],"detail_key":"p40"},{"id":"n33340","layer":"informal","project":"p40","title":"Pure Shannon entropy: refinement monotonicity, the chain rule, conditional subadditivity…","kind":"proof","summary":"Pure Shannon entropy: refinement monotonicity, the chain rule, conditional subadditivity over t…","labels":[],"detail_key":"p40"},{"id":"n33341","layer":"informal","project":"p40","title":"\\varepsilon--\\delta alignment proposition","kind":"theorem","summary":"[\\varepsilon--\\delta alignment proposition] For a measure-continuous flow on a standard Borel s…","labels":["thm:flow-ratioLUB"],"detail_key":"p40"},{"id":"n33342","layer":"informal","project":"p40","title":"The alignment inequality (from \\Crefthm:flow-L1 and \\Crefthm:flow-finJoin) makes each slo…","kind":"proof","summary":"The alignment inequality (from \\Crefthm:flow-L1 and \\Crefthm:flow-finJoin) makes each slope eve…","labels":[],"detail_key":"p40"},{"id":"n33343","layer":"informal","project":"p40","title":"Abstract Abramov homogeneity","kind":"theorem","summary":"[Abstract Abramov homogeneity] For a measure-continuous measure-preserving flow \\varphi on a st…","labels":["thm:flow-abramov-hom"],"detail_key":"p40"},{"id":"n33344","layer":"informal","project":"p40","title":"Interchange the supremum over partitions with the alignment proposition (\\Crefthm:flow-ra…","kind":"proof","summary":"Interchange the supremum over partitions with the alignment proposition (\\Crefthm:flow-ratioLUB…","labels":[],"detail_key":"p40"},{"id":"n33345","layer":"informal","project":"p40","title":"Unit-roof time-s entropy, all s > 0","kind":"theorem","summary":"[Unit-roof time-s entropy, all s > 0] For the unit-roof Bernoulli suspension flow and every s >…","labels":["thm:flow-timeS"],"detail_key":"p40"},{"id":"n33346","layer":"informal","project":"p40","title":"\\Crefthm:flow-abramov-hom for the measure-continuous Bernoulli suspension flow, at the fi…","kind":"proof","summary":"\\Crefthm:flow-abramov-hom for the measure-continuous Bernoulli suspension flow, at the finite v…","labels":[],"detail_key":"p40"},{"id":"n33347","layer":"informal","project":"p40","title":"Constant-roof time-one entropy, all roofs (issue \\#48)","kind":"theorem","summary":"[Constant-roof time-one entropy, all roofs (issue \\#48)] For \\emphevery roof r > 0 (irrational…","labels":["thm:flow-timeOneAll"],"detail_key":"p40"},{"id":"n33348","layer":"informal","project":"p40","title":"Fibre time-rescaling reduces h(\\zeta^(r)_1) to the unit-roof h(\\zeta^(1)_1/r); \\Crefthm:f…","kind":"proof","summary":"Fibre time-rescaling reduces h(\\zeta^(r)_1) to the unit-roof h(\\zeta^(1)_1/r); \\Crefthm:flow-ti…","labels":[],"detail_key":"p40"},{"id":"n33349","layer":"informal","project":"p40","title":"The two-sided Z-indexed matrix cocycle","kind":"definition","summary":"[The two-sided Z-indexed matrix cocycle] Over an invertible measure-preserving base T : X \\sime…","labels":["cocycleZ"],"detail_key":"p40"},{"id":"n33350","layer":"informal","project":"p40","title":"The two-sided cocycle identity","kind":"theorem","summary":"[The two-sided cocycle identity] cocycleZ(m + n)\\,x = cocycleZ m\\,(baseIter n\\,x) \\cdot cocycle…","labels":["cocycleZ_add"],"detail_key":"p40"},{"id":"n33351","layer":"informal","project":"p40","title":"The genuine quotient flow cocycle","kind":"definition","summary":"[The genuine quotient flow cocycle] For the constant unit roof, reading off a measurable canoni…","labels":["quotientFlowCocycle"],"detail_key":"p40"},{"id":"n33352","layer":"informal","project":"p40","title":"Cohomology to the cover cocycle","kind":"theorem","summary":"[Cohomology to the cover cocycle] Over the quotient, \\CrefquotientFlowCocycle is cohomologous t…","labels":["exists_flowCocycle_cohomologous_to_cover"],"detail_key":"p40"},{"id":"n33353","layer":"informal","project":"p40","title":"Take B = quotientFlowCocycle and C(p) = cocycleZ A\\,T \\lfloor p_2\\rfloor p_1; measurabili…","kind":"proof","summary":"Take B = quotientFlowCocycle and C(p) = cocycleZ A\\,T \\lfloor p_2\\rfloor p_1; measurability of…","labels":[],"detail_key":"p40"},{"id":"n33354","layer":"informal","project":"p40","title":"Exponent transport","kind":"theorem","summary":"[Exponent transport] Along the atTop half-line 0 \\le t, the growth rate of \\CrefquotientFlowCoc…","labels":["flowExponentAt_quotientFlowCocycle"],"detail_key":"p40"},{"id":"n33355","layer":"informal","project":"p40","title":"The cat map's quotient flow cocycle","kind":"definition","summary":"[The cat map's quotient flow cocycle] Instantiating \\CrefquotientFlowCocycle at the Arnold cat…","labels":["catQuotientFlowCocycle"],"detail_key":"p40"},{"id":"n33356","layer":"informal","project":"p40","title":"The cat quotient flow-cocycle Lyapunov exponent","kind":"theorem","summary":"[The cat quotient flow-cocycle Lyapunov exponent] For \\hat\\mu-a.e.\\ orbit class q, the growth r…","labels":["catQuotientFlowCocycle_exponent"],"detail_key":"p40"},{"id":"n33357","layer":"informal","project":"p40","title":"Chain the descended cat-suspension exponent (\\Crefthm:susp-catFlowExpEq) through the expo…","kind":"proof","summary":"Chain the descended cat-suspension exponent (\\Crefthm:susp-catFlowExpEq) through the exponent-t…","labels":[],"detail_key":"p40"},{"id":"n33358","layer":"informal","project":"p40","title":"The route gauge is not a metric","kind":"proposition","summary":"[The route gauge is not a metric] The \\emphlow and \\emphhigh routes are essential: a naive mini…","labels":["routeDist_not_metric"],"detail_key":"p40"},{"id":"n33359","layer":"informal","project":"p40","title":"The Bowen--Walters embedding metric","kind":"definition","summary":"[The Bowen--Walters embedding metric] Under diam X \\le 1 the Kuratowski embedding kur a = d(a,…","labels":["embDist"],"detail_key":"p40"},{"id":"n33360","layer":"informal","project":"p40","title":"embDist is a metric","kind":"theorem","summary":"[embDist is a metric] embDist satisfies the triangle inequality and separates points ( ): the t…","labels":["embDist_triangle"],"detail_key":"p40"},{"id":"n33361","layer":"informal","project":"p40","title":"The Kuratowski map is an isometry into X \\to^b R, so both test distances are genuine metr…","kind":"proof","summary":"The Kuratowski map is an isometry into X \\to^b R, so both test distances are genuine metrics; t…","labels":[],"detail_key":"p40"},{"id":"n33362","layer":"informal","project":"p40","title":"Metric-space topology and Polishness","kind":"theorem","summary":"[Metric-space topology and Polishness] For a compact metric base X with diam X\\le 1 and a homeo…","labels":["suspensionPolish"],"detail_key":"p40"},{"id":"n33363","layer":"informal","project":"p40","title":"The metric topology agrees with the quotient topology (an open-ball criterion), so \\textt…","kind":"proof","summary":"The metric topology agrees with the quotient topology (an open-ball criterion), so \\textttMetri…","labels":[],"detail_key":"p40"},{"id":"n33364","layer":"informal","project":"p40","title":"The flow is Lipschitz in time","kind":"theorem","summary":"[The flow is Lipschitz in time] The suspension flow is 5-Lipschitz in the time parameter along…","labels":["embDist_flow_le"],"detail_key":"p40"},{"id":"n33365","layer":"informal","project":"p40","title":"Both flowed points share the base coordinate of the canonical representative; the height…","kind":"proof","summary":"Both flowed points share the base coordinate of the canonical representative; the height advanc…","labels":[],"detail_key":"p40"},{"id":"n33366","layer":"informal","project":"p40","title":"The variable-roof embedding metric","kind":"definition","summary":"[The variable-roof embedding metric] For a roof \\tau\\ge\\rho_\\min > 0, the canonical box represe…","labels":["embDistVar"],"detail_key":"p40"},{"id":"n33367","layer":"informal","project":"p40","title":"Variable-roof flow-Lipschitz bound","kind":"theorem","summary":"[Variable-roof flow-Lipschitz bound] The realisation cost of the variable roof appears only in…","labels":["embDistVar_flow_le"],"detail_key":"p40"},{"id":"n33368","layer":"informal","project":"p40","title":"Repeat the constant-roof estimate \\CrefembDist_flow_le on the normalized coordinate, wher…","kind":"proof","summary":"Repeat the constant-roof estimate \\CrefembDist_flow_le on the normalized coordinate, where the…","labels":[],"detail_key":"p40"},{"id":"n33369","layer":"informal","project":"p40","title":"The descended suspension factor map","kind":"definition","summary":"[The descended suspension factor map] For measurable automorphisms \\(T : X\\simeq X\\), \\(S : Y\\s…","labels":["suspensionFactorMap"],"detail_key":"p40"},{"id":"n33370","layer":"informal","project":"p40","title":"The suspension factor package","kind":"theorem","summary":"[The suspension factor package] \\(suspensionFactorMap\\) intertwines the two suspension flows (…","labels":["isFactorMap_suspensionFactorMap"],"detail_key":"p40"},{"id":"n33371","layer":"informal","project":"p40","title":"The base semiconjugacy \\(\\pi\\circ T = S\\circ\\pi\\) makes the raw fibre map intertwine the…","kind":"proof","summary":"The base semiconjugacy \\(\\pi\\circ T = S\\circ\\pi\\) makes the raw fibre map intertwine the two or…","labels":[],"detail_key":"p40"},{"id":"n33372","layer":"informal","project":"p40","title":"Shannon entropy of a cell family","kind":"definition","summary":"[Shannon entropy of a cell family] The \\emphShannon entropy of a finite family of cells \\(s:\\io…","labels":["entropy"],"detail_key":"p40"},{"id":"n33373","layer":"informal","project":"p40","title":"Finite measurable partition","kind":"definition","summary":"[Finite measurable partition] A \\emphfinite measurable partition of \\((\\alpha,\\mu)\\) is a struc…","labels":["MeasurePartition"],"detail_key":"p40"},{"id":"n33374","layer":"informal","project":"p40","title":"Entropy is at most \\(\\log k\\)","kind":"lemma","summary":"[Entropy is at most \\(\\log k\\)] A finite measurable partition of a probability space into \\(k\\)…","labels":["entropy_le_log_card_partition"],"detail_key":"p40"},{"id":"n33375","layer":"informal","project":"p40","title":"The cell measures \\(p_i=\\mu(A_i)\\) sum to \\(1\\) by finite additivity over the a.e.-disjoi…","kind":"proof","summary":"The cell measures \\(p_i=\\mu(A_i)\\) sum to \\(1\\) by finite additivity over the a.e.-disjoint cov…","labels":[],"detail_key":"p40"},{"id":"n33376","layer":"informal","project":"p40","title":"Subadditivity under joins","kind":"theorem","summary":"[Subadditivity under joins] For two finite measurable partitions \\(\\alpha=(A_i)\\) and \\(\\beta=(…","labels":["entropy_join_le"],"detail_key":"p40"},{"id":"n33377","layer":"informal","project":"p40","title":"The discrete Gibbs inequality \\(\\sum_x p_x\\log p_x \\ge \\sum_x p_x\\log q_x\\) (proved termw…","kind":"proof","summary":"The discrete Gibbs inequality \\(\\sum_x p_x\\log p_x \\ge \\sum_x p_x\\log q_x\\) (proved termwise fr…","labels":[],"detail_key":"p40"},{"id":"n33378","layer":"informal","project":"p40","title":"Invariance under a measure-preserving pullback","kind":"lemma","summary":"[Invariance under a measure-preserving pullback] For a measure-preserving \\(T\\) and a finite me…","labels":["entropy_pullback"],"detail_key":"p40"},{"id":"n33379","layer":"informal","project":"p40","title":"Each cell measure is preserved, \\(\\mu(T^-1B_j)=\\mu(B_j)\\), so the corresponding \\(\\eta\\)-…","kind":"proof","summary":"Each cell measure is preserved, \\(\\mu(T^-1B_j)=\\mu(B_j)\\), so the corresponding \\(\\eta\\)-terms…","labels":[],"detail_key":"p40"},{"id":"n33380","layer":"informal","project":"p40","title":"Conditional Shannon entropy","kind":"definition","summary":"[Conditional Shannon entropy] On a standard Borel probability space, the \\emphconditional Shann…","labels":["condEntropyPartition"],"detail_key":"p40"},{"id":"n33381","layer":"informal","project":"p40","title":"Conditioning does not increase entropy","kind":"theorem","summary":"[Conditioning does not increase entropy] For any finite measurable partition \\(P\\) of a standar…","labels":["condEntropy_le"],"detail_key":"p40"},{"id":"n33382","layer":"informal","project":"p40","title":"Jensen's inequality for the concave \\(\\eta\\), applied cell by cell: the \\(\\mu\\)-average o…","kind":"proof","summary":"Jensen's inequality for the concave \\(\\eta\\), applied cell by cell: the \\(\\mu\\)-average of \\(\\e…","labels":[],"detail_key":"p40"},{"id":"n33383","layer":"informal","project":"p40","title":"Flat iterated join","kind":"definition","summary":"[Flat iterated join] For a measure-preserving \\(T\\) and a finite measurable partition \\(\\alpha\\…","labels":["ksJoin"],"detail_key":"p40"},{"id":"n33384","layer":"informal","project":"p40","title":"Iterated-join entropy sequence","kind":"definition","summary":"[Iterated-join entropy sequence] The \\emphiterated-join entropy sequence of \\((T,\\alpha)\\) is \\…","labels":["ksEntropySeq"],"detail_key":"p40"},{"id":"n33385","layer":"informal","project":"p40","title":"Subadditivity of the entropy sequence","kind":"theorem","summary":"[Subadditivity of the entropy sequence] For a measure-preserving \\(T\\) on a probability space a…","labels":["ksEntropySeq_subadditive"],"detail_key":"p40"},{"id":"n33386","layer":"informal","project":"p40","title":"Splitting the index type along \\(Finn\\oplusFinm\\simeqFin(n+m)\\) exhibits, cell by cell, t…","kind":"proof","summary":"Splitting the index type along \\(Finn\\oplusFinm\\simeqFin(n+m)\\) exhibits, cell by cell, the \\((…","labels":[],"detail_key":"p40"},{"id":"n33387","layer":"informal","project":"p40","title":"Per-partition Kolmogorov--Sinai entropy","kind":"definition","summary":"[Per-partition Kolmogorov--Sinai entropy] The \\emphKolmogorov--Sinai entropy of \\(T\\) relative…","labels":["ksEntropyPartition"],"detail_key":"p40"},{"id":"n33388","layer":"informal","project":"p40","title":"Fekete convergence","kind":"theorem","summary":"[Fekete convergence] The averaged iterated-join entropies converge: \\(\\frac1n H\\bigl(\\bigvee_k<…","labels":["tendsto_ksEntropySeq"],"detail_key":"p40"},{"id":"n33389","layer":"informal","project":"p40","title":"Fekete's subadditivity lemma (\\(\\textttSubadditive.tendsto\\_lim\\)) applied to the sequenc…","kind":"proof","summary":"Fekete's subadditivity lemma (\\(\\textttSubadditive.tendsto\\_lim\\)) applied to the sequence of \\…","labels":[],"detail_key":"p40"},{"id":"n33390","layer":"informal","project":"p40","title":"Kolmogorov--Sinai entropy of the system","kind":"definition","summary":"[Kolmogorov--Sinai entropy of the system] The \\emphKolmogorov--Sinai entropy of the system is t…","labels":["ksEntropy"],"detail_key":"p40"},{"id":"n33391","layer":"informal","project":"p40","title":"One-sided generating partition","kind":"definition","summary":"[One-sided generating partition] A finite measurable partition \\(P\\) is \\emph(one-sided) genera…","labels":["IsGenerating"],"detail_key":"p40"},{"id":"n33392","layer":"informal","project":"p40","title":"Kolmogorov--Sinai generator theorem","kind":"theorem","summary":"[Kolmogorov--Sinai generator theorem] Let \\(T\\) be a measure-preserving transformation of a sta…","labels":["ksEntropy_eq_ksEntropyPartition_of_generating"],"detail_key":"p40"},{"id":"n33393","layer":"informal","project":"p40","title":"By \\(\\le\\)-antisymmetry; \\(h(T,P)\\le h(T)\\) is free from the defining supremum. For the c…","kind":"proof","summary":"By \\(\\le\\)-antisymmetry; \\(h(T,P)\\le h(T)\\) is free from the defining supremum. For the convers…","labels":[],"detail_key":"p40"},{"id":"n33394","layer":"informal","project":"p40","title":"Two-sided generating partition","kind":"definition","summary":"[Two-sided generating partition] For a measure-preserving \\emphautomorphism \\(e:\\alpha\\simeq\\al…","labels":["IsGeneratingTwoSided"],"detail_key":"p40"},{"id":"n33395","layer":"informal","project":"p40","title":"Two-sided generator theorem","kind":"theorem","summary":"[Two-sided generator theorem] Let \\(e\\) be a measure-preserving automorphism of a standard Bore…","labels":["ksEntropy_eq_ksEntropyPartition_of_isGeneratingTwoSided"],"detail_key":"p40"},{"id":"n33396","layer":"informal","project":"p40","title":"Again by reduction to \\(h(e,Q)\\le h(e,P)\\) for arbitrary \\(Q\\), but along the \\emphsymmet…","kind":"proof","summary":"Again by reduction to \\(h(e,Q)\\le h(e,P)\\) for arbitrary \\(Q\\), but along the \\emphsymmetric wi…","labels":[],"detail_key":"p40"},{"id":"n33397","layer":"informal","project":"p40","title":"Relative Kolmogorov--Sinai entropy of a partition","kind":"definition","summary":"[Relative Kolmogorov--Sinai entropy of a partition] For a sub-\\(\\sigma\\)-algebra \\( A\\le m_\\alp…","labels":["condKsEntropyPartition"],"detail_key":"p40"},{"id":"n33398","layer":"informal","project":"p40","title":"Relative Kolmogorov--Sinai entropy of the system","kind":"definition","summary":"[Relative Kolmogorov--Sinai entropy of the system] The \\emphrelative entropy of the system give…","labels":["condKsEntropy"],"detail_key":"p40"},{"id":"n33399","layer":"informal","project":"p40","title":"Abramov--Rokhlin partition identity","kind":"theorem","summary":"[Abramov--Rokhlin partition identity] Let \\(T\\) preserve \\(\\mu\\) on a standard Borel probabilit…","labels":["abramovRokhlin_partition"],"detail_key":"p40"},{"id":"n33400","layer":"informal","project":"p40","title":"Per \\(n\\), the refinement collapses the join \\(H(A_n\\vee B_n)\\) to \\(H(B_n)\\) plus the co…","kind":"proof","summary":"Per \\(n\\), the refinement collapses the join \\(H(A_n\\vee B_n)\\) to \\(H(B_n)\\) plus the conditio…","labels":[],"detail_key":"p40"},{"id":"n33401","layer":"informal","project":"p40","title":"Abramov--Rokhlin addition formula","kind":"theorem","summary":"[Abramov--Rokhlin addition formula] Let \\(\\pi:(\\alpha,T,\\mu)\\to(\\beta,S,\\nu)\\) be a factor map…","labels":["abramov_rokhlin"],"detail_key":"p40"},{"id":"n33402","layer":"informal","project":"p40","title":"Pure \\(\\overline\\R\\)-algebra threading three proved ingredients: rewrite \\(h(T)=h(T,P)\\)…","kind":"proof","summary":"Pure \\(\\overline\\R\\)-algebra threading three proved ingredients: rewrite \\(h(T)=h(T,P)\\) by the…","labels":[],"detail_key":"p40"},{"id":"n33403","layer":"informal","project":"p40","title":"Margulis--Ruelle inequality, abstract reduction","kind":"theorem","summary":"[Margulis--Ruelle inequality, abstract reduction] Let \\(T\\) be an ergodic self-map of \\(R^d\\) w…","labels":["margulisRuelle_le_sumPosExp"],"detail_key":"p40"},{"id":"n33404","layer":"informal","project":"p40","title":"The pure lattice step: unfold the defining double supremum of \\(h(T)\\) over arities \\(n\\)…","kind":"proof","summary":"The pure lattice step: unfold the defining double supremum of \\(h(T)\\) over arities \\(n\\) and \\…","labels":[],"detail_key":"p40"},{"id":"n33405","layer":"informal","project":"p40","title":"Sharp Margulis--Ruelle inequality","kind":"theorem","summary":"[Sharp Margulis--Ruelle inequality] Same setting (ergodic \\(T\\) on \\(R^d\\), nonsingular log-int…","labels":["margulisRuelle_sharp"],"detail_key":"p40"},{"id":"n33406","layer":"informal","project":"p40","title":"The one-step \\emphsharp anisotropic covering count \\( N_\\varepsilon\\bigl(L\\,\\overline B(0…","kind":"proof","summary":"The one-step \\emphsharp anisotropic covering count \\( N_\\varepsilon\\bigl(L\\,\\overline B(0,\\vare…","labels":[],"detail_key":"p40"},{"id":"n33407","layer":"informal","project":"p40","title":"Rokhlin's volume-distortion formula","kind":"theorem","summary":"[Rokhlin's volume-distortion formula] Let \\(T\\) be a measure-preserving, differentiable self-ma…","labels":["ksEntropyPartition_eq_integral_log_abs_det"],"detail_key":"p40"},{"id":"n33408","layer":"informal","project":"p40","title":"Three identities composed. First, the sharp-rate form of the Fekete limit expresses \\(h(T…","kind":"proof","summary":"Three identities composed. First, the sharp-rate form of the Fekete limit expresses \\(h(T,\\xi)\\…","labels":[],"detail_key":"p40"},{"id":"n33409","layer":"informal","project":"p40","title":"Information function of the iterated join","kind":"definition","summary":"[Information function of the iterated join] Every point \\(x\\) has an \\(n\\)-step \\emphitinerary…","labels":["infoFun"],"detail_key":"p40"},{"id":"n33410","layer":"informal","project":"p40","title":"The information function integrates to the join entropy","kind":"theorem","summary":"[The information function integrates to the join entropy] For every \\(n\\), \\[ \\int_\\alpha i_n \\…","labels":["integral_infoFun_eq"],"detail_key":"p40"},{"id":"n33411","layer":"informal","project":"p40","title":"Write \\(i_n\\) as the finite sum of indicators of the itinerary fibers weighted by \\(-\\log…","kind":"proof","summary":"Write \\(i_n\\) as the finite sum of indicators of the itinerary fibers weighted by \\(-\\log\\mu(\\t…","labels":[],"detail_key":"p40"},{"id":"n33412","layer":"informal","project":"p40","title":"Telescoped Breiman chain rule, entropy level","kind":"theorem","summary":"[Telescoped Breiman chain rule, entropy level] On a standard Borel space, the \\(n\\)-step join e…","labels":["ksEntropySeq_eq_sum_condEntropy"],"detail_key":"p40"},{"id":"n33413","layer":"informal","project":"p40","title":"Induction on \\(n\\) from the one-step chain rule: reindexing along \\(Finn \\times \\iota \\si…","kind":"proof","summary":"Induction on \\(n\\) from the one-step chain rule: reindexing along \\(Finn \\times \\iota \\simeq Fi…","labels":[],"detail_key":"p40"},{"id":"n33414","layer":"informal","project":"p40","title":"The sharp KS rate as a conditional entropy","kind":"theorem","summary":"[The sharp KS rate as a conditional entropy] On a standard Borel space (with \\(\\iota\\) nonempty…","labels":["ksEntropyPartition_eq_condEntropy_iSup"],"detail_key":"p40"},{"id":"n33415","layer":"informal","project":"p40","title":"The conditioning \\(\\sigma\\)-algebras increase in \\(k\\), so the fixed-partition L\\'evy the…","kind":"proof","summary":"The conditioning \\(\\sigma\\)-algebras increase in \\(k\\), so the fixed-partition L\\'evy theorem (…","labels":[],"detail_key":"p40"},{"id":"n33416","layer":"informal","project":"p40","title":"Crude name-count bound, Birkhoff-free","kind":"theorem","summary":"[Crude name-count bound, Birkhoff-free] For a measure-preserving \\(T\\) and a finite partition i…","labels":["ae_limsup_div_infoFun_le_log_card"],"detail_key":"p40"},{"id":"n33417","layer":"informal","project":"p40","title":"On each itinerary fiber the integrand \\(\\exp(i_n - n\\log\\#\\iota)\\) is the constant \\(\\mu(…","kind":"proof","summary":"On each itinerary fiber the integrand \\(\\exp(i_n - n\\log\\#\\iota)\\) is the constant \\(\\mu(\\textc…","labels":[],"detail_key":"p40"},{"id":"n33418","layer":"informal","project":"p40","title":"Conditional information function","kind":"definition","summary":"[Conditional information function] For a sub-\\(\\sigma\\)-algebra \\( A\\) of a standard Borel spac…","labels":["condInfoFun"],"detail_key":"p40"},{"id":"n33419","layer":"informal","project":"p40","title":"Chung's maximal inequality","kind":"theorem","summary":"[Chung's maximal inequality] Let \\(g_k\\) be the conditional information function of \\(P\\) given…","labels":["chungTail"],"detail_key":"p40"},{"id":"n33420","layer":"informal","project":"p40","title":"A stopping-time (first-passage) argument: stratify \\(\\g^* > \\lambda\\\\cap P_i_0\\) by the f…","kind":"proof","summary":"A stopping-time (first-passage) argument: stratify \\(\\g^* > \\lambda\\\\cap P_i_0\\) by the first l…","labels":[],"detail_key":"p40"},{"id":"n33421","layer":"informal","project":"p40","title":"Pointwise SMB, from the Breiman telescoping","kind":"theorem","summary":"[Pointwise SMB, from the Breiman telescoping] Let \\(T\\) be ergodic and let \\((i_n)\\) be any seq…","labels":["ae_tendsto_div_infoFun"],"detail_key":"p40"},{"id":"n33422","layer":"informal","project":"p40","title":"ErgodicTheory.Krieger.makerTail","kind":"proof","summary":"Split \\(\\tfrac1n\\sum_j<n g_n-j(T^jx)\\) into the Birkhoff main term \\(\\tfrac1n\\sum_j<n g_\\infty(…","labels":[],"detail_key":"p40"},{"id":"n33423","layer":"informal","project":"p40","title":"Pointwise Shannon--McMillan--Breiman theorem","kind":"theorem","summary":"[Pointwise Shannon--McMillan--Breiman theorem] Let \\(T\\) be an ergodic measure-preserving trans…","labels":["ae_tendsto_div_infoFun_self"],"detail_key":"p40"},{"id":"n33424","layer":"informal","project":"p40","title":"The one-step factorization of the information weight (peeling the first symbol of the iti…","kind":"proof","summary":"The one-step factorization of the information weight (peeling the first symbol of the itinerary…","labels":[],"detail_key":"p40"},{"id":"n33425","layer":"informal","project":"p40","title":"In-measure upper equipartition","kind":"theorem","summary":"[In-measure upper equipartition] For ergodic \\(T\\), the in-measure SMB upper bound holds: for e…","labels":["upperSMBInMeasure_of_ergodic"],"detail_key":"p40"},{"id":"n33426","layer":"informal","project":"p40","title":"Almost-everywhere convergence (\\Crefae_tendsto_div_infoFun_self) makes the deviation sets…","kind":"proof","summary":"Almost-everywhere convergence (\\Crefae_tendsto_div_infoFun_self) makes the deviation sets event…","labels":[],"detail_key":"p40"},{"id":"n33427","layer":"informal","project":"p40","title":"Rokhlin--Kakutani tower lemma","kind":"theorem","summary":"[Rokhlin--Kakutani tower lemma] Let \\(e\\) be an ergodic measure-preserving automorphism of a st…","labels":["rokhlin_tower"],"detail_key":"p40"},{"id":"n33428","layer":"informal","project":"p40","title":"The Kakutani skyscraper construction. Pick a positive measurable set \\(A\\) with \\(\\mu(A)…","kind":"proof","summary":"The Kakutani skyscraper construction. Pick a positive measurable set \\(A\\) with \\(\\mu(A) < \\var…","labels":[],"detail_key":"p40"},{"id":"n33429","layer":"informal","project":"p40","title":"AEP covering bound: few names carry the mass","kind":"theorem","summary":"[AEP covering bound: few names carry the mass] Let \\(T\\) be measure preserving and assume the i…","labels":["exists_cover_names_card_le"],"detail_key":"p40"},{"id":"n33430","layer":"informal","project":"p40","title":"Pigeonhole: cells of measure \\(\\ge e^-N(h+\\varepsilon)\\) number at most \\(e^N(h+\\varepsil…","kind":"proof","summary":"Pigeonhole: cells of measure \\(\\ge e^-N(h+\\varepsilon)\\) number at most \\(e^N(h+\\varepsilon)\\),…","labels":[],"detail_key":"p40"},{"id":"n33431","layer":"informal","project":"p40","title":"Sentinel prefix code","kind":"theorem","summary":"[Sentinel prefix code] Over an alphabet \\(Finl\\) with a reserved sentinel letter \\(s\\), any fin…","labels":["exists_sentinelEncoding"],"detail_key":"p40"},{"id":"n33432","layer":"informal","project":"p40","title":"The name set embeds into the fixed-length data words \\(Finm \\to Fin(l-1)\\) over the non-s…","kind":"proof","summary":"The name set embeds into the fixed-length data words \\(Finm \\to Fin(l-1)\\) over the non-sentine…","labels":[],"detail_key":"p40"},{"id":"n33433","layer":"informal","project":"p40","title":"Two-sided generation mod 0","kind":"definition","summary":"[Two-sided generation mod 0] The \\emphtwo-sided saturation of a finite partition \\(P\\) under an…","labels":["IsGeneratingTwoSidedMod0"],"detail_key":"p40"},{"id":"n33434","layer":"informal","project":"p40","title":"Countable Shannon entropy","kind":"definition","summary":"[Countable Shannon entropy] For a countable family of cells \\(s : \\iota \\to Set\\,\\alpha\\), the…","labels":["cHmu"],"detail_key":"p40"},{"id":"n33435","layer":"informal","project":"p40","title":"Countable finite-entropy two-sided generator (Rokhlin; Keane--Serafin)","kind":"theorem","summary":"[Countable finite-entropy two-sided generator (Rokhlin; Keane--Serafin)] Let \\((\\alpha,\\mu)\\) b…","labels":["exists_countable_twoSided_generator"],"detail_key":"p40"},{"id":"n33436","layer":"informal","project":"p40","title":"Recovering every set of a generating sequence mod \\(0\\) pushes the ambient \\(\\sigma\\)-alg…","kind":"proof","summary":"Recovering every set of a generating sequence mod \\(0\\) pushes the ambient \\(\\sigma\\)-algebra i…","labels":[],"detail_key":"p40"},{"id":"n33437","layer":"informal","project":"p40","title":"Krieger coding data","kind":"definition","summary":"[Krieger coding data] For an automorphism \\(e\\) and \\(k \\in N\\), a \\emphKrieger coding datum bu…","labels":["KriegerCodingData"],"detail_key":"p40"},{"id":"n33438","layer":"informal","project":"p40","title":"Recovery assembly of Krieger's theorem","kind":"theorem","summary":"[Recovery assembly of Krieger's theorem] Given a Krieger coding datum \\(D\\) for \\((e, \\mu, k)\\)…","labels":["krieger_finite_generator_of_coding"],"detail_key":"p40"},{"id":"n33439","layer":"informal","project":"p40","title":"Cross-layer recovery. Mod-\\(0\\) generation by \\(Q\\) places the ambient \\(\\sigma\\)-algebra…","kind":"proof","summary":"Cross-layer recovery. Mod-\\(0\\) generation by \\(Q\\) places the ambient \\(\\sigma\\)-algebra insid…","labels":[],"detail_key":"p40"},{"id":"n33440","layer":"informal","project":"p40","title":"Krieger's finite generator theorem","kind":"theorem","summary":"[Krieger's finite generator theorem] Let \\(e\\) be an ergodic, aperiodic ( : every set of \\(n\\)-…","labels":["krieger_finite_generator"],"detail_key":"p40"},{"id":"n33441","layer":"informal","project":"p40","title":"Immediate from the recovery assembly \\Crefkrieger_finite_generator_of_coding applied to t…","kind":"proof","summary":"Immediate from the recovery assembly \\Crefkrieger_finite_generator_of_coding applied to the sup…","labels":[],"detail_key":"p40"},{"id":"n33442","layer":"informal","project":"p40","title":"Generalized partition function","kind":"definition","summary":"[Generalized partition function] For a finite weight family \\(p : \\iota \\to R\\) and \\(q \\in R\\)…","labels":["partitionFunction"],"detail_key":"p40"},{"id":"n33443","layer":"informal","project":"p40","title":"Mass exponent","kind":"definition","summary":"[Mass exponent] The \\emphmass exponent of the family \\(p\\) at scale \\(\\varepsilon\\) is \\[ \\tau(…","labels":["massExponent"],"detail_key":"p40"},{"id":"n33444","layer":"informal","project":"p40","title":"R\\'enyi / generalized dimension","kind":"definition","summary":"[R\\'enyi / generalized dimension] The \\emphR\\'enyi (generalized) dimension of \\(p\\) at scale \\(…","labels":["renyiDim"],"detail_key":"p40"},{"id":"n33445","layer":"informal","project":"p40","title":"Singularity spectrum","kind":"definition","summary":"[Singularity spectrum] The \\emphsingularity spectrum of \\(p\\) at scale \\(\\varepsilon\\) is the L…","labels":["singularitySpectrum"],"detail_key":"p40"},{"id":"n33446","layer":"informal","project":"p40","title":"Log-convexity of the partition function","kind":"theorem","summary":"[Log-convexity of the partition function] Let \\(p : \\iota \\to R\\) satisfy \\(p_i \\ge 0\\) for all…","labels":["logPartitionFunction_convexOn"],"detail_key":"p40"},{"id":"n33447","layer":"informal","project":"p40","title":"Derivative-free. The midpoint inequality \\(Z_aq_1 + bq_2 \\le Z_q_1^\\,a\\, Z_q_2^\\,b\\) (for…","kind":"proof","summary":"Derivative-free. The midpoint inequality \\(Z_aq_1 + bq_2 \\le Z_q_1^\\,a\\, Z_q_2^\\,b\\) (for \\(a,…","labels":[],"detail_key":"p40"},{"id":"n33448","layer":"informal","project":"p40","title":"Concavity of the mass exponent","kind":"theorem","summary":"[Concavity of the mass exponent] Under the same hypotheses on \\(p\\), for a scale \\(0 < \\varepsi…","labels":["massExponent_concaveOn"],"detail_key":"p40"},{"id":"n33449","layer":"informal","project":"p40","title":"Since \\(0 < \\varepsilon < 1\\), the denominator \\(\\log\\varepsilon\\) is negative; multiplyi…","kind":"proof","summary":"Since \\(0 < \\varepsilon < 1\\), the denominator \\(\\log\\varepsilon\\) is negative; multiplying the…","labels":[],"detail_key":"p40"},{"id":"n33450","layer":"informal","project":"p40","title":"Antitonicity of the R\\'enyi dimension","kind":"theorem","summary":"[Antitonicity of the R\\'enyi dimension] Let \\(p\\) be a probability weight family (\\(p_i \\ge 0\\)…","labels":["renyiDim_antitone"],"detail_key":"p40"},{"id":"n33451","layer":"informal","project":"p40","title":"The classical secant-slope argument. Write \\(h(q) = \\log Z_q\\); it is convex and \\(h(1) =…","kind":"proof","summary":"The classical secant-slope argument. Write \\(h(q) = \\log Z_q\\); it is convex and \\(h(1) = 0\\) f…","labels":[],"detail_key":"p40"},{"id":"n33452","layer":"informal","project":"p40","title":"R\\'enyi dimension of a measure","kind":"definition","summary":"[R\\'enyi dimension of a measure] For a measure \\(\\mu\\) on \\(\\alpha\\), a finite measurable parti…","labels":["renyiDimMeasure"],"detail_key":"p40"},{"id":"n33453","layer":"informal","project":"p40","title":"Antitonicity for a probability measure","kind":"theorem","summary":"[Antitonicity for a probability measure] For a probability measure \\(\\mu\\), a finite measurable…","labels":["renyiDimMeasure_antitone"],"detail_key":"p40"},{"id":"n33454","layer":"informal","project":"p40","title":"Apply Theorem~\\refrenyiDim_antitone to the cell-mass family: nonnegativity is \\(\\textttEN…","kind":"proof","summary":"Apply Theorem~\\refrenyiDim_antitone to the cell-mass family: nonnegativity is \\(\\textttENNReal.…","labels":[],"detail_key":"p40"},{"id":"n33455","layer":"informal","project":"p40","title":"Information dimension is entropy over \\(-\\log\\varepsilon\\)","kind":"theorem","summary":"[Information dimension is entropy over \\(-\\log\\varepsilon\\)] For a probability measure \\(\\mu\\),…","labels":["renyiDimMeasure_one_eq"],"detail_key":"p40"},{"id":"n33456","layer":"informal","project":"p40","title":"Unfold the \\(q = 1\\) branch of \\(D_q\\): its numerator \\(\\sum_i \\mu(P_i)\\log\\mu(P_i)\\) is…","kind":"proof","summary":"Unfold the \\(q = 1\\) branch of \\(D_q\\): its numerator \\(\\sum_i \\mu(P_i)\\log\\mu(P_i)\\) is term-b…","labels":[],"detail_key":"p40"},{"id":"n33457","layer":"informal","project":"p40","title":"R\\'enyi dimension of a flow's invariant measure","kind":"definition","summary":"[R\\'enyi dimension of a flow's invariant measure] For a measure-preserving flow \\(\\varphi\\) wit…","labels":["renyiDimFlow"],"detail_key":"p40"},{"id":"n33458","layer":"informal","project":"p40","title":"Flow-level antitonicity","kind":"corollary","summary":"[Flow-level antitonicity] For a measure-preserving flow \\(\\varphi\\) of a probability measure \\(…","labels":["renyiDimFlow_antitone"],"detail_key":"p40"},{"id":"n33459","layer":"informal","project":"p40","title":"\\textttrenyiDimFlow unfolds to \\textttrenyiDimMeasure, so this is Theorem~\\refrenyiDimMea…","kind":"proof","summary":"\\textttrenyiDimFlow unfolds to \\textttrenyiDimMeasure, so this is Theorem~\\refrenyiDimMeasure_a…","labels":[],"detail_key":"p40"},{"id":"n33460","layer":"informal","project":"p40","title":"Upper local dimension","kind":"definition","summary":"[Upper local dimension] For a measure \\(\\mu\\) on a (pseudo-)metric measurable space \\(E\\) and a…","labels":["localDimension"],"detail_key":"p40"},{"id":"n33461","layer":"informal","project":"p40","title":"Local dimension in the absolutely-continuous case","kind":"theorem","summary":"[Local dimension in the absolutely-continuous case] Let \\(E\\) be a finite-dimensional real inne…","labels":["ae_tendsto_localDimension_of_absolutelyContinuous"],"detail_key":"p40"},{"id":"n33462","layer":"informal","project":"p40","title":"Pure measure differentiation, no dynamics. Besicovitch differentiation gives \\(\\mu(\\bar B…","kind":"proof","summary":"Pure measure differentiation, no dynamics. Besicovitch differentiation gives \\(\\mu(\\bar B(x,r))…","labels":[],"detail_key":"p40"},{"id":"n33463","layer":"informal","project":"p40","title":"A.e.\\ value of the local dimension","kind":"corollary","summary":"[A.e.\\ value of the local dimension] Under the same hypotheses, \\(\\bar d_\\mu(x) = finrank_R E\\)…","labels":["ae_localDimension_eq_finrank"],"detail_key":"p40"},{"id":"n33464","layer":"informal","project":"p40","title":"Where the genuine limit of Theorem~\\refae_tendsto_localDimension_of_absolutelyContinuous…","kind":"proof","summary":"Where the genuine limit of Theorem~\\refae_tendsto_localDimension_of_absolutelyContinuous exists…","labels":[],"detail_key":"p40"},{"id":"n33465","layer":"informal","project":"p40","title":"Local-to-Hausdorff dimension bridge","kind":"theorem","summary":"[Local-to-Hausdorff dimension bridge] Let \\(\\mu\\) be a probability measure on a Borel second-co…","labels":["dimH_eq_of_localDimension_eq"],"detail_key":"p40"},{"id":"n33466","layer":"informal","project":"p40","title":"Two mass-distribution arguments over a bare metric space. \\emphLower bound (Frostman): fo…","kind":"proof","summary":"Two mass-distribution arguments over a bare metric space. \\emphLower bound (Frostman): for each…","labels":[],"detail_key":"p40"},{"id":"n33467","layer":"informal","project":"p40","title":"Hausdorff dimension of full-measure sets, a.c.\\ case","kind":"theorem","summary":"[Hausdorff dimension of full-measure sets, a.c.\\ case] Let \\(\\mu\\) be a probability measure on…","labels":["dimH_eq_finrank_of_ae_full_of_absolutelyContinuous"],"detail_key":"p40"},{"id":"n33468","layer":"informal","project":"p40","title":"The upper bound is monotonicity: \\(\\dim_H s \\le \\dim_H E = finrank_R E\\). The lower bound…","kind":"proof","summary":"The upper bound is monotonicity: \\(\\dim_H s \\le \\dim_H E = finrank_R E\\). The lower bound is th…","labels":[],"detail_key":"p40"},{"id":"n33469","layer":"informal","project":"p40","title":"Entropy = Hausdorff dimension on the full shift","kind":"theorem","summary":"[Entropy = Hausdorff dimension on the full shift] Let \\(\\mu\\) be a shift-invariant probability…","labels":["dimH_eq_ksEntropy_div_log_two"],"detail_key":"p40"},{"id":"n33470","layer":"informal","project":"p40","title":"Atoms are cylinders are dyadic closed balls, so the unconditional pointwise Shannon--McMi…","kind":"proof","summary":"Atoms are cylinders are dyadic closed balls, so the unconditional pointwise Shannon--McMillan--…","labels":[],"detail_key":"p40"},{"id":"n33471","layer":"informal","project":"p40","title":"Unconditional Bernoulli witness","kind":"theorem","summary":"[Unconditional Bernoulli witness] Let \\(bern\\nu\\) be the Bernoulli (i.i.d.\\ product) measure on…","labels":["dimH_bern_eq_Hnu_div_log_two"],"detail_key":"p40"},{"id":"n33472","layer":"informal","project":"p40","title":"Ergodicity of the shift for \\(bern\\nu\\) is Kolmogorov's 0--1 law applied to the tail-meas…","kind":"proof","summary":"Ergodicity of the shift for \\(bern\\nu\\) is Kolmogorov's 0--1 law applied to the tail-measurable…","labels":[],"detail_key":"p40"},{"id":"n33473","layer":"informal","project":"p40","title":"The constant-roof Bernoulli suspension flow","kind":"definition","summary":"[The constant-roof Bernoulli suspension flow] The \\emphBernoulli suspension flow is the time-tr…","labels":["bernSuspensionFlow"],"detail_key":"p40"},{"id":"n33474","layer":"informal","project":"p40","title":"Ergodicity of the suspension flow","kind":"theorem","summary":"[Ergodicity of the suspension flow] Assume the base shift \\(T\\) is ergodic for \\(bernZ\\nu\\). Th…","labels":["ergodic_bernSuspensionFlow"],"detail_key":"p40"},{"id":"n33475","layer":"informal","project":"p40","title":"Lift \\(A\\) through the quotient map \\(\\pi(x,s) = [x,s]\\). Invariance under all vertical t…","kind":"proof","summary":"Lift \\(A\\) through the quotient map \\(\\pi(x,s) = [x,s]\\). Invariance under all vertical transla…","labels":[],"detail_key":"p40"},{"id":"n33476","layer":"informal","project":"p40","title":"Entropy of the suspension flow","kind":"theorem","summary":"[Entropy of the suspension flow] The Kolmogorov--Sinai entropy (Definition~\\refksEntropy) of th…","labels":["ksEntropy_bernSuspensionFlow_one_eq_Hnu"],"detail_key":"p40"},{"id":"n33477","layer":"informal","project":"p40","title":"The fundamental-domain equivalence onto \\(\\alpha_0^Z \\times [0,1)\\) conjugates \\(\\zeta_1\\…","kind":"proof","summary":"The fundamental-domain equivalence onto \\(\\alpha_0^Z \\times [0,1)\\) conjugates \\(\\zeta_1\\) to t…","labels":[],"detail_key":"p40"},{"id":"n33478","layer":"informal","project":"p40","title":"The witness partition","kind":"definition","summary":"[The witness partition] The \\emphwitness partition of the suspension measure \\(\\hat\\mu\\) is the…","labels":["bernSuspensionWitness"],"detail_key":"p40"},{"id":"n33479","layer":"informal","project":"p40","title":"Heterogeneity of the witness","kind":"theorem","summary":"[Heterogeneity of the witness] If \\(\\nu\\) charges two distinct symbols \\(i \\ne j\\) with \\emphdi…","labels":["isHeterogeneous_bernSuspensionWitness"],"detail_key":"p40"},{"id":"n33480","layer":"informal","project":"p40","title":"By the mass identity, the cells indexed by \\(i\\) and \\(j\\) carry masses \\(\\nu\\i\\\\) and \\(…","kind":"proof","summary":"By the mass identity, the cells indexed by \\(i\\) and \\(j\\) carry masses \\(\\nu\\i\\\\) and \\(\\nu\\j\\…","labels":[],"detail_key":"p40"},{"id":"n33481","layer":"informal","project":"p40","title":"\\(q\\)-dependence of the flow's R\\'enyi spectrum","kind":"theorem","summary":"[\\(q\\)-dependence of the flow's R\\'enyi spectrum] Let \\(\\alpha_0\\) consist of exactly two symbo…","labels":["renyiDimFlow_bernSuspension_q_dependent"],"detail_key":"p40"},{"id":"n33482","layer":"informal","project":"p40","title":"A transfer argument. The flow witness's cell masses agree, up to the \\(\\alpha_0 \\simeq Fi…","kind":"proof","summary":"A transfer argument. The flow witness's cell masses agree, up to the \\(\\alpha_0 \\simeq Fin(card…","labels":[],"detail_key":"p40"},{"id":"n33483","layer":"informal","project":"p40","title":"Merged weights","kind":"definition","summary":"[Merged weights] For a finite weight family \\(p : \\iota \\to R\\) and a \\emphmerge map \\(f : \\iot…","labels":["mergedWeights"],"detail_key":"p40"},{"id":"n33484","layer":"informal","project":"p40","title":"R\\'enyi entropy","kind":"definition","summary":"[R\\'enyi entropy] The \\emphR\\'enyi entropy of order \\(q\\) of a family \\(p : \\iota \\to R\\) is \\[…","labels":["renyiEntropy"],"detail_key":"p40"},{"id":"n33485","layer":"informal","project":"p40","title":"Power-sum super/subadditivity under merge","kind":"lemma","summary":"[Power-sum super/subadditivity under merge] Let \\(p : \\iota \\to R\\) with \\(p_a \\ge 0\\). For \\(q…","labels":["partitionFunction_merge"],"detail_key":"p40"},{"id":"n33486","layer":"informal","project":"p40","title":"The two-element bounds \\(x^q + y^q \\le (x+y)^q\\) (for \\(q \\ge 1\\), \\textttReal.add\\_rpow\\…","kind":"proof","summary":"The two-element bounds \\(x^q + y^q \\le (x+y)^q\\) (for \\(q \\ge 1\\), \\textttReal.add\\_rpow\\_le\\_r…","labels":[],"detail_key":"p40"},{"id":"n33487","layer":"informal","project":"p40","title":"Static R\\'enyi DPI","kind":"theorem","summary":"[Static R\\'enyi DPI] For every \\(f : \\iota \\to \\kappa\\), family \\(p \\ge 0\\), and order \\(0 \\le…","labels":["renyiEntropy_merge_le"],"detail_key":"p40"},{"id":"n33488","layer":"informal","project":"p40","title":"For \\(q \\ge 1\\) the partition function grows (\\CrefpartitionFunction_merge) and the prefa…","kind":"proof","summary":"For \\(q \\ge 1\\) the partition function grows (\\CrefpartitionFunction_merge) and the prefactor \\…","labels":[],"detail_key":"p40"},{"id":"n33489","layer":"informal","project":"p40","title":"Cylinder masses and the R\\'enyi rate","kind":"definition","summary":"[Cylinder masses and the R\\'enyi rate] For a probability measure \\(\\mu\\) on \\(Shift\\,A\\), the \\…","labels":["renyiRateSup"],"detail_key":"p40"},{"id":"n33490","layer":"informal","project":"p40","title":"One-block code","kind":"definition","summary":"[One-block code] A symbol map \\(\\varphi : A \\to B\\) lifts coordinatewise to the \\emphone-block…","labels":["blockCode"],"detail_key":"p40"},{"id":"n33491","layer":"informal","project":"p40","title":"Per-length pushforward identity and DPI","kind":"theorem","summary":"[Per-length pushforward identity and DPI] The preimage of a length-\\(n\\) cylinder of \\(Shift\\,B…","labels":["renyiEntropySeq_map_blockCode_le"],"detail_key":"p40"},{"id":"n33492","layer":"informal","project":"p40","title":"The mechanism is entirely per length \\(n\\), so \\emphno stationarity or shift-invariance o…","kind":"proof","summary":"The mechanism is entirely per length \\(n\\), so \\emphno stationarity or shift-invariance of \\(\\m…","labels":[],"detail_key":"p40"},{"id":"n33493","layer":"informal","project":"p40","title":"Rate monotonicity under one-block codes","kind":"theorem","summary":"[Rate monotonicity under one-block codes] For every probability measure \\(\\mu\\) on \\(Shift\\,A\\)…","labels":["renyiRateSup_map_blockCode_le"],"detail_key":"p40"},{"id":"n33494","layer":"informal","project":"p40","title":"Divide the per-length DPI (\\CrefrenyiEntropySeq_map_blockCode_le) by \\(n\\) and pass to \\(…","kind":"proof","summary":"Divide the per-length DPI (\\CrefrenyiEntropySeq_map_blockCode_le) by \\(n\\) and pass to \\(\\limsu…","labels":[],"detail_key":"p40"},{"id":"n33495","layer":"informal","project":"p40","title":"Exact Bernoulli R\\'enyi rate","kind":"theorem","summary":"[Exact Bernoulli R\\'enyi rate] For the i.i.d.\\ (Bernoulli) measure \\(bern\\,\\nu\\) on \\(Shift\\,A\\…","labels":["renyiRateSup_bern"],"detail_key":"p40"},{"id":"n33496","layer":"informal","project":"p40","title":"Everything factorizes over coordinates. The length-\\(n\\) cylinder mass of a word is the p…","kind":"proof","summary":"Everything factorizes over coordinates. The length-\\(n\\) cylinder mass of a word is the product…","labels":[],"detail_key":"p40"},{"id":"n33497","layer":"informal","project":"p40","title":"Strict rate drop under a genuine merge","kind":"theorem","summary":"[Strict rate drop under a genuine merge] Pushing a Bernoulli measure forward along a one-block…","labels":["renyiRate_strict_drop_uniformFin3"],"detail_key":"p40"},{"id":"n33498","layer":"informal","project":"p40","title":"\\textttmap\\_blockCode\\_bern is the coordinatewise pushforward of a product measure, check…","kind":"proof","summary":"\\textttmap\\_blockCode\\_bern is the coordinatewise pushforward of a product measure, checked on…","labels":[],"detail_key":"p40"},{"id":"n33499","layer":"informal","project":"p40","title":"Derivative cocycle generator","kind":"definition","summary":"[Derivative cocycle generator] For a self-map \\(T\\) of \\(E = EuclideanSpace\\,R\\,(Find)\\), the \\…","labels":["derivativeCocycle"],"detail_key":"p40"},{"id":"n33500","layer":"informal","project":"p40","title":"Chain-rule cocycle identity","kind":"theorem","summary":"[Chain-rule cocycle identity] For a differentiable \\(T\\), the \\(n\\)-th cocycle iterate of the d…","labels":["chainRule_cocycle"],"detail_key":"p40"},{"id":"n33501","layer":"informal","project":"p40","title":"Induction on \\(n\\). The base case is \\(D(id) = id\\). For the step, peel the innermost fac…","kind":"proof","summary":"Induction on \\(n\\). The base case is \\(D(id) = id\\). For the step, peel the innermost factor \\(…","labels":[],"detail_key":"p40"},{"id":"n33502","layer":"informal","project":"p40","title":"Oseledets theorem for the derivative cocycle","kind":"theorem","summary":"[Oseledets theorem for the derivative cocycle] Let \\(\\mu\\) be a probability measure on \\(E = Eu…","labels":["oseledets_filtration_derivativeCocycle"],"detail_key":"p40"},{"id":"n33503","layer":"informal","project":"p40","title":"The first conjunct is \\CrefchainRule_cocycle. The second is the one-sided Oseledets theor…","kind":"proof","summary":"The first conjunct is \\CrefchainRule_cocycle. The second is the one-sided Oseledets theorem (\\C…","labels":[],"detail_key":"p40"},{"id":"n33504","layer":"informal","project":"p40","title":"Compounded expansion bound","kind":"lemma","summary":"[Compounded expansion bound] If \\(T\\) is differentiable and uniformly expanding with constant \\…","labels":["cocycle_expanding_bound"],"detail_key":"p40"},{"id":"n33505","layer":"informal","project":"p40","title":"Induction on \\(n\\), using the chain rule \\(D_x(T^[n+1]) = D_Tx(T^[n])\\circ D_xT\\) to peel…","kind":"proof","summary":"Induction on \\(n\\), using the chain rule \\(D_x(T^[n+1]) = D_Tx(T^[n])\\circ D_xT\\) to peel off o…","labels":[],"detail_key":"p40"},{"id":"n33506","layer":"informal","project":"p40","title":"Every singular value is at least \\(K^n\\)","kind":"lemma","summary":"[Every singular value is at least \\(K^n\\)] Under the same hypotheses, every singular value of t…","labels":["expanding_pow_le_singularValues"],"detail_key":"p40"},{"id":"n33507","layer":"informal","project":"p40","title":"By \\CrefchainRule_cocycle the iterate acts as \\(D_x(T^[n])\\), so \\Crefcocycle_expanding_b…","kind":"proof","summary":"By \\CrefchainRule_cocycle the iterate acts as \\(D_x(T^[n])\\), so \\Crefcocycle_expanding_bound g…","labels":[],"detail_key":"p40"},{"id":"n33508","layer":"informal","project":"p40","title":"Every exponent is at least \\(\\log K\\)","kind":"theorem","summary":"[Every exponent is at least \\(\\log K\\)] For an ergodic, log-integrable, differentiable uniforml…","labels":["log_le_exponents_of_expanding"],"detail_key":"p40"},{"id":"n33509","layer":"informal","project":"p40","title":"Pick a base point where the per-index singular-value limit \\(\\frac1n\\log\\sigma_i(A^(n)(x)…","kind":"proof","summary":"Pick a base point where the per-index singular-value limit \\(\\frac1n\\log\\sigma_i(A^(n)(x))\\to\\l…","labels":[],"detail_key":"p40"},{"id":"n33510","layer":"informal","project":"p40","title":"Positivity of the whole spectrum","kind":"corollary","summary":"[Positivity of the whole spectrum] Every Lyapunov exponent of a uniformly expanding map is stri…","labels":["exponents_pos_of_expanding"],"detail_key":"p40"},{"id":"n33511","layer":"informal","project":"p40","title":"\\(K > 1\\) gives \\(\\log K > 0\\); chain with \\Creflog_le_exponents_of_expanding.","kind":"proof","summary":"\\(K > 1\\) gives \\(\\log K > 0\\); chain with \\Creflog_le_exponents_of_expanding.","labels":[],"detail_key":"p40"},{"id":"n33512","layer":"informal","project":"p40","title":"All-positive-spectrum collapse","kind":"proposition","summary":"[All-positive-spectrum collapse] For a uniformly expanding map the positive-part exponent sum i…","labels":["sumPosExp_eq_sumAllExp_of_expanding"],"detail_key":"p40"},{"id":"n33513","layer":"informal","project":"p40","title":"By \\Crefexponents_pos_of_expanding the filter \\(\\i \\mid 0 < \\lambda_i\\\\) is all of the in…","kind":"proof","summary":"By \\Crefexponents_pos_of_expanding the filter \\(\\i \\mid 0 < \\lambda_i\\\\) is all of the index se…","labels":[],"detail_key":"p40"},{"id":"n33514","layer":"informal","project":"p40","title":"The expanding-case right-hand-side identity","kind":"theorem","summary":"[The expanding-case right-hand-side identity] For an ergodic, log-integrable, differentiable un…","labels":["sumPosExp_eq_integral_log_abs_det_of_expanding"],"detail_key":"p40"},{"id":"n33515","layer":"informal","project":"p40","title":"Since all exponents are positive, \\(\\sum\\lambda^+ = \\sum\\lambda\\) (\\CrefsumPosExp_eq_sumA…","kind":"proof","summary":"Since all exponents are positive, \\(\\sum\\lambda^+ = \\sum\\lambda\\) (\\CrefsumPosExp_eq_sumAllExp_…","labels":[],"detail_key":"p40"},{"id":"n33516","layer":"informal","project":"p40","title":"Injectivity partition","kind":"definition","summary":"[Injectivity partition] For a self-map \\(T\\) of \\(EuclideanSpace\\,R\\,(Find)\\) and a finite meas…","labels":["IsInjectivityPartition"],"detail_key":"p40"},{"id":"n33517","layer":"informal","project":"p40","title":"Conditional entropy equals the Jacobian integral","kind":"theorem","summary":"[Conditional entropy equals the Jacobian integral] Let \\(\\mu\\) be an invariant probability meas…","labels":["condEntropy_comap_eq_integral_log_abs_det"],"detail_key":"p40"},{"id":"n33518","layer":"informal","project":"p40","title":"Per cell \\(\\xi_i\\), the change-of-variables crux (\\textttmeasure\\_cell\\_inter\\_preimage\\_…","kind":"proof","summary":"Per cell \\(\\xi_i\\), the change-of-variables crux (\\textttmeasure\\_cell\\_inter\\_preimage\\_eq\\_se…","labels":[],"detail_key":"p40"},{"id":"n33519","layer":"informal","project":"p40","title":"Expanding-map Pesin formula (vacuous on \\(R^d\\), disclosed)","kind":"theorem","summary":"[Expanding-map Pesin formula (vacuous on \\(R^d\\), disclosed)] For an ergodic, absolutely contin…","labels":["pesin_formula_expanding"],"detail_key":"p40"},{"id":"n33520","layer":"informal","project":"p40","title":"Compose three theorems: the Kolmogorov--Sinai generator theorem \\(h_\\mu(T) = h_\\mu(T,\\xi)…","kind":"proof","summary":"Compose three theorems: the Kolmogorov--Sinai generator theorem \\(h_\\mu(T) = h_\\mu(T,\\xi)\\) for…","labels":[],"detail_key":"p40"},{"id":"n33521","layer":"informal","project":"p40","title":"Doubling map: top exponent \\(\\log 2\\)","kind":"theorem","summary":"[Doubling map: top exponent \\(\\log 2\\)] The constant cocycle with generator \\(M = (2)\\) over th…","labels":["doublingMap_topExponent_eq_log_two"],"detail_key":"p40"},{"id":"n33522","layer":"informal","project":"p40","title":"For a constant cocycle with symmetric invertible generator \\(M\\), the sorted Lyapunov spe…","kind":"proof","summary":"For a constant cocycle with symmetric invertible generator \\(M\\), the sorted Lyapunov spectrum…","labels":[],"detail_key":"p40"},{"id":"n33523","layer":"informal","project":"p40","title":"Doubling map: positive-exponent sum \\(\\log 2\\)","kind":"corollary","summary":"[Doubling map: positive-exponent sum \\(\\log 2\\)] The sum of the strictly positive Lyapunov expo…","labels":["doublingMap_sumPosExp_eq_log_two"],"detail_key":"p40"},{"id":"n33524","layer":"informal","project":"p40","title":"The spectrum consists of the single exponent \\(\\log 2 > 0\\), so the positive-part filter…","kind":"proof","summary":"The spectrum consists of the single exponent \\(\\log 2 > 0\\), so the positive-part filter is the…","labels":[],"detail_key":"p40"},{"id":"n33525","layer":"informal","project":"p40","title":"Per-partition Ruelle bound for the doubling map","kind":"theorem","summary":"[Per-partition Ruelle bound for the doubling map] For any finite measurable partition \\(P\\) of…","labels":["doublingMap_ksEntropyPartition_le_sumPosExp"],"detail_key":"p40"},{"id":"n33526","layer":"informal","project":"p40","title":"Specialize the abstract atom-count-growth entropy bound (the arithmetic backbone of the p…","kind":"proof","summary":"Specialize the abstract atom-count-growth entropy bound (the arithmetic backbone of the per-par…","labels":[],"detail_key":"p40"},{"id":"n33527","layer":"informal","project":"p40","title":"Entropy of a uniform-join system","kind":"proposition","summary":"[Entropy of a uniform-join system] Let \\(T\\) preserve a probability measure and let \\(P\\) be a…","labels":["ksEntropyPartition_of_uniform"],"detail_key":"p40"},{"id":"n33528","layer":"informal","project":"p40","title":"The \\(n\\)-fold join is indexed by the \\(b^n\\) formal cell-tuples, each of measure \\(b^-n\\…","kind":"proof","summary":"The \\(n\\)-fold join is indexed by the \\(b^n\\) formal cell-tuples, each of measure \\(b^-n\\), so…","labels":[],"detail_key":"p40"},{"id":"n33529","layer":"informal","project":"p40","title":"Rokhlin equality, entropy side: \\(h(\\alpha,T) = \\log 2\\)","kind":"theorem","summary":"[Rokhlin equality, entropy side: \\(h(\\alpha,T) = \\log 2\\)] For the binary partition \\(\\alpha =…","labels":["ksEntropyPartition_doublingMap_eq_log_two"],"detail_key":"p40"},{"id":"n33530","layer":"informal","project":"p40","title":"The dynamical crux (\\textttvolume\\_binJoinCell) shows every cell of the \\(n\\)-fold join i…","kind":"proof","summary":"The dynamical crux (\\textttvolume\\_binJoinCell) shows every cell of the \\(n\\)-fold join is a dy…","labels":[],"detail_key":"p40"},{"id":"n33531","layer":"informal","project":"p40","title":"Rokhlin equality on the doubling map","kind":"theorem","summary":"[Rokhlin equality on the doubling map] For the doubling map and the binary partition, \\[ h(\\alp…","labels":["rokhlin_equality_doublingMap"],"detail_key":"p40"},{"id":"n33532","layer":"informal","project":"p40","title":"The entropy side is \\CrefksEntropyPartition_doublingMap_eq_log_two. For the integral side…","kind":"proof","summary":"The entropy side is \\CrefksEntropyPartition_doublingMap_eq_log_two. For the integral side, \\(\\d…","labels":[],"detail_key":"p40"},{"id":"n33533","layer":"informal","project":"p40","title":"Cat-map matrix: closed-form Lyapunov spectrum","kind":"theorem","summary":"[Cat-map matrix: closed-form Lyapunov spectrum] Realized as a constant cocycle with generator t…","labels":["catMapMatrix_exponents"],"detail_key":"p40"},{"id":"n33534","layer":"informal","project":"p40","title":"\\(M\\) is symmetric positive definite with trace \\(3\\) and determinant \\(1\\), so its sorte…","kind":"proof","summary":"\\(M\\) is symmetric positive definite with trace \\(3\\) and determinant \\(1\\), so its sorted eige…","labels":[],"detail_key":"p40"},{"id":"n33535","layer":"informal","project":"p40","title":"The cat-map exponents sum to zero","kind":"corollary","summary":"[The cat-map exponents sum to zero] \\(\\lambda_1 + \\lambda_2 = 0\\): the cocycle is conservative…","labels":["catMapMatrix_exponents_sum_eq_zero"],"detail_key":"p40"},{"id":"n33536","layer":"informal","project":"p40","title":"\\(\\log\\lambda_+ + \\log\\lambda_- = \\log(\\lambda_+\\lambda_-) = \\log 1 = 0\\), computing \\(\\l…","kind":"proof","summary":"\\(\\log\\lambda_+ + \\log\\lambda_- = \\log(\\lambda_+\\lambda_-) = \\log 1 = 0\\), computing \\(\\lambda_…","labels":[],"detail_key":"p40"},{"id":"n33537","layer":"informal","project":"p40","title":"The cat-map toral automorphism","kind":"definition","summary":"[The cat-map toral automorphism] The map \\(catTorus:T^2\\toT^2\\), \\((catTorus y)_i = \\sum_j M_ij…","labels":["catTorus"],"detail_key":"p40"},{"id":"n33538","layer":"informal","project":"p40","title":"Measure preservation","kind":"proposition","summary":"[Measure preservation] \\(catTorus\\) preserves the Haar probability measure on \\(T^2\\).","labels":["measurePreserving_catTorus"],"detail_key":"p40"},{"id":"n33539","layer":"informal","project":"p40","title":"\\(catTorus\\) is a continuous surjective additive homomorphism of a compact group; the pus…","kind":"proof","summary":"\\(catTorus\\) is a continuous surjective additive homomorphism of a compact group; the pushforwa…","labels":[],"detail_key":"p40"},{"id":"n33540","layer":"informal","project":"p40","title":"Ergodicity of the Arnold cat map","kind":"theorem","summary":"[Ergodicity of the Arnold cat map] \\(catTorus\\) is ergodic for the Haar probability measure on…","labels":["ergodic_catTorus"],"detail_key":"p40"},{"id":"n33541","layer":"informal","project":"p40","title":"The classical Fourier / character argument. The Koopman operator permutes the multivariat…","kind":"proof","summary":"The classical Fourier / character argument. The Koopman operator permutes the multivariate char…","labels":[],"detail_key":"p40"},{"id":"n33542","layer":"informal","project":"p40","title":"Cat-map spectrum over the genuine ergodic base","kind":"theorem","summary":"[Cat-map spectrum over the genuine ergodic base] Realized as a constant cocycle with generator…","labels":["catTorus_constCocycle_exponents"],"detail_key":"p40"},{"id":"n33543","layer":"informal","project":"p40","title":"The constant-cocycle spectrum theorem applies over any ergodic base; instantiate it over…","kind":"proof","summary":"The constant-cocycle spectrum theorem applies over any ergodic base; instantiate it over \\(catT…","labels":[],"detail_key":"p40"},{"id":"n33544","layer":"informal","project":"p40","title":"The cat map's derivative cocycle has positive top exponent","kind":"theorem","summary":"[The cat map's derivative cocycle has positive top exponent] Let \\(catLift:R^2\\toR^2\\) ( ) be t…","labels":["catLift_derivativeCocycle_topExponent_pos"],"detail_key":"p40"},{"id":"n33545","layer":"informal","project":"p40","title":"The Fr\\'echet derivative of a continuous linear map is the map itself, so \\(derivativeCoc…","kind":"proof","summary":"The Fr\\'echet derivative of a continuous linear map is the map itself, so \\(derivativeCocycle\\,…","labels":[],"detail_key":"p40"},{"id":"n33546","layer":"informal","project":"p40","title":"Per-partition Ruelle bound for the cat map","kind":"theorem","summary":"[Per-partition Ruelle bound for the cat map] For any finite measurable partition \\(P\\) of \\(T^2…","labels":["catTorus_ksEntropyPartition_le_logLambda"],"detail_key":"p40"},{"id":"n33547","layer":"informal","project":"p40","title":"A thin specialization of the abstract atom-count-growth entropy bound (the arithmetic bac…","kind":"proof","summary":"A thin specialization of the abstract atom-count-growth entropy bound (the arithmetic backbone…","labels":[],"detail_key":"p40"},{"id":"n33548","layer":"informal","project":"p40","title":"\\(L^2\\) correlation decay","kind":"theorem","summary":"[\\(L^2\\) correlation decay] Let \\(U_k v = v\\circcatTorus^[k]\\) be the Koopman operator (an \\(L^…","labels":["tendsto_catCorr"],"detail_key":"p40"},{"id":"n33549","layer":"informal","project":"p40","title":"ErgodicTheory.CatMapToral.eventually_pow_mulVec_ne","kind":"proof","summary":"The Koopman operator sends the character \\(\\textttmFourier\\,n\\) to \\(\\textttmFourier(M^kn)\\) (…","labels":[],"detail_key":"p40"},{"id":"n33550","layer":"informal","project":"p40","title":"Strong mixing of the Arnold cat map","kind":"theorem","summary":"[Strong mixing of the Arnold cat map] For arbitrary measurable sets \\(A,B\\subseteqT^2\\), \\[ vol…","labels":["catTorus_mixing"],"detail_key":"p40"},{"id":"n33551","layer":"informal","project":"p40","title":"Feed the indicator functions \\(u = 1_A\\) and \\(v = 1_B\\) to \\Creftendsto_catCorr: the mea…","kind":"proof","summary":"Feed the indicator functions \\(u = 1_A\\) and \\(v = 1_B\\) to \\Creftendsto_catCorr: the means are…","labels":[],"detail_key":"p40"},{"id":"n33552","layer":"informal","project":"p40","title":"Mixing kills unimodular eigenvalues","kind":"lemma","summary":"[Mixing kills unimodular eigenvalues] A reusable spectral interface: for a strongly mixing meas…","labels":["eigenfunction_ae_zero_of_mixing"],"detail_key":"p40"},{"id":"n33553","layer":"informal","project":"p40","title":"If \\(g\\) were not a.e.\\ zero, the eigen-equation \\(g\\circ f^[n] = l^n g\\) forces the corr…","kind":"proof","summary":"If \\(g\\) were not a.e.\\ zero, the eigen-equation \\(g\\circ f^[n] = l^n g\\) forces the correlatio…","labels":[],"detail_key":"p40"},{"id":"n33554","layer":"informal","project":"p40","title":"Eigenfunction rigidity from mixing","kind":"corollary","summary":"[Eigenfunction rigidity from mixing] A measurable \\(g:T^2\\to C\\) with \\(g(catTorus x) = l\\cdot…","labels":["catTorus_eigenfunction_ae_zero_of_mixing"],"detail_key":"p40"},{"id":"n33555","layer":"informal","project":"p40","title":"Discharge \\Crefeigenfunction_ae_zero_of_mixing with the strong mixing of the cat map (\\Cr…","kind":"proof","summary":"Discharge \\Crefeigenfunction_ae_zero_of_mixing with the strong mixing of the cat map (\\CrefcatT…","labels":[],"detail_key":"p40"},{"id":"n33556","layer":"informal","project":"p40","title":"Cat-map eigenfunction rigidity","kind":"theorem","summary":"[Cat-map eigenfunction rigidity] A measurable \\(g : T^2 \\to C\\) with \\(g(catTorus x) = l\\cdot g…","labels":["thm:catTorus-eigZero"],"detail_key":"p40"},{"id":"n33557","layer":"informal","project":"p40","title":"Since \\(\\left\\lVert l \\right\\rVert = 1\\), the modulus \\(\\left\\lVert g \\right\\rVert\\) is \\…","kind":"proof","summary":"Since \\(\\left\\lVert l \\right\\rVert = 1\\), the modulus \\(\\left\\lVert g \\right\\rVert\\) is \\(catTo…","labels":[],"detail_key":"p40"},{"id":"n33558","layer":"informal","project":"p40","title":"Time-one ergodicity of the irrational-roof cat suspension","kind":"theorem","summary":"[Time-one ergodicity of the irrational-roof cat suspension] For the Arnold cat map with Haar \\(…","labels":["thm:catSusp-timeOne"],"detail_key":"p40"},{"id":"n33559","layer":"informal","project":"p40","title":"Discharge the abstract \\Crefthm:susp-timeOneErgodic with base ergodicity (\\Crefergodic_ca…","kind":"proof","summary":"Discharge the abstract \\Crefthm:susp-timeOneErgodic with base ergodicity (\\Crefergodic_catTorus…","labels":[],"detail_key":"p40"},{"id":"n33560","layer":"informal","project":"p40","title":"Concrete non-vacuity witness r = \\sqrt2","kind":"theorem","summary":"[Concrete non-vacuity witness r = \\sqrt2] The irrational roof \\(r := \\sqrt2\\) yields an ergodic…","labels":["thm:catSusp-sqrt2"],"detail_key":"p40"},{"id":"n33561","layer":"informal","project":"p40","title":"Instantiate \\Crefthm:catSusp-timeOne at \\(r = \\sqrt2\\), positive and irrational.","kind":"proof","summary":"Instantiate \\Crefthm:catSusp-timeOne at \\(r = \\sqrt2\\), positive and irrational.","labels":[],"detail_key":"p40"},{"id":"n33562","layer":"informal","project":"p40","title":"Positive-measure atom count","kind":"definition","summary":"[Positive-measure atom count] The number of index families \\(f : Fin n \\to \\iota\\) whose atom o…","labels":["posAtomCount"],"detail_key":"p40"},{"id":"n33563","layer":"informal","project":"p40","title":"The wall lemma","kind":"theorem","summary":"[The wall lemma] Every atom of the \\(n\\)-fold forward join of the \\(5 \\times 5\\) grid partition…","labels":["catTorus_gridJoinAtom_volume_le"],"detail_key":"p40"},{"id":"n33564","layer":"informal","project":"p40","title":"Entropy lower bound","kind":"theorem","summary":"[Entropy lower bound] \\(\\log\\!\\big((3+\\sqrt5)/2\\big) \\le h(catTorus)\\).","labels":["catTorus_ksEntropy_ge"],"detail_key":"p40"},{"id":"n33565","layer":"informal","project":"p40","title":"The wall lemma \\CrefcatTorus_gridJoinAtom_volume_le caps each atom's measure by \\(c\\lambd…","kind":"proof","summary":"The wall lemma \\CrefcatTorus_gridJoinAtom_volume_le caps each atom's measure by \\(c\\lambda\\mu^n…","labels":[],"detail_key":"p40"},{"id":"n33566","layer":"informal","project":"p40","title":"Strict positivity of the cat-map entropy (Tier 1, \\#52)","kind":"corollary","summary":"[Strict positivity of the cat-map entropy (Tier 1, \\#52)] \\(0 < h(catTorus)\\).","labels":["catTorus_ksEntropy_pos"],"detail_key":"p40"},{"id":"n33567","layer":"informal","project":"p40","title":"The Adler--Weiss Markov partition","kind":"definition","summary":"[The Adler--Weiss Markov partition] The golden two-box partition of \\(T^2\\) into the projected…","labels":["catAWPartition"],"detail_key":"p40"},{"id":"n33568","layer":"informal","project":"p40","title":"Exact cover: the junk cell is empty","kind":"lemma","summary":"[Exact cover: the junk cell is empty] The junk cell of \\CrefcatAWPartition is \\emphliterally em…","labels":["awCell_zero_eq_empty"],"detail_key":"p40"},{"id":"n33569","layer":"informal","project":"p40","title":"Blackwell bridge: separating itineraries generate","kind":"theorem","summary":"[Blackwell bridge: separating itineraries generate] For a measurable automorphism of a \\emphsta…","labels":["isGeneratingTwoSided_of_separating"],"detail_key":"p40"},{"id":"n33570","layer":"informal","project":"p40","title":"The Adler--Weiss partition is a two-sided generator","kind":"theorem","summary":"[The Adler--Weiss partition is a two-sided generator] \\(catAWPartition\\) is two-sided generatin…","labels":["isGeneratingTwoSided_catAWPartition"],"detail_key":"p40"},{"id":"n33571","layer":"informal","project":"p40","title":"Golden transfer-matrix entropy bound","kind":"theorem","summary":"[Golden transfer-matrix entropy bound] \\(h(catTorus, catAWPartition) \\le \\log\\!\\big((3+\\sqrt5)/…","labels":["catAW_ksEntropyPartition_le"],"detail_key":"p40"},{"id":"n33572","layer":"informal","project":"p40","title":"Entropy upper bound","kind":"theorem","summary":"[Entropy upper bound] \\(h(catTorus) \\le \\log\\!\\big((3+\\sqrt5)/2\\big)\\).","labels":["catTorus_ksEntropy_le"],"detail_key":"p40"},{"id":"n33573","layer":"informal","project":"p40","title":"Since \\(catAWPartition\\) is a two-sided generator (\\CrefisGeneratingTwoSided_catAWPartiti…","kind":"proof","summary":"Since \\(catAWPartition\\) is a two-sided generator (\\CrefisGeneratingTwoSided_catAWPartition), t…","labels":[],"detail_key":"p40"},{"id":"n33574","layer":"informal","project":"p40","title":"The sharp cat-map Kolmogorov--Sinai entropy","kind":"theorem","summary":"[The sharp cat-map Kolmogorov--Sinai entropy] \\(h(catTorus) = \\log\\!\\big((3+\\sqrt5)/2\\big) = \\l…","labels":["catTorus_ksEntropy_eq"],"detail_key":"p40"},{"id":"n33575","layer":"informal","project":"p40","title":"\\textttle\\_antisymm of the Adler--Weiss generator upper bound \\CrefcatTorus_ksEntropy_le…","kind":"proof","summary":"\\textttle\\_antisymm of the Adler--Weiss generator upper bound \\CrefcatTorus_ksEntropy_le and th…","labels":[],"detail_key":"p40"},{"id":"n33576","layer":"informal","project":"p40","title":"The Fourier-decay class \\( C_s\\)","kind":"definition","summary":"[The Fourier-decay class \\( C_s\\)] Writing \\(\\langle n\\rangle = \\max\\1, \\left\\lvert n_0 \\right\\…","labels":["FourierDecay"],"detail_key":"p40"},{"id":"n33577","layer":"informal","project":"p40","title":"Lattice-sum tail estimate","kind":"lemma","summary":"[Lattice-sum tail estimate] For \\(2 < s\\) the family \\(n\\mapsto\\langle n\\rangle^-s\\) is summabl…","labels":["summable_bracket_rpow"],"detail_key":"p40"},{"id":"n33578","layer":"informal","project":"p40","title":"Split the exponent and dominate the sup norm by a product of one-dimensional factors, red…","kind":"proof","summary":"Split the exponent and dominate the sup norm by a product of one-dimensional factors, reducing…","labels":[],"detail_key":"p40"},{"id":"n33579","layer":"informal","project":"p40","title":"The invariant integer norm form","kind":"definition","summary":"[The invariant integer norm form] The \\emphnorm form \\(Q(p,q) = p^2 - p q - q^2\\) of the cat-ma…","labels":["Qform"],"detail_key":"p40"},{"id":"n33580","layer":"informal","project":"p40","title":"The Diophantine growth bound","kind":"theorem","summary":"[The Diophantine growth bound] Factoring \\(Q\\) over the eigen-line coordinates \\(a_\\pm\\), with…","labels":["lemma_beta"],"detail_key":"p40"},{"id":"n33581","layer":"informal","project":"p40","title":"Write \\(\\left\\lvert Q(n) \\right\\rvert = \\left\\lvert a_+(A^k n) \\right\\rvert\\cdot\\left\\lve…","kind":"proof","summary":"Write \\(\\left\\lvert Q(n) \\right\\rvert = \\left\\lvert a_+(A^k n) \\right\\rvert\\cdot\\left\\lvert a_-…","labels":[],"detail_key":"p40"},{"id":"n33582","layer":"informal","project":"p40","title":"Parseval character expansion of the correlation","kind":"theorem","summary":"[Parseval character expansion of the correlation] For continuous \\(f, g:T^2\\toC\\), Parseval tur…","labels":["hasSum_correlation_fourier_ne_zero"],"detail_key":"p40"},{"id":"n33583","layer":"informal","project":"p40","title":"ErgodicTheory.CatMapToral.mFourierCoeff_comp_iterate","kind":"proof","summary":"Parseval's identity for the multivariate Fourier basis expands the \\(L^2\\) inner product as the…","labels":[],"detail_key":"p40"},{"id":"n33584","layer":"informal","project":"p40","title":"Exponential decay of correlations","kind":"theorem","summary":"[Exponential decay of correlations] For \\(f, g\\in C_s\\) (\\(s > 2\\)) there is a constant \\(C\\ge…","labels":["catCorr_decay"],"detail_key":"p40"},{"id":"n33585","layer":"informal","project":"p40","title":"Split the centred character sum \\CrefhasSum_correlation_fourier_ne_zero at radius \\(\\lang…","kind":"proof","summary":"Split the centred character sum \\CrefhasSum_correlation_fourier_ne_zero at radius \\(\\langle b\\r…","labels":[],"detail_key":"p40"},{"id":"n33586","layer":"informal","project":"p40","title":"Autocovariance decay","kind":"theorem","summary":"[Autocovariance decay] For \\(f\\in C(T^2,R)\\) with \\( C_s\\)-complexification (\\(s > 2\\)), \\(\\lef…","labels":["catAutoCorr_decay"],"detail_key":"p40"},{"id":"n33587","layer":"informal","project":"p40","title":"The autocovariance \\(cov[f, f\\circcatTorus^[k]]\\) equals the centred correlation \\(\\int f…","kind":"proof","summary":"The autocovariance \\(cov[f, f\\circcatTorus^[k]]\\) equals the centred correlation \\(\\int f\\,(f\\c…","labels":[],"detail_key":"p40"},{"id":"n33588","layer":"informal","project":"p40","title":"Green--Kubo variance asymptotics","kind":"theorem","summary":"[Green--Kubo variance asymptotics] For \\(f\\in C_s\\) the rescaled Birkhoff variance converges to…","labels":["catGreenKubo_fourierDecay"],"detail_key":"p40"},{"id":"n33589","layer":"informal","project":"p40","title":"A Toeplitz/Ces\\`aro collapse expresses \\(Var(S_n) = 2\\sum_d<n(n-d)\\rho(d) - n\\rho(0)\\); g…","kind":"proof","summary":"A Toeplitz/Ces\\`aro collapse expresses \\(Var(S_n) = 2\\sum_d<n(n-d)\\rho(d) - n\\rho(0)\\); geometr…","labels":[],"detail_key":"p40"},{"id":"n33590","layer":"informal","project":"p40","title":"Finite-sample concentration","kind":"theorem","summary":"[Finite-sample concentration] For \\(f\\in C_s\\) there is a constant \\(B\\ge 0\\) with, for every s…","labels":["catConcentration_fourierDecay"],"detail_key":"p40"},{"id":"n33591","layer":"informal","project":"p40","title":"Chebyshev's inequality against the linear variance bound of \\CrefcatGreenKubo_fourierDeca…","kind":"proof","summary":"Chebyshev's inequality against the linear variance bound of \\CrefcatGreenKubo_fourierDecay.","labels":[],"detail_key":"p40"},{"id":"n33592","layer":"informal","project":"p40","title":"Exponent-estimator rate","kind":"theorem","summary":"[Exponent-estimator rate] Because the derivative cocycle of the cat map is the \\emphconstant hy…","labels":["catExponent_rate"],"detail_key":"p40"},{"id":"n33593","layer":"informal","project":"p40","title":"ErgodicTheory.CatMapToral.tendsto_catExponent_rate","kind":"proof","summary":"Cayley--Hamilton gives \\(\\textttcat_R^\\,n = a_n\\,\\textttcat_R + b_n\\, 1\\) with coefficients \\((…","labels":[],"detail_key":"p40"},{"id":"n33594","layer":"informal","project":"p40","title":"Suspension-flow correlation decay","kind":"theorem","summary":"[Suspension-flow correlation decay] For base observables \\(f, g\\in C_s\\) with \\(g\\) \\emphcentre…","labels":["catSuspensionDecay_fourierDecay"],"detail_key":"p40"},{"id":"n33595","layer":"informal","project":"p40","title":"On the fundamental domain the fibre product factors by Fubini; the flow \\(\\zeta_t[x,s] =…","kind":"proof","summary":"On the fundamental domain the fibre product factors by Fubini; the flow \\(\\zeta_t[x,s] = [x, s+…","labels":[],"detail_key":"p40"},{"id":"n33596","layer":"informal","project":"p40","title":"The two-sided Adler--Weiss itinerary","kind":"definition","summary":"[The two-sided Adler--Weiss itinerary] \\(awSymbFull : T^2\\to(Fin 5)^Z\\), \\(awSymbFull x\\,k = aw…","labels":["awSymbFull"],"detail_key":"p40"},{"id":"n33597","layer":"informal","project":"p40","title":"The Adler--Weiss coding is an exact factor map","kind":"theorem","summary":"[The Adler--Weiss coding is an exact factor map] \\(awSymbFull\\) is a measure-theoretic factor m…","labels":["isFactorMap_awSymbFull"],"detail_key":"p40"},{"id":"n33598","layer":"informal","project":"p40","title":"Injectivity and conjugacy onto the range","kind":"theorem","summary":"[Injectivity and conjugacy onto the range] Matching two-sided itineraries force equality of poi…","labels":["injective_awSymbFull"],"detail_key":"p40"},{"id":"n33599","layer":"informal","project":"p40","title":"The coarse Adler--Weiss partition","kind":"definition","summary":"[The coarse Adler--Weiss partition] The two-cell \\(MeasurePartition\\) of \\(T^2\\) into the two p…","labels":["coarseAWPartition"],"detail_key":"p40"},{"id":"n33600","layer":"informal","project":"p40","title":"The coarse entropy ceiling","kind":"theorem","summary":"[The coarse entropy ceiling] \\(h(catTorus, coarseAWPartition)\\le\\log 2\\), strictly below the sy…","labels":["coarseAWPartition_ksEntropy_le"],"detail_key":"p40"},{"id":"n33601","layer":"informal","project":"p40","title":"Strict positivity of the coarse entropy","kind":"theorem","summary":"[Strict positivity of the coarse entropy] \\(0 < h(catTorus, coarseAWPartition)\\); indeed \\(\\log…","labels":["coarseAW_ksEntropyPartition_pos"],"detail_key":"p40"},{"id":"n33602","layer":"informal","project":"p40","title":"The substantive estimate is the fine forward-cylinder volume bound: an admissible word of…","kind":"proof","summary":"The substantive estimate is the fine forward-cylinder volume bound: an admissible word of lengt…","labels":[],"detail_key":"p40"},{"id":"n33603","layer":"informal","project":"p40","title":"Stage 1 is entropy-preserving","kind":"theorem","summary":"[Stage 1 is entropy-preserving] The golden-SFT image system \\(Measure.map\\,awSymbFull\\,vol\\) un…","labels":["ksEntropy_mapAwSymbFull_eq"],"detail_key":"p40"},{"id":"n33604","layer":"informal","project":"p40","title":"Stage 2 is a strict-drop lumping","kind":"theorem","summary":"[Stage 2 is a strict-drop lumping] The \\(1\\)-block source merge \\(mergeSrc : (Fin 5)^Z\\to(Fin 2…","labels":["ksEntropy_mapCoarseSymb_le"],"detail_key":"p40"},{"id":"n33605","layer":"informal","project":"p40","title":"The cat symbolic flow tower","kind":"theorem","summary":"[The cat symbolic flow tower] Instantiating the suspension functor twice at time \\(1\\) yields t…","labels":["catSymbolicFlowTower"],"detail_key":"p40"},{"id":"n33606","layer":"informal","project":"p40","title":"Strict future lies below \\(comapT\\)","kind":"lemma","summary":"[Strict future lies below \\(comapT\\)] For a measure-preserving \\(T\\) and a finite measurable pa…","labels":["strictFuture_le_comap"],"detail_key":"p40"},{"id":"n33607","layer":"informal","project":"p40","title":"This is the generator-free half of the equality \\(\\bigvee_k\\sigma(\\dots)=comapT\\,m_E\\) th…","kind":"proof","summary":"This is the generator-free half of the equality \\(\\bigvee_k\\sigma(\\dots)=comapT\\,m_E\\) that hol…","labels":[],"detail_key":"p40"},{"id":"n33608","layer":"informal","project":"p40","title":"Rokhlin's inequality","kind":"theorem","summary":"[Rokhlin's inequality] Let \\(T\\) be a measure-preserving, differentiable self-map of \\(R^d\\) wi…","labels":["integral_log_abs_det_le_ksEntropy"],"detail_key":"p40"},{"id":"n33609","layer":"informal","project":"p40","title":"Four steps chain. First, the injectivity change-of-variables computation (\\CrefcondEntrop…","kind":"proof","summary":"Four steps chain. First, the injectivity change-of-variables computation (\\CrefcondEntropy_coma…","labels":[],"detail_key":"p40"},{"id":"n33610","layer":"informal","project":"p40","title":"The SRB property, volume case","kind":"definition","summary":"[The SRB property, volume case] A map \\(T\\) preserves an \\emphSRB (volume-case) measure \\(\\mu\\)…","labels":["SRBProperty"],"detail_key":"p40"},{"id":"n33611","layer":"informal","project":"p40","title":"Unstable-Jacobian rate","kind":"definition","summary":"[Unstable-Jacobian rate] For an ergodic differentiable \\(T\\) with nonsingular log-integrable de…","labels":["UnstableJacobianRate"],"detail_key":"p40"},{"id":"n33612","layer":"informal","project":"p40","title":"Nonnegative spectrum collapses the positive-part sum","kind":"lemma","summary":"[Nonnegative spectrum collapses the positive-part sum] If every Lyapunov exponent of the deriva…","labels":["sumPosExp_eq_sumAllExp_of_nonneg"],"detail_key":"p40"},{"id":"n33613","layer":"informal","project":"p40","title":"The two finite sums differ only in the summands with \\(\\lambda_i\\le 0\\); under the hypoth…","kind":"proof","summary":"The two finite sums differ only in the summands with \\(\\lambda_i\\le 0\\); under the hypothesis t…","labels":[],"detail_key":"p40"},{"id":"n33614","layer":"informal","project":"p40","title":"SRB reverse inequality, volume case","kind":"theorem","summary":"[SRB reverse inequality, volume case] Let \\(T\\) be ergodic on \\(R^d\\) with a nonsingular, log-i…","labels":["sumPosExp_le_ksEntropy_of_SRB"],"detail_key":"p40"},{"id":"n33615","layer":"informal","project":"p40","title":"The chain is \\(\\sum\\lambda^+\\overset(1)=\\sum\\lambda\\overset(2)=\\int\\log\\left\\lvert \\det D…","kind":"proof","summary":"The chain is \\(\\sum\\lambda^+\\overset(1)=\\sum\\lambda\\overset(2)=\\int\\log\\left\\lvert \\det D_xT \\r…","labels":[],"detail_key":"p40"},{"id":"n33616","layer":"informal","project":"p40","title":"Pesin's entropy formula, spectral form","kind":"theorem","summary":"[Pesin's entropy formula, spectral form] For an ergodic differentiable self-map \\(T\\) of \\(R^d\\…","labels":["pesin_entropy_formula_spectral"],"detail_key":"p40"},{"id":"n33617","layer":"informal","project":"p40","title":"Antisymmetry of the two inequalities. The \\(\\le\\) direction \\(h_\\mu(T)\\le\\sum\\lambda^+\\)…","kind":"proof","summary":"Antisymmetry of the two inequalities. The \\(\\le\\) direction \\(h_\\mu(T)\\le\\sum\\lambda^+\\) is the…","labels":[],"detail_key":"p40"},{"id":"n33618","layer":"informal","project":"p40","title":"Pesin's entropy formula, integral form","kind":"theorem","summary":"[Pesin's entropy formula, integral form] Under the same hypotheses, with \\(\\chi\\) an unstable-J…","labels":["pesin_entropy_formula"],"detail_key":"p40"},{"id":"n33619","layer":"informal","project":"p40","title":"The bridge \\(\\int\\chi\\,d\\mu=sumPosExp\\) holds because \\(\\chi\\) is \\(\\mu\\)-a.e.\\ equal to…","kind":"proof","summary":"The bridge \\(\\int\\chi\\,d\\mu=sumPosExp\\) holds because \\(\\chi\\) is \\(\\mu\\)-a.e.\\ equal to the co…","labels":[],"detail_key":"p40"},{"id":"n33620","layer":"informal","project":"p40","title":"Per-partition Pesin identity for the doubling map","kind":"theorem","summary":"[Per-partition Pesin identity for the doubling map] For the doubling map and its binary partiti…","labels":["pesin_identity_doublingMap_perPartition"],"detail_key":"p40"},{"id":"n33621","layer":"informal","project":"p40","title":"The entropy side is \\(h_\\mu(T,\\alpha)=\\log 2\\) (\\CrefksEntropyPartition_doublingMap_eq_lo…","kind":"proof","summary":"The entropy side is \\(h_\\mu(T,\\alpha)=\\log 2\\) (\\CrefksEntropyPartition_doublingMap_eq_log_two)…","labels":[],"detail_key":"p40"},{"id":"n33622","layer":"informal","project":"p40","title":"The dyadic-arc family","kind":"definition","summary":"[The dyadic-arc family] The set of all \\(n\\)-fold-join cells of the binary partition under the…","labels":["dyadicArcSet"],"detail_key":"p40"},{"id":"n33623","layer":"informal","project":"p40","title":"Binary digit and dyadic partial sum","kind":"definition","summary":"[Binary digit and dyadic partial sum] The \\(n\\)-th binary digit \\(d_n(y)= 1_T^-n(binCell1)(y)\\i…","labels":["binDigit"],"detail_key":"p40"},{"id":"n33624","layer":"informal","project":"p40","title":"One-step doubling recursion","kind":"lemma","summary":"[One-step doubling recursion] On the representative, the doubling map acts by \\(x\\mapsto 2x\\) o…","labels":["rep_doublingMap"],"detail_key":"p40"},{"id":"n33625","layer":"informal","project":"p40","title":"Case split on \\(y\\inbinCell1\\). On the right half \\(d_0=1\\) and \\(2rep(y)-1\\in[0,1)\\), wh…","kind":"proof","summary":"Case split on \\(y\\inbinCell1\\). On the right half \\(d_0=1\\) and \\(2rep(y)-1\\in[0,1)\\), which is…","labels":[],"detail_key":"p40"},{"id":"n33626","layer":"informal","project":"p40","title":"Dynamical partial-sum recursion","kind":"lemma","summary":"[Dynamical partial-sum recursion] The representative splits into its first \\(N\\) digits plus a…","labels":["rep_eq_binPartialSum_add"],"detail_key":"p40"},{"id":"n33627","layer":"informal","project":"p40","title":"Induction on \\(N\\), feeding the one-step recursion \\Crefrep_doublingMap into the tail (us…","kind":"proof","summary":"Induction on \\(N\\), feeding the one-step recursion \\Crefrep_doublingMap into the tail (using th…","labels":[],"detail_key":"p40"},{"id":"n33628","layer":"informal","project":"p40","title":"Convergence of the binary expansion","kind":"lemma","summary":"[Convergence of the binary expansion] \\(binPartialSumN(y)\\torep(y)\\) as \\(N\\to\\infty\\).","labels":["tendsto_binPartialSum"],"detail_key":"p40"},{"id":"n33629","layer":"informal","project":"p40","title":"The tail remainder \\(rep(T^N y)\\,2^-N\\le 2^-N\\to 0\\) is squeezed to zero, so the partial…","kind":"proof","summary":"The tail remainder \\(rep(T^N y)\\,2^-N\\le 2^-N\\to 0\\) is squeezed to zero, so the partial sums c…","labels":[],"detail_key":"p40"},{"id":"n33630","layer":"informal","project":"p40","title":"Digit sets are dyadic-measurable","kind":"lemma","summary":"[Digit sets are dyadic-measurable] Each digit set \\(T^-n(binCell1)\\) is the finite union of the…","labels":["preimage_iterate_binCell_one_mem"],"detail_key":"p40"},{"id":"n33631","layer":"informal","project":"p40","title":"A point lies in \\(T^-n(binCell1)\\) iff its length-\\((n+1)\\) digit string has last digit \\…","kind":"proof","summary":"A point lies in \\(T^-n(binCell1)\\) iff its length-\\((n+1)\\) digit string has last digit \\(1\\);…","labels":[],"detail_key":"p40"},{"id":"n33632","layer":"informal","project":"p40","title":"The representative is dyadic-measurable","kind":"lemma","summary":"[The representative is dyadic-measurable] \\(rep\\) is \\(generateFrom(dyadicArcSet)\\)-measurable.","labels":["measurable_rep_dyadic"],"detail_key":"p40"},{"id":"n33633","layer":"informal","project":"p40","title":"Each digit function is the indicator of a dyadic-measurable digit set (\\Crefpreimage_iter…","kind":"proof","summary":"Each digit function is the indicator of a dyadic-measurable digit set (\\Crefpreimage_iterate_bi…","labels":[],"detail_key":"p40"},{"id":"n33634","layer":"informal","project":"p40","title":"Dyadic arcs generate the Borel structure","kind":"theorem","summary":"[Dyadic arcs generate the Borel structure] The Borel \\(\\sigma\\)-algebra of the circle lies belo…","labels":["borel_le_generateFrom_dyadicArcs"],"detail_key":"p40"},{"id":"n33635","layer":"informal","project":"p40","title":"Since \\((\\uparrow)\\circrep=id\\) factors the identity through the measurable covering proj…","kind":"proof","summary":"Since \\((\\uparrow)\\circrep=id\\) factors the identity through the measurable covering projection…","labels":[],"detail_key":"p40"},{"id":"n33636","layer":"informal","project":"p40","title":"The binary partition generates","kind":"theorem","summary":"[The binary partition generates] The binary partition is a one-sided generator for the doubling…","labels":["binPartition_isGenerating"],"detail_key":"p40"},{"id":"n33637","layer":"informal","project":"p40","title":"The easy inclusion \\(\\le\\) is immediate from measurability of the cells under the iterate…","kind":"proof","summary":"The easy inclusion \\(\\le\\) is immediate from measurability of the cells under the iterates. For…","labels":[],"detail_key":"p40"},{"id":"n33638","layer":"informal","project":"p40","title":"Entropy of the doubling map is \\(\\log 2\\)","kind":"theorem","summary":"[Entropy of the doubling map is \\(\\log 2\\)] The Kolmogorov--Sinai entropy of the doubling map i…","labels":["ksEntropy_doublingMap_eq_log_two"],"detail_key":"p40"},{"id":"n33639","layer":"informal","project":"p40","title":"The generator theorem (\\CrefksEntropy_eq_ksEntropyPartition_of_generating) collapses the…","kind":"proof","summary":"The generator theorem (\\CrefksEntropy_eq_ksEntropyPartition_of_generating) collapses the system…","labels":[],"detail_key":"p40"},{"id":"n33640","layer":"informal","project":"p40","title":"Doubling map: Pesin's formula, witnessed","kind":"theorem","summary":"[Doubling map: Pesin's formula, witnessed] For the doubling map, \\[ h_\\mu(T) \\;=\\; \\sum\\lambda^…","labels":["pesin_formula_doublingMap"],"detail_key":"p40"},{"id":"n33641","layer":"informal","project":"p40","title":"The system entropy is \\(\\log 2\\) (\\CrefksEntropy_doublingMap_eq_log_two) and the positive…","kind":"proof","summary":"The system entropy is \\(\\log 2\\) (\\CrefksEntropy_doublingMap_eq_log_two) and the positive-expon…","labels":[],"detail_key":"p40"},{"id":"n33642","layer":"informal","project":"p40","title":"Coboundary","kind":"definition","summary":"[Coboundary] An observable \\(\\varphi\\) is a \\emphcoboundary for a bare map \\(T:X\\to X\\) if \\(\\v…","labels":["def:livsic-coboundary"],"detail_key":"p40"},{"id":"n33643","layer":"informal","project":"p40","title":"Hölder coboundary","kind":"definition","summary":"[Hölder coboundary] On a metric space, \\(\\varphi\\) is a \\emphHölder coboundary for \\(T\\) if \\(\\…","labels":["def:livsic-holdercoboundary"],"detail_key":"p40"},{"id":"n33644","layer":"informal","project":"p40","title":"Vanishing periodic sums","kind":"definition","summary":"[Vanishing periodic sums] \\(\\varphi\\) \\emphhas vanishing periodic sums for \\(T\\) if every Birkh…","labels":["def:livsic-vps"],"detail_key":"p40"},{"id":"n33645","layer":"informal","project":"p40","title":"Telescoping identity","kind":"lemma","summary":"[Telescoping identity] If \\(\\varphi\\,x = u(T x) - u\\,x\\) for all \\(x\\), then the Birkhoff sum c…","labels":["lem:livsic-telescope"],"detail_key":"p40"},{"id":"n33646","layer":"informal","project":"p40","title":"Induction on \\(n\\) through \\(\\textttbirkhoffSum\\_succ\\): the successor term \\(\\varphi(T^k…","kind":"proof","summary":"Induction on \\(n\\) through \\(\\textttbirkhoffSum\\_succ\\): the successor term \\(\\varphi(T^k x) =…","labels":[],"detail_key":"p40"},{"id":"n33647","layer":"informal","project":"p40","title":"Trivial direction and its obstruction certificate","kind":"theorem","summary":"[Trivial direction and its obstruction certificate] A coboundary has vanishing periodic sums. C…","labels":["thm:livsic-trivial"],"detail_key":"p40"},{"id":"n33648","layer":"informal","project":"p40","title":"Apply the telescoping identity (\\Creflem:livsic-telescope) at a periodic point: \\(S_n\\var…","kind":"proof","summary":"Apply the telescoping identity (\\Creflem:livsic-telescope) at a periodic point: \\(S_n\\varphi\\,p…","labels":[],"detail_key":"p40"},{"id":"n33649","layer":"informal","project":"p40","title":"Exponential closing property","kind":"definition","summary":"[Exponential closing property] \\(T\\) has the \\emphexponential closing property \\(\\textttExpClos…","labels":["def:livsic-expclosing"],"detail_key":"p40"},{"id":"n33650","layer":"informal","project":"p40","title":"The crux estimate","kind":"lemma","summary":"[The crux estimate] Let \\(\\varphi\\) be \\(r\\)-Hölder with constant \\(C_\\varphi\\), let \\(T\\) sati…","labels":["lem:livsic-crux"],"detail_key":"p40"},{"id":"n33651","layer":"informal","project":"p40","title":"Subtract the (vanishing) periodic Birkhoff sum of the shadowing point \\(p\\) from that of…","kind":"proof","summary":"Subtract the (vanishing) periodic Birkhoff sum of the shadowing point \\(p\\) from that of \\(x\\):…","labels":[],"detail_key":"p40"},{"id":"n33652","layer":"informal","project":"p40","title":"McShane--Whitney Hölder extension","kind":"lemma","summary":"[McShane--Whitney Hölder extension] A real-valued function that is \\(\\textttHolderOnWith\\ C\\ r\\…","labels":["lem:livsic-mcshane"],"detail_key":"p40"},{"id":"n33653","layer":"informal","project":"p40","title":"The McShane infimal-convolution formula \\(v(x) = \\inf_y\\in s\\bigl(u(y) + Cdist(x,y)^r\\big…","kind":"proof","summary":"The McShane infimal-convolution formula \\(v(x) = \\inf_y\\in s\\bigl(u(y) + Cdist(x,y)^r\\bigr)\\).…","labels":[],"detail_key":"p40"},{"id":"n33654","layer":"informal","project":"p40","title":"Abstract existence, Katok--Hasselblatt 19.2.1","kind":"theorem","summary":"[Abstract existence, Katok--Hasselblatt 19.2.1] Let \\(T\\) be continuous on a compact metric spa…","labels":["thm:livsic-existence"],"detail_key":"p40"},{"id":"n33655","layer":"informal","project":"p40","title":"The classical dense-orbit construction. Enumerate the orbit \\(e_n = T^n x_0\\) and set the…","kind":"proof","summary":"The classical dense-orbit construction. Enumerate the orbit \\(e_n = T^n x_0\\) and set the runni…","labels":[],"detail_key":"p40"},{"id":"n33656","layer":"informal","project":"p40","title":"The Livšic equivalence","kind":"theorem","summary":"[The Livšic equivalence] Under the standing hypotheses (continuous \\(T\\) on a compact metric sp…","labels":["thm:livsic-iff"],"detail_key":"p40"},{"id":"n33657","layer":"informal","project":"p40","title":"Forward: the pure telescoping obstruction (\\Crefthm:livsic-trivial). Backward: the dense-…","kind":"proof","summary":"Forward: the pure telescoping obstruction (\\Crefthm:livsic-trivial). Backward: the dense-orbit…","labels":[],"detail_key":"p40"},{"id":"n33658","layer":"informal","project":"p40","title":"Shift metric substrate","kind":"lemma","summary":"[Shift metric substrate] The left shift is \\(2\\)-Lipschitz for the \\(\\textttPiNat\\) ultrametric…","labels":["lem:livsic-shiftmetric"],"detail_key":"p40"},{"id":"n33659","layer":"informal","project":"p40","title":"Dropping the first coordinate moves the first-disagreement index down by one, doubling th…","kind":"proof","summary":"Dropping the first coordinate moves the first-disagreement index down by one, doubling the dist…","labels":[],"detail_key":"p40"},{"id":"n33660","layer":"informal","project":"p40","title":"Closing for the full shift","kind":"theorem","summary":"[Closing for the full shift] For every exponent \\(\\alpha > 0\\) the full shift satisfies \\(\\text…","labels":["thm:livsic-shift-closing"],"detail_key":"p40"},{"id":"n33661","layer":"informal","project":"p40","title":"A point that almost \\(n\\)-returns is shadowed by the front-anchored periodization \\(p_i :…","kind":"proof","summary":"A point that almost \\(n\\)-returns is shadowed by the front-anchored periodization \\(p_i := x_i\\…","labels":[],"detail_key":"p40"},{"id":"n33662","layer":"informal","project":"p40","title":"The rich point","kind":"definition","summary":"[The rich point] The \\emphrich point is the sequence obtained by concatenating all finite words…","labels":["def:livsic-richpoint"],"detail_key":"p40"},{"id":"n33663","layer":"informal","project":"p40","title":"Dense orbit for the full shift","kind":"theorem","summary":"[Dense orbit for the full shift] Over a nonempty finite alphabet the forward orbit of the rich…","labels":["thm:livsic-shift-dense"],"detail_key":"p40"},{"id":"n33664","layer":"informal","project":"p40","title":"To match a target on its first \\(N\\) coordinates, shift the rich point to the block equal…","kind":"proof","summary":"To match a target on its first \\(N\\) coordinates, shift the rich point to the block equal to th…","labels":[],"detail_key":"p40"},{"id":"n33665","layer":"informal","project":"p40","title":"Livšic for the one-sided full shift","kind":"theorem","summary":"[Livšic for the one-sided full shift] For a Hölder \\(\\varphi\\) (exponent \\(0 < r\\le 1\\)) on the…","labels":["thm:livsic-fullshift"],"detail_key":"p40"},{"id":"n33666","layer":"informal","project":"p40","title":"Feed \\Crefthm:livsic-iff the substrate (\\Creflem:livsic-shiftmetric), the closing constan…","kind":"proof","summary":"Feed \\Crefthm:livsic-iff the substrate (\\Creflem:livsic-shiftmetric), the closing constant (\\Cr…","labels":[],"detail_key":"p40"},{"id":"n33667","layer":"informal","project":"p40","title":"The \\(Z\\)-indexed \\(\\theta\\)-ultrametric","kind":"definition","summary":"[The \\(Z\\)-indexed \\(\\theta\\)-ultrametric] On \\(\\textttBiShift\\ \\alpha_0 = (Z\\to\\alpha_0)\\), se…","labels":["def:livsic-bishift-metric"],"detail_key":"p40"},{"id":"n33668","layer":"informal","project":"p40","title":"Two-sided geometric bound","kind":"lemma","summary":"[Two-sided geometric bound] For \\(0\\le\\theta<1\\), \\(\\ \\sum_i<n\\theta^\\min(i,\\,n-i) \\le 2\\,(1-\\t…","labels":["lem:livsic-minregime"],"detail_key":"p40"},{"id":"n33669","layer":"informal","project":"p40","title":"The two-sided profile \\(\\theta^\\min(i,\\,n-i)\\) is dominated termwise by \\(\\theta^i + \\the…","kind":"proof","summary":"The two-sided profile \\(\\theta^\\min(i,\\,n-i)\\) is dominated termwise by \\(\\theta^i + \\theta^\\,n…","labels":[],"detail_key":"p40"},{"id":"n33670","layer":"informal","project":"p40","title":"Closing for the two-sided full shift","kind":"theorem","summary":"[Closing for the two-sided full shift] For every \\(\\alpha>0\\) the two-sided full shift \\(\\tilde…","labels":["thm:livsic-bishift-closing"],"detail_key":"p40"},{"id":"n33671","layer":"informal","project":"p40","title":"The shadow of an almost-\\(n\\)-return is the \\emphcentral periodization \\(p_j := x_j\\bmod…","kind":"proof","summary":"The shadow of an almost-\\(n\\)-return is the \\emphcentral periodization \\(p_j := x_j\\bmod n\\) (i…","labels":[],"detail_key":"p40"},{"id":"n33672","layer":"informal","project":"p40","title":"Livšic for the two-sided full shift","kind":"theorem","summary":"[Livšic for the two-sided full shift] For a Hölder \\(\\varphi\\) (\\(0 < r\\le 1\\)) on the two-side…","labels":["thm:livsic-bishift"],"detail_key":"p40"},{"id":"n33673","layer":"informal","project":"p40","title":"Instantiate \\Crefthm:livsic-iff with the \\(Z\\)-ultrametric, the doubled closing constant,…","kind":"proof","summary":"Instantiate \\Crefthm:livsic-iff with the \\(Z\\)-ultrametric, the doubled closing constant, and t…","labels":[],"detail_key":"p40"},{"id":"n33674","layer":"informal","project":"p40","title":"The SFT carrier and shift","kind":"definition","summary":"[The SFT carrier and shift] For \\(M:Fink\\toFink\\to\\textttBool\\), the \\emphcarrier \\(\\textttSFTC…","labels":["def:livsic-sft"],"detail_key":"p40"},{"id":"n33675","layer":"informal","project":"p40","title":"The \\(\\delta = 1/2\\) crux","kind":"lemma","summary":"[The \\(\\delta = 1/2\\) crux] If \\(x\\) almost \\(n\\)-returns within radius \\(1/2\\), then \\(x_0 = x…","labels":["lem:livsic-sft-half"],"detail_key":"p40"},{"id":"n33676","layer":"informal","project":"p40","title":"\\(dist(x,\\sigma^n x)\\le 1/2\\) forces agreement on coordinate \\(0\\), i.e.\\ \\(x_0 = (\\sigma…","kind":"proof","summary":"\\(dist(x,\\sigma^n x)\\le 1/2\\) forces agreement on coordinate \\(0\\), i.e.\\ \\(x_0 = (\\sigma^n x)_…","labels":[],"detail_key":"p40"},{"id":"n33677","layer":"informal","project":"p40","title":"Closing for the SFT, unconditional in \\(M\\)","kind":"theorem","summary":"[Closing for the SFT, unconditional in \\(M\\)] For every \\(\\alpha>0\\) and \\emphevery transition…","labels":["thm:livsic-sft-closing"],"detail_key":"p40"},{"id":"n33678","layer":"informal","project":"p40","title":"The design finding: at radius \\(1/2\\) the periodization \\(p_i := x_i\\bmod n\\) is admissib…","kind":"proof","summary":"The design finding: at radius \\(1/2\\) the periodization \\(p_i := x_i\\bmod n\\) is admissible for…","labels":[],"detail_key":"p40"},{"id":"n33679","layer":"informal","project":"p40","title":"Safe symbol","kind":"definition","summary":"[Safe symbol] A \\emphsafe symbol for \\(M\\) is a symbol \\(s\\) allowed adjacent to every symbol:…","labels":["def:livsic-safesymbol"],"detail_key":"p40"},{"id":"n33680","layer":"informal","project":"p40","title":"Dense orbit for an SFT with a safe symbol","kind":"theorem","summary":"[Dense orbit for an SFT with a safe symbol] If \\(M\\) has a safe symbol, the SFT shift has a poi…","labels":["thm:livsic-sft-dense"],"detail_key":"p40"},{"id":"n33681","layer":"informal","project":"p40","title":"Mirror the full-shift rich point, but first \\emphsanitize each decoded word (keep it if a…","kind":"proof","summary":"Mirror the full-shift rich point, but first \\emphsanitize each decoded word (keep it if admissi…","labels":[],"detail_key":"p40"},{"id":"n33682","layer":"informal","project":"p40","title":"Livšic for a one-sided SFT","kind":"theorem","summary":"[Livšic for a one-sided SFT] For a Hölder \\(\\varphi\\) (\\(0 < r\\le 1\\)) on the SFT and a dense f…","labels":["thm:livsic-sft"],"detail_key":"p40"},{"id":"n33683","layer":"informal","project":"p40","title":"Instantiate \\Crefthm:livsic-iff with the subtype substrate and the unconditional \\(\\delta…","kind":"proof","summary":"Instantiate \\Crefthm:livsic-iff with the subtype substrate and the unconditional \\(\\delta = 1/2…","labels":[],"detail_key":"p40"},{"id":"n33684","layer":"informal","project":"p40","title":"The golden-mean shift, unconditional","kind":"theorem","summary":"[The golden-mean shift, unconditional] For the golden-mean shift ( : forbid the block \\(11\\), L…","labels":["thm:livsic-goldenmean"],"detail_key":"p40"},{"id":"n33685","layer":"informal","project":"p40","title":"ErgodicTheory.goldenMean_proper","kind":"proof","summary":"The symbol \\(0\\) is safe (it forbids nothing), so \\Crefthm:livsic-sft-dense supplies the dense…","labels":[],"detail_key":"p40"},{"id":"n33686","layer":"informal","project":"p40","title":"Dense orbit from ergodicity","kind":"theorem","summary":"[Dense orbit from ergodicity] On a second-countable space with an open-positive probability mea…","labels":["thm:livsic-ergodic-dense"],"detail_key":"p40"},{"id":"n33687","layer":"informal","project":"p40","title":"For each basic open \\(o\\) the visiting set \\(U_o = \\bigcup_n T^-no\\) is almost invariant,…","kind":"proof","summary":"For each basic open \\(o\\) the visiting set \\(U_o = \\bigcup_n T^-no\\) is almost invariant, hence…","labels":[],"detail_key":"p40"},{"id":"n33688","layer":"informal","project":"p40","title":"Closing for the doubling map","kind":"theorem","summary":"[Closing for the doubling map] For every \\(r>0\\) the doubling map satisfies \\(\\textttExpClosing…","labels":["thm:livsic-doubling-closing"],"detail_key":"p40"},{"id":"n33689","layer":"informal","project":"p40","title":"ErgodicTheory.exists_doubling_periodic_shadow","kind":"proof","summary":"Because \\(T^n = (2^n)\\cdot\\) is a group endomorphism, the periodic shadow of \\(x = \\bar a\\) is…","labels":[],"detail_key":"p40"},{"id":"n33690","layer":"informal","project":"p40","title":"Livšic for the doubling map","kind":"theorem","summary":"[Livšic for the doubling map] For a Hölder \\(\\varphi\\) (\\(0 < r\\le 1\\)) on the circle, \\(\\varph…","labels":["thm:livsic-doubling"],"detail_key":"p40"},{"id":"n33691","layer":"informal","project":"p40","title":"ErgodicTheory.const_one_not_isCoboundary_doublingMap","kind":"proof","summary":"The doubling map is continuous, the circle compact, the closing is \\Crefthm:livsic-doubling-clo…","labels":[],"detail_key":"p40"},{"id":"n33692","layer":"informal","project":"p40","title":"Covering projection","kind":"definition","summary":"[Covering projection] \\(\\textttcatProj:R^2\\toT^2\\) is the coordinatewise quotient projection; i…","labels":["def:livsic-catproj"],"detail_key":"p40"},{"id":"n33693","layer":"informal","project":"p40","title":"Exact shadow solution","kind":"definition","summary":"[Exact shadow solution] For a lifted return defect \\(e\\), \\(\\textttcatShadowSol\\ n\\ e\\) solves…","labels":["def:livsic-catshadow"],"detail_key":"p40"},{"id":"n33694","layer":"informal","project":"p40","title":"Anosov exact-solve closing","kind":"theorem","summary":"[Anosov exact-solve closing] For every exponent \\(0 < \\alpha\\le 1\\), \\(\\textttcatTorus\\) has th…","labels":["thm:livsic-cat-closing"],"detail_key":"p40"},{"id":"n33695","layer":"informal","project":"p40","title":"Lift an almost-return to \\(d = \\textttcat_R^n v - v\\), reduce it to its nearest-integer r…","kind":"proof","summary":"Lift an almost-return to \\(d = \\textttcat_R^n v - v\\), reduce it to its nearest-integer represe…","labels":[],"detail_key":"p40"},{"id":"n33696","layer":"informal","project":"p40","title":"Livšic for the Arnold cat map","kind":"theorem","summary":"[Livšic for the Arnold cat map] For a Hölder \\(\\varphi\\) (\\(0 < r\\le 1\\)) on \\(T^2\\), \\(\\varphi…","labels":["thm:livsic-cat"],"detail_key":"p40"},{"id":"n33697","layer":"informal","project":"p40","title":"ErgodicTheory.CatMapToral.const_one_not_isCoboundary_catTorus","kind":"proof","summary":"\\(\\textttcatTorus\\) is continuous, \\(T^2\\) compact, the closing is \\Crefthm:livsic-cat-closing,…","labels":[],"detail_key":"p40"},{"id":"n33698","layer":"informal","project":"p40","title":"Almost-everywhere coboundary","kind":"definition","summary":"[Almost-everywhere coboundary] \\(\\varphi\\) is an \\empha.e.\\ coboundary of \\(u\\) over \\(\\mu\\) if…","labels":["def:livsic-aecob"],"detail_key":"p40"},{"id":"n33699","layer":"informal","project":"p40","title":"Continuous tier","kind":"theorem","summary":"[Continuous tier] If \\(\\varphi\\) is continuous and equals the coboundary of a \\emphcontinuous \\…","labels":["thm:livsic-continuous-tier"],"detail_key":"p40"},{"id":"n33700","layer":"informal","project":"p40","title":"ErgodicTheory.Livsic.isHolderCoboundary_of_continuous_aeCoboundary","kind":"proof","summary":"Full support forces a.e.-equal continuous functions to be equal, so the a.e.\\ equation upgrades…","labels":[],"detail_key":"p40"},{"id":"n33701","layer":"informal","project":"p40","title":"Bounded tier","kind":"theorem","summary":"[Bounded tier] Let \\(T\\) preserve a probability measure \\(\\mu\\), let \\(p\\) be \\(n\\)-periodic, a…","labels":["thm:livsic-bounded-tier"],"detail_key":"p40"},{"id":"n33702","layer":"informal","project":"p40","title":"Around the periodic orbit \\(S_nm\\varphi\\,p = m\\cdot S_n\\varphi\\,p\\). A.e.\\ telescoping gi…","kind":"proof","summary":"Around the periodic orbit \\(S_nm\\varphi\\,p = m\\cdot S_n\\varphi\\,p\\). A.e.\\ telescoping gives \\(…","labels":[],"detail_key":"p40"},{"id":"n33703","layer":"informal","project":"p40","title":"The natural-extension factor","kind":"lemma","summary":"[The natural-extension factor] The restriction \\(\\texttttoShift:\\textttBiShift\\ \\alpha_0\\to\\tex…","labels":["lem:livsic-toshift"],"detail_key":"p40"},{"id":"n33704","layer":"informal","project":"p40","title":"The factor map is \\(1\\)-Lipschitz for the respective \\(\\theta\\)-ultrametrics and pushes t…","kind":"proof","summary":"The factor map is \\(1\\)-Lipschitz for the respective \\(\\theta\\)-ultrametrics and pushes the two…","labels":[],"detail_key":"p40"},{"id":"n33705","layer":"informal","project":"p40","title":"Product structure","kind":"lemma","summary":"[Product structure] The bilateral shift space splits measurably as \\(\\textpast\\otimes\\textfutur…","labels":["lem:livsic-joinpf"],"detail_key":"p40"},{"id":"n33706","layer":"informal","project":"p40","title":"The past (\\(j<0\\)) and future (\\(j\\ge 0\\)) coordinate blocks are independent under the i.…","kind":"proof","summary":"The past (\\(j<0\\)) and future (\\(j\\ge 0\\)) coordinate blocks are independent under the i.i.d.\\…","labels":[],"detail_key":"p40"},{"id":"n33707","layer":"informal","project":"p40","title":"Stable and unstable essential oscillation","kind":"lemma","summary":"[Stable and unstable essential oscillation] For a measurable a.e.\\ transfer function of an \\(r\\…","labels":["lem:livsic-osc"],"detail_key":"p40"},{"id":"n33708","layer":"informal","project":"p40","title":"Points sharing a future have forward orbits that converge, so the telescoped Birkhoff sum…","kind":"proof","summary":"Points sharing a future have forward orbits that converge, so the telescoped Birkhoff sums of \\…","labels":[],"detail_key":"p40"},{"id":"n33709","layer":"informal","project":"p40","title":"Essential boundedness of the transfer","kind":"theorem","summary":"[Essential boundedness of the transfer] A merely measurable (possibly unbounded) a.e.\\ solution…","labels":["thm:livsic-essbdd"],"detail_key":"p40"},{"id":"n33710","layer":"informal","project":"p40","title":"The Fubini glue: over the \\(\\textpast\\otimes\\textfuture\\) product a typical base point \\(…","kind":"proof","summary":"The Fubini glue: over the \\(\\textpast\\otimes\\textfuture\\) product a typical base point \\((a_0,b…","labels":[],"detail_key":"p40"},{"id":"n33711","layer":"informal","project":"p40","title":"Two-sided measurable headline","kind":"theorem","summary":"[Two-sided measurable headline] Over a fully supported two-sided Bernoulli measure, an \\(r\\)-Hö…","labels":["thm:livsic-measurable-bishift"],"detail_key":"p40"},{"id":"n33712","layer":"informal","project":"p40","title":"With the essential bound \\(M\\) from \\Crefthm:livsic-essbdd, the clamp \\(v = \\max(-M',\\min…","kind":"proof","summary":"With the essential bound \\(M\\) from \\Crefthm:livsic-essbdd, the clamp \\(v = \\max(-M',\\min(M',u)…","labels":[],"detail_key":"p40"},{"id":"n33713","layer":"informal","project":"p40","title":"One-sided measurable headline","kind":"theorem","summary":"[One-sided measurable headline] Over a fully supported one-sided Bernoulli measure, an \\(r\\)-Hö…","labels":["thm:livsic-measurable-oneside"],"detail_key":"p40"},{"id":"n33714","layer":"informal","project":"p40","title":"Transport \\(\\varphi,u\\) up the natural-extension factor (\\Creflem:livsic-toshift); apply…","kind":"proof","summary":"Transport \\(\\varphi,u\\) up the natural-extension factor (\\Creflem:livsic-toshift); apply the tw…","labels":[],"detail_key":"p40"},{"id":"n33715","layer":"informal","project":"p40","title":"Full measurable Livšic rigidity, Katok--Hasselblatt 19.2.4","kind":"theorem","summary":"[Full measurable Livšic rigidity, Katok--Hasselblatt 19.2.4] Over a fully supported Bernoulli m…","labels":["thm:livsic-measurable-full"],"detail_key":"p40"},{"id":"n33716","layer":"informal","project":"p40","title":"ErgodicTheory.Livsic.measurable_aeCoboundary_ae_eq_holder","kind":"proof","summary":"Forward: the one-sided measurable headline (\\Crefthm:livsic-measurable-oneside). Backward: vani…","labels":[],"detail_key":"p40"},{"id":"n33717","layer":"informal","project":"p40","title":"Flow coboundary","kind":"definition","summary":"[Flow coboundary] \\(F\\) is a \\emphflow coboundary for \\(\\Phi\\) if it is the flow-time derivativ…","labels":["def:livsic-flowcob"],"detail_key":"p40"},{"id":"n33718","layer":"informal","project":"p40","title":"Flow periodic-orbit obstruction","kind":"theorem","summary":"[Flow periodic-orbit obstruction] If \\(\\Phi_P q = q\\) and the closed-orbit integral \\(\\int_0^P…","labels":["thm:livsic-flow-obstruction"],"detail_key":"p40"},{"id":"n33719","layer":"informal","project":"p40","title":"The fundamental-theorem-of-calculus telescoping around a closed orbit: a transfer functio…","kind":"proof","summary":"The fundamental-theorem-of-calculus telescoping around a closed orbit: a transfer function woul…","labels":[],"detail_key":"p40"},{"id":"n33720","layer":"informal","project":"p40","title":"Induced base observable","kind":"definition","summary":"[Induced base observable] The \\emphinduced base observable of a flow observable \\(F\\) is the on…","labels":["def:livsic-induced"],"detail_key":"p40"},{"id":"n33721","layer":"informal","project":"p40","title":"A flow coboundary induces a base coboundary","kind":"theorem","summary":"[A flow coboundary induces a base coboundary] If \\(F\\) is a flow coboundary of the suspension f…","labels":["thm:livsic-induced-cob"],"detail_key":"p40"},{"id":"n33722","layer":"informal","project":"p40","title":"Evaluate the flow-coboundary equation at the section point \\([x,0]\\) over one lap \\(t = \\…","kind":"proof","summary":"Evaluate the flow-coboundary equation at the section point \\([x,0]\\) over one lap \\(t = \\tau x\\…","labels":[],"detail_key":"p40"},{"id":"n33723","layer":"informal","project":"p40","title":"Tier-1 suspension obstruction","kind":"theorem","summary":"[Tier-1 suspension obstruction] If \\(p\\) is an \\(n\\)-periodic base point and the periodic Birkh…","labels":["thm:livsic-flow-tier1"],"detail_key":"p40"},{"id":"n33724","layer":"informal","project":"p40","title":"Combine \\Crefthm:livsic-induced-cob with the discrete obstruction certificate of \\Crefthm…","kind":"proof","summary":"Combine \\Crefthm:livsic-induced-cob with the discrete obstruction certificate of \\Crefthm:livsi…","labels":[],"detail_key":"p40"},{"id":"n33725","layer":"informal","project":"p40","title":"The lap bridge","kind":"theorem","summary":"[The lap bridge] The integral of \\(F\\) around one full closed flow orbit above an \\(n\\)-periodi…","labels":["thm:livsic-lap-bridge"],"detail_key":"p40"},{"id":"n33726","layer":"informal","project":"p40","title":"ErgodicTheory.not_isFlowCoboundary_suspensionFlowMap_of_periodicOrbitIntegral_ne_zero","kind":"proof","summary":"The lap decomposition: cut the closed orbit at the successive cross-section returns (times \\(S_…","labels":[],"detail_key":"p40"},{"id":"n33727","layer":"informal","project":"p40","title":"Non-vacuity: the cat-map suspension flow","kind":"theorem","summary":"[Non-vacuity: the cat-map suspension flow] The constant observable \\(1\\) is \\emphnot a flow cob…","labels":["thm:livsic-cat-suspension"],"detail_key":"p40"},{"id":"n33728","layer":"informal","project":"p40","title":"Take the fixed point \\(0\\) of the cat map (base period \\(1\\)): the induced base observabl…","kind":"proof","summary":"Take the fixed point \\(0\\) of the cat map (base period \\(1\\)): the induced base observable summ…","labels":[],"detail_key":"p40"},{"id":"n33729","layer":"informal","project":"p40","title":"Seam-glue generator identity","kind":"lemma","summary":"[Seam-glue generator identity] Fix a constant roof \\(\\tau\\equiv c\\), a base transfer function \\…","labels":["uCover_gen"],"detail_key":"p40"},{"id":"n33730","layer":"informal","project":"p40","title":"ErgodicTheory.uCover_act","kind":"proof","summary":"The base cohomological equation \\(\\textttinducedBaseCocycle\\,F\\,x = u_0(T x) - u_0(x)\\) supplie…","labels":[],"detail_key":"p40"},{"id":"n33731","layer":"informal","project":"p40","title":"Tier-III equivalence for constant-roof suspension flows","kind":"theorem","summary":"[Tier-III equivalence for constant-roof suspension flows] Let \\(\\Phi = \\zeta\\) be the suspensio…","labels":["livsic_suspensionFlow_constRoof"],"detail_key":"p40"},{"id":"n33732","layer":"informal","project":"p40","title":"ErgodicTheory.suspTransfer","kind":"proof","summary":"Forward is the general obstruction: a flow coboundary induces a base coboundary (\\Crefthm:livsi…","labels":[],"detail_key":"p40"},{"id":"n33733","layer":"informal","project":"p40","title":"Flow-native form","kind":"theorem","summary":"[Flow-native form] Under the same hypotheses, the obstruction may be phrased directly in flow-n…","labels":["livsic_suspensionFlow_constRoof_orbitIntegral"],"detail_key":"p40"},{"id":"n33734","layer":"informal","project":"p40","title":"Chain \\Creflivsic_suspensionFlow_constRoof through the lap-decomposition bridge (\\Crefthm…","kind":"proof","summary":"Chain \\Creflivsic_suspensionFlow_constRoof through the lap-decomposition bridge (\\Crefthm:livsi…","labels":[],"detail_key":"p40"},{"id":"n33735","layer":"informal","project":"p40","title":"Flow-Livšic for the cat-map suspension flow","kind":"theorem","summary":"[Flow-Livšic for the cat-map suspension flow] For the suspension flow of \\(catTorus\\) under the…","labels":["livsic_catSuspensionFlow"],"detail_key":"p40"},{"id":"n33736","layer":"informal","project":"p40","title":"Instantiate \\Creflivsic_suspensionFlow_constRoof at \\(c = 1\\), discharging the base conve…","kind":"proof","summary":"Instantiate \\Creflivsic_suspensionFlow_constRoof at \\(c = 1\\), discharging the base converse wi…","labels":[],"detail_key":"p40"},{"id":"n33737","layer":"informal","project":"p40","title":"Non-vacuity of the coboundary side","kind":"theorem","summary":"[Non-vacuity of the coboundary side] A \\(1\\)-periodic fibre profile \\(h\\) descends to a flow ob…","labels":["isFlowCoboundary_sinFibreObservable"],"detail_key":"p40"},{"id":"n33738","layer":"informal","project":"p40","title":"The one-lap integral \\(\\int_0^1\\sin(2\\pi s)\\,ds = 0\\) makes \\(\\textttinducedBaseCocycle\\,…","kind":"proof","summary":"The one-lap integral \\(\\int_0^1\\sin(2\\pi s)\\,ds = 0\\) makes \\(\\textttinducedBaseCocycle\\,F\\) id…","labels":[],"detail_key":"p40"},{"id":"n33739","layer":"informal","project":"p40","title":"Hölder flow coboundary","kind":"definition","summary":"[Hölder flow coboundary] On the Bowen--Walters metric space \\(SuspensionSpace T\\,\\tau\\) (consta…","labels":["IsHolderFlowCoboundary"],"detail_key":"p40"},{"id":"n33740","layer":"informal","project":"p40","title":"The induced base observable is Hölder","kind":"lemma","summary":"[The induced base observable is Hölder] If \\(F\\) is \\(embDist\\)-\\(r\\)-Hölder and \\(T\\) is Lipsc…","labels":["holderWith_inducedBaseCocycle"],"detail_key":"p40"},{"id":"n33741","layer":"informal","project":"p40","title":"The one-lap integral \\(\\int_0^1 F[x, \\sigma]\\,d\\sigma\\) inherits the embedding-Hölder mod…","kind":"proof","summary":"The one-lap integral \\(\\int_0^1 F[x, \\sigma]\\,d\\sigma\\) inherits the embedding-Hölder modulus o…","labels":[],"detail_key":"p40"},{"id":"n33742","layer":"informal","project":"p40","title":"Cross-seam Hölder gluing","kind":"theorem","summary":"[Cross-seam Hölder gluing] For an \\(embDist\\)-\\(r\\)-Hölder, bounded \\(F\\) and a base transfer f…","labels":["holderWith_suspTransfer"],"detail_key":"p40"},{"id":"n33743","layer":"informal","project":"p40","title":"Following the exact-seam philosophy of \\CrefuCover_gen: on the fundamental strip the fibr…","kind":"proof","summary":"Following the exact-seam philosophy of \\CrefuCover_gen: on the fundamental strip the fibre inte…","labels":[],"detail_key":"p40"},{"id":"n33744","layer":"informal","project":"p40","title":"Hölder flow-Livšic, constant roof","kind":"theorem","summary":"[Hölder flow-Livšic, constant roof] Let \\(T\\) be a Lipschitz homeomorphism of a compact metric…","labels":["livsic_holderFlow_constRoof"],"detail_key":"p40"},{"id":"n33745","layer":"informal","project":"p40","title":"Forward: a Hölder flow coboundary is a flow coboundary, which induces a base coboundary (…","kind":"proof","summary":"Forward: a Hölder flow coboundary is a flow coboundary, which induces a base coboundary (\\Creft…","labels":[],"detail_key":"p40"},{"id":"n33746","layer":"informal","project":"p40","title":"Hölder flow-Livšic, variable Lipschitz roof","kind":"theorem","summary":"[Hölder flow-Livšic, variable Lipschitz roof] The equivalence extends to a \\emphvariable roof \\…","labels":["livsic_holderFlow_varRoof"],"detail_key":"p40"},{"id":"n33747","layer":"informal","project":"p40","title":"ErgodicTheory.uCover_gen_var","kind":"proof","summary":"Repeat the constant-roof argument on the normalized fibre coordinate. The metric compares point…","labels":[],"detail_key":"p40"},{"id":"n33748","layer":"informal","project":"p40","title":"Hölder flow-Livšic for the cat-map suspension flow","kind":"theorem","summary":"[Hölder flow-Livšic for the cat-map suspension flow] For the suspension flow of \\(catTorus\\) un…","labels":["livsic_catSuspensionHolderFlow"],"detail_key":"p40"},{"id":"n33749","layer":"informal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_dist_le_one","kind":"proof","summary":"Instantiate \\Creflivsic_holderFlow_constRoof at the cat map: the diameter bound \\(diamT^2\\le 1\\…","labels":[],"detail_key":"p40"},{"id":"n33750","layer":"informal","project":"p40","title":"Non-vacuity on both sides","kind":"theorem","summary":"[Non-vacuity on both sides] The equivalence has content on both sides. The \\(\\sin(2\\pi\\cdot)\\)…","labels":["isHolderFlowCoboundary_sinFibreObservable"],"detail_key":"p40"},{"id":"n33751","layer":"informal","project":"p40","title":"Coboundary side: \\(\\int_0^1\\sin(2\\pi s)\\,ds = 0\\) gives vanishing periodic sums, so the b…","kind":"proof","summary":"Coboundary side: \\(\\int_0^1\\sin(2\\pi s)\\,ds = 0\\) gives vanishing periodic sums, so the backwar…","labels":[],"detail_key":"p40"},{"id":"n33752","layer":"informal","project":"p40","title":"Density matrix","kind":"definition","summary":"[Density matrix] For a finite index type \\(n\\), a \\emphdensity matrix is a structure bundling a…","labels":["DensityMatrix"],"detail_key":"p40"},{"id":"n33753","layer":"informal","project":"p40","title":"Eigenvalues form a probability vector","kind":"lemma","summary":"[Eigenvalues form a probability vector] The real eigenvalues \\(\\lambda_i\\) of a density matrix…","labels":["DensityMatrix.sum_eigenvalues_eq_one"],"detail_key":"p40"},{"id":"n33754","layer":"informal","project":"p40","title":"Positive semidefiniteness gives \\(\\lambda_i\\ge 0\\). The Hermitian spectral theorem identi…","kind":"proof","summary":"Positive semidefiniteness gives \\(\\lambda_i\\ge 0\\). The Hermitian spectral theorem identifies \\…","labels":[],"detail_key":"p40"},{"id":"n33755","layer":"informal","project":"p40","title":"Von Neumann entropy","kind":"definition","summary":"[Von Neumann entropy] The \\emphvon Neumann entropy of a density matrix \\(\\rho\\) with eigenvalue…","labels":["vonNeumannEntropy"],"detail_key":"p40"},{"id":"n33756","layer":"informal","project":"p40","title":"Nonnegativity of entropy","kind":"theorem","summary":"[Nonnegativity of entropy] \\(S(\\rho)\\ge 0\\) for every density matrix \\(\\rho\\).","labels":["vonNeumannEntropy_nonneg"],"detail_key":"p40"},{"id":"n33757","layer":"informal","project":"p40","title":"Each eigenvalue satisfies \\(0\\le\\lambda_i\\le 1\\), and on \\([0,1]\\) one has \\(negMulLog(x)…","kind":"proof","summary":"Each eigenvalue satisfies \\(0\\le\\lambda_i\\le 1\\), and on \\([0,1]\\) one has \\(negMulLog(x) = -x\\…","labels":[],"detail_key":"p40"},{"id":"n33758","layer":"informal","project":"p40","title":"Right partial trace","kind":"definition","summary":"[Right partial trace] For an operator \\(M\\) on a bipartite system \\(C^n_A\\otimesC^n_B\\), the \\e…","labels":["partialTraceRight"],"detail_key":"p40"},{"id":"n33759","layer":"informal","project":"p40","title":"Partial trace preserves the trace","kind":"lemma","summary":"[Partial trace preserves the trace] \\(\\Tr(\\Tr_B M) = \\Tr M\\).","labels":["trace_partialTraceRight"],"detail_key":"p40"},{"id":"n33760","layer":"informal","project":"p40","title":"Both sides expand to the full double sum \\(\\sum_i,j M_(i,j),(i,j)\\) over the product inde…","kind":"proof","summary":"Both sides expand to the full double sum \\(\\sum_i,j M_(i,j),(i,j)\\) over the product index set;…","labels":[],"detail_key":"p40"},{"id":"n33761","layer":"informal","project":"p40","title":"Partial trace is completely positive","kind":"lemma","summary":"[Partial trace is completely positive] If \\(M\\) is positive semidefinite then so is \\(\\Tr_B M\\).","labels":["PosSemidef.partialTraceRight"],"detail_key":"p40"},{"id":"n33762","layer":"informal","project":"p40","title":"Write the partial trace in Kraus/compression form \\(\\Tr_B M = \\sum_j E_j^* M E_j\\), the s…","kind":"proof","summary":"Write the partial trace in Kraus/compression form \\(\\Tr_B M = \\sum_j E_j^* M E_j\\), the sum of…","labels":[],"detail_key":"p40"},{"id":"n33763","layer":"informal","project":"p40","title":"Maximum-entropy Jensen bound","kind":"lemma","summary":"[Maximum-entropy Jensen bound] For a probability vector \\(p\\) supported on a nonempty finite se…","labels":["sum_negMulLog_le_log_card"],"detail_key":"p40"},{"id":"n33764","layer":"informal","project":"p40","title":"The concave Jensen inequality (\\(\\textttConcaveOn.le\\_map\\_sum\\)) for \\(negMulLog\\) with…","kind":"proof","summary":"The concave Jensen inequality (\\(\\textttConcaveOn.le\\_map\\_sum\\)) for \\(negMulLog\\) with the un…","labels":[],"detail_key":"p40"},{"id":"n33765","layer":"informal","project":"p40","title":"A state has positive rank","kind":"lemma","summary":"[A state has positive rank] The rank of a density matrix is strictly positive, \\(0 < rank\\rho\\).","labels":["DensityMatrix.rank_pos"],"detail_key":"p40"},{"id":"n33766","layer":"informal","project":"p40","title":"The eigenvalues sum to \\(1\\ne 0\\) (\\CrefDensityMatrix.sum_eigenvalues_eq_one), so at leas…","kind":"proof","summary":"The eigenvalues sum to \\(1\\ne 0\\) (\\CrefDensityMatrix.sum_eigenvalues_eq_one), so at least one…","labels":[],"detail_key":"p40"},{"id":"n33767","layer":"informal","project":"p40","title":"Maximum-entropy bound","kind":"theorem","summary":"[Maximum-entropy bound] The von Neumann entropy of a density matrix is at most the logarithm of…","labels":["vonNeumannEntropy_le_log_rank"],"detail_key":"p40"},{"id":"n33768","layer":"informal","project":"p40","title":"Restrict the entropy sum \\(S(\\rho) = \\sum_inegMulLog(\\lambda_i)\\) to the support \\(s = \\i…","kind":"proof","summary":"Restrict the entropy sum \\(S(\\rho) = \\sum_inegMulLog(\\lambda_i)\\) to the support \\(s = \\i : \\la…","labels":[],"detail_key":"p40"},{"id":"n33769","layer":"informal","project":"p40","title":"Bound by the ambient dimension","kind":"corollary","summary":"[Bound by the ambient dimension] \\(S(\\rho)\\le\\log(\\#n)\\), the logarithm of the ambient Hilbert-…","labels":["vonNeumannEntropy_le_log_card"],"detail_key":"p40"},{"id":"n33770","layer":"informal","project":"p40","title":"The rank never exceeds the number of columns, \\(rank\\rho\\le\\#n\\); monotonicity of \\(\\log\\…","kind":"proof","summary":"The rank never exceeds the number of columns, \\(rank\\rho\\le\\#n\\); monotonicity of \\(\\log\\) appl…","labels":[],"detail_key":"p40"},{"id":"n33771","layer":"informal","project":"p40","title":"Strict positivity off the pure states","kind":"theorem","summary":"[Strict positivity off the pure states] If a density matrix is not idempotent (\\(\\rho^2\\ne\\rho\\…","labels":["vonNeumannEntropy_pos_of_sq_ne"],"detail_key":"p40"},{"id":"n33772","layer":"informal","project":"p40","title":"Entropy is a sum of nonnegative terms (\\CrefvonNeumannEntropy_nonneg), so \\(S(\\rho) = 0\\)…","kind":"proof","summary":"Entropy is a sum of nonnegative terms (\\CrefvonNeumannEntropy_nonneg), so \\(S(\\rho) = 0\\) force…","labels":[],"detail_key":"p40"},{"id":"n33773","layer":"informal","project":"p40","title":"Block-inclusion Kraus operators","kind":"definition","summary":"[Block-inclusion Kraus operators] For each \\(j\\in n_B\\), the \\emphright block-inclusion operato…","labels":["krausInclusionRight"],"detail_key":"p40"},{"id":"n33774","layer":"informal","project":"p40","title":"Kraus conjugation is a block compression","kind":"lemma","summary":"[Kraus conjugation is a block compression] The conjugation \\(E_j M E_j^*\\) is exactly the \\((j,…","labels":["krausConjRight_eq_submatrix"],"detail_key":"p40"},{"id":"n33775","layer":"informal","project":"p40","title":"Entrywise: the two \\(E_j\\)-factors select the row index \\((a,j)\\) and column index \\((a',…","kind":"proof","summary":"Entrywise: the two \\(E_j\\)-factors select the row index \\((a,j)\\) and column index \\((a',j)\\) t…","labels":[],"detail_key":"p40"},{"id":"n33776","layer":"informal","project":"p40","title":"Explicit Kraus form of the partial traces","kind":"theorem","summary":"[Explicit Kraus form of the partial traces] Both partial traces are genuine sums of Kraus conju…","labels":["partialTraceRight_eq_kraus"],"detail_key":"p40"},{"id":"n33777","layer":"informal","project":"p40","title":"By \\CrefkrausConjRight_eq_submatrix each summand \\(E_j M E_j^*\\) is the \\(j\\)-th diagonal…","kind":"proof","summary":"By \\CrefkrausConjRight_eq_submatrix each summand \\(E_j M E_j^*\\) is the \\(j\\)-th diagonal block…","labels":[],"detail_key":"p40"},{"id":"n33778","layer":"informal","project":"p40","title":"Co-isometry of each block inclusion","kind":"lemma","summary":"[Co-isometry of each block inclusion] Each block inclusion is a co-isometry onto its factor: fo…","labels":["krausInclusionRight_mul_conjTranspose"],"detail_key":"p40"},{"id":"n33779","layer":"informal","project":"p40","title":"\\((E_j E_j^*)_a,a' = \\sum_(a'',j')[\\,(a'',j') = (a,j)\\,][\\,(a'',j') = (a',j)\\,] = [\\,a =…","kind":"proof","summary":"\\((E_j E_j^*)_a,a' = \\sum_(a'',j')[\\,(a'',j') = (a,j)\\,][\\,(a'',j') = (a',j)\\,] = [\\,a = a'\\,]\\…","labels":[],"detail_key":"p40"},{"id":"n33780","layer":"informal","project":"p40","title":"Completeness relation","kind":"lemma","summary":"[Completeness relation] The block inclusions satisfy the Kraus completeness / trace-preservatio…","labels":["sum_conjTranspose_mul_krausInclusionRight"],"detail_key":"p40"},{"id":"n33781","layer":"informal","project":"p40","title":"\\(\\bigl(\\sum_j E_j^*E_j\\bigr)_(p),(p') = \\sum_a,j[\\,p = (a,j)\\,][\\,p' = (a,j)\\,] = [\\,p =…","kind":"proof","summary":"\\(\\bigl(\\sum_j E_j^*E_j\\bigr)_(p),(p') = \\sum_a,j[\\,p = (a,j)\\,][\\,p' = (a,j)\\,] = [\\,p = p'\\,]…","labels":[],"detail_key":"p40"},{"id":"n33782","layer":"informal","project":"p40","title":"Umegaki relative entropy","kind":"definition","summary":"[Umegaki relative entropy] The \\emphUmegaki relative entropy of \\(\\rho\\) with respect to \\(\\sig…","labels":["relEntropy"],"detail_key":"p40"},{"id":"n33783","layer":"informal","project":"p40","title":"Trace form of relative entropy","kind":"theorem","summary":"[Trace form of relative entropy] For faithful (positive definite) \\(\\sigma\\), the spectral defi…","labels":["relEntropy_eq_traceLog"],"detail_key":"p40"},{"id":"n33784","layer":"informal","project":"p40","title":"Expand both eigendecompositions. The trace--spectral bridge \\(\\Re\\,\\Tr(\\rho\\,f(\\tau)) = \\…","kind":"proof","summary":"Expand both eigendecompositions. The trace--spectral bridge \\(\\Re\\,\\Tr(\\rho\\,f(\\tau)) = \\sum_k,…","labels":[],"detail_key":"p40"},{"id":"n33785","layer":"informal","project":"p40","title":"Klein / Gibbs nonnegativity","kind":"theorem","summary":"[Klein / Gibbs nonnegativity] For faithful (positive definite) \\(\\sigma\\), \\(\\;0\\le S(\\rho\\|\\si…","labels":["relEntropy_nonneg"],"detail_key":"p40"},{"id":"n33786","layer":"informal","project":"p40","title":"The overlap matrix \\(D_km = \\left\\lvert \\langle e_k\\mid f_m\\rangle \\right\\rvert^2\\) is do…","kind":"proof","summary":"The overlap matrix \\(D_km = \\left\\lvert \\langle e_k\\mid f_m\\rangle \\right\\rvert^2\\) is doubly s…","labels":[],"detail_key":"p40"},{"id":"n33787","layer":"informal","project":"p40","title":"Vanishing on the diagonal","kind":"theorem","summary":"[Vanishing on the diagonal] \\(S(\\rho\\|\\rho) = 0\\) for every density matrix \\(\\rho\\).","labels":["relEntropy_self_eq_zero"],"detail_key":"p40"},{"id":"n33788","layer":"informal","project":"p40","title":"With \\(\\sigma = \\rho\\) the overlap matrix \\(Q = \\rho.eigVec^*\\,\\rho.eigVec = 1\\), so \\(D_…","kind":"proof","summary":"With \\(\\sigma = \\rho\\) the overlap matrix \\(Q = \\rho.eigVec^*\\,\\rho.eigVec = 1\\), so \\(D_km = \\…","labels":[],"detail_key":"p40"},{"id":"n33789","layer":"informal","project":"p40","title":"Unitary invariance","kind":"theorem","summary":"[Unitary invariance] For any unitary \\(W\\), \\(\\;S(W\\rho W^*\\,\\|\\,W\\sigma W^*) = S(\\rho\\|\\sigma)…","labels":["relEntropy_conj_invariant"],"detail_key":"p40"},{"id":"n33790","layer":"informal","project":"p40","title":"Conjugation by \\(W\\) is a \\(*\\)-algebra automorphism, so it commutes with the functional…","kind":"proof","summary":"Conjugation by \\(W\\) is a \\(*\\)-algebra automorphism, so it commutes with the functional calcul…","labels":[],"detail_key":"p40"},{"id":"n33791","layer":"informal","project":"p40","title":"Ancilla invariance","kind":"theorem","summary":"[Ancilla invariance] Tensoring both arguments with a common faithful ancilla \\(\\alpha\\) leaves…","labels":["relEntropy_ancilla_invariant"],"detail_key":"p40"},{"id":"n33792","layer":"informal","project":"p40","title":"Relative entropy is additive over Kronecker products, \\(S(\\rho\\otimes\\alpha\\,\\|\\,\\sigma\\o…","kind":"proof","summary":"Relative entropy is additive over Kronecker products, \\(S(\\rho\\otimes\\alpha\\,\\|\\,\\sigma\\otimes\\…","labels":[],"detail_key":"p40"},{"id":"n33793","layer":"informal","project":"p40","title":"Scalar Klein / Peierls inequality","kind":"theorem","summary":"[Scalar Klein / Peierls inequality] Let \\(D = (D_km)\\) be a doubly stochastic \\(K\\times M\\) mat…","labels":["klein_scalar"],"detail_key":"p40"},{"id":"n33794","layer":"informal","project":"p40","title":"This is the finite scalar core of Klein's inequality (Carlen, \\emphTrace Inequalities and…","kind":"proof","summary":"This is the finite scalar core of Klein's inequality (Carlen, \\emphTrace Inequalities and Quant…","labels":[],"detail_key":"p40"},{"id":"n33795","layer":"informal","project":"p40","title":"Subadditivity of the von Neumann entropy","kind":"theorem","summary":"[Subadditivity of the von Neumann entropy] For a bipartite density matrix \\(\\rho\\) on \\(n_A\\oti…","labels":["vonNeumannEntropy_subadditive"],"detail_key":"p40"},{"id":"n33796","layer":"informal","project":"p40","title":"An elementary route through the scalar Klein inequality, with no matrix logarithm (Carlen…","kind":"proof","summary":"An elementary route through the scalar Klein inequality, with no matrix logarithm (Carlen, \\emp…","labels":[],"detail_key":"p40"},{"id":"n33797","layer":"informal","project":"p40","title":"Operator convexity of \\(-\\log\\)","kind":"theorem","summary":"[Operator convexity of \\(-\\log\\)] The function \\(x\\mapsto -\\log x\\) is operator convex on \\((0,…","labels":["operatorConvexOn_neg_log"],"detail_key":"p40"},{"id":"n33798","layer":"informal","project":"p40","title":"Transport the statement along the \\(R\\)-linear star-algebra equivalence \\(Matrix\\simeq\\te…","kind":"proof","summary":"Transport the statement along the \\(R\\)-linear star-algebra equivalence \\(Matrix\\simeq\\textttCS…","labels":[],"detail_key":"p40"},{"id":"n33799","layer":"informal","project":"p40","title":"Hansen--Pedersen--Jensen operator-Jensen inequality","kind":"theorem","summary":"[Hansen--Pedersen--Jensen operator-Jensen inequality] Let \\(f\\) be operator convex on an interv…","labels":["hpj_affine"],"detail_key":"p40"},{"id":"n33800","layer":"informal","project":"p40","title":"The Effros / Hansen--Pedersen unitary-dilation method. Work in the doubled algebra \\(Matr…","kind":"proof","summary":"The Effros / Hansen--Pedersen unitary-dilation method. Work in the doubled algebra \\(Matrix(Fin…","labels":[],"detail_key":"p40"},{"id":"n33801","layer":"informal","project":"p40","title":"Operator perspective","kind":"definition","summary":"[Operator perspective] For \\(f:R\\toR\\) and matrices \\(L,R\\) with \\(R\\) positive definite, the \\…","labels":["operatorPerspective"],"detail_key":"p40"},{"id":"n33802","layer":"informal","project":"p40","title":"Effros' theorem: joint convexity of the perspective","kind":"theorem","summary":"[Effros' theorem: joint convexity of the perspective] Let \\(f\\) be operator convex on \\(I\\), le…","labels":["operatorPerspective_jointly_convex"],"detail_key":"p40"},{"id":"n33803","layer":"informal","project":"p40","title":"The Effros argument, a single application of \\Crefhpj_affine. Writing \\(R = cR_1 + (1-c)R…","kind":"proof","summary":"The Effros argument, a single application of \\Crefhpj_affine. Writing \\(R = cR_1 + (1-c)R_2\\),…","labels":[],"detail_key":"p40"},{"id":"n33804","layer":"informal","project":"p40","title":"Operator perspective at a general index","kind":"definition","summary":"[Operator perspective at a general index] The same formula \\(P_f(L,R) = R^1/2f(R^-1/2LR^-1/2)R^…","labels":["opPersp"],"detail_key":"p40"},{"id":"n33805","layer":"informal","project":"p40","title":"Effros' theorem at a general finite index","kind":"theorem","summary":"[Effros' theorem at a general finite index] Under the same hypotheses as \\CrefoperatorPerspecti…","labels":["opPersp_jointly_convex"],"detail_key":"p40"},{"id":"n33806","layer":"informal","project":"p40","title":"Transport along the star-algebra equivalence \\(Matrix(m)\\simeqMatrix(Fin(\\left\\lvert m \\r…","kind":"proof","summary":"Transport along the star-algebra equivalence \\(Matrix(m)\\simeqMatrix(Fin(\\left\\lvert m \\right\\r…","labels":[],"detail_key":"p40"},{"id":"n33807","layer":"informal","project":"p40","title":"Logarithm of a Kronecker product","kind":"lemma","summary":"[Logarithm of a Kronecker product] For positive-definite \\(A,B\\), \\(\\ \\log(A\\otimes B) = \\log A…","labels":["cfc_log_kron"],"detail_key":"p40"},{"id":"n33808","layer":"informal","project":"p40","title":"Simultaneously diagonalize by \\(U_A\\otimes U_B\\); the Kronecker of the two eigenvalue dia…","kind":"proof","summary":"Simultaneously diagonalize by \\(U_A\\otimes U_B\\); the Kronecker of the two eigenvalue diagonals…","labels":[],"detail_key":"p40"},{"id":"n33809","layer":"informal","project":"p40","title":"Effros realization of the relative entropy","kind":"lemma","summary":"[Effros realization of the relative entropy] For positive-definite \\(\\rho,\\sigma\\), the perspec…","labels":["opPersp_neg_log_kron"],"detail_key":"p40"},{"id":"n33810","layer":"informal","project":"p40","title":"The sandwiched argument is \\(R^-1/2LR^-1/2 = \\rho^-1\\otimes\\sigma^\\top\\). Apply \\Crefcfc_…","kind":"proof","summary":"The sandwiched argument is \\(R^-1/2LR^-1/2 = \\rho^-1\\otimes\\sigma^\\top\\). Apply \\Crefcfc_log_kr…","labels":[],"detail_key":"p40"},{"id":"n33811","layer":"informal","project":"p40","title":"Scalar Effros functional","kind":"lemma","summary":"[Scalar Effros functional] The positive linear functional \\(M\\mapsto\\langlevec\\,1,\\,M\\,vec\\,1\\r…","labels":["relForm_opPersp_neg_log"],"detail_key":"p40"},{"id":"n33812","layer":"informal","project":"p40","title":"Apply \\(relForm\\) to the closed form of \\CrefopPersp_neg_log_kron. On Kronecker products…","kind":"proof","summary":"Apply \\(relForm\\) to the closed form of \\CrefopPersp_neg_log_kron. On Kronecker products \\(relF…","labels":[],"detail_key":"p40"},{"id":"n33813","layer":"informal","project":"p40","title":"Trace-form relative entropy","kind":"definition","summary":"[Trace-form relative entropy] For matrices \\(\\rho,\\sigma\\), the trace-form relative entropy is…","labels":["relEntropyMat"],"detail_key":"p40"},{"id":"n33814","layer":"informal","project":"p40","title":"Lieb's theorem: joint convexity of relative entropy","kind":"theorem","summary":"[Lieb's theorem: joint convexity of relative entropy] For positive-definite \\(\\rho_1,\\rho_2,\\si…","labels":["relEntropyMat_jointly_convex"],"detail_key":"p40"},{"id":"n33815","layer":"informal","project":"p40","title":"Lieb's theorem (Lieb 1973; Carlen, \\emphTrace Inequalities and Quantum Entropy, Thm.~2.12…","kind":"proof","summary":"Lieb's theorem (Lieb 1973; Carlen, \\emphTrace Inequalities and Quantum Entropy, Thm.~2.12), obt…","labels":[],"detail_key":"p40"},{"id":"n33816","layer":"informal","project":"p40","title":"Finite Jensen form of Lieb's theorem","kind":"lemma","summary":"[Finite Jensen form of Lieb's theorem] For a finite convex combination of faithful states --- w…","labels":["relEntropyMat_convex_sum"],"detail_key":"p40"},{"id":"n33817","layer":"informal","project":"p40","title":"The two-point joint convexity of \\CrefrelEntropyMat_jointly_convex says \\(relEntropyMat\\)…","kind":"proof","summary":"The two-point joint convexity of \\CrefrelEntropyMat_jointly_convex says \\(relEntropyMat\\) is a…","labels":[],"detail_key":"p40"},{"id":"n33818","layer":"informal","project":"p40","title":"Bridge to the spectral relative entropy","kind":"theorem","summary":"[Bridge to the spectral relative entropy] For density matrices \\(\\rho,\\sigma\\) with \\(\\sigma\\)…","labels":["relEntropyMat_eq_relEntropy"],"detail_key":"p40"},{"id":"n33819","layer":"informal","project":"p40","title":"Both sides expand through the spectral theorem: the trace form \\(\\Re\\,\\Tr(\\rho(\\log\\rho -…","kind":"proof","summary":"Both sides expand through the spectral theorem: the trace form \\(\\Re\\,\\Tr(\\rho(\\log\\rho - \\log\\…","labels":[],"detail_key":"p40"},{"id":"n33820","layer":"informal","project":"p40","title":"The partial-trace DPI, as a monomorphic \\textttProp","kind":"definition","summary":"[The partial-trace DPI, as a monomorphic \\textttProp] \\textttRelEntropyMonotoneUnderPartialTrac…","labels":["RelEntropyMonotoneUnderPartialTrace"],"detail_key":"p40"},{"id":"n33821","layer":"informal","project":"p40","title":"Partial-trace DPI, faithful case","kind":"theorem","summary":"[Partial-trace DPI, faithful case] For positive-definite states \\(\\rho,\\sigma\\) on \\(n_A\\times…","labels":["relEntropyMonotone_partialTrace_faithful"],"detail_key":"p40"},{"id":"n33822","layer":"informal","project":"p40","title":"Realize the partial trace as a Weyl twirl: with \\(d = \\#n_E\\) and \\(\\tau = \\tfrac1d 1\\) t…","kind":"proof","summary":"Realize the partial trace as a Weyl twirl: with \\(d = \\#n_E\\) and \\(\\tau = \\tfrac1d 1\\) the max…","labels":[],"detail_key":"p40"},{"id":"n33823","layer":"informal","project":"p40","title":"Partial-trace DPI, unconditional","kind":"theorem","summary":"[Partial-trace DPI, unconditional] The proposition \\textttRelEntropyMonotoneUnderPartialTrace h…","labels":["relEntropyMonotone_partialTrace"],"detail_key":"p40"},{"id":"n33824","layer":"informal","project":"p40","title":"Regularize the first argument along the affine path \\(\\rho_\\varepsilon = (1-\\varepsilon)\\…","kind":"proof","summary":"Regularize the first argument along the affine path \\(\\rho_\\varepsilon = (1-\\varepsilon)\\rho +…","labels":[],"detail_key":"p40"},{"id":"n33825","layer":"informal","project":"p40","title":"Tracing out an adjoined ancilla is the identity","kind":"lemma","summary":"[Tracing out an adjoined ancilla is the identity] For any state \\(\\rho\\) and any ancilla state…","labels":["partialTraceRight_kron"],"detail_key":"p40"},{"id":"n33826","layer":"informal","project":"p40","title":"Entrywise, \\((\\Tr_E(\\rho\\otimes\\alpha))_ii' = \\sum_j\\rho_ii'\\alpha_jj = \\rho_ii'\\cdot\\Tr\\…","kind":"proof","summary":"Entrywise, \\((\\Tr_E(\\rho\\otimes\\alpha))_ii' = \\sum_j\\rho_ii'\\alpha_jj = \\rho_ii'\\cdot\\Tr\\alpha…","labels":[],"detail_key":"p40"},{"id":"n33827","layer":"informal","project":"p40","title":"Isometric-embedding invariance","kind":"theorem","summary":"[Isometric-embedding invariance] For states \\(\\rho,\\sigma\\) on \\(n\\), a faithful ancilla \\(\\alp…","labels":["relEntropy_embed_invariant"],"detail_key":"p40"},{"id":"n33828","layer":"informal","project":"p40","title":"Unitary conjugation leaves relative entropy invariant (\\CrefrelEntropy_conj_invariant), r…","kind":"proof","summary":"Unitary conjugation leaves relative entropy invariant (\\CrefrelEntropy_conj_invariant), reducin…","labels":[],"detail_key":"p40"},{"id":"n33829","layer":"informal","project":"p40","title":"Stinespring reduction of the DPI","kind":"theorem","summary":"[Stinespring reduction of the DPI] \\emphGiven the partial-trace DPI (\\CrefRelEntropyMonotoneUnd…","labels":["stinespring_relEntropy_monotone"],"detail_key":"p40"},{"id":"n33830","layer":"informal","project":"p40","title":"Apply the wall (\\CrefRelEntropyMonotoneUnderPartialTrace) to the dilated states \\(U(\\rho\\…","kind":"proof","summary":"Apply the wall (\\CrefRelEntropyMonotoneUnderPartialTrace) to the dilated states \\(U(\\rho\\otimes…","labels":[],"detail_key":"p40"},{"id":"n33831","layer":"informal","project":"p40","title":"Data-processing inequality for the faithful-ancilla mixed-Stinespring family","kind":"theorem","summary":"[Data-processing inequality for the faithful-ancilla mixed-Stinespring family] Unconditionally,…","labels":["monotonicity_relEntropy_under_stinespring"],"detail_key":"p40"},{"id":"n33832","layer":"informal","project":"p40","title":"Feed the now-discharged partial-trace DPI (\\CrefrelEntropyMonotone_partialTrace) into the…","kind":"proof","summary":"Feed the now-discharged partial-trace DPI (\\CrefrelEntropyMonotone_partialTrace) into the Stine…","labels":[],"detail_key":"p40"},{"id":"n33833","layer":"informal","project":"p40","title":"No faithful-monotone recovery section under a strict drop","kind":"theorem","summary":"[No faithful-monotone recovery section under a strict drop] If a map \\(R\\) satisfies the faithf…","labels":["no_section_of_strict_relEntropy_drop"],"detail_key":"p40"},{"id":"n33834","layer":"informal","project":"p40","title":"Monotonicity of \\(R\\) applied to \\(\\Lambda\\rho,\\Lambda\\sigma\\) gives \\(S(R(\\Lambda\\rho)\\|…","kind":"proof","summary":"Monotonicity of \\(R\\) applied to \\(\\Lambda\\rho,\\Lambda\\sigma\\) gives \\(S(R(\\Lambda\\rho)\\|R(\\Lam…","labels":[],"detail_key":"p40"},{"id":"n33835","layer":"informal","project":"p40","title":"No Stinespring recovery under a strict drop","kind":"corollary","summary":"[No Stinespring recovery under a strict drop] Unconditionally: a strict relative-entropy drop u…","labels":["no_stinespring_section_of_strict_relEntropy_drop"],"detail_key":"p40"},{"id":"n33836","layer":"informal","project":"p40","title":"Specialize \\Crefno_section_of_strict_relEntropy_drop to the Stinespring recovery map, who…","kind":"proof","summary":"Specialize \\Crefno_section_of_strict_relEntropy_drop to the Stinespring recovery map, whose fai…","labels":[],"detail_key":"p40"},{"id":"n33837","layer":"informal","project":"p40","title":"Kraus channel","kind":"definition","summary":"[Kraus channel] A \\emphKraus channel on \\(Matrix_n(C)\\) is a finite family of Kraus operators \\…","labels":["KrausChannel"],"detail_key":"p40"},{"id":"n33838","layer":"informal","project":"p40","title":"Trace preservation","kind":"lemma","summary":"[Trace preservation] A Kraus channel is trace preserving, \\(\\tr(\\Lambda X)=\\tr X\\) for every \\(…","labels":["KrausChannel.toMat_trace"],"detail_key":"p40"},{"id":"n33839","layer":"informal","project":"p40","title":"By trace cyclicity \\(\\tr(K_i X K_i^\\dagger)=\\tr(K_i^\\daggerK_i\\,X)\\); summing over \\(i\\)…","kind":"proof","summary":"By trace cyclicity \\(\\tr(K_i X K_i^\\dagger)=\\tr(K_i^\\daggerK_i\\,X)\\); summing over \\(i\\) and ap…","labels":[],"detail_key":"p40"},{"id":"n33840","layer":"informal","project":"p40","title":"Petz recovery map","kind":"definition","summary":"[Petz recovery map] For a state \\(\\sigma\\) and a channel \\(\\Lambda\\), the \\emphPetz (transpose)…","labels":["petz"],"detail_key":"p40"},{"id":"n33841","layer":"informal","project":"p40","title":"Petz recovery identity","kind":"theorem","summary":"[Petz recovery identity] If the channel output \\(\\Lambda\\sigma\\) is positive definite, then the…","labels":["petz_recovery"],"detail_key":"p40"},{"id":"n33842","layer":"informal","project":"p40","title":"Feeding \\(X=\\Lambda\\sigma\\) into \\(P_\\sigma,\\Lambda\\), the inner conjugation \\((\\Lambda\\s…","kind":"proof","summary":"Feeding \\(X=\\Lambda\\sigma\\) into \\(P_\\sigma,\\Lambda\\), the inner conjugation \\((\\Lambda\\sigma)^…","labels":[],"detail_key":"p40"},{"id":"n33843","layer":"informal","project":"p40","title":"Recovery \\(\\Rightarrow\\) saturation","kind":"theorem","summary":"[Recovery \\(\\Rightarrow\\) saturation] Let \\(\\Lambda\\) and \\(R\\) be maps on states, each monoton…","labels":["petz_recovery_implies_equality"],"detail_key":"p40"},{"id":"n33844","layer":"informal","project":"p40","title":"Monotonicity of \\(\\Lambda\\) gives \\(S(\\Lambda\\rho\\|\\Lambda\\sigma)\\le S(\\rho\\|\\sigma)\\). F…","kind":"proof","summary":"Monotonicity of \\(\\Lambda\\) gives \\(S(\\Lambda\\rho\\|\\Lambda\\sigma)\\le S(\\rho\\|\\sigma)\\). For the…","labels":[],"detail_key":"p40"},{"id":"n33845","layer":"informal","project":"p40","title":"Rectangular isometry \\(-\\log\\) Loewner inequality","kind":"theorem","summary":"[Rectangular isometry \\(-\\log\\) Loewner inequality] Let \\(W:C^q\\toC^p\\) be an isometry (\\(W^\\da…","labels":["rect_isometry_neg_log_loewner"],"detail_key":"p40"},{"id":"n33846","layer":"informal","project":"p40","title":"Extend the orthonormal columns of \\(W\\) to a unitary \\(U\\); conjugating \\(X\\) by \\(U\\) an…","kind":"proof","summary":"Extend the orthonormal columns of \\(W\\) to a unitary \\(U\\); conjugating \\(X\\) by \\(U\\) and rein…","labels":[],"detail_key":"p40"},{"id":"n33847","layer":"informal","project":"p40","title":"Choi \\(-\\log\\) operator inequality","kind":"theorem","summary":"[Choi \\(-\\log\\) operator inequality] For Kraus operators \\(K\\) with \\(\\sum_i K_i^\\daggerK_i=1\\)…","labels":["choi-neg-log-loewner-math"],"detail_key":"p40"},{"id":"n33848","layer":"informal","project":"p40","title":"Stack the Kraus operators into the column isometry \\(V\\) and let \\(X_bd\\) be the block-di…","kind":"proof","summary":"Stack the Kraus operators into the column isometry \\(V\\) and let \\(X_bd\\) be the block-diagonal…","labels":[],"detail_key":"p40"},{"id":"n33849","layer":"informal","project":"p40","title":"Modular form of the relative entropy","kind":"lemma","summary":"[Modular form of the relative entropy] For faithful states \\(\\rho,\\sigma\\), with (vectorised) r…","labels":["kronForm_re_eq_relEntropy"],"detail_key":"p40"},{"id":"n33850","layer":"informal","project":"p40","title":"Compute \\((-\\log)(\\sigma\\otimes(\\rho^-1)^\\top)=-(\\log\\sigma)\\otimes 1+1\\otimes(\\log\\rho)^…","kind":"proof","summary":"Compute \\((-\\log)(\\sigma\\otimes(\\rho^-1)^\\top)=-(\\log\\sigma)\\otimes 1+1\\otimes(\\log\\rho)^\\top\\)…","labels":[],"detail_key":"p40"},{"id":"n33851","layer":"informal","project":"p40","title":"Scalar channel \\(-\\log\\) modular gap","kind":"theorem","summary":"[Scalar channel \\(-\\log\\) modular gap] For a Kraus channel \\(\\Lambda\\) with all four states \\(\\…","labels":["channel_modular_gap"],"detail_key":"p40"},{"id":"n33852","layer":"informal","project":"p40","title":"Since \\(W\\xi=vec(\\rho^1/2)\\) (cyclicity of the channel contraction), the left form is \\(S…","kind":"proof","summary":"Since \\(W\\xi=vec(\\rho^1/2)\\) (cyclicity of the channel contraction), the left form is \\(S(\\rho\\…","labels":[],"detail_key":"p40"},{"id":"n33853","layer":"informal","project":"p40","title":"Injectivity-free gap decomposition","kind":"lemma","summary":"[Injectivity-free gap decomposition] For a contraction \\(W\\) with defect \\((1-W^\\daggerW)\\xi=0\\…","labels":["contraction_gap_decomp"],"detail_key":"p40"},{"id":"n33854","layer":"informal","project":"p40","title":"A pure matrix identity: the isometric proof's \\(Y^-1\\) bridge is replaced by \\(Out^-1YOut…","kind":"proof","summary":"A pure matrix identity: the isometric proof's \\(Y^-1\\) bridge is replaced by \\(Out^-1YOut^-1\\),…","labels":[],"detail_key":"p40"},{"id":"n33855","layer":"informal","project":"p40","title":"Per-\\(t\\) intertwining at gap zero","kind":"lemma","summary":"[Per-\\(t\\) intertwining at gap zero] In the setting of Lemma~\\refcontraction_gap_decomp, with t…","labels":["contraction_perT_intertwine_of_gap_zero"],"detail_key":"p40"},{"id":"n33856","layer":"informal","project":"p40","title":"Both summands of Lemma~\\refcontraction_gap_decomp are nonnegative (\\(X\\succ 0\\) and \\(Out…","kind":"proof","summary":"Both summands of Lemma~\\refcontraction_gap_decomp are nonnegative (\\(X\\succ 0\\) and \\(Out-Y\\suc…","labels":[],"detail_key":"p40"},{"id":"n33857","layer":"informal","project":"p40","title":"Scalar-sourced contraction rigidity spine","kind":"theorem","summary":"[Scalar-sourced contraction rigidity spine] Let \\(W\\) be a contraction (\\(W^\\daggerW\\le 1\\)), \\…","labels":["contraction_resolvent_intertwine_of_re_eq"],"detail_key":"p40"},{"id":"n33858","layer":"informal","project":"p40","title":"Represent \\(-\\log\\) by the integral \\(\\int_0^\\infty\\bigl((1+t)^-1-(\\cdot+t)^-1\\bigr)\\,dt\\…","kind":"proof","summary":"Represent \\(-\\log\\) by the integral \\(\\int_0^\\infty\\bigl((1+t)^-1-(\\cdot+t)^-1\\bigr)\\,dt\\) and…","labels":[],"detail_key":"p40"},{"id":"n33859","layer":"informal","project":"p40","title":"Contraction intertwines every continuous function","kind":"lemma","summary":"[Contraction intertwines every continuous function] Under the hypotheses of Theorem~\\refcontrac…","labels":["contraction_cfc_intertwine"],"detail_key":"p40"},{"id":"n33860","layer":"informal","project":"p40","title":"On the finite union of the two spectra, \\(g\\) is a real-coefficient combination of resolv…","kind":"proof","summary":"On the finite union of the two spectra, \\(g\\) is a real-coefficient combination of resolvents \\…","labels":[],"detail_key":"p40"},{"id":"n33861","layer":"informal","project":"p40","title":"Channel unitary-power intertwining","kind":"theorem","summary":"[Channel unitary-power intertwining] Under entropy saturation, the channel contraction intertwi…","labels":["channel_upow_intertwine"],"detail_key":"p40"},{"id":"n33862","layer":"informal","project":"p40","title":"Apply Lemma~\\refcontraction_cfc_intertwine with \\(g=\\cos(t\\log\\cdot)\\) and \\(g=\\sin(t\\log…","kind":"proof","summary":"Apply Lemma~\\refcontraction_cfc_intertwine with \\(g=\\cos(t\\log\\cdot)\\) and \\(g=\\sin(t\\log\\cdot)…","labels":[],"detail_key":"p40"},{"id":"n33863","layer":"informal","project":"p40","title":"Modular \\(it\\)-intertwining","kind":"definition","summary":"[Modular \\(it\\)-intertwining] The channel adjoint \\emphintertwines the modular \\(it\\)-flows if,…","labels":["IntertwinesIt"],"detail_key":"p40"},{"id":"n33864","layer":"informal","project":"p40","title":"Equality \\(\\Rightarrow\\) modular intertwining (general channel)","kind":"theorem","summary":"[Equality \\(\\Rightarrow\\) modular intertwining (general channel)] For any Kraus channel with al…","labels":["channel_equality_imp_intertwinesIt"],"detail_key":"p40"},{"id":"n33865","layer":"informal","project":"p40","title":"From Theorem~\\refchannel_upow_intertwine, reading off the vec-action \\(\\Delta^itvecX=vec(…","kind":"proof","summary":"From Theorem~\\refchannel_upow_intertwine, reading off the vec-action \\(\\Delta^itvecX=vec(P^itXR…","labels":[],"detail_key":"p40"},{"id":"n33866","layer":"informal","project":"p40","title":"Modular intertwining \\(\\Rightarrow\\) Petz recovery","kind":"theorem","summary":"[Modular intertwining \\(\\Rightarrow\\) Petz recovery] If \\textttIntertwinesIt holds (all four st…","labels":["intertwinesIt_imp_recovery"],"detail_key":"p40"},{"id":"n33867","layer":"informal","project":"p40","title":"Analytic continuation of the \\(it\\)-intertwining to the value \\(t=-i/2\\) (a Kadison-type…","kind":"proof","summary":"Analytic continuation of the \\(it\\)-intertwining to the value \\(t=-i/2\\) (a Kadison-type argume…","labels":[],"detail_key":"p40"},{"id":"n33868","layer":"informal","project":"p40","title":"General Petz recovery from equality --- issue \\#28 headline","kind":"theorem","summary":"[General Petz recovery from equality --- issue \\#28 headline] Let \\(\\Lambda\\) be \\emphany Kraus…","labels":["petz_equality_recovery_general"],"detail_key":"p40"},{"id":"n33869","layer":"informal","project":"p40","title":"Compose Theorem~\\refchannel_equality_imp_intertwinesIt (entropy equality \\(\\Rightarrow\\)…","kind":"proof","summary":"Compose Theorem~\\refchannel_equality_imp_intertwinesIt (entropy equality \\(\\Rightarrow\\) modula…","labels":[],"detail_key":"p40"},{"id":"n33870","layer":"informal","project":"p40","title":"Partial-trace modular gap","kind":"theorem","summary":"[Partial-trace modular gap] If the partial-trace relative entropy is preserved, \\(S(Tr_B\\omega\\…","labels":["partialTrace_modular_gap"],"detail_key":"p40"},{"id":"n33871","layer":"informal","project":"p40","title":"By Theorem~\\refrect_isometry_neg_log_loewner the operator \\(B-A\\ge 0\\), where \\(B=W^\\dagg…","kind":"proof","summary":"By Theorem~\\refrect_isometry_neg_log_loewner the operator \\(B-A\\ge 0\\), where \\(B=W^\\dagger(-\\l…","labels":[],"detail_key":"p40"},{"id":"n33872","layer":"informal","project":"p40","title":"Partial-trace equality \\(\\Rightarrow\\) modular intertwining","kind":"theorem","summary":"[Partial-trace equality \\(\\Rightarrow\\) modular intertwining] Under the same entropy-preservati…","labels":["partialTrace_equality_imp_intertwinesIt"],"detail_key":"p40"},{"id":"n33873","layer":"informal","project":"p40","title":"The gap of Theorem~\\refpartialTrace_modular_gap is upgraded, first to an intertwining of…","kind":"proof","summary":"The gap of Theorem~\\refpartialTrace_modular_gap is upgraded, first to an intertwining of every…","labels":[],"detail_key":"p40"},{"id":"n33874","layer":"informal","project":"p40","title":"Unital \\(*\\)-endomorphism","kind":"definition","summary":"[Unital \\(*\\)-endomorphism] A \\emphfinite quantum dynamics is a map \\(\\Phi:Matrix_d(C)\\toMatrix…","labels":["UnitalStarEndo"],"detail_key":"p40"},{"id":"n33875","layer":"informal","project":"p40","title":"Operational partition of unity","kind":"definition","summary":"[Operational partition of unity] An \\emphoperational partition of unity of size \\(k\\) is a fami…","labels":["OperationalPartition"],"detail_key":"p40"},{"id":"n33876","layer":"informal","project":"p40","title":"Time-ordered refinement","kind":"definition","summary":"[Time-ordered refinement] Given a dynamics \\(\\Phi\\) and an operational partition \\(X=(x_i)\\), t…","labels":["refine"],"detail_key":"p40"},{"id":"n33877","layer":"informal","project":"p40","title":"Telescoping identity","kind":"lemma","summary":"[Telescoping identity] The refinement of an operational partition of unity is again an operatio…","labels":["sum_refine_conjTranspose_mul_refine"],"detail_key":"p40"},{"id":"n33878","layer":"informal","project":"p40","title":"Induct on \\(n\\). Splitting the word as \\(f=(i,g)\\) with \\(i=f_0\\), the summand factors as…","kind":"proof","summary":"Induct on \\(n\\). Splitting the word as \\(f=(i,g)\\) with \\(i=f_0\\), the summand factors as \\(\\Ph…","labels":[],"detail_key":"p40"},{"id":"n33879","layer":"informal","project":"p40","title":"Correlation density matrix","kind":"definition","summary":"[Correlation density matrix] For a dynamics \\(\\Phi\\), a state \\(\\rho\\) (a density matrix on \\(C…","labels":["corrMatrix"],"detail_key":"p40"},{"id":"n33880","layer":"informal","project":"p40","title":"CNT/ALF dynamical entropy","kind":"definition","summary":"[CNT/ALF dynamical entropy] The \\emphentropy of a partition is the infimum von Neumann entropy…","labels":["cntDynamicalEntropy"],"detail_key":"p40"},{"id":"n33881","layer":"informal","project":"p40","title":"Diagonal state","kind":"definition","summary":"[Diagonal state] For a probability vector \\(\\mu:Find\\toR_\\ge 0\\) (\\(\\sum_i\\mu_i=1\\)), the assoc…","labels":["densityOfPMF"],"detail_key":"p40"},{"id":"n33882","layer":"informal","project":"p40","title":"Projection partition","kind":"definition","summary":"[Projection partition] For a cell map \\(c:Find\\toFink\\), the diagonal \\emphprojection partition…","labels":["projPartition"],"detail_key":"p40"},{"id":"n33883","layer":"informal","project":"p40","title":"The permutation dynamics (abelian corner)","kind":"definition","summary":"[The permutation dynamics (abelian corner)] For a permutation \\(\\sigma\\in\\mathfrakS_d\\), the \\e…","labels":["adPerm"],"detail_key":"p40"},{"id":"n33884","layer":"informal","project":"p40","title":"Per-resolution diagonal collapse","kind":"theorem","summary":"[Per-resolution diagonal collapse] On the abelian corner the correlation matrix is diagonal, ca…","labels":["vonNeumannEntropy_corrMatrix_eq_ksEntropySeq"],"detail_key":"p40"},{"id":"n33885","layer":"informal","project":"p40","title":"Because \\(adPerm\\,\\sigma\\) preserves diagonal matrices, the refinement of a projection pa…","kind":"proof","summary":"Because \\(adPerm\\,\\sigma\\) preserves diagonal matrices, the refinement of a projection partitio…","labels":[],"detail_key":"p40"},{"id":"n33886","layer":"informal","project":"p40","title":"Per-partition equality of entropy rates","kind":"theorem","summary":"[Per-partition equality of entropy rates] For each projection partition \\(projPartition\\,c\\), t…","labels":["cntEntropyPartition_eq_ksEntropyPartition"],"detail_key":"p40"},{"id":"n33887","layer":"informal","project":"p40","title":"Both sides are the subadditive limit of the same sequence divided by \\(n\\): the quantum p…","kind":"proof","summary":"Both sides are the subadditive limit of the same sequence divided by \\(n\\): the quantum partiti…","labels":[],"detail_key":"p40"},{"id":"n33888","layer":"informal","project":"p40","title":"Abelian-corner CNT entropy","kind":"definition","summary":"[Abelian-corner CNT entropy] The \\emphabelian-corner CNT dynamical entropy of \\(adPerm\\,\\sigma\\…","labels":["cntDynamicalEntropyAbelian"],"detail_key":"p40"},{"id":"n33889","layer":"informal","project":"p40","title":"Abelian corner \\(=\\) Kolmogorov--Sinai entropy","kind":"theorem","summary":"[Abelian corner \\(=\\) Kolmogorov--Sinai entropy] Suppose every state carries positive mass (\\(\\…","labels":["cntDynamicalEntropyAbelian_eq_ksEntropy"],"detail_key":"p40"},{"id":"n33890","layer":"informal","project":"p40","title":"Two inequalities. For \\(\\le\\), each projection partition's rate equals a classical partit…","kind":"proof","summary":"Two inequalities. For \\(\\le\\), each projection partition's rate equals a classical partition en…","labels":[],"detail_key":"p40"},{"id":"n33891","layer":"informal","project":"p40","title":"Full CNT entropy dominates KS entropy","kind":"theorem","summary":"[Full CNT entropy dominates KS entropy] Under the same positivity hypothesis, the \\emphfull CNT…","labels":["ksEntropy_le_cntDynamicalEntropy"],"detail_key":"p40"},{"id":"n33892","layer":"informal","project":"p40","title":"The abelian dynamical entropy is a supremum over the sub-family of projection partitions,…","kind":"proof","summary":"The abelian dynamical entropy is a supremum over the sub-family of projection partitions, hence…","labels":[],"detail_key":"p40"},{"id":"n33893","layer":"informal","project":"p40","title":"Hilbert--Schmidt Gram vectors","kind":"definition","summary":"[Hilbert--Schmidt Gram vectors] For a dynamics \\(\\Phi\\), a state \\(\\rho\\) on \\(C^d\\) and an ope…","labels":["gramVec"],"detail_key":"p40"},{"id":"n33894","layer":"informal","project":"p40","title":"Gram factorization of the correlation matrix","kind":"theorem","summary":"[Gram factorization of the correlation matrix] The CNT correlation matrix is the Gram matrix of…","labels":["corrVal_eq_conjTranspose_mul_self"],"detail_key":"p40"},{"id":"n33895","layer":"informal","project":"p40","title":"Entrywise this is the trace identity \\(\\tr\\bigl((A\\sqrt\\rho)^\\dagger(B\\sqrt\\rho)\\bigr) =…","kind":"proof","summary":"Entrywise this is the trace identity \\(\\tr\\bigl((A\\sqrt\\rho)^\\dagger(B\\sqrt\\rho)\\bigr) = \\tr(\\r…","labels":[],"detail_key":"p40"},{"id":"n33896","layer":"informal","project":"p40","title":"Uniform rank bound","kind":"theorem","summary":"[Uniform rank bound] The correlation density matrix of an \\(n\\)-fold refinement has rank at mos…","labels":["rank_corrVal_le"],"detail_key":"p40"},{"id":"n33897","layer":"informal","project":"p40","title":"A Gram matrix \\(V^\\daggerV\\) has the same rank as \\(V\\) (\\(\\textttrank\\_conjTranspose\\_mu…","kind":"proof","summary":"A Gram matrix \\(V^\\daggerV\\) has the same rank as \\(V\\) (\\(\\textttrank\\_conjTranspose\\_mul\\_sel…","labels":[],"detail_key":"p40"},{"id":"n33898","layer":"informal","project":"p40","title":"Uniform entropy bound","kind":"theorem","summary":"[Uniform entropy bound] At every resolution \\(n\\), the von Neumann entropy of the correlation m…","labels":["vonNeumannEntropy_corrMatrix_le_log"],"detail_key":"p40"},{"id":"n33899","layer":"informal","project":"p40","title":"Chain the maximum-entropy inequality \\(S\\le\\log(rank)\\) (Theorem~\\refvonNeumannEntropy_le…","kind":"proof","summary":"Chain the maximum-entropy inequality \\(S\\le\\log(rank)\\) (Theorem~\\refvonNeumannEntropy_le_log_r…","labels":[],"detail_key":"p40"},{"id":"n33900","layer":"informal","project":"p40","title":"The entropy rate vanishes for every partition","kind":"theorem","summary":"[The entropy rate vanishes for every partition] For every operational partition \\(X\\), the CNT…","labels":["cntEntropyPartition_eq_zero"],"detail_key":"p40"},{"id":"n33901","layer":"informal","project":"p40","title":"The rate is nonnegative, and by the uniform bound (Theorem~\\refvonNeumannEntropy_corrMatr…","kind":"proof","summary":"The rate is nonnegative, and by the uniform bound (Theorem~\\refvonNeumannEntropy_corrMatrix_le_…","labels":[],"detail_key":"p40"},{"id":"n33902","layer":"informal","project":"p40","title":"The \\(\\inf\\)-rate is a genuine limit","kind":"theorem","summary":"[The \\(\\inf\\)-rate is a genuine limit] For every operational partition, the entropy rate conver…","labels":["tendsto_cntEntropySeq_div"],"detail_key":"p40"},{"id":"n33903","layer":"informal","project":"p40","title":"Squeeze between \\(0\\) and \\(\\log(d^2)/n\\to 0\\); the limit is \\(0\\), which Theorem~\\refcnt…","kind":"proof","summary":"Squeeze between \\(0\\) and \\(\\log(d^2)/n\\to 0\\); the limit is \\(0\\), which Theorem~\\refcntEntrop…","labels":[],"detail_key":"p40"},{"id":"n33904","layer":"informal","project":"p40","title":"Finite-dimensional CNT dynamical entropy vanishes","kind":"theorem","summary":"[Finite-dimensional CNT dynamical entropy vanishes] The full CNT/ALF dynamical entropy of any f…","labels":["cntDynamicalEntropy_eq_zero"],"detail_key":"p40"},{"id":"n33905","layer":"informal","project":"p40","title":"Every term of the defining supremum over operational partitions is \\(0\\) (Theorem~\\refcnt…","kind":"proof","summary":"Every term of the defining supremum over operational partitions is \\(0\\) (Theorem~\\refcntEntrop…","labels":[],"detail_key":"p40"},{"id":"n33906","layer":"informal","project":"p40","title":"Abelian-corner Fekete subadditivity","kind":"theorem","summary":"[Abelian-corner Fekete subadditivity] On the diagonal corner the correlation-entropy sequence \\…","labels":["subadditive_vonNeumannEntropy_corrMatrix_abelian"],"detail_key":"p40"},{"id":"n33907","layer":"informal","project":"p40","title":"By the per-resolution diagonal collapse (Theorem~\\refvonNeumannEntropy_corrMatrix_eq_ksEn…","kind":"proof","summary":"By the per-resolution diagonal collapse (Theorem~\\refvonNeumannEntropy_corrMatrix_eq_ksEntropyS…","labels":[],"detail_key":"p40"},{"id":"n33908","layer":"informal","project":"p40","title":"Abelian-corner Fekete convergence","kind":"theorem","summary":"[Abelian-corner Fekete convergence] On the diagonal corner the averaged entropies converge to t…","labels":["tendsto_cntEntropyPartition_abelian"],"detail_key":"p40"},{"id":"n33909","layer":"informal","project":"p40","title":"Rewriting the rate through the partition-level equality (Theorem~\\refcntEntropyPartition_…","kind":"proof","summary":"Rewriting the rate through the partition-level equality (Theorem~\\refcntEntropyPartition_eq_ksE…","labels":[],"detail_key":"p40"},{"id":"n33910","layer":"informal","project":"p40","title":"Full abelian-corner CNT entropy","kind":"definition","summary":"[Full abelian-corner CNT entropy] The \\emphfull abelian-corner CNT dynamical entropy is the sup…","labels":["cntDynamicalEntropyAbelianFull"],"detail_key":"p40"},{"id":"n33911","layer":"informal","project":"p40","title":"The abelian sups vanish","kind":"theorem","summary":"[The abelian sups vanish] In finite dimension the full diagonal supremum vanishes, \\(h^ab,full_…","labels":["cntDynamicalEntropyAbelianFull_eq_zero"],"detail_key":"p40"},{"id":"n33912","layer":"informal","project":"p40","title":"Each diagonal (resp.\\ projection) partition already has entropy rate \\(0\\) (Theorem~\\refc…","kind":"proof","summary":"Each diagonal (resp.\\ projection) partition already has entropy rate \\(0\\) (Theorem~\\refcntEntr…","labels":[],"detail_key":"p40"},{"id":"n33913","layer":"informal","project":"p40","title":"Free monotonicity","kind":"lemma","summary":"[Free monotonicity] The sharp abelian-corner entropy is dominated by the full diagonal version,…","labels":["cntDynamicalEntropyAbelian_le_full"],"detail_key":"p40"},{"id":"n33914","layer":"informal","project":"p40","title":"Each term of the projection supremum is a term of the wider diagonal supremum (\\(projPart…","kind":"proof","summary":"Each term of the projection supremum is a term of the wider diagonal supremum (\\(projPartition\\…","labels":[],"detail_key":"p40"},{"id":"n33915","layer":"informal","project":"p40","title":"Full abelian corner \\(=\\) KS entropy","kind":"theorem","summary":"[Full abelian corner \\(=\\) KS entropy] For a \\(\\sigma\\)-invariant state with every mass positiv…","labels":["cntDynamicalEntropyAbelianFull_eq_ksEntropy"],"detail_key":"p40"},{"id":"n33916","layer":"informal","project":"p40","title":"The full diagonal sup is \\(0\\) (Theorem~\\refcntDynamicalEntropyAbelianFull_eq_zero); the…","kind":"proof","summary":"The full diagonal sup is \\(0\\) (Theorem~\\refcntDynamicalEntropyAbelianFull_eq_zero); the sharp…","labels":[],"detail_key":"p40"},{"id":"n33917","layer":"informal","project":"p40","title":"KS entropy \\(=\\) full CNT dynamical entropy","kind":"theorem","summary":"[KS entropy \\(=\\) full CNT dynamical entropy] Under the same positivity hypothesis, the classic…","labels":["ksEntropy_eq_cntDynamicalEntropy"],"detail_key":"p40"},{"id":"n33918","layer":"informal","project":"p40","title":"The corner equality (Theorem~\\refcntDynamicalEntropyAbelian_eq_ksEntropy) with the abelia…","kind":"proof","summary":"The corner equality (Theorem~\\refcntDynamicalEntropyAbelian_eq_ksEntropy) with the abelian vani…","labels":[],"detail_key":"p40"},{"id":"n33919","layer":"informal","project":"p40","title":"Resolution \\(1\\): zero entropy","kind":"lemma","summary":"[Resolution \\(1\\): zero entropy] For the pure state \\(\\rho\\), the length-\\(1\\) correlation matr…","labels":["entropy_corrMatrix_one_eq_zero"],"detail_key":"p40"},{"id":"n33920","layer":"informal","project":"p40","title":"Since \\(\\Phi = id\\), the depth-\\(1\\) refinement along \\(f\\) is just \\(x_f(0)\\); the \\(\\rh…","kind":"proof","summary":"Since \\(\\Phi = id\\), the depth-\\(1\\) refinement along \\(f\\) is just \\(x_f(0)\\); the \\(\\rho\\)-we…","labels":[],"detail_key":"p40"},{"id":"n33921","layer":"informal","project":"p40","title":"Resolution \\(2\\): positive entropy","kind":"lemma","summary":"[Resolution \\(2\\): positive entropy] For the same pure state, the length-\\(2\\) correlation matr…","labels":["entropy_corrMatrix_two_pos"],"detail_key":"p40"},{"id":"n33922","layer":"informal","project":"p40","title":"The depth-\\(2\\) refinement along \\(f\\) is \\(x_f(0)x_f(1)\\); computing the \\((w_00,w_00)\\)…","kind":"proof","summary":"The depth-\\(2\\) refinement along \\(f\\) is \\(x_f(0)x_f(1)\\); computing the \\((w_00,w_00)\\) entry…","labels":[],"detail_key":"p40"},{"id":"n33923","layer":"informal","project":"p40","title":"The CNT entropy sequence is not subadditive","kind":"theorem","summary":"[The CNT entropy sequence is not subadditive] The CNT/ALF entropy sequence \\(n\\mapsto S(\\rho[X^…","labels":["not_subadditive_cnt_entropySeq"],"detail_key":"p40"},{"id":"n33924","layer":"informal","project":"p40","title":"If the sequence were subadditive, the instance \\(u_2\\le u_1 + u_1\\) would force \\(S(\\rho[…","kind":"proof","summary":"If the sequence were subadditive, the instance \\(u_2\\le u_1 + u_1\\) would force \\(S(\\rho[X^(2)]…","labels":[],"detail_key":"p40"},{"id":"n33925","layer":"informal","project":"p40","title":"Zero entropy of an idempotent state","kind":"lemma","summary":"[Zero entropy of an idempotent state] If a density matrix is a projection (\\(\\rho^2 = \\rho\\)) t…","labels":["vonNeumannEntropy_eq_zero_of_sq_eq"],"detail_key":"p40"},{"id":"n33926","layer":"informal","project":"p40","title":"Relative entropy against the maximally mixed state","kind":"lemma","summary":"[Relative entropy against the maximally mixed state] \\(D(\\rho\\,\\|\\,I/d) = \\log d - S(\\rho)\\); t…","labels":["relEntropy_maximallyMixed"],"detail_key":"p40"},{"id":"n33927","layer":"informal","project":"p40","title":"Pure-state seal, T2a","kind":"theorem","summary":"[Pure-state seal, T2a] The maximally coherent pure state \\(|+\\rangle\\langle+|\\) dephases to \\(I…","labels":["quantum_seal_dephase"],"detail_key":"p40"},{"id":"n33928","layer":"informal","project":"p40","title":"Faithful-state seal, T2b","kind":"theorem","summary":"[Faithful-state seal, T2b] The faithful one-parameter family \\(\\rho_r = \\tfrac12\\!\\left( 1 & r\\…","labels":["quantum_seal_dephase_faithful"],"detail_key":"p40"},{"id":"n33929","layer":"informal","project":"p40","title":"Strictly above every abelian restriction","kind":"theorem","summary":"[Strictly above every abelian restriction] For \\emphevery abelian (diagonal) operational partit…","labels":["cex_strictly_above_abelian"],"detail_key":"p40"},{"id":"n33930","layer":"informal","project":"p40","title":"No common canonical MASA","kind":"theorem","summary":"[No common canonical MASA] The seal's diagonal MASA is not dynamics-invariant --- \\(U\\cdotdiag(…","labels":["qDynamics_seal_no_common_canonical_masa"],"detail_key":"p40"},{"id":"n33931","layer":"informal","project":"p40","title":"Reservoir cap in \\(2\\log d\\) form","kind":"theorem","summary":"[Reservoir cap in \\(2\\log d\\) form] For every unital \\(*\\)-endomorphism \\(\\Phi\\), state \\(\\rho\\…","labels":["cntCumulativeEntropy_le_reservoir"],"detail_key":"p40"},{"id":"n33932","layer":"informal","project":"p40","title":"Restate the uniform bound \\(S \\le \\log(d^2)\\) (Theorem~\\refvonNeumannEntropy_corrMatrix_l…","kind":"proof","summary":"Restate the uniform bound \\(S \\le \\log(d^2)\\) (Theorem~\\refvonNeumannEntropy_corrMatrix_le_log)…","labels":[],"detail_key":"p40"},{"id":"n33933","layer":"informal","project":"p40","title":"The correlation-entropy sequence is bounded","kind":"theorem","summary":"[The correlation-entropy sequence is bounded] For every partition the sequence \\(n\\mapsto S\\big…","labels":["cntEntropySeq_bddAbove"],"detail_key":"p40"},{"id":"n33934","layer":"informal","project":"p40","title":"The constant \\(\\log(d^2)\\) is an upper bound of the range by the per-resolution cap (Theo…","kind":"proof","summary":"The constant \\(\\log(d^2)\\) is an upper bound of the range by the per-resolution cap (Theorem~\\r…","labels":[],"detail_key":"p40"},{"id":"n33935","layer":"informal","project":"p40","title":"Cumulative-bounded \\(\\Rightarrow\\) per-step rate \\(\\to 0\\)","kind":"theorem","summary":"[Cumulative-bounded \\(\\Rightarrow\\) per-step rate \\(\\to 0\\)] Let \\(a:N\\toR\\) be nonnegative and…","labels":["rate_to_zero_of_cumulative_bounded"],"detail_key":"p40"},{"id":"n33936","layer":"informal","project":"p40","title":"Squeeze between \\(0 \\le a(n)/n\\) and \\(a(n)/n \\le C/n \\to 0\\).","kind":"proof","summary":"Squeeze between \\(0 \\le a(n)/n\\) and \\(a(n)/n \\le C/n \\to 0\\).","labels":[],"detail_key":"p40"},{"id":"n33937","layer":"informal","project":"p40","title":"Pauli operational partition","kind":"definition","summary":"[Pauli operational partition] The \\emphPauli operational partition of \\(Matrix_2(C)\\) is the fa…","labels":["pauliPartition"],"detail_key":"p40"},{"id":"n33938","layer":"informal","project":"p40","title":"Pauli saturation of the correlation matrix at one step","kind":"theorem","summary":"[Pauli saturation of the correlation matrix at one step] For \\emphevery unital \\(*\\)-endomorphi…","labels":["corrMatrix_pauliPartition_one"],"detail_key":"p40"},{"id":"n33939","layer":"informal","project":"p40","title":"The depth-\\(1\\) refinement along a word \\(f\\) is just the selected operator \\((\\tfrac12 P…","kind":"proof","summary":"The depth-\\(1\\) refinement along a word \\(f\\) is just the selected operator \\((\\tfrac12 P_f_0)\\…","labels":[],"detail_key":"p40"},{"id":"n33940","layer":"informal","project":"p40","title":"The reservoir cap is tight at \\(d = 2\\)","kind":"theorem","summary":"[The reservoir cap is tight at \\(d = 2\\)] For every unital \\(*\\)-endomorphism \\(\\Phi\\) of \\(Mat…","labels":["vonNeumannEntropy_corrMatrix_pauliPartition_eq"],"detail_key":"p40"},{"id":"n33941","layer":"informal","project":"p40","title":"By Theorem~\\refcorrMatrix_pauliPartition_one the correlation matrix is maximally mixed on…","kind":"proof","summary":"By Theorem~\\refcorrMatrix_pauliPartition_one the correlation matrix is maximally mixed on \\(4\\)…","labels":[],"detail_key":"p40"},{"id":"n33942","layer":"informal","project":"p40","title":"Growing qubit carrier","kind":"definition","summary":"[Growing qubit carrier] The length-\\(n\\) qubit block index \\(Qbits\\,n\\) is defined by recursion…","labels":["Qbits"],"detail_key":"p40"},{"id":"n33943","layer":"informal","project":"p40","title":"The \\(n\\)-fold product state","kind":"definition","summary":"[The \\(n\\)-fold product state] For a single-qubit state \\(\\rho\\), the product state \\(\\rho^\\oti…","labels":["rhoPow"],"detail_key":"p40"},{"id":"n33944","layer":"informal","project":"p40","title":"Block entropy","kind":"definition","summary":"[Block entropy] The \\emphblock entropy of the tower at level \\(n\\) is the von Neumann entropy o…","labels":["blockEntropy"],"detail_key":"p40"},{"id":"n33945","layer":"informal","project":"p40","title":"The linear block-entropy law","kind":"theorem","summary":"[The linear block-entropy law] The block entropy grows exactly linearly in the level: \\[ blockE…","labels":["blockEntropy_eq"],"detail_key":"p40"},{"id":"n33946","layer":"informal","project":"p40","title":"Induct on \\(n\\): the von Neumann entropy is additive under the Kronecker product, \\(S(\\rh…","kind":"proof","summary":"Induct on \\(n\\): the von Neumann entropy is additive under the Kronecker product, \\(S(\\rho\\otim…","labels":[],"detail_key":"p40"},{"id":"n33947","layer":"informal","project":"p40","title":"Capacity-enlargement embedding","kind":"definition","summary":"[Capacity-enlargement embedding] The capacity-enlargement step is the embedding \\(Matrix_2^n(C)…","labels":["shiftAdjoinQubit"],"detail_key":"p40"},{"id":"n33948","layer":"informal","project":"p40","title":"Maximally mixed corollary","kind":"theorem","summary":"[Maximally mixed corollary] At the maximally mixed single-qubit state \\(I/2\\) the linear law re…","labels":["blockEntropy_maximallyMixed"],"detail_key":"p40"},{"id":"n33949","layer":"informal","project":"p40","title":"Specialise Theorem~\\refblockEntropy_eq at \\(\\rho = I/2\\), where \\(S(I/2) = \\log 2\\).","kind":"proof","summary":"Specialise Theorem~\\refblockEntropy_eq at \\(\\rho = I/2\\), where \\(S(I/2) = \\log 2\\).","labels":[],"detail_key":"p40"},{"id":"n33950","layer":"informal","project":"p40","title":"The spatial (per-step) entropy rate","kind":"theorem","summary":"[The spatial (per-step) entropy rate] The per-step entropy rate converges to the single-qubit e…","labels":["tendsto_blockEntropy_div"],"detail_key":"p40"},{"id":"n33951","layer":"informal","project":"p40","title":"By the linear law (Theorem~\\refblockEntropy_eq), for \\(n\\ge 1\\) the ratio is \\(n\\,S(\\rho)…","kind":"proof","summary":"By the linear law (Theorem~\\refblockEntropy_eq), for \\(n\\ge 1\\) the ratio is \\(n\\,S(\\rho)/n = S…","labels":[],"detail_key":"p40"},{"id":"n33952","layer":"informal","project":"p40","title":"Positive rate on a concrete faithful family","kind":"theorem","summary":"[Positive rate on a concrete faithful family] For the concrete faithful family \\(\\rho_r = \\tfra…","labels":["blockEntropy_rhoR_pos"],"detail_key":"p40"},{"id":"n33953","layer":"informal","project":"p40","title":"By the linear law the block entropy is \\(n\\cdot S(\\rho_r)\\); the single-qubit entropy equ…","kind":"proof","summary":"By the linear law the block entropy is \\(n\\cdot S(\\rho_r)\\); the single-qubit entropy equals th…","labels":[],"detail_key":"p40"},{"id":"n33954","layer":"informal","project":"p40","title":"Partial-dephasing channel","kind":"definition","summary":"[Partial-dephasing channel] For a block index \\(blk\\), the \\emphpartial-dephasing channel \\(\\De…","labels":["dephaseKronId"],"detail_key":"p40"},{"id":"n33955","layer":"informal","project":"p40","title":"The strict drop survives the block tensoring","kind":"theorem","summary":"[The strict drop survives the block tensoring] For a faithful block state \\(\\beta\\) and the ref…","labels":["relEntropy_strict_drop_dephasing_kron"],"detail_key":"p40"},{"id":"n33956","layer":"informal","project":"p40","title":"The block \\(\\beta\\) is a common faithful ancilla, so the relative entropy is unchanged by…","kind":"proof","summary":"The block \\(\\beta\\) is a common faithful ancilla, so the relative entropy is unchanged by it (a…","labels":[],"detail_key":"p40"},{"id":"n33957","layer":"informal","project":"p40","title":"No Stinespring recovery, uniform in the block","kind":"theorem","summary":"[No Stinespring recovery, uniform in the block] For an \\empharbitrary block \\(blk\\) (in particu…","labels":["quantum_seal_dephase_kron_faithful"],"detail_key":"p40"},{"id":"n33958","layer":"informal","project":"p40","title":"The strict relative-entropy drop (Theorem~\\refrelEntropy_strict_drop_dephasing_kron) is u…","kind":"proof","summary":"The strict relative-entropy drop (Theorem~\\refrelEntropy_strict_drop_dephasing_kron) is uniform…","labels":[],"detail_key":"p40"},{"id":"n33959","layer":"informal","project":"p40","title":"Growing quantum world","kind":"definition","summary":"[Growing quantum world] A \\emphgrowing quantum world bundles, for one concrete faithful local s…","labels":["GrowingQuantumWorld"],"detail_key":"p40"},{"id":"n33960","layer":"informal","project":"p40","title":"A growing quantum world exists","kind":"theorem","summary":"[A growing quantum world exists] A concrete alive-and-sealed growing quantum world exists. Take…","labels":["growingQuantumWorld_exists"],"detail_key":"p40"},{"id":"n33961","layer":"informal","project":"p40","title":"Assemble the three faces at \\(r = s = 1/2\\): positivity of the single-qubit entropy from…","kind":"proof","summary":"Assemble the three faces at \\(r = s = 1/2\\): positivity of the single-qubit entropy from \\(h_2(…","labels":[],"detail_key":"p40"},{"id":"n33962","layer":"informal","project":"p40","title":"Directed local, not the completed \\(C^*\\)-chain","kind":"remark","summary":"[Directed local, not the completed \\(C^*\\)-chain] This is the directed \\emphsystem of finite le…","labels":["qbs-scope-directed"],"detail_key":"p40"},{"id":"n33963","layer":"informal","project":"p40","title":"Far-end inclusion","kind":"definition","summary":"[Far-end inclusion] The \\emphinclusion \\(A_n\\hookrightarrow A_n+1\\) appends a fresh qubit at th…","labels":["appendQubit"],"detail_key":"p40"},{"id":"n33964","layer":"informal","project":"p40","title":"The inclusion is injective","kind":"lemma","summary":"[The inclusion is injective] The far-end inclusion is injective: \\(x\\otimes 1\\) (reindexed) det…","labels":["appendQubit_injective"],"detail_key":"p40"},{"id":"n33965","layer":"informal","project":"p40","title":"Read off the \\((0,a),(0,b)\\) entry of \\(x\\otimes 1\\): the identity factor contributes \\(1…","kind":"proof","summary":"Read off the \\((0,a),(0,b)\\) entry of \\(x\\otimes 1\\): the identity factor contributes \\(1_0,0=1…","labels":[],"detail_key":"p40"},{"id":"n33966","layer":"informal","project":"p40","title":"Shift--inclusion commutation","kind":"theorem","summary":"[Shift--inclusion commutation] The capacity-enlargement shift \\(shiftAdjoinQubit\\) (\\(A\\mapsto…","labels":["shiftAdjoinQubit_appendQubit"],"detail_key":"p40"},{"id":"n33967","layer":"informal","project":"p40","title":"Both sides realise the same entrywise triple Kronecker product \\(1\\otimes M\\otimes 1\\): t…","kind":"proof","summary":"Both sides realise the same entrywise triple Kronecker product \\(1\\otimes M\\otimes 1\\): the shi…","labels":[],"detail_key":"p40"},{"id":"n33968","layer":"informal","project":"p40","title":"The tracial state is closed under the tensor step","kind":"lemma","summary":"[The tracial state is closed under the tensor step] The maximally mixed state is closed under t…","labels":["kron_maximallyMixed"],"detail_key":"p40"},{"id":"n33969","layer":"informal","project":"p40","title":"Expand both maximally mixed factors as normalized identities; the Kronecker product of id…","kind":"proof","summary":"Expand both maximally mixed factors as normalized identities; the Kronecker product of identiti…","labels":[],"detail_key":"p40"},{"id":"n33970","layer":"informal","project":"p40","title":"The tracial state is a tower fixed point","kind":"theorem","summary":"[The tracial state is a tower fixed point] The \\(n\\)-fold product of the maximally mixed single…","labels":["rhoPow_maximallyMixed"],"detail_key":"p40"},{"id":"n33971","layer":"informal","project":"p40","title":"Induct on \\(n\\): the tensor step (Definition~\\refrhoPow) tensors a fresh maximally mixed…","kind":"proof","summary":"Induct on \\(n\\): the tensor step (Definition~\\refrhoPow) tensors a fresh maximally mixed qubit…","labels":[],"detail_key":"p40"},{"id":"n33972","layer":"informal","project":"p40","title":"Shift-invariance of every product state","kind":"theorem","summary":"[Shift-invariance of every product state] Writing \\(\\tau_m(y)=\\tr\\bigl((rhoPow\\,\\rho\\,m)\\cdot y…","labels":["rhoPow_shiftAdjoinQubit_pairing"],"detail_key":"p40"},{"id":"n33973","layer":"informal","project":"p40","title":"The product state factors as \\(\\rho\\otimes(rhoPow\\,\\rho\\,n)\\) and the observable as \\(1\\o…","kind":"proof","summary":"The product state factors as \\(\\rho\\otimes(rhoPow\\,\\rho\\,n)\\) and the observable as \\(1\\otimes…","labels":[],"detail_key":"p40"},{"id":"n33974","layer":"informal","project":"p40","title":"Inclusion-compatibility of the tracial state","kind":"theorem","summary":"[Inclusion-compatibility of the tracial state] The far-end site adjoined by \\(appendQubit\\) is…","labels":["appendQubit_maximallyMixed_pairing"],"detail_key":"p40"},{"id":"n33975","layer":"informal","project":"p40","title":"The trace of \\(x\\otimes 1\\) is \\(2\\,\\tr x\\) (the fresh far-end factor contributes \\(\\tr 1…","kind":"proof","summary":"The trace of \\(x\\otimes 1\\) is \\(2\\,\\tr x\\) (the fresh far-end factor contributes \\(\\tr 1=2\\)),…","labels":[],"detail_key":"p40"},{"id":"n33976","layer":"informal","project":"p40","title":"The iterated shift","kind":"definition","summary":"[The iterated shift] The \\emph\\(k\\)-fold shift \\(A_n\\hookrightarrow A_n+k\\) iterates the capaci…","labels":["shiftIter"],"detail_key":"p40"},{"id":"n33977","layer":"informal","project":"p40","title":"The site-translation certificate","kind":"theorem","summary":"[The site-translation certificate] The fixed state \\(rhoPow\\,\\rho\\), read through \\(k\\) shift-i…","labels":["shiftIter_pairing"],"detail_key":"p40"},{"id":"n33978","layer":"informal","project":"p40","title":"Induct on \\(k\\): each shift step peels off one application of Theorem~\\refrhoPow_shiftAdj…","kind":"proof","summary":"Induct on \\(k\\): each shift step peels off one application of Theorem~\\refrhoPow_shiftAdjoinQub…","labels":[],"detail_key":"p40"},{"id":"n33979","layer":"informal","project":"p40","title":"The temporal window entropy","kind":"definition","summary":"[The temporal window entropy] The \\emphwindow entropy \\(windowEntropy\\,\\rho\\,k\\) is the von Neu…","labels":["windowEntropy"],"detail_key":"p40"},{"id":"n33980","layer":"informal","project":"p40","title":"The tracial window entropy","kind":"theorem","summary":"[The tracial window entropy] At the tracial (maximally mixed) state the length-\\(k\\) window car…","labels":["windowEntropy_tracial"],"detail_key":"p40"},{"id":"n33981","layer":"informal","project":"p40","title":"The window entropy is definitionally the block entropy (Definition~\\refwindowEntropy), wh…","kind":"proof","summary":"The window entropy is definitionally the block entropy (Definition~\\refwindowEntropy), which at…","labels":[],"detail_key":"p40"},{"id":"n33982","layer":"informal","project":"p40","title":"The temporal entropy rate is \\(\\log 2\\)","kind":"theorem","summary":"[The temporal entropy rate is \\(\\log 2\\)] The per-window rate \\(windowEntropy\\,mm\\,k / k\\) is t…","labels":["tendsto_windowEntropy_div_tracial"],"detail_key":"p40"},{"id":"n33983","layer":"informal","project":"p40","title":"By Theorem~\\refwindowEntropy_tracial the numerator is \\(k\\cdot\\log 2\\), so for \\(k\\ge 1\\)…","kind":"proof","summary":"By Theorem~\\refwindowEntropy_tracial the numerator is \\(k\\cdot\\log 2\\), so for \\(k\\ge 1\\) the r…","labels":[],"detail_key":"p40"},{"id":"n33984","layer":"informal","project":"p40","title":"Numerically block entropy; a bespoke rate, not the CNT supremum","kind":"remark","summary":"[Numerically block entropy; a bespoke rate, not the CNT supremum] \\(windowEntropy\\) is numerica…","labels":["qbs-scope-rate"],"detail_key":"p40"},{"id":"n33985","layer":"informal","project":"p40","title":"The chain-seal predicate at the tracial blocks","kind":"definition","summary":"[The chain-seal predicate at the tracial blocks] For the faithful local state \\(\\rho_r\\) and di…","labels":["ChainSealed"],"detail_key":"p40"},{"id":"n33986","layer":"informal","project":"p40","title":"The chain seal holds at every level","kind":"theorem","summary":"[The chain seal holds at every level] For all faithful parameters \\(0<r,s<1\\), the chain-seal p…","labels":["chain_seal_dephase_faithful"],"detail_key":"p40"},{"id":"n33987","layer":"informal","project":"p40","title":"Instantiate the uniform-in-the-block seal (Theorem~\\refquantum_seal_dephase_kron_faithful…","kind":"proof","summary":"Instantiate the uniform-in-the-block seal (Theorem~\\refquantum_seal_dephase_kron_faithful) at t…","labels":[],"detail_key":"p40"},{"id":"n33988","layer":"informal","project":"p40","title":"The modular automorphism","kind":"definition","summary":"[The modular automorphism] For a faithful (positive-definite) density matrix \\(\\rho\\), the \\emp…","labels":["modAut"],"detail_key":"p40"},{"id":"n33989","layer":"informal","project":"p40","title":"The one-parameter group law","kind":"theorem","summary":"[The one-parameter group law] The modular flow is a one-parameter group, \\(\\sigma_s\\circ\\sigma_…","labels":["modAut_add"],"detail_key":"p40"},{"id":"n33990","layer":"informal","project":"p40","title":"Expand \\(\\sigma_s(\\sigma_t(a))=\\rho^is\\rho^it\\,a\\,\\rho^-it\\rho^-is\\); the unitary powers…","kind":"proof","summary":"Expand \\(\\sigma_s(\\sigma_t(a))=\\rho^is\\rho^it\\,a\\,\\rho^-it\\rho^-is\\); the unitary powers add, \\…","labels":[],"detail_key":"p40"},{"id":"n33991","layer":"informal","project":"p40","title":"The \\(\\beta=1\\) KMS boundary identity","kind":"theorem","summary":"[The \\(\\beta=1\\) KMS boundary identity] For faithful \\(\\rho\\), with the Bratteli--Robinson II \\…","labels":["kms_boundary"],"detail_key":"p40"},{"id":"n33992","layer":"informal","project":"p40","title":"Substitute \\(\\sigma_-i(y)=\\rho\\,y\\,\\rho^-1\\) and use trace cyclicity together with \\(\\rho…","kind":"proof","summary":"Substitute \\(\\sigma_-i(y)=\\rho\\,y\\,\\rho^-1\\) and use trace cyclicity together with \\(\\rho^-1\\rh…","labels":[],"detail_key":"p40"},{"id":"n33993","layer":"informal","project":"p40","title":"KMS boundary as cyclicity; no Tomita--Takesaki uniqueness","kind":"remark","summary":"[KMS boundary as cyclicity; no Tomita--Takesaki uniqueness] The identity of Theorem~\\refkms_bou…","labels":["qbs-scope-kms"],"detail_key":"p40"},{"id":"n33994","layer":"informal","project":"p40","title":"Tower compatibility of the intrinsic clock","kind":"theorem","summary":"[Tower compatibility of the intrinsic clock] The intrinsic clock is consistent along the chain:…","labels":["modAut_shiftAdjoinQubit"],"detail_key":"p40"},{"id":"n33995","layer":"informal","project":"p40","title":"The unitary power of a Kronecker product factorises, \\((\\rho\\otimes\\rho^\\otimes n)^it =\\r…","kind":"proof","summary":"The unitary power of a Kronecker product factorises, \\((\\rho\\otimes\\rho^\\otimes n)^it =\\rho^it\\…","labels":[],"detail_key":"p40"},{"id":"n33996","layer":"informal","project":"p40","title":"Intrinsic-clock dichotomy: the tracial half","kind":"theorem","summary":"[Intrinsic-clock dichotomy: the tracial half] The maximally mixed (tracial) state has \\emphtriv…","labels":["modAut_maximallyMixed_eq_id"],"detail_key":"p40"},{"id":"n33997","layer":"informal","project":"p40","title":"The maximally mixed state is a scalar multiple of the identity, \\(\\rho=c\\cdot 1\\), so its…","kind":"proof","summary":"The maximally mixed state is a scalar multiple of the identity, \\(\\rho=c\\cdot 1\\), so its unita…","labels":[],"detail_key":"p40"},{"id":"n33998","layer":"informal","project":"p40","title":"Intrinsic-clock dichotomy: the non-tracial half","kind":"theorem","summary":"[Intrinsic-clock dichotomy: the non-tracial half] A faithful non-tracial (Powers-type) product…","labels":["modAut_diagState_ne_id"],"detail_key":"p40"},{"id":"n33999","layer":"informal","project":"p40","title":"On the off-diagonal unit \\(E_01\\) the flow multiplies by the phase \\(\\exp\\!\\bigl(it\\,\\log…","kind":"proof","summary":"On the off-diagonal unit \\(E_01\\) the flow multiplies by the phase \\(\\exp\\!\\bigl(it\\,\\log((1+s)…","labels":[],"detail_key":"p40"},{"id":"n34000","layer":"informal","project":"p40","title":"The quantum Bernoulli shift, directed-local representation","kind":"definition","summary":"[The quantum Bernoulli shift, directed-local representation] A \\emphquantum Bernoulli shift bun…","labels":["QuantumBernoulliShift"],"detail_key":"p40"},{"id":"n34001","layer":"informal","project":"p40","title":"A quantum Bernoulli shift exists","kind":"theorem","summary":"[A quantum Bernoulli shift exists] A concrete quantum Bernoulli shift exists. The witness assem…","labels":["quantumBernoulliShift_exists"],"detail_key":"p40"},{"id":"n34002","layer":"informal","project":"p40","title":"Populate the five fields of Definition~\\refQuantumBernoulliShift with Theorems~\\refshiftA…","kind":"proof","summary":"Populate the five fields of Definition~\\refQuantumBernoulliShift with Theorems~\\refshiftAdjoinQ…","labels":[],"detail_key":"p40"},{"id":"n34003","layer":"informal","project":"p40","title":"Lusin's theorem, continuous-on-a-compact form","kind":"theorem","summary":"[Lusin's theorem, continuous-on-a-compact form] Let \\(u:X\\toR\\) be Borel measurable on a Polish…","labels":["thm:dst-lusin"],"detail_key":"p40"},{"id":"n34004","layer":"informal","project":"p40","title":"The arctan-compression route (Cohn, \\emphMeasure Theory, 2nd ed., Thm.~7.4.4; Rudin, \\emp…","kind":"proof","summary":"The arctan-compression route (Cohn, \\emphMeasure Theory, 2nd ed., Thm.~7.4.4; Rudin, \\emphReal…","labels":[],"detail_key":"p40"},{"id":"n34005","layer":"informal","project":"p40","title":"Binary intersection of analytic sets is analytic","kind":"lemma","summary":"[Binary intersection of analytic sets is analytic] In a Hausdorff space, if \\(s\\) and \\(t\\) are…","labels":["thm:dst-inter"],"detail_key":"p40"},{"id":"n34006","layer":"informal","project":"p40","title":"Write \\(s\\cap t=\\bigcap_b:Boolcond\\,b\\,s\\,t\\) and apply Mathlib's countable intersection…","kind":"proof","summary":"Write \\(s\\cap t=\\bigcap_b:Boolcond\\,b\\,s\\,t\\) and apply Mathlib's countable intersection \\textt…","labels":[],"detail_key":"p40"},{"id":"n34007","layer":"informal","project":"p40","title":"Binary union of analytic sets is analytic","kind":"lemma","summary":"[Binary union of analytic sets is analytic] If \\(s\\) and \\(t\\) are analytic then so is \\(s\\cup…","labels":["thm:dst-union"],"detail_key":"p40"},{"id":"n34008","layer":"informal","project":"p40","title":"Write \\(s\\cup t=\\bigcup_b:Boolcond\\,b\\,s\\,t\\) and apply the countable union \\textttAnalyt…","kind":"proof","summary":"Write \\(s\\cup t=\\bigcup_b:Boolcond\\,b\\,s\\,t\\) and apply the countable union \\textttAnalyticSet.…","labels":[],"detail_key":"p40"},{"id":"n34009","layer":"informal","project":"p40","title":"Product of analytic sets is analytic","kind":"lemma","summary":"[Product of analytic sets is analytic] If \\(s\\subseteq X\\) and \\(t\\subseteq Y\\) are analytic th…","labels":["thm:dst-prod"],"detail_key":"p40"},{"id":"n34010","layer":"informal","project":"p40","title":"Parametrise \\(s=range\\,f\\), \\(t=range\\,g\\) by continuous maps from Polish spaces (\\texttt…","kind":"proof","summary":"Parametrise \\(s=range\\,f\\), \\(t=range\\,g\\) by continuous maps from Polish spaces (\\textttanalyt…","labels":[],"detail_key":"p40"},{"id":"n34011","layer":"informal","project":"p40","title":"Analytic sets are universally measurable","kind":"theorem","summary":"[Analytic sets are universally measurable] Every analytic set in a standard Borel space is \\tex…","labels":["thm:dst-nullmeas"],"detail_key":"p40"},{"id":"n34012","layer":"informal","project":"p40","title":"Choquet's capacitability theorem (Kechris, Thm.~30.13; Srivastava, Thm.~4.3.1): for a fin…","kind":"proof","summary":"Choquet's capacitability theorem (Kechris, Thm.~30.13; Srivastava, Thm.~4.3.1): for a finite Bo…","labels":[],"detail_key":"p40"},{"id":"n34013","layer":"informal","project":"p40","title":"Borel-separated family","kind":"definition","summary":"[Borel-separated family] A countable family \\(E:N\\to P(X)\\) is \\emphBorel separated (\\textttSep…","labels":["thm:dst-sepfam"],"detail_key":"p40"},{"id":"n34014","layer":"informal","project":"p40","title":"The single-entry split, Lemma 4.6.2","kind":"lemma","summary":"[The single-entry split, Lemma 4.6.2] Fix a position \\(j\\) and a countable decomposition \\(E_j\\…","labels":["thm:dst-sepfam-split"],"detail_key":"p40"},{"id":"n34015","layer":"informal","project":"p40","title":"Given separators \\(B^(m)\\) for each refinement, assemble a separator for \\(E\\): at positi…","kind":"proof","summary":"Given separators \\(B^(m)\\) for each refinement, assemble a separator for \\(E\\): at position \\(j…","labels":[],"detail_key":"p40"},{"id":"n34016","layer":"informal","project":"p40","title":"One stage of the Mokobodzki recursion, inductive Lemma 4.6.2","kind":"lemma","summary":"[One stage of the Mokobodzki recursion, inductive Lemma 4.6.2] Applied to images \\(A_n=range\\,f…","labels":["thm:dst-stage-step"],"detail_key":"p40"},{"id":"n34017","layer":"informal","project":"p40","title":"ErgodicTheory.famMix","kind":"proof","summary":"Extend the entries one coordinate at a time along the mixed-length intermediate family \\(famMix…","labels":[],"detail_key":"p40"},{"id":"n34018","layer":"informal","project":"p40","title":"First separation for ranges, Srivastava 4.6.1","kind":"theorem","summary":"[First separation for ranges, Srivastava 4.6.1] If \\(f_n:(N\\toN)\\to X\\) are continuous with \\(\\…","labels":["thm:dst-sepfam-ranges"],"detail_key":"p40"},{"id":"n34019","layer":"informal","project":"p40","title":"Suppose not. Iterating one stage of the recursion (\\Crefthm:dst-stage-step) from the triv…","kind":"proof","summary":"Suppose not. Iterating one stage of the recursion (\\Crefthm:dst-stage-step) from the trivial ce…","labels":[],"detail_key":"p40"},{"id":"n34020","layer":"informal","project":"p40","title":"Generalized first separation theorem, Novikov; Srivastava 4.6.1","kind":"theorem","summary":"[Generalized first separation theorem, Novikov; Srivastava 4.6.1] A countable family \\((A_n)\\)…","labels":["thm:dst-first-separation"],"detail_key":"p40"},{"id":"n34021","layer":"informal","project":"p40","title":"If some \\(A_n_0=\\emptyset\\), take \\(B_n_0=\\emptyset\\) and \\(B_n=univ\\) otherwise. Otherwi…","kind":"proof","summary":"If some \\(A_n_0=\\emptyset\\), take \\(B_n_0=\\emptyset\\) and \\(B_n=univ\\) otherwise. Otherwise eve…","labels":[],"detail_key":"p40"},{"id":"n34022","layer":"informal","project":"p40","title":"Weak reduction principle for coanalytic sets, Srivastava 4.6.5","kind":"theorem","summary":"[Weak reduction principle for coanalytic sets, Srivastava 4.6.5] If \\((S_n)\\) is a countable fa…","labels":["thm:dst-weak-reduction"],"detail_key":"p40"},{"id":"n34023","layer":"informal","project":"p40","title":"Apply \\Crefthm:dst-first-separation to the analytic sets \\((\\bigcup_m S_m)\\cap(S_n)^\\math…","kind":"proof","summary":"Apply \\Crefthm:dst-first-separation to the analytic sets \\((\\bigcup_m S_m)\\cap(S_n)^\\mathsf c\\)…","labels":[],"detail_key":"p40"},{"id":"n34024","layer":"informal","project":"p40","title":"Saint Raymond's theorem, Srivastava 4.7.1","kind":"theorem","summary":"[Saint Raymond's theorem, Srivastava 4.7.1] For disjoint analytic \\(A_0,A_1\\subseteq X\\times Y\\…","labels":["thm:dst-saint-raymond"],"detail_key":"p40"},{"id":"n34025","layer":"informal","project":"p40","title":"Replace \\(A_1\\) by a Borel Lusin separator \\(A_1'\\) from \\(A_0\\) (\\textttAnalyticSet.meas…","kind":"proof","summary":"Replace \\(A_1\\) by a Borel Lusin separator \\(A_1'\\) from \\(A_0\\) (\\textttAnalyticSet.measurably…","labels":[],"detail_key":"p40"},{"id":"n34026","layer":"informal","project":"p40","title":"Kunugui--Novikov theorem, Srivastava 4.7.2","kind":"theorem","summary":"[Kunugui--Novikov theorem, Srivastava 4.7.2] A Borel \\(B\\subseteq X\\times Y\\) all of whose sect…","labels":["thm:dst-kunugui-novikov"],"detail_key":"p40"},{"id":"n34027","layer":"informal","project":"p40","title":"The special case \\(A_0=B^\\mathsf c\\), \\(A_1=B\\) of Saint Raymond's theorem: closed sectio…","kind":"proof","summary":"The special case \\(A_0=B^\\mathsf c\\), \\(A_1=B\\) of Saint Raymond's theorem: closed sections of…","labels":[],"detail_key":"p40"},{"id":"n34028","layer":"informal","project":"p40","title":"Enumerated countable basis","kind":"lemma","summary":"[Enumerated countable basis] A second-countable space admits an \\(N\\)-indexed basis \\((V_n)\\) (…","labels":["thm:dst-nat-basis"],"detail_key":"p40"},{"id":"n34029","layer":"informal","project":"p40","title":"Insert \\(\\emptyset\\) into a countable topological basis to make it a nonempty countable s…","kind":"proof","summary":"Insert \\(\\emptyset\\) into a countable topological basis to make it a nonempty countable set, th…","labels":[],"detail_key":"p40"},{"id":"n34030","layer":"informal","project":"p40","title":"Closed sections become closed after a topology refinement, Srivastava 4.7.4","kind":"theorem","summary":"[Closed sections become closed after a topology refinement, Srivastava 4.7.4] For a Borel \\(B\\s…","labels":["thm:dst-finer-polish"],"detail_key":"p40"},{"id":"n34031","layer":"informal","project":"p40","title":"The complement \\(B^\\mathsf c\\) has open sections, so Kunugui--Novikov (\\Crefthm:dst-kunug…","kind":"proof","summary":"The complement \\(B^\\mathsf c\\) has open sections, so Kunugui--Novikov (\\Crefthm:dst-kunugui-nov…","labels":[],"detail_key":"p40"},{"id":"n34032","layer":"informal","project":"p40","title":"Compact-section projection, compact fibre space","kind":"theorem","summary":"[Compact-section projection, compact fibre space] If \\(Y\\) is compact, a Borel \\(B\\subseteq X\\t…","labels":["thm:dst-proj-compact"],"detail_key":"p40"},{"id":"n34033","layer":"informal","project":"p40","title":"Refine \\(X\\)'s topology to make \\(B\\) closed (\\Crefthm:dst-finer-polish; compact sections…","kind":"proof","summary":"Refine \\(X\\)'s topology to make \\(B\\) closed (\\Crefthm:dst-finer-polish; compact sections are i…","labels":[],"detail_key":"p40"},{"id":"n34034","layer":"informal","project":"p40","title":"Continuous injection into the Hilbert cube","kind":"lemma","summary":"[Continuous injection into the Hilbert cube] Every Polish space \\(Y\\) admits a continuous injec…","labels":["thm:dst-cube"],"detail_key":"p40"},{"id":"n34035","layer":"informal","project":"p40","title":"For \\(Y\\) nonempty, fix a dense sequence \\((u_n)\\) in a compatible complete metric and se…","kind":"proof","summary":"For \\(Y\\) nonempty, fix a dense sequence \\((u_n)\\) in a compatible complete metric and send \\(y…","labels":[],"detail_key":"p40"},{"id":"n34036","layer":"informal","project":"p40","title":"Novikov compact-section projection theorem, Srivastava 4.7.11","kind":"theorem","summary":"[Novikov compact-section projection theorem, Srivastava 4.7.11] For Polish \\(X,Y\\), a Borel \\(B…","labels":["thm:dst-proj"],"detail_key":"p40"},{"id":"n34037","layer":"informal","project":"p40","title":"Compactify the fibre: push \\(B\\) into \\(X\\times(N\\to[0,1])\\) along \\(id\\times e\\) for a c…","kind":"proof","summary":"Compactify the fibre: push \\(B\\) into \\(X\\times(N\\to[0,1])\\) along \\(id\\times e\\) for a continu…","labels":[],"detail_key":"p40"},{"id":"n34038","layer":"informal","project":"p40","title":"Everywhere-Borel distance map via closed balls","kind":"theorem","summary":"[Everywhere-Borel distance map via closed balls] Let \\(V:X\\toSubmodule\\,R\\,(R^d)\\) over a stand…","labels":["thm:dst-infdist"],"detail_key":"p40"},{"id":"n34039","layer":"informal","project":"p40","title":"By \\textttmeasurable\\_of\\_Iic it suffices that each sublevel \\(\\x\\midinfDist\\,c\\,(V x)\\le…","kind":"proof","summary":"By \\textttmeasurable\\_of\\_Iic it suffices that each sublevel \\(\\x\\midinfDist\\,c\\,(V x)\\le r\\\\)…","labels":[],"detail_key":"p40"},{"id":"n34040","layer":"informal","project":"p40","title":"Everywhere-measurable orthogonal projector from a measurable graph","kind":"theorem","summary":"[Everywhere-measurable orthogonal projector from a measurable graph] Over a standard Borel base…","labels":["thm:dst-projmatrix-converter"],"detail_key":"p40"},{"id":"n34041","layer":"informal","project":"p40","title":"Reduce to entrywise measurability (\\textttMatrix carries the Pi structure). Each entry is…","kind":"proof","summary":"Reduce to entrywise measurability (\\textttMatrix carries the Pi structure). Each entry is a pro…","labels":[],"detail_key":"p40"},{"id":"n34042","layer":"informal","project":"p40","title":"A.e.-measurable projector from a measurable graph","kind":"theorem","summary":"[A.e.-measurable projector from a measurable graph] For any s-finite measure \\(\\mu\\) on a stand…","labels":["thm:dst-aeconverter"],"detail_key":"p40"},{"id":"n34043","layer":"informal","project":"p40","title":"The same polarisation reduction, but the distance sublevels are obtained only up to a nul…","kind":"proof","summary":"The same polarisation reduction, but the distance sublevels are obtained only up to a null set,…","labels":[],"detail_key":"p40"},{"id":"n34044","layer":"informal","project":"p40","title":"Issue \\#11 headline: everywhere-Borel forward Lyapunov projector","kind":"theorem","summary":"[Issue \\#11 headline: everywhere-Borel forward Lyapunov projector] Let \\(A:X\\toMatrix_d(R)\\) be…","labels":["thm:dst-headline"],"detail_key":"p40"},{"id":"n34045","layer":"informal","project":"p40","title":"ErgodicTheory.measurableSet_graph_lambdaSublevel","kind":"proof","summary":"The sublevel filtration has a measurable graph built from the measurable data \\(A,T\\) and the e…","labels":[],"detail_key":"p40"},{"id":"n34046","layer":"informal","project":"p40","title":"The a.e.\\ sublevel projector, issue \\#6","kind":"theorem","summary":"[The a.e.\\ sublevel projector, issue \\#6] For the invertible-MET data (ergodic, measure-preserv…","labels":["thm:dst-aemeasurable"],"detail_key":"p40"},{"id":"n34047","layer":"informal","project":"p40","title":"Here the ultrametric-growth gate holds only a.e.\\ (\\textttisUltrametricGrowth\\_lambdaBar,…","kind":"proof","summary":"Here the ultrametric-growth gate holds only a.e.\\ (\\textttisUltrametricGrowth\\_lambdaBar, itsel…","labels":[],"detail_key":"p40"},{"id":"n34048","layer":"informal","project":"p40","title":"The space of probability measures on a compact metric space is Polish","kind":"theorem","summary":"[The space of probability measures on a compact metric space is Polish] For a compact metric Bo…","labels":["thm:dst-pspace-polish"],"detail_key":"p40"},{"id":"n34049","layer":"informal","project":"p40","title":"Prokhorov's theorem makes \\(P(X)\\) compact and the L\\'evy--Prokhorov metric makes it metr…","kind":"proof","summary":"Prokhorov's theorem makes \\(P(X)\\) compact and the L\\'evy--Prokhorov metric makes it metrizable…","labels":[],"detail_key":"p40"},{"id":"n34050","layer":"informal","project":"p40","title":"Joint continuity of the pushforward","kind":"theorem","summary":"[Joint continuity of the pushforward] For a compact metric \\(X\\) and a metric \\(Y\\), the pushfo…","labels":["thm:dst-pushforward-cont"],"detail_key":"p40"},{"id":"n34051","layer":"informal","project":"p40","title":"The Billingsley mapping-theorem argument. Test against a bounded continuous \\(g : Y \\to R…","kind":"proof","summary":"The Billingsley mapping-theorem argument. Test against a bounded continuous \\(g : Y \\to R\\) via…","labels":[],"detail_key":"p40"},{"id":"n34052","layer":"informal","project":"p40","title":"The section relation","kind":"definition","summary":"[The section relation] For \\(p = (T,S,\\pi,\\mu,\\nu) \\in Params\\,X\\) and \\(s \\in C(X,X)\\), the pr…","labels":["thm:dst-sectionrel"],"detail_key":"p40"},{"id":"n34053","layer":"informal","project":"p40","title":"The section relation is closed","kind":"theorem","summary":"[The section relation is closed] \\(\\(p,s) \\mid SectionRel\\,p\\,s\\\\) is a closed subset of \\(Para…","labels":["thm:dst-sectionrel-closed"],"detail_key":"p40"},{"id":"n34054","layer":"informal","project":"p40","title":"Each of the five conjuncts is an equalizer of jointly continuous maps into a Hausdorff sp…","kind":"proof","summary":"Each of the five conjuncts is an equalizer of jointly continuous maps into a Hausdorff space: c…","labels":[],"detail_key":"p40"},{"id":"n34055","layer":"informal","project":"p40","title":"Section-existence is analytic; sealedness coanalytic","kind":"theorem","summary":"[Section-existence is analytic; sealedness coanalytic] The set \\(\\p \\mid \\exists s,\\ SectionRel…","labels":["thm:dst-section-analytic"],"detail_key":"p40"},{"id":"n34056","layer":"informal","project":"p40","title":"A closed subset of a Polish space is analytic and analyticity is preserved by continuous…","kind":"proof","summary":"A closed subset of a Polish space is analytic and analyticity is preserved by continuous images…","labels":[],"detail_key":"p40"},{"id":"n34057","layer":"informal","project":"p40","title":"Honest scope","kind":"remark","summary":"[Honest scope] This is the issue's sanctioned ``restricted class first'' reading: \\emphcontinuo…","labels":["thm:dst-section-scope"],"detail_key":"p40"},{"id":"n34058","layer":"formal","project":"p40","title":"ErgodicTheory.IntegrableLogNorm","kind":"def","summary":"X : Type u_1 → d : Nat → [inst : MeasurableSpace X] → (X → Matrix (Fin d) (Fin d) Real) → Measu…","labels":[],"detail_key":"p40","name":"ErgodicTheory.IntegrableLogNorm","module":"ErgodicTheory.Cocycle.Basic"},{"id":"n34059","layer":"formal","project":"p40","title":"ErgodicTheory.cocycle","kind":"def","summary":"X : Type u_1 → d : Nat → (X → Matrix (Fin d) (Fin d) Real) → (X → X) → Nat → X → Matrix (Fin d)…","labels":[],"detail_key":"p40","name":"ErgodicTheory.cocycle","module":"ErgodicTheory.Cocycle.Basic"},{"id":"n34060","layer":"formal","project":"p40","title":"ErgodicTheory.cocycle_add","kind":"theorem","summary":"∀ X : Type u_1 d : Nat (A : X → Matrix (Fin d) (Fin d) Real) (T : X → X) (m n : Nat) (x : X), E…","labels":[],"detail_key":"p40","name":"ErgodicTheory.cocycle_add","module":"ErgodicTheory.Cocycle.Basic"},{"id":"n34061","layer":"formal","project":"p40","title":"ErgodicTheory.instMeasurableSpaceMatrix","kind":"def","summary":"m : Type u_1 → n : Type u_2 → α : Type u_3 → [MeasurableSpace α] → MeasurableSpace (Matrix m n…","labels":[],"detail_key":"p40","name":"ErgodicTheory.instMeasurableSpaceMatrix","module":"ErgodicTheory.Cocycle.Basic"},{"id":"n34062","layer":"formal","project":"p40","title":"ErgodicTheory.measurable_cocycle","kind":"theorem","summary":"∀ X : Type u_1 d : Nat [inst : MeasurableSpace X] A : X → Matrix (Fin d) (Fin d) Real, Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.measurable_cocycle","module":"ErgodicTheory.Cocycle.Basic"},{"id":"n34063","layer":"formal","project":"p40","title":"ErgodicTheory.det_cocycle_ne_zero","kind":"theorem","summary":"∀ X : Type u_1 T : X → X d : Nat A : X → Matrix (Fin d) (Fin d) Real, (∀ (x : X), Ne (A 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HMul.h…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.hasSum_correlation_fourier_ne_zero","module":"ErgodicTheory.Examples.CatMapCorrExpansion"},{"id":"n34245","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.mFourierCoeff_comp_iterate","kind":"theorem","summary":"∀ (g : ErgodicTheory.CatMapToral.T2 → Complex) (k : Nat) (b : Fin 2 → Int), Eq (UnitAddTorus.mF…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.mFourierCoeff_comp_iterate","module":"ErgodicTheory.Examples.CatMapCorrExpansion"},{"id":"n34246","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.mFourierCoeff_zero_eq_integral","kind":"theorem","summary":"∀ (g : ErgodicTheory.CatMapToral.T2 → Complex), Eq (UnitAddTorus.mFourierCoeff g 0) (MeasureThe…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.mFourierCoeff_zero_eq_integral","module":"ErgodicTheory.Examples.CatMapCorrExpansion"},{"id":"n34247","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catProj","kind":"def","summary":"(Fin 2 → Real) → ErgodicTheory.CatMapToral.T2","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catProj","module":"ErgodicTheory.Examples.CatMapCover"},{"id":"n34248","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catCorr_decay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f g : ContinuousMap ErgodicTheory.CatMapToral.T2 Complex), ErgodicTh…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catCorr_decay","module":"ErgodicTheory.Examples.CatMapDecay"},{"id":"n34249","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catCorr_decay_real","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catCorr_decay_real","module":"ErgodicTheory.Examples.CatMapDecay"},{"id":"n34250","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catLift","kind":"def","summary":"EuclideanSpace Real (Fin 2) → EuclideanSpace Real (Fin 2)","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catLift","module":"ErgodicTheory.Examples.CatMapDerivativeCocycle"},{"id":"n34251","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catLift_derivativeCocycle_topExponent_pos","kind":"theorem","summary":"LT.lt 0 (ErgodicTheory.topExponent ErgodicTheory.CatMapToral.ergodic_catTorus ⋯ ⋯ ⋯ ⋯)","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catLift_derivativeCocycle_topExponent_pos","module":"ErgodicTheory.Examples.CatMapDerivativeCocycle"},{"id":"n34252","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_constCocycle_exponents","kind":"theorem","summary":"And (Eq (ErgodicTheory.exponents ErgodicTheory.CatMapToral.ergodic_catTorus ⋯ ⋯ ⋯ ⋯ ⟨0, ⋯⟩) (Re…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_constCocycle_exponents","module":"ErgodicTheory.Examples.CatMapDerivativeCocycle"},{"id":"n34253","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.coverProj_comp_catLift","kind":"theorem","summary":"∀ (x : EuclideanSpace Real (Fin 2)), Eq (ErgodicTheory.CatMapToral.coverProj (ErgodicTheory.Cat…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.coverProj_comp_catLift","module":"ErgodicTheory.Examples.CatMapDerivativeCocycle"},{"id":"n34254","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.derivativeCocycle_catLift","kind":"theorem","summary":"∀ (x : EuclideanSpace Real (Fin 2)), Eq (ErgodicTheory.derivativeCocycle ErgodicTheory.CatMapTo…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.derivativeCocycle_catLift","module":"ErgodicTheory.Examples.CatMapDerivativeCocycle"},{"id":"n34255","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catShadowSol","kind":"def","summary":"Nat → (Fin 2 → Real) → Fin 2 → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catShadowSol","module":"ErgodicTheory.Examples.CatMapEigenShadow"},{"id":"n34256","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_eigenfunction_ae_zero","kind":"theorem","summary":"∀ g : ErgodicTheory.CatMapToral.T2 → Complex l : Complex, Measurable g → (∀ (x : ErgodicTheory.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_eigenfunction_ae_zero","module":"ErgodicTheory.Examples.CatMapEigenfunction"},{"id":"n34257","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_eq","kind":"theorem","summary":"Eq (ErgodicTheory.Entropy.ksEntropy ErgodicTheory.CatMapToral.measurePreserving_catTorus) ↑(Rea…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_eq","module":"ErgodicTheory.Examples.CatMapEntropy"},{"id":"n34258","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_le","kind":"theorem","summary":"LE.le (ErgodicTheory.Entropy.ksEntropy ErgodicTheory.CatMapToral.measurePreserving_catTorus) ↑(…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_le","module":"ErgodicTheory.Examples.CatMapEntropy"},{"id":"n34259","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_gridJoinAtom_volume_le","kind":"theorem","summary":"∀ (n : Nat) (f : Fin n → Prod (Fin 5) (Fin 5)), LE.le (MeasureTheory.volume (ErgodicTheory.Entr…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_gridJoinAtom_volume_le","module":"ErgodicTheory.Examples.CatMapEntropyLower"},{"id":"n34260","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_ge","kind":"theorem","summary":"LE.le (↑(Real.log (HDiv.hDiv (HAdd.hAdd 3 (Real.sqrt 5)) 2))) (ErgodicTheory.Entropy.ksEntropy…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_ge","module":"ErgodicTheory.Examples.CatMapEntropyLower"},{"id":"n34261","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_pos","kind":"theorem","summary":"LT.lt 0 (ErgodicTheory.Entropy.ksEntropy ErgodicTheory.CatMapToral.measurePreserving_catTorus)","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_ksEntropy_pos","module":"ErgodicTheory.Examples.CatMapEntropyLower"},{"id":"n34262","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catExponent_rate","kind":"theorem","summary":"∀ (n : Nat), LE.le 1 n → LE.le (abs (HSub.hSub (HDiv.hDiv (Real.log (norm (HPow.hPow 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ErgodicTheory.CatMapToral.cat…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.const_one_not_isFlowCoboundary_catSuspension","module":"ErgodicTheory.Examples.CatMapFlowCoboundary"},{"id":"n34265","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.FourierDecay","kind":"def","summary":"Real → (ErgodicTheory.CatMapToral.T2 → Complex) → Prop","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.FourierDecay","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34266","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.bracket","kind":"def","summary":"(Fin 2 → Int) → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.bracket","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34267","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.fourierDecay_finsetSum","kind":"theorem","summary":"∀ ι : Type u_1 (sExp : Real) (s : Finset ι) (c : ι → Complex) (m : ι → Fin 2 → Int), ErgodicThe…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.fourierDecay_finsetSum","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34268","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.fourierDecay_mFourier","kind":"theorem","summary":"∀ (s : Real) (m : Fin 2 → Int), ErgodicTheory.CatMapToral.FourierDecay s ⇑(UnitAddTorus.mFourie…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.fourierDecay_mFourier","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34269","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.summable_bracket_rpow","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → Summable fun n => HPow.hPow (ErgodicTheory.CatMapToral.bracket n) (Neg.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.summable_bracket_rpow","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34270","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.tsum_bracket_rpow_tail_le","kind":"theorem","summary":"∀ s R : Real, LT.lt 2 s → LE.le 1 R → ∀ S : Set (Fin 2 → Int), (∀ (n : Fin 2 → Int), Membership…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.tsum_bracket_rpow_tail_le","module":"ErgodicTheory.Examples.CatMapFourierDecay"},{"id":"n34271","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catComp","kind":"def","summary":"Nat → LinearIsometry (RingHom.id Complex) (Subtype fun x => Membership.mem (MeasureTheory.Lp Co…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catComp","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34272","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catCorr","kind":"def","summary":"Nat → (Subtype fun x => Membership.mem (MeasureTheory.Lp Complex 2 MeasureTheory.volume) x) → (…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catCorr","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34273","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_eigenfunction_ae_zero_of_mixing","kind":"theorem","summary":"∀ g : ErgodicTheory.CatMapToral.T2 → Complex l : Complex, Measurable g → (∀ (x : ErgodicTheory.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_eigenfunction_ae_zero_of_mixing","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34274","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_mixing","kind":"theorem","summary":"∀ (A B : Set ErgodicTheory.CatMapToral.T2), MeasurableSet A → MeasurableSet B → Filter.Tendsto…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_mixing","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34275","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.eventually_pow_mulVec_ne","kind":"theorem","summary":"∀ m : Fin 2 → Int, Ne m 0 → ∀ (n : Fin 2 → Int), Filter.Eventually (fun k => Ne ((HPow.hPow Erg…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.eventually_pow_mulVec_ne","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34276","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.exists_span_approx","kind":"theorem","summary":"∀ (u : Subtype fun x => Membership.mem (MeasureTheory.Lp Complex 2 MeasureTheory.volume) x) ε :…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.exists_span_approx","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34277","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.integral_conj_mFourier_mul","kind":"theorem","summary":"∀ (a b : Fin 2 → Int), Eq (MeasureTheory.integral MeasureTheory.volume fun t => HMul.hMul ((sta…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.integral_conj_mFourier_mul","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34278","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.mFourier_iterate_catTorus","kind":"theorem","summary":"∀ (k : Nat) (n : Fin 2 → Int) (y : ErgodicTheory.CatMapToral.T2), Eq ((UnitAddTorus.mFourier n)…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.mFourier_iterate_catTorus","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34279","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.tendsto_catCorr","kind":"theorem","summary":"∀ (u v : Subtype fun x => Membership.mem (MeasureTheory.Lp Complex 2 MeasureTheory.volume) x),…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.tendsto_catCorr","module":"ErgodicTheory.Examples.CatMapMixing"},{"id":"n34280","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.Qform","kind":"def","summary":"(Fin 2 → Int) → Int","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.Qform","module":"ErgodicTheory.Examples.CatMapNormForm"},{"id":"n34281","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.Qform_ne_zero","kind":"theorem","summary":"∀ n : Fin 2 → Int, Ne n 0 → Ne (ErgodicTheory.CatMapToral.Qform n) 0","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.Qform_ne_zero","module":"ErgodicTheory.Examples.CatMapNormForm"},{"id":"n34282","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.Qform_pow_mulVec","kind":"theorem","summary":"∀ (k : Nat) (n : Fin 2 → Int), Eq (ErgodicTheory.CatMapToral.Qform ((HPow.hPow ErgodicTheory.Ca…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.Qform_pow_mulVec","module":"ErgodicTheory.Examples.CatMapNormForm"},{"id":"n34283","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.aplus_pow","kind":"theorem","summary":"∀ (k : Nat) (v : Fin 2 → Real), Eq (ErgodicTheory.CatMapToral.aplus ((HPow.hPow ErgodicTheory.C…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.aplus_pow","module":"ErgodicTheory.Examples.CatMapNormForm"},{"id":"n34284","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.lemma_beta","kind":"theorem","summary":"∀ n : Fin 2 → Int, Ne n 0 → ∀ (k : Nat), LE.le (HDiv.hDiv (HMul.hMul (HSub.hSub (Real.sqrt 5) 2…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.lemma_beta","module":"ErgodicTheory.Examples.CatMapNormForm"},{"id":"n34285","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catTorus_ksEntropyPartition_le_logLambda","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] [Nonempty ι] (P : ErgodicTheory.Entropy.MeasurePartition Meas…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catTorus_ksEntropyPartition_le_logLambda","module":"ErgodicTheory.Examples.CatMapPerPartition"},{"id":"n34286","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catQuotientFlowCocycle","kind":"def","summary":"ErgodicTheory.FlowCocycle (ErgodicTheory.suspensionFlow ErgodicTheory.CatMapToral.catTorusEquiv…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catQuotientFlowCocycle","module":"ErgodicTheory.Examples.CatMapQuotientFlowCocycle"},{"id":"n34287","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catQuotientFlowCocycle_exponent","kind":"theorem","summary":"Filter.Eventually (fun q => Filter.Tendsto (fun t => HDiv.hDiv (Real.log (norm (ErgodicTheory.C…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catQuotientFlowCocycle_exponent","module":"ErgodicTheory.Examples.CatMapQuotientFlowCocycle"},{"id":"n34288","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catAutoCorr_decay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catAutoCorr_decay","module":"ErgodicTheory.Examples.CatMapStatistics"},{"id":"n34289","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catConcentration_fourierDecay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catConcentration_fourierDecay","module":"ErgodicTheory.Examples.CatMapStatistics"},{"id":"n34290","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catGreenKubo_fourierDecay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catGreenKubo_fourierDecay","module":"ErgodicTheory.Examples.CatMapStatistics"},{"id":"n34291","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catSuspensionDecay_fourierDecay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f g : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheo…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catSuspensionDecay_fourierDecay","module":"ErgodicTheory.Examples.CatMapStatistics"},{"id":"n34292","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catVariance_linear_fourierDecay","kind":"theorem","summary":"∀ s : Real, LT.lt 2 s → ∀ (f : ContinuousMap ErgodicTheory.CatMapToral.T2 Real), (ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catVariance_linear_fourierDecay","module":"ErgodicTheory.Examples.CatMapStatistics"},{"id":"n34293","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catSuspensionFlow_ownExponent_pos","kind":"theorem","summary":"Filter.Eventually (fun q => Exists fun L => And (ErgodicTheory.HasFlowExponent (fun x => Ergodi…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catSuspensionFlow_ownExponent_pos","module":"ErgodicTheory.Examples.CatMapSuspensionFlow"},{"id":"n34294","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catSuspension_ae_hasFlowExponent","kind":"theorem","summary":"Filter.Eventually (fun q => ErgodicTheory.HasFlowExponent (fun x => ErgodicTheory.derivativeCoc…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catSuspension_ae_hasFlowExponent","module":"ErgodicTheory.Examples.CatMapSuspensionFlow"},{"id":"n34295","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catSuspension_flowExponent_eq_base_div_roof","kind":"theorem","summary":"Filter.Eventually (fun q => ErgodicTheory.HasFlowExponent (fun x => ErgodicTheory.derivativeCoc…","labels":[],"detail_key":"p40","name":"ErgodicTheory.CatMapToral.catSuspension_flowExponent_eq_base_div_roof","module":"ErgodicTheory.Examples.CatMapSuspensionFlow"},{"id":"n34296","layer":"formal","project":"p40","title":"ErgodicTheory.CatMapToral.catSuspension_flowExponentAt_eq_base_div_roof","kind":"theorem","summary":"Filter.Eventually (fun q => Eq (ErgodicTheory.flowExponentAt (fun x => 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LT.lt 0 r → ErgodicTheory.ExpClosing ErgodicTheory.doublingMap r 1 (HDiv.hDiv (HPow…","labels":[],"detail_key":"p40","name":"ErgodicTheory.expClosing_doublingMap","module":"ErgodicTheory.Livsic.DoublingClosing"},{"id":"n34418","layer":"formal","project":"p40","title":"ErgodicTheory.ergodic_exists_denseRange_iterate","kind":"theorem","summary":"∀ X : Type u_1 [inst : TopologicalSpace X] [SecondCountableTopology X] [inst_2 : MeasurableSpac…","labels":[],"detail_key":"p40","name":"ErgodicTheory.ergodic_exists_denseRange_iterate","module":"ErgodicTheory.Livsic.ErgodicDenseOrbit"},{"id":"n34419","layer":"formal","project":"p40","title":"ErgodicTheory.IsFlowCoboundary","kind":"def","summary":"Q : Type u_1 → (Real → Q → Q) → (Q → Real) → 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DiscreteTop…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Livsic.isHolderCoboundary_of_continuous_aeCoboundary","module":"ErgodicTheory.Livsic.FullShift"},{"id":"n34422","layer":"formal","project":"p40","title":"ErgodicTheory.Livsic.livsic_fullShift","kind":"theorem","summary":"∀ α₀ : Type u_1 [Nonempty α₀] [Encodable α₀] [inst : TopologicalSpace α₀] [inst_1 : DiscreteTop…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Livsic.livsic_fullShift","module":"ErgodicTheory.Livsic.FullShift"},{"id":"n34423","layer":"formal","project":"p40","title":"ErgodicTheory.Livsic.expClosing_shiftMap","kind":"theorem","summary":"∀ α₀ : Type u_1 [inst : TopologicalSpace α₀] [inst_1 : DiscreteTopology α₀] α : Real, LT.lt 0 α…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Livsic.expClosing_shiftMap","module":"ErgodicTheory.Livsic.FullShiftClosing"},{"id":"n34424","layer":"formal","project":"p40","title":"ErgodicTheory.exists_holderWith_extension","kind":"theorem","summary":"∀ X : 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Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.IsHolderFlowCoboundary.isFlowCoboundary","module":"ErgodicTheory.Livsic.HolderFlowCoboundary"},{"id":"n34427","layer":"formal","project":"p40","title":"ErgodicTheory.holderWith_inducedBaseCocycle","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X] (T : Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.holderWith_inducedBaseCocycle","module":"ErgodicTheory.Livsic.HolderFlowCoboundary"},{"id":"n34428","layer":"formal","project":"p40","title":"ErgodicTheory.holderWith_suspTransfer","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X] (T : Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.holderWith_suspTransfer","module":"ErgodicTheory.Livsic.HolderFlowCoboundary"},{"id":"n34429","layer":"formal","project":"p40","title":"ErgodicTheory.livsic_holderFlow_constRoof","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X] (T : Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.livsic_holderFlow_constRoof","module":"ErgodicTheory.Livsic.HolderFlowCoboundary"},{"id":"n34430","layer":"formal","project":"p40","title":"ErgodicTheory.livsic_holderFlow_constRoof_orbitIntegral","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X] (T : Measurab…","labels":[],"detail_key":"p40","name":"ErgodicTheory.livsic_holderFlow_constRoof_orbitIntegral","module":"ErgodicTheory.Livsic.HolderFlowCoboundary"},{"id":"n34431","layer":"formal","project":"p40","title":"ErgodicTheory.livsic_holderFlow_varRoof","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [inst_2 : BorelSpace X] (T :…","labels":[],"detail_key":"p40","name":"ErgodicTheory.livsic_holderFlow_varRoof","module":"ErgodicTheory.Livsic.HolderFlowCoboundaryVar"},{"id":"n34432","layer":"formal","project":"p40","title":"ErgodicTheory.uCover_gen_var","kind":"theorem","summary":"∀ X : Type u_1 [inst : MetricSpace X] [inst_1 : MeasurableSpace X] [BorelSpace X] (T : 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fun x₀ => DenseRange fu…","labels":[],"detail_key":"p40","name":"ErgodicTheory.exists_denseRange_sftShiftMap_orbit","module":"ErgodicTheory.Livsic.SubshiftDenseOrbit"},{"id":"n34440","layer":"formal","project":"p40","title":"ErgodicTheory.goldenMeanM","kind":"def","summary":"Fin 2 → Fin 2 → Bool","labels":[],"detail_key":"p40","name":"ErgodicTheory.goldenMeanM","module":"ErgodicTheory.Livsic.SubshiftDenseOrbit"},{"id":"n34441","layer":"formal","project":"p40","title":"ErgodicTheory.goldenMean_proper","kind":"theorem","summary":"Not (Membership.mem (ErgodicTheory.SFTCarrier ErgodicTheory.goldenMeanM) fun x => 1)","labels":[],"detail_key":"p40","name":"ErgodicTheory.goldenMean_proper","module":"ErgodicTheory.Livsic.SubshiftDenseOrbit"},{"id":"n34442","layer":"formal","project":"p40","title":"ErgodicTheory.livsic_goldenMean","kind":"theorem","summary":"∀ C r : NNReal φ : ErgodicTheory.SFT ErgodicTheory.goldenMeanM → Real, HolderWith C r φ → 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(ErgodicTheor…","labels":[],"detail_key":"p40","name":"ErgodicTheory.expClosing_sftShiftMap","module":"ErgodicTheory.Livsic.SubshiftFiniteType"},{"id":"n34446","layer":"formal","project":"p40","title":"ErgodicTheory.livsic_sft","kind":"theorem","summary":"∀ k : Nat M : Fin k → Fin k → Bool C r : NNReal φ : ErgodicTheory.SFT M → Real, HolderWith C r…","labels":[],"detail_key":"p40","name":"ErgodicTheory.livsic_sft","module":"ErgodicTheory.Livsic.SubshiftFiniteType"},{"id":"n34447","layer":"formal","project":"p40","title":"ErgodicTheory.sftShiftMap","kind":"def","summary":"k : Nat → (M : Fin k → Fin k → Bool) → ErgodicTheory.SFT M → ErgodicTheory.SFT M","labels":[],"detail_key":"p40","name":"ErgodicTheory.sftShiftMap","module":"ErgodicTheory.Livsic.SubshiftFiniteType"},{"id":"n34448","layer":"formal","project":"p40","title":"ErgodicTheory.sft_head_eq_of_dist_le_half","kind":"theorem","summary":"∀ k : Nat M : Fin k → Fin k → Bool (x : ErgodicTheory.SFT M) (n : Nat), LE.le (dist x 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[MeasureTheory.Is…","labels":[],"detail_key":"p40","name":"ErgodicTheory.oseledets_filtration'","module":"ErgodicTheory.Lyapunov.Extensions.Corollaries"},{"id":"n34457","layer":"formal","project":"p40","title":"ErgodicTheory.oseledets_filtration_with_multiplicities","kind":"theorem","summary":"∀ X : Type u_1 [inst : MeasurableSpace X] d : Nat μ : MeasureTheory.Measure X [MeasureTheory.Is…","labels":[],"detail_key":"p40","name":"ErgodicTheory.oseledets_filtration_with_multiplicities","module":"ErgodicTheory.Lyapunov.Extensions.Corollaries"},{"id":"n34458","layer":"formal","project":"p40","title":"ErgodicTheory.oseledets_top_exponent_eq_furstenbergKesten","kind":"theorem","summary":"∀ X : Type u_1 [inst : MeasurableSpace X] d : Nat μ : MeasureTheory.Measure X T : X → X A : X 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MeasureTh…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDimFlow","module":"ErgodicTheory.Multifractal.Measure"},{"id":"n34586","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiDimFlow_antitone","kind":"theorem","summary":"∀ ι : Type u_2 [inst : Fintype ι] X : Type u_3 [inst_1 : MeasurableSpace X] μ : MeasureTheory.M…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDimFlow_antitone","module":"ErgodicTheory.Multifractal.Measure"},{"id":"n34587","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiDimMeasure","kind":"def","summary":"α : Type u_1 → ι : Type u_2 → [inst : MeasurableSpace α] → [inst_1 : Fintype ι] → (μ : MeasureT…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDimMeasure","module":"ErgodicTheory.Multifractal.Measure"},{"id":"n34588","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiDimMeasure_antitone","kind":"theorem","summary":"∀ α : Type u_1 ι : Type u_2 [inst : MeasurableSpace α] [inst_1 : Fintype ι] μ : MeasureTheory.M…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDimMeasure_antitone","module":"ErgodicTheory.Multifractal.Measure"},{"id":"n34589","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiDimMeasure_one_eq","kind":"theorem","summary":"∀ α : Type u_1 ι : Type u_2 [inst : MeasurableSpace α] [inst_1 : Fintype ι] μ : MeasureTheory.M…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDimMeasure_one_eq","module":"ErgodicTheory.Multifractal.Measure"},{"id":"n34590","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiDim_antitone","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] p : ι → Real, (∀ (i : ι), LE.le 0 (p i)) → (Exists fun i => L…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiDim_antitone","module":"ErgodicTheory.Multifractal.Monotone"},{"id":"n34591","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiRateSup_bern","kind":"theorem","summary":"∀ A : Type u_1 [inst : Fintype A] [Nonempty A] [inst_2 : MeasurableSpace A] [MeasurableSingleto…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiRateSup_bern","module":"ErgodicTheory.Multifractal.RenyiBernoulli"},{"id":"n34592","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiRate_strict_drop_uniformFin3","kind":"theorem","summary":"LT.lt (ErgodicTheory.Multifractal.renyiRateSup (MeasureTheory.Measure.map (ErgodicTheory.Multif…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiRate_strict_drop_uniformFin3","module":"ErgodicTheory.Multifractal.RenyiBernoulli"},{"id":"n34593","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.mergedWeights","kind":"def","summary":"ι : Type u_1 → κ : Type u_2 → [Fintype ι] → [DecidableEq κ] → (ι → κ) → (ι → Real) → κ → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.mergedWeights","module":"ErgodicTheory.Multifractal.RenyiEntropy"},{"id":"n34594","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.partitionFunction_merge_ge","kind":"theorem","summary":"∀ ι : Type u_1 κ : Type u_2 [inst : Fintype ι] [inst_1 : Fintype κ] [inst_2 : DecidableEq κ] (f…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.partitionFunction_merge_ge","module":"ErgodicTheory.Multifractal.RenyiEntropy"},{"id":"n34595","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiEntropy","kind":"def","summary":"ι : Type u_1 → [Fintype ι] → (ι → Real) → Real → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiEntropy","module":"ErgodicTheory.Multifractal.RenyiEntropy"},{"id":"n34596","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiEntropy_merge_le","kind":"theorem","summary":"∀ ι : Type u_1 κ : Type u_2 [inst : Fintype ι] [inst_1 : Fintype κ] [inst_2 : DecidableEq κ] (f…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiEntropy_merge_le","module":"ErgodicTheory.Multifractal.RenyiEntropy"},{"id":"n34597","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.blockCode","kind":"def","summary":"A : Type u_1 → B : Type u_2 → (A → B) → ErgodicTheory.Multifractal.Shift A → ErgodicTheory.Mult…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.blockCode","module":"ErgodicTheory.Multifractal.RenyiRate"},{"id":"n34598","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiEntropySeq_map_blockCode_le","kind":"theorem","summary":"∀ A : Type u_1 [inst : Fintype A] [Nonempty A] [inst_2 : MeasurableSpace A] [MeasurableSingleto…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiEntropySeq_map_blockCode_le","module":"ErgodicTheory.Multifractal.RenyiRate"},{"id":"n34599","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiRateSup","kind":"def","summary":"A : Type u_1 → [Fintype A] → [inst : MeasurableSpace A] → MeasureTheory.Measure (ErgodicTheory.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiRateSup","module":"ErgodicTheory.Multifractal.RenyiRate"},{"id":"n34600","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.renyiRateSup_map_blockCode_le","kind":"theorem","summary":"∀ A : Type u_1 [inst : Fintype A] [Nonempty A] [inst_2 : MeasurableSpace A] [MeasurableSingleto…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.renyiRateSup_map_blockCode_le","module":"ErgodicTheory.Multifractal.RenyiRate"},{"id":"n34601","layer":"formal","project":"p40","title":"ErgodicTheory.Multifractal.dimH_eq_ksEntropy_div_log_two","kind":"theorem","summary":"∀ α₀ : Type u_1 [inst : Fintype α₀] [Nonempty α₀] [inst_2 : TopologicalSpace α₀] [inst_3 : Disc…","labels":[],"detail_key":"p40","name":"ErgodicTheory.Multifractal.dimH_eq_ksEntropy_div_log_two","module":"ErgodicTheory.Multifractal.SymbolicDimension"},{"id":"n34602","layer":"formal","project":"p40","title":"ErgodicTheory.oseledets_filtration","kind":"theorem","summary":"∀ X : Type u_1 [inst : MeasurableSpace X] d : Nat μ : MeasureTheory.Measure X [MeasureTheory.Is…","labels":[],"detail_key":"p40","name":"ErgodicTheory.oseledets_filtration","module":"ErgodicTheory.MultiplicativeErgodic"},{"id":"n34603","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.DensityMatrix","kind":"inductive","summary":"(n : Type u_2) → [Fintype n] → [DecidableEq n] → Type u_2","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.DensityMatrix","module":"ErgodicTheory.OperatorEntropy.Basic"},{"id":"n34604","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.DensityMatrix.sum_eigenvalues_eq_one","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.DensityMatrix.sum_eigenvalues_eq_one","module":"ErgodicTheory.OperatorEntropy.Basic"},{"id":"n34605","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy","kind":"def","summary":"n : Type u_1 → [inst : Fintype n] → [inst_1 : DecidableEq n] → ErgodicTheory.OperatorEntropy.De…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy","module":"ErgodicTheory.OperatorEntropy.Basic"},{"id":"n34606","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_nonneg","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_nonneg","module":"ErgodicTheory.OperatorEntropy.Basic"},{"id":"n34607","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.adPerm","kind":"def","summary":"d : Nat → Equiv.Perm (Fin d) → ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.adPerm","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34608","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian","kind":"def","summary":"d : Nat → (μ : Fin d → ENNReal) → Eq (Finset.univ.sum fun i => μ i) 1 → Equiv.Perm (Fin d) → ER…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34609","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_eq_ksEntropy","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_eq_ksEntropy","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34610","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntEntropyPartition_eq_ksEntropyPartition","kind":"theorem","summary":"∀ d k : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (F…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntEntropyPartition_eq_ksEntropyPartition","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34611","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.densityOfPMF","kind":"def","summary":"d : Nat → (μ : Fin d → ENNReal) → Eq (Finset.univ.sum fun i => μ i) 1 → ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.densityOfPMF","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34612","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.ksEntropy_le_cntDynamicalEntropy","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.ksEntropy_le_cntDynamicalEntropy","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34613","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.projPartition","kind":"def","summary":"d k : Nat → (Fin d → Fin k) → ErgodicTheory.OperatorEntropy.CNT.OperationalPartition d k","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.projPartition","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34614","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_eq_ksEntropySeq","kind":"theorem","summary":"∀ d k : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (F…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_eq_ksEntropySeq","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCorner"},{"id":"n34615","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull","kind":"def","summary":"d : Nat → (μ : Fin d → ENNReal) → Eq (Finset.univ.sum fun i => μ i) 1 → Equiv.Perm (Fin d) → ER…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34616","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull_eq_ksEntropy","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull_eq_ksEntropy","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34617","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull_eq_zero","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelianFull_eq_zero","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34618","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_eq_zero","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_eq_zero","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34619","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_le_full","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropyAbelian_le_full","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34620","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.ksEntropy_eq_cntDynamicalEntropy","kind":"theorem","summary":"∀ d : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (Fin…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.ksEntropy_eq_cntDynamicalEntropy","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianCornerFull"},{"id":"n34621","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.subadditive_vonNeumannEntropy_corrMatrix_abelian","kind":"theorem","summary":"∀ d k : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (F…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.subadditive_vonNeumannEntropy_corrMatrix_abelian","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianFekete"},{"id":"n34622","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.tendsto_cntEntropyPartition_abelian","kind":"theorem","summary":"∀ d k : Nat (μ : Fin d → ENNReal) (hμ : Eq (Finset.univ.sum fun i => μ i) 1) (σ : Equiv.Perm (F…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.tendsto_cntEntropyPartition_abelian","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianFekete"},{"id":"n34623","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cex_abelian_restriction_entropy_zero","kind":"theorem","summary":"∀ k : Nat (X : ErgodicTheory.OperatorEntropy.CNT.OperationalPartition 2 k), (∀ (i : Fin k), (X.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cex_abelian_restriction_entropy_zero","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianRestriction"},{"id":"n34624","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cex_strictly_above_abelian","kind":"theorem","summary":"And (∀ k : Nat (X : ErgodicTheory.OperatorEntropy.CNT.OperationalPartition 2 k), (∀ (i : Fin k)…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cex_strictly_above_abelian","module":"ErgodicTheory.OperatorEntropy.CNT.AbelianRestriction"},{"id":"n34625","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropy","kind":"def","summary":"d : Nat → ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d → ErgodicTheory.OperatorEntropy.De…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropy","module":"ErgodicTheory.OperatorEntropy.CNT.Construction"},{"id":"n34626","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.corrMatrix","kind":"def","summary":"d : Nat → ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d → ErgodicTheory.OperatorEntropy.De…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.corrMatrix","module":"ErgodicTheory.OperatorEntropy.CNT.Construction"},{"id":"n34627","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntCumulativeEntropy_le_reservoir","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntCumulativeEntropy_le_reservoir","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34628","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropy_eq_zero","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntDynamicalEntropy_eq_zero","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34629","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntEntropyPartition_eq_zero","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntEntropyPartition_eq_zero","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34630","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.cntEntropySeq_bddAbove","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.cntEntropySeq_bddAbove","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34631","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.tendsto_cntEntropySeq_div","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.tendsto_cntEntropySeq_div","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34632","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_le_log","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_le_log","module":"ErgodicTheory.OperatorEntropy.CNT.FiniteDimZero"},{"id":"n34633","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.corrVal_eq_conjTranspose_mul_self","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.corrVal_eq_conjTranspose_mul_self","module":"ErgodicTheory.OperatorEntropy.CNT.GramFactorization"},{"id":"n34634","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.gramVec","kind":"def","summary":"d : Nat → ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d → ErgodicTheory.OperatorEntropy.De…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.gramVec","module":"ErgodicTheory.OperatorEntropy.CNT.GramFactorization"},{"id":"n34635","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.rank_corrVal_le","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) (ρ : ErgodicTheory.OperatorE…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.rank_corrVal_le","module":"ErgodicTheory.OperatorEntropy.CNT.GramFactorization"},{"id":"n34636","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.dephase_eigProj_not_commute","kind":"theorem","summary":"Not (Eq (HMul.hMul (Matrix.diagonal ErgodicTheory.OperatorEntropy.CNT.eigProj.diag) ErgodicTheo…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.dephase_eigProj_not_commute","module":"ErgodicTheory.OperatorEntropy.CNT.NonCommutativeCertificate"},{"id":"n34637","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.not_preservesDiag_qDynamics","kind":"theorem","summary":"Not (ErgodicTheory.OperatorEntropy.CNT.PreservesDiag ErgodicTheory.OperatorEntropy.CNT.qDynamic…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.not_preservesDiag_qDynamics","module":"ErgodicTheory.OperatorEntropy.CNT.NonCommutativeCertificate"},{"id":"n34638","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.qDynamics_seal_no_common_canonical_masa","kind":"theorem","summary":"And (Not (ErgodicTheory.OperatorEntropy.CNT.PreservesDiag ErgodicTheory.OperatorEntropy.CNT.qDy…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.qDynamics_seal_no_common_canonical_masa","module":"ErgodicTheory.OperatorEntropy.CNT.NonCommutativeCertificate"},{"id":"n34639","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.OperationalPartition","kind":"inductive","summary":"Nat → Nat → Type","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.OperationalPartition","module":"ErgodicTheory.OperatorEntropy.CNT.Refinement"},{"id":"n34640","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo","module":"ErgodicTheory.OperatorEntropy.CNT.Refinement"},{"id":"n34641","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.refine","kind":"def","summary":"d : Nat → ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d → k : Nat → ErgodicTheory.Operator…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.refine","module":"ErgodicTheory.OperatorEntropy.CNT.Refinement"},{"id":"n34642","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.sum_refine_conjTranspose_mul_refine","kind":"theorem","summary":"∀ d : Nat (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo d) k n : Nat (X : ErgodicTheory…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.sum_refine_conjTranspose_mul_refine","module":"ErgodicTheory.OperatorEntropy.CNT.Refinement"},{"id":"n34643","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.corrMatrix_pauliPartition_one","kind":"theorem","summary":"∀ (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo 2), Eq (ErgodicTheory.OperatorEntropy.C…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.corrMatrix_pauliPartition_one","module":"ErgodicTheory.OperatorEntropy.CNT.ReservoirSaturation"},{"id":"n34644","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.pauliPartition","kind":"def","summary":"ErgodicTheory.OperatorEntropy.CNT.OperationalPartition 2 4","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.pauliPartition","module":"ErgodicTheory.OperatorEntropy.CNT.ReservoirSaturation"},{"id":"n34645","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_pauliPartition_eq","kind":"theorem","summary":"∀ (Φ : ErgodicTheory.OperatorEntropy.CNT.UnitalStarEndo 2), Eq (ErgodicTheory.OperatorEntropy.v…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.vonNeumannEntropy_corrMatrix_pauliPartition_eq","module":"ErgodicTheory.OperatorEntropy.CNT.ReservoirSaturation"},{"id":"n34646","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.entropy_corrMatrix_one_eq_zero","kind":"theorem","summary":"Eq (ErgodicTheory.OperatorEntropy.vonNeumannEntropy (ErgodicTheory.OperatorEntropy.CNT.corrMatr…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.entropy_corrMatrix_one_eq_zero","module":"ErgodicTheory.OperatorEntropy.CNT.SubadditivityCounterexample"},{"id":"n34647","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.entropy_corrMatrix_two_pos","kind":"theorem","summary":"LT.lt 0 (ErgodicTheory.OperatorEntropy.vonNeumannEntropy (ErgodicTheory.OperatorEntropy.CNT.cor…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.entropy_corrMatrix_two_pos","module":"ErgodicTheory.OperatorEntropy.CNT.SubadditivityCounterexample"},{"id":"n34648","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.CNT.not_subadditive_cnt_entropySeq","kind":"theorem","summary":"Not (Subadditive fun n => ErgodicTheory.OperatorEntropy.vonNeumannEntropy (ErgodicTheory.Operat…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.CNT.not_subadditive_cnt_entropySeq","module":"ErgodicTheory.OperatorEntropy.CNT.SubadditivityCounterexample"},{"id":"n34649","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.relEntropy_maximallyMixed","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : Nonempty n] (ρ : ErgodicTh…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.relEntropy_maximallyMixed","module":"ErgodicTheory.OperatorEntropy.EntropyPure"},{"id":"n34650","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_conj","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_conj","module":"ErgodicTheory.OperatorEntropy.EntropyPure"},{"id":"n34651","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_eq_zero_of_sq_eq","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_eq_zero_of_sq_eq","module":"ErgodicTheory.OperatorEntropy.EntropyPure"},{"id":"n34652","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.DensityMatrix.rank_pos","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.DensityMatrix.rank_pos","module":"ErgodicTheory.OperatorEntropy.EntropyRank"},{"id":"n34653","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_le_log_card","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_le_log_card","module":"ErgodicTheory.OperatorEntropy.EntropyRank"},{"id":"n34654","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_le_log_rank","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_le_log_rank","module":"ErgodicTheory.OperatorEntropy.EntropyRank"},{"id":"n34655","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_pos_of_sq_ne","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] (ρ : ErgodicTheory.OperatorEntropy.D…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.vonNeumannEntropy_pos_of_sq_ne","module":"ErgodicTheory.OperatorEntropy.EntropyStrictPos"},{"id":"n34656","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.appendQubit","kind":"def","summary":"n : Nat → Matrix (ErgodicTheory.OperatorEntropy.Qbits n) (ErgodicTheory.OperatorEntropy.Qbits n…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.appendQubit","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainAlgebra"},{"id":"n34657","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.appendQubit_injective","kind":"theorem","summary":"∀ (n : Nat), Function.Injective ErgodicTheory.OperatorEntropy.appendQubit","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.appendQubit_injective","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainAlgebra"},{"id":"n34658","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.appendQubit_maximallyMixed_pairing","kind":"theorem","summary":"∀ (n : Nat) (x : Matrix (ErgodicTheory.OperatorEntropy.Qbits n) (ErgodicTheory.OperatorEntropy.…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.appendQubit_maximallyMixed_pairing","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainAlgebra"},{"id":"n34659","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.shiftAdjoinQubit_appendQubit","kind":"theorem","summary":"∀ n : Nat (M : Matrix (ErgodicTheory.OperatorEntropy.Qbits n) (ErgodicTheory.OperatorEntropy.Qb…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.shiftAdjoinQubit_appendQubit","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainAlgebra"},{"id":"n34660","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.kron_maximallyMixed","kind":"theorem","summary":"∀ nA : Type u_1 [inst : Fintype nA] [inst_1 : DecidableEq nA] [inst_2 : Nonempty nA] nB : Type…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.kron_maximallyMixed","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainState"},{"id":"n34661","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.rhoPow_maximallyMixed","kind":"theorem","summary":"∀ (n : Nat), Eq (ErgodicTheory.OperatorEntropy.rhoPow ErgodicTheory.OperatorEntropy.DensityMatr…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.rhoPow_maximallyMixed","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainState"},{"id":"n34662","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.rhoPow_shiftAdjoinQubit_pairing","kind":"theorem","summary":"∀ (ρ : ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2)) (n : Nat) (x : Matrix (ErgodicTheor…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.rhoPow_shiftAdjoinQubit_pairing","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ChainState"},{"id":"n34663","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.kms_boundary","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] ρ : Matrix n n Complex, ρ.PosDef → ∀…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.kms_boundary","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34664","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.modAut","kind":"def","summary":"n : Type u_1 → [Fintype n] → [DecidableEq n] → ρ : Matrix n n Complex → ρ.PosDef → Real → Matri…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.modAut","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34665","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.modAut_add","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] ρ : Matrix n n Complex (hρ : ρ.PosDe…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.modAut_add","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34666","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.modAut_diagState_ne_id","kind":"theorem","summary":"∀ (s : Real) (hs0 : LT.lt 0 s) (hs1 : LT.lt s 1), Not (∀ (t : Real) (a : Matrix (Fin 2) (Fin 2)…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.modAut_diagState_ne_id","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34667","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.modAut_maximallyMixed_eq_id","kind":"theorem","summary":"∀ n : Type u_1 [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : Nonempty n] (t : Real) (a…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.modAut_maximallyMixed_eq_id","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34668","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.modAut_shiftAdjoinQubit","kind":"theorem","summary":"∀ (ρ : ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2)) (hρ : ρ.val.PosDef) (n : Nat) (t :…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.modAut_shiftAdjoinQubit","module":"ErgodicTheory.OperatorEntropy.GrowingTower.ModularClock"},{"id":"n34669","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.ChainSealed","kind":"def","summary":"(r : Real) → LT.lt 0 r → LT.lt r 1 → (s : Real) → LT.lt 0 s → LT.lt s 1 → Prop","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.ChainSealed","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34670","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.QuantumBernoulliShift","kind":"inductive","summary":"Prop","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.QuantumBernoulliShift","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34671","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.chain_seal_dephase_faithful","kind":"theorem","summary":"∀ (r : Real) (hr0 : LT.lt 0 r) (hr1 : LT.lt r 1) (s : Real) (hs0 : LT.lt 0 s) (hs1 : LT.lt s 1)…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.chain_seal_dephase_faithful","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34672","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.quantumBernoulliShift_exists","kind":"theorem","summary":"Nonempty ErgodicTheory.OperatorEntropy.QuantumBernoulliShift","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.quantumBernoulliShift_exists","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34673","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.shiftIter","kind":"def","summary":"n : Nat → (k : Nat) → Matrix (ErgodicTheory.OperatorEntropy.Qbits n) (ErgodicTheory.OperatorEnt…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.shiftIter","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34674","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.shiftIter_pairing","kind":"theorem","summary":"∀ (ρ : ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2)) (n k : Nat) (x : Matrix (ErgodicThe…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.shiftIter_pairing","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34675","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.tendsto_windowEntropy_div_tracial","kind":"theorem","summary":"Filter.Tendsto (fun k => HDiv.hDiv (ErgodicTheory.OperatorEntropy.windowEntropy ErgodicTheory.O…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.tendsto_windowEntropy_div_tracial","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34676","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.windowEntropy","kind":"def","summary":"ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2) → Nat → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.windowEntropy","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34677","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.windowEntropy_tracial","kind":"theorem","summary":"∀ (k : Nat), Eq (ErgodicTheory.OperatorEntropy.windowEntropy ErgodicTheory.OperatorEntropy.Dens…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.windowEntropy_tracial","module":"ErgodicTheory.OperatorEntropy.GrowingTower.QuantumBernoulli"},{"id":"n34678","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.dephaseKronId","kind":"def","summary":"(blk : Type u_1) → [inst : Fintype blk] → [inst_1 : DecidableEq blk] → ErgodicTheory.OperatorEn…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.dephaseKronId","module":"ErgodicTheory.OperatorEntropy.GrowingTower.SealLift"},{"id":"n34679","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.quantum_seal_dephase_kron_faithful","kind":"theorem","summary":"∀ blk : Type [inst : Fintype blk] [inst_1 : DecidableEq blk] e : Type [inst_2 : Fintype e] [ins…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.quantum_seal_dephase_kron_faithful","module":"ErgodicTheory.OperatorEntropy.GrowingTower.SealLift"},{"id":"n34680","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.relEntropy_strict_drop_dephasing_kron","kind":"theorem","summary":"∀ blk : Type u_1 [inst : Fintype blk] [inst_1 : DecidableEq blk] (r : Real) (hr0 : LT.lt 0 r) (…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.relEntropy_strict_drop_dephasing_kron","module":"ErgodicTheory.OperatorEntropy.GrowingTower.SealLift"},{"id":"n34681","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.Qbits","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.Qbits","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34682","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.blockEntropy","kind":"def","summary":"ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2) → Nat → Real","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.blockEntropy","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34683","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.blockEntropy_eq","kind":"theorem","summary":"∀ (ρ : ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2)) (n : Nat), Eq (ErgodicTheory.Operat…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.blockEntropy_eq","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34684","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.blockEntropy_maximallyMixed","kind":"theorem","summary":"∀ (n : Nat), Eq (ErgodicTheory.OperatorEntropy.blockEntropy ErgodicTheory.OperatorEntropy.Densi…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.blockEntropy_maximallyMixed","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34685","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.blockEntropy_rhoR_pos","kind":"theorem","summary":"∀ (r : Real) (hr0 : LT.lt 0 r) (hr1 : LT.lt r 1) n : Nat, LT.lt 0 n → LT.lt 0 (ErgodicTheory.Op…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.blockEntropy_rhoR_pos","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34686","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.rhoPow","kind":"def","summary":"ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2) → (n : Nat) → ErgodicTheory.OperatorEntropy…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.rhoPow","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34687","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.shiftAdjoinQubit","kind":"def","summary":"n : Nat → Matrix (ErgodicTheory.OperatorEntropy.Qbits n) (ErgodicTheory.OperatorEntropy.Qbits n…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.shiftAdjoinQubit","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34688","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.tendsto_blockEntropy_div","kind":"theorem","summary":"∀ (ρ : ErgodicTheory.OperatorEntropy.DensityMatrix (Fin 2)), Filter.Tendsto (fun n => HDiv.hDiv…","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.tendsto_blockEntropy_div","module":"ErgodicTheory.OperatorEntropy.GrowingTower.Tower"},{"id":"n34689","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.GrowingQuantumWorld","kind":"inductive","summary":"Type","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.GrowingQuantumWorld","module":"ErgodicTheory.OperatorEntropy.GrowingTower.World"},{"id":"n34690","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.growingQuantumWorld_exists","kind":"theorem","summary":"Nonempty ErgodicTheory.OperatorEntropy.GrowingQuantumWorld","labels":[],"detail_key":"p40","name":"ErgodicTheory.OperatorEntropy.growingQuantumWorld_exists","module":"ErgodicTheory.OperatorEntropy.GrowingTower.World"},{"id":"n34691","layer":"formal","project":"p40","title":"ErgodicTheory.OperatorEntropy.klein_scalar","kind":"theorem","summary":"∀ K : Type u_1 M : Type u_2 [inst : Fintype K] 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(H…","labels":[],"detail_key":"p41","name":"HigherCategoryTheory.SingleSorted.UnderlyingFunctor","module":"HigherCategoryTheory.SingleSorted.Underlying.Functor"},{"id":"n34875","layer":"informal","project":"p41","title":"Types versus sets","kind":"notation","summary":"[Types versus sets] Following the convention of~\\citevidal2024higher, we use the word \\emphset…","labels":["not:types_vs_sets"],"detail_key":"p41"},{"id":"n34876","layer":"informal","project":"p41","title":"Standard categorical notation","kind":"notation","summary":"[Standard categorical notation] We adopt the following standard categorical notation: \\item \\ma…","labels":["not:standard_category_theory"],"detail_key":"p41"},{"id":"n34877","layer":"informal","project":"p41","title":"Index set","kind":"notation","summary":"[Index set] If not otherwise specified, we use I to denote a set of indices equipped with a pre…","labels":["not:index_set"],"detail_key":"p41"},{"id":"n34878","layer":"informal","project":"p41","title":"Extended natural numbers","kind":"notation","summary":"[Extended natural numbers] We write \\overlineN= N\\cup \\left\\ \\omega \\right\\ for the extended na…","labels":["not:extended_naturals"],"detail_key":"p41"},{"id":"n34879","layer":"informal","project":"p41","title":"Single-sorted category structure","kind":"definition","summary":"[Single-sorted category structure] Let C be a set whose elements will be called \\emphmorphisms.…","labels":["def:SingleSortedCategoryStruct","ax:ss_sc","ax:ss_tg","ax:ss_comp"],"detail_key":"p41"},{"id":"n34880","layer":"informal","project":"p41","title":"Pre-single-sorted category","kind":"definition","summary":"[Pre-single-sorted category] A \\emphpre-single-sorted I-category is a tuple \\[ \\mathsfC = (C, (…","labels":["def:PreSingleSortedCategory","ax:ss_glob","ax:ss_sc_tg_comp","ax:ss_right_id","ax:ss_left_id","ax:ss_assoc"],"detail_key":"p41"},{"id":"n34881","layer":"informal","project":"p41","title":"Single-sorted category","kind":"definition","summary":"[Single-sorted category] A \\emphsingle-sorted I-category is a pre-single-sorted I-category \\mat…","labels":["def:SingleSortedCategory","ax:ss_cross_glob","ax:ss_cross_sc_tg_comp","ax:ss_interchange"],"detail_key":"p41"},{"id":"n34882","layer":"informal","project":"p41","title":"Single-sorted finite category","kind":"definition","summary":"[Single-sorted finite category] Given a natural number n \\in N, a \\emphsingle-sorted n-category…","labels":["def:SingleSortedNCategory"],"detail_key":"p41"},{"id":"n34883","layer":"informal","project":"p41","title":"Pre-single-sorted one-categories lifting","kind":"definition","summary":"[Pre-single-sorted one-categories lifting] Any pre-single-sorted 1-category is, in fact, a sing…","labels":["def:PreSingleSortedCategoryLift"],"detail_key":"p41"},{"id":"n34884","layer":"informal","project":"p41","title":"Cell","kind":"definition","summary":"[Cell] Let \\mathsfC be a single-sorted I-category and k \\in I. We say that a morphism f \\in C i…","labels":["def:cell"],"detail_key":"p41"},{"id":"n34885","layer":"informal","project":"p41","title":"Target-based cell","kind":"definition","summary":"[Target-based cell] Let \\mathsfC be a single-sorted I-category and k \\in I. A morphism f \\in C…","labels":["def:cell_tg"],"detail_key":"p41"},{"id":"n34886","layer":"informal","project":"p41","title":"Equivalence of cell definitions","kind":"theorem","summary":"[Equivalence of cell definitions] Let \\mathsfC be a single-sorted I-category and k \\in I. A mor…","labels":["thm:cell_sc_iff_cell_tg"],"detail_key":"p41"},{"id":"n34887","layer":"informal","project":"p41","title":"Let f \\in C. Suppose first that f is a k-cell, i.e., \\Sc^k (f) = f. By Axiom~\\hyperref[ax…","kind":"proof","summary":"Let f \\in C. Suppose first that f is a k-cell, i.e., \\Sc^k (f) = f. By Axiom~\\hyperref[ax:ss_gl…","labels":[],"detail_key":"p41"},{"id":"n34888","layer":"informal","project":"p41","title":"Single-sorted omega category","kind":"definition","summary":"[Single-sorted omega category] A \\emphsingle-sorted \\omega-category is a single-sorted category…","labels":["def:SingleSortedOmegaCategory"],"detail_key":"p41"},{"id":"n34889","layer":"informal","project":"p41","title":"Single-sorted functor","kind":"definition","summary":"[Single-sorted functor] Let \\mathsfC = (C, (\\Sc^k, \\Tg^k, \\Pcomp^k)_k \\in I) and \\mathsfD = (D,…","labels":["def:SingleSortedFunctor","ax:ss_fun_sc_tg","ax:ss_fun_comp"],"detail_key":"p41"},{"id":"n34890","layer":"informal","project":"p41","title":"Composition of single-sorted functors","kind":"construction","summary":"[Composition of single-sorted functors] Let \\mathsfC, \\mathsfD, \\mathsfE be single-sorted I-cat…","labels":["def:SingleSortedFunctor_comp"],"detail_key":"p41"},{"id":"n34891","layer":"informal","project":"p41","title":"Let k \\in I. For all f \\in C, since F and G are single-sorted functors, we have that \\[ (…","kind":"proof","summary":"Let k \\in I. For all f \\in C, since F and G are single-sorted functors, we have that \\[ (G \\cir…","labels":[],"detail_key":"p41"},{"id":"n34892","layer":"informal","project":"p41","title":"Identity single-sorted functor","kind":"construction","summary":"[Identity single-sorted functor] Let \\mathsfC be a single-sorted I-category. Then, the identity…","labels":["def:SingleSortedFunctor_id"],"detail_key":"p41"},{"id":"n34893","layer":"informal","project":"p41","title":"It trivially follows from reflexivity.","kind":"proof","summary":"It trivially follows from reflexivity.","labels":[],"detail_key":"p41"},{"id":"n34894","layer":"informal","project":"p41","title":"Single-sorted finite functor","kind":"definition","summary":"[Single-sorted finite functor] Given a natural number n \\in N, a \\emphsingle-sorted n-functor i…","labels":["def:SingleSortedNFunctor"],"detail_key":"p41"},{"id":"n34895","layer":"informal","project":"p41","title":"Single-sorted omega functor","kind":"definition","summary":"[Single-sorted omega functor] A \\emphsingle-sorted \\omega-functor is a single-sorted functor be…","labels":["def:SingleSortedOmegaFunctor"],"detail_key":"p41"},{"id":"n34896","layer":"informal","project":"p41","title":"Category of single-sorted categories","kind":"construction","summary":"[Category of single-sorted categories] The \\emphcategory of single-sorted I-categories, denoted…","labels":["def:SingleSortedCategory_category","ax:ss_cat_obj","ax:ss_cat_mor","ax:ss_cat_comp","ax:ss_cat_id"],"detail_key":"p41"},{"id":"n34897","layer":"informal","project":"p41","title":"Composition and the identity functor are well defined by Definitions~\\refdef:SingleSorted…","kind":"proof","summary":"Composition and the identity functor are well defined by Definitions~\\refdef:SingleSortedFuncto…","labels":[],"detail_key":"p41"},{"id":"n34898","layer":"informal","project":"p41","title":"Category of single-sorted finite categories","kind":"construction","summary":"[Category of single-sorted finite categories] Given a natural number n \\in N, the \\emphcategory…","labels":["def:SingleSortedNCategory_category"],"detail_key":"p41"},{"id":"n34899","layer":"informal","project":"p41","title":"This is a particular case of Definition~\\refdef:SingleSortedCategory_category, so the res…","kind":"proof","summary":"This is a particular case of Definition~\\refdef:SingleSortedCategory_category, so the result fo…","labels":[],"detail_key":"p41"},{"id":"n34900","layer":"informal","project":"p41","title":"Category of single-sorted omega categories","kind":"construction","summary":"[Category of single-sorted omega categories] The \\emphcategory of single-sorted \\omega-categori…","labels":["def:SingleSortedOmegaCategory_category"],"detail_key":"p41"},{"id":"n34901","layer":"informal","project":"p41","title":"This is a particular case of Definition~\\refdef:SingleSortedCategory_category, so the res…","kind":"proof","summary":"This is a particular case of Definition~\\refdef:SingleSortedCategory_category, so the result fo…","labels":[],"detail_key":"p41"},{"id":"n34902","layer":"informal","project":"p41","title":"We extend the notation n\\mathsfCat to all n \\in \\overlineN, understanding that, \\item if…","kind":"notation","summary":"We extend the notation n\\mathsfCat to all n \\in \\overlineN, understanding that, \\item if n \\in…","labels":[],"detail_key":"p41"},{"id":"n34903","layer":"informal","project":"p41","title":"Discrete single-sorted finite category","kind":"construction","summary":"[Discrete single-sorted finite category] Let \\mathsfC be a single-sorted n-category, and let m…","labels":["def:SingleSortedNCategory_discrete"],"detail_key":"p41"},{"id":"n34904","layer":"informal","project":"p41","title":"Each axiom either involves only dimensions below n, reducing to the corresponding axiom o…","kind":"proof","summary":"Each axiom either involves only dimensions below n, reducing to the corresponding axiom of \\mat…","labels":[],"detail_key":"p41"},{"id":"n34905","layer":"informal","project":"p41","title":"Discrete single-sorted omega category","kind":"construction","summary":"[Discrete single-sorted omega category] Let \\mathsfC be a single-sorted n-category. The \\emphdi…","labels":["def:SingleSortedNCategory_discreteOmega"],"detail_key":"p41"},{"id":"n34906","layer":"informal","project":"p41","title":"The single-sorted category axioms follow by the same argument as in Definition~\\refdef:Si…","kind":"proof","summary":"The single-sorted category axioms follow by the same argument as in Definition~\\refdef:SingleSo…","labels":[],"detail_key":"p41"},{"id":"n34907","layer":"informal","project":"p41","title":"Discrete single-sorted finite functor","kind":"construction","summary":"[Discrete single-sorted finite functor] Let \\mathsfC and \\mathsfD be single-sorted n-categories…","labels":["def:SingleSortedNFunctor_discrete"],"detail_key":"p41"},{"id":"n34908","layer":"informal","project":"p41","title":"Each preservation axiom either involves only dimensions below n, where it follows from F…","kind":"proof","summary":"Each preservation axiom either involves only dimensions below n, where it follows from F being…","labels":[],"detail_key":"p41"},{"id":"n34909","layer":"informal","project":"p41","title":"Discrete single-sorted omega functor","kind":"construction","summary":"[Discrete single-sorted omega functor] Let \\mathsfC and \\mathsfD be single-sorted n-categories,…","labels":["def:SingleSortedNFunctor_discreteOmega"],"detail_key":"p41"},{"id":"n34910","layer":"informal","project":"p41","title":"The argument is identical to that of Definition~\\refdef:SingleSortedNFunctor_discrete.","kind":"proof","summary":"The argument is identical to that of Definition~\\refdef:SingleSortedNFunctor_discrete.","labels":[],"detail_key":"p41"},{"id":"n34911","layer":"informal","project":"p41","title":"Single-sorted discretization functor","kind":"construction","summary":"[Single-sorted discretization functor] Let n, m \\in \\overlineN with n \\leq m. The \\emphdiscreti…","labels":["def:SingleSortedDiscretizationFunctor"],"detail_key":"p41"},{"id":"n34912","layer":"informal","project":"p41","title":"Functoriality follows from the fact that the discrete constructions preserve the underlyi…","kind":"proof","summary":"Functoriality follows from the fact that the discrete constructions preserve the underlying map…","labels":[],"detail_key":"p41"},{"id":"n34913","layer":"informal","project":"p41","title":"lem:SingleSorted_underlying_source_is_cell","kind":"lemma","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every f \\in \\mathsfC and ever…","labels":["lem:SingleSorted_underlying_source_is_cell"],"detail_key":"p41"},{"id":"n34914","layer":"informal","project":"p41","title":"Applying Axiom~\\hyperref[ax:ss_cross_glob]\\ref*def:SingleSortedCategory.\\ref*ax:ss_cross_…","kind":"proof","summary":"Applying Axiom~\\hyperref[ax:ss_cross_glob]\\ref*def:SingleSortedCategory.\\ref*ax:ss_cross_glob,…","labels":[],"detail_key":"p41"},{"id":"n34915","layer":"informal","project":"p41","title":"cor:SingleSorted_underlying_source_birestriction","kind":"corollary","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every k \\in I such that k < m…","labels":["cor:SingleSorted_underlying_source_birestriction"],"detail_key":"p41"},{"id":"n34916","layer":"informal","project":"p41","title":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_source_is_cell.","kind":"proof","summary":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_source_is_cell.","labels":[],"detail_key":"p41"},{"id":"n34917","layer":"informal","project":"p41","title":"lem:SingleSorted_underlying_target_is_cell","kind":"lemma","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every f \\in \\mathsfC and ever…","labels":["lem:SingleSorted_underlying_target_is_cell"],"detail_key":"p41"},{"id":"n34918","layer":"informal","project":"p41","title":"Applying Axiom~\\hyperref[ax:ss_cross_glob]\\ref*def:SingleSortedCategory.\\ref*ax:ss_cross_…","kind":"proof","summary":"Applying Axiom~\\hyperref[ax:ss_cross_glob]\\ref*def:SingleSortedCategory.\\ref*ax:ss_cross_glob,…","labels":[],"detail_key":"p41"},{"id":"n34919","layer":"informal","project":"p41","title":"cor:SingleSorted_underlying_target_birestriction","kind":"corollary","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every k \\in I such that k < m…","labels":["cor:SingleSorted_underlying_target_birestriction"],"detail_key":"p41"},{"id":"n34920","layer":"informal","project":"p41","title":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_target_is_cell.","kind":"proof","summary":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_target_is_cell.","labels":[],"detail_key":"p41"},{"id":"n34921","layer":"informal","project":"p41","title":"lem:SingleSorted_underlying_comp_is_cell","kind":"lemma","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every f, g \\in C_m and every…","labels":["lem:SingleSorted_underlying_comp_is_cell"],"detail_key":"p41"},{"id":"n34922","layer":"informal","project":"p41","title":"Applying Axiom~\\hyperref[ax:ss_cross_sc_tg_comp]\\ref*def:SingleSortedCategory.\\ref*ax:ss_…","kind":"proof","summary":"Applying Axiom~\\hyperref[ax:ss_cross_sc_tg_comp]\\ref*def:SingleSortedCategory.\\ref*ax:ss_cross_…","labels":[],"detail_key":"p41"},{"id":"n34923","layer":"informal","project":"p41","title":"cor:SingleSorted_underlying_comp_birestriction","kind":"corollary","summary":"Let \\mathsfC be a single-sorted I-category and m \\in I. Then, for every k \\in I such that k < m…","labels":["cor:SingleSorted_underlying_comp_birestriction"],"detail_key":"p41"},{"id":"n34924","layer":"informal","project":"p41","title":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_comp_is_cell.","kind":"proof","summary":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_comp_is_cell.","labels":[],"detail_key":"p41"},{"id":"n34925","layer":"informal","project":"p41","title":"Underlying single-sorted finite category","kind":"construction","summary":"[Underlying single-sorted finite category] Let \\mathsfC be a single-sorted n-category, and let…","labels":["def:SingleSortedNCategory_underlying"],"detail_key":"p41"},{"id":"n34926","layer":"informal","project":"p41","title":"The birestrictions are well-defined by Corollaries~\\refcor:SingleSorted_underlying_source…","kind":"proof","summary":"The birestrictions are well-defined by Corollaries~\\refcor:SingleSorted_underlying_source_bires…","labels":[],"detail_key":"p41"},{"id":"n34927","layer":"informal","project":"p41","title":"Underlying single-sorted omega category","kind":"construction","summary":"[Underlying single-sorted omega category] Let \\mathsfC be a single-sorted \\omega-category, and…","labels":["def:SingleSortedOmegaCategory_underlying"],"detail_key":"p41"},{"id":"n34928","layer":"informal","project":"p41","title":"The single-sorted category axioms follow by the same argument as in Definition~\\refdef:Si…","kind":"proof","summary":"The single-sorted category axioms follow by the same argument as in Definition~\\refdef:SingleSo…","labels":[],"detail_key":"p41"},{"id":"n34929","layer":"informal","project":"p41","title":"lem:SingleSorted_underlying_functor_is_cell","kind":"lemma","summary":"Let \\mathsfC and \\mathsfD be single-sorted I-categories, F \\colon \\mathsfC \\to \\mathsfD be a si…","labels":["lem:SingleSorted_underlying_functor_is_cell"],"detail_key":"p41"},{"id":"n34930","layer":"informal","project":"p41","title":"Applying Axiom~\\hyperref[ax:ss_fun_sc_tg]\\ref*def:SingleSortedFunctor.\\ref*ax:ss_fun_sc_t…","kind":"proof","summary":"Applying Axiom~\\hyperref[ax:ss_fun_sc_tg]\\ref*def:SingleSortedFunctor.\\ref*ax:ss_fun_sc_tg and…","labels":[],"detail_key":"p41"},{"id":"n34931","layer":"informal","project":"p41","title":"cor:SingleSorted_underlying_functor_birestriction","kind":"corollary","summary":"Let \\mathsfC and \\mathsfD be single-sorted I-categories, F \\colon \\mathsfC \\to \\mathsfD be a si…","labels":["cor:SingleSorted_underlying_functor_birestriction"],"detail_key":"p41"},{"id":"n34932","layer":"informal","project":"p41","title":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_functor_is_cell.","kind":"proof","summary":"Follows directly from Lemma~\\reflem:SingleSorted_underlying_functor_is_cell.","labels":[],"detail_key":"p41"},{"id":"n34933","layer":"informal","project":"p41","title":"Underlying single-sorted finite functor","kind":"construction","summary":"[Underlying single-sorted finite functor] Let \\mathsfC and \\mathsfD be single-sorted n-categori…","labels":["def:SingleSortedNFunctor_underlying"],"detail_key":"p41"},{"id":"n34934","layer":"informal","project":"p41","title":"The birestriction is well-defined by Corollary~\\refcor:SingleSorted_underlying_functor_bi…","kind":"proof","summary":"The birestriction is well-defined by Corollary~\\refcor:SingleSorted_underlying_functor_birestri…","labels":[],"detail_key":"p41"},{"id":"n34935","layer":"informal","project":"p41","title":"Underlying single-sorted omega functor","kind":"construction","summary":"[Underlying single-sorted omega functor] Let \\mathsfC and \\mathsfD be single-sorted \\omega-cate…","labels":["def:SingleSortedOmegaFunctor_underlying"],"detail_key":"p41"},{"id":"n34936","layer":"informal","project":"p41","title":"The argument is identical to that of Definition~\\refdef:SingleSortedNFunctor_underlying.","kind":"proof","summary":"The argument is identical to that of Definition~\\refdef:SingleSortedNFunctor_underlying.","labels":[],"detail_key":"p41"},{"id":"n34937","layer":"informal","project":"p41","title":"Single-sorted underlying functor","kind":"construction","summary":"[Single-sorted underlying functor] Let n, m \\in \\overlineN with m \\leq n. The \\emphunderlying f…","labels":["def:SingleSortedUnderlyingFunctor"],"detail_key":"p41"},{"id":"n34938","layer":"informal","project":"p41","title":"Functoriality follows from the fact that the underlying constructions preserve the birest…","kind":"proof","summary":"Functoriality follows from the fact that the underlying constructions preserve the birestrictio…","labels":[],"detail_key":"p41"},{"id":"n34939","layer":"informal","project":"p41","title":"Many-sorted category structure","kind":"definition","summary":"[Many-sorted category structure] Let (C_k)_k \\in I be a family of sets whose elements will be c…","labels":["def:ManySortedCategoryStruct","ax:ms_sc","ax:ms_tg","ax:ms_idm","ax:ms_comp"],"detail_key":"p41"},{"id":"n34940","layer":"informal","project":"p41","title":"Pre-many-sorted category","kind":"definition","summary":"[Pre-many-sorted category] A \\emphpre-many-sorted I-category is a tuple \\[ \\mathsfC = ((C_k)_k…","labels":["def:PreManySortedCategory","ax:ms_sc_tg_comp","ax:ms_sc_tg_idm","ax:ms_id_laws","ax:ms_assoc"],"detail_key":"p41"},{"id":"n34941","layer":"informal","project":"p41","title":"prop:PreManySortedCategory.injetive_idm","kind":"proposition","summary":"Let \\mathsfC be a pre-many-sorted I-category and let j < k in I. Then the identity map \\Idm^(k,…","labels":["prop:PreManySortedCategory.injetive_idm"],"detail_key":"p41"},{"id":"n34942","layer":"informal","project":"p41","title":"Let f, g \\in C_j with \\Idm^(k,j)(f) = \\Idm^(k,j)(g). Then \\[ f = \\Sc^(k,j)(\\Idm^(k,j)(f))…","kind":"proof","summary":"Let f, g \\in C_j with \\Idm^(k,j)(f) = \\Idm^(k,j)(g). Then \\[ f = \\Sc^(k,j)(\\Idm^(k,j)(f)) = \\Sc…","labels":[],"detail_key":"p41"},{"id":"n34943","layer":"informal","project":"p41","title":"Many-sorted category","kind":"definition","summary":"[Many-sorted category] A \\emphmany-sorted I-category is a pre-many-sorted I-category \\mathsfC =…","labels":["def:ManySortedCategory","ax:ms_cross_glob","ax:ms_cross_sc_tg_comp","ax:ms_idm_idm","ax:ms_idm_comp","ax:ms_interchange"],"detail_key":"p41"},{"id":"n34944","layer":"informal","project":"p41","title":"Many-sorted finite category","kind":"definition","summary":"[Many-sorted finite category] Given a natural number n \\in N, a \\emphmany-sorted n-category is…","labels":["def:ManySortedNCategory"],"detail_key":"p41"},{"id":"n34945","layer":"informal","project":"p41","title":"Pre-many-sorted one-categories lifting","kind":"definition","summary":"[Pre-many-sorted one-categories lifting] Any pre-many-sorted 1-category is, in fact, a many-sor…","labels":["def:PreManySortedCategoryLift"],"detail_key":"p41"},{"id":"n34946","layer":"informal","project":"p41","title":"Many-sorted omega category","kind":"definition","summary":"[Many-sorted omega category] A \\emphmany-sorted \\omega-category is a many-sorted category index…","labels":["def:ManySortedOmegaCategory"],"detail_key":"p41"},{"id":"n34947","layer":"informal","project":"p41","title":"Many-sorted functor","kind":"definition","summary":"[Many-sorted functor] Let \\mathsfC = ((C_k)_k \\in I, (\\Sc^(k,j), \\Tg^(k,j), \\Idm^(k,j), \\circ^(…","labels":["def:ManySortedFunctor","ax:ms_fun_sc_tg","ax:ms_fun_idm","ax:ms_fun_comp"],"detail_key":"p41"},{"id":"n34948","layer":"informal","project":"p41","title":"Composition of many-sorted functors","kind":"construction","summary":"[Composition of many-sorted functors] Let \\mathsfC, \\mathsfD, \\mathsfE be many-sorted I-categor…","labels":["def:ManySortedFunctor_comp"],"detail_key":"p41"},{"id":"n34949","layer":"informal","project":"p41","title":"Let k, j \\in I with j < k. For all f \\in C_k, since F and G are many-sorted functors, we…","kind":"proof","summary":"Let k, j \\in I with j < k. For all f \\in C_k, since F and G are many-sorted functors, we have t…","labels":[],"detail_key":"p41"},{"id":"n34950","layer":"informal","project":"p41","title":"Identity many-sorted functor","kind":"construction","summary":"[Identity many-sorted functor] Let \\mathsfC be a many-sorted I-category. Then, the family of id…","labels":["def:ManySortedFunctor_id"],"detail_key":"p41"},{"id":"n34951","layer":"informal","project":"p41","title":"It trivially follows from reflexivity.","kind":"proof","summary":"It trivially follows from reflexivity.","labels":[],"detail_key":"p41"},{"id":"n34952","layer":"informal","project":"p41","title":"Many-sorted finite functor","kind":"definition","summary":"[Many-sorted finite functor] Given a natural number n \\in N, a \\emphmany-sorted n-functor is a…","labels":["def:ManySortedNFunctor"],"detail_key":"p41"},{"id":"n34953","layer":"informal","project":"p41","title":"Many-sorted omega functor","kind":"definition","summary":"[Many-sorted omega functor] A \\emphmany-sorted \\omega-functor is a many-sorted functor between…","labels":["def:ManySortedOmegaFunctor"],"detail_key":"p41"},{"id":"n34954","layer":"informal","project":"p41","title":"Category of many-sorted categories","kind":"construction","summary":"[Category of many-sorted categories] The \\emphcategory of many-sorted I-categories, denoted by…","labels":["def:ManySortedCategory_category","ax:ms_cat_obj","ax:ms_cat_mor","ax:ms_cat_comp","ax:ms_cat_id"],"detail_key":"p41"},{"id":"n34955","layer":"informal","project":"p41","title":"Composition and the identity functor are well defined by Definitions~\\refdef:ManySortedFu…","kind":"proof","summary":"Composition and the identity functor are well defined by Definitions~\\refdef:ManySortedFunctor_…","labels":[],"detail_key":"p41"},{"id":"n34956","layer":"informal","project":"p41","title":"Category of many-sorted finite categories","kind":"construction","summary":"[Category of many-sorted finite categories] Given a natural number n \\in N, the \\emphcategory o…","labels":["def:ManySortedNCategory_category"],"detail_key":"p41"},{"id":"n34957","layer":"informal","project":"p41","title":"This is a particular case of Definition~\\refdef:ManySortedCategory_category, so the resul…","kind":"proof","summary":"This is a particular case of Definition~\\refdef:ManySortedCategory_category, so the result foll…","labels":[],"detail_key":"p41"},{"id":"n34958","layer":"informal","project":"p41","title":"Category of many-sorted omega categories","kind":"construction","summary":"[Category of many-sorted omega categories] The \\emphcategory of many-sorted \\omega-categories,…","labels":["def:ManySortedOmegaCategory_category"],"detail_key":"p41"},{"id":"n34959","layer":"informal","project":"p41","title":"This is a particular case of Definition~\\refdef:ManySortedCategory_category, so the resul…","kind":"proof","summary":"This is a particular case of Definition~\\refdef:ManySortedCategory_category, so the result foll…","labels":[],"detail_key":"p41"},{"id":"n34960","layer":"informal","project":"p41","title":"We extend the notation n\\mathsfCat_ms to all n \\in \\overlineN, understanding that, \\item…","kind":"notation","summary":"We extend the notation n\\mathsfCat_ms to all n \\in \\overlineN, understanding that, \\item if n \\…","labels":[],"detail_key":"p41"},{"id":"n34961","layer":"informal","project":"p41","title":"Discrete many-sorted finite family","kind":"definition","summary":"[Discrete many-sorted finite family] Let \\mathsfC be a many-sorted n-category on a family (C_k)…","labels":["def:ManySortedDiscreteFamily"],"detail_key":"p41"},{"id":"n34962","layer":"informal","project":"p41","title":"Discrete many-sorted finite category","kind":"construction","summary":"[Discrete many-sorted finite category] Let \\mathsfC be a many-sorted n-category on a family (C_…","labels":["def:ManySortedNCategory_discrete"],"detail_key":"p41"},{"id":"n34963","layer":"informal","project":"p41","title":"Each axiom, whether a pre-category axiom at a pair (k, j) or a cross-dimensional axiom at…","kind":"proof","summary":"Each axiom, whether a pre-category axiom at a pair (k, j) or a cross-dimensional axiom at a tri…","labels":[],"detail_key":"p41"},{"id":"n34964","layer":"informal","project":"p41","title":"Discrete many-sorted omega family","kind":"definition","summary":"[Discrete many-sorted omega family] Let \\mathsfC be a many-sorted n-category on a family (C_k)_…","labels":["def:ManySortedDiscreteOmegaFamily"],"detail_key":"p41"},{"id":"n34965","layer":"informal","project":"p41","title":"Discrete many-sorted omega category","kind":"construction","summary":"[Discrete many-sorted omega category] Let \\mathsfC be a many-sorted n-category on a family (C_k…","labels":["def:ManySortedNCategory_discreteOmega"],"detail_key":"p41"},{"id":"n34966","layer":"informal","project":"p41","title":"The many-sorted category axioms follow by the same argument as in Definition~\\refdef:Many…","kind":"proof","summary":"The many-sorted category axioms follow by the same argument as in Definition~\\refdef:ManySorted…","labels":[],"detail_key":"p41"},{"id":"n34967","layer":"informal","project":"p41","title":"Discrete many-sorted finite functor","kind":"construction","summary":"[Discrete many-sorted finite functor] Let \\mathsfC and \\mathsfD be many-sorted n-categories, an…","labels":["def:ManySortedNFunctor_discrete"],"detail_key":"p41"},{"id":"n34968","layer":"informal","project":"p41","title":"Each preservation axiom at a pair (k, j) with j < k either involves only indices at most…","kind":"proof","summary":"Each preservation axiom at a pair (k, j) with j < k either involves only indices at most n, whe…","labels":[],"detail_key":"p41"},{"id":"n34969","layer":"informal","project":"p41","title":"Discrete many-sorted omega functor","kind":"construction","summary":"[Discrete many-sorted omega functor] Let \\mathsfC and \\mathsfD be many-sorted n-categories, and…","labels":["def:ManySortedNFunctor_discreteOmega"],"detail_key":"p41"},{"id":"n34970","layer":"informal","project":"p41","title":"The argument is identical to that of Definition~\\refdef:ManySortedNFunctor_discrete.","kind":"proof","summary":"The argument is identical to that of Definition~\\refdef:ManySortedNFunctor_discrete.","labels":[],"detail_key":"p41"},{"id":"n34971","layer":"informal","project":"p41","title":"Many-sorted discretization functor","kind":"construction","summary":"[Many-sorted discretization functor] Let n, m \\in \\overlineN with n \\leq m. The \\emphmany-sorte…","labels":["def:ManySortedDiscretizationFunctor"],"detail_key":"p41"},{"id":"n34972","layer":"informal","project":"p41","title":"Functoriality follows from the fact that the discrete constructions preserve the underlyi…","kind":"proof","summary":"Functoriality follows from the fact that the discrete constructions preserve the underlying map…","labels":[],"detail_key":"p41"},{"id":"n34973","layer":"informal","project":"p41","title":"Underlying many-sorted finite category","kind":"construction","summary":"[Underlying many-sorted finite category] Let \\mathsfC be a many-sorted n-category on a family (…","labels":["def:ManySortedNCategory_underlying"],"detail_key":"p41"},{"id":"n34974","layer":"informal","project":"p41","title":"Every axiom of \\mathsfC^<m is an axiom of \\mathsfC restricted to lower dimensions.","kind":"proof","summary":"Every axiom of \\mathsfC^<m is an axiom of \\mathsfC restricted to lower dimensions.","labels":[],"detail_key":"p41"},{"id":"n34975","layer":"informal","project":"p41","title":"Underlying many-sorted omega category","kind":"construction","summary":"[Underlying many-sorted omega category] Let \\mathsfC be a many-sorted \\omega-category on a fami…","labels":["def:ManySortedOmegaCategory_underlying"],"detail_key":"p41"},{"id":"n34976","layer":"informal","project":"p41","title":"The many-sorted category axioms follow by the same argument as in Definition~\\refdef:Many…","kind":"proof","summary":"The many-sorted category axioms follow by the same argument as in Definition~\\refdef:ManySorted…","labels":[],"detail_key":"p41"},{"id":"n34977","layer":"informal","project":"p41","title":"Underlying many-sorted finite functor","kind":"construction","summary":"[Underlying many-sorted finite functor] Let \\mathsfC and \\mathsfD be many-sorted n-categories,…","labels":["def:ManySortedNFunctor_underlying"],"detail_key":"p41"},{"id":"n34978","layer":"informal","project":"p41","title":"Every preservation axiom of F^<m is a preservation axiom of F restricted to lower dimensi…","kind":"proof","summary":"Every preservation axiom of F^<m is a preservation axiom of F restricted to lower dimensions.","labels":[],"detail_key":"p41"},{"id":"n34979","layer":"informal","project":"p41","title":"Underlying many-sorted omega functor","kind":"construction","summary":"[Underlying many-sorted omega functor] Let \\mathsfC and \\mathsfD be many-sorted \\omega-categori…","labels":["def:ManySortedOmegaFunctor_underlying"],"detail_key":"p41"},{"id":"n34980","layer":"informal","project":"p41","title":"The argument is identical to that of Definition~\\refdef:ManySortedNFunctor_underlying.","kind":"proof","summary":"The argument is identical to that of Definition~\\refdef:ManySortedNFunctor_underlying.","labels":[],"detail_key":"p41"},{"id":"n34981","layer":"informal","project":"p41","title":"Many-sorted underlying functor","kind":"construction","summary":"[Many-sorted underlying functor] Let n, m \\in \\overlineN with m \\leq n. The \\emphmany-sorted un…","labels":["def:ManySortedUnderlyingFunctor"],"detail_key":"p41"},{"id":"n34982","layer":"informal","project":"p41","title":"Functoriality follows from the fact that restricting the family preserves identities and…","kind":"proof","summary":"Functoriality follows from the fact that restricting the family preserves identities and compos…","labels":[],"detail_key":"p41"},{"id":"n34983","layer":"formal","project":"p42","title":"HypergraphLowerBound.H","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.H","module":"FrontierMathOpenHypergraphs.Basic"},{"id":"n34984","layer":"formal","project":"p42","title":"HypergraphLowerBound.k","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.k","module":"FrontierMathOpenHypergraphs.Basic"},{"id":"n34985","layer":"formal","project":"p42","title":"HypergraphLowerBound.partition_iff_uniqueCoverage","kind":"theorem","summary":"∀ V : Type u_1 [inst : DecidableEq V] (edges : HypergraphLowerBound.Hypergraph V) (n : Nat), If…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.partition_iff_uniqueCoverage","module":"FrontierMathOpenHypergraphs.Basic"},{"id":"n34986","layer":"formal","project":"p42","title":"HypergraphLowerBound.uniqueCoverage","kind":"def","summary":"V : Type u_1 → [DecidableEq V] → HypergraphLowerBound.Hypergraph V → HypergraphLowerBound.Hyper…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.uniqueCoverage","module":"FrontierMathOpenHypergraphs.Basic"},{"id":"n34987","layer":"formal","project":"p42","title":"HypergraphLowerBound.M","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.M","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34988","layer":"formal","project":"p42","title":"HypergraphLowerBound.asymptotic_lower_bound","kind":"theorem","summary":"LE.le (HMul.hMul 2 (Real.log 2)) (Filter.liminf (fun n => HDiv.hDiv ↑(HypergraphLowerBound.H n)…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.asymptotic_lower_bound","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34989","layer":"formal","project":"p42","title":"HypergraphLowerBound.digit_sum_formula","kind":"theorem","summary":"∀ (n : Nat), LE.le 1 n → Eq (HypergraphLowerBound.k n) (HAdd.hAdd n (HypergraphLowerBound.digit…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.digit_sum_formula","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34990","layer":"formal","project":"p42","title":"HypergraphLowerBound.fixed_t_bound","kind":"theorem","summary":"∀ (t : Nat), LE.le 2 t → Exists fun C => And (LT.lt 0 C) (∀ (n : Nat), LE.le 1 n → GE.ge (↑(Hyp…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.fixed_t_bound","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34991","layer":"formal","project":"p42","title":"HypergraphLowerBound.lubellFrame","kind":"def","summary":"(t : Nat) → Multiset (HypergraphLowerBound.SupportPattern t)","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.lubellFrame","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34992","layer":"formal","project":"p42","title":"HypergraphLowerBound.lubell_is_frame","kind":"theorem","summary":"∀ (t : Nat), LE.le 2 t → And (HypergraphLowerBound.IsFrame (HypergraphLowerBound.lubellFrame t)…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.lubell_is_frame","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34993","layer":"formal","project":"p42","title":"HypergraphLowerBound.thm_asymptotic","kind":"theorem","summary":"And (∀ (t : Nat), LE.le 2 t → Exists fun C => And (LT.lt 0 C) (∀ (n : Nat), LE.le 1 n → GE.ge (…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.thm_asymptotic","module":"FrontierMathOpenHypergraphs.Lubell"},{"id":"n34994","layer":"formal","project":"p42","title":"HypergraphLowerBound.IsFrame","kind":"def","summary":"t : Nat → Multiset (HypergraphLowerBound.SupportPattern t) → (Fin t → Nat) → Prop","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.IsFrame","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n34995","layer":"formal","project":"p42","title":"HypergraphLowerBound.SupportPattern","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.SupportPattern","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n34996","layer":"formal","project":"p42","title":"HypergraphLowerBound.frame_recurrence","kind":"theorem","summary":"∀ t : Nat (F : Multiset (HypergraphLowerBound.SupportPattern t)) (edgeCounts vertexCounts : Fin…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.frame_recurrence","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n34997","layer":"formal","project":"p42","title":"HypergraphLowerBound.omega_count","kind":"def","summary":"t : Nat → Multiset (HypergraphLowerBound.SupportPattern t) → Finset (Fin t) → Finset (Fin t) →…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.omega_count","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n34998","layer":"formal","project":"p42","title":"HypergraphLowerBound.substitutionHypergraph","kind":"def","summary":"t : Nat → (F : Multiset (HypergraphLowerBound.SupportPattern t)) → HypergraphLowerBound.BlockFa…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.substitutionHypergraph","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n34999","layer":"formal","project":"p42","title":"HypergraphLowerBound.substitution_theorem","kind":"theorem","summary":"∀ t : Nat (F : Multiset (HypergraphLowerBound.SupportPattern t)) (cap : Fin t → Nat) (blocks :…","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.substitution_theorem","module":"FrontierMathOpenHypergraphs.Substitution"},{"id":"n35000","layer":"formal","project":"p42","title":"HypergraphLowerBound.A","kind":"def","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.A","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35001","layer":"formal","project":"p42","title":"HypergraphLowerBound.bootstrap_26_25","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.bootstrap_26_25","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35002","layer":"formal","project":"p42","title":"HypergraphLowerBound.constructive_An","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.constructive_An","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35003","layer":"formal","project":"p42","title":"HypergraphLowerBound.finite_bank_valid","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.finite_bank_valid","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35004","layer":"formal","project":"p42","title":"HypergraphLowerBound.floor_26_25","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.floor_26_25","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35005","layer":"formal","project":"p42","title":"HypergraphLowerBound.k_four_way","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.k_four_way","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35006","layer":"formal","project":"p42","title":"HypergraphLowerBound.thm_main","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.thm_main","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35007","layer":"formal","project":"p42","title":"HypergraphLowerBound.thm_main_H_lower_bound","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.thm_main_H_lower_bound","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35008","layer":"formal","project":"p42","title":"HypergraphLowerBound.uniform_26_25","kind":"theorem","summary":"","labels":[],"detail_key":"p42","name":"HypergraphLowerBound.uniform_26_25","module":"FrontierMathOpenHypergraphs.Uniform"},{"id":"n35009","layer":"informal","project":"p42","title":"thm:main","kind":"theorem","summary":"There exists a sequence of hypergraphs \\left(G_n\\right)_n\\ge 1 with the following properties. F…","labels":["thm:main"],"detail_key":"p42"},{"id":"n35010","layer":"informal","project":"p42","title":"cor:main_H_lower_bound","kind":"corollary","summary":"For every n\\ge 15, \\[ H(n)\\ge \\frac2625\\,k_n. \\]","labels":["cor:main_H_lower_bound"],"detail_key":"p42"},{"id":"n35011","layer":"informal","project":"p42","title":"This is the final conclusion packaged into Theorem~\\refthm:main.","kind":"proof","summary":"This is the final conclusion packaged into Theorem~\\refthm:main.","labels":[],"detail_key":"p42"},{"id":"n35012","layer":"informal","project":"p42","title":"thm:asymptotic","kind":"theorem","summary":"The formal theorem linked here packages the two asymptotic theorems proved later in the file. F…","labels":["thm:asymptotic"],"detail_key":"p42"},{"id":"n35013","layer":"informal","project":"p42","title":"def:H_n","kind":"definition","summary":"Write H(n) for the largest k\\inN such that there exists a finite edge family H with |\\bigcup H|…","labels":["def:H_n"],"detail_key":"p42"},{"id":"n35014","layer":"informal","project":"p42","title":"def:k_n","kind":"definition","summary":"Define the sequence k:N\\toN by k(0)=0, k(1)=1, and \\[ k(n)=\\lfloor n/2\\rfloor+k(\\lfloor n/2\\rfl…","labels":["def:k_n"],"detail_key":"p42"},{"id":"n35015","layer":"informal","project":"p42","title":"def:u_G","kind":"definition","summary":"For a hypergraph G=(V,H) and a chosen edge family P\\subseteq H, define \\[ u_G(P):=|\\v\\in V:\\tex…","labels":["def:u_G"],"detail_key":"p42"},{"id":"n35016","layer":"informal","project":"p42","title":"lem:partition_u","kind":"lemma","summary":"A hypergraph G contains no partition of size greater than n if and only if \\[ u_G(P)\\le n \\qqua…","labels":["lem:partition_u"],"detail_key":"p42"},{"id":"n35017","layer":"informal","project":"p42","title":"If (D,P) is a partition of size m, then every vertex of D is counted by u_G(P), so u_G(P)…","kind":"proof","summary":"If (D,P) is a partition of size m, then every vertex of D is counted by u_G(P), so u_G(P)\\ge m.…","labels":[],"detail_key":"p42"},{"id":"n35018","layer":"informal","project":"p42","title":"def:support_pattern","kind":"definition","summary":"Fix t\\ge 2. A \\emphsupport pattern on [t]:=\\1,\\dots,t\\ is a subset of [t] of size at least 2.","labels":["def:support_pattern"],"detail_key":"p42"},{"id":"n35019","layer":"informal","project":"p42","title":"def:omega","kind":"definition","summary":"Let F be a finite multiset of support patterns on [t]. For I\\subseteq T\\subseteq [t], define \\[…","labels":["def:omega"],"detail_key":"p42"},{"id":"n35020","layer":"informal","project":"p42","title":"def:n_frame","kind":"definition","summary":"Let n=(n_1,\\dots,n_t)\\inN^t. We say that F is an \\emphn-frame if \\[ \\omega_F(T,I)\\le \\sum_j\\in…","labels":["def:n_frame"],"detail_key":"p42"},{"id":"n35021","layer":"informal","project":"p42","title":"def:substitution","kind":"definition","summary":"Let F be a support multiset on [t]. Let G_i=(V_i,H_i), 1\\le i\\le t, be hypergraphs with pairwis…","labels":["def:substitution"],"detail_key":"p42"},{"id":"n35022","layer":"informal","project":"p42","title":"thm:substitution","kind":"theorem","summary":"Let n=(n_1,\\dots,n_t)\\inN^t, and let F be an n-frame on [t]. Assume that each G_i contains no p…","labels":["thm:substitution"],"detail_key":"p42"},{"id":"n35023","layer":"informal","project":"p42","title":"Fix P\\subseteq E(G), and write P_i:=P\\cap H_i. Define \\[ T:=\\i:|P_i|\\le 1\\, \\qquad I:=\\i:…","kind":"proof","summary":"Fix P\\subseteq E(G), and write P_i:=P\\cap H_i. Define \\[ T:=\\i:|P_i|\\le 1\\, \\qquad I:=\\i:|P_i|=…","labels":[],"detail_key":"p42"},{"id":"n35024","layer":"informal","project":"p42","title":"cor:frame_recurrence","kind":"corollary","summary":"Suppose that for 1\\le i\\le t there is a hypergraph G_i with exactly n_i edges, no partition of…","labels":["cor:frame_recurrence"],"detail_key":"p42"},{"id":"n35025","layer":"informal","project":"p42","title":"The edge set is the disjoint union of the edge sets of the blocks, so the edge count is a…","kind":"proof","summary":"The edge set is the disjoint union of the edge sets of the blocks, so the edge count is additiv…","labels":[],"detail_key":"p42"},{"id":"n35026","layer":"informal","project":"p42","title":"prop:finite_bank","kind":"proposition","summary":"Every exact small frame listed in the paper, every booster, and every residue gadget R_r is a v…","labels":["prop:finite_bank"],"detail_key":"p42"},{"id":"n35027","layer":"informal","project":"p42","title":"This is a finite exhaustive check of the frame inequalities \\omega_F(T,I)\\le \\sum_j\\in T\\…","kind":"proof","summary":"This is a finite exhaustive check of the frame inequalities \\omega_F(T,I)\\le \\sum_j\\in T\\setmin…","labels":[],"detail_key":"p42"},{"id":"n35028","layer":"informal","project":"p42","title":"def:A_n","kind":"definition","summary":"Define integers A_n recursively as follows. Set A_0:=0 and A_1:=1. For 2\\le n<60, let A_n be th…","labels":["def:A_n"],"detail_key":"p42"},{"id":"n35029","layer":"informal","project":"p42","title":"thm:constructive_An","kind":"theorem","summary":"For every n\\ge 1 there exists a hypergraph G with exactly n distinct edges, no partition of siz…","labels":["thm:constructive_An"],"detail_key":"p42"},{"id":"n35030","layer":"informal","project":"p42","title":"We argue by induction on n. For n=1, take a single edge containing a single vertex. For 2…","kind":"proof","summary":"We argue by induction on n. For n=1, take a single edge containing a single vertex. For 2\\le n<…","labels":[],"detail_key":"p42"},{"id":"n35031","layer":"informal","project":"p42","title":"lem:k_four_way","kind":"lemma","summary":"For every m\\ge 1, k_4m & =4k_m+4m, \\\\ k_4m+1 & =3k_m+k_m+1+4m, \\\\ k_4m+2 & =2k_m+2k_m+1+4m+1, \\…","labels":["lem:k_four_way"],"detail_key":"p42"},{"id":"n35032","layer":"informal","project":"p42","title":"The defining recurrence for k_n immediately gives k_2m=m+2k_m and k_2m+1=m+k_m+k_m+1. App…","kind":"proof","summary":"The defining recurrence for k_n immediately gives k_2m=m+2k_m and k_2m+1=m+k_m+k_m+1. Applying…","labels":[],"detail_key":"p42"},{"id":"n35033","layer":"informal","project":"p42","title":"lem:floor_26_25","kind":"lemma","summary":"For every integer m\\ge 15, e_0(m) & \\ge \\frac2625\\cdot 4m, \\\\ e_1(m) & \\ge \\frac2625\\cdot 4m, \\…","labels":["lem:floor_26_25"],"detail_key":"p42"},{"id":"n35034","layer":"informal","project":"p42","title":"Using \\lfloor x\\rfloor\\ge x-1 and direct calculation, each difference is at least (13m-75…","kind":"proof","summary":"Using \\lfloor x\\rfloor\\ge x-1 and direct calculation, each difference is at least (13m-75)/75,…","labels":[],"detail_key":"p42"},{"id":"n35035","layer":"informal","project":"p42","title":"prop:bootstrap_26_25","kind":"proposition","summary":"For every integer n with 15\\le n<60, \\[ 25A_n\\ge 26k_n, \\] and equality holds if and only if n=…","labels":["prop:bootstrap_26_25"],"detail_key":"p42"},{"id":"n35036","layer":"informal","project":"p42","title":"This is the finite table check.","kind":"proof","summary":"This is the finite table check.","labels":[],"detail_key":"p42"},{"id":"n35037","layer":"informal","project":"p42","title":"thm:uniform_26_25","kind":"theorem","summary":"For every integer n\\ge 15, \\[ A_n\\ge \\frac2625\\,k_n. \\] Together with Theorem~\\refthm:construct…","labels":["thm:uniform_26_25"],"detail_key":"p42"},{"id":"n35038","layer":"informal","project":"p42","title":"The range 15\\le n<60 is Proposition~\\refprop:bootstrap_26_25. For n\\ge 60, write n=4m+r w…","kind":"proof","summary":"The range 15\\le n<60 is Proposition~\\refprop:bootstrap_26_25. For n\\ge 60, write n=4m+r with 0\\…","labels":[],"detail_key":"p42"},{"id":"n35039","layer":"informal","project":"p42","title":"def:M_t","kind":"definition","summary":"For t\\ge 2, define \\[ M_t:=lcm\\left\\\\binomt-1r:1\\le r\\le t-1\\right\\. \\]","labels":["def:M_t"],"detail_key":"p42"},{"id":"n35040","layer":"informal","project":"p42","title":"def:lubell_frame","kind":"definition","summary":"Let L_t be the support multiset on [t] that contains every j-subset of [t] with multiplicity \\[…","labels":["def:lubell_frame"],"detail_key":"p42"},{"id":"n35041","layer":"informal","project":"p42","title":"prop:lubell","kind":"proposition","summary":"For every t\\ge 2, the support multiset L_t is an (M_t,\\dots,M_t)-frame. Moreover, \\[ |L_t|=M_t\\…","labels":["prop:lubell"],"detail_key":"p42"},{"id":"n35042","layer":"informal","project":"p42","title":"Fix I\\subseteq T\\subseteq [t] with i:=|I| and z:=|T\\setminus I|. A support pattern of siz…","kind":"proof","summary":"Fix I\\subseteq T\\subseteq [t] with i:=|I| and z:=|T\\setminus I|. A support pattern of size r+1…","labels":[],"detail_key":"p42"},{"id":"n35043","layer":"informal","project":"p42","title":"prop:digit_sum","kind":"proposition","summary":"Let s_2(m) denote the sum of the binary digits of m. Then for every n\\ge 1, \\[ k_n=n+\\sum_j=0^n…","labels":["prop:digit_sum"],"detail_key":"p42"},{"id":"n35044","layer":"informal","project":"p42","title":"Set k_0:=0 and \\Delta_n:=k_n-k_n-1 for n\\ge 1. From the recurrence, \\Delta_2m=1+\\Delta_m…","kind":"proof","summary":"Set k_0:=0 and \\Delta_n:=k_n-k_n-1 for n\\ge 1. From the recurrence, \\Delta_2m=1+\\Delta_m and \\D…","labels":[],"detail_key":"p42"},{"id":"n35045","layer":"informal","project":"p42","title":"thm:fixed_t","kind":"theorem","summary":"Fix t\\ge 2. There exists a constant C_t>0 such that \\[ H(n)\\ge \\frach_t-1\\log_2 t\\,n\\log_2 n -…","labels":["thm:fixed_t"],"detail_key":"p42"},{"id":"n35046","layer":"informal","project":"p42","title":"Fix t, set M:=M_t and B:=|L_t|=Mt(h_t-1). Define L^(t)_n recursively by L^(t)_1:=1, and f…","kind":"proof","summary":"Fix t, set M:=M_t and B:=|L_t|=Mt(h_t-1). 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The second Lean declaration exposes the vertex type, three…","labels":[],"detail_key":"p43"},{"id":"n35437","layer":"informal","project":"p43","title":"Finite simplicial complex in the plane","kind":"definition","summary":"[Finite simplicial complex in the plane] A \\emphplane complex is a finite family of affine simp…","labels":["moise:def:plane_complex"],"detail_key":"p43"},{"id":"n35438","layer":"informal","project":"p43","title":"Subdivisions and PL maps","kind":"definition","summary":"[Subdivisions and PL maps] K' subdivides K when they have the same support and every face carri…","labels":["moise:def:pl_maps"],"detail_key":"p43"},{"id":"n35439","layer":"informal","project":"p43","title":"Realization bridge","kind":"lemma","summary":"[Realization bridge] A purely two-dimensional plane complex induces a geometric triangulation o…","labels":["moise:lem:plane_complex_realization"],"detail_key":"p43"},{"id":"n35440","layer":"informal","project":"p43","title":"Elementary affine geometry and uniqueness of barycentric coordinates on affinely independ…","kind":"proof","summary":"Elementary affine geometry and uniqueness of barycentric coordinates on affinely independent fa…","labels":[],"detail_key":"p43"},{"id":"n35441","layer":"informal","project":"p43","title":"Polygonal simple closed curve","kind":"definition","summary":"[Polygonal simple closed curve] A \\emphpolygonal circle is a cyclically indexed family of at le…","labels":["moise:def:polygonal_circle"],"detail_key":"p43"},{"id":"n35442","layer":"informal","project":"p43","title":"Crossing index","kind":"definition","summary":"[Crossing index] The \\emphindex of a plane point is the parity of the number of polygon edges c…","labels":["moise:def:crossing_index"],"detail_key":"p43"},{"id":"n35443","layer":"informal","project":"p43","title":"The index vanishes far away","kind":"lemma","summary":"[The index vanishes far away] A point strictly above every vertex is off the polygon and has in…","labels":["moise:lem:index_zero_high"],"detail_key":"p43"},{"id":"n35444","layer":"informal","project":"p43","title":"No edge satisfies the height condition; points of a segment have heights bounded by the e…","kind":"proof","summary":"No edge satisfies the height condition; points of a segment have heights bounded by the endpoin…","labels":[],"detail_key":"p43"},{"id":"n35445","layer":"informal","project":"p43","title":"Local constancy of the index","kind":"lemma","summary":"[Local constancy of the index] Off the polygon the index is locally constant.","labels":["moise:lem:index_locally_constant"],"detail_key":"p43"},{"id":"n35446","layer":"informal","project":"p43","title":"Casework: near a point off the polygon, the crossing status of each edge is unchanged unl…","kind":"proof","summary":"Casework: near a point off the polygon, the crossing status of each edge is unchanged unless th…","labels":[],"detail_key":"p43"},{"id":"n35447","layer":"informal","project":"p43","title":"An index-one point exists","kind":"lemma","summary":"[An index-one point exists] Some point off the polygon has index one.","labels":["moise:lem:index_one_exists"],"detail_key":"p43"},{"id":"n35448","layer":"informal","project":"p43","title":"Choose a non-vertex height attained by the polygon, take the leftmost polygon point at th…","kind":"proof","summary":"Choose a non-vertex height attained by the polygon, take the leftmost polygon point at that hei…","labels":[],"detail_key":"p43"},{"id":"n35449","layer":"informal","project":"p43","title":"Moise Ch.~2, Thm.~1, Lemma 2","kind":"lemma","summary":"[Moise Ch.~2, Thm.~1, Lemma 2] The complement of a polygon is disconnected.","labels":["moise:lem:complement_disconnected"],"detail_key":"p43"},{"id":"n35450","layer":"informal","project":"p43","title":"The index-0 and index-1 loci are relatively open by local constancy, cover the complement…","kind":"proof","summary":"The index-0 and index-1 loci are relatively open by local constancy, cover the complement, and…","labels":[],"detail_key":"p43"},{"id":"n35451","layer":"informal","project":"p43","title":"Jordan curve theorem for polygons; Moise Ch.~2, Thms.~1, 5, 6","kind":"theorem","summary":"[Jordan curve theorem for polygons; Moise Ch.~2, Thms.~1, 5, 6] The complement of a polygon has…","labels":["moise:thm:polygonal_jordan"],"detail_key":"p43"},{"id":"n35452","layer":"informal","project":"p43","title":"The crossing-index argument, boundedness distinction, visibility segment to the strip, an…","kind":"proof","summary":"The crossing-index argument, boundedness distinction, visibility segment to the strip, and the…","labels":[],"detail_key":"p43"},{"id":"n35453","layer":"informal","project":"p43","title":"Interior, exterior, closed region","kind":"definition","summary":"[Interior, exterior, closed region] The interior and exterior regions of a polygon are the boun…","labels":["moise:def:polygon_regions"],"detail_key":"p43"},{"id":"n35454","layer":"informal","project":"p43","title":"Polygonal disks are polyhedra; Moise Ch.~2, Thm.~2","kind":"theorem","summary":"[Polygonal disks are polyhedra; Moise Ch.~2, Thm.~2] The closed region bounded by a polygon is…","labels":["moise:thm:region_polyhedron"],"detail_key":"p43"},{"id":"n35455","layer":"informal","project":"p43","title":"Place the compact closed region in a large triangle, subdivide that triangle successively…","kind":"proof","summary":"Place the compact closed region in a large triangle, subdivide that triangle successively by th…","labels":[],"detail_key":"p43"},{"id":"n35456","layer":"informal","project":"p43","title":"Polygonal disks are triangulable","kind":"lemma","summary":"[Polygonal disks are triangulable] The closed region bounded by a polygon admits a geometric tr…","labels":["moise:lem:region_triangulable"],"detail_key":"p43"},{"id":"n35457","layer":"informal","project":"p43","title":"Combine \\refmoise:thm:region_polyhedron with the realization bridge.","kind":"proof","summary":"Combine \\refmoise:thm:region_polyhedron with the realization bridge.","labels":[],"detail_key":"p43"},{"id":"n35458","layer":"informal","project":"p43","title":"Schoenflies for polygons; Moise Ch.~3, Thm.~5","kind":"theorem","summary":"[Schoenflies for polygons; Moise Ch.~3, Thm.~5] Any two polygons in the plane are equivalent un…","labels":["moise:thm:polygonal_schoenflies"],"detail_key":"p43"},{"id":"n35459","layer":"informal","project":"p43","title":"Apply relative straightening to both polygons and compose the resulting maps with the aff…","kind":"proof","summary":"Apply relative straightening to both polygons and compose the resulting maps with the affine am…","labels":[],"detail_key":"p43"},{"id":"n35460","layer":"informal","project":"p43","title":"Relative Schoenflies for polygons; Moise Ch.~3, Thm.~7","kind":"theorem","summary":"[Relative Schoenflies for polygons; Moise Ch.~3, Thm.~7] The straightening homeomorphism can be…","labels":["moise:thm:polygonal_schoenflies_rel"],"detail_key":"p43"},{"id":"n35461","layer":"informal","project":"p43","title":"Choose the homeomorphisms in Moise's Thm.~4 induction supported in the open set.","kind":"proof","summary":"Choose the homeomorphisms in Moise's Thm.~4 induction supported in the open set.","labels":[],"detail_key":"p43"},{"id":"n35462","layer":"informal","project":"p43","title":"Broken-line joinedness","kind":"definition","summary":"[Broken-line joinedness] Two points of a set are joined by a broken line when a finite chain of…","labels":["moise:def:broken_line"],"detail_key":"p43"},{"id":"n35463","layer":"informal","project":"p43","title":"Open connected sets are broken-line connected","kind":"lemma","summary":"[Open connected sets are broken-line connected] In an open preconnected subset of the plane, an…","labels":["moise:lem:broken_line_connected"],"detail_key":"p43"},{"id":"n35464","layer":"informal","project":"p43","title":"Clopen-chain argument: the locus joined to a fixed point is open (extend by one segment i…","kind":"proof","summary":"Clopen-chain argument: the locus joined to a fixed point is open (extend by one segment inside…","labels":[],"detail_key":"p43"},{"id":"n35465","layer":"informal","project":"p43","title":"Cone extension; Moise Ch.~5, Thms.~3--6","kind":"theorem","summary":"[Cone extension; Moise Ch.~5, Thms.~3--6] A map that is PL and injective on the frontier of a t…","labels":["moise:thm:cone_extension"],"detail_key":"p43"},{"id":"n35466","layer":"informal","project":"p43","title":"Cone from an interior point: the extension is affine on each segment from the cone point…","kind":"proof","summary":"Cone from an interior point: the extension is affine on each segment from the cone point to the…","labels":[],"detail_key":"p43"},{"id":"n35467","layer":"informal","project":"p43","title":"One-skeleton approximation; Moise Ch.~6, Thm.~2","kind":"theorem","summary":"[One-skeleton approximation; Moise Ch.~6, Thm.~2] An embedding of the support of a finite one-d…","labels":["moise:thm:one_skeleton_approx"],"detail_key":"p43"},{"id":"n35468","layer":"informal","project":"p43","title":"Subdivide finely, replace each small arc by a broken line in a small neighborhood (\\refmo…","kind":"proof","summary":"Subdivide finely, replace each small arc by a broken line in a small neighborhood (\\refmoise:le…","labels":[],"detail_key":"p43"},{"id":"n35469","layer":"informal","project":"p43","title":"PL approximation of embedded 2-manifolds; Moise Ch.~6, Thm.~3","kind":"theorem","summary":"[PL approximation of embedded 2-manifolds; Moise Ch.~6, Thm.~3] An embedding of the support of…","labels":["moise:thm:pl_approximation"],"detail_key":"p43"},{"id":"n35470","layer":"informal","project":"p43","title":"Approximate on the one-skeleton by \\refmoise:thm:one_skeleton_approx, extend across each…","kind":"proof","summary":"Approximate on the one-skeleton by \\refmoise:thm:one_skeleton_approx, extend across each 2-cell…","labels":[],"detail_key":"p43"},{"id":"n35471","layer":"informal","project":"p43","title":"Moise charts","kind":"definition","summary":"[Moise charts] A \\emphMoise chart is an open domain homeomorphic to the standard open unit disk…","labels":["moise:def:moise_chart"],"detail_key":"p43"},{"id":"n35472","layer":"informal","project":"p43","title":"Boundary faithfulness","kind":"definition","summary":"[Boundary faithfulness] A chart is boundary-faithful when disk charts contain no manifold-bound…","labels":["moise:def:boundary_faithful"],"detail_key":"p43"},{"id":"n35473","layer":"informal","project":"p43","title":"Local chart extraction","kind":"lemma","summary":"[Local chart extraction] Every point of an Eval surface has a boundary-faithful Moise chart who…","labels":["moise:lem:chart_extraction"],"detail_key":"p43"},{"id":"n35474","layer":"informal","project":"p43","title":"Take the preferred mathlib chart; a small ball (interior case) or relative half-ball (edg…","kind":"proof","summary":"Take the preferred mathlib chart; a small ball (interior case) or relative half-ball (edge case…","labels":[],"detail_key":"p43"},{"id":"n35475","layer":"informal","project":"p43","title":"Finite chart cover","kind":"lemma","summary":"[Finite chart cover] A compact Eval surface has a finite cover by boundary-faithful Moise chart…","labels":["moise:lem:finite_chart_cover"],"detail_key":"p43"},{"id":"n35476","layer":"informal","project":"p43","title":"Finite subcover of the core interiors.","kind":"proof","summary":"Finite subcover of the core interiors.","labels":[],"detail_key":"p43"},{"id":"n35477","layer":"informal","project":"p43","title":"Partial triangulation","kind":"definition","summary":"[Partial triangulation] A \\emphpartial triangulation of S is a finite face family together with…","labels":["moise:def:partial_triangulation"],"detail_key":"p43"},{"id":"n35478","layer":"informal","project":"p43","title":"The Rad\\'o invariant","kind":"definition","summary":"[The Rad\\'o invariant] The invariant carried through the bordered induction records that every…","labels":["moise:def:rado_invariant"],"detail_key":"p43"},{"id":"n35479","layer":"informal","project":"p43","title":"Faithful intrinsic fine subdivision","kind":"lemma","summary":"[Faithful intrinsic fine subdivision] Every finite intrinsic two-complex has a faithful subdivi…","labels":["moise:lem:intrinsic_fine_subdivision"],"detail_key":"p43"},{"id":"n35480","layer":"informal","project":"p43","title":"Construct the midpoint complex with old vertices and one vertex for each old edge. Explic…","kind":"proof","summary":"Construct the midpoint complex with old vertices and one vertex for each old edge. Explicit bar…","labels":[],"detail_key":"p43"},{"id":"n35481","layer":"informal","project":"p43","title":"Finite compact collar","kind":"lemma","summary":"[Finite compact collar] If a compact subset of a partial triangulation lies in an ambient open…","labels":["moise:lem:finite_compact_collar"],"detail_key":"p43"},{"id":"n35482","layer":"informal","project":"p43","title":"Apply the Lebesgue-number lemma to the two-set cover consisting of the open set and the c…","kind":"proof","summary":"Apply the Lebesgue-number lemma to the two-set cover consisting of the open set and the complem…","labels":[],"detail_key":"p43"},{"id":"n35483","layer":"informal","project":"p43","title":"Vanishing-error frontier glue","kind":"lemma","summary":"[Vanishing-error frontier glue] If a continuous map is replaced on an open set by a continuous…","labels":["moise:lem:frontier_glue"],"detail_key":"p43"},{"id":"n35484","layer":"informal","project":"p43","title":"At interior and exterior points continuity is local. At a frontier point, combine the app…","kind":"proof","summary":"At interior and exterior points continuity is local. At a frontier point, combine the approxima…","labels":[],"detail_key":"p43"},{"id":"n35485","layer":"informal","project":"p43","title":"Frontier-controlled chart straightening","kind":"theorem","summary":"[Frontier-controlled chart straightening] On an open chart overlap, an adaptive locally finite…","labels":["moise:thm:controlled_chart_straightening"],"detail_key":"p43"},{"id":"n35486","layer":"informal","project":"p43","title":"Choose an adaptive open cover subordinate to the pointwise error and target region contro…","kind":"proof","summary":"Choose an adaptive open cover subordinate to the pointwise error and target region controls. Po…","labels":[],"detail_key":"p43"},{"id":"n35487","layer":"informal","project":"p43","title":"Gluing along a common presentation; Moise Thm.~7.6","kind":"theorem","summary":"[Gluing along a common presentation; Moise Thm.~7.6] Two partial triangulations presented on a…","labels":["moise:thm:partial_glue"],"detail_key":"p43"},{"id":"n35488","layer":"informal","project":"p43","title":"The realization of the union family is the set-union of the two realizations, so the glue…","kind":"proof","summary":"The realization of the union family is the set-union of the two realizations, so the glued embe…","labels":[],"detail_key":"p43"},{"id":"n35489","layer":"informal","project":"p43","title":"Crossing-case welding; Moise Ch.~8, Thm.~3 preparation","kind":"theorem","summary":"[Crossing-case welding; Moise Ch.~8, Thm.~3 preparation] In the genuine crossing case, the stra…","labels":["moise:thm:crossing_weld"],"detail_key":"p43"},{"id":"n35490","layer":"informal","project":"p43","title":"Straighten the old complex over the chart overlap by the locally finite controlled polygo…","kind":"proof","summary":"Straighten the old complex over the chart overlap by the locally finite controlled polygonal re…","labels":[],"detail_key":"p43"},{"id":"n35491","layer":"informal","project":"p43","title":"The induction step; Moise Ch.~8, Thm.~3, step","kind":"theorem","summary":"[The induction step; Moise Ch.~8, Thm.~3, step] Given a partial triangulation satisfying the in…","labels":["moise:thm:induction_step"],"detail_key":"p43"},{"id":"n35492","layer":"informal","project":"p43","title":"If the new core already lies in the interior of the built support, absorb it directly; if…","kind":"proof","summary":"If the new core already lies in the interior of the built support, absorb it directly; if the a…","labels":[],"detail_key":"p43"},{"id":"n35493","layer":"informal","project":"p43","title":"Assembled Rad\\'o induction","kind":"theorem","summary":"[Assembled Rad\\'o induction] A compact connected Eval surface admits a geometric triangulation.","labels":["moise:thm:rado_assembly"],"detail_key":"p43"},{"id":"n35494","layer":"informal","project":"p43","title":"Starting from the empty complex, absorb the finitely many chart cores one at a time by \\r…","kind":"proof","summary":"Starting from the empty complex, absorb the finitely many chart cores one at a time by \\refmois…","labels":[],"detail_key":"p43"},{"id":"n35495","layer":"informal","project":"p43","title":"Finite surface cell complex","kind":"definition","summary":"[Finite surface cell complex] A finite combinatorial presentation of a compact bordered surface…","labels":["moise:def:surface_cell_complex"],"detail_key":"p43"},{"id":"n35496","layer":"informal","project":"p43","title":"Finite surface triangulation incidence package","kind":"definition","summary":"[Finite surface triangulation incidence package] The finite triangulation package consumed by t…","labels":["moise:def:finite_surface_triangulation"],"detail_key":"p43"},{"id":"n35497","layer":"informal","project":"p43","title":"Raw triangulation-to-cell-presentation bridge","kind":"definition","summary":"[Raw triangulation-to-cell-presentation bridge] The compatibility bridge copies each triangle t…","labels":["moise:def:triangulation_cell_complex"],"detail_key":"p43"},{"id":"n35498","layer":"informal","project":"p43","title":"Faithful polygonal-realization handoff","kind":"theorem","summary":"[Faithful polygonal-realization handoff] The valid finite-cyclic presentation enumerated from t…","labels":["moise:thm:compact_surface_cellulable"],"detail_key":"p43"},{"id":"n35499","layer":"informal","project":"p43","title":"Triangulate by \\refmoise:thm:moise_triangulation, enumerate the cyclic signed face bounda…","kind":"proof","summary":"Triangulate by \\refmoise:thm:moise_triangulation, enumerate the cyclic signed face boundaries,…","labels":[],"detail_key":"p43"},{"id":"n35500","layer":"informal","project":"p43","title":"Polygonal pre-realization and gluing relation","kind":"definition","summary":"[Polygonal pre-realization and gluing relation] The pre-realization is the disjoint union of st…","labels":["moise:def:gluing_relation"],"detail_key":"p43"},{"id":"n35501","layer":"informal","project":"p43","title":"Polygonal occurrence adapter","kind":"definition","summary":"[Polygonal occurrence adapter] Replace every face by an indexed closed disc and every boundary…","labels":["moise:def:polygonal_occurrence_adapter"],"detail_key":"p43"},{"id":"n35502","layer":"informal","project":"p43","title":"Quotient congruence","kind":"lemma","summary":"[Quotient congruence] A homeomorphism of pre-realizations carrying one gluing relation to the o…","labels":["moise:lem:cell_realization_quotient_congr"],"detail_key":"p43"},{"id":"n35503","layer":"informal","project":"p43","title":"Apply mathlib's quotient-homeomorphism congruence to the two generated setoids.","kind":"proof","summary":"Apply mathlib's quotient-homeomorphism congruence to the two generated setoids.","labels":[],"detail_key":"p43"},{"id":"n35504","layer":"informal","project":"p43","title":"Elementary finite-cyclic moves","kind":"definition","summary":"[Elementary finite-cyclic moves] The validity-bundled equivalence closure is generated by faith…","labels":["moise:def:gx_elementary_moves"],"detail_key":"p43"},{"id":"n35505","layer":"informal","project":"p43","title":"Elementary moves preserve realization","kind":"lemma","summary":"[Elementary moves preserve realization] If L is obtained from K by one elementary Gallier--Xu m…","labels":["moise:lem:elementary_moves_preserve_realization"],"detail_key":"p43"},{"id":"n35506","layer":"informal","project":"p43","title":"The P1 realization theorem reparametrizes a subdivided boundary side. The P2 realization…","kind":"proof","summary":"The P1 realization theorem reparametrizes a subdivided boundary side. The P2 realization theore…","labels":[],"detail_key":"p43"},{"id":"n35507","layer":"informal","project":"p43","title":"Eval quotient representatives","kind":"definition","summary":"[Eval quotient representatives] The orientable and nonorientable relations are the exact raw bo…","labels":["moise:def:eval_quotient_representatives"],"detail_key":"p43"},{"id":"n35508","layer":"informal","project":"p43","title":"One-face carrier and quotient bridge","kind":"lemma","summary":"[One-face carrier and quotient bridge] Every one-face polygonal pre-realization is homeomorphic…","labels":["moise:lem:representative_carrier_bridge"],"detail_key":"p43"},{"id":"n35509","layer":"informal","project":"p43","title":"Forget the side-count index on the polygon cell, collapse the unique \\textttPUnit face in…","kind":"proof","summary":"Forget the side-count index on the polygon cell, collapse the unique \\textttPUnit face in the s…","labels":[],"detail_key":"p43"},{"id":"n35510","layer":"informal","project":"p43","title":"One-face pairing-position characterization","kind":"lemma","summary":"[One-face pairing-position characterization] A polygonal identification in a one-face presentat…","labels":["moise:lem:one_face_pairing_positions"],"detail_key":"p43"},{"id":"n35511","layer":"informal","project":"p43","title":"Unpack membership in the range of boundary-pairing identifications, eliminate the unique…","kind":"proof","summary":"Unpack membership in the range of boundary-pairing identifications, eliminate the unique \\textt…","labels":[],"detail_key":"p43"},{"id":"n35512","layer":"informal","project":"p43","title":"Canonical normal-form incidence presentations","kind":"definition","summary":"[Canonical normal-form incidence presentations] The orientable word concatenates commutator blo…","labels":["moise:def:canonical_normal_form_complexes"],"detail_key":"p43"},{"id":"n35513","layer":"informal","project":"p43","title":"Canonical nonorientable pairing classification","kind":"lemma","summary":"[Canonical nonorientable pairing classification] Every directed polygonal identification genera…","labels":["moise:lem:canonical_nonorientable_pairings"],"detail_key":"p43"},{"id":"n35514","layer":"informal","project":"p43","title":"Use the exact occurrence finsets supplied by the canonical block positions. The two cross…","kind":"proof","summary":"Use the exact occurrence finsets supplied by the canonical block positions. The two crosscap oc…","labels":[],"detail_key":"p43"},{"id":"n35515","layer":"informal","project":"p43","title":"Canonical orientable pairing classification","kind":"lemma","summary":"[Canonical orientable pairing classification] Every directed polygonal identification generated…","labels":["moise:lem:canonical_orientable_pairings"],"detail_key":"p43"},{"id":"n35516","layer":"informal","project":"p43","title":"Compute the exact occurrence finsets for the three twice-occurring edge families and the…","kind":"proof","summary":"Compute the exact occurrence finsets for the three twice-occurring edge families and the single…","labels":[],"detail_key":"p43"},{"id":"n35517","layer":"informal","project":"p43","title":"Canonical carrier coordinates and forward generator inclusions","kind":"lemma","summary":"[Canonical carrier coordinates and forward generator inclusions] The closed-disc carrier sends…","labels":["moise:lem:canonical_forward_generator_coordinates"],"detail_key":"p43"},{"id":"n35518","layer":"informal","project":"p43","title":"Substitute the certified block positions into the generic side-coordinate formula. Handle…","kind":"proof","summary":"Substitute the certified block positions into the generic side-coordinate formula. Handle and c…","labels":[],"detail_key":"p43"},{"id":"n35519","layer":"informal","project":"p43","title":"Canonical generator-closure comparison","kind":"lemma","summary":"[Canonical generator-closure comparison] The polygonal generators of the canonical orientable a…","labels":["moise:lem:canonical_generator_comparison"],"detail_key":"p43"},{"id":"n35520","layer":"informal","project":"p43","title":"Use the exhaustive pairing classifications to package the forward coordinate lemmas for a…","kind":"proof","summary":"Use the exhaustive pairing classifications to package the forward coordinate lemmas for an arbi…","labels":[],"detail_key":"p43"},{"id":"n35521","layer":"informal","project":"p43","title":"Certified finite-cyclic normalization result","kind":"definition","summary":"[Certified finite-cyclic normalization result] A normalization result records an admissible nor…","labels":["moise:def:normal_form_witness"],"detail_key":"p43"},{"id":"n35522","layer":"informal","project":"p43","title":"Faithful finite-cyclic normal-form reduction","kind":"theorem","summary":"[Faithful finite-cyclic normal-form reduction] Every valid connected finite-cyclic surface pres…","labels":["moise:thm:gx_normal_form"],"detail_key":"p43"},{"id":"n35523","layer":"informal","project":"p43","title":"Following Gallier--Xu Ch.~6, remove inverse pairs, reduce vertices, merge faces, introduc…","kind":"proof","summary":"Following Gallier--Xu Ch.~6, remove inverse pairs, reduce vertices, merge faces, introduce cros…","labels":[],"detail_key":"p43"},{"id":"n35524","layer":"informal","project":"p43","title":"Canonical endpoint realization homeomorphisms","kind":"lemma","summary":"[Canonical endpoint realization homeomorphisms] Each admissible canonical finite-cyclic present…","labels":["moise:lem:canonical_endpoint_realizations"],"detail_key":"p43"},{"id":"n35525","layer":"informal","project":"p43","title":"For the sphere, identify the two monogons with the two hemispheres and descend the bounda…","kind":"proof","summary":"For the sphere, identify the two monogons with the two hemispheres and descend the boundary glu…","labels":[],"detail_key":"p43"},{"id":"n35526","layer":"informal","project":"p43","title":"Finite-cyclic presentation gives an Eval branch","kind":"lemma","summary":"[Finite-cyclic presentation gives an Eval branch] A valid connected finite-cyclic presentation…","labels":["moise:lem:normal_form_witness_eval_branch"],"detail_key":"p43"},{"id":"n35527","layer":"informal","project":"p43","title":"Compose the normalization homeomorphism with the corresponding canonical sphere, orientab…","kind":"proof","summary":"Compose the normalization homeomorphism with the corresponding canonical sphere, orientable, or…","labels":[],"detail_key":"p43"},{"id":"n35528","layer":"informal","project":"p43","title":"Topological classification of compact Eval surfaces","kind":"theorem","summary":"[Topological classification of compact Eval surfaces] Every compact connected Hausdorff topolog…","labels":["moise:thm:eval_classification_of_surfaces"],"detail_key":"p43"},{"id":"n35529","layer":"informal","project":"p43","title":"Apply the faithful geometric-to-polygonal bridge, normalize its finite-cyclic presentatio…","kind":"proof","summary":"Apply the faithful geometric-to-polygonal bridge, normalize its finite-cyclic presentation, sel…","labels":[],"detail_key":"p43"},{"id":"n35530","layer":"informal","project":"p44","title":"NeuralNetwork","kind":"definition","summary":"An (artificial) neural network is a directed graph \\( G = (U, C) \\), where neurons \\( u \\in U \\…","labels":["NeuralNetwork"],"detail_key":"p44"},{"id":"n35531","layer":"informal","project":"p44","title":"HopfieldNetwork","kind":"definition","summary":"A Hopfield network is a neural network with graph G = (U,C) as described in the previous sectio…","labels":["HopfieldNetwork"],"detail_key":"p44"},{"id":"n35532","layer":"informal","project":"p44","title":"HopfieldNet_convergence_fair","kind":"theorem","summary":"If the activations of the neurons of a Hopfield network are updated asynchronously, a stable st…","labels":["HopfieldNet_convergence_fair"],"detail_key":"p44"},{"id":"n35533","layer":"informal","project":"p44","title":"HopfieldNet_stabilize","kind":"theorem","summary":"A function that returns the stabilized state after updating.","labels":["HopfieldNet_stabilize"],"detail_key":"p44"},{"id":"n35534","layer":"informal","project":"p44","title":"Corollary of convergence Theorem for Hopfield networks (Theorem 8.1, \\citecomp)","kind":"theorem","summary":"[Corollary of convergence Theorem for Hopfield networks (Theorem 8.1, \\citecomp)] If the neuron…","labels":["HopfieldNet_convergence_fairCor"],"detail_key":"p44"},{"id":"n35535","layer":"informal","project":"p44","title":"Asymmetric HopfieldNetwork","kind":"definition","summary":"We define asymmetric Hopfield networks as general neural networks with the same graph514 struct…","labels":["Asymmetric HopfieldNetwork"],"detail_key":"p44"},{"id":"n35536","layer":"informal","project":"p44","title":"PotentialFunction","kind":"definition","summary":"The potential function for asymmetric Hopfield networks at time step k represents the energy of…","labels":["PotentialFunction"],"detail_key":"p44"},{"id":"n35537","layer":"informal","project":"p44","title":"Potential","kind":"lemma","summary":"The significance of this potential function lies in its relationship to Lyapunov stability566 t…","labels":["{Potential"],"detail_key":"p44"},{"id":"n35538","layer":"informal","project":"p44","title":"boltzmannDistribution","kind":"definition","summary":"The Boltzmann distribution: P(s) = \\frace^-E(s)/TZ where E(s) is the energy of state s, T is th…","labels":["boltzmannDistribution"],"detail_key":"p44"},{"id":"n35539","layer":"informal","project":"p44","title":"GibbsSampling","kind":"definition","summary":"For neuron updates, we use Gibbs sampling, as introduced by Geman and Geman \\citegeman, where a…","labels":["GibbsSampling"],"detail_key":"p44"},{"id":"n35540","layer":"informal","project":"p44","title":"SimulatedAnnealing","kind":"definition","summary":"We also implement simulated annealing, as introduced by Kirkpatrick et al. \\citekirk, which sys…","labels":["SimulatedAnnealing"],"detail_key":"p44"},{"id":"n35541","layer":"informal","project":"p44","title":"Metropolis-Hastings","kind":"definition","summary":"Another sampling method we formalize is the Metropolis-Hastings algorithm, introduced by Metrop…","labels":["Metropolis-Hastings"],"detail_key":"p44"},{"id":"n35542","layer":"informal","project":"p44","title":"Total variation","kind":"definition","summary":"To measure convergence to the equilibrium Boltzmann distribution, we use the total variation di…","labels":["Total variation"],"detail_key":"p44"},{"id":"n35543","layer":"informal","project":"p44","title":"stochasticHopfieldMarkovProcess","kind":"definition","summary":"The stochastic Hopfield Markov process, which models the evolution of Hopfield network states o…","labels":["stochasticHopfieldMarkovProcess"],"detail_key":"p44"},{"id":"n35544","layer":"informal","project":"p44","title":"PhaseSpacePoint","kind":"definition","summary":"A `PhaseSpacePoint` represents a state in the phase space of the Hopfield system. In the paper,…","labels":["PhaseSpacePoint"],"detail_key":"p44"},{"id":"n35545","layer":"informal","project":"p44","title":"localField","kind":"definition","summary":"The `localField` for neuron i in state s is the weighted sum of inputs from other neurons, minu…","labels":["localField"],"detail_key":"p44"},{"id":"n35546","layer":"informal","project":"p44","title":"updateRule","kind":"definition","summary":"The `updateRule` defines the neural state update according to the paper's Equation 1: Vi \\right…","labels":["updateRule"],"detail_key":"p44"},{"id":"n35547","layer":"informal","project":"p44","title":"PhaseSpaceFlow","kind":"definition","summary":"A `PhaseSpaceFlow` describes how the system state evolves over time. It maps each point in phas…","labels":["PhaseSpaceFlow"],"detail_key":"p44"},{"id":"n35548","layer":"informal","project":"p44","title":"FixedPoint","kind":"definition","summary":"A `FixedPoint` of the phase space flow is a state that does not change under evolution. In the…","labels":["FixedPoint"],"detail_key":"p44"},{"id":"n35549","layer":"informal","project":"p44","title":"BasinOfAttraction","kind":"definition","summary":"A `BasinOfAttraction` of a fixed point is the set of all states that converge to it. In the pap…","labels":["BasinOfAttraction"],"detail_key":"p44"},{"id":"n35550","layer":"informal","project":"p44","title":"EnergyLandscape","kind":"definition","summary":"The `EnergyLandscape` of a Hopfield network is the energy function defined over all possible st…","labels":["EnergyLandscape"],"detail_key":"p44"},{"id":"n35551","layer":"informal","project":"p44","title":"energy_decrease","kind":"definition","summary":"The `energy\\_decrease` when updating neuron i is always non-positive, as proven in the paper wi…","labels":["energy_decrease"],"detail_key":"p44"},{"id":"n35552","layer":"informal","project":"p44","title":"convergence_to_fixed_point","kind":"definition","summary":"This theorem captures the convergence result from the paper: \"Every initial state flows to a li…","labels":["convergence_to_fixed_point"],"detail_key":"p44"},{"id":"n35553","layer":"informal","project":"p44","title":"normalizedPattern","kind":"definition","summary":"The `normalizedPattern` converts a neural state to a vector with -1/+1 values, matching the (2V…","labels":["normalizedPattern"],"detail_key":"p44"},{"id":"n35554","layer":"informal","project":"p44","title":"hebbian","kind":"definition","summary":"The `hebbian` function computes the weight matrix according to Equation 2 of Hopfield's paper:…","labels":["hebbian"],"detail_key":"p44"},{"id":"n35555","layer":"informal","project":"p44","title":"isPseudoOrthogonal","kind":"definition","summary":"The `pseudoOrthogonality` property from Hopfield's paper (Equations 3-4) states: For random pat…","labels":["isPseudoOrthogonal"],"detail_key":"p44"},{"id":"n35556","layer":"informal","project":"p44","title":"spin_glass_analogy","kind":"definition","summary":"The `spin\\_glass\\_analogy` formalizes the connection between Hopfield networks and physical spi…","labels":["spin_glass_analogy"],"detail_key":"p44"},{"id":"n35557","layer":"informal","project":"p44","title":"energy_convergence","kind":"definition","summary":"The `energy\\_convergence` theorem formalizes the connection between energy minimization and the…","labels":["energy_convergence"],"detail_key":"p44"},{"id":"n35558","layer":"informal","project":"p44","title":"retrievalDistance","kind":"definition","summary":"The `retrievalDistance` function measures how far from a pattern we can initialize the network…","labels":["retrievalDistance"],"detail_key":"p44"},{"id":"n35559","layer":"informal","project":"p44","title":"ContentAddressableMemory","kind":"definition","summary":"A Content Addressable Memory is a system that can retrieve a complete pattern from a partial or…","labels":["ContentAddressableMemory"],"detail_key":"p44"},{"id":"n35560","layer":"informal","project":"p44","title":"MetricDecayFunction","kind":"definition","summary":"A generic function type representing how a metric (like completion probability or familiarity)…","labels":["MetricDecayFunction"],"detail_key":"p44"},{"id":"n35561","layer":"informal","project":"p44","title":"ExponentialDecayMetric","kind":"definition","summary":"A specific exponential decay model, often used to model probabilities or familiarity scores. Th…","labels":["ExponentialDecayMetric"],"detail_key":"p44"},{"id":"n35562","layer":"informal","project":"p44","title":"AbstractCompletionProbability","kind":"definition","summary":"The `AbstractCompletionProbability` measures the likelihood of correctly completing a pattern a…","labels":["AbstractCompletionProbability"],"detail_key":"p44"},{"id":"n35563","layer":"informal","project":"p44","title":"ErrorCorrection","kind":"definition","summary":"`ErrorCorrection` quantifies the network's ability to correct errors in the input pattern. It's…","labels":["ErrorCorrection"],"detail_key":"p44"},{"id":"n35564","layer":"informal","project":"p44","title":"error_correction_guarantee","kind":"definition","summary":"The `error\\_correction\\_guarantee` theorem establishes that Hopfield networks can correct a sub…","labels":["error_correction_guarantee"],"detail_key":"p44"},{"id":"n35565","layer":"informal","project":"p44","title":"BasinOfAttraction'","kind":"definition","summary":"The `BasinOfAttraction'` of a pattern is the set of all states that converge to it. This concep…","labels":["BasinOfAttraction'"],"detail_key":"p44"},{"id":"n35566","layer":"informal","project":"p44","title":"BasinVolume","kind":"definition","summary":"The `BasinVolume` is the \"size\" of the basin of attraction, measured as the fraction of the sta…","labels":["BasinVolume"],"detail_key":"p44"},{"id":"n35567","layer":"informal","project":"p44","title":"basin_volume_bound","kind":"theorem","summary":"The `basin\\_volume\\_bound` theorem establishes that the basin volume decreases exponentially wi…","labels":["basin_volume_bound"],"detail_key":"p44"},{"id":"n35568","layer":"informal","project":"p44","title":"AbstractFamiliarityMeasure","kind":"definition","summary":"`AbstractFamiliarityMeasure` quantifies how familiar a given state `s` is to the network, based…","labels":["AbstractFamiliarityMeasure"],"detail_key":"p44"},{"id":"n35569","layer":"informal","project":"p44","title":"StorageCapacity","kind":"definition","summary":"The `StorageCapacity` of a Hopfield network is the maximum number of patterns that can be store…","labels":["StorageCapacity"],"detail_key":"p44"},{"id":"n35570","layer":"informal","project":"p44","title":"Hopfield82.Real.erf","kind":"definition","summary":"The error function is defined as: \\[ erf(x) = \\frac2\\sqrt\\pi \\int_0^x e^-t^2 \\, dt \\] This func…","labels":["Hopfield82.Real.erf"],"detail_key":"p44"},{"id":"n35571","layer":"informal","project":"p44","title":"PatternRetrievalError","kind":"definition","summary":"The `PatternRetrievalError` function computes the probability of an error in pattern retrieval…","labels":["PatternRetrievalError"],"detail_key":"p44"},{"id":"n35572","layer":"informal","project":"p44","title":"storage_capacity_bound","kind":"theorem","summary":"The result from the paper that a Hopfield network can store approximately 0.15N patterns with h…","labels":["storage_capacity_bound"],"detail_key":"p44"},{"id":"n35573","layer":"informal","project":"p44","title":"DeleteNeuron","kind":"definition","summary":"The `DeleteNeuron` function simulates the failure of a neuron by removing its connections. This…","labels":["DeleteNeuron"],"detail_key":"p44"},{"id":"n35574","layer":"informal","project":"p44","title":"FaultTolerance","kind":"definition","summary":"The `FaultTolerance` of a Hopfield network is its ability to maintain function despite the fail…","labels":["FaultTolerance"],"detail_key":"p44"},{"id":"n35575","layer":"informal","project":"p44","title":"pattern_stability_in_hebbian","kind":"definition","summary":"When m is at most a tenth of total neurons, each pattern is fixed point in the undamaged network","labels":["pattern_stability_in_hebbian"],"detail_key":"p44"},{"id":"n35576","layer":"informal","project":"p44","title":"delete_single_neuron_step","kind":"lemma","summary":"When deleting a single neuron from a network, the resulting weighted sum for a neuron u that's…","labels":["delete_single_neuron_step"],"detail_key":"p44"},{"id":"n35577","layer":"informal","project":"p44","title":"delete_empty_neurons_step","kind":"definition","summary":"When deleting neurons from an empty list, the result is the original network","labels":["delete_empty_neurons_step"],"detail_key":"p44"},{"id":"n35578","layer":"informal","project":"p44","title":"delete_cons_neuron_step","kind":"definition","summary":"When deleting a list of neurons with a new neuron added at the front, the effect on the weighte…","labels":["delete_cons_neuron_step"],"detail_key":"p44"},{"id":"n35579","layer":"informal","project":"p44","title":"delete_singleton_neuron_step","kind":"definition","summary":"For a singleton list, the effect matches the single neuron deletion case.","labels":["delete_singleton_neuron_step"],"detail_key":"p44"},{"id":"n35580","layer":"informal","project":"p44","title":"delete_neuron_from_deleted_network","kind":"definition","summary":"The effect of deleting a neuron from an already deleted network on a neuron u that is not in th…","labels":["delete_neuron_from_deleted_network"],"detail_key":"p44"},{"id":"n35581","layer":"informal","project":"p44","title":"commute_delete_foldl","kind":"definition","summary":"Helper lemma: DeleteNeuron commutes with foldl of DeleteNeuron if the neuron is not in the list.","labels":["commute_delete_foldl"],"detail_key":"p44"},{"id":"n35582","layer":"informal","project":"p44","title":"foldl_delete_preserves_weights","kind":"definition","summary":"Helper lemma: Weights are preserved by foldl if indices are not in the list","labels":["foldl_delete_preserves_weights"],"detail_key":"p44"},{"id":"n35583","layer":"informal","project":"p44","title":"delete_neurons_recursive","kind":"definition","summary":"Helper lemma: Deleting a list of neurons recursively subtracts their contributions. Requires th…","labels":["delete_neurons_recursive"],"detail_key":"p44"},{"id":"n35584","layer":"informal","project":"p44","title":"deleted_neurons_field_effect","kind":"definition","summary":"DeleteNeurons removes weights connected to deleted neurons.","labels":["deleted_neurons_field_effect"],"detail_key":"p44"},{"id":"n35585","layer":"informal","project":"p44","title":"hebbian_weight_deleted_neurons_l_eq_k_term","kind":"definition","summary":"Calculates the contribution to the deleted field sum from the target pattern `k` itself in the…","labels":["hebbian_weight_deleted_neurons_l_eq_k_term"],"detail_key":"p44"},{"id":"n35586","layer":"informal","project":"p44","title":"hebbian_weight_deleted_neurons_cross_talk_term","kind":"definition","summary":"Defines the cross-talk contribution to the deleted field sum. This term arises from the interac…","labels":["hebbian_weight_deleted_neurons_cross_talk_term"],"detail_key":"p44"},{"id":"n35587","layer":"informal","project":"p44","title":"cross_talk_term_abs_bound_assumption","kind":"lemma","summary":"Axiom stating the bound on the absolute value of the cross-talk term. This encapsulates the sta…","labels":["cross_talk_term_abs_bound_assumption"],"detail_key":"p44"},{"id":"n35588","layer":"informal","project":"p44","title":"Hebbian_stable","kind":"lemma","summary":"Decomposes the total sum representing the field reduction into the contribution from the target…","labels":["Hebbian_stable"],"detail_key":"p44"},{"id":"n35589","layer":"informal","project":"p44","title":"bound_cross_talk_term","kind":"lemma","summary":"Placeholder lemma for bounding the cross-talk term. Proving a tight bound likely requires assum…","labels":["bound_cross_talk_term"],"detail_key":"p44"},{"id":"n35590","layer":"informal","project":"p44","title":"deleted_field_bound","kind":"lemma","summary":"The field reduction from deleting neurons has a bounded effect. This version uses the decomposi…","labels":["deleted_field_bound"],"detail_key":"p44"},{"id":"n35591","layer":"informal","project":"p44","title":"field_remains_sufficient","kind":"lemma","summary":"With constrained m and limited deleted neurons, the field remains strong enough","labels":["field_remains_sufficient"],"detail_key":"p44"},{"id":"n35592","layer":"informal","project":"p44","title":"bound_cross_talk_term_abs","kind":"lemma","summary":"For random orthogonal patterns, the cross-talk term has a bounded absolute value. This is a fun…","labels":["bound_cross_talk_term_abs"],"detail_key":"p44"},{"id":"n35593","layer":"informal","project":"p44","title":"deleted_field_product_bound","kind":"lemma","summary":"TO DO","labels":["deleted_field_product_bound"],"detail_key":"p44"},{"id":"n35594","layer":"informal","project":"p44","title":"field_remains_sufficient_for_N_div_5","kind":"lemma","summary":"With bounded numbers of patterns and deleted neurons, the field remains strong enough to mainta…","labels":["field_remains_sufficient_for_N_div_5"],"detail_key":"p44"},{"id":"n35595","layer":"informal","project":"p44","title":"hebbian_deleted_threshold_is_zero","kind":"lemma","summary":"\\notready TO DO","labels":["hebbian_deleted_threshold_is_zero"],"detail_key":"p44"},{"id":"n35596","layer":"informal","project":"p44","title":"net_input_at_non_deleted_neuron","kind":"lemma","summary":"\\notready TO DO","labels":["net_input_at_non_deleted_neuron"],"detail_key":"p44"},{"id":"n35597","layer":"informal","project":"p44","title":"product_net_input_activation_at_non_deleted_neuron","kind":"lemma","summary":"\\notready TO DO","labels":["product_net_input_activation_at_non_deleted_neuron"],"detail_key":"p44"},{"id":"n35598","layer":"informal","project":"p44","title":"non_deleted_neuron_maintains_sign_of_activation","kind":"definition","summary":"\\notready TO DO","labels":["non_deleted_neuron_maintains_sign_of_activation"],"detail_key":"p44"},{"id":"n35599","layer":"informal","project":"p44","title":"DeleteNeurons_with_Finset","kind":"definition","summary":"When deleting neurons from a Finset, we can use Finset.toList to convert the Finset to a List.…","labels":["DeleteNeurons_with_Finset"],"detail_key":"p44"},{"id":"n35600","layer":"informal","project":"p44","title":"fault_tolerance_bound","kind":"definition","summary":"A Hopfield network can tolerate the failure of up to 10\\% of its neurons while maintaining all…","labels":["fault_tolerance_bound"],"detail_key":"p44"},{"id":"n35601","layer":"formal","project":"p44","title":"Hopfield82.energy_convergence","kind":"theorem","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.energy_convergence","module":"HopfieldNet.Attic.Hopfield82.EnergyConvergence"},{"id":"n35602","layer":"formal","project":"p44","title":"Hopfield82.spin_glass_analogy","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.spin_glass_analogy","module":"HopfieldNet.Attic.Hopfield82.EnergyConvergence"},{"id":"n35603","layer":"formal","project":"p44","title":"Hopfield82.PatternRetrievalError","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.PatternRetrievalError","module":"HopfieldNet.Attic.Hopfield82.MemoryConfusion"},{"id":"n35604","layer":"formal","project":"p44","title":"Hopfield82.Real.erf","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.Real.erf","module":"HopfieldNet.Attic.Hopfield82.MemoryConfusion"},{"id":"n35605","layer":"formal","project":"p44","title":"Hopfield82.StorageCapacity","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.StorageCapacity","module":"HopfieldNet.Attic.Hopfield82.MemoryConfusion"},{"id":"n35606","layer":"formal","project":"p44","title":"Hopfield82.storage_capacity_bound","kind":"theorem","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.storage_capacity_bound","module":"HopfieldNet.Attic.Hopfield82.MemoryConfusion"},{"id":"n35607","layer":"formal","project":"p44","title":"Hopfield82.hebbian","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.hebbian","module":"HopfieldNet.Attic.Hopfield82.MemoryStorage"},{"id":"n35608","layer":"formal","project":"p44","title":"Hopfield82.isPseudoOrthogonal","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.isPseudoOrthogonal","module":"HopfieldNet.Attic.Hopfield82.MemoryStorage"},{"id":"n35609","layer":"formal","project":"p44","title":"Hopfield82.normalizedPattern","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.normalizedPattern","module":"HopfieldNet.Attic.Hopfield82.MemoryStorage"},{"id":"n35610","layer":"formal","project":"p44","title":"Hopfield82.BasinOfAttraction","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.BasinOfAttraction","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35611","layer":"formal","project":"p44","title":"Hopfield82.EnergyLandscape","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.EnergyLandscape","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35612","layer":"formal","project":"p44","title":"Hopfield82.FixedPoint","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.FixedPoint","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35613","layer":"formal","project":"p44","title":"Hopfield82.PhaseSpaceFlow","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.PhaseSpaceFlow","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35614","layer":"formal","project":"p44","title":"Hopfield82.PhaseSpacePoint","kind":"abbrev","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.PhaseSpacePoint","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35615","layer":"formal","project":"p44","title":"Hopfield82.convergence_to_fixed_point","kind":"theorem","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.convergence_to_fixed_point","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35616","layer":"formal","project":"p44","title":"Hopfield82.energy_decrease","kind":"theorem","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.energy_decrease","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35617","layer":"formal","project":"p44","title":"Hopfield82.localField","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.localField","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35618","layer":"formal","project":"p44","title":"Hopfield82.updateRule","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"Hopfield82.updateRule","module":"HopfieldNet.Attic.Hopfield82.PhaseSpaceFlow"},{"id":"n35619","layer":"formal","project":"p44","title":"MarkovChain.stochasticHopfieldMarkovProcess","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"MarkovChain.stochasticHopfieldMarkovProcess","module":"HopfieldNet.Markov"},{"id":"n35620","layer":"formal","project":"p44","title":"MarkovChain.totalVariation","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"MarkovChain.totalVariation","module":"HopfieldNet.Markov"},{"id":"n35621","layer":"formal","project":"p44","title":"NN.State.gibbsUpdateSingleNeuron","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"NN.State.gibbsUpdateSingleNeuron","module":"HopfieldNet.Stochastic"},{"id":"n35622","layer":"formal","project":"p44","title":"NN.State.metropolisHastingsStep","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"NN.State.metropolisHastingsStep","module":"HopfieldNet.Stochastic"},{"id":"n35623","layer":"formal","project":"p44","title":"NN.State.simulatedAnnealing","kind":"def","summary":"","labels":[],"detail_key":"p44","name":"NN.State.simulatedAnnealing","module":"HopfieldNet.Stochastic"},{"id":"n35624","layer":"informal","project":"p45","title":"Euclid's proof","kind":"theorem","summary":"[Euclid's proof] A finite set \\p_1, \\dots, p_r\\ cannot be the collection of \\emphall prime numb…","labels":["thm:euclids_proof"],"detail_key":"p45"},{"id":"n35625","layer":"informal","project":"p45","title":"For any finite set \\p_1, \\dots, p_r\\ of primes, consider the number n = p_1 p_2 \\cdots p_…","kind":"proof","summary":"For any finite set \\p_1, \\dots, p_r\\ of primes, consider the number n = p_1 p_2 \\cdots p_r + 1.…","labels":[],"detail_key":"p45"},{"id":"n35626","layer":"informal","project":"p45","title":"Second Proof","kind":"theorem","summary":"[Second Proof] Any two Fermat numbers F_n := 2^2^n + 1 are relatively prime.","labels":["thm:second_proof"],"detail_key":"p45"},{"id":"n35627","layer":"informal","project":"p45","title":"Let us first look at the Fermat numbers \\(F_n = 2^2^n +1\\) for \\(n = 0,1,2,\\dots\\). We wi…","kind":"proof","summary":"Let us first look at the Fermat numbers \\(F_n = 2^2^n +1\\) for \\(n = 0,1,2,\\dots\\). We will sho…","labels":[],"detail_key":"p45"},{"id":"n35628","layer":"informal","project":"p45","title":"Third Proof","kind":"theorem","summary":"[Third Proof] There is no largest prime.","labels":["thm:third_proof"],"detail_key":"p45"},{"id":"n35629","layer":"informal","project":"p45","title":"Suppose \\(P\\) is finite and \\(p\\) is the largest prime. We consider the so-called \\emphMe…","kind":"proof","summary":"Suppose \\(P\\) is finite and \\(p\\) is the largest prime. We consider the so-called \\emphMersenne…","labels":[],"detail_key":"p45"},{"id":"n35630","layer":"informal","project":"p45","title":"Fourth Proof","kind":"theorem","summary":"[Fourth Proof] The prime counting function is unbounded","labels":["thm:fourth_proof"],"detail_key":"p45"},{"id":"n35631","layer":"informal","project":"p45","title":"Let \\(\\pi(x) := \\#\\p \\leq x : p \\in P\\\\) be the number of primes that are less than or eq…","kind":"proof","summary":"Let \\(\\pi(x) := \\#\\p \\leq x : p \\in P\\\\) be the number of primes that are less than or equal to…","labels":[],"detail_key":"p45"},{"id":"n35632","layer":"informal","project":"p45","title":"Fifth Proof","kind":"theorem","summary":"[Fifth Proof] The set of primes \\(P\\) is infinite.","labels":["thm:fifth_proof"],"detail_key":"p45"},{"id":"n35633","layer":"informal","project":"p45","title":"Consider the following curious topology on the set \\(Z\\) of integers. For \\(a, b \\in Z, b…","kind":"proof","summary":"Consider the following curious topology on the set \\(Z\\) of integers. For \\(a, b \\in Z, b > 0\\)…","labels":[],"detail_key":"p45"},{"id":"n35634","layer":"informal","project":"p45","title":"Sixth Proof","kind":"theorem","summary":"[Sixth Proof] The series \\(\\sum_p\\inP\\frac 1 p\\) diverges.","labels":["thm:sixth_proof"],"detail_key":"p45"},{"id":"n35635","layer":"informal","project":"p45","title":"Our final proof goes a considerable step further and demonstrates not only that there are…","kind":"proof","summary":"Our final proof goes a considerable step further and demonstrates not only that there are infin…","labels":[],"detail_key":"p45"},{"id":"n35636","layer":"informal","project":"p45","title":"thm:infty_proof","kind":"theorem","summary":"If the sequence \\(S = (s_1, s_2, s_3, \\dots)\\) is almost injective and of subexponential growth…","labels":["thm:infty_proof"],"detail_key":"p45"},{"id":"n35637","layer":"informal","project":"p45","title":"We may assume that \\(f(n)\\) is monotonely increasing. Otherwise, replace \\(f(n)\\) by \\(F(…","kind":"proof","summary":"We may assume that \\(f(n)\\) is monotonely increasing. Otherwise, replace \\(f(n)\\) by \\(F(n) = \\…","labels":[],"detail_key":"p45"},{"id":"n35638","layer":"informal","project":"p45","title":"Infinity of primes","kind":"theorem","summary":"[Infinity of primes] There are infinitely many primes. (Six + infinitely many proofs)","labels":["thm:infinity_of_primes"],"detail_key":"p45"},{"id":"n35639","layer":"informal","project":"p45","title":"See theorems in this chapter.","kind":"proof","summary":"See theorems in this chapter.","labels":[],"detail_key":"p45"},{"id":"n35640","layer":"informal","project":"p45","title":"Bertrand's postulate","kind":"theorem","summary":"[Bertrand's postulate] For any positive natural number, there is a prime which is greater than…","labels":["thm:bertrands_postulate"],"detail_key":"p45"},{"id":"n35641","layer":"informal","project":"p45","title":"eq:prod_prime_le","kind":"proof","summary":"We will estimate the size of the binomial coefficient \\binom2nn carefully enough to see that if…","labels":["eq:prod_prime_le"],"detail_key":"p45"},{"id":"n35642","layer":"informal","project":"p45","title":"thm:estimate_integral","kind":"theorem","summary":"For all \\(n \\in N\\) \\[ \\log n + \\frac 1 n < H_n < \\log n + 1. \\]","labels":["thm:estimate_integral"],"detail_key":"p45"},{"id":"n35643","layer":"informal","project":"p45","title":"There is a very simple-but-effective method of estimating sums by integrals. For estimati…","kind":"proof","summary":"There is a very simple-but-effective method of estimating sums by integrals. For estimating the…","labels":[],"detail_key":"p45"},{"id":"n35644","layer":"informal","project":"p45","title":"thm:estimate_factorials","kind":"theorem","summary":"For all \\(n \\in N\\) \\[ e\\left(\\fracne\\right)^n < n! < en\\left(\\fracne\\right)^n. \\]","labels":["thm:estimate_factorials"],"detail_key":"p45"},{"id":"n35645","layer":"informal","project":"p45","title":"The same method applied to \\[ \\log(n!) = \\log 2 + \\log 3 + \\dots + \\log n = \\sum_k=2^n \\l…","kind":"proof","summary":"The same method applied to \\[ \\log(n!) = \\log 2 + \\log 3 + \\dots + \\log n = \\sum_k=2^n \\log k \\…","labels":[],"detail_key":"p45"},{"id":"n35646","layer":"informal","project":"p45","title":"thm:estimate_binomial_coefficient","kind":"theorem","summary":"\\[\\binomnk \\le \\fracn^kk! \\le \\fracn^k2^k - 1\\]","labels":["thm:estimate_binomial_coefficient"],"detail_key":"p45"},{"id":"n35647","layer":"informal","project":"p45","title":"\\[ \\binomnk = \\fracn (n-1) \\cdots (n-k+1)k! \\le \\fracn^kk! \\le \\fracn^k2^k-1. \\qedhere \\]","kind":"proof","summary":"\\[ \\binomnk = \\fracn (n-1) \\cdots (n-k+1)k! \\le \\fracn^kk! \\le \\fracn^k2^k-1. \\qedhere \\]","labels":[],"detail_key":"p45"},{"id":"n35648","layer":"informal","project":"p45","title":"Sylvester's theorem","kind":"theorem","summary":"[Sylvester's theorem] For all positive natural \\(n, k\\) such that \\(n \\ge 2k\\), at least one of…","labels":["sylvester"],"detail_key":"p45"},{"id":"n35649","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35650","layer":"informal","project":"p45","title":"Binomial coefficients are (almost) never powers","kind":"theorem","summary":"[Binomial coefficients are (almost) never powers] The equation \\(\\binom n k = m ^ l\\) has no in…","labels":["binomial_never_powers"],"detail_key":"p45"},{"id":"n35651","layer":"informal","project":"p45","title":"Note first that we may assume \\(n \\ge 2k\\) because of \\(\\binomnk = \\binomnn-k\\). Suppose…","kind":"proof","summary":"Note first that we may assume \\(n \\ge 2k\\) because of \\(\\binomnk = \\binomnn-k\\). Suppose the th…","labels":[],"detail_key":"p45"},{"id":"n35652","layer":"informal","project":"p45","title":"Lemma 1","kind":"lemma","summary":"[Lemma 1] For primes \\(p = 4m + 1\\) the equation \\(s^2 \\equiv -1 (\\mod p)\\) has two solutions \\…","labels":["ch04.lemma₁"],"detail_key":"p45"},{"id":"n35653","layer":"informal","project":"p45","title":"For p = 2 take s = 1. For odd p, we construct the equivalence relation on \\1, 2, \\dots, p…","kind":"proof","summary":"For p = 2 take s = 1. For odd p, we construct the equivalence relation on \\1, 2, \\dots, p - 1\\…","labels":[],"detail_key":"p45"},{"id":"n35654","layer":"informal","project":"p45","title":"Lemma 2","kind":"lemma","summary":"[Lemma 2] No number \\(n = 4m + 3\\) is a sum of two squares.","labels":["ch04.lemma2"],"detail_key":"p45"},{"id":"n35655","layer":"informal","project":"p45","title":"The square of any even number is (2k)^2 = 4k^2 \\equiv 0 \\pmod4, while squares of odd numb…","kind":"proof","summary":"The square of any even number is (2k)^2 = 4k^2 \\equiv 0 \\pmod4, while squares of odd numbers yi…","labels":[],"detail_key":"p45"},{"id":"n35656","layer":"informal","project":"p45","title":"First proof","kind":"proposition","summary":"[First proof] Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can b…","labels":["ch04.proposition1"],"detail_key":"p45"},{"id":"n35657","layer":"informal","project":"p45","title":"Consider the pairs (x', y') of integers with 0 \\leq x', y' \\leq \\sqrtp, that is, x', y' \\…","kind":"proof","summary":"Consider the pairs (x', y') of integers with 0 \\leq x', y' \\leq \\sqrtp, that is, x', y' \\in \\0,…","labels":[],"detail_key":"p45"},{"id":"n35658","layer":"informal","project":"p45","title":"Second proof","kind":"proposition","summary":"[Second proof] Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can…","labels":["ch04.proposition2"],"detail_key":"p45"},{"id":"n35659","layer":"informal","project":"p45","title":"We study the set \\[ S := \\(x, y, z) \\in Z^3 : 4xy + z^2 = p, \\ x > 0, \\ y > 0\\. \\] This s…","kind":"proof","summary":"We study the set \\[ S := \\(x, y, z) \\in Z^3 : 4xy + z^2 = p, \\ x > 0, \\ y > 0\\. \\] This set is…","labels":[],"detail_key":"p45"},{"id":"n35660","layer":"informal","project":"p45","title":"Third proof","kind":"proposition","summary":"[Third proof] Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can b…","labels":["ch04.proposition3"],"detail_key":"p45"},{"id":"n35661","layer":"informal","project":"p45","title":"Again we fix a prime number p = 4n + 1 and consider the set of solutions \\[ T = \\(x, y, z…","kind":"proof","summary":"Again we fix a prime number p = 4n + 1 and consider the set of solutions \\[ T = \\(x, y, z) \\in…","labels":[],"detail_key":"p45"},{"id":"n35662","layer":"informal","project":"p45","title":"sum_of_two_squares","kind":"theorem","summary":"A natural number \\(n\\) can be represented as a sum of two squares if and only if every prime fa…","labels":["sum_of_two_squares"],"detail_key":"p45"},{"id":"n35663","layer":"informal","project":"p45","title":"Call a number n representable if it is a sum of two squares, that is, if n = x^2 + y^2 fo…","kind":"proof","summary":"Call a number n representable if it is a sum of two squares, that is, if n = x^2 + y^2 for some…","labels":[],"detail_key":"p45"},{"id":"n35664","layer":"informal","project":"p45","title":"Fermat's little theorem","kind":"theorem","summary":"[Fermat's little theorem] For \\(a \\not\\equiv 0 \\pmodp\\), \\[ a^p - 1 \\equiv 1 \\pmodp \\]","labels":["fermats_little"],"detail_key":"p45"},{"id":"n35665","layer":"informal","project":"p45","title":"Since Z_p^* = Z_p \\setminus \\0\\ is a group with multiplication, the set \\1a, 2a, 3a, \\dot…","kind":"proof","summary":"Since Z_p^* = Z_p \\setminus \\0\\ is a group with multiplication, the set \\1a, 2a, 3a, \\dots, (p-…","labels":[],"detail_key":"p45"},{"id":"n35666","layer":"informal","project":"p45","title":"Euler's criterion","kind":"theorem","summary":"[Euler's criterion] For \\(a \\not\\equiv 0 \\pmodp\\), \\[ (\\fracap) \\equiv a ^\\fracp-12 \\pmodp \\]","labels":["euler_criterion"],"detail_key":"p45"},{"id":"n35667","layer":"informal","project":"p45","title":"From Fermat's little theorem, the polynomial x^p-1 - 1 \\in Z_p[x] has as roots all nonzer…","kind":"proof","summary":"From Fermat's little theorem, the polynomial x^p-1 - 1 \\in Z_p[x] has as roots all nonzero resi…","labels":[],"detail_key":"p45"},{"id":"n35668","layer":"informal","project":"p45","title":"Product Rule","kind":"theorem","summary":"[Product Rule] (\\fracabp) = (\\fracap) \\cdot (\\fracbp)","labels":["product_rule","eq:product_rule"],"detail_key":"p45"},{"id":"n35669","layer":"informal","project":"p45","title":"This obviously holds for the right-hand side of Euler's criterion.","kind":"proof","summary":"This obviously holds for the right-hand side of Euler's criterion.","labels":[],"detail_key":"p45"},{"id":"n35670","layer":"informal","project":"p45","title":"Lemma of Gauss","kind":"theorem","summary":"[Lemma of Gauss] Suppose a \\not\\equiv 0 \\pmodp. Take the numbers 1a, 2a, \\dots, \\fracp-12a and…","labels":["gauss_lemma"],"detail_key":"p45"},{"id":"n35671","layer":"informal","project":"p45","title":"Suppose u_1, \\dots, u_s are the residues smaller than 0, and that v_1, \\dots, v_\\fracp-12…","kind":"proof","summary":"Suppose u_1, \\dots, u_s are the residues smaller than 0, and that v_1, \\dots, v_\\fracp-12 - s a…","labels":[],"detail_key":"p45"},{"id":"n35672","layer":"informal","project":"p45","title":"Quadratic reciprocity I","kind":"theorem","summary":"[Quadratic reciprocity I] Let p and q be different odd primes. Then \\[ (\\fracqp)(\\fracpq) = (-1…","labels":["quadratic_reciprocity1"],"detail_key":"p45"},{"id":"n35673","layer":"informal","project":"p45","title":"eq:s","kind":"proof","summary":"The key to our first proof is a counting formula given by Lemma of Gauss. Let p and q be odd pr…","labels":["eq:s","eq:t"],"detail_key":"p45"},{"id":"n35674","layer":"informal","project":"p45","title":"mult_cyclic","kind":"theorem","summary":"The multiplicative group of a finite field is cyclic.","labels":["mult_cyclic"],"detail_key":"p45"},{"id":"n35675","layer":"informal","project":"p45","title":"eq:sum_psi","kind":"proof","summary":"Let F^* be the multiplicative group of the field F, with |F^*| = n. Writing \\ord(a) for the ord…","labels":["eq:sum_psi","eq:sum_phi"],"detail_key":"p45"},{"id":"n35676","layer":"informal","project":"p45","title":"A","kind":"theorem","summary":"[A] Let p and q be distinct odd primes, and consider the finite field F with q^p-1 elements. Th…","labels":["fact_A"],"detail_key":"p45"},{"id":"n35677","layer":"informal","project":"p45","title":"The prime field of F is Z_q, whence qa = 0 for any a \\in F. This implies that (a + b)^q =…","kind":"proof","summary":"The prime field of F is Z_q, whence qa = 0 for any a \\in F. This implies that (a + b)^q = a^q +…","labels":[],"detail_key":"p45"},{"id":"n35678","layer":"informal","project":"p45","title":"B","kind":"theorem","summary":"[B] For the field F defined in (A), there exists an element \\zeta \\in F of multiplicative order…","labels":["fact_B"],"detail_key":"p45"},{"id":"n35679","layer":"informal","project":"p45","title":"The multi","kind":"proof","summary":"The multi","labels":[],"detail_key":"p45"},{"id":"n35680","layer":"informal","project":"p45","title":"Quadratic reciprocity II","kind":"theorem","summary":"[Quadratic reciprocity II] Let p and q be different odd primes. Then \\[ (\\fracqp)(\\fracpq) = (-…","labels":["quadratic_reciprocity2"],"detail_key":"p45"},{"id":"n35681","layer":"informal","project":"p45","title":"eq:first-expression","kind":"proof","summary":"The second proof does not use Gauss' lemma, instead it employs so-called ``Gauss sums'' in fini…","labels":["eq:first-expression","{eq:G","eq:second-expression"],"detail_key":"p45"},{"id":"n35682","layer":"informal","project":"p45","title":"Roots of unity","kind":"theorem","summary":"[Roots of unity] The n-th roots of unity are \\[ \\lambda_k = e^\\frac2k\\pi in = \\cos(2k\\pi/n) + i…","labels":["root_of_unity"],"detail_key":"p45"},{"id":"n35683","layer":"informal","project":"p45","title":"Any complex number z = x + iy may be written in the ``polar'' form \\[ z = r e^i\\varphi =…","kind":"proof","summary":"Any complex number z = x + iy may be written in the ``polar'' form \\[ z = r e^i\\varphi = r(\\cos…","labels":[],"detail_key":"p45"},{"id":"n35684","layer":"informal","project":"p45","title":"Wedderburn's theorem","kind":"theorem","summary":"[Wedderburn's theorem] Every finite division ring is commutative.","labels":["wedderburn"],"detail_key":"p45"},{"id":"n35685","layer":"informal","project":"p45","title":"eq:class_formula","kind":"proof","summary":"Our first ingredient comes from a blend of linear algebra and basic group theory. For an arbitr…","labels":["eq:class_formula","eq:nk_divides_n","eq:prod_phi","eq:poly_div","eq:div_relations"],"detail_key":"p45"},{"id":"n35686","layer":"informal","project":"p45","title":"ch7.lemma","kind":"lemma","summary":"If A is a real symmetric n \\times n matrix that is not diagonal, that is, \\Od(A) > 0, then ther…","labels":["ch7.lemma"],"detail_key":"p45"},{"id":"n35687","layer":"informal","project":"p45","title":"eq:b_kl","kind":"proof","summary":"We use a very clever method attributed to Carl Gustav Jacob Jacobi. Suppose that a_rs \\ne 0 for…","labels":["eq:b_kl"],"detail_key":"p45"},{"id":"n35688","layer":"informal","project":"p45","title":"diagonalize_real_symmetric","kind":"theorem","summary":"For every real symmetric matrix A there is a real orthogonal matrix Q such that Q^TAQ is diagon…","labels":["diagonalize_real_symmetric"],"detail_key":"p45"},{"id":"n35689","layer":"informal","project":"p45","title":"The theorem follows in three quick steps. Let A be a real symmetric n \\times n matrix. \\i…","kind":"proof","summary":"The theorem follows in three quick steps. Let A be a real symmetric n \\times n matrix. \\item[(A…","labels":[],"detail_key":"p45"},{"id":"n35690","layer":"informal","project":"p45","title":"Hadamard's inequality","kind":"theorem","summary":"[Hadamard's inequality] For any real n \\times n matrix A = (a_ij) with |a_ij| \\le 1, \\[ |\\det A…","labels":["thm:hadamard_inequality"],"detail_key":"p45"},{"id":"n35691","layer":"informal","project":"p45","title":"eq:trace_B","kind":"proof","summary":"The problem to find the maximum value of \\det A on the set of all real n \\times n matrices A =…","labels":["eq:trace_B","eq:QBQ","eq:AM_GM","eq:hadamard_bound"],"detail_key":"p45"},{"id":"n35692","layer":"informal","project":"p45","title":"thm:hadamard_order","kind":"theorem","summary":"If a Hadamard matrix of size n \\times n exists for n > 2, then n must be a multiple of 4.","labels":["thm:hadamard_order"],"detail_key":"p45"},{"id":"n35693","layer":"informal","project":"p45","title":"A short argument shows that if n is greater than 2, then it must be a multiple of 4. Inde…","kind":"proof","summary":"A short argument shows that if n is greater than 2, then it must be a multiple of 4. Indeed, su…","labels":[],"detail_key":"p45"},{"id":"n35694","layer":"informal","project":"p45","title":"thm:hadamard_existence","kind":"theorem","summary":"Hadamard matrices exist for all n = 2^m.","labels":["thm:hadamard_existence"],"detail_key":"p45"},{"id":"n35695","layer":"informal","project":"p45","title":"Consider an m-set X and index the 2^m subsets C \\subseteq X in any way C_1, \\dots, C_2^m.…","kind":"proof","summary":"Consider an m-set X and index the 2^m subsets C \\subseteq X in any way C_1, \\dots, C_2^m. The m…","labels":[],"detail_key":"p45"},{"id":"n35696","layer":"informal","project":"p45","title":"thm:det_greater_n_fact","kind":"theorem","summary":"There exists an n \\times n matrix with entries \\pm 1 whose determinant is greater than \\sqrtn!.","labels":["thm:det_greater_n_fact"],"detail_key":"p45"},{"id":"n35697","layer":"informal","project":"p45","title":"eq:sum_A","kind":"proof","summary":"Let us look at all 2^n^2 matrices with \\pm 1-entries and consider some averages of the determin…","labels":["eq:sum_A"],"detail_key":"p45"},{"id":"n35698","layer":"informal","project":"p45","title":"e_irrational","kind":"theorem","summary":"e is irrational.","labels":["e_irrational"],"detail_key":"p45"},{"id":"n35699","layer":"informal","project":"p45","title":"To start with, it is rather easy to see (as did Fourier in 1815) that e = \\sum_k \\ge 0 \\f…","kind":"proof","summary":"To start with, it is rather easy to see (as did Fourier in 1815) that e = \\sum_k \\ge 0 \\frac1k!…","labels":[],"detail_key":"p45"},{"id":"n35700","layer":"informal","project":"p45","title":"e_pow_2_irrational","kind":"theorem","summary":"e^2 is irrational.","labels":["e_pow_2_irrational"],"detail_key":"p45"},{"id":"n35701","layer":"informal","project":"p45","title":"Now one might be led to think that this simple multiply-by-n! trick is not sufficient to…","kind":"proof","summary":"Now one might be led to think that this simple multiply-by-n! trick is not sufficient to show t…","labels":[],"detail_key":"p45"},{"id":"n35702","layer":"informal","project":"p45","title":"Little Lemma","kind":"theorem","summary":"[Little Lemma] For any n \\ge 1 the integer n! contains the prime factor 2 at most n-1 times ---…","labels":["little_lemma"],"detail_key":"p45"},{"id":"n35703","layer":"informal","project":"p45","title":"This lemma is not hard to show: \\lfloor \\fracn2 \\rfloor of the factors of n! are even, \\l…","kind":"proof","summary":"This lemma is not hard to show: \\lfloor \\fracn2 \\rfloor of the factors of n! are even, \\lfloor…","labels":[],"detail_key":"p45"},{"id":"n35704","layer":"informal","project":"p45","title":"e_pow_4_irrational","kind":"theorem","summary":"e^4 is irrational.","labels":["e_pow_4_irrational"],"detail_key":"p45"},{"id":"n35705","layer":"informal","project":"p45","title":"eq:e4_irrational","kind":"proof","summary":"In order to show that e^4 is irrational, we now courageously assume that e^4 = \\fracab were rat…","labels":["eq:e4_irrational"],"detail_key":"p45"},{"id":"n35706","layer":"informal","project":"p45","title":"lem_aux_i","kind":"lemma","summary":"\\upshape For some fixed n \\ge 1, let \\[ f(x) = \\fracx^n (1-x)^nn!. \\] \\item[(i)] \\itshape The f…","labels":["lem_aux_i"],"detail_key":"p45"},{"id":"n35707","layer":"informal","project":"p45","title":"Part (i) is clear.","kind":"proof","summary":"Part (i) is clear.","labels":[],"detail_key":"p45"},{"id":"n35708","layer":"informal","project":"p45","title":"lem_aux_ii","kind":"lemma","summary":"\\item[(ii)] For 0 < x < 1 we have 0 < f(x) < \\frac1n!.","labels":["lem_aux_ii"],"detail_key":"p45"},{"id":"n35709","layer":"informal","project":"p45","title":"Part (ii) is also clear.","kind":"proof","summary":"Part (ii) is also clear.","labels":[],"detail_key":"p45"},{"id":"n35710","layer":"informal","project":"p45","title":"lem_aux_iii","kind":"lemma","summary":"\\item[(iii)] The derivatives f^(k)(0) and f^(k)(1) are integers for all k \\ge 0.","labels":["lem_aux_iii"],"detail_key":"p45"},{"id":"n35711","layer":"informal","project":"p45","title":"For (iii) note that by (i) the k-th derivative f^(k) vanishes at x=0 unless n \\le k \\le 2…","kind":"proof","summary":"For (iii) note that by (i) the k-th derivative f^(k) vanishes at x=0 unless n \\le k \\le 2n, and…","labels":[],"detail_key":"p45"},{"id":"n35712","layer":"informal","project":"p45","title":"e_pow_irrational","kind":"theorem","summary":"e^r is irrational for every r \\in Q \\setminus \\0\\.","labels":["e_pow_irrational","book.irrational.Theorem_1"],"detail_key":"p45"},{"id":"n35713","layer":"informal","project":"p45","title":"It suffices to show that e^s cannot be rational for a positive integer s (if e^\\fracst we…","kind":"proof","summary":"It suffices to show that e^s cannot be rational for a positive integer s (if e^\\fracst were rat…","labels":[],"detail_key":"p45"},{"id":"n35714","layer":"informal","project":"p45","title":"pi_pow_2_irrational","kind":"theorem","summary":"\\pi^2 is irrational.","labels":["pi_pow_2_irrational","book.irrational.Theorem_2"],"detail_key":"p45"},{"id":"n35715","layer":"informal","project":"p45","title":"Assume that \\pi^2 = \\fracab for integers a, b > 0. We now use the polynomial \\[ F(x) := b…","kind":"proof","summary":"Assume that \\pi^2 = \\fracab for integers a, b > 0. We now use the polynomial \\[ F(x) := b^n \\le…","labels":[],"detail_key":"p45"},{"id":"n35716","layer":"informal","project":"p45","title":"arccos_irrational","kind":"theorem","summary":"For every odd integer \\(n \\ge 3\\), the number \\[ A(n) := \\frac1\\pi \\arccos \\left(\\frac1\\sqrtn\\r…","labels":["arccos_irrational","book.irrational.Theorem_3"],"detail_key":"p45"},{"id":"n35717","layer":"informal","project":"p45","title":"eq:cos_add","kind":"proof","summary":"We use the addition theorem \\[ \\cos \\alpha + \\cos \\beta = 2 \\cos \\frac\\alpha+\\beta2 \\cos \\frac\\…","labels":["eq:cos_add"],"detail_key":"p45"},{"id":"n35718","layer":"informal","project":"p45","title":"Euler's series: Proof 1","kind":"theorem","summary":"[Euler's series: Proof 1] \\[ \\sum_n \\ge 1 \\frac1n^2 = \\frac\\pi^26 \\]","labels":["euler_series"],"detail_key":"p45"},{"id":"n35719","layer":"informal","project":"p45","title":"The proof consists in two different evaluations of the double integral \\[ I := \\int_0^1 \\…","kind":"proof","summary":"The proof consists in two different evaluations of the double integral \\[ I := \\int_0^1 \\int_0^…","labels":[],"detail_key":"p45"},{"id":"n35720","layer":"informal","project":"p45","title":"Euler's series: Proof 2","kind":"theorem","summary":"[Euler's series: Proof 2] \\[ \\sum_k \\ge 0\\frac1(2k+1)^2 = \\frac\\pi^28 \\]","labels":["euler_series_2"],"detail_key":"p45"},{"id":"n35721","layer":"informal","project":"p45","title":"As above, we may express this as a double integral, namely \\[ J = \\int_0^1 \\int_0^1 \\frac…","kind":"proof","summary":"As above, we may express this as a double integral, namely \\[ J = \\int_0^1 \\int_0^1 \\frac11-x^2…","labels":[],"detail_key":"p45"},{"id":"n35722","layer":"informal","project":"p45","title":"Euler's series: Proof 3","kind":"theorem","summary":"[Euler's series: Proof 3] \\[ \\sum_n\\ge 1\\frac1n^2 = \\frac\\pi^26 \\]","labels":["euler_series_3"],"detail_key":"p45"},{"id":"n35723","layer":"informal","project":"p45","title":"eq:cot_sum","kind":"proof","summary":"The first step is to establish a remarkable relation between values of the (squared) cotangent…","labels":["eq:cot_sum","eq:sin_nx","eq:csc_sum"],"detail_key":"p45"},{"id":"n35724","layer":"informal","project":"p45","title":"Euler's series: Proof 4","kind":"theorem","summary":"[Euler's series: Proof 4] \\[ \\sum_n\\ge 1\\frac1n^2 = \\frac\\pi^26 \\]","labels":["euler_series_4"],"detail_key":"p45"},{"id":"n35725","layer":"informal","project":"p45","title":"The first trick in this proof is to consider the Gregory–Leibniz series in doubly-infinit…","kind":"proof","summary":"The first trick in this proof is to consider the Gregory–Leibniz series in doubly-infinite form…","labels":[],"detail_key":"p45"},{"id":"n35726","layer":"informal","project":"p45","title":"Four proofs of Euler's series","kind":"theorem","summary":"[Four proofs of Euler's series] Collecting the proofs from the chapter.","labels":["four_proofs_euler_series"],"detail_key":"p45"},{"id":"n35727","layer":"informal","project":"p45","title":"See theorems in this chapter.","kind":"proof","summary":"See theorems in this chapter.","labels":[],"detail_key":"p45"},{"id":"n35728","layer":"informal","project":"p45","title":"Pearl Lemma","kind":"lemma","summary":"[Pearl Lemma] If P and Q are equidecomposable, then one can place a positive number of pearls (…","labels":["pearl_lemma"],"detail_key":"p45"},{"id":"n35729","layer":"informal","project":"p45","title":"Assign a variable x_i to each segment in the decomposition of P and a variable y_j to eac…","kind":"proof","summary":"Assign a variable x_i to each segment in the decomposition of P and a variable y_j to each segm…","labels":[],"detail_key":"p45"},{"id":"n35730","layer":"informal","project":"p45","title":"Cone Lemma","kind":"lemma","summary":"[Cone Lemma] If a system of homogeneous linear equations with integer 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Then \\(G\\) has at most \\(3*n - 6\\) e…","labels":["euler_consequence_a"],"detail_key":"p45"},{"id":"n35755","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35756","layer":"informal","project":"p45","title":"euler_consequence_b","kind":"proposition","summary":"Let \\(G\\) be any simple plane graph with \\(n>2\\) vertices. Then \\(G\\) has a vertex of degree at…","labels":["euler_consequence_b"],"detail_key":"p45"},{"id":"n35757","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35758","layer":"informal","project":"p45","title":"euler_consequence_c","kind":"proposition","summary":"Let \\(G\\) be any simple plane graph with \\(n>2\\) vertices. 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Let V = \\1, \\dots, n\\ be the ver…","kind":"proof","summary":"This proof, using Cauchy's inequality, is due to Mantel. Let V = \\1, \\dots, n\\ be the vertex se…","labels":[],"detail_key":"p45"},{"id":"n35824","layer":"informal","project":"p45","title":"ch20theorem3proof2","kind":"theorem","summary":"Suppose G is a graph on n vertices without triangles. Then G has at most \\fracn^24 edges, and e…","labels":["ch20theorem3proof2"],"detail_key":"p45"},{"id":"n35825","layer":"informal","project":"p45","title":"The following proof of Theorem~3, using the inequality of the arithmetic and the geometri…","kind":"proof","summary":"The following proof of Theorem~3, using the inequality of the arithmetic and the geometric mean…","labels":[],"detail_key":"p45"},{"id":"n35826","layer":"informal","project":"p45","title":"argand_inequality","kind":"lemma","summary":"Let p(z) = \\sum_k=0^n c_k z^k be a complex polynomial of degree n\\ge 1. 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Clearly, p(z)z^-n approaches the leading coefficient c_n of p(z) as |z| goes…","labels":[],"detail_key":"p45"},{"id":"n35830","layer":"informal","project":"p45","title":"valutaion on R","kind":"definition","summary":"[valutaion on R]","labels":["valuation"],"detail_key":"p45"},{"id":"n35831","layer":"informal","project":"p45","title":"Three-coloring of plane","kind":"definition","summary":"[Three-coloring of plane] TODO","labels":["three_coloring"],"detail_key":"p45"},{"id":"n35832","layer":"informal","project":"p45","title":"Rainbow triangle","kind":"definition","summary":"[Rainbow triangle] TODO","labels":["rainbow_triangle"],"detail_key":"p45"},{"id":"n35833","layer":"informal","project":"p45","title":"ch22lemma1","kind":"lemma","summary":"For any blue point p_0 = (x_b, y_b), green point (x_g, y_g), and red point (x_r, y_r), the v-va…","labels":["ch22lemma1"],"detail_key":"p45"},{"id":"n35834","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35835","layer":"informal","project":"p45","title":"ch22corollary","kind":"corollary","summary":"Any line of the plane receives at most two different colors. The area of a rainbow triangle can…","labels":["ch22corollary"],"detail_key":"p45"},{"id":"n35836","layer":"informal","project":"p45","title":"Follow from \\refch22lemma1","kind":"proof","summary":"Follow from \\refch22lemma1","labels":[],"detail_key":"p45"},{"id":"n35837","layer":"informal","project":"p45","title":"ch22lemma2","kind":"lemma","summary":"Every dissection of the unit square S = [0, 1]^2 into finitely many triangles contains an odd n…","labels":["ch22lemma2"],"detail_key":"p45"},{"id":"n35838","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35839","layer":"informal","project":"p45","title":"Monsky's theorem","kind":"theorem","summary":"[Monsky's theorem] It is not possible to dissect a square into an odd number of triangles of eq…","labels":["monsky_theorem"],"detail_key":"p45"},{"id":"n35840","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35841","layer":"informal","project":"p45","title":"valuation_lemma","kind":"lemma","summary":"A proper subring R\\subset K is a valuation ring with respect to some valuation v into some orde…","labels":["valuation_lemma"],"detail_key":"p45"},{"id":"n35842","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35843","layer":"informal","project":"p45","title":"valuation_on_reals","kind":"theorem","summary":"The field of real numbers R has a non-Archimedean valuation to an ordered abelian group \\[v: R…","labels":["valuation_on_reals"],"detail_key":"p45"},{"id":"n35844","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35845","layer":"informal","project":"p45","title":"ch23theorem1","kind":"theorem","summary":"Let f(z) be a complex polynomial of degree at least 1 and leading coefficient 1. Set C = \\ z \\i…","labels":["ch23theorem1"],"detail_key":"p45"},{"id":"n35846","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35847","layer":"informal","project":"p45","title":"ch23theorem2","kind":"theorem","summary":"Let p(x) be a real polynomial of degree n \\geq 1 with leading coefficient 1, and all roots real…","labels":["ch23theorem2"],"detail_key":"p45"},{"id":"n35848","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35849","layer":"informal","project":"p45","title":"ch23corollary","kind":"corollary","summary":"Let p(x) be a real polynomial of degree n \\geq 1 with leading coefficient 1, and suppose that |…","labels":["ch23corollary"],"detail_key":"p45"},{"id":"n35850","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35851","layer":"informal","project":"p45","title":"Chebyshev's theorem","kind":"theorem","summary":"[Chebyshev's theorem] Let p(x) be a real polynomial of degree n \\geq 1 with leading coefficient…","labels":["chebyshev"],"detail_key":"p45"},{"id":"n35852","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35853","layer":"informal","project":"p45","title":"Fact 1","kind":"theorem","summary":"[Fact 1] If b is a multiple root of p'(x), then b is also a root of p(x).","labels":["ch23fact1"],"detail_key":"p45"},{"id":"n35854","layer":"informal","project":"p45","title":"Let b_1 < \\cdots < b_r be the roots of p(x) with multiplicities s_1, \\ldots, s_r, \\sum_j=…","kind":"proof","summary":"Let b_1 < \\cdots < b_r be the roots of p(x) with multiplicities s_1, \\ldots, s_r, \\sum_j=1^r s_…","labels":[],"detail_key":"p45"},{"id":"n35855","layer":"informal","project":"p45","title":"Fact 2","kind":"theorem","summary":"[Fact 2] We have p'(x)^2 \\geq p(x)p''(x) for all x \\in R.","labels":["ch23fact2"],"detail_key":"p45"},{"id":"n35856","layer":"informal","project":"p45","title":"If x = a_i is a root of p(x), then there is nothing to show. Assume then x is not a root.…","kind":"proof","summary":"If x = a_i is a root of p(x), then there is nothing to show. Assume then x is not a root. The p…","labels":[],"detail_key":"p45"},{"id":"n35857","layer":"informal","project":"p45","title":"vanderwaerden","kind":"theorem","summary":"Let M = (m_ij) be a doubly stochastic n \\times n matrix. Then \\[\\per M \\ge \\fracn!n^n\\] and equ…","labels":["vanderwaerden"],"detail_key":"p45"},{"id":"n35858","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35859","layer":"informal","project":"p45","title":"Gurvit's proposition","kind":"proposition","summary":"[Gurvit's proposition] If p(x)\\in R_+[x_1, \\dots, x_n] is a H-stable and homogeneous of degree…","labels":["gurvit"],"detail_key":"p45"},{"id":"n35860","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35861","layer":"informal","project":"p45","title":"ch25theorem","kind":"theorem","summary":"Let a_1, \\dots, a_n be vectors in R^d, each of length at least 1, and let R_1, \\dots, R_k be k…","labels":["ch25theorem"],"detail_key":"p45"},{"id":"n35862","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35863","layer":"informal","project":"p45","title":"A","kind":"lemma","summary":"[A] The functions f and g are defined for all non-integral values and are continuous there.","labels":["ch26lemma_a"],"detail_key":"p45"},{"id":"n35864","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35865","layer":"informal","project":"p45","title":"B","kind":"lemma","summary":"[B] Both f and g are \\emphperiodic of period 1, that is f(x + 1) = f(x) and g(x + 1) = g(x) hol…","labels":["ch26lemma_b"],"detail_key":"p45"},{"id":"n35866","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35867","layer":"informal","project":"p45","title":"C","kind":"lemma","summary":"[C] Both f and g are \\emphodd functions, that is we have f(-x) = -f(x) and g(-x) = -g(x) for al…","labels":["ch26lemma_c"],"detail_key":"p45"},{"id":"n35868","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35869","layer":"informal","project":"p45","title":"D","kind":"lemma","summary":"[D] The two functions f and g satisfy the same functional equation: f(\\fracx2) + f(\\fracx + 12)…","labels":["ch26lemma_d"],"detail_key":"p45"},{"id":"n35870","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35871","layer":"informal","project":"p45","title":"E","kind":"lemma","summary":"[E] By setting h(x) := 0 for x \\in Z, h becomes a continuous function on all of R that shares t…","labels":["ch26lemma_e"],"detail_key":"p45"},{"id":"n35872","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35873","layer":"informal","project":"p45","title":"ch26","kind":"theorem","summary":"\\[ \\pi\\cot\\pi x = \\frac1x + \\sum_n = 1^\\infty \\left(\\frac1x + n + \\frac1x - n\\right) \\] for x\\i…","labels":["ch26"],"detail_key":"p45"},{"id":"n35874","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35875","layer":"informal","project":"p45","title":"Buffon's needle problem","kind":"theorem","summary":"[Buffon's needle problem] If a short needle, of length \\ell, is dropped on paper that is ruled…","labels":["buffon_needle"],"detail_key":"p45"},{"id":"n35876","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35877","layer":"informal","project":"p45","title":"Pigeon-hole principle","kind":"theorem","summary":"[Pigeon-hole principle] If n objects are placed in r boxes, where r < n, then at least one of t…","labels":["pigeon_hole_principle"],"detail_key":"p45"},{"id":"n35878","layer":"informal","project":"p45","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p45"},{"id":"n35879","layer":"informal","project":"p45","title":"Claim","kind":"theorem","summary":"[Claim] Consider the numbers 1, 2, 3, \\dots, 2n, and take any n + 1 of them. Then there are two…","labels":["ch28claim1"],"detail_key":"p45"},{"id":"n35880","layer":"informal","project":"p45","title":"This is again obvious. There must be two numbers which are only 1 apart, and hence relati…","kind":"proof","summary":"This is again obvious. There must be two numbers which are only 1 apart, and hence relatively p…","labels":[],"detail_key":"p45"},{"id":"n35881","layer":"informal","project":"p45","title":"Claim","kind":"theorem","summary":"[Claim] Suppose again A \\subset \\1, 2, \\dots, 2n\\ with |A| = n+1. Then there are always two num…","labels":["ch28claim2"],"detail_key":"p45"},{"id":"n35882","layer":"informal","project":"p45","title":"Write every number a \\in A in the form a = 2^k m, where m is an odd number between 1 and…","kind":"proof","summary":"Write every number a \\in A in the form a = 2^k m, where m is an odd number between 1 and 2n - 1…","labels":[],"detail_key":"p45"},{"id":"n35883","layer":"informal","project":"p45","title":"Claim (Erd\\Hos--Szekeres)","kind":"theorem","summary":"[Claim (Erd\\Hos--Szekeres)] In any sequence a_1, a_2, \\ldots, a_mn+1 of mn+1 distinct real numb…","labels":["ch28claim3"],"detail_key":"p45"},{"id":"n35884","layer":"informal","project":"p45","title":"This time the application of the pigeon-hole principle is not immediate. Associate to eac…","kind":"proof","summary":"This time the application of the pigeon-hole principle is not immediate. Associate to each a_i…","labels":[],"detail_key":"p45"},{"id":"n35885","layer":"informal","project":"p45","title":"Claim","kind":"theorem","summary":"[Claim] Suppose we are given n integers a_1, \\dots, a_n, which need not be distinct. Then there…","labels":["ch28claim4"],"detail_key":"p45"},{"id":"n35886","layer":"informal","project":"p45","title":"For the proof we set N = \\0, 1, \\dots, n\\ and R = \\0, 1, \\dots, n-1\\. Consider the map f…","kind":"proof","summary":"For the proof we set N = \\0, 1, \\dots, n\\ and R = \\0, 1, \\dots, n-1\\. Consider the map f : N \\t…","labels":[],"detail_key":"p45"},{"id":"n35887","layer":"informal","project":"p45","title":"Double counting","kind":"theorem","summary":"[Double counting] Suppose that we are given two finite sets R and C and a subset S \\subseteq R…","labels":["double_counting"],"detail_key":"p45"},{"id":"n35888","layer":"informal","project":"p45","title":"Again, there is nothing to prove. The first sum classifies the pairs in S according to th…","kind":"proof","summary":"Again, there is nothing to prove. The first sum classifies the pairs in S according to the firs…","labels":[],"detail_key":"p45"},{"id":"n35889","layer":"informal","project":"p45","title":"Average number of divisors","kind":"theorem","summary":"[Average number of divisors] For any positive integer n, \\[ \\sum_j=1^n t(j) = \\sum_i=1^n \\left\\…","labels":["ch28_avg_divisors"],"detail_key":"p45"},{"id":"n35890","layer":"informal","project":"p45","title":"Consider the matrix A (as above) for the integers 1 up to n. Counting by columns we get \\…","kind":"proof","summary":"Consider the matrix A (as above) for the integers 1 up to n. Counting by columns we get \\sum_j=…","labels":[],"detail_key":"p45"},{"id":"n35891","layer":"informal","project":"p45","title":"Handshaking","kind":"lemma","summary":"[Handshaking] Let G be a finite simple graph with vertex set V and edge set E and let d(v) deno…","labels":["handshaking"],"detail_key":"p45"},{"id":"n35892","layer":"informal","project":"p45","title":"For the proof consider S \\subseteq V \\times E, where S is the set of pairs (v, e) such th…","kind":"proof","summary":"For the proof consider S \\subseteq V \\times E, where S is the set of pairs (v, e) such that v \\…","labels":[],"detail_key":"p45"},{"id":"n35893","layer":"informal","project":"p45","title":"Reiman","kind":"theorem","summary":"[Reiman] If the graph G on n vertices contains no 4-cycles, then \\[ |E| \\leq \\left\\lfloor \\frac…","labels":["ch28theorem"],"detail_key":"p45"},{"id":"n35894","layer":"informal","project":"p45","title":"The proof follows from the chain of inequalities above: the key combinatorial step \\sum \\…","kind":"proof","summary":"The proof follows from the chain of inequalities above: the key combinatorial step \\sum \\binomd…","labels":[],"detail_key":"p45"},{"id":"n35895","layer":"informal","project":"p45","title":"Cherry counting","kind":"theorem","summary":"[Cherry counting] If the graph G on n vertices contains no 4-cycles, then \\[ \\sum_v \\in V \\bino…","labels":["ch28_sum_choose"],"detail_key":"p45"},{"id":"n35896","layer":"informal","project":"p45","title":"Each pair (v, \\u, w\\) with u, w \\in N(v) maps to \\u, w\\. The C_4-free condition ensures t…","kind":"proof","summary":"Each pair (v, \\u, w\\) with u, w \\in N(v) maps to \\u, w\\. The C_4-free condition ensures this ma…","labels":[],"detail_key":"p45"},{"id":"n35897","layer":"informal","project":"p45","title":"Sperner's Lemma","kind":"lemma","summary":"[Sperner's Lemma] Suppose that some ``big'' triangle with vertices V_1, V_2, V_3 is triangulate…","labels":["sperner"],"detail_key":"p45"},{"id":"n35898","layer":"informal","project":"p45","title":"We will prove a stronger statement: The number of tricolored triangles is not only nonzer…","kind":"proof","summary":"We will prove a stronger statement: The number of tricolored triangles is not only nonzero, it…","labels":[],"detail_key":"p45"},{"id":"n35899","layer":"informal","project":"p45","title":"Brouwer's Fixed Point Theorem (for n = 2)","kind":"theorem","summary":"[Brouwer's Fixed Point Theorem (for n = 2)] Every continuous function f : B^2 \\to B^2 of a 2-di…","labels":["brouwer"],"detail_key":"p45"},{"id":"n35900","layer":"informal","project":"p45","title":"Let \\Delta be the triangle in R^3 with vertices e_1 = (1,0,0), e_2 = (0,1,0), and e_3 = (…","kind":"proof","summary":"Let \\Delta be the triangle in R^3 with vertices e_1 = (1,0,0), e_2 = (0,1,0), and e_3 = (0,0,1)…","labels":[],"detail_key":"p45"},{"id":"n35901","layer":"informal","project":"p45","title":"First proof","kind":"theorem","summary":"[First proof] Whenever a rectangle is tiled by rectangles all of which have at least one side o…","labels":["tiling_rectangles1"],"detail_key":"p45"},{"id":"n35902","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35903","layer":"informal","project":"p45","title":"Second proof","kind":"theorem","summary":"[Second proof] Whenever a rectangle is tiled by rectangles all of which have at least one side…","labels":["tiling_rectangles2"],"detail_key":"p45"},{"id":"n35904","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35905","layer":"informal","project":"p45","title":"Third proof","kind":"theorem","summary":"[Third proof] Whenever a rectangle is tiled by rectangles all of which have at least one side o…","labels":["tiling_rectangles3"],"detail_key":"p45"},{"id":"n35906","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35907","layer":"informal","project":"p45","title":"Sperner's theorem","kind":"theorem","summary":"[Sperner's theorem] The size of a largest antichain of an n-set is \\binomn\\lfloor n/2\\rfloor.","labels":["ch30theorem1"],"detail_key":"p45"},{"id":"n35908","layer":"informal","project":"p45","title":"We follow Lubell's elegant proof via the LYM inequality. Consider all n! permutations \\si…","kind":"proof","summary":"We follow Lubell's elegant proof via the LYM inequality. Consider all n! permutations \\sigma of…","labels":[],"detail_key":"p45"},{"id":"n35909","layer":"informal","project":"p45","title":"Katona's arc lemma","kind":"lemma","summary":"[Katona's arc lemma] Let n \\ge 2k, and suppose we are given t distinct arcs A_1, \\dots A_t of l…","labels":["ch30lemma"],"detail_key":"p45"},{"id":"n35910","layer":"informal","project":"p45","title":"Consider the n points arranged on a circle. Each arc of length k consists of k consecutiv…","kind":"proof","summary":"Consider the n points arranged on a circle. Each arc of length k consists of k consecutive poin…","labels":[],"detail_key":"p45"},{"id":"n35911","layer":"informal","project":"p45","title":"Erd\\Hos--Ko--Rado","kind":"theorem","summary":"[Erd\\Hos--Ko--Rado] Let n \\ge 2k. The largest size of an intersecting k-uniform family in an n-…","labels":["ch30theorem2"],"detail_key":"p45"},{"id":"n35912","layer":"informal","project":"p45","title":"We follow Katona's cyclic permutation argument. Arrange \\1,\\dots,n\\ on a circle. Among th…","kind":"proof","summary":"We follow Katona's cyclic permutation argument. Arrange \\1,\\dots,n\\ on a circle. Among the n ar…","labels":[],"detail_key":"p45"},{"id":"n35913","layer":"informal","project":"p45","title":"Hall's marriage theorem","kind":"theorem","summary":"[Hall's marriage theorem] Let A_1, \\dots A_n be a collection of subsets of a finite set X. Then…","labels":["ch30theorem3"],"detail_key":"p45"},{"id":"n35914","layer":"informal","project":"p45","title":"Necessity is clear: the m distinct representatives of any m sets must lie in their union.…","kind":"proof","summary":"Necessity is clear: the m distinct representatives of any m sets must lie in their union. For s…","labels":[],"detail_key":"p45"},{"id":"n35915","layer":"informal","project":"p45","title":"k systems of distinct representatives","kind":"corollary","summary":"[k systems of distinct representatives] Suppose the sets A_1, \\dots, A_n all have size k \\ge 1…","labels":["ch30corollary"],"detail_key":"p45"},{"id":"n35916","layer":"informal","project":"p45","title":"ch31lemma","kind":"lemma","summary":"Let Q : \\mathfrakS_n \\longrightarrow R be any probability distribution that defines a shuffling…","labels":["ch31lemma"],"detail_key":"p45"},{"id":"n35917","layer":"informal","project":"p45","title":"If X is a random variable with values in \\mathfrakS_n, with probability distribution Q, t…","kind":"proof","summary":"If X is a random variable with values in \\mathfrakS_n, with probability distribution Q, then we…","labels":[],"detail_key":"p45"},{"id":"n35918","layer":"informal","project":"p45","title":"ch31theorem1","kind":"theorem","summary":"Let c \\geq 0 and k := \\lceil n \\log n + cn \\rceil. Then after performing k top-in-at-random shu…","labels":["ch31theorem1"],"detail_key":"p45"},{"id":"n35919","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35920","layer":"informal","project":"p45","title":"ch31theorem2","kind":"theorem","summary":"After performing k riffle shuffles on a deck of n cards, the variation distance from a uniform…","labels":["ch31theorem2"],"detail_key":"p45"},{"id":"n35921","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35922","layer":"informal","project":"p45","title":"ch32lemma","kind":"lemma","summary":"Let G = (V, E) be a finite weighted acyclic directed graph, A = \\A_1, \\dots, A_n\\ and B = \\B_1,…","labels":["ch32lemma"],"detail_key":"p45"},{"id":"n35923","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35924","layer":"informal","project":"p45","title":"ch32theorem","kind":"theorem","summary":"Let G = (V, E) be a finite weighted acyclic directed graph, A = \\A_1, \\dots, A_n\\ and B = \\B_1,…","labels":["ch32theorem"],"detail_key":"p45"},{"id":"n35925","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35926","layer":"informal","project":"p45","title":"First proof (bijection)","kind":"theorem","summary":"[First proof (bijection)] There are n^n - 2 different labeled trees on n nodes.","labels":["cayley_formala_proof1"],"detail_key":"p45"},{"id":"n35927","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35928","layer":"informal","project":"p45","title":"Second proof (Linear Algebra)","kind":"theorem","summary":"[Second proof (Linear Algebra)] There are n^n - 2 different labeled trees on n nodes.","labels":["cayley_formala_proof2"],"detail_key":"p45"},{"id":"n35929","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35930","layer":"informal","project":"p45","title":"Second proof (Recursion)","kind":"theorem","summary":"[Second proof (Recursion)] There are n^n - 2 different labeled trees on n nodes.","labels":["cayley_formala_proof3"],"detail_key":"p45"},{"id":"n35931","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35932","layer":"informal","project":"p45","title":"Second proof (Double Counting)","kind":"theorem","summary":"[Second proof (Double Counting)] There are n^n - 2 different labeled trees on n nodes.","labels":["cayley_formala_proof4"],"detail_key":"p45"},{"id":"n35933","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35934","layer":"informal","project":"p45","title":"ch34theorem","kind":"theorem","summary":"\\[ \\prod_k\\ge 1(1 - x^k) = 1 + \\sum_j\\ge 1(-1)^j(x^\\frac3j^2 - j2 + x^3j^2 + j2). \\]","labels":["ch34theorem"],"detail_key":"p45"},{"id":"n35935","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35936","layer":"informal","project":"p45","title":"ch35lemma1","kind":"lemma","summary":"Every nonzero polynomial p(x) \\in F[x_1, \\dots, x_n] of degree d has at most dq^n-1 roots in F^…","labels":["ch35lemma1"],"detail_key":"p45"},{"id":"n35937","layer":"informal","project":"p45","title":"We use induction on n, with fact (1) above as the starting case n = 1. Let us split p(x)…","kind":"proof","summary":"We use induction on n, with fact (1) above as the starting case n = 1. Let us split p(x) into s…","labels":[],"detail_key":"p45"},{"id":"n35938","layer":"informal","project":"p45","title":"ch35lemma2","kind":"lemma","summary":"For every set E \\subseteq F^n of size |E| < \\binomn+dd there is a nonzero polynomial p(x) \\in F…","labels":["ch35lemma2"],"detail_key":"p45"},{"id":"n35939","layer":"informal","project":"p45","title":"Consider the vector space V_d of all polynomials in F[x_1, \\dots, x_n] of degree at most…","kind":"proof","summary":"Consider the vector space V_d of all polynomials in F[x_1, \\dots, x_n] of degree at most d. A b…","labels":[],"detail_key":"p45"},{"id":"n35940","layer":"informal","project":"p45","title":"finite Kakeya problem","kind":"theorem","summary":"[finite Kakeya problem] Let K \\subseteq F^n be a Kakeya set. Then \\[ |K| \\geq \\binom|F| + n - 1…","labels":["kakeya"],"detail_key":"p45"},{"id":"n35941","layer":"informal","project":"p45","title":"The second inequality is clear from the definition of binomial coefficients. For the firs…","kind":"proof","summary":"The second inequality is clear from the definition of binomial coefficients. For the first, set…","labels":[],"detail_key":"p45"},{"id":"n35942","layer":"informal","project":"p45","title":"ch36lemma1","kind":"lemma","summary":"Any (r \\times n)-Latin rectangle, r < n, can be extended to an ((r+1) \\times n)-Latin rectangle…","labels":["ch36lemma1"],"detail_key":"p45"},{"id":"n35943","layer":"informal","project":"p45","title":"We apply Hall's theorem \\refch30theorem3 (see Chapter \\refchapter30). Let A_j be the set…","kind":"proof","summary":"We apply Hall's theorem \\refch30theorem3 (see Chapter \\refchapter30). Let A_j be the set of num…","labels":[],"detail_key":"p45"},{"id":"n35944","layer":"informal","project":"p45","title":"ch36lemma2","kind":"lemma","summary":"Let P be a partial Latin square of order n with at most n - 1 cells filled and at most \\fracn2…","labels":["ch36lemma2"],"detail_key":"p45"},{"id":"n35945","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35946","layer":"informal","project":"p45","title":"Smetaniuk's theorem","kind":"theorem","summary":"[Smetaniuk's theorem] Any partial Latin square of order n with at most n - 1 filled cells can b…","labels":["smetaniuk_theorem"],"detail_key":"p45"},{"id":"n35947","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35948","layer":"informal","project":"p45","title":"ch37theorem1","kind":"theorem","summary":"Let M = (m_ij) be an n \\times n matrix with entries in \\0, 1\\, and let d_1, \\dots, d_n be the r…","labels":["ch37theorem1"],"detail_key":"p45"},{"id":"n35949","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35950","layer":"informal","project":"p45","title":"ch37theorem2","kind":"theorem","summary":"The number L(n) of Latin squares of order n is bounded by \\[ \\fracn!^2nn^n^2 \\le L(n) \\le \\prod…","labels":["ch37theorem2"],"detail_key":"p45"},{"id":"n35951","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35952","layer":"informal","project":"p45","title":"Fact A","kind":"theorem","summary":"[Fact A] \\[H(X)\\le \\log_2(|\\suppX).\\]","labels":["ch37fact_a"],"detail_key":"p45"},{"id":"n35953","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35954","layer":"informal","project":"p45","title":"Fact B","kind":"theorem","summary":"[Fact B] \\[H(X, Y) = H(X) + H(Y|X).\\]","labels":["ch37fact_b"],"detail_key":"p45"},{"id":"n35955","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35956","layer":"informal","project":"p45","title":"Fact B","kind":"theorem","summary":"[Fact B] \\[H(Y|X)\\le \\sum_j=1^d \\Prop(X\\in E_j)\\log_2 j.\\]","labels":["ch37fact_c"],"detail_key":"p45"},{"id":"n35957","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35958","layer":"informal","project":"p45","title":"ch38definition1","kind":"definition","summary":"Let \\vecG = (V, E) be a directed graph. A kernel K \\subseteq V is a subset of the vertices such…","labels":["ch38definition1"],"detail_key":"p45"},{"id":"n35959","layer":"informal","project":"p45","title":"ch38lemma1","kind":"lemma","summary":"Let \\vecG = (V, E) be a directed graph, and suppose that for each vertex v \\in V we have a colo…","labels":["ch38lemma1"],"detail_key":"p45"},{"id":"n35960","layer":"informal","project":"p45","title":"We proceed by induction on |V|. For |V| = 1 there is nothing to prove. Choose a color c \\…","kind":"proof","summary":"We proceed by induction on |V|. For |V| = 1 there is nothing to prove. Choose a color c \\in C =…","labels":[],"detail_key":"p45"},{"id":"n35961","layer":"informal","project":"p45","title":"ch38definition2","kind":"definition","summary":"A matching M of G = (X \\cup Y, E) is called stable if the following condition holds: Whenever u…","labels":["ch38definition2"],"detail_key":"p45"},{"id":"n35962","layer":"informal","project":"p45","title":"ch38lemma2","kind":"lemma","summary":"A stable matching always exists.","labels":["ch38lemma2"],"detail_key":"p45"},{"id":"n35963","layer":"informal","project":"p45","title":"Consider the following algorithm. In the first stage all men u \\in X propose to their top…","kind":"proof","summary":"Consider the following algorithm. In the first stage all men u \\in X propose to their top choic…","labels":[],"detail_key":"p45"},{"id":"n35964","layer":"informal","project":"p45","title":"ch38theorem","kind":"theorem","summary":"We have \\chi_\\ell(S_n) = n for all n.","labels":["ch38theorem"],"detail_key":"p45"},{"id":"n35965","layer":"informal","project":"p45","title":"As before we denote the vertices of S_n by (i, j), 1 \\leq i, j \\leq n. Thus (i, j) and (r…","kind":"proof","summary":"As before we denote the vertices of S_n by (i, j), 1 \\leq i, j \\leq n. Thus (i, j) and (r, s) a…","labels":[],"detail_key":"p45"},{"id":"n35966","layer":"informal","project":"p45","title":"five_colorable","kind":"theorem","summary":"All planar graphs G can be 5-colored: \\[ \\chi_\\ell(G)\\le 5. \\]","labels":["five_colorable"],"detail_key":"p45"},{"id":"n35967","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35968","layer":"informal","project":"p45","title":"museum_guards","kind":"theorem","summary":"For any museum with n walls, \\lfloor \\fracn3 \\rfloor guards suffice.","labels":["museum_guards"],"detail_key":"p45"},{"id":"n35969","layer":"informal","project":"p45","title":"First of all, let us draw n - 3 noncrossing diagonals between corners of the walls until…","kind":"proof","summary":"First of all, let us draw n - 3 noncrossing diagonals between corners of the walls until the in…","labels":[],"detail_key":"p45"},{"id":"n35970","layer":"informal","project":"p45","title":"First Proof","kind":"theorem","summary":"[First Proof] If a graph G = (V, E) on n vertices has no p-clique, p \\geq 2, then \\[ |E| \\leq \\…","labels":["ch41proof1"],"detail_key":"p45"},{"id":"n35971","layer":"informal","project":"p45","title":"We use induction on n. One easily computes that (1) is true for n < p. Let G be a graph o…","kind":"proof","summary":"We use induction on n. One easily computes that (1) is true for n < p. Let G be a graph on V =…","labels":[],"detail_key":"p45"},{"id":"n35972","layer":"informal","project":"p45","title":"Second Proof","kind":"theorem","summary":"[Second Proof] If a graph G = (V, E) on n vertices has no p-clique, p \\geq 2, then \\[ |E| \\leq…","labels":["ch41proof2"],"detail_key":"p45"},{"id":"n35973","layer":"informal","project":"p45","title":"This proof makes use of the structure of the Turán graphs. Let v_m \\in V be a vertex of m…","kind":"proof","summary":"This proof makes use of the structure of the Turán graphs. Let v_m \\in V be a vertex of maximal…","labels":[],"detail_key":"p45"},{"id":"n35974","layer":"informal","project":"p45","title":"Third Proof","kind":"theorem","summary":"[Third Proof] If a graph G = (V, E) on n vertices has no p-clique, p \\geq 2, then \\[ |E| \\leq \\…","labels":["ch41proof3"],"detail_key":"p45"},{"id":"n35975","layer":"informal","project":"p45","title":"Consider a \\emphprobability distribution w = (w_1, \\dots, w_n) on the vertices, that is,…","kind":"proof","summary":"Consider a \\emphprobability distribution w = (w_1, \\dots, w_n) on the vertices, that is, an ass…","labels":[],"detail_key":"p45"},{"id":"n35976","layer":"informal","project":"p45","title":"Fourth Proof","kind":"theorem","summary":"[Fourth Proof] If a graph G = (V, E) on n vertices has no p-clique, p \\geq 2, then \\[ |E| \\leq…","labels":["ch41proof4"],"detail_key":"p45"},{"id":"n35977","layer":"informal","project":"p45","title":"This time we use some concepts from probability theory. Let G be an arbitrary graph on th…","kind":"proof","summary":"This time we use some concepts from probability theory. Let G be an arbitrary graph on the vert…","labels":[],"detail_key":"p45"},{"id":"n35978","layer":"informal","project":"p45","title":"Fifth Proof","kind":"theorem","summary":"[Fifth Proof] If a graph G = (V, E) on n vertices has no p-clique, p \\geq 2, then \\[ |E| \\leq \\…","labels":["ch41proof5"],"detail_key":"p45"},{"id":"n35979","layer":"informal","project":"p45","title":"Let G be a graph on n vertices without a p-clique and with a maximal number of edges. Cla…","kind":"proof","summary":"Let G be a graph on n vertices without a p-clique and with a maximal number of edges. Claim. G…","labels":[],"detail_key":"p45"},{"id":"n35980","layer":"informal","project":"p45","title":"Five proofs of Turán's graph theorem","kind":"theorem","summary":"[Five proofs of Turán's graph theorem] Collecting the proofs from the chapter...","labels":["turan_graph"],"detail_key":"p45"},{"id":"n35981","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35982","layer":"informal","project":"p45","title":"ch42theorem","kind":"theorem","summary":"Whenever T = \\v^(1), \\dots, v^(m)\\ is an orthonormal representation of G with constant \\sigma_T…","labels":["ch42theorem"],"detail_key":"p45"},{"id":"n35983","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35984","layer":"informal","project":"p45","title":"Lyusternik–Shnirel'man","kind":"theorem","summary":"[Lyusternik–Shnirel'man] If the d-sphere S^d is covered by d + 1 sets, \\[ S^d = U_1 \\cup \\dots…","labels":["lyusternik_shnirelman"],"detail_key":"p45"},{"id":"n35985","layer":"informal","project":"p45","title":"Let a covering S^d = U_1 \\cup \\dots \\cup U_d \\cup U_d+1 be given as specified, and assume…","kind":"proof","summary":"Let a covering S^d = U_1 \\cup \\dots \\cup U_d \\cup U_d+1 be given as specified, and assume that…","labels":[],"detail_key":"p45"},{"id":"n35986","layer":"informal","project":"p45","title":"Gale's theorem","kind":"theorem","summary":"[Gale's theorem] There is an arrangement of 2k + d points on S^d such that every open hemispher…","labels":["gale_theorem"],"detail_key":"p45"},{"id":"n35987","layer":"informal","project":"p45","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p45"},{"id":"n35988","layer":"informal","project":"p45","title":"Kneser's conjecture","kind":"theorem","summary":"[Kneser's conjecture] We have \\[ \\chi(K(2k + d, k)) = d + 2. \\]","labels":["kneser_conjecture"],"detail_key":"p45"},{"id":"n35989","layer":"informal","project":"p45","title":"For our ground set let us take 2k + d points in general position on the sphere S^d+1. Sup…","kind":"proof","summary":"For our ground set let us take 2k + d points in general position on the sphere S^d+1. Suppose t…","labels":[],"detail_key":"p45"},{"id":"n35990","layer":"informal","project":"p45","title":"borsuk_ulam","kind":"theorem","summary":"For every continuous map f : S^d \\to R^d from d-sphere to d-space, there are antipodal points x…","labels":["borsuk_ulam"],"detail_key":"p45"},{"id":"n35991","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35992","layer":"informal","project":"p45","title":"friendship","kind":"theorem","summary":"Suppose that G is a finite graph in which any two vertices have precisely one common neighbor.…","labels":["friendship"],"detail_key":"p45"},{"id":"n35993","layer":"informal","project":"p45","title":"Suppose the assertion is false, and G is a counterexample, that is, no vertex of G is adj…","kind":"proof","summary":"Suppose the assertion is false, and G is a counterexample, that is, no vertex of G is adjacent…","labels":[],"detail_key":"p45"},{"id":"n35994","layer":"informal","project":"p45","title":"ch45theorem1","kind":"theorem","summary":"Every family of at most 2^d-1 d-sets is 2-colorable, that is, m(d) > 2^d-1.","labels":["ch45theorem1"],"detail_key":"p45"},{"id":"n35995","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35996","layer":"informal","project":"p45","title":"ch45theorem2","kind":"theorem","summary":"Every family of at most 2^d-1 d-sets is 2-colorable, that is, m(d) > 2^d-1.","labels":["ch45theorem2"],"detail_key":"p45"},{"id":"n35997","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n35998","layer":"informal","project":"p45","title":"ch45theorem3","kind":"theorem","summary":"For every k \\geq 2, there exists a graph G with chromatic number \\chi(G) > k and girth \\gamma(G…","labels":["ch45theorem3"],"detail_key":"p45"},{"id":"n35999","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n36000","layer":"informal","project":"p45","title":"ch45theorem4","kind":"theorem","summary":"Let G be a simple graph with n vertices and m edges, where m \\ge 4n. Then \\[ \\mycr(G) \\ge \\frac…","labels":["ch45theorem4"],"detail_key":"p45"},{"id":"n36001","layer":"informal","project":"p45","title":"TODO","kind":"proof","summary":"TODO","labels":[],"detail_key":"p45"},{"id":"n36002","layer":"formal","project":"p45","title":"chapter28.sperner_lemma","kind":"theorem","summary":"∀ m : Nat (T : chapter28.Triangulation m) (c : Fin m → Fin 3) (count12 : Finset (Fin m) → Nat),…","labels":[],"detail_key":"p45","name":"chapter28.sperner_lemma","module":"FormalBook.Ch28.SpernerBrouwer"},{"id":"n36003","layer":"formal","project":"p45","title":"Asymptotics.infinitely_many_more_proofs","kind":"theorem","summary":"∀ (S : Nat → Int), AlmostInjective S → Asymptotics.ofSubexponentialGrowth S → (Set.ofPred fun p…","labels":[],"detail_key":"p45","name":"Asymptotics.infinitely_many_more_proofs","module":"FormalBook.Chapter_01"},{"id":"n36004","layer":"formal","project":"p45","title":"infinity_of_primes₁","kind":"theorem","summary":"∀ (S : Finset Nat), (∀ (q : Nat), Membership.mem S q → Nat.Prime q) → Exists fun p => And (Nat.…","labels":[],"detail_key":"p45","name":"infinity_of_primes₁","module":"FormalBook.Chapter_01"},{"id":"n36005","layer":"formal","project":"p45","title":"infinity_of_primes₂","kind":"theorem","summary":"∀ (k n : Nat), LT.lt k n → n.fermatNumber.Coprime k.fermatNumber","labels":[],"detail_key":"p45","name":"infinity_of_primes₂","module":"FormalBook.Chapter_01"},{"id":"n36006","layer":"formal","project":"p45","title":"infinity_of_primes₃","kind":"theorem","summary":"Not (Exists fun p => And (Nat.Prime p) (∀ (q : Nat), Nat.Prime q → LE.le q p))","labels":[],"detail_key":"p45","name":"infinity_of_primes₃","module":"FormalBook.Chapter_01"},{"id":"n36007","layer":"formal","project":"p45","title":"infinity_of_primes₄","kind":"theorem","summary":"Filter.Tendsto Nat.primeCounting Filter.atTop Filter.atTop","labels":[],"detail_key":"p45","name":"infinity_of_primes₄","module":"FormalBook.Chapter_01"},{"id":"n36008","layer":"formal","project":"p45","title":"infinity_of_primes₅","kind":"theorem","summary":"(Set.ofPred fun p => Nat.Prime p).Infinite","labels":[],"detail_key":"p45","name":"infinity_of_primes₅","module":"FormalBook.Chapter_01"},{"id":"n36009","layer":"formal","project":"p45","title":"infinity_of_primes₆","kind":"theorem","summary":"Filter.Tendsto (fun n => (Finset.filter (fun p => Nat.Prime p) (Finset.range n)).sum fun p => H…","labels":[],"detail_key":"p45","name":"infinity_of_primes₆","module":"FormalBook.Chapter_01"},{"id":"n36010","layer":"formal","project":"p45","title":"chapter2.bound_binomial_coeff","kind":"theorem","summary":"∀ k n : Nat, And (LE.le (↑(n.choose k)) (HDiv.hDiv (HPow.hPow (↑n) k) ↑k.factorial)) (LE.le (HD…","labels":[],"detail_key":"p45","name":"chapter2.bound_binomial_coeff","module":"FormalBook.Chapter_02"},{"id":"n36011","layer":"formal","project":"p45","title":"chapter2.bound_factorial","kind":"theorem","summary":"∀ n : Nat, LT.lt 1 n → GT.gt (↑n.factorial) (HMul.hMul chapter2.e (HPow.hPow (HDiv.hDiv (↑n) ch…","labels":[],"detail_key":"p45","name":"chapter2.bound_factorial","module":"FormalBook.Chapter_02"},{"id":"n36012","layer":"formal","project":"p45","title":"chapter2.exists_prime_lt_and_le_two_mul","kind":"theorem","summary":"∀ (n : Nat), Ne n 0 → Exists fun p => And (Nat.Prime p) (And (LT.lt n p) (LE.le p (HMul.hMul 2…","labels":[],"detail_key":"p45","name":"chapter2.exists_prime_lt_and_le_two_mul","module":"FormalBook.Chapter_02"},{"id":"n36013","layer":"formal","project":"p45","title":"chapter2.harmonic_number_bounds","kind":"theorem","summary":"∀ n : Nat, LT.lt 1 n → And (LT.lt (HAdd.hAdd (Real.log ↑n) (HDiv.hDiv 1 ↑n)) ↑(harmonic n)) (LT…","labels":[],"detail_key":"p45","name":"chapter2.harmonic_number_bounds","module":"FormalBook.Chapter_02"},{"id":"n36014","layer":"formal","project":"p45","title":"chapter3.binomials_coefficients_never_powers","kind":"theorem","summary":"∀ (k l m n : Nat), LE.le 2 l → LE.le 4 k → LE.le k (HSub.hSub n 4) → Ne (n.choose k) (HPow.hPow…","labels":[],"detail_key":"p45","name":"chapter3.binomials_coefficients_never_powers","module":"FormalBook.Chapter_03"},{"id":"n36015","layer":"formal","project":"p45","title":"chapter3.sylvester","kind":"theorem","summary":"∀ (k n : Nat), GE.ge n (HMul.hMul 2 k) → GT.gt k 0 → Exists fun p => And (GT.gt p k) (And (Nat.…","labels":[],"detail_key":"p45","name":"chapter3.sylvester","module":"FormalBook.Chapter_03"},{"id":"n36016","layer":"formal","project":"p45","title":"ch04.lemma₁","kind":"theorem","summary":"∀ p : Nat [h : Fact (Nat.Prime p)], have num_solutions := (Finset.filter (fun s => Eq (HPow.hPo…","labels":[],"detail_key":"p45","name":"ch04.lemma₁","module":"FormalBook.Chapter_04"},{"id":"n36017","layer":"formal","project":"p45","title":"ch04.lemma₂","kind":"theorem","summary":"∀ (n m : Nat), Eq n (HAdd.hAdd (HMul.hMul 4 m) 3) → Not (Exists fun a => Exists fun b => Eq n (…","labels":[],"detail_key":"p45","name":"ch04.lemma₂","module":"FormalBook.Chapter_04"},{"id":"n36018","layer":"formal","project":"p45","title":"ch04.theorem₁","kind":"theorem","summary":"∀ p : Nat [h : Fact (Nat.Prime p)], Eq (HMod.hMod p 4) 1 → Exists fun a => Exists fun b => Eq (…","labels":[],"detail_key":"p45","name":"ch04.theorem₁","module":"FormalBook.Chapter_04"},{"id":"n36019","layer":"formal","project":"p45","title":"ch04.theorem₂","kind":"theorem","summary":"∀ p : Nat [h : Fact (Nat.Prime p)], Eq (HMod.hMod p 4) 1 → Exists fun a => Exists fun b => Eq (…","labels":[],"detail_key":"p45","name":"ch04.theorem₂","module":"FormalBook.Chapter_04"},{"id":"n36020","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.euler_criterion","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (a : Int), Ne (↑a) 0 → Eq (↑(book.quadratic_reciprocity…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.euler_criterion","module":"FormalBook.Chapter_05"},{"id":"n36021","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.fact_A","kind":"theorem","summary":"∀ (p q : Nat), Ne p 2 → Ne q 2 → ∀ [Fact (Nat.Prime p)] [Fact (Nat.Prime q)], Ne p q → ∀ (K : T…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.fact_A","module":"FormalBook.Chapter_05"},{"id":"n36022","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.fact_B","kind":"theorem","summary":"∀ (p : Nat) [Fact (Prime p)] (K : Type u_1) [inst : Field K] (ζ : Units K), Eq (HPow.hPow ζ p)…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.fact_B","module":"FormalBook.Chapter_05"},{"id":"n36023","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.fermat_little","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (a : Int), Ne (↑a) 0 → Eq (HPow.hPow (↑a) (HSub.hSub p…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.fermat_little","module":"FormalBook.Chapter_05"},{"id":"n36024","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.lemma_of_Gauss","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (a : Int), Ne (↑a) 0 → ∀ (r : Int → Int), (∀ (i : Int),…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.lemma_of_Gauss","module":"FormalBook.Chapter_05"},{"id":"n36025","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.mult_cyclic","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [Fintype K], Exists fun ζ => ∀ (α : Units K), Exists fun k =>…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.mult_cyclic","module":"FormalBook.Chapter_05"},{"id":"n36026","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.product_rule","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (a b : Int), Eq (book.quadratic_reciprocity.legendre_sy…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.product_rule","module":"FormalBook.Chapter_05"},{"id":"n36027","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.quadratic_reciprocity_1","kind":"theorem","summary":"∀ (p q : Nat), Ne p 2 → Ne q 2 → ∀ [inst : Fact (Nat.Prime p)] [inst_1 : Fact (Nat.Prime q)], E…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.quadratic_reciprocity_1","module":"FormalBook.Chapter_05"},{"id":"n36028","layer":"formal","project":"p45","title":"book.quadratic_reciprocity.quadratic_reciprocity_2","kind":"theorem","summary":"∀ (p q : Nat), Ne p 2 → Ne q 2 → ∀ [inst : Fact (Nat.Prime p)] [inst_1 : Fact (Nat.Prime q)], E…","labels":[],"detail_key":"p45","name":"book.quadratic_reciprocity.quadratic_reciprocity_2","module":"FormalBook.Chapter_05"},{"id":"n36029","layer":"formal","project":"p45","title":"wedderburn","kind":"theorem","summary":"∀ R : Type u_1 [DecidableEq R] [inst : DivisionRing R] (h : Fintype R), IsField R","labels":[],"detail_key":"p45","name":"wedderburn","module":"FormalBook.Chapter_06"},{"id":"n36030","layer":"formal","project":"p45","title":"chapter7.Theorem₁","kind":"theorem","summary":"∀ (n : Nat) (A : Matrix (Fin n) (Fin n) Real), A.IsHermitian → Exists fun Q => And (Membership.…","labels":[],"detail_key":"p45","name":"chapter7.Theorem₁","module":"FormalBook.Chapter_07"},{"id":"n36031","layer":"formal","project":"p45","title":"chapter7.Theorem₂","kind":"theorem","summary":"∀ (n : Nat), LT.lt 1 n → Exists fun M => And (∀ (i j : Fin n), Or (Eq (M i j) (-1)) (Eq (M i 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\\textttPphi2.asymCa…","labels":["Pphi2-asymCanonicalRoughWickPower-two-site-marginal-eq-zero-of-ne"],"detail_key":"p46"},{"id":"n38021","layer":"informal","project":"p46","title":"\\textttasymCanonicalRoughWickPower\\_two\\_site\\_marginal\\_integrable","kind":"theorem","summary":"[\\textttasymCanonicalRoughWickPower\\_two\\_site\\_marginal\\_integrable] \\textttPphi2.asymCanonica…","labels":["Pphi2-asymCanonicalRoughWickPower-two-site-marginal-integrable"],"detail_key":"p46"},{"id":"n38022","layer":"informal","project":"p46","title":"\\textttasymCanonicalSmoothGamma","kind":"definition","summary":"[\\textttasymCanonicalSmoothGamma] \\textttPphi2.asymCanonicalSmoothGamma","labels":["Pphi2-asymCanonicalSmoothGamma"],"detail_key":"p46"},{"id":"n38023","layer":"informal","project":"p46","title":"\\textttasymCanonicalRoughWickPower\\_two\\_site\\_marginal\\_eq\\_diag","kind":"theorem","summary":"[\\textttasymCanonicalRoughWickPower\\_two\\_site\\_marginal\\_eq\\_diag] 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\\textttPphi2.abs\\_asymC…","labels":["Pphi2-abs-asymCanonicalSmoothCovariance-le-asymSmoothWickConstant"],"detail_key":"p46"},{"id":"n38034","layer":"informal","project":"p46","title":"\\textttasymCanonicalSmoothCovariance","kind":"definition","summary":"[\\textttasymCanonicalSmoothCovariance] \\textttPphi2.asymCanonicalSmoothCovariance","labels":["Pphi2-asymCanonicalSmoothCovariance"],"detail_key":"p46"},{"id":"n38035","layer":"informal","project":"p46","title":"\\textttasymCanonicalCrossTerm\\_sq\\_integrable","kind":"theorem","summary":"[\\textttasymCanonicalCrossTerm\\_sq\\_integrable] \\textttPphi2.asymCanonicalCrossTerm\\_sq\\_integr…","labels":["Pphi2-asymCanonicalCrossTerm-sq-integrable"],"detail_key":"p46"},{"id":"n38036","layer":"informal","project":"p46","title":"\\textttasymCanonicalRoughFieldFunctionOfSnd\\_eq\\_sum\\_gamma","kind":"theorem","summary":"[\\textttasymCanonicalRoughFieldFunctionOfSnd\\_eq\\_sum\\_gamma] \\textttPphi2.asymCanonicalRoughFi…","labels":["Pphi2-asymCanonicalRoughFieldFunctionOfSnd-eq-sum-gamma"],"detail_key":"p46"},{"id":"n38037","layer":"informal","project":"p46","title":"\\textttasymCanonicalSmoothWickPower\\_two\\_site\\_marginal\\_integrable","kind":"theorem","summary":"[\\textttasymCanonicalSmoothWickPower\\_two\\_site\\_marginal\\_integrable] \\textttPphi2.asymCanonic…","labels":["Pphi2-asymCanonicalSmoothWickPower-two-site-marginal-integrable"],"detail_key":"p46"},{"id":"n38038","layer":"informal","project":"p46","title":"\\textttasymCanonicalRoughWickConstant","kind":"definition","summary":"[\\textttasymCanonicalRoughWickConstant] \\textttPphi2.asymCanonicalRoughWickConstant","labels":["Pphi2-asymCanonicalRoughWickConstant"],"detail_key":"p46"},{"id":"n38039","layer":"informal","project":"p46","title":"\\textttasymCanonicalRoughCovariance\\_eq\\_sum\\_gamma\\_mul\\_gamma","kind":"theorem","summary":"[\\textttasymCanonicalRoughCovariance\\_eq\\_sum\\_gamma\\_mul\\_gamma] \\textttPphi2.asymCanonicalRou…","labels":["Pphi2-asymCanonicalRoughCovariance-eq-sum-gamma-mul-gamma"],"detail_key":"p46"},{"id":"n38040","layer":"informal","project":"p46","title":"\\textttasymCanonicalSmoothGamma\\_sq\\_sum\\_eq\\_asymSmoothWickConstant","kind":"theorem","summary":"[\\textttasymCanonicalSmoothGamma\\_sq\\_sum\\_eq\\_asymSmoothWickConstant] \\textttPphi2.asymCanonic…","labels":["Pphi2-asymCanonicalSmoothGamma-sq-sum-eq-asymSmoothWickConstant"],"detail_key":"p46"},{"id":"n38041","layer":"informal","project":"p46","title":"\\textttasymCanonicalCrossTerm\\_l2\\_sq\\_eq\\_covSum","kind":"theorem","summary":"[\\textttasymCanonicalCrossTerm\\_l2\\_sq\\_eq\\_covSum] \\textttPphi2.asymCanonicalCrossTerm\\_l2\\_sq…","labels":["Pphi2-asymCanonicalCrossTerm-l2-sq-eq-covSum"],"detail_key":"p46"},{"id":"n38042","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] \\textttPphi2.asymCanonicalRoughWickConstant.congr\\_simp","labels":["Pphi2-asymCanonicalRoughWickConstant-congr-simp"],"detail_key":"p46"},{"id":"n38043","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] 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\\textttPphi2.EuclideanOS.SatisfiesFullOS.os3","labels":["Pphi2-EuclideanOS-SatisfiesFullOS-os3"],"detail_key":"p46"},{"id":"n38057","layer":"informal","project":"p46","title":"\\textttSatisfiesFullOS","kind":"definition","summary":"[\\textttSatisfiesFullOS] Structure bundling all five axioms.","labels":["Pphi2-EuclideanOS-SatisfiesFullOS"],"detail_key":"p46"},{"id":"n38058","layer":"informal","project":"p46","title":"\\textttgeneratingFunctional\\_im\\_eq\\_integral\\_sin","kind":"theorem","summary":"[\\textttgeneratingFunctional\\_im\\_eq\\_integral\\_sin] \\textttPphi2.EuclideanOS.generatingFunctio…","labels":["Pphi2-EuclideanOS-generatingFunctional-im-eq-integral-sin"],"detail_key":"p46"},{"id":"n38059","layer":"informal","project":"p46","title":"\\textttos1","kind":"theorem","summary":"[\\textttos1] \\textttPphi2.EuclideanOS.SatisfiesFullOS.os1","labels":["Pphi2-EuclideanOS-SatisfiesFullOS-os1"],"detail_key":"p46"},{"id":"n38060","layer":"informal","project":"p46","title":"\\textttschwartzRe","kind":"definition","summary":"[\\textttschwartzRe] Extract real/imaginary parts of complex Schwartz functions as real Schwartz…","labels":["Pphi2-EuclideanOS-schwartzRe"],"detail_key":"p46"},{"id":"n38061","layer":"informal","project":"p46","title":"\\textttOS3\\_ReflectionPositivity","kind":"definition","summary":"[\\textttOS3\\_ReflectionPositivity] \\textttPphi2.EuclideanOS.OS3\\_ReflectionPositivity","labels":["Pphi2-EuclideanOS-OS3-ReflectionPositivity"],"detail_key":"p46"},{"id":"n38062","layer":"informal","project":"p46","title":"\\textttOS4\\_Clustering","kind":"definition","summary":"[\\textttOS4\\_Clustering] 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\\i…","labels":["Pphi2-torusGaussianLimit-isGaussian"],"detail_key":"p46"},{"id":"n38066","layer":"informal","project":"p46","title":"\\textttgaussian\\_measure\\_unique\\_of\\_covariance","kind":"theorem","summary":"[\\textttgaussian\\_measure\\_unique\\_of\\_covariance] Main theorem: Two centered Gaussian probabil…","labels":["Pphi2-gaussian-measure-unique-of-covariance"],"detail_key":"p46"},{"id":"n38067","layer":"informal","project":"p46","title":"\\textttisGaussian","kind":"theorem","summary":"[\\textttisGaussian] \\textttPphi2.IsTorusGaussianContinuumLimit.isGaussian","labels":["Pphi2-IsTorusGaussianContinuumLimit-isGaussian"],"detail_key":"p46"},{"id":"n38068","layer":"informal","project":"p46","title":"\\textttconfiguration\\_neg\\_apply","kind":"theorem","summary":"[\\textttconfiguration\\_neg\\_apply] Negation in configuration space: (-\\omega)(f) = 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\\textttPphi2.lattice\\_second\\_moment\\_le\\_mass\\…","labels":["Pphi2-lattice-second-moment-le-mass-inv"],"detail_key":"p46"},{"id":"n38254","layer":"informal","project":"p46","title":"\\textttmassEigenvalues\\_ge\\_mass\\_sq","kind":"theorem","summary":"[\\textttmassEigenvalues\\_ge\\_mass\\_sq] \\lambda_k \\ge m^2 for all k. Proved by spectral decompos…","labels":["Pphi2-massEigenvalues-ge-mass-sq"],"detail_key":"p46"},{"id":"n38255","layer":"informal","project":"p46","title":"\\textttevalTorusAtSite\\_basisVec","kind":"theorem","summary":"[\\textttevalTorusAtSite\\_basisVec] \\textttPphi2.evalTorusAtSite\\_basisVec","labels":["Pphi2-evalTorusAtSite-basisVec"],"detail_key":"p46"},{"id":"n38256","layer":"informal","project":"p46","title":"\\texttttorusEmbeddedTwoPoint\\_eq\\_lattice\\_cross\\_moment","kind":"theorem","summary":"[\\texttttorusEmbeddedTwoPoint\\_eq\\_lattice\\_cross\\_moment] \\textttPphi2.torusEmbeddedTwoPoint\\_…","labels":["Pphi2-torusEmbeddedTwoPoint-eq-lattice-cross-moment"],"detail_key":"p46"},{"id":"n38257","layer":"informal","project":"p46","title":"\\texttttorusEmbeddedTwoPoint\\_eq\\_spectral\\_sum","kind":"theorem","summary":"[\\texttttorusEmbeddedTwoPoint\\_eq\\_spectral\\_sum] \\langle\\Phi_N(f), \\Phi_N(g)\\rangle = \\sum_k \\…","labels":["Pphi2-torusEmbeddedTwoPoint-eq-spectral-sum"],"detail_key":"p46"},{"id":"n38258","layer":"informal","project":"p46","title":"\\textttlatticeTestFn\\_norm\\_sq\\_bounded","kind":"theorem","summary":"[\\textttlatticeTestFn\\_norm\\_sq\\_bounded] \\sum_x (\\iota^* f)(x)^2 \\le C uniformly in N. 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\\textttPphi2.torusContinuumGreen\\_nonneg","labels":["Pphi2-torusContinuumGreen-nonneg"],"detail_key":"p46"},{"id":"n38262","layer":"informal","project":"p46","title":"\\texttttorusEmbeddedTwoPoint\\_uniform\\_bound","kind":"theorem","summary":"[\\texttttorusEmbeddedTwoPoint\\_uniform\\_bound] E[\\Phi_N(f)^2] \\le m^-2 C_f uniformly in N.","labels":["Pphi2-torusEmbeddedTwoPoint-uniform-bound"],"detail_key":"p46"},{"id":"n38263","layer":"informal","project":"p46","title":"\\texttttorusContinuumGreen\\_pos","kind":"theorem","summary":"[\\texttttorusContinuumGreen\\_pos] G_L(f,f) > 0 for f \\ne 0; G_L(f,f) \\ge 0 for all f.","labels":["Pphi2-torusContinuumGreen-pos"],"detail_key":"p46"},{"id":"n38264","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] \\textttPphi2.torusEmbedLift.congr\\_simp","labels":["Pphi2-torusEmbedLift-congr-simp"],"detail_key":"p46"},{"id":"n38265","layer":"informal","project":"p46","title":"\\texttttorusEmbedLift\\_eval\\_eq","kind":"theorem","summary":"[\\texttttorusEmbedLift\\_eval\\_eq] (\\tilde\\iota_N \\omega)(f) = \\omega(\\iota^* f) where \\iota^* f…","labels":["Pphi2-torusEmbedLift-eval-eq"],"detail_key":"p46"},{"id":"n38266","layer":"informal","project":"p46","title":"\\texttttorus\\_propagator\\_convergence","kind":"theorem","summary":"[\\texttttorus\\_propagator\\_convergence] \\texttorusEmbeddedTwoPoint(f,g) \\to G_L(f,g) as N \\to \\…","labels":["Pphi2-torus-propagator-convergence"],"detail_key":"p46"},{"id":"n38267","layer":"informal","project":"p46","title":"\\textttlatticeTestFn\\_expand","kind":"theorem","summary":"[\\textttlatticeTestFn\\_expand] \\textttPphi2.latticeTestFn\\_expand","labels":["Pphi2-latticeTestFn-expand"],"detail_key":"p46"},{"id":"n38268","layer":"informal","project":"p46","title":"\\textttdm\\_basis\\_eq\\_fourierBasis","kind":"theorem","summary":"[\\textttdm\\_basis\\_eq\\_fourierBasis] \\textttPphi2.dm\\_basis\\_eq\\_fourierBasis","labels":["Pphi2-dm-basis-eq-fourierBasis"],"detail_key":"p46"},{"id":"n38269","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] \\textttPphi2.torusEmbeddedTwoPoint.congr\\_simp","labels":["Pphi2-torusEmbeddedTwoPoint-congr-simp"],"detail_key":"p46"},{"id":"n38270","layer":"informal","project":"p46","title":"\\textttlatticeConfigEuclideanTimeShift","kind":"definition","summary":"[\\textttlatticeConfigEuclideanTimeShift] \\textttPphi2.latticeConfigEuclideanTimeShift","labels":["Pphi2-latticeConfigEuclideanTimeShift"],"detail_key":"p46"},{"id":"n38271","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeShift\\_mod","kind":"theorem","summary":"[\\textttlatticeEuclideanTimeShift\\_mod] \\textttPphi2.latticeEuclideanTimeShift\\_mod","labels":["Pphi2-latticeEuclideanTimeShift-mod"],"detail_key":"p46"},{"id":"n38272","layer":"informal","project":"p46","title":"\\textttlatticeConfigEuclideanTimeShift\\_mod","kind":"theorem","summary":"[\\textttlatticeConfigEuclideanTimeShift\\_mod] \\textttPphi2.latticeConfigEuclideanTimeShift\\_mod","labels":["Pphi2-latticeConfigEuclideanTimeShift-mod"],"detail_key":"p46"},{"id":"n38273","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeShift","kind":"definition","summary":"[\\textttlatticeEuclideanTimeShift] \\textttPphi2.latticeEuclideanTimeShift","labels":["Pphi2-latticeEuclideanTimeShift"],"detail_key":"p46"},{"id":"n38274","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeSeparation","kind":"definition","summary":"[\\textttlatticeEuclideanTimeSeparation] \\textttPphi2.latticeEuclideanTimeSeparation","labels":["Pphi2-latticeEuclideanTimeSeparation"],"detail_key":"p46"},{"id":"n38275","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeSeparation\\_eq\\_min","kind":"theorem","summary":"[\\textttlatticeEuclideanTimeSeparation\\_eq\\_min] \\textttPphi2.latticeEuclideanTimeSeparation\\_e…","labels":["Pphi2-latticeEuclideanTimeSeparation-eq-min"],"detail_key":"p46"},{"id":"n38276","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeSeparation\\_val","kind":"theorem","summary":"[\\textttlatticeEuclideanTimeSeparation\\_val] \\textttPphi2.latticeEuclideanTimeSeparation\\_val","labels":["Pphi2-latticeEuclideanTimeSeparation-val"],"detail_key":"p46"},{"id":"n38277","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] \\textttPphi2.latticeEuclideanTimeShift.congr\\_simp","labels":["Pphi2-latticeEuclideanTimeShift-congr-simp"],"detail_key":"p46"},{"id":"n38278","layer":"informal","project":"p46","title":"\\textttlatticeEuclideanTimeSeparation\\_mod","kind":"theorem","summary":"[\\textttlatticeEuclideanTimeSeparation\\_mod] \\textttPphi2.latticeEuclideanTimeSeparation\\_mod","labels":["Pphi2-latticeEuclideanTimeSeparation-mod"],"detail_key":"p46"},{"id":"n38279","layer":"informal","project":"p46","title":"\\texttttimeCoupling\\_eq\\_zero\\_iff","kind":"theorem","summary":"[\\texttttimeCoupling\\_eq\\_zero\\_iff] K(\\psi,\\psi') = 0 \\iff \\psi = \\psi'. *Proof*: From \\texttt…","labels":["Pphi2-timeCoupling-eq-zero-iff"],"detail_key":"p46"},{"id":"n38280","layer":"informal","project":"p46","title":"\\textttcongr\\_simp","kind":"theorem","summary":"[\\textttcongr\\_simp] \\textttPphi2.spatialAction.congr\\_simp","labels":["Pphi2-spatialAction-congr-simp"],"detail_key":"p46"},{"id":"n38281","layer":"informal","project":"p46","title":"\\textttSpatialField","kind":"definition","summary":"[\\textttSpatialField] \\textttPphi2.SpatialField","labels":["Pphi2-SpatialField"],"detail_key":"p46"},{"id":"n38282","layer":"informal","project":"p46","title":"\\texttttimeCoupling\\_nonneg","kind":"theorem","summary":"[\\texttttimeCoupling\\_nonneg] K(\\psi,\\psi') \\ge 0 (sum of squares). *Proof*: Direct from \\textt…","labels":["Pphi2-timeCoupling-nonneg"],"detail_key":"p46"},{"id":"n38283","layer":"informal","project":"p46","title":"\\texttttimeCoupling","kind":"definition","summary":"[\\texttttimeCoupling] ``\\textttlean def timeCoupling (psi psi' : SpatialField 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:…","labels":["Pphi2-transferKernel"],"detail_key":"p46"},{"id":"n38287","layer":"informal","project":"p46","title":"\\textttspatialPotential","kind":"definition","summary":"[\\textttspatialPotential] ``\\textttlean def spatialPotential (P : InteractionPolynomial) (\\_a m…","labels":["Pphi2-spatialPotential"],"detail_key":"p46"},{"id":"n38288","layer":"informal","project":"p46","title":"\\texttttransferKernel\\_pos","kind":"theorem","summary":"[\\texttttransferKernel\\_pos] T(\\psi,\\psi') > 0 for all \\psi,\\psi'. *Proof*: Immediate from \\tex…","labels":["Pphi2-transferKernel-pos"],"detail_key":"p46"},{"id":"n38289","layer":"informal","project":"p46","title":"\\textttspatialAction","kind":"definition","summary":"[\\textttspatialAction] ``\\textttlean def spatialAction (P : InteractionPolynomial) (a mass : 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:y^2:_c_2.","labels":["Pphi2-wickMonomial-two-add"],"detail_key":"p46"},{"id":"n38293","layer":"informal","project":"p46","title":"\\textttwickMonomial\\_four\\_add","kind":"theorem","summary":"[\\textttwickMonomial\\_four\\_add] n=4: explicit 5-term expansion for the \\varphi^4 case.","labels":["Pphi2-wickMonomial-four-add"],"detail_key":"p46"},{"id":"n38294","layer":"informal","project":"p46","title":"\\textttwickPolynomial\\_add\\_sub\\_self","kind":"theorem","summary":"[\\textttwickPolynomial\\_add\\_sub\\_self] \\textttPphi2.wickPolynomial\\_add\\_sub\\_self","labels":["Pphi2-wickPolynomial-add-sub-self"],"detail_key":"p46"},{"id":"n38295","layer":"informal","project":"p46","title":"\\textttwickMonomial\\_four\\_lower\\_bound","kind":"theorem","summary":"[\\textttwickMonomial\\_four\\_lower\\_bound] :x^4:_c \\ge -6c^2 for all x and c \\ge 0. Proof: x^4 -…","labels":["Pphi2-wickMonomial-four-lower-bound"],"detail_key":"p46"},{"id":"n38296","layer":"informal","project":"p46","title":"\\textttnormalizedMoment\\_hasDerivAt\\_explicit","kind":"theorem","summary":"[\\textttnormalizedMoment\\_hasDerivAt\\_explicit] \\textttPphi2.normalizedMoment\\_hasDerivAt\\_expl…","labels":["Pphi2-normalizedMoment-hasDerivAt-explicit"],"detail_key":"p46"},{"id":"n38297","layer":"informal","project":"p46","title":"\\textttu4\\_hasDerivAt2","kind":"theorem","summary":"[\\textttu4\\_hasDerivAt2] \\textttPphi2.u4\\_hasDerivAt2","labels":["Pphi2-u4-hasDerivAt2"],"detail_key":"p46"},{"id":"n38298","layer":"informal","project":"p46","title":"\\textttpartitionFnDeriv","kind":"definition","summary":"[\\textttpartitionFnDeriv] \\textttPphi2.partitionFnDeriv","labels":["Pphi2-partitionFnDeriv"],"detail_key":"p46"},{"id":"n38299","layer":"informal","project":"p46","title":"\\textttu4Deriv","kind":"definition","summary":"[\\textttu4Deriv] 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\\textttPphi2.torus\\_interacting\\_fourth\\_m…","labels":["Pphi2-torus-interacting-fourth-moment-tendsto"],"detail_key":"p46"},{"id":"n38651","layer":"informal","project":"p46","title":"\\texttttorus\\_interacting\\_fourth\\_moment\\_uniform","kind":"theorem","summary":"[\\texttttorus\\_interacting\\_fourth\\_moment\\_uniform] \\textttPphi2.torus\\_interacting\\_fourth\\_m…","labels":["Pphi2-torus-interacting-fourth-moment-uniform"],"detail_key":"p46"},{"id":"n38652","layer":"informal","project":"p46","title":"\\texttttorus\\_interacting\\_eighth\\_moment\\_uniform","kind":"theorem","summary":"[\\texttttorus\\_interacting\\_eighth\\_moment\\_uniform] \\textttPphi2.torus\\_interacting\\_eighth\\_m…","labels":["Pphi2-torus-interacting-eighth-moment-uniform"],"detail_key":"p46"},{"id":"n38653","layer":"informal","project":"p46","title":"\\texttttorusContinuumMeasure\\_isProbability","kind":"theorem","summary":"[\\texttttorusContinuumMeasure\\_isProbability] 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\\textttPphi2.cylinderPullback\\_expMoment\\_eq","labels":["Pphi2-cylinderPullback-expMoment-eq"],"detail_key":"p46"},{"id":"n38688","layer":"informal","project":"p46","title":"\\textttcylinderPullback\\_expMoment\\_le\\_green","kind":"theorem","summary":"[\\textttcylinderPullback\\_expMoment\\_le\\_green] \\textttPphi2.cylinderPullback\\_expMoment\\_le\\_g…","labels":["Pphi2-cylinderPullback-expMoment-le-green"],"detail_key":"p46"},{"id":"n38689","layer":"informal","project":"p46","title":"\\textttMeasureHasGreenMomentBound","kind":"definition","summary":"[\\textttMeasureHasGreenMomentBound] \\textttPphi2.MeasureHasGreenMomentBound","labels":["Pphi2-MeasureHasGreenMomentBound"],"detail_key":"p46"},{"id":"n38690","layer":"informal","project":"p46","title":"\\textttasymTransferWeight\\_gaussian\\_decay","kind":"theorem","summary":"[\\textttasymTransferWeight\\_gaussian\\_decay] 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(self.realTo…","labels":[],"detail_key":"p46","name":"Common.QFT.Euclidean.SchwingerFormulation.realToComplex_add","module":"Common.QFT.Euclidean.Formulations"},{"id":"n38957","layer":"formal","project":"p46","title":"Common.QFT.Euclidean.SchwingerFormulation.realToComplex_continuous","kind":"theorem","summary":"∀ (self : Common.QFT.Euclidean.SchwingerFormulation), Continuous self.realToComplex","labels":[],"detail_key":"p46","name":"Common.QFT.Euclidean.SchwingerFormulation.realToComplex_continuous","module":"Common.QFT.Euclidean.Formulations"},{"id":"n38958","layer":"formal","project":"p46","title":"Common.QFT.Euclidean.SchwingerFormulation.realToComplex_smul","kind":"theorem","summary":"∀ (self : Common.QFT.Euclidean.SchwingerFormulation) (r : Real) (f : self.TestFunction), Eq 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Iff…","labels":[],"detail_key":"p46","name":"Common.QFT.Euclidean.TensorSchwingerModel.ext_iff","module":"Common.QFT.Euclidean.Formulations"},{"id":"n38964","layer":"formal","project":"p46","title":"Common.QFT.Euclidean.TensorSchwingerModel.schwinger","kind":"def","summary":"F : Common.QFT.Euclidean.Formulation → Common.QFT.Euclidean.TensorSchwingerModel F → (n : Nat)…","labels":[],"detail_key":"p46","name":"Common.QFT.Euclidean.TensorSchwingerModel.schwinger","module":"Common.QFT.Euclidean.Formulations"},{"id":"n38965","layer":"formal","project":"p46","title":"Common.QFT.Euclidean.TensorToDistributionalBridge","kind":"inductive","summary":"(F : Common.QFT.Euclidean.SchwingerFormulation) → Common.QFT.Euclidean.TensorSchwingerModel 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d : Nat (g : Pphi2.ContinuumMotion d) (f : Pphi2.ContinuumTestFunction d) (x : Pphi2.Continuu…","labels":[],"detail_key":"p46","name":"Pphi2.continuumEuclideanAction_apply","module":"Pphi2.Backgrounds.EuclideanPlane"},{"id":"n39338","layer":"formal","project":"p46","title":"Pphi2.planeBackground","kind":"def","summary":"Nat → Pphi2.EuclideanPlaneBackground","labels":[],"detail_key":"p46","name":"Pphi2.planeBackground","module":"Pphi2.Backgrounds.EuclideanPlane"},{"id":"n39339","layer":"formal","project":"p46","title":"Pphi2.planeBackground_dim","kind":"theorem","summary":"∀ (d : Nat), Eq (Pphi2.planeBackground d).dim d","labels":[],"detail_key":"p46","name":"Pphi2.planeBackground_dim","module":"Pphi2.Backgrounds.EuclideanPlane"},{"id":"n39340","layer":"formal","project":"p46","title":"Pphi2.schwartzTranslate","kind":"def","summary":"(d : Nat) → Pphi2.ContinuumSpaceTime d → ContinuousLinearMap (RingHom.id Real) 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[inst : NeZero N] (P : InteractionPolynomial) (mass : Real) (hmass : LT.lt 0 mass…","labels":[],"detail_key":"p46","name":"Pphi2.continuum_second_moment_uniform","module":"Pphi2.ContinuumLimit.Tightness"},{"id":"n39419","layer":"formal","project":"p46","title":"Pphi2.continuumTimeReflection","kind":"def","summary":"ContinuousLinearMap (RingHom.id Real) (Pphi2.ContinuumTestFunction 2) (Pphi2.ContinuumTestFunct…","labels":[],"detail_key":"p46","name":"Pphi2.continuumTimeReflection","module":"Pphi2.ContinuumLimit.TimeReflection"},{"id":"n39420","layer":"formal","project":"p46","title":"Pphi2.continuumTimeReflection_apply_coord","kind":"theorem","summary":"∀ (f : Pphi2.ContinuumTestFunction 2) (p : EuclideanSpace Real (Fin 2)), Eq 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[i…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughCovariance_pow_one_sum_le_uniform_in_aN_proved","module":"Pphi2.NelsonEstimate.CovarianceBoundsGJ"},{"id":"n39889","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughCovariance_pow_sum_le_uniform_in_aN","kind":"theorem","summary":"∀ d : Nat, Eq d 2 → ∀ (mass L : Real), LT.lt 0 L → LT.lt 0 mass → ∀ (m : Nat), LE.le 1 m → Exis…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughCovariance_pow_sum_le_uniform_in_aN","module":"Pphi2.NelsonEstimate.CovarianceBoundsGJ"},{"id":"n39890","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughCovariance_pow_sum_le_uniform_in_aN_of_three_le","kind":"theorem","summary":"∀ d : Nat, Eq d 2 → ∀ (mass L : Real), LT.lt 0 L → LT.lt 0 mass → ∀ (m : Nat), LE.le 3 m → 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B…","labels":[],"detail_key":"p46","name":"Pphi2.smoothWickConstant_le_log_uniform_in_aN","module":"Pphi2.NelsonEstimate.CovarianceBoundsGJ"},{"id":"n39893","layer":"formal","project":"p46","title":"Pphi2.covariance_split","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (m : Nat), Eq (Inv.inv (GaussianField.lattice…","labels":[],"detail_key":"p46","name":"Pphi2.covariance_split","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39894","layer":"formal","project":"p46","title":"Pphi2.roughCovEigenvalue","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real → Nat → Real","labels":[],"detail_key":"p46","name":"Pphi2.roughCovEigenvalue","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39895","layer":"formal","project":"p46","title":"Pphi2.roughCovEigenvalue_le_T","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), LE.le 0 T → ∀ (m : Nat), LT.lt 0 a → LT.lt 0…","labels":[],"detail_key":"p46","name":"Pphi2.roughCovEigenvalue_le_T","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39896","layer":"formal","project":"p46","title":"Pphi2.roughCovEigenvalue_le_inv","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (m : Nat), LT.lt 0 a → LT.lt 0 mass → LE.le (…","labels":[],"detail_key":"p46","name":"Pphi2.roughCovEigenvalue_le_inv","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39897","layer":"formal","project":"p46","title":"Pphi2.roughCovEigenvalue_pos","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), LT.lt 0 T → ∀ (m : Nat), LT.lt 0 a → LT.lt 0…","labels":[],"detail_key":"p46","name":"Pphi2.roughCovEigenvalue_pos","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39898","layer":"formal","project":"p46","title":"Pphi2.roughCovariance_sq_summable","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), LT.lt 0 T → LT.lt 0 a → LT.lt 0 mass → LE.le…","labels":[],"detail_key":"p46","name":"Pphi2.roughCovariance_sq_summable","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39899","layer":"formal","project":"p46","title":"Pphi2.roughWickConstant","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.roughWickConstant","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39900","layer":"formal","project":"p46","title":"Pphi2.smoothCovEigenvalue","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real → Nat → Real","labels":[],"detail_key":"p46","name":"Pphi2.smoothCovEigenvalue","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39901","layer":"formal","project":"p46","title":"Pphi2.smoothCovEigenvalue_pos","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), LT.lt 0 T → ∀ (m : Nat), LT.lt 0 a → LT.lt 0…","labels":[],"detail_key":"p46","name":"Pphi2.smoothCovEigenvalue_pos","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39902","layer":"formal","project":"p46","title":"Pphi2.smoothVarianceConstant","kind":"def","summary":"Nat → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.smoothVarianceConstant","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39903","layer":"formal","project":"p46","title":"Pphi2.smoothVarianceConstant_pos","kind":"theorem","summary":"∀ (d : Nat) (a mass : Real), LT.lt 0 a → LT.lt 0 mass → LT.lt 0 (Pphi2.smoothVarianceConstant d…","labels":[],"detail_key":"p46","name":"Pphi2.smoothVarianceConstant_pos","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39904","layer":"formal","project":"p46","title":"Pphi2.smoothVariance_le_log","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), Eq d 2 → ∀ (T : Real), LT.lt 0 T → LT.lt 0 a →…","labels":[],"detail_key":"p46","name":"Pphi2.smoothVariance_le_log","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39905","layer":"formal","project":"p46","title":"Pphi2.smoothVariance_le_log_uniform","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), Eq d 2 → ∀ (T : Real), LT.lt 0 T → LT.lt 0 a →…","labels":[],"detail_key":"p46","name":"Pphi2.smoothVariance_le_log_uniform","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39906","layer":"formal","project":"p46","title":"Pphi2.smoothWickConstant","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.smoothWickConstant","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39907","layer":"formal","project":"p46","title":"Pphi2.wickConstant_split","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), Eq (Pphi2.wickConstant d N a mass) (HAdd.hAd…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_split","module":"Pphi2.NelsonEstimate.CovarianceSplit"},{"id":"n39908","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.degreeCutoffPower","kind":"def","summary":"InteractionPolynomial → Nat","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.degreeCutoffPower","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39909","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.degreeCutoffPower_pos","kind":"theorem","summary":"∀ (P : InteractionPolynomial), LT.lt 0 (Pphi2.DynamicalCutoff.degreeCutoffPower P)","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.degreeCutoffPower_pos","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39910","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale","kind":"def","summary":"InteractionPolynomial → Real → Real → 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(Pphi2.DynamicalCutoff…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale_eq_one","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39913","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale_log_pow_le","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (C M : Real), LT.lt 0 C → LE.le (HMul.hMul 2 C) M → LE.le (HMul.h…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale_log_pow_le","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39914","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale_pos","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (C M : Real), LT.lt 0 (Pphi2.DynamicalCutoff.degreeDynamicalCutof…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.degreeDynamicalCutoffScale_pos","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39915","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale","kind":"def","summary":"Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39916","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_eq_exp","kind":"theorem","summary":"∀ (C M : Real), LT.lt (HMul.hMul 2 C) M → Eq (Pphi2.DynamicalCutoff.dynamicalCutoffScale C M) (…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_eq_exp","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39917","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_eq_one","kind":"theorem","summary":"∀ (C M : Real), LE.le M (HMul.hMul 2 C) → Eq (Pphi2.DynamicalCutoff.dynamicalCutoffScale C M) 1","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_eq_one","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39918","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_le_one","kind":"theorem","summary":"∀ (C M : Real), LT.lt 0 C → LE.le (Pphi2.DynamicalCutoff.dynamicalCutoffScale C M) 1","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_le_one","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39919","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_log_sq_le","kind":"theorem","summary":"∀ (C M : Real), LT.lt 0 C → LE.le (HMul.hMul 2 C) M → LE.le (HMul.hMul C (HPow.hPow (HAdd.hAdd…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_log_sq_le","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39920","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_pos","kind":"theorem","summary":"∀ (C M : Real), LT.lt 0 (Pphi2.DynamicalCutoff.dynamicalCutoffScale C M)","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.dynamicalCutoffScale_pos","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39921","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.measure_le_neg_of_smooth_rough_split","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) (V : α → Real) (V_S E_R…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.measure_le_neg_of_smooth_rough_split","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39922","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.one_add_abs_log_degreeDynamicalCutoff_eq_rpow","kind":"theorem","summary":"∀ (P : InteractionPolynomial) C M : Real, LT.lt 0 C → LT.lt (HMul.hMul 2 C) M → Eq (HAdd.hAdd 1…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.one_add_abs_log_degreeDynamicalCutoff_eq_rpow","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39923","layer":"formal","project":"p46","title":"Pphi2.DynamicalCutoff.smooth_lower_bound_at_cutoff","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → Eq d 2 → ∀ (L : Rea…","labels":[],"detail_key":"p46","name":"Pphi2.DynamicalCutoff.smooth_lower_bound_at_cutoff","module":"Pphi2.NelsonEstimate.DynamicalCutoff"},{"id":"n39924","layer":"formal","project":"p46","title":"GaussianField.latticeHeatKernelMatrix.congr_simp","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a a_1 : Real), Eq a a_1 → ∀ (t t_1 : Real), Eq t t_1 → ∀ (a_2…","labels":[],"detail_key":"p46","name":"GaussianField.latticeHeatKernelMatrix.congr_simp","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39925","layer":"formal","project":"p46","title":"Pphi2.CanonicalJoint","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Type","labels":[],"detail_key":"p46","name":"Pphi2.CanonicalJoint","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39926","layer":"formal","project":"p46","title":"Pphi2.CanonicalJoint.congr_simp","kind":"theorem","summary":"∀ (d d_1 : Nat), Eq d d_1 → ∀ (N N_1 : Nat) (e_N : Eq N N_1) [inst : NeZero N], Eq (Pphi2.Canon…","labels":[],"detail_key":"p46","name":"Pphi2.CanonicalJoint.congr_simp","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39927","layer":"formal","project":"p46","title":"Pphi2.CanonicalJointSumIndex","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Type","labels":[],"detail_key":"p46","name":"Pphi2.CanonicalJointSumIndex","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39928","layer":"formal","project":"p46","title":"Pphi2.FieldDecomposition","kind":"inductive","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real → Type 1","labels":[],"detail_key":"p46","name":"Pphi2.FieldDecomposition","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39929","layer":"formal","project":"p46","title":"Pphi2.FieldDecomposition.Joint","kind":"def","summary":"d N : Nat → [inst : NeZero N] → a mass T : Real → Pphi2.FieldDecomposition d N a mass T → 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→ [inst : NeZero N] → a mass T : Real → (self : Pphi2.FieldDecomposition d N a mass T…","labels":[],"detail_key":"p46","name":"Pphi2.FieldDecomposition.φ_R","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39936","layer":"formal","project":"p46","title":"Pphi2.FieldDecomposition.φ_R_measurable","kind":"theorem","summary":"∀ d N : Nat [inst : NeZero N] a mass T : Real (self : Pphi2.FieldDecomposition d N a mass T), M…","labels":[],"detail_key":"p46","name":"Pphi2.FieldDecomposition.φ_R_measurable","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39937","layer":"formal","project":"p46","title":"Pphi2.FieldDecomposition.φ_S","kind":"def","summary":"d N : Nat → [inst : NeZero N] → a mass T : Real → (self : Pphi2.FieldDecomposition d N a mass 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[NeZero N] → Real → Real → (Fin d → Fin N) → Real","labels":[],"detail_key":"p46","name":"Pphi2.canonicalEigenvalue","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39941","layer":"formal","project":"p46","title":"Pphi2.canonicalEigenvalue.congr_simp","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a a_1 : Real), Eq a a_1 → ∀ (mass mass_1 : Real), Eq mass mass…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalEigenvalue.congr_simp","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39942","layer":"formal","project":"p46","title":"Pphi2.canonicalJointMeasure","kind":"def","summary":"(d N : Nat) → [inst : NeZero N] → MeasureTheory.Measure (Pphi2.CanonicalJoint d N)","labels":[],"detail_key":"p46","name":"Pphi2.canonicalJointMeasure","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n39943","layer":"formal","project":"p46","title":"Pphi2.canonicalJointMeasure_isProbability","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N], 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Meas…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldFunction_pointwise_measurable","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40017","layer":"formal","project":"p46","title":"Pphi2.canonicalSumFieldFunction_smul","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T c : Real) (η : Pphi2.CanonicalJoint d N) (x : Gaussia…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldFunction_smul","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40018","layer":"formal","project":"p46","title":"Pphi2.canonicalSumFieldLaw","kind":"def","summary":"(d N : Nat) → [NeZero N] → Real → Real → Real → MeasureTheory.Measure (GaussianField.FinLattice…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldLaw","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40019","layer":"formal","project":"p46","title":"Pphi2.canonicalSumFieldLaw_eq_latticeGaussianFieldLaw","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (ha : LT.lt 0 a) (hmass : LT.lt 0 mass) (T : Re…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldLaw_eq_latticeGaussianFieldLaw","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40020","layer":"formal","project":"p46","title":"Pphi2.canonicalSumFieldLaw_eq_map_canonicalSumFieldFunction","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), Eq (Pphi2.canonicalSumFieldLaw d N a mass T)…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldLaw_eq_map_canonicalSumFieldFunction","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40021","layer":"formal","project":"p46","title":"Pphi2.canonicalSumFieldLaw_isGaussian","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real), ProbabilityTheory.IsGaussian (Pphi2.canonica…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSumFieldLaw_isGaussian","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40022","layer":"formal","project":"p46","title":"Pphi2.heatKernel_diagonal_mass_weighted_eq_eigenvalue_average","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a : Real), LT.lt 0 a → ∀ (mass t : Real) (x : GaussianField.Fi…","labels":[],"detail_key":"p46","name":"Pphi2.heatKernel_diagonal_mass_weighted_eq_eigenvalue_average","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40023","layer":"formal","project":"p46","title":"Pphi2.heatKernel_row_pairing_eq_diagonal","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a s t : Real) (x : GaussianField.FinLatticeSites d N), Eq (Fin…","labels":[],"detail_key":"p46","name":"Pphi2.heatKernel_row_pairing_eq_diagonal","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40024","layer":"formal","project":"p46","title":"Pphi2.heatKernel_row_sum_eq_one","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a : Real), LT.lt 0 a → ∀ (t : Real) (x : GaussianField.FinLatt…","labels":[],"detail_key":"p46","name":"Pphi2.heatKernel_row_sum_eq_one","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40025","layer":"formal","project":"p46","title":"Pphi2.integral_comp_canonicalSumConfig","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (ha : LT.lt 0 a) (hmass : LT.lt 0 mass) (T : Re…","labels":[],"detail_key":"p46","name":"Pphi2.integral_comp_canonicalSumConfig","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40026","layer":"formal","project":"p46","title":"Pphi2.latticeFieldToConfig","kind":"def","summary":"(d N : Nat) → [NeZero N] → GaussianField.FinLatticeField 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(d N : Nat) [inst : NeZero N] (a : Real), Ne a 0 → ∀ (t : Real) (x y : GaussianField.FinLatti…","labels":[],"detail_key":"p46","name":"Pphi2.latticeHeatKernel_entry_eq_eigenvalue_average","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40030","layer":"formal","project":"p46","title":"Pphi2.roughWickConstant.congr_simp","kind":"theorem","summary":"∀ (d d_1 : Nat), Eq d d_1 → ∀ (N N_1 : Nat) (e_N : Eq N N_1) [inst : NeZero N] (a a_1 : Real),…","labels":[],"detail_key":"p46","name":"Pphi2.roughWickConstant.congr_simp","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40031","layer":"formal","project":"p46","title":"Pphi2.sitePairingCLM","kind":"def","summary":"(d N : Nat) → [NeZero N] → GaussianField.FinLatticeField d N → ContinuousLinearMap (RingHom.id…","labels":[],"detail_key":"p46","name":"Pphi2.sitePairingCLM","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40032","layer":"formal","project":"p46","title":"Pphi2.wickPower_two_site_pi_gaussianReal_eq_diag","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] (γ_x γ_y : ι → Real) (n : Nat), Eq (MeasureTheory.integral (M…","labels":[],"detail_key":"p46","name":"Pphi2.wickPower_two_site_pi_gaussianReal_eq_diag","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40033","layer":"formal","project":"p46","title":"Pphi2.wickPower_two_site_pi_gaussianReal_eq_zero_of_ne","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] (γ_x γ_y : ι → Real) n m : Nat, Ne n m → Eq (MeasureTheory.in…","labels":[],"detail_key":"p46","name":"Pphi2.wickPower_two_site_pi_gaussianReal_eq_zero_of_ne","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40034","layer":"formal","project":"p46","title":"Pphi2.wickPower_two_site_pi_gaussianReal_integrable","kind":"theorem","summary":"∀ ι : Type u_1 [inst : Fintype ι] (γ_x γ_y : ι → Real) (n m : Nat), MeasureTheory.Integrable (f…","labels":[],"detail_key":"p46","name":"Pphi2.wickPower_two_site_pi_gaussianReal_integrable","module":"Pphi2.NelsonEstimate.FieldDecomposition"},{"id":"n40035","layer":"formal","project":"p46","title":"Pphi2.BridgeFromTail.bridgeAxiom_of_setup_real_generic","kind":"theorem","summary":"∀ X : Type u_1 [inst : MeasurableSpace X] (μ : MeasureTheory.Measure X) [MeasureTheory.IsProbab…","labels":[],"detail_key":"p46","name":"Pphi2.BridgeFromTail.bridgeAxiom_of_setup_real_generic","module":"Pphi2.NelsonEstimate.GenericBridge"},{"id":"n40036","layer":"formal","project":"p46","title":"Pphi2.gaussian_sum_bound","kind":"theorem","summary":"∀ (α : Real), LT.lt 0 α → ∀ (M : Nat), LE.le ((Finset.Icc (Neg.neg ↑M) ↑M).sum fun k => Real.ex…","labels":[],"detail_key":"p46","name":"Pphi2.gaussian_sum_bound","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40037","layer":"formal","project":"p46","title":"Pphi2.heat_kernel_1d_bound","kind":"theorem","summary":"∀ (N : Nat) [NeZero N] (a : Real), LT.lt 0 a → ∀ (t : Real), LT.lt 0 t → Exists fun C => And (L…","labels":[],"detail_key":"p46","name":"Pphi2.heat_kernel_1d_bound","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40038","layer":"formal","project":"p46","title":"Pphi2.heat_kernel_1d_bound_uniform","kind":"theorem","summary":"∀ (L : Real), LT.lt 0 L → Exists fun C => And (LT.lt 0 C) (∀ (N : Nat) [NeZero N] (a : Real), L…","labels":[],"detail_key":"p46","name":"Pphi2.heat_kernel_1d_bound_uniform","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40039","layer":"formal","project":"p46","title":"Pphi2.heat_kernel_trace_bound","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (t : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.heat_kernel_trace_bound","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40040","layer":"formal","project":"p46","title":"Pphi2.heat_kernel_trace_bound_uniform","kind":"theorem","summary":"∀ (d : Nat) (L : Real), LT.lt 0 L → ∀ (mass : Real), LT.lt 0 mass → Exists fun C => And (LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.heat_kernel_trace_bound_uniform","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40041","layer":"formal","project":"p46","title":"Pphi2.roughVariance_from_heat_kernel","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a 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0 T → Eq (HDiv.hDiv (Real.exp (HMul.hMul (Neg…","labels":[],"detail_key":"p46","name":"Pphi2.schwinger_smooth","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40045","layer":"formal","project":"p46","title":"Pphi2.schwinger_smooth_Ioi","kind":"theorem","summary":"∀ (lam : Real), LT.lt 0 lam → ∀ (T : Real), Eq (HDiv.hDiv (Real.exp (HMul.hMul (Neg.neg T) lam)…","labels":[],"detail_key":"p46","name":"Pphi2.schwinger_smooth_Ioi","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40046","layer":"formal","project":"p46","title":"Pphi2.sin_sq_lower_bound","kind":"theorem","summary":"∀ (x : Real), LE.le (abs x) (HDiv.hDiv Real.pi 2) → LE.le (HMul.hMul (HPow.hPow (HDiv.hDiv 2 Re…","labels":[],"detail_key":"p46","name":"Pphi2.sin_sq_lower_bound","module":"Pphi2.NelsonEstimate.HeatKernelBound"},{"id":"n40047","layer":"formal","project":"p46","title":"Pphi2.smoothVariance_from_heat_kernel","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 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fun…","labels":[],"detail_key":"p46","name":"Pphi2.IntegrabilityHelpers.lintegral_exp_neg_lt_top","module":"Pphi2.NelsonEstimate.IntegrabilityHelpers"},{"id":"n40050","layer":"formal","project":"p46","title":"Pphi2.IntegrabilityHelpers.lintegral_layer_cake_lt_top_of_eventual_decay","kind":"theorem","summary":"∀ (ψ : Real → ENNReal) (T₀ : Real), LT.lt 0 T₀ → LT.lt (MeasureTheory.lintegral (MeasureTheory.…","labels":[],"detail_key":"p46","name":"Pphi2.IntegrabilityHelpers.lintegral_layer_cake_lt_top_of_eventual_decay","module":"Pphi2.NelsonEstimate.IntegrabilityHelpers"},{"id":"n40051","layer":"formal","project":"p46","title":"Pphi2.LatticeBridge.LatticeRoughErrorSetup","kind":"inductive","summary":"Nat → InteractionPolynomial → Real → 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Re…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeBridge.bridgeAxiom_of_varying_coupled_setup_real","module":"Pphi2.NelsonEstimate.LatticeBridge"},{"id":"n40065","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.interactionFunctional_eq_smooth_plus_rough","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial) (T : Real) (ω : Gau…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.interactionFunctional_eq_smooth_plus_rough","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40066","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeRoughError","kind":"def","summary":"(d N : Nat) → [NeZero N] → Real → Real → InteractionPolynomial → Real → 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Measura…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeRoughError_measurable","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40069","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeRoughError_pure_quartic_chaos2_form","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n),…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeRoughError_pure_quartic_chaos2_form","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40070","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeRoughError_pure_quartic_summed","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n),…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeRoughError_pure_quartic_summed","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40071","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeSmoothInteraction","kind":"def","summary":"(d N : Nat) → [NeZero N] → Real → Real → InteractionPolynomial → Real → GaussianField.Configura…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeSmoothInteraction","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40072","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeSmoothInteraction_lower_bound_at_cutoff_quartic","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n),…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeSmoothInteraction_lower_bound_at_cutoff_quartic","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40073","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeSmoothInteraction_measurable","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial) (T : Real), Measura…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeSmoothInteraction_measurable","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40074","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeSmoothInteraction_of_pure","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n),…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeSmoothInteraction_of_pure","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40075","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeWickSquareSum","kind":"def","summary":"(d N : Nat) → [NeZero N] → Real → Real → GaussianField.Configuration (GaussianField.FinLatticeF…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeWickSquareSum","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40076","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.latticeWickSquareSum_measurable","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), Measurable (Pphi2.LatticeSetup.latticeWickSqua…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeWickSquareSum_measurable","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40077","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.wickMonomial_four_diff","kind":"theorem","summary":"∀ (c c' x : Real), Eq (HSub.hSub (Pphi2.wickMonomial 4 c x) (Pphi2.wickMonomial 4 c' x)) (HAdd.…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.wickMonomial_four_diff","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40078","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.wickMonomial_two_shift","kind":"theorem","summary":"∀ (c c' x : Real), Eq (Pphi2.wickMonomial 2 c x) (HAdd.hAdd (Pphi2.wickMonomial 2 c' x) (HSub.h…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.wickMonomial_two_shift","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40079","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.wickPolynomial_of_pure","kind":"theorem","summary":"∀ (P : InteractionPolynomial), (∀ (m : Fin P.n), Eq (P.coeff m) 0) → ∀ (c x : Real), Eq (Pphi2.…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.wickPolynomial_of_pure","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40080","layer":"formal","project":"p46","title":"Pphi2.LatticeSetup.wickPolynomial_pure_quartic_diff","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n),…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.wickPolynomial_pure_quartic_diff","module":"Pphi2.NelsonEstimate.LatticeSetup"},{"id":"n40081","layer":"formal","project":"p46","title":"Pphi2.LayerCake.exp_neg_sq_le_exp_two_max","kind":"theorem","summary":"∀ (V : Real), LE.le (HPow.hPow (Real.exp (Neg.neg V)) 2) (Real.exp (HMul.hMul 2 (max 0 (Neg.neg…","labels":[],"detail_key":"p46","name":"Pphi2.LayerCake.exp_neg_sq_le_exp_two_max","module":"Pphi2.NelsonEstimate.LayerCake"},{"id":"n40082","layer":"formal","project":"p46","title":"Pphi2.LayerCake.lintegral_expSq_neg_le_layer_cake","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite…","labels":[],"detail_key":"p46","name":"Pphi2.LayerCake.lintegral_expSq_neg_le_layer_cake","module":"Pphi2.NelsonEstimate.LayerCake"},{"id":"n40083","layer":"formal","project":"p46","title":"Pphi2.LayerCake.lintegral_expSq_neg_le_of_tail","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite…","labels":[],"detail_key":"p46","name":"Pphi2.LayerCake.lintegral_expSq_neg_le_of_tail","module":"Pphi2.NelsonEstimate.LayerCake"},{"id":"n40084","layer":"formal","project":"p46","title":"Pphi2.LayerCake.setOf_le_max_eq_setOf_le_neg","kind":"theorem","summary":"∀ α : Type u_1 (V : α → Real) t : Real, LT.lt 0 t → Eq (setOf fun a => LE.le t (max 0 (Neg.neg…","labels":[],"detail_key":"p46","name":"Pphi2.LayerCake.setOf_le_max_eq_setOf_le_neg","module":"Pphi2.NelsonEstimate.LayerCake"},{"id":"n40085","layer":"formal","project":"p46","title":"Pphi2.nelson_exponential_estimate_lattice","kind":"theorem","summary":"∀ (L : Real) [hL : Fact (LT.lt 0 L)] (P : InteractionPolynomial) (mass : Real) (hmass : LT.lt 0…","labels":[],"detail_key":"p46","name":"Pphi2.nelson_exponential_estimate_lattice","module":"Pphi2.NelsonEstimate.NelsonEstimate"},{"id":"n40086","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_cutoff_tail_quartic_uniform","kind":"theorem","summary":"∀ (P : InteractionPolynomial), (∀ (m : Fin P.n), Eq (P.coeff m) 0) → Eq P.n 4 → ∀ (mass L : Rea…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_cutoff_tail_quartic_uniform","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40087","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_cutoff_tail_uniform","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → LT.lt 0 mass → Exists fun K => And (…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_cutoff_tail_uniform","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40088","layer":"formal","project":"p46","title":"Pphi2.degreeCutoffTail","kind":"def","summary":"InteractionPolynomial → Real → Real → Real → ENNReal","labels":[],"detail_key":"p46","name":"Pphi2.degreeCutoffTail","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40089","layer":"formal","project":"p46","title":"Pphi2.degreePiecewiseTail","kind":"def","summary":"InteractionPolynomial → Real → Real → Real → ENNReal","labels":[],"detail_key":"p46","name":"Pphi2.degreePiecewiseTail","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40090","layer":"formal","project":"p46","title":"Pphi2.degreePiecewiseTail_layerCake_lt_top","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (K C : Real), LT.lt 0 K → LT.lt 0 C → LT.lt (MeasureTheory.linteg…","labels":[],"detail_key":"p46","name":"Pphi2.degreePiecewiseTail_layerCake_lt_top","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40091","layer":"formal","project":"p46","title":"Pphi2.nelson_exponential_estimate_master","kind":"theorem","summary":"∀ (d : Nat), Eq d 2 → ∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → ∀ (hmass : LT.…","labels":[],"detail_key":"p46","name":"Pphi2.nelson_exponential_estimate_master","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40092","layer":"formal","project":"p46","title":"Pphi2.nelson_exponential_estimate_master_bounded","kind":"axiom","summary":"∀ (d : Nat) (P : InteractionPolynomial) (mass : Real) (hmass : LT.lt 0 mass), Exists fun K => A…","labels":[],"detail_key":"p46","name":"Pphi2.nelson_exponential_estimate_master_bounded","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40093","layer":"formal","project":"p46","title":"Pphi2.polynomial_chaos_exp_moment_bridge","kind":"theorem","summary":"∀ (d : Nat), Eq d 2 → ∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → ∀ (hmass : LT.…","labels":[],"detail_key":"p46","name":"Pphi2.polynomial_chaos_exp_moment_bridge","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40094","layer":"formal","project":"p46","title":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → ∀ (hmass : LT.lt 0 mass), Exists fun…","labels":[],"detail_key":"p46","name":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40095","layer":"formal","project":"p46","title":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded_of_cutoffTail","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → ∀ (hmass : LT.lt 0 mass) (K C : Real…","labels":[],"detail_key":"p46","name":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded_of_cutoffTail","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40096","layer":"formal","project":"p46","title":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded_of_tail","kind":"theorem","summary":"∀ d : Nat, Eq d 2 → ∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → ∀ (hmass : LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.polynomial_chaos_exp_moment_bridge_bounded_of_tail","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40097","layer":"formal","project":"p46","title":"Pphi2.quarticPiecewiseTail_layerCake_lt_top","kind":"theorem","summary":"∀ (K C : Real), LT.lt 0 K → LT.lt 0 C → LT.lt (MeasureTheory.lintegral (MeasureTheory.volume.re…","labels":[],"detail_key":"p46","name":"Pphi2.quarticPiecewiseTail_layerCake_lt_top","module":"Pphi2.NelsonEstimate.PolynomialChaosBridge"},{"id":"n40098","layer":"formal","project":"p46","title":"Pphi2.two_div_degree_eq_inv_cutoffPower","kind":"theorem","summary":"∀ (P : InteractionPolynomial), Eq (HDiv.hDiv 2 ↑P.n) (HDiv.hDiv 1 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mass…","labels":[],"detail_key":"p46","name":"Pphi2.LatticeSetup.latticeSmoothInteraction.congr_simp","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40101","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm","kind":"def","summary":"(d N : Nat) → [inst : NeZero N] → Real → Real → Real → Pphi2.CanonicalJoint d N → Nat → Nat → R…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40102","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm_inner_eq_zero","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm_inner_eq_zero","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40103","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm_l2_sq_eq_covSum","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm_l2_sq_eq_covSum","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40104","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm_l2_sq_le","kind":"theorem","summary":"∀ d : Nat, Eq d 2 → ∀ (mass L : Real), LT.lt 0 L → LT.lt 0 mass → ∀ (k j : Nat), LE.le 1 (HSub.…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm_l2_sq_le","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40105","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm_pair_integrable","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm_pair_integrable","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40106","layer":"formal","project":"p46","title":"Pphi2.canonicalCrossTerm_scaled_inner_sum_l2_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalCrossTerm_scaled_inner_sum_l2_sq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40107","layer":"formal","project":"p46","title":"Pphi2.canonicalFullInteractionJoint","kind":"def","summary":"(d N : Nat) → [inst : NeZero N] → Real → Real → Real → InteractionPolynomial → Pphi2.CanonicalJ…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalFullInteractionJoint","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40108","layer":"formal","project":"p46","title":"Pphi2.canonicalFullInteractionJoint_eq_interactionFunctional","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial) (η : Pphi2.Canoni…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalFullInteractionJoint_eq_interactionFunctional","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40109","layer":"formal","project":"p46","title":"Pphi2.canonicalFullInteractionJoint_memLp","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalFullInteractionJoint_memLp","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40110","layer":"formal","project":"p46","title":"Pphi2.canonicalFullInteractionJoint_moment_le","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalFullInteractionJoint_moment_le","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40111","layer":"formal","project":"p46","title":"Pphi2.canonicalLeading_perCoeff_inner_eq_zero","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial), (∀ (j : Nat), Me…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalLeading_perCoeff_inner_eq_zero","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40112","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError","kind":"def","summary":"(d N : Nat) → [inst : NeZero N] → Real → Real → Real → InteractionPolynomial → Pphi2.CanonicalJ…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40113","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_eq_sum_diff","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : 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LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_l2_sq_eq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40116","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_l2_sq_eq_lead_plus_perCoef_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial), (∀ (j : Nat), Me…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_l2_sq_eq_lead_plus_perCoef_sq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40117","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_leading_l2_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_leading_l2_sq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40118","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_moment_le","kind":"theorem","summary":"∀ (d N : 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1…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_neg_tail_of_stdGaussian_explicit","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40121","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_neg_tail_of_stdGaussian_explicit_ae","kind":"theorem","summary":"∀ d N : Nat [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial) (m : Nat), LE.le 1…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_neg_tail_of_stdGaussian_explicit_ae","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40122","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_neg_tail_uniform_in_aN","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (mass L : Real), LT.lt 0 L → LT.lt 0 mass → Exists fun K => And (…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_neg_tail_uniform_in_aN","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40123","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_perCoef_outer_l2_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial), (∀ (m : Fin P.n)…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_perCoef_outer_l2_sq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40124","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_perCoeff_l2_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_perCoeff_l2_sq","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40125","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_pointwise_decomposition","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass T : Real) (P : InteractionPolynomial) (η : Pphi2.Canoni…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_pointwise_decomposition","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40126","layer":"formal","project":"p46","title":"Pphi2.canonicalRoughError_sq_integrable","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalRoughError_sq_integrable","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40127","layer":"formal","project":"p46","title":"Pphi2.canonicalSmoothInteraction","kind":"def","summary":"(d N : Nat) → [inst : NeZero N] → Real → Real → Real → InteractionPolynomial → 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E…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSmoothInteraction_lower_bound_log_uniform_in_aN","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40132","layer":"formal","project":"p46","title":"Pphi2.canonicalSmoothInteraction_moment_le","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (T : Real), LT.lt…","labels":[],"detail_key":"p46","name":"Pphi2.canonicalSmoothInteraction_moment_le","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40133","layer":"formal","project":"p46","title":"Pphi2.expMoment_bound_of_cutoff_degree_tail","kind":"theorem","summary":"∀ d : Nat (P : InteractionPolynomial) (mass L : Real) (hmass : LT.lt 0 mass) (C : Real), (∀ (N…","labels":[],"detail_key":"p46","name":"Pphi2.expMoment_bound_of_cutoff_degree_tail","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40134","layer":"formal","project":"p46","title":"Pphi2.expMoment_bound_of_cutoff_quartic_tail","kind":"theorem","summary":"∀ d : Nat (P : InteractionPolynomial) (mass L : Real) (hmass : LT.lt 0 mass) (C : Real), (∀ (N…","labels":[],"detail_key":"p46","name":"Pphi2.expMoment_bound_of_cutoff_quartic_tail","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40135","layer":"formal","project":"p46","title":"Pphi2.finite_indexed_wick_sum_mem_wienerChaosLE","kind":"theorem","summary":"∀ ι : Type u_1 n d : Nat (s : Finset ι) (β : ι → Fin n → Nat) (c : ι → Real), (∀ (i : ι), Membe…","labels":[],"detail_key":"p46","name":"Pphi2.finite_indexed_wick_sum_mem_wienerChaosLE","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40136","layer":"formal","project":"p46","title":"Pphi2.integrable_sq_real_sum_of_pairwise","kind":"theorem","summary":"∀ α : Type u_1 ι : Type u_2 [inst : MeasurableSpace α] μ : MeasureTheory.Measure α (s : Finset…","labels":[],"detail_key":"p46","name":"Pphi2.integrable_sq_real_sum_of_pairwise","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40137","layer":"formal","project":"p46","title":"Pphi2.integrable_sum_mul_sum_of_pairwise","kind":"theorem","summary":"∀ α : Type u_1 ιA : Type u_2 ιB : Type u_3 [inst : MeasurableSpace α] μ : MeasureTheory.Measure…","labels":[],"detail_key":"p46","name":"Pphi2.integrable_sum_mul_sum_of_pairwise","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40138","layer":"formal","project":"p46","title":"Pphi2.integral_exp_neg_interaction_sq_eq_canonicalJoint","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real) (P : InteractionPolynomial) (ha : LT.lt 0 a) (h…","labels":[],"detail_key":"p46","name":"Pphi2.integral_exp_neg_interaction_sq_eq_canonicalJoint","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40139","layer":"formal","project":"p46","title":"Pphi2.integral_sq_add_of_inner_eq_zero","kind":"theorem","summary":"∀ α : Type u_1 [inst : MeasurableSpace α] μ : MeasureTheory.Measure α (L R : α → Real), Eq (Mea…","labels":[],"detail_key":"p46","name":"Pphi2.integral_sq_add_of_inner_eq_zero","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40140","layer":"formal","project":"p46","title":"Pphi2.integral_sq_real_sum_of_pairwise_orthogonal","kind":"theorem","summary":"∀ α : Type u_1 ι : Type u_2 [inst : MeasurableSpace α] μ : MeasureTheory.Measure α (s : Finset…","labels":[],"detail_key":"p46","name":"Pphi2.integral_sq_real_sum_of_pairwise_orthogonal","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40141","layer":"formal","project":"p46","title":"Pphi2.integral_sum_mul_sum_eq_zero_of_orth","kind":"theorem","summary":"∀ α : Type u_1 ιA : Type u_2 ιB : Type u_3 [inst : MeasurableSpace α] μ : MeasureTheory.Measure…","labels":[],"detail_key":"p46","name":"Pphi2.integral_sum_mul_sum_eq_zero_of_orth","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40142","layer":"formal","project":"p46","title":"Pphi2.latticeFieldToConfig.congr_simp","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (φ φ_1 : GaussianField.FinLatticeField d N), Eq φ φ_1 → Eq (Pph…","labels":[],"detail_key":"p46","name":"Pphi2.latticeFieldToConfig.congr_simp","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40143","layer":"formal","project":"p46","title":"Pphi2.multiWickMonomial_eq_hermiteMultiEval","kind":"theorem","summary":"∀ n : Nat (α : Fin n → Nat) (ξ : Fin n → Real), Eq (Finset.univ.prod fun i => Pphi2.wickMonomia…","labels":[],"detail_key":"p46","name":"Pphi2.multiWickMonomial_eq_hermiteMultiEval","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40144","layer":"formal","project":"p46","title":"Pphi2.rough_error_variance","kind":"theorem","summary":"∀ d : Nat, Eq d 2 → ∀ (P : InteractionPolynomial) (L mass : Real), LT.lt 0 L → LT.lt 0 mass → E…","labels":[],"detail_key":"p46","name":"Pphi2.rough_error_variance","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40145","layer":"formal","project":"p46","title":"Pphi2.wickExpansionCoeff","kind":"def","summary":"ι : Type u_1 → [Fintype ι] → Nat → (ι → Real) → (ι → Nat) → Real","labels":[],"detail_key":"p46","name":"Pphi2.wickExpansionCoeff","module":"Pphi2.NelsonEstimate.RoughErrorBound"},{"id":"n40146","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_eq_root_local","kind":"theorem","summary":"∀ (n : Nat) (c x : Real), Eq (Pphi2.wickMonomial n 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(HDiv.hD…","labels":[],"detail_key":"p46","name":"Pphi2.smooth_interaction_lower_bound_volume","module":"Pphi2.NelsonEstimate.SmoothLowerBound"},{"id":"n40155","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_add_binomial","kind":"theorem","summary":"∀ (n : Nat) (c₁ c₂ x y : Real), Eq (Pphi2.wickMonomial n (HAdd.hAdd c₁ c₂) (HAdd.hAdd x y)) ((F…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_add_binomial","module":"Pphi2.NelsonEstimate.WickBinomial"},{"id":"n40156","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_add_sub_self","kind":"theorem","summary":"∀ (n : Nat) (c₁ c₂ x y : Real), Eq (HSub.hSub (Pphi2.wickMonomial n (HAdd.hAdd c₁ c₂) (HAdd.hAd…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_add_sub_self","module":"Pphi2.NelsonEstimate.WickBinomial"},{"id":"n40157","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_four_add","kind":"theorem","summary":"∀ (c₁ c₂ x y : Real), Eq (Pphi2.wickMonomial 4 (HAdd.hAdd c₁ c₂) (HAdd.hAdd x y)) (HAdd.hAdd (H…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_four_add","module":"Pphi2.NelsonEstimate.WickBinomial"},{"id":"n40158","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_four_lower_bound","kind":"theorem","summary":"∀ (c : Real), LE.le 0 c → ∀ (x : Real), LE.le (HMul.hMul (-6) (HPow.hPow c 2)) (Pphi2.wickMonom…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_four_lower_bound","module":"Pphi2.NelsonEstimate.WickBinomial"},{"id":"n40159","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_two_add","kind":"theorem","summary":"∀ (c₁ c₂ x y : Real), Eq (Pphi2.wickMonomial 2 (HAdd.hAdd c₁ c₂) (HAdd.hAdd x y)) (HAdd.hAdd (H…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_two_add","module":"Pphi2.NelsonEstimate.WickBinomial"},{"id":"n40160","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_add_sub_self","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (c₁ c₂ x y : Real), Eq (HSub.hSub 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: Nat) [inst : NeZero N], Eq (GaussianField.evalMapMeasurableEquiv d N) (GaussianField.e…","labels":[],"detail_key":"p46","name":"GaussianField.evalMapMeasurableEquiv.congr_simp","module":"Pphi2.OSProofs.OS2_WardIdentity"},{"id":"n40174","layer":"formal","project":"p46","title":"Pphi2.anomaly_bound_from_superrenormalizability","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (P : InteractionPolynomial) (mass : Real) (hmass : LT.lt 0 mass)…","labels":[],"detail_key":"p46","name":"Pphi2.anomaly_bound_from_superrenormalizability","module":"Pphi2.OSProofs.OS2_WardIdentity"},{"id":"n40175","layer":"formal","project":"p46","title":"Pphi2.anomaly_scaling_dimension","kind":"theorem","summary":"∀ (_P : InteractionPolynomial) (a mass : Real), LT.lt 0 a → LT.lt 0 mass → ∀ (k : Fin 2 → Real)…","labels":[],"detail_key":"p46","name":"Pphi2.anomaly_scaling_dimension","module":"Pphi2.OSProofs.OS2_WardIdentity"},{"id":"n40176","layer":"formal","project":"p46","title":"Pphi2.anomaly_vanishes","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (P : InteractionPolynomial) (mass : Real) (hmass : LT.lt 0 mass)…","labels":[],"detail_key":"p46","name":"Pphi2.anomaly_vanishes","module":"Pphi2.OSProofs.OS2_WardIdentity"},{"id":"n40177","layer":"formal","project":"p46","title":"Pphi2.continuumLimit_satisfies_fullOS","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (mass : Real), LT.lt 0 mass → ∀ (μ : MeasureTheory.Measure (Gauss…","labels":[],"detail_key":"p46","name":"Pphi2.continuumLimit_satisfies_fullOS","module":"Pphi2.OSProofs.OS2_WardIdentity"},{"id":"n40178","layer":"formal","project":"p46","title":"Pphi2.generatingFunctional.congr_simp","kind":"theorem","summary":"∀ (μ μ_1 : MeasureTheory.Measure Pphi2.FieldConfig2) (e_μ : Eq μ μ_1) [inst : 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(h…","labels":[],"detail_key":"p46","name":"Pphi2.spectral_gap_pos","module":"Pphi2.TransferMatrix.SpectralGap"},{"id":"n40424","layer":"formal","project":"p46","title":"Pphi2.SpatialField","kind":"def","summary":"Nat → Type","labels":[],"detail_key":"p46","name":"Pphi2.SpatialField","module":"Pphi2.TransferMatrix.TransferMatrix"},{"id":"n40425","layer":"formal","project":"p46","title":"Pphi2.spatialAction","kind":"def","summary":"(Ns : Nat) → [NeZero Ns] → InteractionPolynomial → Real → Real → Real → Pphi2.SpatialField Ns →…","labels":[],"detail_key":"p46","name":"Pphi2.spatialAction","module":"Pphi2.TransferMatrix.TransferMatrix"},{"id":"n40426","layer":"formal","project":"p46","title":"Pphi2.spatialAction.congr_simp","kind":"theorem","summary":"∀ (Ns : Nat) [inst : NeZero Ns] (P P_1 : InteractionPolynomial), Eq P P_1 → ∀ (a a_1 : Real), 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Pphi2.…","labels":[],"detail_key":"p46","name":"Pphi2.transferKernel","module":"Pphi2.TransferMatrix.TransferMatrix"},{"id":"n40433","layer":"formal","project":"p46","title":"Pphi2.transferKernel_pos","kind":"theorem","summary":"∀ (Ns : Nat) [inst : NeZero Ns] (P : InteractionPolynomial) (a mass : Real) (ψ ψ' : Pphi2.Spati…","labels":[],"detail_key":"p46","name":"Pphi2.transferKernel_pos","module":"Pphi2.TransferMatrix.TransferMatrix"},{"id":"n40434","layer":"formal","project":"p46","title":"Pphi2.transferKernel_symmetric","kind":"theorem","summary":"∀ (Ns : Nat) [inst : NeZero Ns] (P : InteractionPolynomial) (a mass : Real) (ψ ψ' : Pphi2.Spati…","labels":[],"detail_key":"p46","name":"Pphi2.transferKernel_symmetric","module":"Pphi2.TransferMatrix.TransferMatrix"},{"id":"n40435","layer":"formal","project":"p46","title":"Pphi2.wickConstant","kind":"def","summary":"Nat → (N : Nat) → [NeZero N] → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40436","layer":"formal","project":"p46","title":"Pphi2.wickConstant_antitone_mass","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a : Real), LT.lt 0 a → ∀ (m₁ m₂ : Real), LT.lt 0 m₁ → LE.le m₁…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_antitone_mass","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40437","layer":"formal","project":"p46","title":"Pphi2.wickConstant_eq_latticeEigenvalue1d_family_average","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), Eq (Pphi2.wickConstant d N a mass) (HMul.hMul…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_eq_latticeEigenvalue1d_family_average","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40438","layer":"formal","project":"p46","title":"Pphi2.wickConstant_le_inv_mass_sq","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → LE.le (Pphi2.wickCo…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_le_inv_mass_sq","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40439","layer":"formal","project":"p46","title":"Pphi2.wickConstant_pos","kind":"theorem","summary":"∀ (d N : Nat) [inst : NeZero N] (a mass : Real), LT.lt 0 a → LT.lt 0 mass → LT.lt 0 (Pphi2.wick…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_pos","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40440","layer":"formal","project":"p46","title":"Pphi2.wickConstant_two_eq_dft_eigenvalue_average","kind":"theorem","summary":"∀ (N : Nat) [inst : NeZero N] (a mass : Real), Eq (Pphi2.wickConstant 2 N a mass) (HMul.hMul (I…","labels":[],"detail_key":"p46","name":"Pphi2.wickConstant_two_eq_dft_eigenvalue_average","module":"Pphi2.WickOrdering.Counterterm"},{"id":"n40441","layer":"formal","project":"p46","title":"Pphi2.poly_even_degree_bounded_below","kind":"theorem","summary":"∀ (p : Polynomial Real), LT.lt 0 p.natDegree → Even p.natDegree → LT.lt 0 p.leadingCoeff → Exis…","labels":[],"detail_key":"p46","name":"Pphi2.poly_even_degree_bounded_below","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40442","layer":"formal","project":"p46","title":"Pphi2.wickMonomial","kind":"def","summary":"Nat → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40443","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_continuous₂","kind":"theorem","summary":"∀ (n : Nat), Continuous fun p => Pphi2.wickMonomial n p.1 p.2","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_continuous₂","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40444","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_eq_hermite","kind":"theorem","summary":"∀ (n : Nat) (c : Real), LT.lt 0 c → ∀ (x : Real), Eq (Pphi2.wickMonomial n c x) (HMul.hMul (HPo…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_eq_hermite","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40445","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_four","kind":"theorem","summary":"∀ (c x : Real), Eq (Pphi2.wickMonomial 4 c x) (HAdd.hAdd (HSub.hSub (HPow.hPow x 4) (HMul.hMul…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_four","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40446","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_one","kind":"theorem","summary":"∀ (c x : Real), Eq (Pphi2.wickMonomial 1 c x) x","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_one","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40447","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_succ_succ","kind":"theorem","summary":"∀ (n : Nat) (c x : Real), Eq (Pphi2.wickMonomial (HAdd.hAdd n 2) c x) (HSub.hSub (HMul.hMul x (…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_succ_succ","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40448","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_three","kind":"theorem","summary":"∀ (c x : Real), Eq (Pphi2.wickMonomial 3 c x) (HSub.hSub (HPow.hPow x 3) (HMul.hMul (HMul.hMul…","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_three","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40449","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_two","kind":"theorem","summary":"∀ (c x : Real), Eq (Pphi2.wickMonomial 2 c x) (HSub.hSub (HPow.hPow x 2) c)","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_two","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40450","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_zero","kind":"theorem","summary":"∀ (c x : Real), Eq (Pphi2.wickMonomial 0 c x) 1","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_zero","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40451","layer":"formal","project":"p46","title":"Pphi2.wickMonomial_zero_variance","kind":"theorem","summary":"∀ (n : Nat) (x : Real), Eq (Pphi2.wickMonomial n 0 x) (HPow.hPow x n)","labels":[],"detail_key":"p46","name":"Pphi2.wickMonomial_zero_variance","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40452","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial","kind":"def","summary":"InteractionPolynomial → Real → Real → Real","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40453","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_bounded_below","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (c : Real), Exists fun A => And (LT.lt 0 A) (∀ (x : Real), GE.ge…","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial_bounded_below","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40454","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_continuous₂","kind":"theorem","summary":"∀ (P : InteractionPolynomial), Continuous fun p => Pphi2.wickPolynomial P p.1 p.2","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial_continuous₂","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40455","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_lower_bound_general","kind":"theorem","summary":"∀ (P : InteractionPolynomial), Exists fun A => And (LE.le 0 A) (∀ (c x : Real), LE.le 0 c → GE.…","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial_lower_bound_general","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40456","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_uniform_bounded_below","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (C : Real), LE.le 0 C → Exists fun A => And (LT.lt 0 A) (∀ (c : R…","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial_uniform_bounded_below","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40457","layer":"formal","project":"p46","title":"Pphi2.wickPolynomial_zero_variance","kind":"theorem","summary":"∀ (P : InteractionPolynomial) (x : Real), Eq (Pphi2.wickPolynomial P 0 x) (P.eval x)","labels":[],"detail_key":"p46","name":"Pphi2.wickPolynomial_zero_variance","module":"Pphi2.WickOrdering.WickPolynomial"},{"id":"n40458","layer":"informal","project":"p47","title":"theorem-1","kind":"theorem","summary":"Let X be a set with at most \\fracq^n+1-qq-1 elements. If f_1, \\ldots, f_k \\in Map(X,F_q) for so…","labels":["theorem-1"],"detail_key":"p47"},{"id":"n40459","layer":"informal","project":"p47","title":"prop-2","kind":"proposition","summary":"For every field F_q and every positive integer n there are a set X_n of cardinality \\fracq^n+1-…","labels":["prop-2"],"detail_key":"p47"},{"id":"n40460","layer":"informal","project":"p47","title":"corollary-3","kind":"corollary","summary":"Let n > 0 and let \\phi \\colon F \\to Map(A^n(F_q),F_q) be a homomorphism of vector spaces over F…","labels":["corollary-3"],"detail_key":"p47"},{"id":"n40461","layer":"informal","project":"p47","title":"corollary-4","kind":"corollary","summary":"Let n \\ge 0 and let \\phi \\colon F \\to Map(P^n(F_q),F_q) be a homomorphism of vector spaces over…","labels":["corollary-4"],"detail_key":"p47"},{"id":"n40462","layer":"informal","project":"p47","title":"lemma-5","kind":"lemma","summary":"The map \\[ c M_n \\to P^n(K)\\\\ M \\mapsto (N(M)\\setminus\\left\\lbrace \\theta\\right\\rbrace)_\\sim \\]…","labels":["lemma-5"],"detail_key":"p47"},{"id":"n40463","layer":"informal","project":"p47","title":"Denote by N_n the set of all matrices in M_n, n+1(K) having the rank equal to n. For ever…","kind":"proof","summary":"Denote by N_n the set of all matrices in M_n, n+1(K) having the rank equal to n. For every M \\i…","labels":[],"detail_key":"p47"},{"id":"n40464","layer":"informal","project":"p47","title":"Proof of Theorem \\reftheorem-1","kind":"proof","summary":"[Proof of Theorem \\reftheorem-1] It is enough to prove the statement for k=n+1 since we may app…","labels":[],"detail_key":"p47"},{"id":"n40465","layer":"informal","project":"p47","title":"lemma-6","kind":"lemma","summary":"Let K be an arbitrary field. For any matrix A \\in M_n, m(K) where n\\le m there exist a matrix M…","labels":["lemma-6"],"detail_key":"p47"},{"id":"n40466","layer":"informal","project":"p47","title":"Denote by I_r, k, l the matrix in M_k, l(K) having x_11=\\ldots=x_rr=1 and all remaining e…","kind":"proof","summary":"Denote by I_r, k, l the matrix in M_k, l(K) having x_11=\\ldots=x_rr=1 and all remaining entries…","labels":[],"detail_key":"p47"},{"id":"n40467","layer":"informal","project":"p47","title":"Proof of Proposition \\refprop-2","kind":"proof","summary":"[Proof of Proposition \\refprop-2] For every point P \\in P^n(F_q) choose a set of homogeneous co…","labels":[],"detail_key":"p47"},{"id":"n40468","layer":"informal","project":"p47","title":"Proof of Corollary \\refcorollary-3","kind":"proof","summary":"[Proof of Corollary \\refcorollary-3] For any positive integer n we have \\fracq^n+1-qq-1\\ge q^n=…","labels":[],"detail_key":"p47"},{"id":"n40469","layer":"informal","project":"p47","title":"remark-7","kind":"remark","summary":"It has been suggested by the reviewer of this paper to include the following example to demonst…","labels":["remark-7"],"detail_key":"p47"},{"id":"n40470","layer":"informal","project":"p47","title":"Proof of Corollary \\refcorollary-4","kind":"proof","summary":"[Proof of Corollary \\refcorollary-4] Let \\left\\lbrace f_1, \\ldots, f_k\\right\\rbrace be the imag…","labels":[],"detail_key":"p47"},{"id":"n40471","layer":"formal","project":"p47","title":"corollary_2","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [Fintype F] (n : Nat), GT.gt n 0 → ∀ (V : Type u_7) [inst_2 : A…","labels":[],"detail_key":"p47","name":"corollary_2","module":"fineqs.Main"},{"id":"n40472","layer":"formal","project":"p47","title":"corollary_3","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [Fintype F] (n : Nat) (V : Type u_7) [inst_2 : AddCommGroup V]…","labels":[],"detail_key":"p47","name":"corollary_3","module":"fineqs.Main"},{"id":"n40473","layer":"formal","project":"p47","title":"matrix_rank_lemma","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [Fintype F] (n m : Nat), LE.le n m → ∀ (A : Matrix (Fin n) (Fin…","labels":[],"detail_key":"p47","name":"matrix_rank_lemma","module":"fineqs.Main"},{"id":"n40474","layer":"formal","project":"p47","title":"prop_1","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [inst_1 : Fintype F] (n : Nat), GT.gt n 0 → Exists fun X => Exi…","labels":[],"detail_key":"p47","name":"prop_1","module":"fineqs.Main"},{"id":"n40475","layer":"formal","project":"p47","title":"remark_example","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [inst_1 : Fintype F] (n : Nat), GT.gt n 0 → have f := fun i x =…","labels":[],"detail_key":"p47","name":"remark_example","module":"fineqs.Main"},{"id":"n40476","layer":"formal","project":"p47","title":"theorem_1","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [inst_1 : Fintype F] X : Type u_6 [Finite X] [Finite X] (n : Na…","labels":[],"detail_key":"p47","name":"theorem_1","module":"fineqs.Main"},{"id":"n40477","layer":"formal","project":"p47","title":"theorem_1_aux","kind":"theorem","summary":"∀ F : Type u_5 [inst : Field F] [inst_1 : Fintype F] X : Type u_6 [Finite X] (n : Nat) (f : X →…","labels":[],"detail_key":"p47","name":"theorem_1_aux","module":"fineqs.Main"},{"id":"n40478","layer":"informal","project":"p48","title":"Variation of a Vector-Valued Measure","kind":"definition","summary":"[Variation of a Vector-Valued Measure] Let (X, A) be a measurable space and let Y be a Banach s…","labels":["def:variation"],"detail_key":"p48"},{"id":"n40479","layer":"informal","project":"p48","title":"Rudin 6.12 (polar representation of a complex measure)","kind":"theorem","summary":"[Rudin 6.12 (polar representation of a complex measure)] Let \\mu be a complex measure on a \\sig…","labels":["thm:polar_rep"],"detail_key":"p48"},{"id":"n40480","layer":"informal","project":"p48","title":"This rather depends on how the integral with respect to a complex measure is defined. See…","kind":"proof","summary":"This rather depends on how the integral with respect to a complex measure is defined. See [Rudi…","labels":[],"detail_key":"p48"},{"id":"n40481","layer":"informal","project":"p48","title":"def:riesz_measure","kind":"definition","summary":"\\notready Let X be a locally compact Hausdorff space. Associated to every bounded linear functi…","labels":["def:riesz_measure"],"detail_key":"p48"},{"id":"n40482","layer":"informal","project":"p48","title":"Rudin 3.14","kind":"theorem","summary":"[Rudin 3.14] For 1 \\leq p < \\infty, C_c(X) is dense in L^p(\\mu).","labels":["thm:Cc_dense_Lp"],"detail_key":"p48"},{"id":"n40483","layer":"informal","project":"p48","title":"Define S as in Theorem 3.13. If s \\in S and \\varepsilon > 0, there exists a g \\in C_c(X)…","kind":"proof","summary":"Define S as in Theorem 3.13. If s \\in S and \\varepsilon > 0, there exists a g \\in C_c(X) such t…","labels":[],"detail_key":"p48"},{"id":"n40484","layer":"informal","project":"p48","title":"Rudin 6.13","kind":"theorem","summary":"[Rudin 6.13] Suppose \\mu is a positive measure on \\mathfrakM, g \\in L^1(\\mu), and \\lambda(E) =…","labels":["thm:abs_integral"],"detail_key":"p48"},{"id":"n40485","layer":"informal","project":"p48","title":"By Theorem \\refthm:polar_rep, there is a function h, of absolute value 1, such that d\\lam…","kind":"proof","summary":"By Theorem \\refthm:polar_rep, there is a function h, of absolute value 1, such that d\\lambda =…","labels":[],"detail_key":"p48"},{"id":"n40486","layer":"informal","project":"p48","title":"Rudin 6.16","kind":"theorem","summary":"[Rudin 6.16] Suppose 1 \\leq p < \\infty, \\mu is a \\sigma-finite positive measure on X, and \\Phi…","labels":["thm:Lp_duality"],"detail_key":"p48"},{"id":"n40487","layer":"informal","project":"p48","title":"Rudin 6.16: Duality of L^1 and L^∞ (not in Mathlib \\urlhttps://leanprover.zulipchat.com/#…","kind":"proof","summary":"Rudin 6.16: Duality of L^1 and L^∞ (not in Mathlib \\urlhttps://leanprover.zulipchat.com/#narrow…","labels":[],"detail_key":"p48"},{"id":"n40488","layer":"informal","project":"p48","title":"lem:rieszMeasure_unique","kind":"lemma","summary":"Let X be a locally compact Hausdorff space, and let \\Phi be a bounded linear functional on C_0(…","labels":["lem:rieszMeasure_unique"],"detail_key":"p48"},{"id":"n40489","layer":"informal","project":"p48","title":"Suppose \\mu is a regular complex Borel measure on X and \\int f \\, d\\mu = 0 for all f \\in…","kind":"proof","summary":"Suppose \\mu is a regular complex Borel measure on X and \\int f \\, d\\mu = 0 for all f \\in C_0(X)…","labels":[],"detail_key":"p48"},{"id":"n40490","layer":"informal","project":"p48","title":"lem:exists_pos_lin_func","kind":"lemma","summary":"Consider a given bounded linear functional \\Phi on C_0(X). Assume \\|\\Phi\\| = 1. (Update stateme…","labels":["lem:exists_pos_lin_func"],"detail_key":"p48"},{"id":"n40491","layer":"informal","project":"p48","title":"Assume \\|\\Phi\\| = 1, without loss of generality. So all depends on finding a positive lin…","kind":"proof","summary":"Assume \\|\\Phi\\| = 1, without loss of generality. So all depends on finding a positive linear fu…","labels":[],"detail_key":"p48"},{"id":"n40492","layer":"informal","project":"p48","title":"Rudin 6.19","kind":"theorem","summary":"[Rudin 6.19] If X is a locally compact Hausdorff space, then every bounded linear functional \\P…","labels":["integral_rieszMeasure"],"detail_key":"p48"},{"id":"n40493","layer":"informal","project":"p48","title":"Once we have the \\Lambda from Lemma~\\reflem:exists_pos_lin_func, we associate with it a p…","kind":"proof","summary":"Once we have the \\Lambda from Lemma~\\reflem:exists_pos_lin_func, we associate with it a positiv…","labels":[],"detail_key":"p48"},{"id":"n40494","layer":"informal","project":"p48","title":"Rudin 6.19","kind":"lemma","summary":"[Rudin 6.19] Moreover, the norm of \\Phi is the total variation of \\mu: \\|\\Phi\\| = |\\mu|(X). \\ta…","labels":["lem:norm_eq_variation"],"detail_key":"p48"},{"id":"n40495","layer":"informal","project":"p48","title":"Since \\|\\Phi\\| = 1, (6) shows that \\int_X |g| \\, d\\lambda \\geq \\sup \\|\\Phi(f)| : f \\in C_…","kind":"proof","summary":"Since \\|\\Phi\\| = 1, (6) shows that \\int_X |g| \\, d\\lambda \\geq \\sup \\|\\Phi(f)| : f \\in C_0(X),…","labels":[],"detail_key":"p48"},{"id":"n40496","layer":"informal","project":"p48","title":"thm:ComplexRMK","kind":"theorem","summary":"Placeholder to combine the three results which make up The Reisz Theorem.","labels":["thm:ComplexRMK"],"detail_key":"p48"},{"id":"n40497","layer":"informal","project":"p48","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p48"},{"id":"n40498","layer":"informal","project":"p48","title":"lem:projection_projection","kind":"lemma","summary":"It holds that p(K) = p(K)^2 = p(K)^*.","labels":["lem:projection_projection"],"detail_key":"p48"},{"id":"n40499","layer":"informal","project":"p48","title":"The first equality follows by the uniqueness of the orthogonal decomposition. The second…","kind":"proof","summary":"The first equality follows by the uniqueness of the orthogonal decomposition. The second equali…","labels":[],"detail_key":"p48"},{"id":"n40500","layer":"informal","project":"p48","title":"lem:exists_subspace","kind":"lemma","summary":"For p \\in B(H) such that p = p^2 = p^*, there is a closed subspace K such that p = p(K).","labels":["lem:exists_subspace"],"detail_key":"p48"},{"id":"n40501","layer":"informal","project":"p48","title":"By p = p^2, it is a projection. Let K be the image of p. Note that x = (p + (1-p))x = px…","kind":"proof","summary":"By p = p^2, it is a projection. Let K be the image of p. Note that x = (p + (1-p))x = px + (1-p…","labels":[],"detail_key":"p48"},{"id":"n40502","layer":"informal","project":"p48","title":"Rudin 12.6, part 1","kind":"lemma","summary":"[Rudin 12.6, part 1] Let \\x_n\\ be a sequence of pairwise orthogonal vectors in H. 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\\leanfilePhysicslib4/Spacetime/Causality.lean…","labels":["thrm:causal-order-refinements"],"detail_key":"p49"},{"id":"n40598","layer":"informal","project":"p49","title":"For irreflexivity of chronological precedence, p \\ll p would give p \\prec p because chron…","kind":"proof","summary":"For irreflexivity of chronological precedence, p \\ll p would give p \\prec p because chronologic…","labels":[],"detail_key":"p49"},{"id":"n40599","layer":"informal","project":"p49","title":"Chronological Future and Chronological Past","kind":"definition","summary":"[Chronological Future and Chronological Past] \\leanfilePhysicslib4/Spacetime/Causality.lean For…","labels":["def:chronological-future-and-chronological-past"],"detail_key":"p49"},{"id":"n40600","layer":"informal","project":"p49","title":"Causal Future and Causal Past","kind":"definition","summary":"[Causal Future and Causal Past] \\leanfilePhysicslib4/Spacetime/Causality.lean For a spacetime M…","labels":["def:causal-future-and-causal-past"],"detail_key":"p49"},{"id":"n40601","layer":"informal","project":"p49","title":"Chronological Precedence Implies Causal Precedence","kind":"lemma","summary":"[Chronological Precedence Implies Causal Precedence] \\leanfilePhysicslib4/Spacetime/Causality.l…","labels":["lmm:chronological-implies-causal"],"detail_key":"p49"},{"id":"n40602","layer":"informal","project":"p49","title":"A timelike tangent vector is in particular causal (timelike or null), so a future-oriente…","kind":"proof","summary":"A timelike tangent vector is in particular causal (timelike or null), so a future-oriented time…","labels":[],"detail_key":"p49"},{"id":"n40603","layer":"informal","project":"p49","title":"Monotonicity of Futures and Pasts","kind":"lemma","summary":"[Monotonicity of Futures and Pasts] \\leanfilePhysicslib4/Spacetime/Causality.lean The 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\\leanfilePhysicslib4/Spacetime/Causali…","labels":["lmm:completely-spacelike-structural"],"detail_key":"p49"},{"id":"n40615","layer":"informal","project":"p49","title":"Each follows directly from the pointwise definition: monotonicity by restricting the univ…","kind":"proof","summary":"Each follows directly from the pointwise definition: monotonicity by restricting the universall…","labels":[],"detail_key":"p49"},{"id":"n40616","layer":"informal","project":"p49","title":"Spacelike Complement of a Region","kind":"definition","summary":"[Spacelike Complement of a Region] \\leanfilePhysicslib4/Spacetime/CausalComplement.lean The \\em…","labels":["def:spacelike-complement"],"detail_key":"p49"},{"id":"n40617","layer":"informal","project":"p49","title":"Order Structure of the Spacelike Complement","kind":"lemma","summary":"[Order Structure of the Spacelike Complement] \\leanfilePhysicslib4/Spacetime/CausalComplement.l…","labels":["lmm:spacelike-complement-order"],"detail_key":"p49"},{"id":"n40618","layer":"informal","project":"p49","title":"Antitonicity and the Galois bridge are the pointwise monotonicity of complete spacelike s…","kind":"proof","summary":"Antitonicity and the Galois bridge are the pointwise monotonicity of complete spacelike separat…","labels":[],"detail_key":"p49"},{"id":"n40619","layer":"informal","project":"p49","title":"Causal closure operator","kind":"definition","summary":"[Causal closure operator] \\leanfilePhysicslib4/Spacetime/CausalComplement.lean The \\emphcausal…","labels":["def:causal-closure"],"detail_key":"p49"},{"id":"n40620","layer":"informal","project":"p49","title":"The Causal Closure is a Closure Operator","kind":"lemma","summary":"[The Causal Closure is a Closure Operator] The causal closure B \\mapsto B^\\perp\\perp is a \\emph…","labels":["lmm:causal-closure-is-closure-operator"],"detail_key":"p49"},{"id":"n40621","layer":"informal","project":"p49","title":"All thr","kind":"proof","summary":"All thr","labels":[],"detail_key":"p49"},{"id":"n40622","layer":"informal","project":"p49","title":"Causally complete region","kind":"definition","summary":"[Causally complete region] \\leanfilePhysicslib4/Spacetime/CausalComplement.lean A region is \\em…","labels":["def:causally-complete-region"],"detail_key":"p49"},{"id":"n40623","layer":"informal","project":"p49","title":"Lattice of causally complete regions","kind":"theorem","summary":"[Lattice of causally complete regions] \\leanfilePhysicslib4/Spacetime/CausalComplement.lean The…","labels":["thrm:causally-complete-lattice"],"detail_key":"p49"},{"id":"n40624","layer":"informal","project":"p49","title":"The lat","kind":"proof","summary":"The lat","labels":[],"detail_key":"p49"},{"id":"n40625","layer":"informal","project":"p49","title":"De Morgan Laws for the Spacelike Complement","kind":"lemma","summary":"[De Morgan Laws for the Spacelike Complement] \\leanfilePhysicslib4/Spacetime/CausalComplement.l…","labels":["lmm:spacelike-complement-de-morgan"],"detail_key":"p49"},{"id":"n40626","layer":"informal","project":"p49","title":"Unfold","kind":"proof","summary":"Unfold","labels":[],"detail_key":"p49"},{"id":"n40627","layer":"informal","project":"p49","title":"De Morgan Laws for the Causal Complement","kind":"theorem","summary":"[De Morgan Laws for the Causal Complement] \\leanfilePhysicslib4/Spacetime/CausalComplement.lean…","labels":["thrm:causal-complement-de-morgan"],"detail_key":"p49"},{"id":"n40628","layer":"informal","project":"p49","title":"Recall from the lattice structure (\\refthrm:causally-complete-lattice) that on causally c…","kind":"proof","summary":"Recall from the lattice structure (\\refthrm:causally-complete-lattice) that on causally 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regions are causally convex] Let M be a Lorentzian spacetime. \\item Every sp…","labels":["thrm:causally-complete-convex"],"detail_key":"p49"},{"id":"n40633","layer":"informal","project":"p49","title":"For (i), suppose p, r \\in B^\\perp and p \\prec q \\prec r; we must show q \\in B^\\perp, i.e.…","kind":"proof","summary":"For (i), suppose p, r \\in B^\\perp and p \\prec q \\prec r; we must show q \\in B^\\perp, i.e. that…","labels":[],"detail_key":"p49"},{"id":"n40634","layer":"informal","project":"p49","title":"Causally convex regions form a closure system","kind":"lemma","summary":"[Causally convex regions form a closure system] Let M be a spacetime with time orientation t. T…","labels":["lmm:causally-convex-closure-ops"],"detail_key":"p49"},{"id":"n40635","layer":"informal","project":"p49","title":"Each cl","kind":"proof","summary":"Each cl","labels":[],"detail_key":"p49"},{"id":"n40636","layer":"informal","project":"p49","title":"Causal-convex hull","kind":"definition","summary":"[Causal-convex hull] Let M be a spacetime with time orientation t and let B \\subseteq M be an a…","labels":["def:causal-convex-hull"],"detail_key":"p49"},{"id":"n40637","layer":"informal","project":"p49","title":"The Causal-Convex Hull is Extensive and Causally Convex","kind":"lemma","summary":"[The Causal-Convex Hull is Extensive and Causally Convex] Let M be a spacetime with time orient…","labels":["lmm:causal-convex-hull-extensive"],"detail_key":"p49"},{"id":"n40638","layer":"informal","project":"p49","title":"For (i)","kind":"proof","summary":"For (i)","labels":[],"detail_key":"p49"},{"id":"n40639","layer":"informal","project":"p49","title":"The causal-convex hull is a closure operator","kind":"theorem","summary":"[The causal-convex hull is a closure operator] Let M be a spacetime with time orientation t. Th…","labels":["thrm:causal-convex-hull-closure"],"detail_key":"p49"},{"id":"n40640","layer":"informal","project":"p49","title":"(i) If B \\subseteq C and C is causally convex, then C is a member of the intersected fami…","kind":"proof","summary":"(i) If B \\subseteq C and C is causally convex, then C is a member of the intersected family, so…","labels":[],"detail_key":"p49"},{"id":"n40641","layer":"informal","project":"p49","title":"Alexandrov Topology","kind":"definition","summary":"[Alexandrov Topology] \\leanfilePhysicslib4/Spacetime/Minkowski.lean Alexandrov topology on a sp…","labels":["def:alexandrov-topology"],"detail_key":"p49"},{"id":"n40642","layer":"informal","project":"p49","title":"Basis Sets Are Alexandrov-Open","kind":"lemma","summary":"[Basis Sets Are Alexandrov-Open] \\leanfilePhysicslib4/Spacetime/Causality.lean Every basis set…","labels":["lmm:alexandrov-basis-open"],"detail_key":"p49"},{"id":"n40643","layer":"informal","project":"p49","title":"By construction the Alexandrov topology is generated by these basis sets, and any generat…","kind":"proof","summary":"By construction the Alexandrov topology is generated by these basis sets, and any generating se…","labels":[],"detail_key":"p49"},{"id":"n40644","layer":"informal","project":"p49","title":"Openness of Chronological Futures and Pasts","kind":"lemma","summary":"[Openness of Chronological Futures and Pasts] \\leanfilePhysicslib4/Spacetime/Causality.lean Let…","labels":["lmm:chronological-future-past-open"],"detail_key":"p49"},{"id":"n40645","layer":"informal","project":"p49","title":"The first item is the basis lemma (\\reflmm:alexandrov-basis-open) directly: I^+(p) \\cap I…","kind":"proof","summary":"The first item is the basis lemma (\\reflmm:alexandrov-basis-open) directly: I^+(p) \\cap I^-(q)…","labels":[],"detail_key":"p49"},{"id":"n40646","layer":"informal","project":"p49","title":"Unconditional Openness of Chronological Futures and Pasts on Standard 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\\leanfilePhysicslib4/Spacetime/C…","labels":["lmm:alexandrov-nbhd-univ-of-no-diamond"],"detail_key":"p49"},{"id":"n40653","layer":"informal","project":"p49","title":"The Ale","kind":"proof","summary":"The Ale","labels":[],"detail_key":"p49"},{"id":"n40654","layer":"informal","project":"p49","title":"Covering from the Hausdorff Assumption","kind":"lemma","summary":"[Covering from the Hausdorff Assumption] On a Lorentzian spacetime with at least two points, th…","labels":["lmm:alexandrov-covering-hausdorff"],"detail_key":"p49"},{"id":"n40655","layer":"informal","project":"p49","title":"Suppose","kind":"proof","summary":"Suppose","labels":[],"detail_key":"p49"},{"id":"n40656","layer":"informal","project":"p49","title":"The Alexandrov Diamonds Form a Topological Basis","kind":"theorem","summary":"[The Alexandrov Diamonds Form a Topological Basis] \\leanfilePhysicslib4/Spacetime/LorentzianSpa…","labels":["thrm:alexandrov-topological-basis"],"detail_key":"p49"},{"id":"n40657","layer":"informal","project":"p49","title":"The thr","kind":"proof","summary":"The thr","labels":[],"detail_key":"p49"},{"id":"n40658","layer":"informal","project":"p49","title":"Past Interpolation on Standard Minkowski","kind":"lemma","summary":"[Past Interpolation on Standard Minkowski] \\leanfilePhysicslib4/Spacetime/MinkowskiDirected.lea…","labels":["lmm:minkowski-past-between"],"detail_key":"p49"},{"id":"n40659","layer":"informal","project":"p49","title":"Take a","kind":"proof","summary":"Take a","labels":[],"detail_key":"p49"},{"id":"n40660","layer":"informal","project":"p49","title":"Future Interpolation on Standard Minkowski","kind":"lemma","summary":"[Future Interpolation on Standard Minkowski] \\leanfilePhysicslib4/Spacetime/MinkowskiDirected.l…","labels":["lmm:minkowski-future-between"],"detail_key":"p49"},{"id":"n40661","layer":"informal","project":"p49","title":"Dual to","kind":"proof","summary":"Dual to","labels":[],"detail_key":"p49"},{"id":"n40662","layer":"informal","project":"p49","title":"Standard Minkowski Diamonds Are Downward-Directed","kind":"lemma","summary":"[Standard Minkowski Diamonds Are Downward-Directed] \\leanfilePhysicslib4/Spacetime/MinkowskiDir…","labels":["lmm:minkowski-diamonds-downward-directed"],"detail_key":"p49"},{"id":"n40663","layer":"informal","project":"p49","title":"From x","kind":"proof","summary":"From x","labels":[],"detail_key":"p49"},{"id":"n40664","layer":"informal","project":"p49","title":"Common Chronological Predecessor on Standard Minkowski","kind":"lemma","summary":"[Common Chronological Predecessor on Standard Minkowski] On standard Minkowski spacetime, for a…","labels":["lmm:minkowski-exists-common-past"],"detail_key":"p49"},{"id":"n40665","layer":"informal","project":"p49","title":"Take p","kind":"proof","summary":"Take p","labels":[],"detail_key":"p49"},{"id":"n40666","layer":"informal","project":"p49","title":"Common Chronological Successor on Standard Minkowski","kind":"lemma","summary":"[Common Chronological Successor on Standard Minkowski] On standard Minkowski spacetime, for any…","labels":["lmm:minkowski-exists-common-future"],"detail_key":"p49"},{"id":"n40667","layer":"informal","project":"p49","title":"Dual to","kind":"proof","summary":"Dual to","labels":[],"detail_key":"p49"},{"id":"n40668","layer":"informal","project":"p49","title":"Standard Minkowski Diamonds Are Upward-Directed","kind":"lemma","summary":"[Standard Minkowski Diamonds Are Upward-Directed] On standard Minkowski spacetime the Alexandro…","labels":["lmm:minkowski-diamonds-upward-directed"],"detail_key":"p49"},{"id":"n40669","layer":"informal","project":"p49","title":"Write B_1 = I^+(p_1) \\cap I^-(q_1) and B_2 = I^+(p_2) \\cap I^-(q_2). Apply \\reflmm:minkow…","kind":"proof","summary":"Write B_1 = I^+(p_1) \\cap I^-(q_1) and B_2 = I^+(p_2) \\cap I^-(q_2). Apply \\reflmm:minkowski-ex…","labels":[],"detail_key":"p49"},{"id":"n40670","layer":"informal","project":"p49","title":"The Alexandrov Diamonds are a Basis on Standard Minkowski","kind":"theorem","summary":"[The Alexandrov Diamonds are a Basis on Standard Minkowski] \\leanfilePhysicslib4/Spacetime/Mink…","labels":["thrm:minkowski-alexandrov-basis"],"detail_key":"p49"},{"id":"n40671","layer":"informal","project":"p49","title":"Apply \\","kind":"proof","summary":"Apply \\","labels":[],"detail_key":"p49"},{"id":"n40672","layer":"informal","project":"p49","title":"Dilations Preserve the Minkowski Cones","kind":"lemma","summary":"[Dilations Preserve the Minkowski Cones] For \\lambda > 0, the dilation x \\mapsto \\lambda x pres…","labels":["lmm:minkowski-dilation-cone"],"detail_key":"p49"},{"id":"n40673","layer":"informal","project":"p49","title":"Scaling multiplies the defining quadratic form by \\lambda^2 > 0 and the time-order differ…","kind":"proof","summary":"Scaling multiplies the 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\\leanfilePhysicslib4/Spacetime/Curves.lean The parameter sp…","labels":["lmm:path-parameter-unique-diff"],"detail_key":"p49"},{"id":"n40681","layer":"informal","project":"p49","title":"A connected subset of R is convex, and a closed convex set with at least two points conta…","kind":"proof","summary":"A connected subset of R is convex, and a closed convex set with at least two points contains a…","labels":[],"detail_key":"p49"},{"id":"n40682","layer":"informal","project":"p49","title":"Pushforward of a Path Under an Isometry","kind":"lemma","summary":"[Pushforward of a Path Under an Isometry] \\leanfilePhysicslib4/Spacetime/IsometryCausality.lean…","labels":["lmm:pushforward-path"],"detail_key":"p49"},{"id":"n40683","layer":"informal","project":"p49","title":"Smoothness and non-vanishing of the derivative of \\varphi \\circ \\mu follow from the chain…","kind":"proof","summary":"Smoothness and non-vanishing of the derivative of \\varphi \\circ \\mu follow from the chain rule…","labels":[],"detail_key":"p49"},{"id":"n40684","layer":"informal","project":"p49","title":"Isometries Preserve Chronology","kind":"lemma","summary":"[Isometries Preserve Chronology] \\leanfilePhysicslib4/Spacetime/IsometryCausality.lean Say an i…","labels":["lmm:isometry-preserves-chronology"],"detail_key":"p49"},{"id":"n40685","layer":"informal","project":"p49","title":"Future-orientation preservation makes the pushforward of a future-oriented trip a future-…","kind":"proof","summary":"Future-orientation preservation makes the pushforward of a future-oriented trip a future-orient…","labels":[],"detail_key":"p49"},{"id":"n40686","layer":"informal","project":"p49","title":"Isometries Preserve Basis Sets","kind":"lemma","summary":"[Isometries Preserve Basis Sets] \\leanfilePhysicslib4/Spacetime/IsometryCausality.lean The fut","labels":["lmm:isometry-preserves-basis-sets"],"detail_key":"p49"},{"id":"n40687","layer":"informal","project":"p49","title":"Bundling the inverse into the predicate makes the subgroup axioms follow from the identit…","kind":"proof","summary":"Bundling the inverse into the predicate makes the subgroup axioms follow from the identity and…","labels":[],"detail_key":"p49"},{"id":"n40688","layer":"informal","project":"p49","title":"Axiom 5 Basis-Set Preservation","kind":"lemma","summary":"[Axiom 5 Basis-Set Preservation] \\leanfilePhysicslib4/AQFT/HaagKastlerCurved/IdentityComponent.…","labels":["lmm:axiom5-basis-preservation"],"detail_key":"p49"},{"id":"n40689","layer":"informal","project":"p49","title":"The abstract isometry group of the bridge is, by definition, the oriented identity compon…","kind":"proof","summary":"The abstract isometry group of the bridge is, by definition, the oriented identity component, s…","labels":[],"detail_key":"p49"},{"id":"n40690","layer":"informal","project":"p49","title":"Pullback of a Spacetime Metric","kind":"definition","summary":"[Pullback of a Spacetime Metric] Let (M,g) be a spacetime (\\refdef:spacetime) and let \\psi : M…","labels":["def:pullback-metric"],"detail_key":"p49"},{"id":"n40691","layer":"informal","project":"p49","title":"The Differential of a Diffeomorphism is a Linear Equivalence","kind":"lemma","summary":"[The Differential of a Diffeomorphism is a Linear Equivalence] For a C^\\infty diffeomorphism \\p…","labels":["lmm:mfderiv-diffeo-linear-equiv"],"detail_key":"p49"},{"id":"n40692","layer":"informal","project":"p49","title":"This is Mathlib's \\textttDiffeomorph.mfderivToContinuousLinearEquiv, which packages the d…","kind":"proof","summary":"This is Mathlib's \\textttDiffeomorph.mfderivToContinuousLinearEquiv, which packages the differe…","labels":[],"detail_key":"p49"},{"id":"n40693","layer":"informal","project":"p49","title":"Round-Trip Cancellation: d\\psi After d(\\psi^-1)","kind":"lemma","summary":"[Round-Trip Cancellation: d\\psi After d(\\psi^-1)] Let \\psi be a C^\\infty diffeomorphism of M an…","labels":["lmm:mfderiv-symm-cancel-left"],"detail_key":"p49"},{"id":"n40694","layer":"informal","project":"p49","title":"Differentiate the composite \\psi \\circ \\psi^-1 at the point \\psi(x), and apply the result…","kind":"proof","summary":"Differentiate the composite \\psi \\circ \\psi^-1 at the point \\psi(x), and apply the result to u.…","labels":[],"detail_key":"p49"},{"id":"n40695","layer":"informal","project":"p49","title":"Round-Trip Cancellation: d(\\psi^-1) After d\\psi","kind":"lemma","summary":"[Round-Trip Cancellation: d(\\psi^-1) After d\\psi] Let \\psi be a C^\\infty diffeomorphism of M an…","labels":["lmm:mfderiv-symm-cancel-right"],"detail_key":"p49"},{"id":"n40696","layer":"informal","project":"p49","title":"The mirror of \\reflmm:mfderiv-symm-cancel-left, run on \\psi^-1 \\circ \\psi at x: \\textttmf…","kind":"proof","summary":"The mirror of \\reflmm:mfderiv-symm-cancel-left, run on \\psi^-1 \\circ \\psi at x: \\textttmfderiv\\…","labels":[],"detail_key":"p49"},{"id":"n40697","layer":"informal","project":"p49","title":"The Formal Inverse of d\\psi_x is the Inverse Equivalence","kind":"lemma","summary":"[The Formal Inverse of d\\psi_x is the Inverse Equivalence] For a C^\\infty diffeomorphism \\psi o…","labels":["lmm:mfderiv-inverse-eq-symm"],"detail_key":"p49"},{"id":"n40698","layer":"informal","project":"p49","title":"Mathlib","kind":"proof","summary":"Mathlib","labels":[],"detail_key":"p49"},{"id":"n40699","layer":"informal","project":"p49","title":"The Pullback Metric is Symmetric","kind":"lemma","summary":"[The Pullback Metric is Symmetric] For every x \\in M the form (\\psi^*g)_x of \\refdef:pullback-m…","labels":["lmm:pullback-metric-symm"],"detail_key":"p49"},{"id":"n40700","layer":"informal","project":"p49","title":"Unfold \\refdef:pullback-metric on both sides and apply the symmetry of g_\\psi(x) to the p…","kind":"proof","summary":"Unfold \\refdef:pullback-metric on both sides and apply the symmetry of g_\\psi(x) to the pair (d…","labels":[],"detail_key":"p49"},{"id":"n40701","layer":"informal","project":"p49","title":"The Pullback Metric is Non-Degenerate","kind":"lemma","summary":"[The Pullback Metric is Non-Degenerate] For every x \\in M the form (\\psi^*g)_x of \\refdef:pullb…","labels":["lmm:pullback-metric-nondegenerate"],"detail_key":"p49"},{"id":"n40702","layer":"informal","project":"p49","title":"Since d\\psi_x is surjective (\\reflmm:mfderiv-diffeo-linear-equiv), every u \\in TM|_\\psi(x…","kind":"proof","summary":"Since d\\psi_x is surjective (\\reflmm:mfderiv-diffeo-linear-equiv), every u \\in TM|_\\psi(x) is d…","labels":[],"detail_key":"p49"},{"id":"n40703","layer":"informal","project":"p49","title":"The Pullback Metric is Lorentzian","kind":"lemma","summary":"[The Pullback Metric is Lorentzian] For every x \\in M there is a basis of TM|_x whose Gram 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\\longmaps…","labels":[],"detail_key":"p49"},{"id":"n40707","layer":"informal","project":"p49","title":"The Pullback of a Spacetime is a Spacetime","kind":"theorem","summary":"[The Pullback of a Spacetime is a Spacetime] For a spacetime (M,g) and a C^\\infty diffeomorphis…","labels":["thrm:pullback-is-spacetime"],"detail_key":"p49"},{"id":"n40708","layer":"informal","project":"p49","title":"The man","kind":"proof","summary":"The man","labels":[],"detail_key":"p49"},{"id":"n40709","layer":"informal","project":"p49","title":"Two-Sided Preservation of Future Orientation","kind":"definition","summary":"[Two-Sided Preservation of Future Orientation] Let g_1 and g_2 be metrics on M with time orient…","labels":["def:preserves-future-orientation"],"detail_key":"p49"},{"id":"n40710","layer":"informal","project":"p49","title":"The Pullback of a Bundle-Smooth Vector Field Along a Diffeomorphism is Bundle-Smooth","kind":"lemma","summary":"[The Pullback of a Bundle-Smooth Vector Field Along a Diffeomorphism is Bundle-Smooth] Let \\psi…","labels":["lmm:mpullback-vectorField-contMDiff-of-diffeo"],"detail_key":"p49"},{"id":"n40711","layer":"informal","project":"p49","title":"This is \\textttContMDiff.mpullback\\_vectorField, whose four hypotheses must all be suppli…","kind":"proof","summary":"This is \\textttContMDiff.mpullback\\_vectorField, whose four hypotheses must all be supplied ---…","labels":[],"detail_key":"p49"},{"id":"n40712","layer":"informal","project":"p49","title":"The Pullback Time Orientation is Nowhere Vanishing","kind":"lemma","summary":"[The Pullback Time Orientation is Nowhere Vanishing] For every x \\in M, (\\psi^*t)_x = (d\\psi_x)…","labels":["lmm:pullback-time-orientation-ne-zero"],"detail_key":"p49"},{"id":"n40713","layer":"informal","project":"p49","title":"By \\ref","kind":"proof","summary":"By \\ref","labels":[],"detail_key":"p49"},{"id":"n40714","layer":"informal","project":"p49","title":"The Pullback Time Orientation is Everywhere Timelike","kind":"lemma","summary":"[The Pullback Time Orientation is Everywhere Timelike] For every x \\in M, (\\psi^*g)_x\\big((\\psi…","labels":["lmm:pullback-time-orientation-timelike"],"detail_key":"p49"},{"id":"n40715","layer":"informal","project":"p49","title":"Unfold","kind":"proof","summary":"Unfold","labels":[],"detail_key":"p49"},{"id":"n40716","layer":"informal","project":"p49","title":"Pullback of a Time Orientation","kind":"lemma","summary":"[Pullback of a Time Orientation] Let t be a time orientation of (M,g) (\\refdef:time-orientable)…","labels":["lmm:pullback-time-orientation"],"detail_key":"p49"},{"id":"n40717","layer":"informal","project":"p49","title":"Assembl","kind":"proof","summary":"Assembl","labels":[],"detail_key":"p49"},{"id":"n40718","layer":"informal","project":"p49","title":"Transport of Future-Pointing Timelike Vectors","kind":"lemma","summary":"[Transport of Future-Pointing Timelike Vectors] For v \\in TM|_x timelike for \\psi^*g, (\\psi^*g)…","labels":["lmm:pullback-future-pointing-timelike"],"detail_key":"p49"},{"id":"n40719","layer":"informal","project":"p49","title":"Unfold","kind":"proof","summary":"Unfold","labels":[],"detail_key":"p49"},{"id":"n40720","layer":"informal","project":"p49","title":"Transport of Future-Pointing Null Vectors","kind":"lemma","summary":"[Transport of Future-Pointing Null Vectors] For v \\in TM|_x null for \\psi^*g, v is future-point…","labels":["lmm:pullback-future-pointing-null"],"detail_key":"p49"},{"id":"n40721","layer":"informal","project":"p49","title":"Future-pointing for a null vector is \\emphnot a sign condition: by \\refdef:future-and-pas…","kind":"proof","summary":"Future-pointing for a null vector is \\emphnot a sign condition: by \\refdef:future-and-past-poin…","labels":[],"detail_key":"p49"},{"id":"n40722","layer":"informal","project":"p49","title":"The Pullback Preserves the Future Orientation Two-Sidedly","kind":"lemma","summary":"[The Pullback Preserves the Future Orientation Two-Sidedly] The diffeomorphism \\psi, regarded a…","labels":["lmm:pullback-preserves-future-orientation"],"detail_key":"p49"},{"id":"n40723","layer":"informal","project":"p49","title":"A future-pointing vector is either timelike or null (\\refdef:future-and-past-pointing-vec…","kind":"proof","summary":"A future-pointing vector is either timelike or null (\\refdef:future-and-past-pointing-vectors).…","labels":[],"detail_key":"p49"},{"id":"n40724","layer":"informal","project":"p49","title":"Isometry Between Two Metrics on One Manifold","kind":"definition","summary":"[Isometry Between Two Metrics on One Manifold] Let g_1 and g_2 be metrics on the same manifold…","labels":["def:cross-metric-isometry"],"detail_key":"p49"},{"id":"n40725","layer":"informal","project":"p49","title":"The Inverse of a Cross-Metric Isometry is a Cross-Metric Isometry","kind":"lemma","summary":"[The Inverse of a Cross-Metric Isometry is a Cross-Metric Isometry] Let \\psi be an isometry fro…","labels":["lmm:cross-metric-isometry-symm"],"detail_key":"p49"},{"id":"n40726","layer":"informal","project":"p49","title":"Given y, put x := \\psi^-1(y), so that y = \\psi(x) by \\textttDiffeomorph.apply\\_symm\\_appl…","kind":"proof","summary":"Given y, put x := \\psi^-1(y), so that y = \\psi(x) by \\textttDiffeomorph.apply\\_symm\\_apply; eve…","labels":[],"detail_key":"p49"},{"id":"n40727","layer":"informal","project":"p49","title":"Cross-Metric Isometries Preserve the Causal Classification","kind":"lemma","summary":"[Cross-Metric Isometries Preserve the Causal Classification] Let \\psi be an isometry from (M,g_…","labels":["lmm:cross-metric-isometry-preserves-classification"],"detail_key":"p49"},{"id":"n40728","layer":"informal","project":"p49","title":"Specialise the defining equation of \\refdef:cross-metric-isometry to w = v; the three equ…","kind":"proof","summary":"Specialise the defining equation of \\refdef:cross-metric-isometry to w = v; the three equivalen…","labels":[],"detail_key":"p49"},{"id":"n40729","layer":"informal","project":"p49","title":"The Tangent Chain Rule Along a Path","kind":"lemma","summary":"[The Tangent Chain Rule Along a Path] Let \\psi be a C^\\infty diffeomorphism of M and \\mu a smoo…","labels":["lmm:cross-metric-pushforward-path-tangent"],"detail_key":"p49"},{"id":"n40730","layer":"informal","project":"p49","title":"This is \\textttmfderivWithin\\_comp for the composite of \\mu with \\psi. Write P for the pa…","kind":"proof","summary":"This is \\textttmfderivWithin\\_comp for the composite of \\mu with \\psi. Write P for the paramete…","labels":[],"detail_key":"p49"},{"id":"n40731","layer":"informal","project":"p49","title":"Pushforward of a Path Under a Cross-Metric Isometry","kind":"lemma","summary":"[Pushforward of a Path Under a Cross-Metric Isometry] An isometry \\psi from (M,g_1) to (M,g_2)…","labels":["lmm:cross-metric-pushforward-path"],"detail_key":"p49"},{"id":"n40732","layer":"informal","project":"p49","title":"The parameter space and its properties are copied. Continuity and smoothness of \\psi \\cir…","kind":"proof","summary":"The parameter space and its properties are copied. Continuity and smoothness of \\psi \\circ \\mu…","labels":[],"detail_key":"p49"},{"id":"n40733","layer":"informal","project":"p49","title":"The Pushforward Preserves the Timelike and Causal Conditions","kind":"lemma","summary":"[The Pushforward Preserves the Timelike and Causal Conditions] If \\mu is timelike (respectively…","labels":["lmm:cross-metric-pushforward-path-causal"],"detail_key":"p49"},{"id":"n40734","layer":"informal","project":"p49","title":"Fix s","kind":"proof","summary":"Fix s","labels":[],"detail_key":"p49"},{"id":"n40735","layer":"informal","project":"p49","title":"The Pushforward Transports Endpoints","kind":"lemma","summary":"[The Pushforward Transports Endpoints] If p is a past (respectively future) endpoint of \\mu, th…","labels":["lmm:cross-metric-pushforward-path-endpoints"],"detail_key":"p49"},{"id":"n40736","layer":"informal","project":"p49","title":"Being a","kind":"proof","summary":"Being a","labels":[],"detail_key":"p49"},{"id":"n40737","layer":"informal","project":"p49","title":"Cross-Metric Isometries Transport Chronological Precedence","kind":"lemma","summary":"[Cross-Metric Isometries Transport Chronological Precedence] Let \\psi be an isometry from (M,g_…","labels":["lmm:cross-metric-isometry-preserves-chronology"],"detail_key":"p49"},{"id":"n40738","layer":"informal","project":"p49","title":"A witne","kind":"proof","summary":"A witne","labels":[],"detail_key":"p49"},{"id":"n40739","layer":"informal","project":"p49","title":"Image of the Chronological Future","kind":"lemma","summary":"[Image of the Chronological Future] Let \\psi be an isometry from (M,g_1) to (M,g_2) satisfying…","labels":["lmm:cross-metric-chronological-future-image"],"detail_key":"p49"},{"id":"n40740","layer":"informal","project":"p49","title":"The inclusion \\subseteq is \\reflmm:cross-metric-isometry-preserves-chronology applied to…","kind":"proof","summary":"The inclusion \\subseteq is \\reflmm:cross-metric-isometry-preserves-chronology applied to p \\ll_…","labels":[],"detail_key":"p49"},{"id":"n40741","layer":"informal","project":"p49","title":"Image of the Chronological Past","kind":"lemma","summary":"[Image of the Chronological Past] Under the hypotheses of \\reflmm:cross-metric-chronological-fu…","labels":["lmm:cross-metric-chronological-past-image"],"detail_key":"p49"},{"id":"n40742","layer":"informal","project":"p49","title":"Identic","kind":"proof","summary":"Identic","labels":[],"detail_key":"p49"},{"id":"n40743","layer":"informal","project":"p49","title":"Cross-Metric Isometries Preserve Basis Sets","kind":"lemma","summary":"[Cross-Metric Isometries Preserve Basis Sets] Let \\psi be an isometry from (M,g_1) to (M,g_2) s…","labels":["lmm:cross-metric-isometry-preserves-basis-sets"],"detail_key":"p49"},{"id":"n40744","layer":"informal","project":"p49","title":"The ima","kind":"proof","summary":"The ima","labels":[],"detail_key":"p49"},{"id":"n40745","layer":"informal","project":"p49","title":"A Bijection Matching Generating Families is a Homeomorphism","kind":"lemma","summary":"[A Bijection Matching Generating Families is a Homeomorphism] Let f","labels":["lmm:bijection-generated-topology-homeomorphism"],"detail_key":"p49"},{"id":"n40746","layer":"informal","project":"p49","title":"Purely topological, with no geometry involved. By \\textttcontinuous\\_generateFrom\\_iff co…","kind":"proof","summary":"Purely topological, with no geometry involved. By \\textttcontinuous\\_generateFrom\\_iff continui…","labels":[],"detail_key":"p49"},{"id":"n40747","layer":"informal","project":"p49","title":"The Pullback Alexandrov Topology","kind":"lemma","summary":"[The Pullback Alexandrov Topology] Let (M,g,t) be a spacetime with time orientation and \\psi a…","labels":["lmm:pullback-alexandrov-homeomorphism"],"detail_key":"p49"},{"id":"n40748","layer":"informal","project":"p49","title":"Apply \\","kind":"proof","summary":"Apply \\","labels":[],"detail_key":"p49"},{"id":"n40749","layer":"informal","project":"p49","title":"The Pullback of a Lorentzian Spacetime is a Lorentzian Spacetime","kind":"theorem","summary":"[The Pullback of a Lorentzian Spacetime is a Lorentzian Spacetime] Let (M,g,t) be a Lorentzian…","labels":["thrm:pullback-is-lorentzian-spacetime"],"detail_key":"p49"},{"id":"n40750","layer":"informal","project":"p49","title":"The underlying spacetime is \\refthrm:pullback-is-spacetime and the time orientation is \\r…","kind":"proof","summary":"The underlying spacetime is \\refthrm:pullback-is-spacetime and the time orientation is \\reflmm:…","labels":[],"detail_key":"p49"},{"id":"n40751","layer":"informal","project":"p49","title":"Axiom 1: Local Algebras","kind":"definition","summary":"[Axiom 1: Local Algebras] \\leanfilePhysicslib4/AQFT/HaagKastler/LocalAlgebras.lean For any basi…","labels":["def:local-algebras"],"detail_key":"p49"},{"id":"n40752","layer":"informal","project":"p49","title":"Axiom 2: Isotony","kind":"definition","summary":"[Axiom 2: Isotony] \\leanfilePhysicslib4/AQFT/HaagKastler/Isotony.lean Let B_1 and B_2 be any tw…","labels":["def:isotony"],"detail_key":"p49"},{"id":"n40753","layer":"informal","project":"p49","title":"Alexandrov Diamonds are Directed under Inclusion","kind":"lemma","summary":"[Alexandrov Diamonds are Directed under Inclusion] The order-theoretic repackaging of \\reflmm:m…","labels":["lmm:alexandrov-diamonds-isDirected"],"detail_key":"p49"},{"id":"n40754","layer":"informal","project":"p49","title":"Unfoldi","kind":"proof","summary":"Unfoldi","labels":[],"detail_key":"p49"},{"id":"n40755","layer":"informal","project":"p49","title":"The Isotony Family is a Directed System","kind":"lemma","summary":"[The Isotony Family is a Directed System] The isotony family i_B_1B_2 of Axiom 2 (\\refdef:isoto…","labels":["lmm:isotony-directed-system"],"detail_key":"p49"},{"id":"n40756","layer":"informal","project":"p49","title":"Mathlib","kind":"proof","summary":"Mathlib","labels":[],"detail_key":"p49"},{"id":"n40757","layer":"informal","project":"p49","title":"The Colimit Norm is Well Defined","kind":"lemma","summary":"[The Colimit Norm is Well Defined] Setting \\|[a]\\| := \\|a\\| for a representative a \\in \\mathfra…","labels":["lmm:quasilocal-colimit-norm-well-defined"],"detail_key":"p49"},{"id":"n40758","layer":"informal","project":"p49","title":"Two rep","kind":"proof","summary":"Two rep","labels":[],"detail_key":"p49"},{"id":"n40759","layer":"informal","project":"p49","title":"Common Representatives for Two Colimit Elements","kind":"lemma","summary":"[Common Representatives for Two Colimit Elements] For any two elements x, y of \\varinjlim_B \\ma…","labels":["lmm:quasilocal-colimit-common-representatives"],"detail_key":"p49"},{"id":"n40760","layer":"informal","project":"p49","title":"This is Mathlib's \\textttDirectLimit.exists\\_eq\\_mk\\ensuremath_2, \\texttttheorem exists\\_…","kind":"proof","summary":"This is Mathlib's \\textttDirectLimit.exists\\_eq\\_mk\\ensuremath_2, \\texttttheorem exists\\_eq\\_mk…","labels":[],"detail_key":"p49"},{"id":"n40761","layer":"informal","project":"p49","title":"The Colimit Norm is a Ring Norm and a Normed-Space Norm","kind":"lemma","summary":"[The Colimit Norm is a Ring Norm and a Normed-Space Norm] The norm of \\reflmm:quasilocal-colimi…","labels":["lmm:quasilocal-colimit-norm-axioms"],"detail_key":"p49"},{"id":"n40762","layer":"informal","project":"p49","title":"Every clause is transport along a representative, which is why they share a node. For the…","kind":"proof","summary":"Every clause is transport along a representative, which is why they share a node. For the two b…","labels":[],"detail_key":"p49"},{"id":"n40763","layer":"informal","project":"p49","title":"The Quasilocal Union is a Normed *-Algebra","kind":"lemma","summary":"[The Quasilocal Union is a Normed *-Algebra] The col","labels":["lmm:quasilocal-union-normed-star-algebra"],"detail_key":"p49"},{"id":"n40764","layer":"informal","project":"p49","title":"Assemble. The *-algebra structure over C is supplied by Mathlib's \\textttDirectLimit inst…","kind":"proof","summary":"Assemble. The *-algebra structure over C is supplied by Mathlib's \\textttDirectLimit instances,…","labels":[],"detail_key":"p49"},{"id":"n40765","layer":"informal","project":"p49","title":"The Colimit Satisfies the C*-Inequality","kind":"lemma","summary":"[The Colimit Satisfies the C*-Inequality] The normed *-algebra of \\reflmm:quasilocal-union-norm…","labels":["lmm:quasilocal-colimit-cstar-identity"],"detail_key":"p49"},{"id":"n40766","layer":"informal","project":"p49","title":"A singl","kind":"proof","summary":"A singl","labels":[],"detail_key":"p49"},{"id":"n40767","layer":"informal","project":"p49","title":"Standing Hypotheses for the Completion Results","kind":"definition","summary":"[Standing Hypotheses for the Completion Results] Throughout the remainder of this subsection, A…","labels":["def:completion-standing-hypotheses"],"detail_key":"p49"},{"id":"n40768","layer":"informal","project":"p49","title":"The Involution on a Completion","kind":"definition","summary":"[The Involution on a Completion] Let A be as in \\refdef:completion-standing-hypotheses. Define…","labels":["def:completion-star"],"detail_key":"p49"},{"id":"n40769","layer":"informal","project":"p49","title":"The Involution Extends to the Completion","kind":"lemma","summary":"[The Involution Extends to the Completion] Let A be as in \\refdef:completion-standing-hypothese…","labels":["lmm:star-extends-to-completion"],"detail_key":"p49"},{"id":"n40770","layer":"informal","project":"p49","title":"Four component laws are asserted, and they are proved as four \\emphstandalone named lemma…","kind":"proof","summary":"Four component laws are asserted, and they are proved as four \\emphstandalone named lemmas, not…","labels":[],"detail_key":"p49"},{"id":"n40771","layer":"informal","project":"p49","title":"The Completion Coercion as a Bundled *-Algebra Homomorphism","kind":"lemma","summary":"[The Completion Coercion as a Bundled *-Algebra Homomorphism] Let A be as in \\refdef:completion…","labels":["lmm:completion-coe-star-alg-hom"],"detail_key":"p49"},{"id":"n40772","layer":"informal","project":"p49","title":"Every law needed is already available unbundled; the content of this node is that the \\em…","kind":"proof","summary":"Every law needed is already available unbundled; the content of this node is that the \\emphbund…","labels":[],"detail_key":"p49"},{"id":"n40773","layer":"informal","project":"p49","title":"The C*-Inequality Passes to the Completion","kind":"lemma","summary":"[The C*-Inequality Passes to the Completion] Let A be as in \\refdef:completion-standing-hypothe…","labels":["lmm:completion-cstar-identity"],"detail_key":"p49"},{"id":"n40774","layer":"informal","project":"p49","title":"Routine, not hard, and worth saying why. \\textttCStarRing is a \\textttProp-valued class w…","kind":"proof","summary":"Routine, not hard, and worth saying why. \\textttCStarRing is a \\textttProp-valued class with ex…","labels":[],"detail_key":"p49"},{"id":"n40775","layer":"informal","project":"p49","title":"The Completion is a Normed C-Algebra","kind":"lemma","summary":"[The Completion is a Normed C-Algebra] Let A be as in \\refdef:completion-standing-hypotheses. T…","labels":["lmm:completion-normed-algebra"],"detail_key":"p49"},{"id":"n40776","layer":"informal","project":"p49","title":"This node exists to work around one specific trap, and the trap is the whole of its conte…","kind":"proof","summary":"This node exists to work around one specific trap, and the trap is the whole of its content. Ma…","labels":[],"detail_key":"p49"},{"id":"n40777","layer":"informal","project":"p49","title":"The Completion of a C*-Normed *-Algebra is a C*-Algebra","kind":"lemma","summary":"[The Completion of a C*-Normed *-Algebra is a C*-Algebra] Let A be as in \\refdef:completion-sta…","labels":["lmm:completion-of-cstar-normed-star-algebra"],"detail_key":"p49"},{"id":"n40778","layer":"informal","project":"p49","title":"Pure bundling, and the audit confirms it is literally that: \\textttCStarAlgebra is declar…","kind":"proof","summary":"Pure bundling, and the audit confirms it is literally that: \\textttCStarAlgebra is declared as…","labels":[],"detail_key":"p49"},{"id":"n40779","layer":"informal","project":"p49","title":"The Completion of the Quasilocal Colimit is a C*-Algebra","kind":"lemma","summary":"[The Completion of the Quasilocal Colimit is a C*-Algebra] The completion of the normed 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Images of the Canonical Embeddings are Dense] The union of the ranges of the \\iota_B of \\r…","labels":["lmm:quasilocal-embeddings-dense"],"detail_key":"p49"},{"id":"n40796","layer":"informal","project":"p49","title":"The ran","kind":"proof","summary":"The ran","labels":[],"detail_key":"p49"},{"id":"n40797","layer":"informal","project":"p49","title":"Quasilocal Observables are Strongly Dense in the Bicommutant","kind":"theorem","summary":"[Quasilocal Observables are Strongly Dense in the Bicommutant] Let \\pi be a \\emphunital *-repre…","labels":["thrm:quasilocal-strongly-dense"],"detail_key":"p49"},{"id":"n40798","layer":"informal","project":"p49","title":"\\pi(\\mathfrakU) is a *-subalgebra of B(H) containing the identity, which is exactly the u…","kind":"proof","summary":"\\pi(\\mathfrakU) is a *-subalgebra of B(H) containing the identity, which is exactly the unitali…","labels":[],"detail_key":"p49"},{"id":"n40799","layer":"informal","project":"p49","title":"Axiom 5: Lorentz 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For…","labels":["thrm:von-neumann-isotony"],"detail_key":"p49"},{"id":"n40810","layer":"informal","project":"p49","title":"The local observables of B_1 embed into those of B_2 via the quasilocal isotony coherence…","kind":"proof","summary":"The local observables of B_1 embed into those of B_2 via the quasilocal isotony coherence, and…","labels":[],"detail_key":"p49"},{"id":"n40811","layer":"informal","project":"p49","title":"Bundled von Neumann Microcausality and Isotony","kind":"theorem","summary":"[Bundled von Neumann Microcausality and Isotony] \\leanfilePhysicslib4/AQFT/HaagKastler/LocalVon…","labels":["thrm:von-neumann-bundled-order"],"detail_key":"p49"},{"id":"n40812","layer":"informal","project":"p49","title":"Both reduce to the set-level statements through the coercion \\uparrow R(B) = \\pi(\\mathfra…","kind":"proof","summary":"Both reduce to the set-level statements through the coercion \\uparrow R(B) = \\pi(\\mathfrakU(B))…","labels":[],"detail_key":"p49"},{"id":"n40813","layer":"informal","project":"p49","title":"The Net of von Neumann Algebras","kind":"definition","summary":"[The Net of von Neumann Algebras] \\leanfilePhysicslib4/AQFT/HaagKastler/LocalVonNeumann.lean Pa…","labels":["def:von-neumann-net"],"detail_key":"p49"},{"id":"n40814","layer":"informal","project":"p49","title":"Statistical Independence (Schlieder Property)","kind":"theorem","summary":"[Statistical Independence (Schlieder Property)] \\leanfilePhysicslib4/AQFT/HaagKastler/LocalVonN…","labels":["thrm:statistical-independence"],"detail_key":"p49"},{"id":"n40815","layer":"informal","project":"p49","title":"The implication is elementary (R vanishes on the dense set of vectors A\\Omega for A a loc…","kind":"proof","summary":"The implication is elementary (R vanishes on the dense set of vectors A\\Omega for A a local 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\\su…","labels":["thrm:irreducible-inclusion-factor"],"detail_key":"p49"},{"id":"n40837","layer":"informal","project":"p49","title":"Indeed the center R(B_2) \\cap R(B_2)' is contained in the relative commutant (\\refthrm:re…","kind":"proof","summary":"Indeed the center R(B_2) \\cap R(B_2)' is contained in the relative commutant (\\refthrm:relative…","labels":[],"detail_key":"p49"},{"id":"n40838","layer":"informal","project":"p49","title":"Self-Inclusion is Irreducible iff Factor","kind":"theorem","summary":"[Self-Inclusion is Irreducible iff Factor] The trivial self-inclusion R(B) \\subseteq R(B) is ir…","labels":["thrm:self-inclusion-factor"],"detail_key":"p49"},{"id":"n40839","layer":"informal","project":"p49","title":"The relative commutant of the self-inclusion is R(B)' \\cap R(B), i.e. the center of R(B)…","kind":"proof","summary":"The relative commutant of the self-inclusion is R(B)' \\cap R(B), i.e. the center of R(B) (up to…","labels":[],"detail_key":"p49"},{"id":"n40840","layer":"informal","project":"p49","title":"Irreducible Representation","kind":"definition","summary":"[Irreducible Representation] \\leanfilePhysicslib4/GNS/Irreducibility.lean A *-representation \\p…","labels":["def:irreducible-representation"],"detail_key":"p49"},{"id":"n40841","layer":"informal","project":"p49","title":"Topological Schur Lemma","kind":"theorem","summary":"[Topological Schur Lemma] \\leanfilePhysicslib4/GNS/Irreducibility.lean Let \\Omega be a cyclic v…","labels":["thrm:schur-lemma"],"detail_key":"p49"},{"id":"n40842","layer":"informal","project":"p49","title":"The proof is pure Hilbert-space analysis: using the *-representation property and the com…","kind":"proof","summary":"The proof is pure Hilbert-space analysis: using the *-representation property and the commutati…","labels":[],"detail_key":"p49"},{"id":"n40843","layer":"informal","project":"p49","title":"Commutant Scalar iff Proportional 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(\\refthrm:pure-iff-irreduci…","labels":[],"detail_key":"p49"},{"id":"n40864","layer":"informal","project":"p49","title":"Norm of a Positive Functional","kind":"theorem","summary":"[Norm of a Positive Functional] \\leanfilePhysicslib4/GNS/ExtremeState.lean For a positive linea…","labels":["thrm:norm-positive-functional"],"detail_key":"p49"},{"id":"n40865","layer":"informal","project":"p49","title":"The bound Re\\,\\varphi(1) \\le \\Vert\\varphi\\Vert is immediate from \\Vert 1\\Vert = 1; conver…","kind":"proof","summary":"The bound Re\\,\\varphi(1) \\le \\Vert\\varphi\\Vert is immediate from \\Vert 1\\Vert = 1; conversely C…","labels":[],"detail_key":"p49"},{"id":"n40866","layer":"informal","project":"p49","title":"Extreme Point of the State Space","kind":"definition","summary":"[Extreme Point of the State Space] \\leanfilePhysicslib4/GNS/ExtremeState.lean A state \\omega 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exhibits a real convex combination of…","labels":[],"detail_key":"p49"},{"id":"n40871","layer":"informal","project":"p49","title":"Pullback of a State","kind":"definition","summary":"[Pullback of a State] For a unital","labels":["def:state-pullback"],"detail_key":"p49"},{"id":"n40872","layer":"informal","project":"p49","title":"Functoriality of the State Pullback","kind":"theorem","summary":"[Functoriality of the State Pullback] The pullback is a contravariant functor on C*-algebras: p…","labels":["thrm:state-pullback-functorial"],"detail_key":"p49"},{"id":"n40873","layer":"informal","project":"p49","title":"Both identities are immediate from the defining equation (\\omega \\circ \\pi)(a) = \\omega(\\…","kind":"proof","summary":"Both identities are immediate from the defining equation (\\omega \\circ \\pi)(a) = \\omega(\\pi(a))…","labels":[],"detail_key":"p49"},{"id":"n40874","layer":"informal","project":"p49","title":"Purity is Invariant under a *-Isomorphism","kind":"theorem","summary":"[Purity is Invariant under a *-Isomorphism] For a *-isomorphism \\Phi : A \\simeq B of C*-algebra…","labels":["thrm:pure-pullback-invariant"],"detail_key":"p49"},{"id":"n40875","layer":"informal","project":"p49","title":"A dominated positive functional \\psi \\le \\omega \\circ \\Phi transports to \\psi \\circ \\Phi^…","kind":"proof","summary":"A dominated positive functional \\psi \\le \\omega \\circ \\Phi transports to \\psi \\circ \\Phi^-1 \\le…","labels":[],"detail_key":"p49"},{"id":"n40876","layer":"informal","project":"p49","title":"Weak-* Compactness of the State Space","kind":"theorem","summary":"[Weak-* Compactness of the State Space] \\leanfilePhysicslib4/GNS/PureStateExists.lean The state…","labels":["thrm:state-space-weak-compact"],"detail_key":"p49"},{"id":"n40877","layer":"informal","project":"p49","title":"By Banach-Alaoglu it suffices that it is a weak-* closed subset of the closed unit ball:…","kind":"proof","summary":"By Banach-Alaoglu it suffices that it is a weak-* closed subset of the closed unit ball: the po…","labels":[],"detail_key":"p49"},{"id":"n40878","layer":"informal","project":"p49","title":"Pure \\iff Extreme Point on the Quasilocal Algebra","kind":"theorem","summary":"[Pure \\iff Extreme Point on the Quasilocal Algebra] \\leanfilePhysicslib4/AQFT/HaagKastler/Purit…","labels":["thrm:pure-iff-extreme-quasilocal"],"detail_key":"p49"},{"id":"n40879","layer":"informal","project":"p49","title":"This is the abstract equivalence \\refthrm:pure-iff-extreme applied to the C*-algebra \\mat…","kind":"proof","summary":"This is the abstract equivalence \\refthrm:pure-iff-extreme applied to the C*-algebra \\mathfrakU.","labels":[],"detail_key":"p49"},{"id":"n40880","layer":"informal","project":"p49","title":"Pure \\iff Irreducible GNS on the Quasilocal Algebra","kind":"theorem","summary":"[Pure \\iff Irreducible GNS on the Quasilocal Algebra] \\leanfilePhysicslib4/AQFT/HaagKastler/Pur…","labels":["thrm:pure-iff-irreducible-quasilocal"],"detail_key":"p49"},{"id":"n40881","layer":"informal","project":"p49","title":"This combines the GNS construction with the abstract \\refthrm:pure-iff-irreducible.","kind":"proof","summary":"This combines the GNS construction with the abstract \\refthrm:pure-iff-irreducible.","labels":[],"detail_key":"p49"},{"id":"n40882","layer":"informal","project":"p49","title":"Unitary Equivalence of Representations","kind":"definition","summary":"[Unitary Equivalence of Representations] \\leanfilePhysicslib4/GNS/UnitaryEquiv.lean Two *-repre…","labels":["def:unitary-equivalence"],"detail_key":"p49"},{"id":"n40883","layer":"informal","project":"p49","title":"Irreducibility and Factoriality are Unitary Invariants","kind":"theorem","summary":"[Irreducibility and Factoriality are Unitary Invariants] \\leanfilePhysicslib4/GNS/UnitaryEquiv.…","labels":["thrm:unitary-equiv-invariants"],"detail_key":"p49"},{"id":"n40884","layer":"informal","project":"p49","title":"The transport is packaged through the cross-space conjugation T \\mapsto U T U^-1, a multi…","kind":"proof","summary":"The transport is packaged through the cross-space conjugation T \\mapsto U T U^-1, a multiplicat…","labels":[],"detail_key":"p49"},{"id":"n40885","layer":"informal","project":"p49","title":"Cyclicity pulls back along a surjective *-homomorphism","kind":"lemma","summary":"[Cyclicity pulls back along a surjective *-homomorphism] Let \\pi : B \\to B(H) be a *-representa…","labels":["lmm:cyclic-pullback-surjective"],"detail_key":"p49"},{"id":"n40886","layer":"informal","project":"p49","title":"Indeed surjectivity of \\Phi gives \\\\pi(\\Phi(a))\\Omega : a \\in A\\ = \\\\pi(b)\\Omega : b \\in…","kind":"proof","summary":"Indeed surjectivity of \\Phi gives \\\\pi(\\Phi(a))\\Omega : a \\in A\\ = \\\\pi(b)\\Omega : b \\in B\\, so…","labels":[],"detail_key":"p49"},{"id":"n40887","layer":"informal","project":"p49","title":"GNS covariance under a *-isomorphism","kind":"theorem","summary":"[GNS covariance under a *-isomorphism] Let \\Phi","labels":["thrm:gns-covariance"],"detail_key":"p49"},{"id":"n40888","layer":"informal","project":"p49","title":"The pro","kind":"proof","summary":"The pro","labels":[],"detail_key":"p49"},{"id":"n40889","layer":"informal","project":"p49","title":"The GNS representation of a pullback state","kind":"theorem","summary":"[The GNS representation of a pullback state] Restated","labels":["thrm:gns-covariance-unitary-equiv"],"detail_key":"p49"},{"id":"n40890","layer":"informal","project":"p49","title":"The intertwining unitary of \\refthrm:gns-covariance is exactly the required witness. Toge…","kind":"proof","summary":"The intertwining unitary of \\refthrm:gns-covariance is exactly the required witness. Together w…","labels":[],"detail_key":"p49"},{"id":"n40891","layer":"informal","project":"p49","title":"Pullback along a surjection preserves the image algebra","kind":"lemma","summary":"[Pullback along a surjection preserves the image algebra] Let \\p","labels":["lmm:pullback-image-invariants"],"detail_key":"p49"},{"id":"n40892","layer":"informal","project":"p49","title":"The ima","kind":"proof","summary":"The ima","labels":[],"detail_key":"p49"},{"id":"n40893","layer":"informal","project":"p49","title":"Superselection type transports along a *-isomorphism","kind":"theorem","summary":"[Superselection type transports along a *-isomorphism] Let \\Phi","labels":["thrm:gns-sector-transport"],"detail_key":"p49"},{"id":"n40894","layer":"informal","project":"p49","title":"Compose","kind":"proof","summary":"Compose","labels":[],"detail_key":"p49"},{"id":"n40895","layer":"informal","project":"p49","title":"GNS covariance for local algebras","kind":"theorem","summary":"[GNS covariance for local 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S^\\ast S = a\\cdot 1 and S^\\ast T = b \\cdot 1 (commuting with \\pi_1, hence scalar),…","kind":"proof","summary":"Indeed S^\\ast S = a\\cdot 1 and S^\\ast T = b \\cdot 1 (commuting with \\pi_1, hence scalar), and S…","labels":[],"detail_key":"p49"},{"id":"n40903","layer":"informal","project":"p49","title":"Endomorphism Algebra of an Irreducible Representation","kind":"lemma","summary":"[Endomorphism Algebra of an Irreducible Representation] \\leanfilePhysicslib4/GNS/Superselection…","labels":["lmm:endomorphism-scalar"],"detail_key":"p49"},{"id":"n40904","layer":"informal","project":"p49","title":"This is the commutant form of irreducibility read through the intertwiner language, and i…","kind":"proof","summary":"This is the commutant form of irreducibility read through the intertwiner language, and it iden…","labels":[],"detail_key":"p49"},{"id":"n40905","layer":"informal","project":"p49","title":"The commutant (self-intertwiner) von Neumann algebra","kind":"theorem","summary":"[The 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von Neumann algebra","kind":"definition","summary":"[Center of a von Neumann algebra] The center o","labels":["def:von-neumann-center"],"detail_key":"p49"},{"id":"n40914","layer":"informal","project":"p49","title":"The center is a von Neumann algebra","kind":"lemma","summary":"[The center is a von Neumann algebra] Let R be a bundled von Neumann algebra on a Hilbert space…","labels":["lmm:von-neumann-inter-is-von-neumann"],"detail_key":"p49"},{"id":"n40915","layer":"informal","project":"p49","title":"The instance M = R, N = R' of \\reflmm:von-neumann-inter-general.","kind":"proof","summary":"The instance M = R, N = R' of \\reflmm:von-neumann-inter-general.","labels":[],"detail_key":"p49"},{"id":"n40916","layer":"informal","project":"p49","title":"The intersection of two von Neumann algebras is a von Neumann algebra","kind":"lemma","summary":"[The intersection of two von Neumann algebras is a von Neumann algebra] Let M and N be 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\\bigopl…","labels":["def:direct-sum-representation"],"detail_key":"p49"},{"id":"n40931","layer":"informal","project":"p49","title":"Subrepresentations and Commutant of a Direct Sum","kind":"theorem","summary":"[Subrepresentations and Commutant of a Direct Sum] \\leanfilePhysicslib4/GNS/DirectSum.lean Each…","labels":["thrm:direct-sum-subrepresentation"],"detail_key":"p49"},{"id":"n40932","layer":"informal","project":"p49","title":"Both claims are coordinatewise computations with the diagonal action of \\refdef:direct-su…","kind":"proof","summary":"Both claims are coordinatewise computations with the diagonal action of \\refdef:direct-sum-repr…","labels":[],"detail_key":"p49"},{"id":"n40933","layer":"informal","project":"p49","title":"Amplification","kind":"definition","summary":"[Amplification] \\leanfilePhysicslib4/GNS/Amplification.lean The \\emph\\iota-fold amplification \\…","labels":["def:amplification"],"detail_key":"p49"},{"id":"n40934","layer":"informal","project":"p49","title":"Reducibility of a Direct Sum","kind":"theorem","summary":"[Reducibility of a Direct Sum] \\leanfilePhysicslib4/GNS/Amplification.lean A direct sum with tw…","labels":["thrm:direct-sum-reducible"],"detail_key":"p49"},{"id":"n40935","layer":"informal","project":"p49","title":"Were \\bigoplus_i \\pi_i irreducible, every summand projection would be a scalar (its commu…","kind":"proof","summary":"Were \\bigoplus_i \\pi_i irreducible, every summand projection would be a scalar (its commutant b…","labels":[],"detail_key":"p49"},{"id":"n40936","layer":"informal","project":"p49","title":"Covariant Family of Local States","kind":"definition","summary":"[Covariant Family of Local States] \\leanfilePhysicslib4/AQFT/HaagKastler/CovariantState.lean Gi…","labels":["def:covariant-state-family"],"detail_key":"p49"},{"id":"n40937","layer":"informal","project":"p49","title":"Composition of Covariance","kind":"lemma","summary":"[Composition of Covariance] \\leanfilePhysicslib4/AQFT/HaagKastler/CovariantState.lean For a cov…","labels":["lmm:covariant-state-family-compose"],"detail_key":"p49"},{"id":"n40938","layer":"informal","project":"p49","title":"This reflects the multiplicativity of the Lorentz action: the covariance relation of \\ref…","kind":"proof","summary":"This reflects the multiplicativity of the Lorentz action: the covariance relation of \\refdef:co…","labels":[],"detail_key":"p49"},{"id":"n40939","layer":"informal","project":"p49","title":"Quasilocal Covariance Automorphism","kind":"definition","summary":"[Quasilocal Covariance Automorphism] \\leanfilePhysicslib4/AQFT/HaagKastler/QuasilocalAction.lea…","labels":["def:quasilocal-lift"],"detail_key":"p49"},{"id":"n40940","layer":"informal","project":"p49","title":"Uniqueness of the Quasilocal Lift","kind":"lemma","summary":"[Uniqueness of the Quasilocal Lift] \\leanfilePhysicslib4/AQFT/HaagKastler/QuasilocalAction.lean…","labels":["lmm:quasilocal-lift-unique"],"detail_key":"p49"},{"id":"n40941","layer":"informal","project":"p49","title":"They agree on the union of the local images, which is dense in \\mathfrakU, and *-automorp…","kind":"proof","summary":"They agree on the union of the local images, which is dense in \\mathfrakU, and *-automorphisms…","labels":[],"detail_key":"p49"},{"id":"n40942","layer":"informal","project":"p49","title":"Existence of the Quasilocal Lift","kind":"theorem","summary":"[Existence of the Quasilocal Lift] \\leanfilePhysicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.…","labels":["thrm:quasilocal-lift-exists"],"detail_key":"p49"},{"id":"n40943","layer":"informal","project":"p49","title":"The intertwiner is defined on the directed union of local images (a dense *-subalgebra) a…","kind":"proof","summary":"The intertwiner is defined on the directed union of local images (a dense *-subalgebra) and ext…","labels":[],"detail_key":"p49"},{"id":"n40944","layer":"informal","project":"p49","title":"Existence for the Trivial Net","kind":"theorem","summary":"[Existence for the Trivial Net] \\leanfilePhysicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lea…","labels":["thrm:quasilocal-lift-trivial"],"detail_key":"p49"},{"id":"n40945","layer":"informal","project":"p49","title":"Covariance-compatibility holds because every *-automorphism of C is the identity; the lif…","kind":"proof","summary":"Covariance-compatibility holds because every *-automorphism of C is the identity; the lift is t…","labels":[],"detail_key":"p49"},{"id":"n40946","layer":"informal","project":"p49","title":"Covariant Quasilocal Algebra","kind":"definition","summary":"[Covariant Quasilocal Algebra] \\leanfilePhysicslib4/AQFT/HaagKastler/QuasilocalIntertwiner.lean…","labels":["def:covariant-quasilocal-algebra"],"detail_key":"p49"},{"id":"n40947","layer":"informal","project":"p49","title":"Group-Action Coherence of the 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\\leanfilePhysicslib4/AQFT/HaagKastler/Quasil…","labels":["thrm:invariant-state-gns-unitary"],"detail_key":"p49"},{"id":"n40951","layer":"informal","project":"p49","title":"The unitaries are obtained by extending the densely-defined isometry \\pi(a)\\Omega \\mapsto…","kind":"proof","summary":"The unitaries are obtained by extending the densely-defined isometry \\pi(a)\\Omega \\mapsto \\pi(\\…","labels":[],"detail_key":"p49"},{"id":"n40952","layer":"informal","project":"p49","title":"Irreducible Covariant Representation of a Pure Invariant State","kind":"theorem","summary":"[Irreducible Covariant Representation of a Pure Invariant State] \\leanfilePhysicslib4/AQFT/Haag…","labels":["thrm:irreducible-covariant-representation"],"detail_key":"p49"},{"id":"n40953","layer":"informal","project":"p49","title":"This combines the covariant GNS triple of \\refthrm:invariant-state-gns-unitary with purit…","kind":"proof","summary":"This combines the covariant GNS triple of \\refthrm:invariant-state-gns-unitary with purity \\Rig…","labels":[],"detail_key":"p49"},{"id":"n40954","layer":"informal","project":"p49","title":"Positive Energy (bounded-generator scaffold)","kind":"definition","summary":"[Positive Energy (bounded-generator scaffold)] \\leanfilePhysicslib4/AQFT/HaagKastler/VacuumStat…","labels":["def:positive-energy"],"detail_key":"p49"},{"id":"n40955","layer":"informal","project":"p49","title":"Positive-Energy API","kind":"theorem","summary":"[Positive-Energy API] \\leanfilePhysicslib4/AQFT/HaagKastler/VacuumState.lean Structura","labels":["thrm:positive-energy-api"],"detail_key":"p49"},{"id":"n40956","layer":"informal","project":"p49","title":"(ii) Differentiating at t = 0 gives i P = i Q; positivity is not needed. (iii) The genera…","kind":"proof","summary":"(ii) Differentiating at t = 0 gives i P = i Q; positivity is not needed. (iii) The generator W…","labels":[],"detail_key":"p49"},{"id":"n40957","layer":"informal","project":"p49","title":"Vacuum State (generator-parameterized scaffold)","kind":"definition","summary":"[Vacuum State (generator-parameterized scaffold)] \\leanfilePhysicslib4/AQFT/HaagKastler/VacuumS…","labels":["def:vacuum-state"],"detail_key":"p49"},{"id":"n40958","layer":"informal","project":"p49","title":"No-Stone Consequences of a Vacuum State","kind":"theorem","summary":"[No-Stone Consequences of a Vacuum State] \\leanfilePhysicslib4/AQFT/HaagKastler/VacuumState.lea…","labels":["thrm:vacuum-no-stone"],"detail_key":"p49"},{"id":"n40959","layer":"informal","project":"p49","title":"Both follow by projecting to invariance and chaining with purity \\Rightarrow irreducibili…","kind":"proof","summary":"Both follow by projecting to invariance and chaining with purity \\Rightarrow irreducibility; 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\\leanfilePhysicslib4/A…","labels":["thrm:additive-free-locality-in-curved-spacetime"],"detail_key":"p49"},{"id":"n41015","layer":"informal","project":"p49","title":"The Galois bridge discharges the spacelike hypothesis, and L.\\mathtttoAbstract identifies…","kind":"proof","summary":"The Galois bridge discharges the spacelike hypothesis, and L.\\mathtttoAbstract identifies the c…","labels":[],"detail_key":"p49"},{"id":"n41016","layer":"informal","project":"p49","title":"Geometric Covariance of the von Neumann Net (Curved Spacetime)","kind":"theorem","summary":"[Geometric Covariance of the von Neumann Net (Curved Spacetime)] \\leanfilePhysicslib4/AQFT/Haag…","labels":["thrm:von-neumann-geometric-covariance-in-curved-spacetime"],"detail_key":"p49"},{"id":"n41017","layer":"informal","project":"p49","title":"There i","kind":"proof","summary":"There i","labels":[],"detail_key":"p49"},{"id":"n41018","layer":"informal","project":"p49","title":"Orbit-Invariance of Factoriality (Curved 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(\\textttSet.inter\\_subset\\_right).","kind":"proof","summary":"Discharged through the coercion to underlying sets (\\textttSet.inter\\_subset\\_right).","labels":[],"detail_key":"p49"},{"id":"n41025","layer":"informal","project":"p49","title":"Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime)","kind":"theorem","summary":"[Relative Commutant Commutes with the Smaller Algebra (Curved Spacetime)] Every element of the…","labels":["thrm:relative-commutant-coe-commutant-in-curved-spacetime"],"detail_key":"p49"},{"id":"n41026","layer":"informal","project":"p49","title":"This is \\textttSet.inter\\_subset\\_left for the intersection defining the relative commuta…","kind":"proof","summary":"This is \\textttSet.inter\\_subset\\_left for the intersection defining the relative commutant.","labels":[],"detail_key":"p49"},{"id":"n41027","layer":"informal","project":"p49","title":"Relative Commutant Contains the Center (Curved Spacetime)","kind":"theorem","summary":"[Relative 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N.quasilocal.car…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.pure_iff_extreme","module":"Physicslib4.AQFT.HaagKastler.Purity"},{"id":"n41113","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv","kind":"theorem","summary":"∀ (N : Physicslib4.AQFT.HaagKastler.HaagKastlerNet) (L : Physicslib4.AQFT.HaagKastler.Inhomogen…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.unitaryEquiv_gns_covEquiv","module":"Physicslib4.AQFT.HaagKastler.Purity"},{"id":"n41114","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift","kind":"inductive","summary":"(N : Physicslib4.AQFT.HaagKastler.HaagKastlerNet) → Physicslib4.AQFT.HaagKastler.QuasilocalAlge…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift","module":"Physicslib4.AQFT.HaagKastler.QuasilocalAction"},{"id":"n41115","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique","kind":"theorem","summary":"∀ N : Physicslib4.AQFT.HaagKastler.HaagKastlerNet Q : Physicslib4.AQFT.HaagKastler.QuasilocalAl…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.HaagKastlerNet.QuasilocalLift.unique","module":"Physicslib4.AQFT.HaagKastler.QuasilocalAction"},{"id":"n41116","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra","kind":"inductive","summary":"(U : Physicslib4.AQFT.HaagKastler.LocalNet) → Physicslib4.AQFT.HaagKastler.Isotony U → Type (u…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra","module":"Physicslib4.AQFT.HaagKastler.QuasilocalAlgebra"},{"id":"n41117","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.Diamond","kind":"def","summary":"Type","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.Diamond","module":"Physicslib4.AQFT.HaagKastler.QuasilocalColimit"},{"id":"n41118","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.QuasilocalColimit","kind":"def","summary":"(U : Physicslib4.AQFT.HaagKastler.LocalNet) → Physicslib4.AQFT.HaagKastler.Isotony U → Type u","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.QuasilocalColimit","module":"Physicslib4.AQFT.HaagKastler.QuasilocalColimit"},{"id":"n41119","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.QuasilocalCompletion","kind":"def","summary":"(U : Physicslib4.AQFT.HaagKastler.LocalNet) → Physicslib4.AQFT.HaagKastler.Isotony U → 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Physicslib4.AQFT.HaagKa…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.action_mul","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41148","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.action_one","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra), Eq (C.action 1) (StarAlgEquiv.…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.action_one","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41149","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isFactor_iff_gns_action","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (L : Physicslib4.AQFT.HaagKastl…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isFactor_iff_gns_action","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41150","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isIrreducible_iff_gns_action","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (L : Physicslib4.AQFT.HaagKastl…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isIrreducible_iff_gns_action","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41151","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isPure_precomp_action_iff","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (ω : Physicslib4.GNS.State C.qu…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isPure_precomp_action_iff","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41152","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.unitaryEquiv_gns_action","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (L : Physicslib4.AQFT.HaagKastl…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.unitaryEquiv_gns_action","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41153","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift","kind":"theorem","summary":"∀ N : Physicslib4.AQFT.HaagKastler.HaagKastlerNet Q : Physicslib4.AQFT.HaagKastler.QuasilocalAl…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.nonempty_quasilocalLift","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41154","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift","kind":"theorem","summary":"∀ (L : Physicslib4.AQFT.HaagKastler.InhomogeneousLorentzGroup), Nonempty (Physicslib4.AQFT.Haag…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.nonempty_trivialQuasilocalLift","module":"Physicslib4.AQFT.HaagKastler.QuasilocalIntertwiner"},{"id":"n41155","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow","kind":"def","summary":"(C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) → (Real → Physicslib4.AQFT.HaagKa…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41156","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow.exists_stron…","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (flow : Real → Physicslib4.AQFT…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow.exists_strongContinuous_unitary","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41157","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow.invariant","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (flow : Real → Physicslib4.AQFT…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsGroundStateForFlow.invariant","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41158","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsKMSStateForFlow","kind":"def","summary":"(C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) → (Real → Physicslib4.AQFT.HaagKa…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsKMSStateForFlow","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41159","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsKMSStateForFlow.convexCombo","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (flow : Real → Physicslib4.AQFT…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsKMSStateForFlow.convexCombo","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41160","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.flowAut","kind":"def","summary":"(C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) → (Real → Physicslib4.AQFT.HaagKa…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.flowAut","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41161","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isOneParameterAut_flowAut","kind":"theorem","summary":"∀ (C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) (flow : Real → Physicslib4.AQFT…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.isOneParameterAut_flowAut","module":"Physicslib4.AQFT.HaagKastler.QuasilocalKMS"},{"id":"n41162","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable","kind":"def","summary":"U : Physicslib4.AQFT.HaagKastler.LocalNet → i : Physicslib4.AQFT.HaagKastler.Isotony U → (Q : P…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.IsQuasilocalObservable","module":"Physicslib4.AQFT.HaagKastler.QuasilocalObservable"},{"id":"n41163","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsVacuumState","kind":"def","summary":"(C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) → ((Real → 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Physicslib4.AQFT.Ha…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsVacuumState.invariant","module":"Physicslib4.AQFT.HaagKastler.VacuumState"},{"id":"n41166","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsVacuumStateConcrete","kind":"def","summary":"(C : Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra) → Physicslib4.GNS.State C.quasilo…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastler.CovariantQuasilocalAlgebra.IsVacuumStateConcrete","module":"Physicslib4.AQFT.HaagKastler.VacuumState"},{"id":"n41167","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastler.IsFutureTimelikeTranslation","kind":"def","summary":"(Real → Physicslib4.AQFT.HaagKastler.InhomogeneousLorentzGroup) → 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Physicslib4.AQFT.HaagKastler…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.NetEquivalence","module":"Physicslib4.AQFT.HaagKastlerCurved.GeneralCovariance"},{"id":"n41178","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.NetTheory","kind":"def","summary":"Type 1","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.NetTheory","module":"Physicslib4.AQFT.HaagKastlerCurved.GeneralCovariance"},{"id":"n41179","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.pullbackCarrierEquiv","kind":"def","summary":"(L : Physicslib4.Spacetime.LorentzianSpacetime) → (ψ : L.toSpacetime.Diffeo L.toSpacetime) → 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:…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.toAbstract_pullback_isBasisSet","module":"Physicslib4.AQFT.HaagKastlerCurved.GeneralCovariance"},{"id":"n41182","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localOperators","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localOperators","module":"Physicslib4.AQFT.HaagKastlerCurved.GeometricCovariance"},{"id":"n41183","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.lieConj_image_localVonNeumann","module":"Physicslib4.AQFT.HaagKastlerCurved.GeometricCovariance"},{"id":"n41184","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannEquiv","module":"Physicslib4.AQFT.HaagKastlerCurved.GeometricCovariance"},{"id":"n41185","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_isFactor_smul","module":"Physicslib4.AQFT.HaagKastlerCurved.GeometricCovariance"},{"id":"n41186","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.toAbstractIdentityComponent_isBasisSet_smul","kind":"theorem","summary":"∀ (L : Physicslib4.Spacetime.LorentzianSpacetime) (φ : L.toAbstractIdentityComponent.Isom) B :…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.toAbstractIdentityComponent_isBasisSet_smul","module":"Physicslib4.AQFT.HaagKastlerCurved.IdentityComponent"},{"id":"n41187","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.IsometricCovariance","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → 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Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsIrreducibleInclusion","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41194","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.center_le_relativeCommutant","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41195","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.eq_zero_of_commute_of_cyclic","kind":"theorem","summary":"∀ H : Type u_1 [inst : NormedAddCommGroup H] [inst_1 : InnerProductSpace Complex H] S : Set (Co…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.eq_zero_of_commute_of_cyclic","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41196","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_of_isIrreducibleInclusion","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41197","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducibleInclusion_self_iff_isFactor","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41198","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localOperators","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41199","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41200","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41201","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commut…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_eq_commutant","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41202","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_center_isAbelian","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41203","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_isAbelian_iff_eq…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_isAbelian_iff_eq_scalars_of_isFactor","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41204","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_isFactor_iff_cen…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_isFactor_iff_center_eq_scalars","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41205","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_le_commutant","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41206","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_mono","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_mono","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41207","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumannAlgebra_separating","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41208","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_mono","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41209","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_separating","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41210","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.localVonNeumann_subset_centralizer","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41211","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41212","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_coe_subset_commutant","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41213","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.relativeCommutant_le_right","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41214","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.vonNeumannNet","module":"Physicslib4.AQFT.HaagKastlerCurved.LocalVonNeumann"},{"id":"n41215","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet","kind":"inductive","summary":"Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → Type (max 1 u_1)","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet","module":"Physicslib4.AQFT.HaagKastlerCurved.Net"},{"id":"n41216","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrredu…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.areDisjoint_or_unitaryEquiv_of_isIrreducible","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41217","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_factor_of_isPure","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41218","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_generates_all_of_isPure","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_generates_all_of_isPure","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41219","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_pure_iff_irreducible","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41220","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_iff_gns_covEquiv","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_iff_gns_covEquiv","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41221","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducible_iff_gns_covEquiv","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducible_iff_gns_covEquiv","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41222","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.pure_iff_extreme","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41223","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_covEquiv","module":"Physicslib4.AQFT.HaagKastlerCurved.Purity"},{"id":"n41224","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabil…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_irreducible_covariant_stabilizer","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41225","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41226","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongCon…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.exists_gns_unitary_stabilizer_strongContinuous","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41227","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_iff_gns_stabAut","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isFactor_iff_gns_stabAut","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41228","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducible_iff_gns_stabAut","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isIrreducible_iff_gns_stabAut","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41229","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.isPure_precomp_stabAut_iff","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41230","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41231","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_mul","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_mul","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41232","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.stabAut_one","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41233","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.unitaryEquiv_gns_stabAut","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerAction"},{"id":"n41234","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow","kind":"def","summary":"M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime → (N : Physicslib4.AQFT.HaagKastlerC…","labels":[],"detail_key":"p49","name":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow","module":"Physicslib4.AQFT.HaagKastlerCurved.StabilizerKMS"},{"id":"n41235","layer":"formal","project":"p49","title":"Physicslib4.AQFT.HaagKastlerCurved.HaagKastlerNet.IsGroundStateForFlow.exists_strongConti…","kind":"theorem","summary":"∀ M : Physicslib4.AQFT.HaagKastlerCurved.LorentzianSpacetime (N : 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(M.cau…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_inf","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41359","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_sup","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (B₁ B₂ : M.CausallyCompleteRegion), Eq (M.cau…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_sup","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41360","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.causalComplement_top","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime), Eq (M.causalComplement Top.top) 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(M.IsCausallyComplete…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyComplete_iff_isClosed","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41363","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyComplete_inter","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) B₁ B₂ : Set M.Carrier, M.IsCausallyComplete B…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyComplete_inter","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41364","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyComplete_spacelikeComplement","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (B : Set M.Carrier), M.IsCausallyComplete (M.…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyComplete_spacelikeComplement","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41365","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyConvex_of_isCausallyComplete","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) B : Set M.Carrier, M.IsCausallyComplete B → M…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCausallyConvex_of_isCausallyComplete","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41366","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement","kind":"def","summary":"(M : Physicslib4.Spacetime.LorentzianSpacetime) → Set M.Carrier → Set 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(M.s…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_iUnion","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41369","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_spacelikeComplement_spaceli…","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (B : Set M.Carrier), Eq (M.spacelikeComplemen…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_spacelikeComplement_spacelikeComplement","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41370","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_union","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (B₁ B₂ : Set M.Carrier), Eq (M.spacelikeCompl…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.spacelikeComplement_union","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41371","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.subset_spacelikeComplement_iff","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) B₁ B₂ : Set M.Carrier, Iff (LE.le B₁ (M.space…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.subset_spacelikeComplement_iff","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41372","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.subset_spacelikeComplement_spacelikeComplement","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (B : Set M.Carrier), LE.le B (M.spacelikeComp…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.subset_spacelikeComplement_spacelikeComplement","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41373","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.spacelikeComplement_isCausallyConvex","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (t : M.TimeOrientation) (B : Set M.Carrier), M.IsCausallyConvex t…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.spacelikeComplement_isCausallyConvex","module":"Physicslib4.Spacetime.CausalComplement"},{"id":"n41374","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.IsFuturePointing","kind":"def","summary":"(M : Physicslib4.Spacetime) → M.TimeOrientation → x : M.Carrier → TangentSpace M.model x → 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(M…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.chronologicalFutureSet_mono","module":"Physicslib4.Spacetime.Causality"},{"id":"n41427","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.chronologicalFuture_subset_causalFuture","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (t : M.TimeOrientation) (p : M.Carrier), LE.le (M.chronologicalFu…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.chronologicalFuture_subset_causalFuture","module":"Physicslib4.Spacetime.Causality"},{"id":"n41428","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.chronologicalPast","kind":"def","summary":"(M : Physicslib4.Spacetime) → M.TimeOrientation → M.Carrier → Set M.Carrier","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.chronologicalPast","module":"Physicslib4.Spacetime.Causality"},{"id":"n41429","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.chronologicalPastSet","kind":"def","summary":"(M : 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(M.chronologicalPa…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.chronologicalPast_subset_causalPast","module":"Physicslib4.Spacetime.Causality"},{"id":"n41432","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.chronologicallyPrecedes_irrefl","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (t : M.TimeOrientation), M.NoClosedCausalCurve t → ∀ (p : M.Carri…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.chronologicallyPrecedes_irrefl","module":"Physicslib4.Spacetime.Causality"},{"id":"n41433","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.chronologicallyPrecedes_trans","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (t : M.TimeOrientation) p q r : M.Carrier, 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N.TimeOrie…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.CrossIsometry.chronologicallyPrecedes","module":"Physicslib4.Spacetime.CrossMetricIsometry"},{"id":"n41462","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.CrossIsometry.isNull_mfderiv_iff","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime N : Physicslib4.Spacetime (Ψ : M.CrossIsometry N) (x : M.Carrier) (…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.CrossIsometry.isNull_mfderiv_iff","module":"Physicslib4.Spacetime.CrossMetricIsometry"},{"id":"n41463","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.CrossIsometry.isSpacelike_mfderiv_iff","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime N : Physicslib4.Spacetime (Ψ : M.CrossIsometry N) (x : M.Carrier) (…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.CrossIsometry.isSpacelike_mfderiv_iff","module":"Physicslib4.Spacetime.CrossMetricIsometry"},{"id":"n41464","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.CrossIsometry.isTimelike_mfderiv_iff","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime N : Physicslib4.Spacetime (Ψ : M.CrossIsometry N) (x : M.Carrier) (…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.CrossIsometry.isTimelike_mfderiv_iff","module":"Physicslib4.Spacetime.CrossMetricIsometry"},{"id":"n41465","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.CrossIsometry.preserves_self","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime N : Physicslib4.Spacetime (Ψ : M.CrossIsometry N) (x : M.Carrier) 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(s'…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.mem_frontier_of_isMax","module":"Physicslib4.Spacetime.Curves"},{"id":"n41511","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.mem_frontier_of_isMin","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (μ : M.Path) s : Real, Membership.mem μ.parameterSpace s → (∀ (s'…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.mem_frontier_of_isMin","module":"Physicslib4.Spacetime.Curves"},{"id":"n41512","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.parameterSpace_eq_Icc_of_endpoints","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (μ : M.SmoothPath) p q : M.Carrier, M.IsPastEndpoint μ p → M.IsFu…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.parameterSpace_eq_Icc_of_endpoints","module":"Physicslib4.Spacetime.Curves"},{"id":"n41513","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Diffeo","kind":"def","summary":"Physicslib4.Spacetime → 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: Physicslib4.Spacetime (g : M.Isometry) (x : M.Carrier) (v : TangentSpace M.model x), Iff…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.isNull_mfderiv_iff","module":"Physicslib4.Spacetime.Isometry"},{"id":"n41531","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.isSpacelike_mfderiv_iff","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (x : M.Carrier) (v : TangentSpace M.model x), Iff…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.isSpacelike_mfderiv_iff","module":"Physicslib4.Spacetime.Isometry"},{"id":"n41532","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.isTimelike_mfderiv_iff","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (x : M.Carrier) (v : TangentSpace M.model x), 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g.PreservesFutureOrientat…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.chronologicalFuture_image","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41540","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.chronologicalFuture_image_subset","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (t : M.TimeOrientation), g.PreservesFutureOrientat…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.chronologicalFuture_image_subset","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41541","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.chronologicalPast_image","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (t : M.TimeOrientation), g.PreservesFutureOrientat…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.chronologicalPast_image","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41542","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.chronologicalPast_image_subset","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (t : M.TimeOrientation), g.PreservesFutureOrientat…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.chronologicalPast_image_subset","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41543","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.chronologicallyPrecedes_pushforward","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (t : M.TimeOrientation), g.PreservesFutureOrientat…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.chronologicallyPrecedes_pushforward","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41544","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.futureOrientationPreserving","kind":"def","summary":"(M : Physicslib4.Spacetime) → M.TimeOrientation → Subgroup M.Isometry","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.futureOrientationPreserving","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41545","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.orientedIdentityComponent","kind":"def","summary":"(M : Physicslib4.Spacetime) → M.TimeOrientation → Subgroup 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g.Pres…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.pushforwardPath_isFutureOriented","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41552","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.pushforwardPath_isPastEndpoint","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (μ : M.SmoothPath) p : M.Carrier, M.IsPastEndpoint…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.Isometry.pushforwardPath_isPastEndpoint","module":"Physicslib4.Spacetime.IsometryCausality"},{"id":"n41553","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.Isometry.pushforwardPath_isTimelike","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime (g : M.Isometry) (μ : M.SmoothPath), 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M.IsCompletelySpacelike…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_empty_left","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41586","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_empty_right","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (O : Set M.Carrier), M.IsCompletelySpacelike…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_empty_right","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41587","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_mono","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) O₁ O₁' O₂ O₂' : Set M.Carrier, LE.le O₁' O₁ →…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_mono","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41588","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_union_left","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (O₁ O₁' O₂ : Set M.Carrier), Iff (M.IsComplet…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_union_left","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41589","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_union_right","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) (O₁ O₂ O₂' : Set M.Carrier), Iff (M.IsComplet…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isCompletelySpacelike_union_right","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41590","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isOpen_alexandrov_of_isBasisSet","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) B : Set M.Carrier, M.IsBasisSet B → IsOpen B","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isOpen_alexandrov_of_isBasisSet","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41591","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.isTopologicalBasis_alexandrovBasis","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) [Nontrivial M.Carrier], (∀ (B₁ : Set M.toSpac…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.isTopologicalBasis_alexandrovBasis","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41592","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.LorentzianSpacetime.sUnion_alexandrovBasis_eq_univ","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime.LorentzianSpacetime) [Nontrivial M.Carrier], Eq (M.toSpacetime.ale…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.LorentzianSpacetime.sUnion_alexandrovBasis_eq_univ","module":"Physicslib4.Spacetime.LorentzianSpacetime"},{"id":"n41593","layer":"formal","project":"p49","title":"Physicslib4.MinkowskiSpacetime","kind":"def","summary":"Physicslib4.Spacetime","labels":[],"detail_key":"p49","name":"Physicslib4.MinkowskiSpacetime","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41594","layer":"formal","project":"p49","title":"Physicslib4.StandardMinkowskiSpacetime","kind":"def","summary":"Physicslib4.Spacetime","labels":[],"detail_key":"p49","name":"Physicslib4.StandardMinkowskiSpacetime","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41595","layer":"formal","project":"p49","title":"Physicslib4.exists_chronologicalFuture_standardMinkowski","kind":"theorem","summary":"∀ (x : Physicslib4.SpacetimeModel), Exists fun b => Membership.mem (Physicslib4.StandardMinkows…","labels":[],"detail_key":"p49","name":"Physicslib4.exists_chronologicalFuture_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41596","layer":"formal","project":"p49","title":"Physicslib4.exists_chronologicalPast_standardMinkowski","kind":"theorem","summary":"∀ (x : Physicslib4.SpacetimeModel), Exists fun a => Membership.mem (Physicslib4.StandardMinkows…","labels":[],"detail_key":"p49","name":"Physicslib4.exists_chronologicalPast_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41597","layer":"formal","project":"p49","title":"Physicslib4.isOpen_alexandrov_chronologicalFuture_standardMinkowski","kind":"theorem","summary":"∀ (p : Physicslib4.SpacetimeModel), IsOpen (Physicslib4.StandardMinkowskiSpacetime.chronologica…","labels":[],"detail_key":"p49","name":"Physicslib4.isOpen_alexandrov_chronologicalFuture_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41598","layer":"formal","project":"p49","title":"Physicslib4.isOpen_alexandrov_chronologicalPast_standardMinkowski","kind":"theorem","summary":"∀ (q : Physicslib4.SpacetimeModel), IsOpen (Physicslib4.StandardMinkowskiSpacetime.chronologica…","labels":[],"detail_key":"p49","name":"Physicslib4.isOpen_alexandrov_chronologicalPast_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41599","layer":"formal","project":"p49","title":"Physicslib4.isOpen_chronologicalFuture_standardMinkowski","kind":"theorem","summary":"∀ (p : Physicslib4.SpacetimeModel), IsOpen (Physicslib4.StandardMinkowskiSpacetime.chronologica…","labels":[],"detail_key":"p49","name":"Physicslib4.isOpen_chronologicalFuture_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41600","layer":"formal","project":"p49","title":"Physicslib4.isOpen_chronologicalPast_standardMinkowski","kind":"theorem","summary":"∀ (q : Physicslib4.SpacetimeModel), IsOpen (Physicslib4.StandardMinkowskiSpacetime.chronologica…","labels":[],"detail_key":"p49","name":"Physicslib4.isOpen_chronologicalPast_standardMinkowski","module":"Physicslib4.Spacetime.Minkowski"},{"id":"n41601","layer":"formal","project":"p49","title":"Physicslib4.alexandrovBasis_image_smul","kind":"theorem","summary":"∀ (lam : Real), LT.lt 0 lam → ∀ B : Set Physicslib4.SpacetimeModel, Membership.mem (Physicslib4…","labels":[],"detail_key":"p49","name":"Physicslib4.alexandrovBasis_image_smul","module":"Physicslib4.Spacetime.MinkowskiDilation"},{"id":"n41602","layer":"formal","project":"p49","title":"Physicslib4.exists_minkowskiForm_smul_ne","kind":"theorem","summary":"∀ (lam : Real), Ne (HPow.hPow lam 2) 1 → Exists fun v => Exists fun w => Ne ((Physicslib4.minko…","labels":[],"detail_key":"p49","name":"Physicslib4.exists_minkowskiForm_smul_ne","module":"Physicslib4.Spacetime.MinkowskiDilation"},{"id":"n41603","layer":"formal","project":"p49","title":"Physicslib4.minkowskiBackwardCone_smul","kind":"theorem","summary":"∀ (lam : Real), LT.lt 0 lam → ∀ (p q : Physicslib4.SpacetimeModel), Iff (Membership.mem (Physic…","labels":[],"detail_key":"p49","name":"Physicslib4.minkowskiBackwardCone_smul","module":"Physicslib4.Spacetime.MinkowskiDilation"},{"id":"n41604","layer":"formal","project":"p49","title":"Physicslib4.minkowskiForm_smul","kind":"theorem","summary":"∀ (lam : Real) (v w : Physicslib4.SpacetimeModel), Eq ((Physicslib4.minkowskiForm (HSMul.hSMul…","labels":[],"detail_key":"p49","name":"Physicslib4.minkowskiForm_smul","module":"Physicslib4.Spacetime.MinkowskiDilation"},{"id":"n41605","layer":"formal","project":"p49","title":"Physicslib4.minkowskiForwardCone_smul","kind":"theorem","summary":"∀ (lam : Real), LT.lt 0 lam → ∀ (p q : Physicslib4.SpacetimeModel), Iff (Membership.mem (Physic…","labels":[],"detail_key":"p49","name":"Physicslib4.minkowskiForwardCone_smul","module":"Physicslib4.Spacetime.MinkowskiDilation"},{"id":"n41606","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.alexandrovBasis_directed","kind":"theorem","summary":"∀ B₁ B₂ : Set Physicslib4.SpacetimeModel, Membership.mem (Physicslib4.StandardMinkowskiSpacetim…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.alexandrovBasis_directed","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41607","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.alexandrovBasis_exists_subset_inter_standardMinkowski","kind":"theorem","summary":"∀ (B₁ : Set Physicslib4.SpacetimeModel), Membership.mem (Physicslib4.StandardMinkowskiSpacetime…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.alexandrovBasis_exists_subset_inter_standardMinkowski","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41608","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.exists_common_future","kind":"theorem","summary":"∀ (q₁ q₂ : Physicslib4.SpacetimeModel), Exists fun q => And (Membership.mem (Physicslib4.minkow…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.exists_common_future","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41609","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.exists_common_past","kind":"theorem","summary":"∀ (p₁ p₂ : Physicslib4.SpacetimeModel), Exists fun p => And (Membership.mem (Physicslib4.minkow…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.exists_common_past","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41610","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.exists_future_between_standardMinkowski","kind":"theorem","summary":"∀ q₁ q₂ x : Physicslib4.SpacetimeModel, Membership.mem (Physicslib4.minkowskiBackwardCone q₁) x…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.exists_future_between_standardMinkowski","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41611","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.exists_past_between_standardMinkowski","kind":"theorem","summary":"∀ p₁ p₂ x : Physicslib4.SpacetimeModel, Membership.mem (Physicslib4.minkowskiForwardCone p₁) x…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.exists_past_between_standardMinkowski","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41612","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.isTopologicalBasis_alexandrovBasis_standardMinkowski","kind":"theorem","summary":"TopologicalSpace.IsTopologicalBasis (Physicslib4.StandardMinkowskiSpacetime.alexandrovBasis Phy…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.isTopologicalBasis_alexandrovBasis_standardMinkowski","module":"Physicslib4.Spacetime.MinkowskiDirected"},{"id":"n41613","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.PreservesFutureOrientation","kind":"def","summary":"M : Physicslib4.Spacetime → N : Physicslib4.Spacetime → M.Diffeo N → M.TimeOrientation → N.Time…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.PreservesFutureOrientation","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41614","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.PreservesFutureOrientationTwoSided","kind":"def","summary":"M : Physicslib4.Spacetime → N : Physicslib4.Spacetime → M.Diffeo N → M.TimeOrientation → N.Time…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.PreservesFutureOrientationTwoSided","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41615","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.bilinearPrecomp","kind":"def","summary":"ContinuousLinearMap (RingHom.id Real) Physicslib4.SpacetimeModel (ContinuousLinearMap (RingHom.…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.bilinearPrecomp","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41616","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.bilinearPrecomp_apply","kind":"theorem","summary":"∀ (g : ContinuousLinearMap (RingHom.id Real) Physicslib4.SpacetimeModel (ContinuousLinearMap (R…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.bilinearPrecomp_apply","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41617","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.contMDiff_mpullback_vectorField","kind":"theorem","summary":"∀ M : Physicslib4.Spacetime N : Physicslib4.Spacetime (ψ : M.Diffeo N) (V : (x : N.Carrier) → T…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.contMDiff_mpullback_vectorField","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41618","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isNull","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) x : M.Carrier v : Tangen…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isNull","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41619","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isTimelike","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) x : M.Carrier v : Tangen…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.isFuturePointing_pullback_iff_of_isTimelike","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41620","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.isTimelike_mpullback_field","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) (x : M.Carrier), (M.pull…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.isTimelike_mpullback_field","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41621","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.mpullback_field_ne_zero","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) (x : M.Carrier), Ne (Vec…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.mpullback_field_ne_zero","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41622","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullback","kind":"def","summary":"(M : 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Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation), Eq (M.pullbackTimeOrien…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackTimeOrientation_field","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41626","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal","kind":"def","summary":"(M : Physicslib4.Spacetime) → M.Diffeo M → (x : M.Carrier) → ContinuousLinearMap (RingHom.id Re…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41627","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_apply","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (x : M.Carrier) (v w : TangentSpace M.model x),…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_apply","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41628","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_contMDiff","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M), ContMDiff M.model (M.model.prod (modelWithCorne…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_contMDiff","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41629","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_lorentzian","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (x : M.Carrier), Physicslib4.LorentzianAt fun v…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_lorentzian","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41630","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_mpullback_field_apply","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) (x : M.Carrier) (v : Tan…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_mpullback_field_apply","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41631","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_mpullback_field_self","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation) (x : M.Carrier), Eq (((M…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_mpullback_field_self","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41632","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_nondegenerate","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (x : M.Carrier) (v : TangentSpace M.model x), (∀…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_nondegenerate","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41633","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullbackVal_symm","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (x : M.Carrier) (v w : TangentSpace M.model x),…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullbackVal_symm","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41634","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullback_Carrier","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M), Eq (M.pullback ψ).Carrier M.Carrier","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullback_Carrier","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41635","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullback_model","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M), Eq (M.pullback ψ).model M.model","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullback_model","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41636","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullback_preservesFutureOrientationTwoSided","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M) (t : M.TimeOrientation), Physicslib4.Spacetime.P…","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullback_preservesFutureOrientationTwoSided","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41637","layer":"formal","project":"p49","title":"Physicslib4.Spacetime.pullback_val","kind":"theorem","summary":"∀ (M : Physicslib4.Spacetime) (ψ : M.Diffeo M), Eq (M.pullback ψ).val (M.pullbackVal ψ)","labels":[],"detail_key":"p49","name":"Physicslib4.Spacetime.pullback_val","module":"Physicslib4.Spacetime.Pullback"},{"id":"n41638","layer":"informal","project":"p50","title":"field","kind":"lemma","summary":"For all rational r, we have r^2 \\neq d, so K is a field.","labels":["field"],"detail_key":"p50"},{"id":"n41639","layer":"informal","project":"p50","title":"Clear since we assume that d is squarefree.","kind":"proof","summary":"Clear since we assume that d is squarefree.","labels":[],"detail_key":"p50"},{"id":"n41640","layer":"informal","project":"p50","title":"d_congr","kind":"lemma","summary":"We have that d = \\pm 1 \\bmod 4 or d = 2 \\bmod 4.","labels":["d_congr"],"detail_key":"p50"},{"id":"n41641","layer":"informal","project":"p50","title":"If d = 0 \\bmod 4 then d would not be squarefree.","kind":"proof","summary":"If d = 0 \\bmod 4 then d would not be squarefree.","labels":[],"detail_key":"p50"},{"id":"n41642","layer":"informal","project":"p50","title":"easy_incl","kind":"lemma","summary":"We have that \\sqrtd is an integral element of K.","labels":["easy_incl"],"detail_key":"p50"},{"id":"n41643","layer":"informal","project":"p50","title":"Clear since \\sqrtd is a root of x^2-d.","kind":"proof","summary":"Clear since \\sqrtd is a root of x^2-d.","labels":[],"detail_key":"p50"},{"id":"n41644","layer":"informal","project":"p50","title":"rational_iff","kind":"lemma","summary":"We have that z \\in Q if and only if b = 0.","labels":["rational_iff"],"detail_key":"p50"},{"id":"n41645","layer":"informal","project":"p50","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p50"},{"id":"n41646","layer":"informal","project":"p50","title":"minpoly","kind":"lemma","summary":"If b \\neq 0 then the minimal polynomial of z over Q is \\[ X^2-2aX+(a^2-db^2) \\]","labels":["minpoly"],"detail_key":"p50"},{"id":"n41647","layer":"informal","project":"p50","title":"It's clear that z is a root of P and that P \\in Q[X] is monic. Irreducibility follows by…","kind":"proof","summary":"It's clear that z is a root of P and that P \\in Q[X] is monic. Irreducibility follows by the fa…","labels":[],"detail_key":"p50"},{"id":"n41648","layer":"informal","project":"p50","title":"trace","kind":"lemma","summary":"We have that the trace of z is 2a.","labels":["trace"],"detail_key":"p50"},{"id":"n41649","layer":"informal","project":"p50","title":"If b = 0 then z = a \\in Q and the trace is 2a since [K : Q] = 2. Otherwise this is clear…","kind":"proof","summary":"If b = 0 then z = a \\in Q and the trace is 2a since [K : Q] = 2. Otherwise this is clear by Lem…","labels":[],"detail_key":"p50"},{"id":"n41650","layer":"informal","project":"p50","title":"norm","kind":"lemma","summary":"We have that the norm of z is a^2-db^2.","labels":["norm"],"detail_key":"p50"},{"id":"n41651","layer":"informal","project":"p50","title":"If b = 0 then z = a \\in Q and the norm is a^2 since [K : Q] = 2. Otherwise this is clear…","kind":"proof","summary":"If b = 0 then z = a \\in Q and the norm is a^2 since [K : Q] = 2. Otherwise this is clear by Lem…","labels":[],"detail_key":"p50"},{"id":"n41652","layer":"informal","project":"p50","title":"trace_int","kind":"lemma","summary":"We have that 2a \\in Z.","labels":["trace_int"],"detail_key":"p50"},{"id":"n41653","layer":"informal","project":"p50","title":"Since the trace of an algebraic integer is an integers, this follows by Lemma \\reftrace.","kind":"proof","summary":"Since the trace of an algebraic integer is an integers, this follows by Lemma \\reftrace.","labels":[],"detail_key":"p50"},{"id":"n41654","layer":"informal","project":"p50","title":"t_spec","kind":"definition","summary":"We write t (for trace) to denote 2a as an integer. Mathematically we have t = 2a.","labels":["t_spec"],"detail_key":"p50"},{"id":"n41655","layer":"informal","project":"p50","title":"norm_int","kind":"lemma","summary":"We have that a^2-db^2 \\in Z.","labels":["norm_int"],"detail_key":"p50"},{"id":"n41656","layer":"informal","project":"p50","title":"Since the norm of an algebraic integer is an integers, this follows by Lemma \\refnorm.","kind":"proof","summary":"Since the norm of an algebraic integer is an integers, this follows by Lemma \\refnorm.","labels":[],"detail_key":"p50"},{"id":"n41657","layer":"informal","project":"p50","title":"n_spec","kind":"definition","summary":"We write n (for norm) to denote a^2-db^2 as an integer. Mathematically we have n = a^2-db^2.","labels":["n_spec"],"detail_key":"p50"},{"id":"n41658","layer":"informal","project":"p50","title":"four_n","kind":"lemma","summary":"We have that 4n = (2a)^2 - d(2b)^2.","labels":["four_n"],"detail_key":"p50"},{"id":"n41659","layer":"informal","project":"p50","title":"Obvious by applying \\refn_spec.","kind":"proof","summary":"Obvious by applying \\refn_spec.","labels":[],"detail_key":"p50"},{"id":"n41660","layer":"informal","project":"p50","title":"squarefree_mul","kind":"lemma","summary":"Let n be a squarefree integer and let r be a rational such that b r^2 is an integer. Then r is…","labels":["squarefree_mul"],"detail_key":"p50"},{"id":"n41661","layer":"informal","project":"p50","title":"Easy.","kind":"proof","summary":"Easy.","labels":[],"detail_key":"p50"},{"id":"n41662","layer":"informal","project":"p50","title":"two_b_int","kind":"lemma","summary":"We have that 2b \\in Z.","labels":["two_b_int"],"detail_key":"p50"},{"id":"n41663","layer":"informal","project":"p50","title":"By Lemma \\reffour_n, (2a)^2 - d(2b)^2 is an integer and so, by Lemma \\reftrace_int, we kn…","kind":"proof","summary":"By Lemma \\reffour_n, (2a)^2 - d(2b)^2 is an integer and so, by Lemma \\reftrace_int, we know tha…","labels":[],"detail_key":"p50"},{"id":"n41664","layer":"informal","project":"p50","title":"B₂_spec","kind":"definition","summary":"We write B_2 to denote 2b as an integer. Mathematically we have B_2 = 2b.","labels":["B₂_spec"],"detail_key":"p50"},{"id":"n41665","layer":"informal","project":"p50","title":"b_int_of_a_int","kind":"lemma","summary":"If a \\in Z then b \\in Z.","labels":["b_int_of_a_int"],"detail_key":"p50"},{"id":"n41666","layer":"informal","project":"p50","title":"By Lemma \\reffour_n and our assumption, both (2a)^2 and (2a)^2 - d(2b)^2 are integers div…","kind":"proof","summary":"By Lemma \\reffour_n and our assumption, both (2a)^2 and (2a)^2 - d(2b)^2 are integers divisible…","labels":[],"detail_key":"p50"},{"id":"n41667","layer":"informal","project":"p50","title":"B_spec","kind":"definition","summary":"If a is an integer, we write B to denote b as an integer. Mathematically we have B = b.","labels":["B_spec"],"detail_key":"p50"},{"id":"n41668","layer":"informal","project":"p50","title":"a_not_int","kind":"lemma","summary":"If a \\not\\in Z then d = 1 \\bmod4.","labels":["a_not_int"],"detail_key":"p50"},{"id":"n41669","layer":"informal","project":"p50","title":"We have that 2a, that is an integer, must be odd. By Lemmas \\reffour_n and \\reftwo_b_int,…","kind":"proof","summary":"We have that 2a, that is an integer, must be odd. By Lemmas \\reffour_n and \\reftwo_b_int, we ha…","labels":[],"detail_key":"p50"},{"id":"n41670","layer":"informal","project":"p50","title":"d_2_or_3","kind":"theorem","summary":"Assume that d = 2 \\bmod4 or d = 3 \\bmod4. Then \\[ O_K = Z[\\sqrtd] \\]","labels":["d_2_or_3"],"detail_key":"p50"},{"id":"n41671","layer":"informal","project":"p50","title":"By Lemma \\refeasy_incl we know that Z[\\sqrtd] \\subseteq O_K. Let z = a + b \\sqrtd \\in O_K…","kind":"proof","summary":"By Lemma \\refeasy_incl we know that Z[\\sqrtd] \\subseteq O_K. Let z = a + b \\sqrtd \\in O_K, with…","labels":[],"detail_key":"p50"},{"id":"n41672","layer":"informal","project":"p50","title":"e_spec","kind":"lemma","summary":"We have that e is an integer and 4e = d - 1.","labels":["e_spec"],"detail_key":"p50"},{"id":"n41673","layer":"informal","project":"p50","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p50"},{"id":"n41674","layer":"informal","project":"p50","title":"algebra_R_S","kind":"lemma","summary":"We have that \\[ \\left(2 \\left( \\frac1+\\sqrtd2 \\right) - 1 \\right)^2 = d \\] so that S is an R-al…","labels":["algebra_R_S"],"detail_key":"p50"},{"id":"n41675","layer":"informal","project":"p50","title":"Obvious by Lemma \\refe_spec.","kind":"proof","summary":"Obvious by Lemma \\refe_spec.","labels":[],"detail_key":"p50"},{"id":"n41676","layer":"informal","project":"p50","title":"algebra_S_K","kind":"lemma","summary":"We have that \\[ \\left( \\frac1+\\sqrtd2 \\right)^2 = \\left( \\frac1+\\sqrtd2 \\right) + e \\] so that…","labels":["algebra_S_K"],"detail_key":"p50"},{"id":"n41677","layer":"informal","project":"p50","title":"Obvious by Lemma \\refe_spec.","kind":"proof","summary":"Obvious by Lemma \\refe_spec.","labels":[],"detail_key":"p50"},{"id":"n41678","layer":"informal","project":"p50","title":"commutes_R_S_K","kind":"lemma","summary":"The obvious diagram between R, S and K commutes.","labels":["commutes_R_S_K"],"detail_key":"p50"},{"id":"n41679","layer":"informal","project":"p50","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p50"},{"id":"n41680","layer":"informal","project":"p50","title":"easy_incl_d_1","kind":"lemma","summary":"We have that \\frac1+\\sqrtd2 \\in O_K.","labels":["easy_incl_d_1"],"detail_key":"p50"},{"id":"n41681","layer":"informal","project":"p50","title":"Clear since \\frac1+\\sqrtd2 is a root of X^2 - X - e \\in Z[X].","kind":"proof","summary":"Clear since \\frac1+\\sqrtd2 is a root of X^2 - X - e \\in Z[X].","labels":[],"detail_key":"p50"},{"id":"n41682","layer":"informal","project":"p50","title":"d_1_int","kind":"lemma","summary":"Take z = a + b \\sqrtd \\in O_K with a, b \\in Q. If a \\in Z then z \\in Z\\left[ \\frac1+\\sqrtd2 \\ri…","labels":["d_1_int"],"detail_key":"p50"},{"id":"n41683","layer":"informal","project":"p50","title":"By Lemma \\refb_int_of_a_int we have that b \\in Z and so z \\in Z[\\sqrtd] \\subseteq Z\\left[…","kind":"proof","summary":"By Lemma \\refb_int_of_a_int we have that b \\in Z and so z \\in Z[\\sqrtd] \\subseteq Z\\left[ \\frac…","labels":[],"detail_key":"p50"},{"id":"n41684","layer":"informal","project":"p50","title":"d_1","kind":"theorem","summary":"We have \\[ O_K = Z\\left[ \\frac1+\\sqrtd2 \\right] \\]","labels":["d_1"],"detail_key":"p50"},{"id":"n41685","layer":"informal","project":"p50","title":"By Lemma \\refeasy_incl_d_1 we know that Z\\left[ \\frac1+\\sqrtd2 \\right] \\subseteq O_K. Let…","kind":"proof","summary":"By Lemma \\refeasy_incl_d_1 we know that Z\\left[ \\frac1+\\sqrtd2 \\right] \\subseteq O_K. Let z = a…","labels":[],"detail_key":"p50"},{"id":"n41686","layer":"informal","project":"p50","title":"human_d_congr","kind":"human_lemma","summary":"We have that d = \\pm 1 \\bmod 4 or d = 2 \\bmod 4.","labels":["human_d_congr"],"detail_key":"p50"},{"id":"n41687","layer":"informal","project":"p50","title":"If d = 0 \\bmod 4 then d would not be squarefree.","kind":"proof","summary":"If d = 0 \\bmod 4 then d would not be squarefree.","labels":[],"detail_key":"p50"},{"id":"n41688","layer":"informal","project":"p50","title":"human_easy_incl","kind":"human_lemma","summary":"We have that \\sqrtd \\in O_K.","labels":["human_easy_incl"],"detail_key":"p50"},{"id":"n41689","layer":"informal","project":"p50","title":"Clear since \\sqrtd is a root of x^2-d.","kind":"proof","summary":"Clear since \\sqrtd is a root of x^2-d.","labels":[],"detail_key":"p50"},{"id":"n41690","layer":"informal","project":"p50","title":"human_easy_incl_d_1","kind":"human_lemma","summary":"If d = 1 \\bmod4 then \\frac1+\\sqrtd2 \\in O_K.","labels":["human_easy_incl_d_1"],"detail_key":"p50"},{"id":"n41691","layer":"informal","project":"p50","title":"Write d = 4a + 1, with a \\in Z. Then \\frac1+\\sqrtd2 is a root of x^2 - x - a \\in Z[x].","kind":"proof","summary":"Write d = 4a + 1, with a \\in Z. Then \\frac1+\\sqrtd2 is a root of x^2 - x - a \\in Z[x].","labels":[],"detail_key":"p50"},{"id":"n41692","layer":"informal","project":"p50","title":"human_minpoly","kind":"human_lemma","summary":"The minimal polynomial of t over Q is \\[ P(x) = x^2-2at+(a^2-db^2) \\]","labels":["human_minpoly"],"detail_key":"p50"},{"id":"n41693","layer":"informal","project":"p50","title":"It's clear that t is a root of P and that P \\in Q[x] is monic. Irreducibility follows by…","kind":"proof","summary":"It's clear that t is a root of P and that P \\in Q[x] is monic. Irreducibility follows by the fa…","labels":[],"detail_key":"p50"},{"id":"n41694","layer":"informal","project":"p50","title":"human_trace","kind":"human_lemma","summary":"We have that the trace of t is 2a.","labels":["human_trace"],"detail_key":"p50"},{"id":"n41695","layer":"informal","project":"p50","title":"Clear by Lemma \\refhuman_minpoly.","kind":"proof","summary":"Clear by Lemma \\refhuman_minpoly.","labels":[],"detail_key":"p50"},{"id":"n41696","layer":"informal","project":"p50","title":"human_norm","kind":"human_lemma","summary":"We have that the norm of t is a^2-db^2.","labels":["human_norm"],"detail_key":"p50"},{"id":"n41697","layer":"informal","project":"p50","title":"Clear by Lemma \\refhuman_minpoly.","kind":"proof","summary":"Clear by Lemma \\refhuman_minpoly.","labels":[],"detail_key":"p50"},{"id":"n41698","layer":"informal","project":"p50","title":"human_trace_int","kind":"human_lemma","summary":"We have that 2a \\in Z","labels":["human_trace_int"],"detail_key":"p50"},{"id":"n41699","layer":"informal","project":"p50","title":"Since the trace of an algebraic integer is an integers, this follows by Lemma \\refhuman_t…","kind":"proof","summary":"Since the trace of an algebraic integer is an integers, this follows by Lemma \\refhuman_trace.","labels":[],"detail_key":"p50"},{"id":"n41700","layer":"informal","project":"p50","title":"human_norm_int","kind":"human_lemma","summary":"We have that a^2-db^2 \\in Z","labels":["human_norm_int"],"detail_key":"p50"},{"id":"n41701","layer":"informal","project":"p50","title":"Since the norm of an algebraic integer is an integers, this follows by Lemma \\refhuman_no…","kind":"proof","summary":"Since the norm of an algebraic integer is an integers, this follows by Lemma \\refhuman_norm.","labels":[],"detail_key":"p50"},{"id":"n41702","layer":"informal","project":"p50","title":"human_divisible","kind":"human_lemma","summary":"We have that (2a)^2 - d(2b)^2 is an integer divisible by 4.","labels":["human_divisible"],"detail_key":"p50"},{"id":"n41703","layer":"informal","project":"p50","title":"Clear since (2a)^2 - d(2b)^2 = 4(a^2-db^2) and a^2-db^2 \\in Z by Lemma \\refhuman_norm_int.","kind":"proof","summary":"Clear since (2a)^2 - d(2b)^2 = 4(a^2-db^2) and a^2-db^2 \\in Z by Lemma \\refhuman_norm_int.","labels":[],"detail_key":"p50"},{"id":"n41704","layer":"informal","project":"p50","title":"human_2b_int","kind":"human_lemma","summary":"We have that 2b \\in Z.","labels":["human_2b_int"],"detail_key":"p50"},{"id":"n41705","layer":"informal","project":"p50","title":"By Lemma \\refhuman_divisible, (2a)^2 - d(2b)^2 is an integer and so, by Lemma \\refhuman_t…","kind":"proof","summary":"By Lemma \\refhuman_divisible, (2a)^2 - d(2b)^2 is an integer and so, by Lemma \\refhuman_trace_i…","labels":[],"detail_key":"p50"},{"id":"n41706","layer":"informal","project":"p50","title":"human_b_int_of_a_int","kind":"human_lemma","summary":"If a \\in Z then b \\in Z.","labels":["human_b_int_of_a_int"],"detail_key":"p50"},{"id":"n41707","layer":"informal","project":"p50","title":"By Lemma \\refhuman_divisible and our assumption, both (2a)^2 and (2a)^2 - d(2b)^2 are int…","kind":"proof","summary":"By Lemma \\refhuman_divisible and our assumption, both (2a)^2 and (2a)^2 - d(2b)^2 are integers…","labels":[],"detail_key":"p50"},{"id":"n41708","layer":"informal","project":"p50","title":"human_a_not_int","kind":"human_lemma","summary":"If a \\not\\in Z then d = 1 \\bmod4.","labels":["human_a_not_int"],"detail_key":"p50"},{"id":"n41709","layer":"informal","project":"p50","title":"We have that 2a, that is an integer, must be odd. By Lemmas \\refhuman_divisible and \\refh…","kind":"proof","summary":"We have that 2a, that is an integer, must be odd. By Lemmas \\refhuman_divisible and \\refhuman_2…","labels":[],"detail_key":"p50"},{"id":"n41710","layer":"informal","project":"p50","title":"By Lemma \\refhuman_easy_incl we know that Z[\\sqrtd] \\subseteq O_K. Let t = a + b \\sqrtd \\…","kind":"proof","summary":"By Lemma \\refhuman_easy_incl we know that Z[\\sqrtd] \\subseteq O_K. Let t = a + b \\sqrtd \\in O_K…","labels":[],"detail_key":"p50"},{"id":"n41711","layer":"informal","project":"p50","title":"human_d_1_int","kind":"human_lemma","summary":"Assume that d = 1 \\bmod4 and take t = a + b \\sqrtd \\in O_K with a, b \\in Q. If a \\in Z then t \\…","labels":["human_d_1_int"],"detail_key":"p50"},{"id":"n41712","layer":"informal","project":"p50","title":"By Lemma \\refhuman_b_int_of_a_int we have that b \\in Z and so t \\in Z[\\sqrtd] \\subseteq Z…","kind":"proof","summary":"By Lemma \\refhuman_b_int_of_a_int we have that b \\in Z and so t \\in Z[\\sqrtd] \\subseteq Z\\left[…","labels":[],"detail_key":"p50"},{"id":"n41713","layer":"informal","project":"p50","title":"By Lemma \\refhuman_easy_incl_d_1 we know that Z\\left[ \\frac1+\\sqrtd2 \\right] \\subseteq O_…","kind":"proof","summary":"By Lemma \\refhuman_easy_incl_d_1 we know that Z\\left[ \\frac1+\\sqrtd2 \\right] \\subseteq O_K. Let…","labels":[],"detail_key":"p50"},{"id":"n41714","layer":"informal","project":"p51","title":"def:regular-prime","kind":"definition","summary":"An odd prime p is \\emphregular when p does not divide the class number of the pth cyclotomic fi…","labels":["def:regular-prime"],"detail_key":"p51"},{"id":"n41715","layer":"informal","project":"p51","title":"def:cm-real-subfield","kind":"definition","summary":"For a number field K, let K^+ denote its maximal real subfield, the largest subfield of K all o…","labels":["def:cm-real-subfield"],"detail_key":"p51"},{"id":"n41716","layer":"informal","project":"p51","title":"def:class-number-h","kind":"definition","summary":"For a number field K, the class number h(K) is the cardinality of the ideal class group of its…","labels":["def:class-number-h"],"detail_key":"p51"},{"id":"n41717","layer":"informal","project":"p51","title":"def:class-number-hplus","kind":"definition","summary":"For a CM field K, the plus class number is the class number of the maximal real subfield: \\[ h^…","labels":["def:class-number-hplus"],"detail_key":"p51"},{"id":"n41718","layer":"informal","project":"p51","title":"thm:map-comap-ring-of-integers","kind":"theorem","summary":"If J is an ideal of O_L^+, then extension to O_L followed by contraction is the identity: \\[ (J…","labels":["thm:map-comap-ring-of-integers"],"detail_key":"p51"},{"id":"n41719","layer":"informal","project":"p51","title":"For a faithfully flat ring map A \\to B, every ideal J \\subseteq A satisfies JB \\cap A = J…","kind":"proof","summary":"For a faithfully flat ring map A \\to B, every ideal J \\subseteq A satisfies JB \\cap A = J: the…","labels":[],"detail_key":"p51"},{"id":"n41720","layer":"informal","project":"p51","title":"thm:is-principal-map-span-singleton","kind":"theorem","summary":"Let I be an ideal of O_K^+. If I O_K is generated by the image of an element b_0\\in O_K^+, then…","labels":["thm:is-principal-map-span-singleton"],"detail_key":"p51"},{"id":"n41721","layer":"informal","project":"p51","title":"Contract the equality I O_K=(b_0) back to O_K^+. The left-hand side contracts to I by \\cr…","kind":"proof","summary":"Contract the equality I O_K=(b_0) back to O_K^+. The left-hand side contracts to I by \\crefthm:…","labels":[],"detail_key":"p51"},{"id":"n41722","layer":"informal","project":"p51","title":"thm:ideal-map-conj","kind":"theorem","summary":"If I is an ideal of O_K^+, then the extended ideal I O_K is fixed by complex conjugation.","labels":["thm:ideal-map-conj"],"detail_key":"p51"},{"id":"n41723","layer":"informal","project":"p51","title":"The extension I O_K is generated by elements coming from K^+, and those elements are fixe…","kind":"proof","summary":"The extension I O_K is generated by elements coming from K^+, and those elements are fixed by c…","labels":[],"detail_key":"p51"},{"id":"n41724","layer":"informal","project":"p51","title":"thm:conj-generator-associated","kind":"theorem","summary":"Suppose I \\ne 0 and I O_K=(a). Then, by \\crefthm:ideal-map-conj, also I O_K=(\\overline a), and…","labels":["thm:conj-generator-associated"],"detail_key":"p51"},{"id":"n41725","layer":"informal","project":"p51","title":"Two generators of the same nonzero principal ideal in a domain differ by a unit: from (a)…","kind":"proof","summary":"Two generators of the same nonzero principal ideal in a domain differ by a unit: from (a) = (\\o…","labels":[],"detail_key":"p51"},{"id":"n41726","layer":"informal","project":"p51","title":"thm:conj-unit-mul-one","kind":"theorem","summary":"The unit u obtained from \\overline a=ua in \\crefthm:conj-generator-associated satisfies \\[ u\\ov…","labels":["thm:conj-unit-mul-one"],"detail_key":"p51"},{"id":"n41727","layer":"informal","project":"p51","title":"Apply complex conjugation to \\overline a=ua: since conjugation is an involution, a = \\ove…","kind":"proof","summary":"Apply complex conjugation to \\overline a=ua: since conjugation is an involution, a = \\overline…","labels":[],"detail_key":"p51"},{"id":"n41728","layer":"informal","project":"p51","title":"thm:antisymmetric-unit-classification","kind":"theorem","summary":"Let K be the pth cyclotomic field with p odd. If u\\in O_K^\\times satisfies u\\overline u=1, then…","labels":["thm:antisymmetric-unit-classification"],"detail_key":"p51"},{"id":"n41729","layer":"informal","project":"p51","title":"At every complex embedding \\sigma of K, conjugation corresponds to complex conjugation of…","kind":"proof","summary":"At every complex embedding \\sigma of K, conjugation corresponds to complex conjugation of the i…","labels":[],"detail_key":"p51"},{"id":"n41730","layer":"informal","project":"p51","title":"thm:conj-fixed-associate","kind":"theorem","summary":"In the situation of \\crefthm:conj-generator-associated, the generator a may be replaced by an a…","labels":["thm:conj-fixed-associate"],"detail_key":"p51"},{"id":"n41731","layer":"informal","project":"p51","title":"By \\crefthm:antisymmetric-unit-classification, the unit u = \\overline a / a has the form…","kind":"proof","summary":"By \\crefthm:antisymmetric-unit-classification, the unit u = \\overline a / a has the form (-1)^n…","labels":[],"detail_key":"p51"},{"id":"n41732","layer":"informal","project":"p51","title":"thm:mem-ring-of-integers-conj-self","kind":"theorem","summary":"If b\\in O_K is fixed by complex conjugation, then it is the image of an element of O_K^+.","labels":["thm:mem-ring-of-integers-conj-self"],"detail_key":"p51"},{"id":"n41733","layer":"informal","project":"p51","title":"For a CM field, K^+ is exactly the fixed field of conjugation, so the element b lies in K…","kind":"proof","summary":"For a CM field, K^+ is exactly the fixed field of conjugation, so the element b lies in K^+. Be…","labels":[],"detail_key":"p51"},{"id":"n41734","layer":"informal","project":"p51","title":"thm:is-principal-after-extension","kind":"theorem","summary":"Let K be the pth cyclotomic field, with p odd. If an ideal of O_K^+ becomes principal after ext…","labels":["thm:is-principal-after-extension"],"detail_key":"p51"},{"id":"n41735","layer":"informal","project":"p51","title":"Let I be such an ideal; we may assume I \\ne 0. Choose a generator a of I O_K. The extende…","kind":"proof","summary":"Let I be such an ideal; we may assume I \\ne 0. Choose a generator a of I O_K. The extended idea…","labels":[],"detail_key":"p51"},{"id":"n41736","layer":"informal","project":"p51","title":"def:class-group-map","kind":"definition","summary":"The inclusion O_K^+\\subseteq O_K induces, by extension of fractional ideals, the class-group ho…","labels":["def:class-group-map"],"detail_key":"p51"},{"id":"n41737","layer":"informal","project":"p51","title":"thm:class-group-map-injective","kind":"theorem","summary":"For an odd prime cyclotomic field, the map \\[ Cl( O_K^+)\\longrightarrow Cl( O_K) \\] is injectiv…","labels":["thm:class-group-map-injective"],"detail_key":"p51"},{"id":"n41738","layer":"informal","project":"p51","title":"A group homomorphism is injective when its kernel is trivial. If an ideal class of O_K^+…","kind":"proof","summary":"A group homomorphism is injective when its kernel is trivial. If an ideal class of O_K^+ maps t…","labels":[],"detail_key":"p51"},{"id":"n41739","layer":"informal","project":"p51","title":"thm:hplus-divides-h","kind":"theorem","summary":"The plus class number divides the full class number: \\[ h^+(K)\\mid h(K). \\]","labels":["thm:hplus-divides-h"],"detail_key":"p51"},{"id":"n41740","layer":"informal","project":"p51","title":"By \\crefthm:class-group-map-injective, Cl( O_K^+) embeds as a subgroup of the finite grou…","kind":"proof","summary":"By \\crefthm:class-group-map-injective, Cl( O_K^+) embeds as a subgroup of the finite group Cl(…","labels":[],"detail_key":"p51"},{"id":"n41741","layer":"informal","project":"p51","title":"def:class-number-hminus","kind":"definition","summary":"The relative class number is the natural-number quotient \\[ h^-(K)=h(K)/h^+(K), \\] which is wel…","labels":["def:class-number-hminus"],"detail_key":"p51"},{"id":"n41742","layer":"informal","project":"p51","title":"thm:h-factorisation","kind":"theorem","summary":"The class number factors as \\[ h(K)=h^+(K)\\,h^-(K). \\]","labels":["thm:h-factorisation"],"detail_key":"p51"},{"id":"n41743","layer":"informal","project":"p51","title":"Multiply the defining quotient of \\crefdef:class-number-hminus by h^+(K); this is legitim…","kind":"proof","summary":"Multiply the defining quotient of \\crefdef:class-number-hminus by h^+(K); this is legitimate in…","labels":[],"detail_key":"p51"},{"id":"n41744","layer":"informal","project":"p51","title":"thm:dvd-h-iff-dvd-hminus-from-plus-to-minus","kind":"theorem","summary":"Assume that a separate argument proves the implication \\[ p\\mid h^+(K)\\Longrightarrow p\\mid h^-…","labels":["thm:dvd-h-iff-dvd-hminus-from-plus-to-minus"],"detail_key":"p51"},{"id":"n41745","layer":"informal","project":"p51","title":"Write h(K) = h^+(K)\\,h^-(K) by \\crefthm:h-factorisation. If p \\mid h(K), then by primalit…","kind":"proof","summary":"Write h(K) = h^+(K)\\,h^-(K) by \\crefthm:h-factorisation. If p \\mid h(K), then by primality p\\mi…","labels":[],"detail_key":"p51"},{"id":"n41746","layer":"informal","project":"p51","title":"def:dirichlet-character","kind":"definition","summary":"A \\emphDirichlet character modulo N with values in a commutative ring R is a multiplicative cha…","labels":["def:dirichlet-character"],"detail_key":"p51"},{"id":"n41747","layer":"informal","project":"p51","title":"def:generalized-bernoulli","kind":"definition","summary":"For a Dirichlet character \\chi modulo N and n \\ge 0, the \\emphgeneralised Bernoulli number is \\…","labels":["def:generalized-bernoulli"],"detail_key":"p51"},{"id":"n41748","layer":"informal","project":"p51","title":"def:teichmuller","kind":"definition","summary":"The \\emphTeichm\\\"uller character of the prime p is the unique character \\[ \\omega\\colon\\left(Z/…","labels":["def:teichmuller"],"detail_key":"p51"},{"id":"n41749","layer":"informal","project":"p51","title":"thm:bernoulli-padicInt-below","kind":"theorem","summary":"For 0<n<p-1, the classical Bernoulli number B_n is p-adically integral: B_n \\in Z_p.","labels":["thm:bernoulli-padicInt-below"],"detail_key":"p51"},{"id":"n41750","layer":"informal","project":"p51","title":"By the von Staudt--Clausen theorem, the denominator of B_n (for n even, the only nontrivi…","kind":"proof","summary":"By the von Staudt--Clausen theorem, the denominator of B_n (for n even, the only nontrivial cas…","labels":[],"detail_key":"p51"},{"id":"n41751","layer":"informal","project":"p51","title":"thm:bernoulli-den-coprime","kind":"theorem","summary":"For 0<n<p-1, the prime p does not divide the denominator of B_n.","labels":["thm:bernoulli-den-coprime"],"detail_key":"p51"},{"id":"n41752","layer":"informal","project":"p51","title":"This is the rational form of \\crefthm:bernoulli-padicInt-below: a rational number lies in…","kind":"proof","summary":"This is the rational form of \\crefthm:bernoulli-padicInt-below: a rational number lies in Z_p e…","labels":[],"detail_key":"p51"},{"id":"n41753","layer":"informal","project":"p51","title":"lem:padic-bernoulli-factor","kind":"lemma","summary":"For 0<n<p-1, the quotient B_n/n lies in Z_p, and it is a p-adic unit if and only if p does not…","labels":["lem:padic-bernoulli-factor"],"detail_key":"p51"},{"id":"n41754","layer":"informal","project":"p51","title":"The denominator of B_n is prime to p by \\crefthm:bernoulli-den-coprime, and n < p-1 < p i…","kind":"proof","summary":"The denominator of B_n is prime to p by \\crefthm:bernoulli-den-coprime, and n < p-1 < p is also…","labels":[],"detail_key":"p51"},{"id":"n41755","layer":"informal","project":"p51","title":"thm:teichmuller-bernoulli-congruence","kind":"theorem","summary":"For the odd exponents j occurring in the relative class-number formula (odd j with j + 1 divisi…","labels":["thm:teichmuller-bernoulli-congruence"],"detail_key":"p51"},{"id":"n41756","layer":"informal","project":"p51","title":"This is the standard congruence linking generalised Bernoulli values of Teichm\\\"uller pow…","kind":"proof","summary":"This is the standard congruence linking generalised Bernoulli values of Teichm\\\"uller powers to…","labels":[],"detail_key":"p51"},{"id":"n41757","layer":"informal","project":"p51","title":"thm:hplus-even-lvalues","kind":"theorem","summary":"The plus class number is given by the even part of the cyclotomic L-value product: \\[ h^+(K)= c…","labels":["thm:hplus-even-lvalues"],"detail_key":"p51"},{"id":"n41758","layer":"informal","project":"p51","title":"The analytic class-number formula for the totally real field K^+ expresses h^+ through th…","kind":"proof","summary":"The analytic class-number formula for the totally real field K^+ expresses h^+ through the resi…","labels":[],"detail_key":"p51"},{"id":"n41759","layer":"informal","project":"p51","title":"thm:raw-gauss-product","kind":"theorem","summary":"The product of the Gauss sums over the odd characters modulo p is \\[ \\prod_\\chi\\ odd \\tau(\\chi)…","labels":["thm:raw-gauss-product"],"detail_key":"p51"},{"id":"n41760","layer":"informal","project":"p51","title":"The proof splits into the cases p\\equiv1\\pmod4 and p\\equiv3\\pmod4. A non-quadratic odd ch…","kind":"proof","summary":"The proof splits into the cases p\\equiv1\\pmod4 and p\\equiv3\\pmod4. A non-quadratic odd characte…","labels":[],"detail_key":"p51"},{"id":"n41761","layer":"informal","project":"p51","title":"thm:cyclotomic-gauss-goal","kind":"theorem","summary":"The explicit Gauss-product evaluation of \\crefthm:raw-gauss-product satisfies the packaged Gaus…","labels":["thm:cyclotomic-gauss-goal"],"detail_key":"p51"},{"id":"n41762","layer":"informal","project":"p51","title":"Purely formal: the assembly theorem consumes the Gauss product in a normalised shape (wit…","kind":"proof","summary":"Purely formal: the assembly theorem consumes the Gauss product in a normalised shape (with the…","labels":[],"detail_key":"p51"},{"id":"n41763","layer":"informal","project":"p51","title":"thm:hminus-formula-from-inputs","kind":"theorem","summary":"Assume the factorisation of the Dedekind zeta residue of K into the even and odd L-value produc…","labels":["thm:hminus-formula-from-inputs"],"detail_key":"p51"},{"id":"n41764","layer":"informal","project":"p51","title":"Divide the analytic class-number formula of K (in residue form) by that of K^+. Using h =…","kind":"proof","summary":"Divide the analytic class-number formula of K (in residue form) by that of K^+. Using h = h^+ h…","labels":[],"detail_key":"p51"},{"id":"n41765","layer":"informal","project":"p51","title":"thm:hminus-formula","kind":"theorem","summary":"For an odd prime cyclotomic field, \\[ h^-(K) = 2p\\prod_\\chi\\ odd \\left(-\\frac12 B_1,\\chi^-1\\rig…","labels":["thm:hminus-formula"],"detail_key":"p51"},{"id":"n41766","layer":"informal","project":"p51","title":"Supply the three inputs of \\crefthm:hminus-formula-from-inputs: the residue of \\zeta_K fa…","kind":"proof","summary":"Supply the three inputs of \\crefthm:hminus-formula-from-inputs: the residue of \\zeta_K factors…","labels":[],"detail_key":"p51"},{"id":"n41767","layer":"informal","project":"p51","title":"thm:hminus-teichmuller-mod-p","kind":"theorem","summary":"There exists z\\inZ_p such that, inside Q_p, \\[ h^-(K) = \\prod_\\substackj<p-2\\\\ j\\ odd \\left(-\\f…","labels":["thm:hminus-teichmuller-mod-p"],"detail_key":"p51"},{"id":"n41768","layer":"informal","project":"p51","title":"Starting from \\crefthm:hminus-formula, index the odd characters modulo p as \\omega^j with…","kind":"proof","summary":"Starting from \\crefthm:hminus-formula, index the odd characters modulo p as \\omega^j with j odd…","labels":[],"detail_key":"p51"},{"id":"n41769","layer":"informal","project":"p51","title":"thm:hminus-bernoulli-mod-p","kind":"theorem","summary":"There exists z\\inZ_p such that, inside Q_p, \\[ h^-(K) = \\prod_\\substackj<p-2\\\\ j\\ odd \\left(-\\f…","labels":["thm:hminus-bernoulli-mod-p"],"detail_key":"p51"},{"id":"n41770","layer":"informal","project":"p51","title":"Replace each factor B_1,\\omega^j in \\crefthm:hminus-teichmuller-mod-p by the congruent cl…","kind":"proof","summary":"Replace each factor B_1,\\omega^j in \\crefthm:hminus-teichmuller-mod-p by the congruent classica…","labels":[],"detail_key":"p51"},{"id":"n41771","layer":"informal","project":"p51","title":"lem:odd-index-to-k","kind":"lemma","summary":"For any predicate Q on natural numbers, \\[ \\bigl(\\exists j,\\ j<p-2,\\ j\\ odd,\\ Q(j+1)\\bigr) \\qua…","labels":["lem:odd-index-to-k"],"detail_key":"p51"},{"id":"n41772","layer":"informal","project":"p51","title":"If j is odd, write j=2k-1 with k \\ge 1, so j+1=2k, and j < p-2 becomes 2k \\le p-3 after p…","kind":"proof","summary":"If j is odd, write j=2k-1 with k \\ge 1, so j+1=2k, and j < p-2 becomes 2k \\le p-3 after parity…","labels":[],"detail_key":"p51"},{"id":"n41773","layer":"informal","project":"p51","title":"thm:hminus-bernoulli-final","kind":"theorem","summary":"For p odd, \\[ p\\mid h^-(K) \\quad\\Longleftrightarrow\\quad \\exists k,\\ 1\\le k,\\ 2k\\le p-3,\\quad p…","labels":["thm:hminus-bernoulli-final"],"detail_key":"p51"},{"id":"n41774","layer":"informal","project":"p51","title":"By \\crefthm:hminus-bernoulli-mod-p, h^- is congruent modulo p to the finite product of th…","kind":"proof","summary":"By \\crefthm:hminus-bernoulli-mod-p, h^- is congruent modulo p to the finite product of the p-ad…","labels":[],"detail_key":"p51"},{"id":"n41775","layer":"informal","project":"p51","title":"thm:hminus-nondiv-to-bernoulli-nondiv","kind":"theorem","summary":"If p\\nmid h^-(K), then for every k with 1\\le k and 2k\\le p-3, \\[ p\\nmidnum(B_2k). \\]","labels":["thm:hminus-nondiv-to-bernoulli-nondiv"],"detail_key":"p51"},{"id":"n41776","layer":"informal","project":"p51","title":"Contrapositive of one direction of \\crefthm:hminus-bernoulli-final: a numerator divisible…","kind":"proof","summary":"Contrapositive of one direction of \\crefthm:hminus-bernoulli-final: a numerator divisible by p…","labels":[],"detail_key":"p51"},{"id":"n41777","layer":"informal","project":"p51","title":"def:real-cyclotomic-unit","kind":"definition","summary":"For 2\\le a\\le (p-1)/2, the \\emphreal cyclotomic unit \\varepsilon_a is the unit of O_K^+ obtaine…","labels":["def:real-cyclotomic-unit"],"detail_key":"p51"},{"id":"n41778","layer":"informal","project":"p51","title":"def:CPlus-generator","kind":"definition","summary":"The \\emphstandard generators are the (p-3)/2 real cyclotomic units \\[ \\varepsilon_2,\\ \\varepsil…","labels":["def:CPlus-generator"],"detail_key":"p51"},{"id":"n41779","layer":"informal","project":"p51","title":"def:CPlus","kind":"definition","summary":"The subgroup C^+\\subseteq( O_K^+)^\\times is generated by -1 and the standard real cyclotomic un…","labels":["def:CPlus"],"detail_key":"p51"},{"id":"n41780","layer":"informal","project":"p51","title":"def:normalizedCPlus","kind":"definition","summary":"For 2 \\le a \\le (p-1)/2, the \\emphnormalised cyclotomic unit is \\[ \\xi_a \\;=\\; \\zeta_p^\\,(1-a)/…","labels":["def:normalizedCPlus"],"detail_key":"p51"},{"id":"n41781","layer":"informal","project":"p51","title":"thm:normalized-square","kind":"theorem","summary":"Each normalised generator squares to the corresponding standard generator: \\[ \\xi_a^\\,2 = \\vare…","labels":["thm:normalized-square"],"detail_key":"p51"},{"id":"n41782","layer":"informal","project":"p51","title":"Using the identity 1-\\zeta_p^-a = -\\zeta_p^-a(1-\\zeta_p^a) in the numerator and denominat…","kind":"proof","summary":"Using the identity 1-\\zeta_p^-a = -\\zeta_p^-a(1-\\zeta_p^a) in the numerator and denominator of…","labels":[],"detail_key":"p51"},{"id":"n41783","layer":"informal","project":"p51","title":"thm:CPlus-normalized-index","kind":"theorem","summary":"For an odd prime p, \\[ p\\mid[( O_K^+)^\\times:C^+] \\Longleftrightarrow p\\mid[( O_K^+)^\\times:C^+…","labels":["thm:CPlus-normalized-index"],"detail_key":"p51"},{"id":"n41784","layer":"informal","project":"p51","title":"By \\crefthm:normalized-square, C^+ \\subseteq C^+_norm and every square of a normalised ge…","kind":"proof","summary":"By \\crefthm:normalized-square, C^+ \\subseteq C^+_norm and every square of a normalised generato…","labels":[],"detail_key":"p51"},{"id":"n41785","layer":"informal","project":"p51","title":"def:EPlus","kind":"definition","summary":"The ambient group for the saturation argument is the full unit group \\[ E^+ = ( O_K^+)^\\times .…","labels":["def:EPlus"],"detail_key":"p51"},{"id":"n41786","layer":"informal","project":"p51","title":"def:pPowerSubgroup","kind":"definition","summary":"For a subgroup H of an abelian group, H^p denotes the exact image of the pth-power map: \\[ H^p=…","labels":["def:pPowerSubgroup"],"detail_key":"p51"},{"id":"n41787","layer":"informal","project":"p51","title":"def:pSaturated","kind":"definition","summary":"A subgroup H\\le E is \\emphp-saturated in E when \\[ H\\cap E^p\\subseteq H^p, \\] that is, when eve…","labels":["def:pSaturated"],"detail_key":"p51"},{"id":"n41788","layer":"informal","project":"p51","title":"def:CPlusExponentProduct","kind":"definition","summary":"An \\emphexponent product is an expression \\[ (-1)^s\\prod_a=2^(p-1)/2\\varepsilon_a^\\,e_a \\] with…","labels":["def:CPlusExponentProduct"],"detail_key":"p51"},{"id":"n41789","layer":"informal","project":"p51","title":"thm:exists-CPlusExponentProduct","kind":"theorem","summary":"Every element of C^+ admits an exponent-product expression.","labels":["thm:exists-CPlusExponentProduct"],"detail_key":"p51"},{"id":"n41790","layer":"informal","project":"p51","title":"Induction over membership in the subgroup generated by -1 and the \\varepsilon_a: each gen…","kind":"proof","summary":"Induction over membership in the subgroup generated by -1 and the \\varepsilon_a: each generator…","labels":[],"detail_key":"p51"},{"id":"n41791","layer":"informal","project":"p51","title":"thm:CPlus-pSaturated-exponents","kind":"theorem","summary":"Assume that whenever an exponent product \\[ (-1)^s\\prod_a\\varepsilon_a^\\,e_a \\] is a pth power…","labels":["thm:CPlus-pSaturated-exponents"],"detail_key":"p51"},{"id":"n41792","layer":"informal","project":"p51","title":"Let x \\in C^+ be a pth power in E^+, and write x = (-1)^s \\prod_a \\varepsilon_a^e_a by \\c…","kind":"proof","summary":"Let x \\in C^+ be a pth power in E^+, and write x = (-1)^s \\prod_a \\varepsilon_a^e_a by \\crefthm…","labels":[],"detail_key":"p51"},{"id":"n41793","layer":"informal","project":"p51","title":"thm:not-dvd-index-of-pSaturated","kind":"theorem","summary":"If C^+ is p-saturated in E^+, then \\[ p\\nmid[E^+:C^+]. \\]","labels":["thm:not-dvd-index-of-pSaturated"],"detail_key":"p51"},{"id":"n41794","layer":"informal","project":"p51","title":"The group-theoretic statement is: a finite-index p-saturated subgroup H \\le E whose inclu…","kind":"proof","summary":"The group-theoretic statement is: a finite-index p-saturated subgroup H \\le E whose inclusion c…","labels":[],"detail_key":"p51"},{"id":"n41795","layer":"informal","project":"p51","title":"def:kummerLogRank","kind":"definition","summary":"Set \\[ r \\;=\\; \\fracp-32, \\] the common cardinality of the set of standard generators \\varepsil…","labels":["def:kummerLogRank"],"detail_key":"p51"},{"id":"n41796","layer":"informal","project":"p51","title":"def:concreteKummerLogMatrix","kind":"definition","summary":"The \\emphKummer logarithm matrix M is the r \\times r matrix over F_p whose ath column consists…","labels":["def:concreteKummerLogMatrix"],"detail_key":"p51"},{"id":"n41797","layer":"informal","project":"p51","title":"thm:kummer-log-vandermonde","kind":"theorem","summary":"The Kummer logarithm matrix factors as \\[ M = D \\cdot V, \\] where D is the diagonal matrix of B…","labels":["thm:kummer-log-vandermonde"],"detail_key":"p51"},{"id":"n41798","layer":"informal","project":"p51","title":"Expand the completed logarithm of each generator in the Dwork basis and compute the coeff…","kind":"proof","summary":"Expand the completed logarithm of each generator in the Dwork basis and compute the coefficient…","labels":[],"detail_key":"p51"},{"id":"n41799","layer":"informal","project":"p51","title":"thm:vandermonde-nonzero","kind":"theorem","summary":"The Teichm\\\"uller Vandermonde determinant occurring in \\crefthm:kummer-log-vandermonde is nonze…","labels":["thm:vandermonde-nonzero"],"detail_key":"p51"},{"id":"n41800","layer":"informal","project":"p51","title":"The nodes are the values \\eta_a^2 for 2 \\le a \\le (p-1)/2; they are pairwise distinct and…","kind":"proof","summary":"The nodes are the values \\eta_a^2 for 2 \\le a \\le (p-1)/2; they are pairwise distinct and disti…","labels":[],"detail_key":"p51"},{"id":"n41801","layer":"informal","project":"p51","title":"thm:kummer-det-bernoulli-factor","kind":"theorem","summary":"\\det M \\ne 0 if and only if every diagonal Bernoulli factor is nonzero.","labels":["thm:kummer-det-bernoulli-factor"],"detail_key":"p51"},{"id":"n41802","layer":"informal","project":"p51","title":"By \\crefthm:kummer-log-vandermonde, \\det M = \\det D \\cdot \\det V, and \\det V \\ne 0 by \\cr…","kind":"proof","summary":"By \\crefthm:kummer-log-vandermonde, \\det M = \\det D \\cdot \\det V, and \\det V \\ne 0 by \\crefthm:…","labels":[],"detail_key":"p51"},{"id":"n41803","layer":"informal","project":"p51","title":"thm:bernoulliFactor-num","kind":"theorem","summary":"The diagonal Bernoulli factor attached to j is nonzero in F_p exactly when \\[ p\\nmidnum(B_2j).…","labels":["thm:bernoulliFactor-num"],"detail_key":"p51"},{"id":"n41804","layer":"informal","project":"p51","title":"The factor is the reduction modulo p of a p-integral rational expression which is a p-adi…","kind":"proof","summary":"The factor is the reduction modulo p of a p-integral rational expression which is a p-adic unit…","labels":[],"detail_key":"p51"},{"id":"n41805","layer":"informal","project":"p51","title":"thm:kummer-log-det-bernoulli","kind":"theorem","summary":"The Kummer logarithm determinant is nonzero precisely when no Bernoulli numerator in Kummer's r…","labels":["thm:kummer-log-det-bernoulli"],"detail_key":"p51"},{"id":"n41806","layer":"informal","project":"p51","title":"\\Crefthm:kummer-det-bernoulli-factor reduces the determinant condition to the nonvanishin…","kind":"proof","summary":"\\Crefthm:kummer-det-bernoulli-factor reduces the determinant condition to the nonvanishing of e…","labels":[],"detail_key":"p51"},{"id":"n41807","layer":"informal","project":"p51","title":"thm:completed-log-relation","kind":"theorem","summary":"If an exponent product in C^+ is a pth power in the full unit group, then its completed logarit…","labels":["thm:completed-log-relation"],"detail_key":"p51"},{"id":"n41808","layer":"informal","project":"p51","title":"The completed logarithm is a homomorphism on the relevant units: it turns the exponent pr…","kind":"proof","summary":"The completed logarithm is a homomorphism on the relevant units: it turns the exponent product…","labels":[],"detail_key":"p51"},{"id":"n41809","layer":"informal","project":"p51","title":"thm:kummer-mulvec-zero","kind":"theorem","summary":"In the situation of \\crefthm:completed-log-relation, \\[ M e=0 \\quad\\textover F_p, \\] where e is…","labels":["thm:kummer-mulvec-zero"],"detail_key":"p51"},{"id":"n41810","layer":"informal","project":"p51","title":"Take coordinates in the Dwork basis. The jth coordinate of the combination \\sum_a e_a \\lo…","kind":"proof","summary":"Take coordinates in the Dwork basis. The jth coordinate of the combination \\sum_a e_a \\log\\vare…","labels":[],"detail_key":"p51"},{"id":"n41811","layer":"informal","project":"p51","title":"thm:kummer-linear-algebra","kind":"theorem","summary":"If \\det M\\ne0 and Me=0, then e = 0 in F_p^\\,r.","labels":["thm:kummer-linear-algebra"],"detail_key":"p51"},{"id":"n41812","layer":"informal","project":"p51","title":"Over the field F_p, a square matrix with nonzero determinant is invertible, so its kernel…","kind":"proof","summary":"Over the field F_p, a square matrix with nonzero determinant is invertible, so its kernel is tr…","labels":[],"detail_key":"p51"},{"id":"n41813","layer":"informal","project":"p51","title":"thm:kummer-log-exponents-zero","kind":"theorem","summary":"If \\det M \\ne 0, then every exponent product in C^+ which is a pth power in the full unit group…","labels":["thm:kummer-log-exponents-zero"],"detail_key":"p51"},{"id":"n41814","layer":"informal","project":"p51","title":"Apply \\crefthm:completed-log-relation to the pth-power relation, then \\crefthm:kummer-mul…","kind":"proof","summary":"Apply \\crefthm:completed-log-relation to the pth-power relation, then \\crefthm:kummer-mulvec-ze…","labels":[],"detail_key":"p51"},{"id":"n41815","layer":"informal","project":"p51","title":"thm:kummer-log-det-saturation","kind":"theorem","summary":"If \\det M \\ne 0, then C^+ is p-saturated in ( O_K^+)^\\times.","labels":["thm:kummer-log-det-saturation"],"detail_key":"p51"},{"id":"n41816","layer":"informal","project":"p51","title":"\\Crefthm:kummer-log-exponents-zero is exactly the exponent-vanishing hypothesis of the sa…","kind":"proof","summary":"\\Crefthm:kummer-log-exponents-zero is exactly the exponent-vanishing hypothesis of the saturati…","labels":[],"detail_key":"p51"},{"id":"n41817","layer":"informal","project":"p51","title":"def:cyclotomicUnitIndexSubgroup","kind":"definition","summary":"The subgroup featuring in the prime-conductor index theorem is the normalised cyclotomic-unit s…","labels":["def:cyclotomicUnitIndexSubgroup"],"detail_key":"p51"},{"id":"n41818","layer":"informal","project":"p51","title":"thm:index-subgroup-eq-CPlus","kind":"theorem","summary":"Up to the normalisation of \\crefthm:CPlus-normalized-index, the subgroup used in the index theo…","labels":["thm:index-subgroup-eq-CPlus"],"detail_key":"p51"},{"id":"n41819","layer":"informal","project":"p51","title":"Both inclusions are checked on generators. A normalised generator \\xi_a times a root of u…","kind":"proof","summary":"Both inclusions are checked on generators. A normalised generator \\xi_a times a root of unity r…","labels":[],"detail_key":"p51"},{"id":"n41820","layer":"informal","project":"p51","title":"thm:deleted-fourier-kummer-dirichlet","kind":"theorem","summary":"The deleted-Fourier determinant identity supplies the Kummer--Dirichlet determinant hypothesis…","labels":["thm:deleted-fourier-kummer-dirichlet"],"detail_key":"p51"},{"id":"n41821","layer":"informal","project":"p51","title":"The logarithmic embeddings of the cyclotomic units form a matrix whose group-theoretic st…","kind":"proof","summary":"The logarithmic embeddings of the cyclotomic units form a matrix whose group-theoretic structur…","labels":[],"detail_key":"p51"},{"id":"n41822","layer":"informal","project":"p51","title":"thm:index-pprimary-aux","kind":"theorem","summary":"For p\\ge5, the normalised cyclotomic-unit index has the same p-divisibility as h^+: \\[ p\\mid[(…","labels":["thm:index-pprimary-aux"],"detail_key":"p51"},{"id":"n41823","layer":"informal","project":"p51","title":"This is Sinnott's prime-conductor index theorem: for prime conductor the index of the cir…","kind":"proof","summary":"This is Sinnott's prime-conductor index theorem: for prime conductor the index of the circular…","labels":[],"detail_key":"p51"},{"id":"n41824","layer":"informal","project":"p51","title":"thm:sinnott-index-pprimary","kind":"theorem","summary":"For every odd prime conductor, \\[ p\\mid[( O_K^+)^\\times:C^+_norm] \\Longleftrightarrow p\\mid h^+…","labels":["thm:sinnott-index-pprimary"],"detail_key":"p51"},{"id":"n41825","layer":"informal","project":"p51","title":"For p=3 the field K^+ is Q: the normalised subgroup is the whole unit group \\\\pm1\\ and h^…","kind":"proof","summary":"For p=3 the field K^+ is Q: the normalised subgroup is the whole unit group \\\\pm1\\ and h^+=1, s…","labels":[],"detail_key":"p51"},{"id":"n41826","layer":"informal","project":"p51","title":"thm:bernoulli-nonzero-index","kind":"theorem","summary":"Let p be an odd prime. Suppose that \\[ \\forall j,\\ 1\\le j,\\ 2j\\le p-3,\\quad p\\nmidnum(B_2j). \\]…","labels":["thm:bernoulli-nonzero-index"],"detail_key":"p51"},{"id":"n41827","layer":"informal","project":"p51","title":"If p=3, Kummer's range is empty, K^+ = Q, the group C^+ is the full unit group \\\\pm1\\, an…","kind":"proof","summary":"If p=3, Kummer's range is empty, K^+ = Q, the group C^+ is the full unit group \\\\pm1\\, and the…","labels":[],"detail_key":"p51"},{"id":"n41828","layer":"informal","project":"p51","title":"thm:not-dvd-hplus-of-not-dvd-hminus-units","kind":"theorem","summary":"If p\\nmid h^-(K), then p\\nmid h^+(K).","labels":["thm:not-dvd-hplus-of-not-dvd-hminus-units"],"detail_key":"p51"},{"id":"n41829","layer":"informal","project":"p51","title":"By \\crefthm:hminus-nondiv-to-bernoulli-nondiv, the hypothesis p\\nmid h^-(K) implies that…","kind":"proof","summary":"By \\crefthm:hminus-nondiv-to-bernoulli-nondiv, the hypothesis p\\nmid h^-(K) implies that no Ber…","labels":[],"detail_key":"p51"},{"id":"n41830","layer":"informal","project":"p51","title":"thm:hplus-to-hminus-units","kind":"theorem","summary":"If p\\mid h^+(K), then p\\mid h^-(K).","labels":["thm:hplus-to-hminus-units"],"detail_key":"p51"},{"id":"n41831","layer":"informal","project":"p51","title":"Contrapositive of \\crefthm:not-dvd-hplus-of-not-dvd-hminus-units: if p\\nmid h^-(K), that…","kind":"proof","summary":"Contrapositive of \\crefthm:not-dvd-hplus-of-not-dvd-hminus-units: if p\\nmid h^-(K), that theore…","labels":[],"detail_key":"p51"},{"id":"n41832","layer":"informal","project":"p51","title":"thm:dvd-h-iff-bernoulli-units","kind":"theorem","summary":"For an odd prime p and the pth cyclotomic field K, \\[ p\\mid h(K) \\quad\\Longleftrightarrow\\quad…","labels":["thm:dvd-h-iff-bernoulli-units"],"detail_key":"p51"},{"id":"n41833","layer":"informal","project":"p51","title":"The implication p\\mid h^+(K)\\Rightarrow p\\mid h^-(K) is \\crefthm:hplus-to-hminus-units; f…","kind":"proof","summary":"The implication p\\mid h^+(K)\\Rightarrow p\\mid h^-(K) is \\crefthm:hplus-to-hminus-units; feeding…","labels":[],"detail_key":"p51"},{"id":"n41834","layer":"informal","project":"p51","title":"thm:kummer-criterion","kind":"theorem","summary":"Let p be an odd prime. Then \\[ p\\ \\textis regular \\quad\\Longleftrightarrow\\quad \\forall k,\\ 1\\l…","labels":["thm:kummer-criterion"],"detail_key":"p51"},{"id":"n41835","layer":"informal","project":"p51","title":"By \\crefdef:regular-prime, p is regular if and only if p is coprime to the class number o…","kind":"proof","summary":"By \\crefdef:regular-prime, p is regular if and only if p is coprime to the class number of the…","labels":[],"detail_key":"p51"},{"id":"n41836","layer":"informal","project":"p51","title":"thm:irregular-of-bernoulli-witness","kind":"theorem","summary":"Let p be an odd prime. If there is a k with 1\\le k, 2k\\le p-3 and p\\midnum(B_2k), then p is not…","labels":["thm:irregular-of-bernoulli-witness"],"detail_key":"p51"},{"id":"n41837","layer":"informal","project":"p51","title":"Immediate from \\crefthm:kummer-criterion.","kind":"proof","summary":"Immediate from \\crefthm:kummer-criterion.","labels":[],"detail_key":"p51"},{"id":"n41838","layer":"informal","project":"p51","title":"thm:bernoulli-witness-of-irregular","kind":"theorem","summary":"Conversely, if an odd prime p is not regular, then there is a k with 1\\le k, 2k\\le p-3 and p\\mi…","labels":["thm:bernoulli-witness-of-irregular"],"detail_key":"p51"},{"id":"n41839","layer":"informal","project":"p51","title":"Immediate from \\crefthm:kummer-criterion.","kind":"proof","summary":"Immediate from \\crefthm:kummer-criterion.","labels":[],"detail_key":"p51"},{"id":"n41840","layer":"informal","project":"p51","title":"lem:infinite-of-escape","kind":"lemma","summary":"Let P be a predicate on the natural numbers. If no finite set contains every n with P(n), then…","labels":["lem:infinite-of-escape"],"detail_key":"p51"},{"id":"n41841","layer":"informal","project":"p51","title":"If the set were finite, it would itself be a finite covering set.","kind":"proof","summary":"If the set were finite, it would itself be a finite covering set.","labels":[],"detail_key":"p51"},{"id":"n41842","layer":"informal","project":"p51","title":"lem:num-prime-of-large","kind":"lemma","summary":"If a rational number q has |q| > 1, then some prime divides num(q).","labels":["lem:num-prime-of-large"],"detail_key":"p51"},{"id":"n41843","layer":"informal","project":"p51","title":"Since |q| > 1, the numerator of |q| strictly exceeds its denominator; in particular |num(…","kind":"proof","summary":"Since |q| > 1, the numerator of |q| strictly exceeds its denominator; in particular |num(q)| \\g…","labels":[],"detail_key":"p51"},{"id":"n41844","layer":"informal","project":"p51","title":"lem:num-transport","kind":"lemma","summary":"Let q, r be rationals with q \\equiv r \\pmodp. If p\\midnum(q), then p\\midnum(r).","labels":["lem:num-transport"],"detail_key":"p51"},{"id":"n41845","layer":"informal","project":"p51","title":"Divisibility of the numerator by p is equivalent to q \\in pZ_p, which is invariant under…","kind":"proof","summary":"Divisibility of the numerator by p is equivalent to q \\in pZ_p, which is invariant under pertur…","labels":[],"detail_key":"p51"},{"id":"n41846","layer":"informal","project":"p51","title":"lem:num-of-divided","kind":"lemma","summary":"Let q be a rational and n a positive natural number. If p\\midnum(q/n), then p\\midnum(q).","labels":["lem:num-of-divided"],"detail_key":"p51"},{"id":"n41847","layer":"informal","project":"p51","title":"Dividing by n can only cancel factors of the numerator.","kind":"proof","summary":"Dividing by n can only cancel factors of the numerator.","labels":[],"detail_key":"p51"},{"id":"n41848","layer":"informal","project":"p51","title":"thm:vonstaudt-not-dvd-divided","kind":"theorem","summary":"Let p be a prime and n a positive even index with (p-1)\\mid n. Then p\\nmidnum(B_n/n).","labels":["thm:vonstaudt-not-dvd-divided"],"detail_key":"p51"},{"id":"n41849","layer":"informal","project":"p51","title":"By the von Staudt--Clausen theorem, \\[ B_n + \\sum_(q-1)\\mid n \\frac1q \\in Z, \\] the sum r…","kind":"proof","summary":"By the von Staudt--Clausen theorem, \\[ B_n + \\sum_(q-1)\\mid n \\frac1q \\in Z, \\] the sum running…","labels":[],"detail_key":"p51"},{"id":"n41850","layer":"informal","project":"p51","title":"lem:vonstaudt-p-integral","kind":"lemma","summary":"Let p be a prime and n an even index. Then pB_n \\in Z_p.","labels":["lem:vonstaudt-p-integral"],"detail_key":"p51"},{"id":"n41851","layer":"informal","project":"p51","title":"By von Staudt--Clausen, B_n differs from an integer by the correction sum \\sum_(q-1)\\mid…","kind":"proof","summary":"By von Staudt--Clausen, B_n differs from an integer by the correction sum \\sum_(q-1)\\mid n 1/q.…","labels":[],"detail_key":"p51"},{"id":"n41852","layer":"informal","project":"p51","title":"thm:bernoulli-growth-bound","kind":"theorem","summary":"For every k \\ge 1, \\[ \\left|\\fracB_2k2k\\right| \\ \\ge\\ \\frac(2k-1)!2^2k-1\\,\\pi^2k. \\]","labels":["thm:bernoulli-growth-bound"],"detail_key":"p51"},{"id":"n41853","layer":"informal","project":"p51","title":"Euler's formula gives \\[ \\zeta(2k) \\;=\\; (-1)^k+1\\,\\fracB_2k\\,(2\\pi)^2k2\\,(2k)!, \\] and \\…","kind":"proof","summary":"Euler's formula gives \\[ \\zeta(2k) \\;=\\; (-1)^k+1\\,\\fracB_2k\\,(2\\pi)^2k2\\,(2k)!, \\] and \\zeta(2…","labels":[],"detail_key":"p51"},{"id":"n41854","layer":"informal","project":"p51","title":"thm:bernoulli-growth-tendsto","kind":"theorem","summary":"\\left|B_2k/2k\\right| \\to \\infty as k \\to \\infty.","labels":["thm:bernoulli-growth-tendsto"],"detail_key":"p51"},{"id":"n41855","layer":"informal","project":"p51","title":"The lower bound of \\crefthm:bernoulli-growth-bound tends to infinity: factorial growth be…","kind":"proof","summary":"The lower bound of \\crefthm:bernoulli-growth-bound tends to infinity: factorial growth beats th…","labels":[],"detail_key":"p51"},{"id":"n41856","layer":"informal","project":"p51","title":"cor:bernoulli-large-multiple","kind":"corollary","summary":"For every positive even C there is a t with \\left|B_m/m\\right| > 1 for m = C\\cdot 2^t.","labels":["cor:bernoulli-large-multiple"],"detail_key":"p51"},{"id":"n41857","layer":"informal","project":"p51","title":"The indices C \\cdot 2^t are even and tend to infinity, so \\crefthm:bernoulli-growth-tends…","kind":"proof","summary":"The indices C \\cdot 2^t are even and tend to infinity, so \\crefthm:bernoulli-growth-tendsto app…","labels":[],"detail_key":"p51"},{"id":"n41858","layer":"informal","project":"p51","title":"def:irregular-base","kind":"definition","summary":"For a finite set S of natural numbers put M(S) = \\max(3, \\max S) and \\[ C(S) \\;=\\; 2 \\cdot M(S)…","labels":["def:irregular-base"],"detail_key":"p51"},{"id":"n41859","layer":"informal","project":"p51","title":"lem:irregular-base-mem","kind":"lemma","summary":"C(S) is even and positive, and for every prime q \\in S one has (q-1)\\mid C(S).","labels":["lem:irregular-base-mem"],"detail_key":"p51"},{"id":"n41860","layer":"informal","project":"p51","title":"q \\le M(S), so q - 1 is a positive integer at most M(S) and therefore divides M(S)!.","kind":"proof","summary":"q \\le M(S), so q - 1 is a positive integer at most M(S) and therefore divides M(S)!.","labels":[],"detail_key":"p51"},{"id":"n41861","layer":"informal","project":"p51","title":"lem:irregular-base-closed","kind":"lemma","summary":"Let m = C(S)\\cdot 2^t. Every prime q dividing m satisfies (q-1)\\mid m.","labels":["lem:irregular-base-closed"],"detail_key":"p51"},{"id":"n41862","layer":"informal","project":"p51","title":"If q = 2 this is trivial. Otherwise q divides M(S)!, hence q \\le M(S), and as before (q-1…","kind":"proof","summary":"If q = 2 this is trivial. Otherwise q divides M(S)!, hence q \\le M(S), and as before (q-1) \\mid…","labels":[],"detail_key":"p51"},{"id":"n41863","layer":"informal","project":"p51","title":"thm:voronoi-congruence","kind":"theorem","summary":"Let p \\ge 5 be a prime, a an integer prime to p, and k a positive even index with (p-1)\\nmid k.…","labels":["thm:voronoi-congruence"],"detail_key":"p51"},{"id":"n41864","layer":"informal","project":"p51","title":"This is Voronoi's congruence (see e.g.\\ \\cite[Chapter~15]ireland-rosen). Writing ja = p\\l…","kind":"proof","summary":"This is Voronoi's congruence (see e.g.\\ \\cite[Chapter~15]ireland-rosen). Writing ja = p\\lfloor…","labels":[],"detail_key":"p51"},{"id":"n41865","layer":"informal","project":"p51","title":"thm:kummer-congruence-gefive","kind":"theorem","summary":"Let p \\ge 5 be a prime and m, n positive even indices with m \\equiv n \\pmodp-1 and (p-1)\\nmid n…","labels":["thm:kummer-congruence-gefive"],"detail_key":"p51"},{"id":"n41866","layer":"informal","project":"p51","title":"Choose a primitive root a modulo p and apply \\crefthm:voronoi-congruence to both indices.…","kind":"proof","summary":"Choose a primitive root a modulo p and apply \\crefthm:voronoi-congruence to both indices. Since…","labels":[],"detail_key":"p51"},{"id":"n41867","layer":"informal","project":"p51","title":"thm:kummer-congruence-full","kind":"theorem","summary":"Let p be an odd prime and m, n positive even indices with m \\equiv n \\pmodp-1 and (p-1)\\nmid n.…","labels":["thm:kummer-congruence-full"],"detail_key":"p51"},{"id":"n41868","layer":"informal","project":"p51","title":"It remains to lift the restriction p \\ge 5 of \\crefthm:kummer-congruence-gefive to all od…","kind":"proof","summary":"It remains to lift the restriction p \\ge 5 of \\crefthm:kummer-congruence-gefive to all odd prim…","labels":[],"detail_key":"p51"},{"id":"n41869","layer":"informal","project":"p51","title":"lem:positive-residue","kind":"lemma","summary":"Let p be an odd prime and m an even index with (p-1)\\nmid m. Then m' := m \\bmod (p-1) satisfies…","labels":["lem:positive-residue"],"detail_key":"p51"},{"id":"n41870","layer":"informal","project":"p51","title":"The residue is nonzero precisely because (p-1)\\nmid m, and it inherits parity from m sinc…","kind":"proof","summary":"The residue is nonzero precisely because (p-1)\\nmid m, and it inherits parity from m since p-1…","labels":[],"detail_key":"p51"},{"id":"n41871","layer":"informal","project":"p51","title":"thm:carlitz-criterion","kind":"theorem","summary":"An odd prime p is irregular if and only if p \\mid num(B_m/m) for some positive even index m.","labels":["thm:carlitz-criterion"],"detail_key":"p51"},{"id":"n41872","layer":"informal","project":"p51","title":"If p is irregular, \\crefthm:bernoulli-witness-of-irregular produces a witness B_2k in Kum…","kind":"proof","summary":"If p is irregular, \\crefthm:bernoulli-witness-of-irregular produces a witness B_2k in Kummer's…","labels":[],"detail_key":"p51"},{"id":"n41873","layer":"informal","project":"p51","title":"thm:carlitz-numerator-prime","kind":"theorem","summary":"For every finite set S of natural numbers there are M and a prime p such that M is even and pos…","labels":["thm:carlitz-numerator-prime"],"detail_key":"p51"},{"id":"n41874","layer":"informal","project":"p51","title":"Put M = C(S)\\cdot 2^t with t chosen by \\crefcor:bernoulli-large-multiple so that |B_M/M|…","kind":"proof","summary":"Put M = C(S)\\cdot 2^t with t chosen by \\crefcor:bernoulli-large-multiple so that |B_M/M| > 1. B…","labels":[],"detail_key":"p51"},{"id":"n41875","layer":"informal","project":"p51","title":"thm:carlitz-escape","kind":"theorem","summary":"For every finite set S of natural numbers there is an irregular prime p \\notin S.","labels":["thm:carlitz-escape"],"detail_key":"p51"},{"id":"n41876","layer":"informal","project":"p51","title":"Take the prime p and the index M of \\crefthm:carlitz-numerator-prime. Then p is irregular…","kind":"proof","summary":"Take the prime p and the index M of \\crefthm:carlitz-numerator-prime. Then p is irregular by \\c…","labels":[],"detail_key":"p51"},{"id":"n41877","layer":"informal","project":"p51","title":"thm:infinitely-many-irregular","kind":"theorem","summary":"There are infinitely many irregular primes.","labels":["thm:infinitely-many-irregular"],"detail_key":"p51"},{"id":"n41878","layer":"informal","project":"p51","title":"By \\crefthm:carlitz-escape no finite set contains all irregular primes, so the set is inf…","kind":"proof","summary":"By \\crefthm:carlitz-escape no finite set contains all irregular primes, so the set is infinite…","labels":[],"detail_key":"p51"},{"id":"n41879","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.bernoulliFactor_ne_zero_iff_not_dvd_bernoulli_num","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] j : Nat, LE.le 1 j → LE.le (HMul.hMul 2 j) (HSub.hSub p…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.bernoulliFactor_ne_zero_iff_not_dvd_bernoulli_num","module":"KummerCriterion.CyclotomicUnits.KummerLogDeterminant"},{"id":"n41880","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_det_ne_zero_iff_forall_bernoulliF…","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_det_ne_zero_iff_forall_bernoulliFactor_ne_zero","module":"KummerCriterion.CyclotomicUnits.KummerLogDeterminant"},{"id":"n41881","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_eq_diagonal_mul_vandermonde","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_eq_diagonal_mul_vandermonde","module":"KummerCriterion.CyclotomicUnits.KummerLogDeterminant"},{"id":"n41882","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.kummerLogMatrix_det_ne_zero_iff_bernoulli_nonzero","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.kummerLogMatrix_det_ne_zero_iff_bernoulli_nonzero","module":"KummerCriterion.CyclotomicUnits.KummerLogDeterminant"},{"id":"n41883","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix","kind":"def","summary":"(p : Nat) → [Fact (Nat.Prime p)] → (K : Type u_1) → [inst : Field K] → [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix","module":"KummerCriterion.CyclotomicUnits.KummerLogTrace"},{"id":"n41884","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.CPlusGenerator_exponents_modP_zero_of_kummerLog_det_ne_ze…","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.CPlusGenerator_exponents_modP_zero_of_kummerLog_det_ne_zero","module":"KummerCriterion.CyclotomicUnits.LogDomain"},{"id":"n41885","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.completedLog_relation_of_CPlus_product_mem_powers","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.completedLog_relation_of_CPlus_product_mem_powers","module":"KummerCriterion.CyclotomicUnits.LogDomain"},{"id":"n41886","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_mulVec_exponents_eq_zero","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.concreteKummerLogMatrix_mulVec_exponents_eq_zero","module":"KummerCriterion.CyclotomicUnits.LogDomain"},{"id":"n41887","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.cyclotomicUnits_pSaturated_of_kummerLog_det_ne_zero","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.cyclotomicUnits_pSaturated_of_kummerLog_det_ne_zero","module":"KummerCriterion.CyclotomicUnits.LogDomain"},{"id":"n41888","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.exponents_modP_eq_zero_of_kummerLogMatrix_relation","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (K : Type u_1) [inst_1 : Field K] [inst_2 : NumberField…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.exponents_modP_eq_zero_of_kummerLogMatrix_relation","module":"KummerCriterion.CyclotomicUnits.LogDomain"},{"id":"n41889","layer":"formal","project":"p51","title":"KummerCriterion.cyclotomicUnitIndexSubgroup","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type → [inst : Field K] → [inst_1 : NumberField K] → [IsCy…","labels":[],"detail_key":"p51","name":"KummerCriterion.cyclotomicUnitIndexSubgroup","module":"KummerCriterion.CyclotomicUnits.NormalizedIndex"},{"id":"n41890","layer":"formal","project":"p51","title":"KummerCriterion.cyclotomicUnitIndexSubgroup_eq_CPlus","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.cyclotomicUnitIndexSubgroup_eq_CPlus","module":"KummerCriterion.CyclotomicUnits.NormalizedIndex"},{"id":"n41891","layer":"formal","project":"p51","title":"KummerCriterion.cyclotomicUnitIndex_primeConductor_pPrimary","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.cyclotomicUnitIndex_primeConductor_pPrimary","module":"KummerCriterion.CyclotomicUnits.NormalizedIndex"},{"id":"n41892","layer":"formal","project":"p51","title":"KummerCriterion.cyclotomicUnitIndex_primeConductor_pPrimary_aux","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.cyclotomicUnitIndex_primeConductor_pPrimary_aux","module":"KummerCriterion.CyclotomicUnits.NormalizedIndex"},{"id":"n41893","layer":"formal","project":"p51","title":"KummerCriterion.kummerDirichletDeterminant_of_deletedFourier","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.kummerDirichletDeterminant_of_deletedFourier","module":"KummerCriterion.CyclotomicUnits.NormalizedIndex"},{"id":"n41894","layer":"formal","project":"p51","title":"KummerCriterion.CPlus_index_prime_dvd_iff_normalizedCPlus_index_prime_dvd","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CPlus_index_prime_dvd_iff_normalizedCPlus_index_prime_dvd","module":"KummerCriterion.CyclotomicUnits.NormalizedSubgroup"},{"id":"n41895","layer":"formal","project":"p51","title":"KummerCriterion.normalizedCPlus","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type u_1 → [inst : Field K] → [inst_1 : NumberField K] → […","labels":[],"detail_key":"p51","name":"KummerCriterion.normalizedCPlus","module":"KummerCriterion.CyclotomicUnits.NormalizedSubgroup"},{"id":"n41896","layer":"formal","project":"p51","title":"KummerCriterion.normalizedCPlusGenerator_sq_eq_CPlusGenerator","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.normalizedCPlusGenerator_sq_eq_CPlusGenerator","module":"KummerCriterion.CyclotomicUnits.NormalizedSubgroup"},{"id":"n41897","layer":"formal","project":"p51","title":"KummerCriterion.CPlus","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type u_1 → [inst : Field K] → [inst_1 : NumberField K] → […","labels":[],"detail_key":"p51","name":"KummerCriterion.CPlus","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41898","layer":"formal","project":"p51","title":"KummerCriterion.CPlusExponentProduct","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type u_1 → [inst : Field K] → [inst_1 : NumberField K] → […","labels":[],"detail_key":"p51","name":"KummerCriterion.CPlusExponentProduct","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41899","layer":"formal","project":"p51","title":"KummerCriterion.CPlusGenerator","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type u_1 → [inst : Field K] → [inst_1 : NumberField K] → […","labels":[],"detail_key":"p51","name":"KummerCriterion.CPlusGenerator","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41900","layer":"formal","project":"p51","title":"KummerCriterion.CPlus_pSaturated_of_generator_exponents_modP_zero","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.CPlus_pSaturated_of_generator_exponents_modP_zero","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41901","layer":"formal","project":"p51","title":"KummerCriterion.EPlus","kind":"def","summary":"K : Type u_1 → [inst : Field K] → Subgroup (Units (NumberField.RingOfIntegers (Subtype fun x =>…","labels":[],"detail_key":"p51","name":"KummerCriterion.EPlus","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41902","layer":"formal","project":"p51","title":"KummerCriterion.exists_CPlusExponentProduct_of_mem_CPlus","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type u_1 [inst_1 : Field K] [inst_2 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_CPlusExponentProduct_of_mem_CPlus","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41903","layer":"formal","project":"p51","title":"KummerCriterion.pPowerSubgroup","kind":"def","summary":"G : Type u_2 → [inst : CommGroup G] → Subgroup G → Nat → Subgroup G","labels":[],"detail_key":"p51","name":"KummerCriterion.pPowerSubgroup","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41904","layer":"formal","project":"p51","title":"KummerCriterion.pSaturated","kind":"def","summary":"G : Type u_2 → [inst : CommGroup G] → Subgroup G → Subgroup G → Nat → Prop","labels":[],"detail_key":"p51","name":"KummerCriterion.pSaturated","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41905","layer":"formal","project":"p51","title":"KummerCriterion.realCyclotomicUnit","kind":"def","summary":"p : Nat → [Fact (Nat.Prime p)] → K : Type u_1 → [inst : Field K] → [inst_1 : NumberField K] → […","labels":[],"detail_key":"p51","name":"KummerCriterion.realCyclotomicUnit","module":"KummerCriterion.CyclotomicUnits.Saturation"},{"id":"n41906","layer":"formal","project":"p51","title":"KummerCriterion.not_dvd_index_of_pSaturated","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_dvd_index_of_pSaturated","module":"KummerCriterion.CyclotomicUnits.SaturationIndex"},{"id":"n41907","layer":"formal","project":"p51","title":"KummerCriterion.bernoulli_nonzero_of_not_dvd_hMinus","kind":"theorem","summary":"∀ p : Nat [Fact (Nat.Prime p)] K : Type u_1 [inst : Field K] [inst_1 : NumberField K] [IsCyclot…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulli_nonzero_of_not_dvd_hMinus","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41908","layer":"formal","project":"p51","title":"KummerCriterion.dvd_h_iff_dvd_hMinus_of_dvd_hPlus_imp","kind":"theorem","summary":"∀ p : Nat [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ K : Type u_1 [inst : Field K] [inst_1 : NumberF…","labels":[],"detail_key":"p51","name":"KummerCriterion.dvd_h_iff_dvd_hMinus_of_dvd_hPlus_imp","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41909","layer":"formal","project":"p51","title":"KummerCriterion.dvd_h_iff_exists_dvd_bernoulli_units","kind":"theorem","summary":"∀ p : Nat [Fact (Nat.Prime p)] K : Type [inst : Field K] [inst_1 : NumberField K] [IsCyclotomic…","labels":[],"detail_key":"p51","name":"KummerCriterion.dvd_h_iff_exists_dvd_bernoulli_units","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41910","layer":"formal","project":"p51","title":"KummerCriterion.not_dvd_cyclotomicUnitIndex_of_bernoulli_nonzero","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] K : Type [inst_1 : Field K] [inst_2 : NumberField K] [ins…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_dvd_cyclotomicUnitIndex_of_bernoulli_nonzero","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41911","layer":"formal","project":"p51","title":"KummerCriterion.not_dvd_hPlus_of_not_dvd_hMinus_units","kind":"theorem","summary":"∀ p : Nat [Fact (Nat.Prime p)] K : Type [inst : Field K] [inst_1 : NumberField K] [IsCyclotomic…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_dvd_hPlus_of_not_dvd_hMinus_units","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41912","layer":"formal","project":"p51","title":"KummerCriterion.weakReflection_dvd_hMinus_of_dvd_hPlus_units","kind":"theorem","summary":"∀ p : Nat [Fact (Nat.Prime p)] K : Type [inst : Field K] [inst_1 : NumberField K] [IsCyclotomic…","labels":[],"detail_key":"p51","name":"KummerCriterion.weakReflection_dvd_hMinus_of_dvd_hPlus_units","module":"KummerCriterion.CyclotomicUnits.UnitsReflection"},{"id":"n41913","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.kummerLogRank","kind":"def","summary":"Nat → Nat","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.kummerLogRank","module":"KummerCriterion.CyclotomicUnits.Vandermonde"},{"id":"n41914","layer":"formal","project":"p51","title":"KummerCriterion.CyclotomicUnits.vandermonde_teichmuller_even_sub_one_det_ne_zero","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (hp_three : LE.le 3 p), Ne (KummerCriterion.CyclotomicU…","labels":[],"detail_key":"p51","name":"KummerCriterion.CyclotomicUnits.vandermonde_teichmuller_even_sub_one_det_ne_zero","module":"KummerCriterion.CyclotomicUnits.Vandermonde"},{"id":"n41915","layer":"formal","project":"p51","title":"KummerCriterion.exists_odd_index_iff_exists_k","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (Q : Nat → Prop), Iff (Exists fun j => And (M…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_odd_index_iff_exists_k","module":"KummerCriterion.HMinus.HMinusCriterion"},{"id":"n41916","layer":"formal","project":"p51","title":"KummerCriterion.p_dvd_hMinus_iff_p_dvd_some_bernoulli","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.p_dvd_hMinus_iff_p_dvd_some_bernoulli","module":"KummerCriterion.HMinus.HMinusCriterion"},{"id":"n41917","layer":"formal","project":"p51","title":"KummerCriterion.cyclotomicHGaussGoal_holds","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.cyclotomicHGaussGoal_holds","module":"KummerCriterion.HMinus.LValueReduction.GaussProduct"},{"id":"n41918","layer":"formal","project":"p51","title":"KummerCriterion.rawGaussProduct","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → Eq ((KummerCriterion.oddCharacters p).prod fun…","labels":[],"detail_key":"p51","name":"KummerCriterion.rawGaussProduct","module":"KummerCriterion.HMinus.LValueReduction.GaussProduct"},{"id":"n41919","layer":"formal","project":"p51","title":"KummerCriterion.hPlus_formula_of_evenLValues","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.hPlus_formula_of_evenLValues","module":"KummerCriterion.HMinus.LValueReduction.LValues"},{"id":"n41920","layer":"formal","project":"p51","title":"KummerCriterion.hMinus_formula","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.hMinus_formula","module":"KummerCriterion.HMinus.LValueReduction.Teichmuller"},{"id":"n41921","layer":"formal","project":"p51","title":"KummerCriterion.hMinus_formula_of_residue_and_hPlus_cyclotomic_and_gauss","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.hMinus_formula_of_residue_and_hPlus_cyclotomic_and_gauss","module":"KummerCriterion.HMinus.LValueReduction.Teichmuller"},{"id":"n41922","layer":"formal","project":"p51","title":"KummerCriterion.hMinus_formula_bernoulli_mod_p","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.hMinus_formula_bernoulli_mod_p","module":"KummerCriterion.HMinus.PadicCorollaries"},{"id":"n41923","layer":"formal","project":"p51","title":"KummerCriterion.hMinus_formula_teichmuller_mod_p","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)] (K : Type u_1) [inst : Field K] [inst_1 : NumberField K]…","labels":[],"detail_key":"p51","name":"KummerCriterion.hMinus_formula_teichmuller_mod_p","module":"KummerCriterion.HMinus.PadicCorollaries"},{"id":"n41924","layer":"formal","project":"p51","title":"KummerCriterion.exists_bernoulli_num_dvd_of_not_isRegularPrime","kind":"theorem","summary":"∀ p : Nat, Nat.Prime p → Ne p 2 → Not (IsRegularPrime p) → Exists fun k => And (LE.le 1 k) (And…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_bernoulli_num_dvd_of_not_isRegularPrime","module":"KummerCriterion.IrregularPrimes.Basic"},{"id":"n41925","layer":"formal","project":"p51","title":"KummerCriterion.infinite_of_forall_finite_set_not_cover","kind":"theorem","summary":"∀ P : Nat → Prop, (∀ (S : Finset Nat), (∀ (p : Nat), P p → Membership.mem S p) → False) → (Set.…","labels":[],"detail_key":"p51","name":"KummerCriterion.infinite_of_forall_finite_set_not_cover","module":"KummerCriterion.IrregularPrimes.Basic"},{"id":"n41926","layer":"formal","project":"p51","title":"KummerCriterion.not_isRegularPrime_of_bernoulli_num_dvd","kind":"theorem","summary":"∀ p : Nat, Nat.Prime p → Ne p 2 → (Exists fun k => And (LE.le 1 k) (And (LE.le (HMul.hMul 2 k)…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_isRegularPrime_of_bernoulli_num_dvd","module":"KummerCriterion.IrregularPrimes.Basic"},{"id":"n41927","layer":"formal","project":"p51","title":"KummerCriterion.bernoulli_div_self_abs_lower_bound","kind":"theorem","summary":"∀ k : Nat, Ne k 0 → LE.le (HDiv.hDiv (↑(HSub.hSub (HMul.hMul 2 k) 1).factorial) (HMul.hMul (HPo…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulli_div_self_abs_lower_bound","module":"KummerCriterion.IrregularPrimes.BernoulliGrowth"},{"id":"n41928","layer":"formal","project":"p51","title":"KummerCriterion.exists_large_even_multiple_abs_bernoulli_div_self_gt_one","kind":"theorem","summary":"∀ C : Nat, LT.lt 0 C → Even C → Exists fun t => LT.lt 1 (abs ↑(HDiv.hDiv (bernoulli (HMul.hMul…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_large_even_multiple_abs_bernoulli_div_self_gt_one","module":"KummerCriterion.IrregularPrimes.BernoulliGrowth"},{"id":"n41929","layer":"formal","project":"p51","title":"KummerCriterion.tendsto_abs_bernoulli_div_self_even","kind":"theorem","summary":"Filter.Tendsto (fun n => abs ↑(HDiv.hDiv (bernoulli (HMul.hMul 2 n)) (HMul.hMul 2 ↑n))) Filter.…","labels":[],"detail_key":"p51","name":"KummerCriterion.tendsto_abs_bernoulli_div_self_even","module":"KummerCriterion.IrregularPrimes.BernoulliGrowth"},{"id":"n41930","layer":"formal","project":"p51","title":"KummerCriterion.irregularBase","kind":"def","summary":"Finset Nat → Nat","labels":[],"detail_key":"p51","name":"KummerCriterion.irregularBase","module":"KummerCriterion.IrregularPrimes.DivisorClosedBase"},{"id":"n41931","layer":"formal","project":"p51","title":"KummerCriterion.sub_one_dvd_irregularBase_of_mem","kind":"theorem","summary":"∀ S : Finset Nat p : Nat, Membership.mem S p → Nat.Prime p → Dvd.dvd (HSub.hSub p 1) (KummerCri…","labels":[],"detail_key":"p51","name":"KummerCriterion.sub_one_dvd_irregularBase_of_mem","module":"KummerCriterion.IrregularPrimes.DivisorClosedBase"},{"id":"n41932","layer":"formal","project":"p51","title":"KummerCriterion.sub_one_dvd_m_of_prime_dvd_m","kind":"theorem","summary":"∀ S : Finset Nat q t : Nat, Nat.Prime q → Dvd.dvd q (HMul.hMul (KummerCriterion.irregularBase S…","labels":[],"detail_key":"p51","name":"KummerCriterion.sub_one_dvd_m_of_prime_dvd_m","module":"KummerCriterion.IrregularPrimes.DivisorClosedBase"},{"id":"n41933","layer":"formal","project":"p51","title":"KummerCriterion.exists_not_isRegularPrime_not_mem_carlitz","kind":"theorem","summary":"∀ (S : Finset Nat), Exists fun p => Exists fun hp => And (Not (IsRegularPrime p)) (Not (Members…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_not_isRegularPrime_not_mem_carlitz","module":"KummerCriterion.IrregularPrimes.Infinitude"},{"id":"n41934","layer":"formal","project":"p51","title":"KummerCriterion.exists_numerator_prime_for_carlitz_base","kind":"theorem","summary":"∀ (S : Finset Nat), Exists fun M => Exists fun p => And (Nat.Prime p) (And (Ne p 2) (And (Even…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_numerator_prime_for_carlitz_base","module":"KummerCriterion.IrregularPrimes.Infinitude"},{"id":"n41935","layer":"formal","project":"p51","title":"KummerCriterion.infinite_not_isRegularPrime","kind":"theorem","summary":"(Set.ofPred fun p => Exists fun hp => Not (IsRegularPrime p)).Infinite","labels":[],"detail_key":"p51","name":"KummerCriterion.infinite_not_isRegularPrime","module":"KummerCriterion.IrregularPrimes.Infinitude"},{"id":"n41936","layer":"formal","project":"p51","title":"KummerCriterion.not_isRegularPrime_iff_exists_dvd_bernoulli_div_self_num","kind":"theorem","summary":"∀ q : Nat, Nat.Prime q → Ne q 2 → Iff (Not (IsRegularPrime q)) (Exists fun m => And (LT.lt 0 m)…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_isRegularPrime_iff_exists_dvd_bernoulli_div_self_num","module":"KummerCriterion.IrregularPrimes.Infinitude"},{"id":"n41937","layer":"formal","project":"p51","title":"KummerCriterion.positiveResidue_properties","kind":"theorem","summary":"∀ p m : Nat, Nat.Prime p → Ne p 2 → Even m → Not (Dvd.dvd (HSub.hSub p 1) m) → have m' := Kumme…","labels":[],"detail_key":"p51","name":"KummerCriterion.positiveResidue_properties","module":"KummerCriterion.IrregularPrimes.Infinitude"},{"id":"n41938","layer":"formal","project":"p51","title":"KummerCriterion.bernoulli_div_sModEq_of_modEq_full","kind":"theorem","summary":"∀ p m n : Nat [inst : Fact (Nat.Prime p)], Ne p 2 → LT.lt 0 m → LT.lt 0 n → Even m → Even n → N…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulli_div_sModEq_of_modEq_full","module":"KummerCriterion.IrregularPrimes.KummerCongruenceFull"},{"id":"n41939","layer":"formal","project":"p51","title":"KummerCriterion.bernoulli_div_sModEq_of_modEq_full_geFive","kind":"theorem","summary":"∀ p m n : Nat [inst : Fact (Nat.Prime p)], LE.le 5 p → LT.lt 0 m → LT.lt 0 n → Even m → Even n…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulli_div_sModEq_of_modEq_full_geFive","module":"KummerCriterion.IrregularPrimes.KummerCongruenceFull"},{"id":"n41940","layer":"formal","project":"p51","title":"KummerCriterion.voronoi_congruence_mod_p_strong","kind":"theorem","summary":"∀ p a h : Nat [inst : Fact (Nat.Prime p)], LE.le 5 p → Not (Dvd.dvd p a) → LT.lt 0 h → Even h →…","labels":[],"detail_key":"p51","name":"KummerCriterion.voronoi_congruence_mod_p_strong","module":"KummerCriterion.IrregularPrimes.KummerCongruenceFull"},{"id":"n41941","layer":"formal","project":"p51","title":"KummerCriterion.dvd_num_of_dvd_div_nat_num","kind":"theorem","summary":"∀ p n : Nat q : Rat, LT.lt 0 n → Dvd.dvd (↑p) (HDiv.hDiv q ↑n).num → Dvd.dvd (↑p) q.num","labels":[],"detail_key":"p51","name":"KummerCriterion.dvd_num_of_dvd_div_nat_num","module":"KummerCriterion.IrregularPrimes.RatNumerator"},{"id":"n41942","layer":"formal","project":"p51","title":"KummerCriterion.exists_prime_dvd_num_of_one_lt_abs","kind":"theorem","summary":"∀ q : Rat, LT.lt 1 (abs ↑q) → Exists fun p => And (Nat.Prime p) (Dvd.dvd (↑p) q.num)","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_prime_dvd_num_of_one_lt_abs","module":"KummerCriterion.IrregularPrimes.RatNumerator"},{"id":"n41943","layer":"formal","project":"p51","title":"KummerCriterion.prime_dvd_num_of_padic_sub_eq_p_mul","kind":"theorem","summary":"∀ p : Nat [inst : Fact (Nat.Prime p)] q r : Rat, Dvd.dvd (↑p) q.num → (Exists fun z => Eq (HSub…","labels":[],"detail_key":"p51","name":"KummerCriterion.prime_dvd_num_of_padic_sub_eq_p_mul","module":"KummerCriterion.IrregularPrimes.RatNumerator"},{"id":"n41944","layer":"formal","project":"p51","title":"KummerCriterion.not_dvd_num_bernoulli_div_self_of_sub_one_dvd","kind":"theorem","summary":"∀ p n : Nat, Nat.Prime p → LT.lt 0 n → Even n → Dvd.dvd (HSub.hSub p 1) n → Not (Dvd.dvd (↑p) (…","labels":[],"detail_key":"p51","name":"KummerCriterion.not_dvd_num_bernoulli_div_self_of_sub_one_dvd","module":"KummerCriterion.IrregularPrimes.VonStaudtConsequences"},{"id":"n41945","layer":"formal","project":"p51","title":"KummerCriterion.p_mul_bernoulli_mem_padicInt_vonStaudt","kind":"theorem","summary":"∀ p n : Nat [inst : Fact (Nat.Prime p)], Even n → Exists fun z => Eq (HMul.hMul ↑p ↑(bernoulli…","labels":[],"detail_key":"p51","name":"KummerCriterion.p_mul_bernoulli_mem_padicInt_vonStaudt","module":"KummerCriterion.IrregularPrimes.VonStaudtConsequences"},{"id":"n41946","layer":"formal","project":"p51","title":"KummerCriterion.BernoulliGen","kind":"def","summary":"N : Nat → R : Type u_1 → [inst : CommRing R] → [Algebra Rat R] → [NeZero N] → DirichletCharacte…","labels":[],"detail_key":"p51","name":"KummerCriterion.BernoulliGen","module":"KummerCriterion.KummerCongruence.BernoulliGeneralized"},{"id":"n41947","layer":"formal","project":"p51","title":"KummerCriterion.bernoulli_mem_padicInt_of_lt_sub_one","kind":"theorem","summary":"∀ p : Nat [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (k : Nat), LT.lt k (HSub.hSub p 1) → Exists fun…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulli_mem_padicInt_of_lt_sub_one","module":"KummerCriterion.KummerCongruence.BernoulliGeneralized"},{"id":"n41948","layer":"formal","project":"p51","title":"KummerCriterion.exists_padicInt_bernoulli_factor","kind":"theorem","summary":"∀ p : Nat [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ n : Nat, LT.lt 0 n → LT.lt n p → Not (Dvd.dvd p…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_padicInt_bernoulli_factor","module":"KummerCriterion.KummerCongruence.BernoulliGeneralized"},{"id":"n41949","layer":"formal","project":"p51","title":"KummerCriterion.prime_not_dvd_bernoulli_den_of_lt_sub_one","kind":"theorem","summary":"∀ p n : Nat [hp : Fact (Nat.Prime p)], Ne p 2 → LT.lt n (HSub.hSub p 1) → Not (Dvd.dvd p (berno…","labels":[],"detail_key":"p51","name":"KummerCriterion.prime_not_dvd_bernoulli_den_of_lt_sub_one","module":"KummerCriterion.KummerCongruence.BernoulliGeneralized"},{"id":"n41950","layer":"formal","project":"p51","title":"KummerCriterion.teichmullerCharQp","kind":"def","summary":"(p : Nat) → [inst : Fact (Nat.Prime p)] → DirichletCharacter (Padic p) p","labels":[],"detail_key":"p51","name":"KummerCriterion.teichmullerCharQp","module":"KummerCriterion.KummerCongruence.BernoulliGeneralized"},{"id":"n41951","layer":"formal","project":"p51","title":"KummerCriterion.bernoulliGen_teichmuller_pow_sModEq_div","kind":"theorem","summary":"∀ p : Nat [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ n : Nat, Odd n → LT.lt 0 n → Not (Dvd.dvd (HSub…","labels":[],"detail_key":"p51","name":"KummerCriterion.bernoulliGen_teichmuller_pow_sModEq_div","module":"KummerCriterion.KummerCongruence.Bridge"},{"id":"n41952","layer":"formal","project":"p51","title":"KummerCriterion.classGroupMap","kind":"def","summary":"(K : Type u_1) → [inst : Field K] → [inst_1 : NumberField K] → [NumberField.IsCMField K] → Mono…","labels":[],"detail_key":"p51","name":"KummerCriterion.classGroupMap","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41953","layer":"formal","project":"p51","title":"KummerCriterion.classGroupMap_injective","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.classGroupMap_injective","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41954","layer":"formal","project":"p51","title":"KummerCriterion.h","kind":"def","summary":"(K : Type u_1) → [inst : Field K] → [NumberField K] → Nat","labels":[],"detail_key":"p51","name":"KummerCriterion.h","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41955","layer":"formal","project":"p51","title":"KummerCriterion.hMinus","kind":"def","summary":"(K : Type u_1) → [inst : Field K] → [inst_1 : NumberField K] → [NumberField.IsCMField K] → Nat","labels":[],"detail_key":"p51","name":"KummerCriterion.hMinus","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41956","layer":"formal","project":"p51","title":"KummerCriterion.hPlus","kind":"def","summary":"(K : Type u_1) → [inst : Field K] → [inst_1 : NumberField K] → [NumberField.IsCMField K] → Nat","labels":[],"detail_key":"p51","name":"KummerCriterion.hPlus","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41957","layer":"formal","project":"p51","title":"KummerCriterion.hPlus_dvd_h","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.hPlus_dvd_h","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41958","layer":"formal","project":"p51","title":"KummerCriterion.h_eq_hPlus_mul_hMinus","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.h_eq_hPlus_mul_hMinus","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41959","layer":"formal","project":"p51","title":"KummerCriterion.isPrincipal_of_isPrincipal_map_Kplus","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.isPrincipal_of_isPrincipal_map_Kplus","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41960","layer":"formal","project":"p51","title":"KummerCriterion.isPrincipal_of_map_eq_span_singleton_of_mem","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [NumberField K] (I : Ideal (NumberField.RingOfIntegers (Subty…","labels":[],"detail_key":"p51","name":"KummerCriterion.isPrincipal_of_map_eq_span_singleton_of_mem","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41961","layer":"formal","project":"p51","title":"KummerCriterion.map_comap_eq_ringOfIntegers","kind":"theorem","summary":"∀ (L : Type u_2) [inst : Field L] [NumberField L] (J : Ideal (NumberField.RingOfIntegers (Subty…","labels":[],"detail_key":"p51","name":"KummerCriterion.map_comap_eq_ringOfIntegers","module":"KummerCriterion.TotallyRealSubfield.ClassGroup"},{"id":"n41962","layer":"formal","project":"p51","title":"KummerCriterion.antisymmetric_unit_eq_neg_one_pow_mul_zeta_pow","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.antisymmetric_unit_eq_neg_one_pow_mul_zeta_pow","module":"KummerCriterion.TotallyRealSubfield.Conjugation"},{"id":"n41963","layer":"formal","project":"p51","title":"KummerCriterion.conj_generator_associated","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] [inst_2 : NumberField.IsCMField K] (…","labels":[],"detail_key":"p51","name":"KummerCriterion.conj_generator_associated","module":"KummerCriterion.TotallyRealSubfield.Conjugation"},{"id":"n41964","layer":"formal","project":"p51","title":"KummerCriterion.conj_unit_mul_eq_one","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] [inst_2 : NumberField.IsCMField K] (…","labels":[],"detail_key":"p51","name":"KummerCriterion.conj_unit_mul_eq_one","module":"KummerCriterion.TotallyRealSubfield.Conjugation"},{"id":"n41965","layer":"formal","project":"p51","title":"KummerCriterion.ideal_map_conj_eq","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] [inst_2 : NumberField.IsCMField K] (…","labels":[],"detail_key":"p51","name":"KummerCriterion.ideal_map_conj_eq","module":"KummerCriterion.TotallyRealSubfield.Conjugation"},{"id":"n41966","layer":"formal","project":"p51","title":"KummerCriterion.exists_conj_fixed_associate_of_classification","kind":"theorem","summary":"∀ (p : Nat) [hp : Fact (Nat.Prime p)], Ne p 2 → ∀ (K : Type u_1) [inst : Field K] [inst_1 : Num…","labels":[],"detail_key":"p51","name":"KummerCriterion.exists_conj_fixed_associate_of_classification","module":"KummerCriterion.TotallyRealSubfield.FixedAssociate"},{"id":"n41967","layer":"formal","project":"p51","title":"KummerCriterion.mem_ringOfIntegers_of_conj_eq_self","kind":"theorem","summary":"∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K] [inst_2 : NumberField.IsCMField K] (…","labels":[],"detail_key":"p51","name":"KummerCriterion.mem_ringOfIntegers_of_conj_eq_self","module":"KummerCriterion.TotallyRealSubfield.FixedAssociate"},{"id":"n41968","layer":"informal","project":"p52","title":"MyNat","kind":"definition","summary":"We define MyNat as the inductive type with two constructors: \\item a term 0 (called \\textttzero…","labels":["MyNat"],"detail_key":"p52"},{"id":"n41969","layer":"informal","project":"p52","title":"MyNat.add","kind":"definition","summary":"We define \\emphaddition on MyNat by recursion on the second argument: \\[ a + 0 = a, \\qquad a +…","labels":["MyNat.add"],"detail_key":"p52"},{"id":"n41970","layer":"informal","project":"p52","title":"MyNat.mul","kind":"definition","summary":"We define \\emphmultiplication on MyNat by recursion on the second argument: \\[ a * 0 = 0, \\qqua…","labels":["MyNat.mul"],"detail_key":"p52"},{"id":"n41971","layer":"informal","project":"p52","title":"MyNat.zero_add","kind":"lemma","summary":"For all a in MyNat we have 0 + a = a.","labels":["MyNat.zero_add"],"detail_key":"p52"},{"id":"n41972","layer":"informal","project":"p52","title":"By induction on a.","kind":"proof","summary":"By induction on a.","labels":[],"detail_key":"p52"},{"id":"n41973","layer":"informal","project":"p52","title":"MyNat.add_assoc","kind":"lemma","summary":"Addition on MyNat is associative.","labels":["MyNat.add_assoc"],"detail_key":"p52"},{"id":"n41974","layer":"informal","project":"p52","title":"By induction on the last variable.","kind":"proof","summary":"By induction on the last variable.","labels":[],"detail_key":"p52"},{"id":"n41975","layer":"informal","project":"p52","title":"MyNat.add_comm","kind":"lemma","summary":"Addition on MyNat is commutative.","labels":["MyNat.add_comm"],"detail_key":"p52"},{"id":"n41976","layer":"informal","project":"p52","title":"By induction.","kind":"proof","summary":"By induction.","labels":[],"detail_key":"p52"},{"id":"n41977","layer":"informal","project":"p52","title":"MyNat.left_distrib","kind":"lemma","summary":"For all a, b and c in MyNat we have a * (b + c) = a * b + a * c.","labels":["MyNat.left_distrib"],"detail_key":"p52"},{"id":"n41978","layer":"informal","project":"p52","title":"By induction on c.","kind":"proof","summary":"By induction on c.","labels":[],"detail_key":"p52"},{"id":"n41979","layer":"informal","project":"p52","title":"MyNat.right_distrib","kind":"lemma","summary":"For all a, b and c in MyNat we have (a + b) * c = a * c + b * c.","labels":["MyNat.right_distrib"],"detail_key":"p52"},{"id":"n41980","layer":"informal","project":"p52","title":"By induction on c.","kind":"proof","summary":"By induction on c.","labels":[],"detail_key":"p52"},{"id":"n41981","layer":"informal","project":"p52","title":"MyNat.mul_assoc","kind":"lemma","summary":"Multiplication on MyNat is associative.","labels":["MyNat.mul_assoc"],"detail_key":"p52"},{"id":"n41982","layer":"informal","project":"p52","title":"By induction, using left distributivity.","kind":"proof","summary":"By induction, using left distributivity.","labels":[],"detail_key":"p52"},{"id":"n41983","layer":"informal","project":"p52","title":"MyNat.mul_comm","kind":"lemma","summary":"Multiplication on MyNat is commutative.","labels":["MyNat.mul_comm"],"detail_key":"p52"},{"id":"n41984","layer":"informal","project":"p52","title":"By induction.","kind":"proof","summary":"By induction.","labels":[],"detail_key":"p52"},{"id":"n41985","layer":"informal","project":"p52","title":"MyNat.one_mul","kind":"lemma","summary":"For all a in MyNat we have 1 * a = a.","labels":["MyNat.one_mul"],"detail_key":"p52"},{"id":"n41986","layer":"informal","project":"p52","title":"By induction, recalling that 1 = succ(0).","kind":"proof","summary":"By induction, recalling that 1 = succ(0).","labels":[],"detail_key":"p52"},{"id":"n41987","layer":"informal","project":"p52","title":"MyNat.instCommSemiring","kind":"proposition","summary":"MyNat with addition and multiplication is a commutative semiring.","labels":["MyNat.instCommSemiring"],"detail_key":"p52"},{"id":"n41988","layer":"informal","project":"p52","title":"We collect all the lemmas above: addition is associative and commutative with neutral ele…","kind":"proof","summary":"We collect all the lemmas above: addition is associative and commutative with neutral element 0…","labels":[],"detail_key":"p52"},{"id":"n41989","layer":"informal","project":"p52","title":"MyNat.zero_ne_one","kind":"lemma","summary":"In MyNat we have 0 \\neq 1.","labels":["MyNat.zero_ne_one"],"detail_key":"p52"},{"id":"n41990","layer":"informal","project":"p52","title":"The constructors zero and succ are distinct, and 1 = succ(0).","kind":"proof","summary":"The constructors zero and succ are distinct, and 1 = succ(0).","labels":[],"detail_key":"p52"},{"id":"n41991","layer":"informal","project":"p52","title":"MyNat.mul_ne_zero","kind":"lemma","summary":"Let a and b in MyNat be such that a \\neq 0 and b \\neq 0. Then a * b \\neq 0.","labels":["MyNat.mul_ne_zero"],"detail_key":"p52"},{"id":"n41992","layer":"informal","project":"p52","title":"If a \\neq 0 and b \\neq 0 we can write a = succ(a') and b = succ(b'), and then a * b = suc…","kind":"proof","summary":"If a \\neq 0 and b \\neq 0 we can write a = succ(a') and b = succ(b'), and then a * b = succ(a' *…","labels":[],"detail_key":"p52"},{"id":"n41993","layer":"informal","project":"p52","title":"MyNat.le","kind":"definition","summary":"Let a and b in MyNat. We write a \\leq b if there exists a natural number x such that \\[ b = a +…","labels":["MyNat.le"],"detail_key":"p52"},{"id":"n41994","layer":"informal","project":"p52","title":"MyNat.le_refl","kind":"lemma","summary":"The relation \\leq on MyNat is reflexive.","labels":["MyNat.le_refl"],"detail_key":"p52"},{"id":"n41995","layer":"informal","project":"p52","title":"We can take x = 0.","kind":"proof","summary":"We can take x = 0.","labels":[],"detail_key":"p52"},{"id":"n41996","layer":"informal","project":"p52","title":"MyNat.le_trans","kind":"lemma","summary":"The relation \\leq on MyNat is transitive.","labels":["MyNat.le_trans"],"detail_key":"p52"},{"id":"n41997","layer":"informal","project":"p52","title":"If b = a + x and c = b + y then c = a + (x + y).","kind":"proof","summary":"If b = a + x and c = b + y then c = a + (x + y).","labels":[],"detail_key":"p52"},{"id":"n41998","layer":"informal","project":"p52","title":"MyNat.le_antisymm","kind":"lemma","summary":"The relation \\leq on MyNat is antisymmetric.","labels":["MyNat.le_antisymm"],"detail_key":"p52"},{"id":"n41999","layer":"informal","project":"p52","title":"If b = a + x and a = b + y then a = a + (x + y), so x + y = 0 and hence x = y = 0, giving…","kind":"proof","summary":"If b = a + x and a = b + y then a = a + (x + y), so x + y = 0 and hence x = y = 0, giving a = b.","labels":[],"detail_key":"p52"},{"id":"n42000","layer":"informal","project":"p52","title":"MyNat.le_total","kind":"lemma","summary":"The order \\leq on MyNat is total.","labels":["MyNat.le_total"],"detail_key":"p52"},{"id":"n42001","layer":"informal","project":"p52","title":"By induction one shows that for all a and b either a \\leq b or b \\leq a.","kind":"proof","summary":"By induction one shows that for all a and b either a \\leq b or b \\leq a.","labels":[],"detail_key":"p52"},{"id":"n42002","layer":"informal","project":"p52","title":"MyNat.linearOrder","kind":"lemma","summary":"We have that MyNat with \\leq is a \\emphlinear order.","labels":["MyNat.linearOrder"],"detail_key":"p52"},{"id":"n42003","layer":"informal","project":"p52","title":"Clear from the lemmas above.","kind":"proof","summary":"Clear from the lemmas above.","labels":[],"detail_key":"p52"},{"id":"n42004","layer":"informal","project":"p52","title":"MyNat.add_le_add_left","kind":"lemma","summary":"Let a, b and c in MyNat be such that a \\leq b. Then a + c \\leq b + c.","labels":["MyNat.add_le_add_left"],"detail_key":"p52"},{"id":"n42005","layer":"informal","project":"p52","title":"Let x be such that b = a + x. The same x shows that a + c \\leq b + c.","kind":"proof","summary":"Let x be such that b = a + x. The same x shows that a + c \\leq b + c.","labels":[],"detail_key":"p52"},{"id":"n42006","layer":"informal","project":"p52","title":"MyNat.mul_lt_mul_of_pos_left","kind":"lemma","summary":"Let a, b and c in MyNat be such that 0 < c and a < b. Then c * a < c * b.","labels":["MyNat.mul_lt_mul_of_pos_left"],"detail_key":"p52"},{"id":"n42007","layer":"informal","project":"p52","title":"Write b = a + x with x \\neq 0. Then c * b = c * a + c * x with c * x \\neq 0, so c * a < c…","kind":"proof","summary":"Write b = a + x with x \\neq 0. Then c * b = c * a + c * x with c * x \\neq 0, so c * a < c * b.","labels":[],"detail_key":"p52"},{"id":"n42008","layer":"informal","project":"p52","title":"MyNat.mul_lt_mul_of_pos_right","kind":"lemma","summary":"Let a, b and c in MyNat be such that 0 < c and a < b. Then a * c < b * c.","labels":["MyNat.mul_lt_mul_of_pos_right"],"detail_key":"p52"},{"id":"n42009","layer":"informal","project":"p52","title":"Identical to the previous lemma using commutativity of multiplication.","kind":"proof","summary":"Identical to the previous lemma using commutativity of multiplication.","labels":[],"detail_key":"p52"},{"id":"n42010","layer":"informal","project":"p52","title":"MyPreint","kind":"definition","summary":"Let MyPreint be MyNat\\times MyNat.","labels":["MyPreint"],"detail_key":"p52"},{"id":"n42011","layer":"informal","project":"p52","title":"MyPreint.R","kind":"definition","summary":"We define a relation R on MyPreint as follows: (a,b) and (c, d) are related if and only if \\[ a…","labels":["MyPreint.R"],"detail_key":"p52"},{"id":"n42012","layer":"informal","project":"p52","title":"MyPreint.R_refl","kind":"lemma","summary":"R is a reflexive relation.","labels":["MyPreint.R_refl"],"detail_key":"p52"},{"id":"n42013","layer":"informal","project":"p52","title":"This follows by commutativity of addition in MyNat.","kind":"proof","summary":"This follows by commutativity of addition in MyNat.","labels":[],"detail_key":"p52"},{"id":"n42014","layer":"informal","project":"p52","title":"MyPreint.R_symm","kind":"lemma","summary":"R is a symmetric relation.","labels":["MyPreint.R_symm"],"detail_key":"p52"},{"id":"n42015","layer":"informal","project":"p52","title":"This follows by commutativity of addition in MyNat.","kind":"proof","summary":"This follows by commutativity of addition in MyNat.","labels":[],"detail_key":"p52"},{"id":"n42016","layer":"informal","project":"p52","title":"MyPreint.R_trans","kind":"lemma","summary":"R is a transitive relation.","labels":["MyPreint.R_trans"],"detail_key":"p52"},{"id":"n42017","layer":"informal","project":"p52","title":"Let x, y and z in MyPreint such that x R y and y R z. We can write x = (a,b) and similarl…","kind":"proof","summary":"Let x, y and z in MyPreint such that x R y and y R z. We can write x = (a,b) and similarly for…","labels":[],"detail_key":"p52"},{"id":"n42018","layer":"informal","project":"p52","title":"MyPreint.R_equiv","kind":"lemma","summary":"We have that R is an equivalence relation. From now on, we will write x \\approx y for x R y.","labels":["MyPreint.R_equiv"],"detail_key":"p52"},{"id":"n42019","layer":"informal","project":"p52","title":"Clear from Lemma~\\refMyPreint.R_refl, Lemma~\\refMyPreint.R_symm and Lemma~\\refMyPreint.R_…","kind":"proof","summary":"Clear from Lemma~\\refMyPreint.R_refl, Lemma~\\refMyPreint.R_symm and Lemma~\\refMyPreint.R_trans.","labels":[],"detail_key":"p52"},{"id":"n42020","layer":"informal","project":"p52","title":"MyPreint.neg","kind":"definition","summary":"We define an operation, called \\emphnegation on MyPreint as follows: the negation of x = (a,b)…","labels":["MyPreint.neg"],"detail_key":"p52"},{"id":"n42021","layer":"informal","project":"p52","title":"MyPreint.neg_quotient","kind":"lemma","summary":"If x \\approx x', then -x \\approx -x'.","labels":["MyPreint.neg_quotient"],"detail_key":"p52"},{"id":"n42022","layer":"informal","project":"p52","title":"Let x = (a,b) and x' = (a',b'), so by assumption a + b' = b + a'. By definition we have \\…","kind":"proof","summary":"Let x = (a,b) and x' = (a',b'), so by assumption a + b' = b + a'. By definition we have \\[ -x=-…","labels":[],"detail_key":"p52"},{"id":"n42023","layer":"informal","project":"p52","title":"MyPreint.add","kind":"definition","summary":"We define an operation, called \\emphaddition on MyPreint as follows: the addition of x = (a,b)…","labels":["MyPreint.add"],"detail_key":"p52"},{"id":"n42024","layer":"informal","project":"p52","title":"MyPreint.add_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x + y \\approx x' + y'.","labels":["MyPreint.add_quotient"],"detail_key":"p52"},{"id":"n42025","layer":"informal","project":"p52","title":"Let x = (a,b), y = (c,d), x' = (a',b') and y' = (c',d') such that x \\approx x' and y \\app…","kind":"proof","summary":"Let x = (a,b), y = (c,d), x' = (a',b') and y' = (c',d') such that x \\approx x' and y \\approx y'…","labels":[],"detail_key":"p52"},{"id":"n42026","layer":"informal","project":"p52","title":"MyPreint.mul","kind":"definition","summary":"We define an operation, called \\emphmultiplication on MyPreint as follows: the multiplication o…","labels":["MyPreint.mul"],"detail_key":"p52"},{"id":"n42027","layer":"informal","project":"p52","title":"MyPreint.mul_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x * y \\approx x' * y'.","labels":["MyPreint.mul_quotient"],"detail_key":"p52"},{"id":"n42028","layer":"informal","project":"p52","title":"H1","kind":"proof","summary":"Let x = (a,b), y = (c,d), x' = (a',b') and y' = (c',d') such that x \\approx x' and y \\approx y'…","labels":["H1","H2","H3","H4"],"detail_key":"p52"},{"id":"n42029","layer":"informal","project":"p52","title":"MyInt","kind":"definition","summary":"We define our integers MyInt as \\[ MyInt= MyPreint\\, / \\approx \\] We will write ⟦ (a, b) ⟧ for…","labels":["MyInt"],"detail_key":"p52"},{"id":"n42030","layer":"informal","project":"p52","title":"MyInt.zero","kind":"definition","summary":"We define the zero of MyInt, denoted 0 as the class of (0,0).","labels":["MyInt.zero"],"detail_key":"p52"},{"id":"n42031","layer":"informal","project":"p52","title":"MyInt.one","kind":"definition","summary":"We define the one of MyInt, denoted 1 as the class of (1,0).","labels":["MyInt.one"],"detail_key":"p52"},{"id":"n42032","layer":"informal","project":"p52","title":"MyInt.neg","kind":"definition","summary":"We define the negation of x = ⟦ (a, b) ⟧ in MyInt as \\[ -x = ⟦ -(a, b) ⟧ \\] Thanks to Lemma~\\re…","labels":["MyInt.neg"],"detail_key":"p52"},{"id":"n42033","layer":"informal","project":"p52","title":"MyInt.add","kind":"definition","summary":"We define the addition of x = ⟦ (a, b) ⟧ and y = ⟦ (c, d) ⟧ in MyInt as \\[ x + y = ⟦ (a, b)+(c,…","labels":["MyInt.add"],"detail_key":"p52"},{"id":"n42034","layer":"informal","project":"p52","title":"MyInt.mul","kind":"definition","summary":"We define the multiplication of x = ⟦ (a, b) ⟧ and y = ⟦ (c, d) ⟧ in MyInt as \\[ x * y = ⟦ (a,…","labels":["MyInt.mul"],"detail_key":"p52"},{"id":"n42035","layer":"informal","project":"p52","title":"MyInt.add_assoc","kind":"lemma","summary":"Addition on MyInt is associative.","labels":["MyInt.add_assoc"],"detail_key":"p52"},{"id":"n42036","layer":"informal","project":"p52","title":"To prove the lemma it is enough to prove that, for all a, b, c, d, e and f in MyNat, we h…","kind":"proof","summary":"To prove the lemma it is enough to prove that, for all a, b, c, d, e and f in MyNat, we have \\[…","labels":[],"detail_key":"p52"},{"id":"n42037","layer":"informal","project":"p52","title":"MyInt.commRing","kind":"proposition","summary":"MyInt with addition and multiplication is a commutative ring.","labels":["MyInt.commRing"],"detail_key":"p52"},{"id":"n42038","layer":"informal","project":"p52","title":"We have to prove various properties, namely: \\item addition is associative (already done…","kind":"proof","summary":"We have to prove various properties, namely: \\item addition is associative (already done in Lem…","labels":[],"detail_key":"p52"},{"id":"n42039","layer":"informal","project":"p52","title":"MyInt.zero_ne_one","kind":"lemma","summary":"In MyInt we have 0 \\neq 1.","labels":["MyInt.zero_ne_one"],"detail_key":"p52"},{"id":"n42040","layer":"informal","project":"p52","title":"If 0 = 1 by definition we would have ⟦ (0,0) ⟧ = ⟦ (1,0) ⟧ so 0+1=0+0 in MyNat, that is a…","kind":"proof","summary":"If 0 = 1 by definition we would have ⟦ (0,0) ⟧ = ⟦ (1,0) ⟧ so 0+1=0+0 in MyNat, that is absurd.","labels":[],"detail_key":"p52"},{"id":"n42041","layer":"informal","project":"p52","title":"MyInt.mul_ne_zero","kind":"lemma","summary":"Let x and y in MyInt such that x \\neq 0 and y \\neq 0. Then x*y \\neq 0.","labels":["MyInt.mul_ne_zero"],"detail_key":"p52"},{"id":"n42042","layer":"informal","project":"p52","title":"It is enough to prove that, for all a, b, c and d in MyNat such that a \\neq b and c \\neq…","kind":"proof","summary":"It is enough to prove that, for all a, b, c and d in MyNat such that a \\neq b and c \\neq d we h…","labels":[],"detail_key":"p52"},{"id":"n42043","layer":"informal","project":"p52","title":"MyInt.eq_of_mul_eq_mul_right","kind":"lemma","summary":"Let x, y and z in MyInt such that x \\neq 0 and y * x = z * x. Then y = z.","labels":["MyInt.eq_of_mul_eq_mul_right"],"detail_key":"p52"},{"id":"n42044","layer":"informal","project":"p52","title":"We have 0 = y*x-z*x=(y-z)*x. Since x \\neq 0 we have by Lemma~\\refMyInt.mul_ne_zero that y…","kind":"proof","summary":"We have 0 = y*x-z*x=(y-z)*x. Since x \\neq 0 we have by Lemma~\\refMyInt.mul_ne_zero that y-z=0 a…","labels":[],"detail_key":"p52"},{"id":"n42045","layer":"informal","project":"p52","title":"MyInt.i","kind":"definition","summary":"We define a map i \\colon MyNat\\to MyInt\\\\ n \\mapsto ⟦ (n,0) ⟧","labels":["MyInt.i"],"detail_key":"p52"},{"id":"n42046","layer":"informal","project":"p52","title":"MyInt.i_zero","kind":"lemma","summary":"We have that i(0) = 0.","labels":["MyInt.i_zero"],"detail_key":"p52"},{"id":"n42047","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42048","layer":"informal","project":"p52","title":"MyInt.i_one","kind":"lemma","summary":"We have that i(1) = 1.","labels":["MyInt.i_one"],"detail_key":"p52"},{"id":"n42049","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42050","layer":"informal","project":"p52","title":"MyInt.i_add","kind":"lemma","summary":"For all a and b in MyNat we have that \\[ i(a+b) = i(a) + i(b) \\]","labels":["MyInt.i_add"],"detail_key":"p52"},{"id":"n42051","layer":"informal","project":"p52","title":"We have i(a+b) = ⟦ (a+b, 0) ⟧, i(a) = ⟦ (a, 0) ⟧ and i(b) = ⟦ (b, 0) ⟧, so we need to pro…","kind":"proof","summary":"We have i(a+b) = ⟦ (a+b, 0) ⟧, i(a) = ⟦ (a, 0) ⟧ and i(b) = ⟦ (b, 0) ⟧, so we need to prove tha…","labels":[],"detail_key":"p52"},{"id":"n42052","layer":"informal","project":"p52","title":"MyInt.i_mul","kind":"lemma","summary":"For all a and b in MyNat we have that \\[ i(a*b) = i(a) * i(b) \\]","labels":["MyInt.i_mul"],"detail_key":"p52"},{"id":"n42053","layer":"informal","project":"p52","title":"We have i(a*b) = ⟦ (a*b, 0) ⟧, i(a) = ⟦ (a, 0) ⟧ and i(b) = ⟦ (b, 0) ⟧, so we need to pro…","kind":"proof","summary":"We have i(a*b) = ⟦ (a*b, 0) ⟧, i(a) = ⟦ (a, 0) ⟧ and i(b) = ⟦ (b, 0) ⟧, so we need to prove tha…","labels":[],"detail_key":"p52"},{"id":"n42054","layer":"informal","project":"p52","title":"MyInt.i_injective","kind":"lemma","summary":"We have that i is injective.","labels":["MyInt.i_injective"],"detail_key":"p52"},{"id":"n42055","layer":"informal","project":"p52","title":"Let a and b such that i(a)=i(b). This means ⟦ (a,0) ⟧ = ⟦ (b,0) ⟧ so a + 0 = 0 + b and he…","kind":"proof","summary":"Let a and b such that i(a)=i(b). This means ⟦ (a,0) ⟧ = ⟦ (b,0) ⟧ so a + 0 = 0 + b and hence a…","labels":[],"detail_key":"p52"},{"id":"n42056","layer":"informal","project":"p52","title":"MyInt.le","kind":"definition","summary":"Let x and y in MyInt. We write x \\leq y if there exist a natural number n such that \\[ y = x +…","labels":["MyInt.le"],"detail_key":"p52"},{"id":"n42057","layer":"informal","project":"p52","title":"MyInt.le_refl","kind":"lemma","summary":"The relation \\leq on MyInt is reflexive.","labels":["MyInt.le_refl"],"detail_key":"p52"},{"id":"n42058","layer":"informal","project":"p52","title":"We can just take n = 0.","kind":"proof","summary":"We can just take n = 0.","labels":[],"detail_key":"p52"},{"id":"n42059","layer":"informal","project":"p52","title":"MyInt.le_trans","kind":"lemma","summary":"The relation \\leq on MyInt is transitive.","labels":["MyInt.le_trans"],"detail_key":"p52"},{"id":"n42060","layer":"informal","project":"p52","title":"Let x, y and z such that x \\leq y and y \\leq z. It follows that there exist p and q such…","kind":"proof","summary":"Let x, y and z such that x \\leq y and y \\leq z. It follows that there exist p and q such that y…","labels":[],"detail_key":"p52"},{"id":"n42061","layer":"informal","project":"p52","title":"MyInt.le_antisymm","kind":"lemma","summary":"The relation \\leq on MyInt is antisymmetric.","labels":["MyInt.le_antisymm"],"detail_key":"p52"},{"id":"n42062","layer":"informal","project":"p52","title":"Let x and y such that x \\leq y and y \\leq x. It follows that there exist p and q such tha…","kind":"proof","summary":"Let x and y such that x \\leq y and y \\leq x. It follows that there exist p and q such that y =…","labels":[],"detail_key":"p52"},{"id":"n42063","layer":"informal","project":"p52","title":"MyInt.le_total","kind":"lemma","summary":"The order \\leq on MyInt is a total order.","labels":["MyInt.le_total"],"detail_key":"p52"},{"id":"n42064","layer":"informal","project":"p52","title":"Let x and y be in MyInt. We can write x = ⟦ (a,b) ⟧ and y = ⟦ (c,d) ⟧ and we need to prov…","kind":"proof","summary":"Let x and y be in MyInt. We can write x = ⟦ (a,b) ⟧ and y = ⟦ (c,d) ⟧ and we need to prove that…","labels":[],"detail_key":"p52"},{"id":"n42065","layer":"informal","project":"p52","title":"MyInt.linearOrder","kind":"lemma","summary":"We have that MyInt with \\leq is a \\emphlinear order","labels":["MyInt.linearOrder"],"detail_key":"p52"},{"id":"n42066","layer":"informal","project":"p52","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p52"},{"id":"n42067","layer":"informal","project":"p52","title":"MyInt.zero_le_one","kind":"lemma","summary":"In MyInt we have that 0 \\leq 1.","labels":["MyInt.zero_le_one"],"detail_key":"p52"},{"id":"n42068","layer":"informal","project":"p52","title":"We use 1 (as natural number). We need to prove that 0 + i(1) = 1. Unravelling the definit…","kind":"proof","summary":"We use 1 (as natural number). We need to prove that 0 + i(1) = 1. Unravelling the definitions t…","labels":[],"detail_key":"p52"},{"id":"n42069","layer":"informal","project":"p52","title":"MyInt.i_le_iff","kind":"lemma","summary":"Given two natural numbers a and b, we have i(a) \\leq i(b) if and only if a \\leq b.","labels":["MyInt.i_le_iff"],"detail_key":"p52"},{"id":"n42070","layer":"informal","project":"p52","title":"\\item If i(a) \\leq i(b), let n be such that i(b) = i(a)+i(n)=i(a+n). We obtain b = a + n…","kind":"proof","summary":"\\item If i(a) \\leq i(b), let n be such that i(b) = i(a)+i(n)=i(a+n). We obtain b = a + n by inj…","labels":[],"detail_key":"p52"},{"id":"n42071","layer":"informal","project":"p52","title":"MyInt.add_le_add_left","kind":"lemma","summary":"Let x, y and z in MyInt be such that x \\leq y. Then x + z \\leq y + z.","labels":["MyInt.add_le_add_left"],"detail_key":"p52"},{"id":"n42072","layer":"informal","project":"p52","title":"Let n be such that y = x + i(n). It's immediate that n also works to show that x + z \\leq…","kind":"proof","summary":"Let n be such that y = x + i(n). It's immediate that n also works to show that x + z \\leq y + z.","labels":[],"detail_key":"p52"},{"id":"n42073","layer":"informal","project":"p52","title":"MyInt.mul_pos","kind":"lemma","summary":"Let x and y in MyInt be such that 0 < x and 0 < y. Then 0 < x * y.","labels":["MyInt.mul_pos"],"detail_key":"p52"},{"id":"n42074","layer":"informal","project":"p52","title":"By Lemma~\\refMyInt.mul_ne_zero we already know that x*y \\neq 0, so it is enough to prove…","kind":"proof","summary":"By Lemma~\\refMyInt.mul_ne_zero we already know that x*y \\neq 0, so it is enough to prove that 0…","labels":[],"detail_key":"p52"},{"id":"n42075","layer":"informal","project":"p52","title":"MyInt.archimedean","kind":"lemma","summary":"For every x in MyInt there exists n in MyNat such that x \\leq i(n).","labels":["MyInt.archimedean"],"detail_key":"p52"},{"id":"n42076","layer":"informal","project":"p52","title":"Write x = ⟦(a,b)⟧. Then i(a) = x + i(b) in MyInt, so x \\leq i(a).","kind":"proof","summary":"Write x = ⟦(a,b)⟧. Then i(a) = x + i(b) in MyInt, so x \\leq i(a).","labels":[],"detail_key":"p52"},{"id":"n42077","layer":"informal","project":"p52","title":"MyPrerat","kind":"definition","summary":"Let MyPrerat be MyInt\\times (MyInt\\setminus \\0\\).","labels":["MyPrerat"],"detail_key":"p52"},{"id":"n42078","layer":"informal","project":"p52","title":"MyPrerat.R","kind":"definition","summary":"We define a relation R on MyPrerat as follows: (a,b) and (c, d) are related if and only if \\[ a…","labels":["MyPrerat.R"],"detail_key":"p52"},{"id":"n42079","layer":"informal","project":"p52","title":"MyPrerat.R_refl","kind":"lemma","summary":"R is a reflexive relation.","labels":["MyPrerat.R_refl"],"detail_key":"p52"},{"id":"n42080","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42081","layer":"informal","project":"p52","title":"MyPrerat.R_symm","kind":"lemma","summary":"R is a symmetric relation.","labels":["MyPrerat.R_symm"],"detail_key":"p52"},{"id":"n42082","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42083","layer":"informal","project":"p52","title":"MyPrerat.R_trans","kind":"lemma","summary":"R is a transitive relation.","labels":["MyPrerat.R_trans"],"detail_key":"p52"},{"id":"n42084","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42085","layer":"informal","project":"p52","title":"MyPrerat.R_equiv","kind":"lemma","summary":"We have that R is an equivalence relation. From now on, we will write x \\approx y for x R y.","labels":["MyPrerat.R_equiv"],"detail_key":"p52"},{"id":"n42086","layer":"informal","project":"p52","title":"Clear from Lemma~\\refMyPrerat.R_refl, Lemma~\\refMyPrerat.R_symm and Lemma~\\refMyPrerat.R_…","kind":"proof","summary":"Clear from Lemma~\\refMyPrerat.R_refl, Lemma~\\refMyPrerat.R_symm and Lemma~\\refMyPrerat.R_trans.","labels":[],"detail_key":"p52"},{"id":"n42087","layer":"informal","project":"p52","title":"MyPrerat.neg","kind":"definition","summary":"We define an operation, called \\emphnegation on MyPrerat as follows: the negation of x = (a,b)…","labels":["MyPrerat.neg"],"detail_key":"p52"},{"id":"n42088","layer":"informal","project":"p52","title":"MyPrerat.neg_quotient","kind":"lemma","summary":"If x \\approx x', then -x \\approx -x'.","labels":["MyPrerat.neg_quotient"],"detail_key":"p52"},{"id":"n42089","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42090","layer":"informal","project":"p52","title":"MyPrerat.add","kind":"definition","summary":"We define an operation, called \\emphaddition on MyPrerat as follows: the addition of x = (a,b)…","labels":["MyPrerat.add"],"detail_key":"p52"},{"id":"n42091","layer":"informal","project":"p52","title":"MyPrerat.add_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x + y \\approx x' + y'.","labels":["MyPrerat.add_quotient"],"detail_key":"p52"},{"id":"n42092","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42093","layer":"informal","project":"p52","title":"MyPrerat.mul","kind":"definition","summary":"We define an operation, called \\emphmultiplication on MyPrerat as follows: the multiplication o…","labels":["MyPrerat.mul"],"detail_key":"p52"},{"id":"n42094","layer":"informal","project":"p52","title":"MyPrerat.mul_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x * y \\approx x' * y'.","labels":["MyPrerat.mul_quotient"],"detail_key":"p52"},{"id":"n42095","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42096","layer":"informal","project":"p52","title":"MyPrerat.inv","kind":"definition","summary":"We define an operation, called \\emphinversion on MyPrerat as follows: the inverse of x = (a,b)…","labels":["MyPrerat.inv"],"detail_key":"p52"},{"id":"n42097","layer":"informal","project":"p52","title":"MyPrerat.inv_quotient","kind":"lemma","summary":"If x \\approx x', then x^-1 \\approx x'^-1.","labels":["MyPrerat.inv_quotient"],"detail_key":"p52"},{"id":"n42098","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42099","layer":"informal","project":"p52","title":"MyRat","kind":"definition","summary":"We define our rationals MyRat as \\[ MyRat= MyPrerat\\, / \\approx \\] We will write ⟦ (a, b) ⟧ for…","labels":["MyRat"],"detail_key":"p52"},{"id":"n42100","layer":"informal","project":"p52","title":"MyRat.zero","kind":"definition","summary":"We define the zero of MyRat, denoted 0 as the class of (0,1) (note that 1 \\neq 0 in MyInt).","labels":["MyRat.zero"],"detail_key":"p52"},{"id":"n42101","layer":"informal","project":"p52","title":"MyRat.one","kind":"definition","summary":"We define the one of MyRat, denoted 1 as the class of (1,1) (note that 1 \\neq 0 in MyInt).","labels":["MyRat.one"],"detail_key":"p52"},{"id":"n42102","layer":"informal","project":"p52","title":"MyRat.neg","kind":"definition","summary":"We define the negation of x = ⟦ (a, b) ⟧ in MyRat as \\[ -x = ⟦ -(a, b) ⟧ \\] Thanks to Lemma~\\re…","labels":["MyRat.neg"],"detail_key":"p52"},{"id":"n42103","layer":"informal","project":"p52","title":"MyRat.add","kind":"definition","summary":"We define the addition of x = ⟦ (a, b) ⟧ and y = ⟦ (c, d) ⟧ in MyRat as \\[ x + y = ⟦ (a,b)+(c,d…","labels":["MyRat.add"],"detail_key":"p52"},{"id":"n42104","layer":"informal","project":"p52","title":"MyRat.mul","kind":"definition","summary":"We define the multiplication of x = ⟦ (a, b) ⟧ and y = ⟦ (c, d) ⟧ in MyRat as \\[ x * y = ⟦ (a,…","labels":["MyRat.mul"],"detail_key":"p52"},{"id":"n42105","layer":"informal","project":"p52","title":"MyRat.inv","kind":"definition","summary":"We define the inverse of x = ⟦ (a, b) ⟧ in MyRat as \\[ x^-1 = ⟦ (a, b)^-1 ⟧ \\] Thanks to Lemma~…","labels":["MyRat.inv"],"detail_key":"p52"},{"id":"n42106","layer":"informal","project":"p52","title":"MyRat.commRing","kind":"proposition","summary":"MyRat with addition and multiplication is a commutative ring.","labels":["MyRat.commRing"],"detail_key":"p52"},{"id":"n42107","layer":"informal","project":"p52","title":"We have to prove various properties, namely: \\item addition is associative \\item 0 works…","kind":"proof","summary":"We have to prove various properties, namely: \\item addition is associative \\item 0 works as neu…","labels":[],"detail_key":"p52"},{"id":"n42108","layer":"informal","project":"p52","title":"MyRat.zero_ne_one","kind":"lemma","summary":"In MyRat we have 0 \\neq 1.","labels":["MyRat.zero_ne_one"],"detail_key":"p52"},{"id":"n42109","layer":"informal","project":"p52","title":"If 0 = 1 by definition we would have ⟦ (0,1) ⟧ = ⟦ (1,1) ⟧ so 0*1=1*0 in MyInt, that is a…","kind":"proof","summary":"If 0 = 1 by definition we would have ⟦ (0,1) ⟧ = ⟦ (1,1) ⟧ so 0*1=1*0 in MyInt, that is absurd.","labels":[],"detail_key":"p52"},{"id":"n42110","layer":"informal","project":"p52","title":"MyRat.mul_inv_cancel","kind":"lemma","summary":"Let x \\neq 0 be in MyRat. Then x * x^-1 = 1.","labels":["MyRat.mul_inv_cancel"],"detail_key":"p52"},{"id":"n42111","layer":"informal","project":"p52","title":"Let x = ⟦ (a,b) ⟧, with b \\neq 0. Since x \\neq 0 we have a \\neq 0 and so x^-1 = ⟦ (b,a) ⟧…","kind":"proof","summary":"Let x = ⟦ (a,b) ⟧, with b \\neq 0. Since x \\neq 0 we have a \\neq 0 and so x^-1 = ⟦ (b,a) ⟧. The…","labels":[],"detail_key":"p52"},{"id":"n42112","layer":"informal","project":"p52","title":"MyRat.field","kind":"proposition","summary":"MyRat with addition and multiplication is a field.","labels":["MyRat.field"],"detail_key":"p52"},{"id":"n42113","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyRat.mul_inv_cancel.","kind":"proof","summary":"Clear because of Lemma~\\refMyRat.mul_inv_cancel.","labels":[],"detail_key":"p52"},{"id":"n42114","layer":"informal","project":"p52","title":"MyRat.i","kind":"definition","summary":"We define a map i \\colon MyNat\\to MyRat\\\\ n \\mapsto ⟦ (MyInt.i n,1) ⟧","labels":["MyRat.i"],"detail_key":"p52"},{"id":"n42115","layer":"informal","project":"p52","title":"MyRat.i_zero","kind":"lemma","summary":"We have that i(0) = 0.","labels":["MyRat.i_zero"],"detail_key":"p52"},{"id":"n42116","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42117","layer":"informal","project":"p52","title":"MyRat.i_one","kind":"lemma","summary":"We have that i(1) = 1.","labels":["MyRat.i_one"],"detail_key":"p52"},{"id":"n42118","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42119","layer":"informal","project":"p52","title":"MyRat.i_add","kind":"lemma","summary":"For all a and b in MyNat we have that \\[ i(a+b) = i(a) + i(b) \\]","labels":["MyRat.i_add"],"detail_key":"p52"},{"id":"n42120","layer":"informal","project":"p52","title":"We have i(a+b) = ⟦ (MyInt.i (a+b), 1) ⟧ = ⟦ (MyInt.i (a), 1) + (MyInt.i (b), 1) ⟧, i(a) =…","kind":"proof","summary":"We have i(a+b) = ⟦ (MyInt.i (a+b), 1) ⟧ = ⟦ (MyInt.i (a), 1) + (MyInt.i (b), 1) ⟧, i(a) = ⟦ (My…","labels":[],"detail_key":"p52"},{"id":"n42121","layer":"informal","project":"p52","title":"MyRat.i_mul","kind":"lemma","summary":"For all a and b in MyNat we have that \\[ i(a*b) = i(a) * i(b) \\]","labels":["MyRat.i_mul"],"detail_key":"p52"},{"id":"n42122","layer":"informal","project":"p52","title":"Similar to the proof of Lemma~\\refMyRat.i_add.","kind":"proof","summary":"Similar to the proof of Lemma~\\refMyRat.i_add.","labels":[],"detail_key":"p52"},{"id":"n42123","layer":"informal","project":"p52","title":"MyRat.i_injective","kind":"lemma","summary":"We have that i is injective.","labels":["MyRat.i_injective"],"detail_key":"p52"},{"id":"n42124","layer":"informal","project":"p52","title":"Let a and b be such that i(a)=i(b). This means ⟦ (MyInt.i a,1) ⟧ = ⟦ (MyInt.i b,1) ⟧, so…","kind":"proof","summary":"Let a and b be such that i(a)=i(b). This means ⟦ (MyInt.i a,1) ⟧ = ⟦ (MyInt.i b,1) ⟧, so (MyInt…","labels":[],"detail_key":"p52"},{"id":"n42125","layer":"informal","project":"p52","title":"MyRat.j","kind":"definition","summary":"We define a map j \\colon MyInt\\to MyRat\\\\ n \\mapsto ⟦ (n,1) ⟧","labels":["MyRat.j"],"detail_key":"p52"},{"id":"n42126","layer":"informal","project":"p52","title":"MyRat.j_zero","kind":"lemma","summary":"We have that j(0) = 0.","labels":["MyRat.j_zero"],"detail_key":"p52"},{"id":"n42127","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42128","layer":"informal","project":"p52","title":"MyRat.j_one","kind":"lemma","summary":"We have that j(1) = 1.","labels":["MyRat.j_one"],"detail_key":"p52"},{"id":"n42129","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42130","layer":"informal","project":"p52","title":"MyRat.j_add","kind":"lemma","summary":"For all a and b in MyInt we have that \\[ j(a+b) = j(a) + j(b) \\]","labels":["MyRat.j_add"],"detail_key":"p52"},{"id":"n42131","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42132","layer":"informal","project":"p52","title":"MyRat.j_mul","kind":"lemma","summary":"For all a and b in MyInt we have that \\[ j(a*b) = j(a) * j(b) \\]","labels":["MyRat.j_mul"],"detail_key":"p52"},{"id":"n42133","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42134","layer":"informal","project":"p52","title":"MyRat.j_injective","kind":"lemma","summary":"We have that j is injective.","labels":["MyRat.j_injective"],"detail_key":"p52"},{"id":"n42135","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42136","layer":"informal","project":"p52","title":"MyRat.j_comp_eq_i","kind":"lemma","summary":"Let n be a natural number. Then MyRat.j (MyInt.i (n)) = MyRat.i (n).","labels":["MyRat.j_comp_eq_i"],"detail_key":"p52"},{"id":"n42137","layer":"informal","project":"p52","title":"It follows from unravelling all the definitions.","kind":"proof","summary":"It follows from unravelling all the definitions.","labels":[],"detail_key":"p52"},{"id":"n42138","layer":"informal","project":"p52","title":"MyRat.Quotient.mk_def","kind":"lemma","summary":"Let a and b be in MyInt with b \\neq 0. Then ⟦ (a, b) ⟧ = j(a)*j(b)^-1.","labels":["MyRat.Quotient.mk_def"],"detail_key":"p52"},{"id":"n42139","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42140","layer":"informal","project":"p52","title":"MyRat.IsNonneg","kind":"definition","summary":"Given x in MyRat, we say that x is \\emphnonnegative if it has a representative with nonnegative…","labels":["MyRat.IsNonneg"],"detail_key":"p52"},{"id":"n42141","layer":"informal","project":"p52","title":"MyRat.zero_nonneg","kind":"lemma","summary":"We have that 0 in MyRat is nonnegative.","labels":["MyRat.zero_nonneg"],"detail_key":"p52"},{"id":"n42142","layer":"informal","project":"p52","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p52"},{"id":"n42143","layer":"informal","project":"p52","title":"MyRat.one_nonneg","kind":"lemma","summary":"We have that 1 in MyRat is nonnegative.","labels":["MyRat.one_nonneg"],"detail_key":"p52"},{"id":"n42144","layer":"informal","project":"p52","title":"Obvious.","kind":"proof","summary":"Obvious.","labels":[],"detail_key":"p52"},{"id":"n42145","layer":"informal","project":"p52","title":"MyRat.nonneg_neg","kind":"lemma","summary":"Let x be in MyRat such that both x and -x are nonnegative. Then x = 0.","labels":["MyRat.nonneg_neg"],"detail_key":"p52"},{"id":"n42146","layer":"informal","project":"p52","title":"Unravelling all the definitions we end up with a, b, c and d in MyInt such that 0 \\leq a,…","kind":"proof","summary":"Unravelling all the definitions we end up with a, b, c and d in MyInt such that 0 \\leq a, 0 < b…","labels":[],"detail_key":"p52"},{"id":"n42147","layer":"informal","project":"p52","title":"MyRat.nonneg_neg_of_not_nonneg","kind":"lemma","summary":"Let x be in MyRat such that x is not nonnegative. Then -x is nonnegative.","labels":["MyRat.nonneg_neg_of_not_nonneg"],"detail_key":"p52"},{"id":"n42148","layer":"informal","project":"p52","title":"Annoying but easy, left as an exercise.","kind":"proof","summary":"Annoying but easy, left as an exercise.","labels":[],"detail_key":"p52"},{"id":"n42149","layer":"informal","project":"p52","title":"MyRat.isNonneg_add_isNonneg","kind":"lemma","summary":"Let x and y be in MyRat both nonnegative. Then x+y is nonnegative.","labels":["MyRat.isNonneg_add_isNonneg"],"detail_key":"p52"},{"id":"n42150","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42151","layer":"informal","project":"p52","title":"MyRat.isNonneg_mul_isNonneg","kind":"lemma","summary":"Let x and y be in MyRat both nonnegative. Then x*y is nonnegative.","labels":["MyRat.isNonneg_mul_isNonneg"],"detail_key":"p52"},{"id":"n42152","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42153","layer":"informal","project":"p52","title":"MyRat.isNonneg_inv_isNonneg","kind":"lemma","summary":"Let x be in MyRat be nonnegative. Then x^-1 is nonnegative.","labels":["MyRat.isNonneg_inv_isNonneg"],"detail_key":"p52"},{"id":"n42154","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42155","layer":"informal","project":"p52","title":"MyRat.le","kind":"definition","summary":"Let x and y in MyRat. We write x \\leq y if y - x is nonnegative.","labels":["MyRat.le"],"detail_key":"p52"},{"id":"n42156","layer":"informal","project":"p52","title":"MyRat.zero_le_iff_IsNonneg","kind":"lemma","summary":"We have that 0 \\leq x if and only if x is nonnegative.","labels":["MyRat.zero_le_iff_IsNonneg"],"detail_key":"p52"},{"id":"n42157","layer":"informal","project":"p52","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p52"},{"id":"n42158","layer":"informal","project":"p52","title":"MyRat.zero_le_one","kind":"lemma","summary":"In MyRat we have that 0 \\leq 1.","labels":["MyRat.zero_le_one"],"detail_key":"p52"},{"id":"n42159","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyRat.one_nonneg.","kind":"proof","summary":"Clear because of Lemma~\\refMyRat.one_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42160","layer":"informal","project":"p52","title":"MyRat.le_refl","kind":"lemma","summary":"The relation \\leq on MyRat is reflexive.","labels":["MyRat.le_refl"],"detail_key":"p52"},{"id":"n42161","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyRat.zero_nonneg.","kind":"proof","summary":"Clear because of Lemma~\\refMyRat.zero_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42162","layer":"informal","project":"p52","title":"MyRat.le_trans","kind":"lemma","summary":"The relation \\leq on MyRat is transitive.","labels":["MyRat.le_trans"],"detail_key":"p52"},{"id":"n42163","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyRat.isNonneg_add_isNonneg.","kind":"proof","summary":"It follows from Lemma~\\refMyRat.isNonneg_add_isNonneg.","labels":[],"detail_key":"p52"},{"id":"n42164","layer":"informal","project":"p52","title":"MyRat.le_antisymm","kind":"lemma","summary":"The relation \\leq on MyRat is antisymmetric.","labels":["MyRat.le_antisymm"],"detail_key":"p52"},{"id":"n42165","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyRat.nonneg_neg.","kind":"proof","summary":"It follows from Lemma~\\refMyRat.nonneg_neg.","labels":[],"detail_key":"p52"},{"id":"n42166","layer":"informal","project":"p52","title":"MyRat.add_le_add_left","kind":"lemma","summary":"Let x, y and z in MyRat be such that x \\leq y. Then x + z \\leq y + z.","labels":["MyRat.add_le_add_left"],"detail_key":"p52"},{"id":"n42167","layer":"informal","project":"p52","title":"Clear from the definitions.","kind":"proof","summary":"Clear from the definitions.","labels":[],"detail_key":"p52"},{"id":"n42168","layer":"informal","project":"p52","title":"MyRat.mul_nonneg","kind":"lemma","summary":"Let x and y in MyRat be such that 0 \\leq x and 0 \\leq y. Then 0 \\leq x * y.","labels":["MyRat.mul_nonneg"],"detail_key":"p52"},{"id":"n42169","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyRat.isNonneg_mul_isNonneg.","kind":"proof","summary":"It follows from Lemma~\\refMyRat.isNonneg_mul_isNonneg.","labels":[],"detail_key":"p52"},{"id":"n42170","layer":"informal","project":"p52","title":"MyRat.j_le_iff","kind":"lemma","summary":"Let x and y in MyInt. We have that j(x) \\leq j(y) if and only if x \\leq y.","labels":["MyRat.j_le_iff"],"detail_key":"p52"},{"id":"n42171","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42172","layer":"informal","project":"p52","title":"MyRat.i_le_iff","kind":"lemma","summary":"Let x and y in MyNat. We have that i(x) \\leq i(y) if and only if x \\leq y.","labels":["MyRat.i_le_iff"],"detail_key":"p52"},{"id":"n42173","layer":"informal","project":"p52","title":"It follows immediately by Lemma~\\refMyInt.i_le_iff, Lemma~\\refMyRat.j_le_iff and Lemma~\\r…","kind":"proof","summary":"It follows immediately by Lemma~\\refMyInt.i_le_iff, Lemma~\\refMyRat.j_le_iff and Lemma~\\refMyRa…","labels":[],"detail_key":"p52"},{"id":"n42174","layer":"informal","project":"p52","title":"MyRat.le_total","kind":"lemma","summary":"The order \\leq on MyRat is a total order.","labels":["MyRat.le_total"],"detail_key":"p52"},{"id":"n42175","layer":"informal","project":"p52","title":"This follows by Lemma~\\refMyRat.nonneg_neg_of_not_nonneg.","kind":"proof","summary":"This follows by Lemma~\\refMyRat.nonneg_neg_of_not_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42176","layer":"informal","project":"p52","title":"MyRat.linearOrder","kind":"lemma","summary":"We have that MyRat with \\leq is a \\emphlinear order","labels":["MyRat.linearOrder"],"detail_key":"p52"},{"id":"n42177","layer":"informal","project":"p52","title":"Clear from the lemma above.","kind":"proof","summary":"Clear from the lemma above.","labels":[],"detail_key":"p52"},{"id":"n42178","layer":"informal","project":"p52","title":"MyRat.mul_pos","kind":"lemma","summary":"Let x and y in MyRat be such that 0 < x and 0 < y. Then 0 < x * y.","labels":["MyRat.mul_pos"],"detail_key":"p52"},{"id":"n42179","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42180","layer":"informal","project":"p52","title":"MyRat.archimedean","kind":"lemma","summary":"For every x in MyRat there exists n in MyNat such that x \\leq i(n).","labels":["MyRat.archimedean"],"detail_key":"p52"},{"id":"n42181","layer":"informal","project":"p52","title":"Choose a representative x = a/b with 0 < b. By Lemma~\\refMyInt.archimedean, choose a natu…","kind":"proof","summary":"Choose a representative x = a/b with 0 < b. By Lemma~\\refMyInt.archimedean, choose a natural nu…","labels":[],"detail_key":"p52"},{"id":"n42182","layer":"informal","project":"p52","title":"IsCauchy","kind":"definition","summary":"A sequence x \\colon MyNat\\to MyRat is a \\emphCauchy sequence if \\[ \\forall \\varepsilon > 0, \\ \\…","labels":["IsCauchy"],"detail_key":"p52"},{"id":"n42183","layer":"informal","project":"p52","title":"IsCauchy.bounded","kind":"lemma","summary":"Every Cauchy sequence x is bounded: there exists B > 0 such that |x_n| \\leq B for all n.","labels":["IsCauchy.bounded"],"detail_key":"p52"},{"id":"n42184","layer":"informal","project":"p52","title":"Apply the definition with \\varepsilon = 1 to get an N, and take B to be the maximum of 1…","kind":"proof","summary":"Apply the definition with \\varepsilon = 1 to get an N, and take B to be the maximum of 1 + |x_N…","labels":[],"detail_key":"p52"},{"id":"n42185","layer":"informal","project":"p52","title":"MyPrereal","kind":"definition","summary":"Let MyPrereal be the type of Cauchy sequences, that is the subtype of x \\colon MyNat\\to MyRat s…","labels":["MyPrereal"],"detail_key":"p52"},{"id":"n42186","layer":"informal","project":"p52","title":"MyPrereal.R","kind":"definition","summary":"We define a relation R on MyPrereal as follows: x and y are related if and only if \\[ \\forall \\…","labels":["MyPrereal.R"],"detail_key":"p52"},{"id":"n42187","layer":"informal","project":"p52","title":"MyPrereal.R_refl","kind":"lemma","summary":"R is a reflexive relation.","labels":["MyPrereal.R_refl"],"detail_key":"p52"},{"id":"n42188","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42189","layer":"informal","project":"p52","title":"MyPrereal.R_symm","kind":"lemma","summary":"R is a symmetric relation.","labels":["MyPrereal.R_symm"],"detail_key":"p52"},{"id":"n42190","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42191","layer":"informal","project":"p52","title":"MyPrereal.R_trans","kind":"lemma","summary":"R is a transitive relation.","labels":["MyPrereal.R_trans"],"detail_key":"p52"},{"id":"n42192","layer":"informal","project":"p52","title":"Exercise, using the triangle inequality and splitting \\varepsilon into \\varepsilon/2 + \\v…","kind":"proof","summary":"Exercise, using the triangle inequality and splitting \\varepsilon into \\varepsilon/2 + \\varepsi…","labels":[],"detail_key":"p52"},{"id":"n42193","layer":"informal","project":"p52","title":"MyPrereal.R_equiv","kind":"lemma","summary":"We have that R is an equivalence relation. From now on, we will write x \\approx y for x R y.","labels":["MyPrereal.R_equiv"],"detail_key":"p52"},{"id":"n42194","layer":"informal","project":"p52","title":"Clear from Lemma~\\refMyPrereal.R_refl, Lemma~\\refMyPrereal.R_symm and Lemma~\\refMyPrereal…","kind":"proof","summary":"Clear from Lemma~\\refMyPrereal.R_refl, Lemma~\\refMyPrereal.R_symm and Lemma~\\refMyPrereal.R_tra…","labels":[],"detail_key":"p52"},{"id":"n42195","layer":"informal","project":"p52","title":"MyPrereal.IsCauchy.const","kind":"lemma","summary":"For any rational x, the constant sequence n \\mapsto x is a Cauchy sequence.","labels":["MyPrereal.IsCauchy.const"],"detail_key":"p52"},{"id":"n42196","layer":"informal","project":"p52","title":"Obvious, since |x - x| = 0 \\leq \\varepsilon.","kind":"proof","summary":"Obvious, since |x - x| = 0 \\leq \\varepsilon.","labels":[],"detail_key":"p52"},{"id":"n42197","layer":"informal","project":"p52","title":"MyPrereal.zero","kind":"definition","summary":"We define the zero of MyPrereal as the constant sequence 0.","labels":["MyPrereal.zero"],"detail_key":"p52"},{"id":"n42198","layer":"informal","project":"p52","title":"MyPrereal.one","kind":"definition","summary":"We define the one of MyPrereal as the constant sequence 1.","labels":["MyPrereal.one"],"detail_key":"p52"},{"id":"n42199","layer":"informal","project":"p52","title":"MyPrereal.IsCauchy.neg","kind":"lemma","summary":"If x is a Cauchy sequence, then -x (the termwise negation) is a Cauchy sequence. In particular…","labels":["MyPrereal.IsCauchy.neg"],"detail_key":"p52"},{"id":"n42200","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42201","layer":"informal","project":"p52","title":"MyPrereal.neg_quotient","kind":"lemma","summary":"If x \\approx x', then -x \\approx -x'.","labels":["MyPrereal.neg_quotient"],"detail_key":"p52"},{"id":"n42202","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42203","layer":"informal","project":"p52","title":"MyPrereal.IsCauchy.add","kind":"lemma","summary":"If x and y are Cauchy sequences, then x + y (the termwise sum) is a Cauchy sequence. In particu…","labels":["MyPrereal.IsCauchy.add"],"detail_key":"p52"},{"id":"n42204","layer":"informal","project":"p52","title":"Exercise, splitting \\varepsilon into \\varepsilon/2 + \\varepsilon/2.","kind":"proof","summary":"Exercise, splitting \\varepsilon into \\varepsilon/2 + \\varepsilon/2.","labels":[],"detail_key":"p52"},{"id":"n42205","layer":"informal","project":"p52","title":"MyPrereal.add_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x + y \\approx x' + y'.","labels":["MyPrereal.add_quotient"],"detail_key":"p52"},{"id":"n42206","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42207","layer":"informal","project":"p52","title":"MyPrereal.IsCauchy.mul","kind":"lemma","summary":"If x and y are Cauchy sequences, then x * y (the termwise product) is a Cauchy sequence. In par…","labels":["MyPrereal.IsCauchy.mul"],"detail_key":"p52"},{"id":"n42208","layer":"informal","project":"p52","title":"Write x_p y_p - x_q y_q = x_p(y_p - y_q) + (x_p - x_q)y_q and use that Cauchy sequences a…","kind":"proof","summary":"Write x_p y_p - x_q y_q = x_p(y_p - y_q) + (x_p - x_q)y_q and use that Cauchy sequences are bou…","labels":[],"detail_key":"p52"},{"id":"n42209","layer":"informal","project":"p52","title":"MyPrereal.mul_quotient","kind":"lemma","summary":"If x \\approx x' and y \\approx y', then x * y \\approx x' * y'.","labels":["MyPrereal.mul_quotient"],"detail_key":"p52"},{"id":"n42210","layer":"informal","project":"p52","title":"Exercise, again using boundedness.","kind":"proof","summary":"Exercise, again using boundedness.","labels":[],"detail_key":"p52"},{"id":"n42211","layer":"informal","project":"p52","title":"MyPrereal.pos_of_not_equiv_zero","kind":"lemma","summary":"Let x be in MyPrereal with x \\not\\approx 0. Then there exist \\delta > 0 and N such that \\delta…","labels":["MyPrereal.pos_of_not_equiv_zero"],"detail_key":"p52"},{"id":"n42212","layer":"informal","project":"p52","title":"Since x \\not\\approx 0 there is some \\varepsilon > 0 for which the defining condition fail…","kind":"proof","summary":"Since x \\not\\approx 0 there is some \\varepsilon > 0 for which the defining condition fails; com…","labels":[],"detail_key":"p52"},{"id":"n42213","layer":"informal","project":"p52","title":"MyPrereal.IsCauchy.inv","kind":"lemma","summary":"Let x be in MyPrereal with x \\not\\approx 0. Then the termwise inverse x^-1 is a Cauchy sequence.","labels":["MyPrereal.IsCauchy.inv"],"detail_key":"p52"},{"id":"n42214","layer":"informal","project":"p52","title":"Using Lemma~\\refMyPrereal.pos_of_not_equiv_zero the terms are eventually bounded away fro…","kind":"proof","summary":"Using Lemma~\\refMyPrereal.pos_of_not_equiv_zero the terms are eventually bounded away from 0, a…","labels":[],"detail_key":"p52"},{"id":"n42215","layer":"informal","project":"p52","title":"MyPrereal.inv","kind":"definition","summary":"We define the inverse of x in MyPrereal as the termwise inverse x^-1 if x \\not\\approx 0, and as…","labels":["MyPrereal.inv"],"detail_key":"p52"},{"id":"n42216","layer":"informal","project":"p52","title":"MyPrereal.inv_quotient","kind":"lemma","summary":"If x \\approx x', then x^-1 \\approx x'^-1.","labels":["MyPrereal.inv_quotient"],"detail_key":"p52"},{"id":"n42217","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42218","layer":"informal","project":"p52","title":"MyReal","kind":"definition","summary":"We define our reals MyReal as \\[ MyReal= MyPrereal\\, / \\approx \\] We will write ⟦ x ⟧ for the c…","labels":["MyReal"],"detail_key":"p52"},{"id":"n42219","layer":"informal","project":"p52","title":"MyReal.zero","kind":"definition","summary":"We define the zero of MyReal, denoted 0, as the class of the zero of MyPrereal.","labels":["MyReal.zero"],"detail_key":"p52"},{"id":"n42220","layer":"informal","project":"p52","title":"MyReal.one","kind":"definition","summary":"We define the one of MyReal, denoted 1, as the class of the one of MyPrereal.","labels":["MyReal.one"],"detail_key":"p52"},{"id":"n42221","layer":"informal","project":"p52","title":"MyReal.neg","kind":"definition","summary":"We define the negation of ⟦ x ⟧ in MyReal as ⟦ -x ⟧. Thanks to Lemma~\\refMyPrereal.neg_quotient…","labels":["MyReal.neg"],"detail_key":"p52"},{"id":"n42222","layer":"informal","project":"p52","title":"MyReal.add","kind":"definition","summary":"We define the addition of ⟦ x ⟧ and ⟦ y ⟧ in MyReal as ⟦ x + y ⟧. Thanks to Lemma~\\refMyPrereal…","labels":["MyReal.add"],"detail_key":"p52"},{"id":"n42223","layer":"informal","project":"p52","title":"MyReal.mul","kind":"definition","summary":"We define the multiplication of ⟦ x ⟧ and ⟦ y ⟧ in MyReal as ⟦ x * y ⟧. Thanks to Lemma~\\refMyP…","labels":["MyReal.mul"],"detail_key":"p52"},{"id":"n42224","layer":"informal","project":"p52","title":"MyReal.inv","kind":"definition","summary":"We define the inverse of ⟦ x ⟧ in MyReal as ⟦ x^-1 ⟧. Thanks to Lemma~\\refMyPrereal.inv_quotien…","labels":["MyReal.inv"],"detail_key":"p52"},{"id":"n42225","layer":"informal","project":"p52","title":"MyReal.commRing","kind":"proposition","summary":"MyReal with addition and multiplication is a commutative ring.","labels":["MyReal.commRing"],"detail_key":"p52"},{"id":"n42226","layer":"informal","project":"p52","title":"All the ring axioms are checked by reducing to representatives and using that the operati…","kind":"proof","summary":"All the ring axioms are checked by reducing to representatives and using that the operations ar…","labels":[],"detail_key":"p52"},{"id":"n42227","layer":"informal","project":"p52","title":"MyReal.zero_ne_one","kind":"lemma","summary":"In MyReal we have 0 \\neq 1.","labels":["MyReal.zero_ne_one"],"detail_key":"p52"},{"id":"n42228","layer":"informal","project":"p52","title":"If 0 = 1 then the constant sequences 0 and 1 would be related, which is false since |0 -…","kind":"proof","summary":"If 0 = 1 then the constant sequences 0 and 1 would be related, which is false since |0 - 1| = 1…","labels":[],"detail_key":"p52"},{"id":"n42229","layer":"informal","project":"p52","title":"MyReal.mul_inv_cancel","kind":"lemma","summary":"Let x \\neq 0 be in MyReal. Then x * x^-1 = 1.","labels":["MyReal.mul_inv_cancel"],"detail_key":"p52"},{"id":"n42230","layer":"informal","project":"p52","title":"Let x = ⟦ a ⟧ with a \\not\\approx 0. By Lemma~\\refMyPrereal.pos_of_not_equiv_zero the term…","kind":"proof","summary":"Let x = ⟦ a ⟧ with a \\not\\approx 0. By Lemma~\\refMyPrereal.pos_of_not_equiv_zero the terms a_n…","labels":[],"detail_key":"p52"},{"id":"n42231","layer":"informal","project":"p52","title":"MyReal.field","kind":"proposition","summary":"MyReal with addition and multiplication is a field.","labels":["MyReal.field"],"detail_key":"p52"},{"id":"n42232","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyReal.zero_ne_one and Lemma~\\refMyReal.mul_inv_cancel.","kind":"proof","summary":"Clear because of Lemma~\\refMyReal.zero_ne_one and Lemma~\\refMyReal.mul_inv_cancel.","labels":[],"detail_key":"p52"},{"id":"n42233","layer":"informal","project":"p52","title":"MyReal.k","kind":"definition","summary":"We define a map k \\colon MyRat\\to MyReal\\\\ x \\mapsto ⟦ (n \\mapsto x) ⟧ sending a rational to th…","labels":["MyReal.k"],"detail_key":"p52"},{"id":"n42234","layer":"informal","project":"p52","title":"MyReal.k_zero","kind":"lemma","summary":"We have that k(0) = 0.","labels":["MyReal.k_zero"],"detail_key":"p52"},{"id":"n42235","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42236","layer":"informal","project":"p52","title":"MyReal.k_one","kind":"lemma","summary":"We have that k(1) = 1.","labels":["MyReal.k_one"],"detail_key":"p52"},{"id":"n42237","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42238","layer":"informal","project":"p52","title":"MyReal.k_neg","kind":"lemma","summary":"For all x in MyRat we have that k(-x) = -k(x).","labels":["MyReal.k_neg"],"detail_key":"p52"},{"id":"n42239","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42240","layer":"informal","project":"p52","title":"MyReal.k_add","kind":"lemma","summary":"For all x and y in MyRat we have that k(x + y) = k(x) + k(y).","labels":["MyReal.k_add"],"detail_key":"p52"},{"id":"n42241","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42242","layer":"informal","project":"p52","title":"MyReal.k_sub","kind":"lemma","summary":"For all x and y in MyRat we have that k(x - y) = k(x) - k(y).","labels":["MyReal.k_sub"],"detail_key":"p52"},{"id":"n42243","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42244","layer":"informal","project":"p52","title":"MyReal.k_mul","kind":"lemma","summary":"For all x and y in MyRat we have that k(x * y) = k(x) * k(y).","labels":["MyReal.k_mul"],"detail_key":"p52"},{"id":"n42245","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42246","layer":"informal","project":"p52","title":"MyReal.k_inv","kind":"lemma","summary":"For all x in MyRat we have that k(x^-1) = k(x)^-1.","labels":["MyReal.k_inv"],"detail_key":"p52"},{"id":"n42247","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42248","layer":"informal","project":"p52","title":"MyReal.k_injective","kind":"lemma","summary":"We have that k is injective.","labels":["MyReal.k_injective"],"detail_key":"p52"},{"id":"n42249","layer":"informal","project":"p52","title":"If k(x) = k(y) then the constant sequences x and y are related, so |x - y| is arbitrarily…","kind":"proof","summary":"If k(x) = k(y) then the constant sequences x and y are related, so |x - y| is arbitrarily small…","labels":[],"detail_key":"p52"},{"id":"n42250","layer":"informal","project":"p52","title":"MyPrereal.IsPos","kind":"definition","summary":"Given x in MyPrereal, we say that x is \\emphpositive if there exist \\delta > 0 and N such that…","labels":["MyPrereal.IsPos"],"detail_key":"p52"},{"id":"n42251","layer":"informal","project":"p52","title":"MyPrereal.pos_of_isPos","kind":"lemma","summary":"If x is positive, then there exists N such that 0 < x_n for all n \\geq N.","labels":["MyPrereal.pos_of_isPos"],"detail_key":"p52"},{"id":"n42252","layer":"informal","project":"p52","title":"Immediate from the definition.","kind":"proof","summary":"Immediate from the definition.","labels":[],"detail_key":"p52"},{"id":"n42253","layer":"informal","project":"p52","title":"MyPrereal.one_pos","kind":"lemma","summary":"The one of MyPrereal is positive.","labels":["MyPrereal.one_pos"],"detail_key":"p52"},{"id":"n42254","layer":"informal","project":"p52","title":"Take \\delta = 1.","kind":"proof","summary":"Take \\delta = 1.","labels":[],"detail_key":"p52"},{"id":"n42255","layer":"informal","project":"p52","title":"MyPrereal.not_isPos_zero","kind":"lemma","summary":"If x \\approx 0, then x is not positive.","labels":["MyPrereal.not_isPos_zero"],"detail_key":"p52"},{"id":"n42256","layer":"informal","project":"p52","title":"If x were positive its terms would eventually be at least \\delta > 0, contradicting x \\ap…","kind":"proof","summary":"If x were positive its terms would eventually be at least \\delta > 0, contradicting x \\approx 0.","labels":[],"detail_key":"p52"},{"id":"n42257","layer":"informal","project":"p52","title":"MyPrereal.not_equiv_zero_of_isPos","kind":"lemma","summary":"If x is positive, then x \\not\\approx 0.","labels":["MyPrereal.not_equiv_zero_of_isPos"],"detail_key":"p52"},{"id":"n42258","layer":"informal","project":"p52","title":"Contrapositive of Lemma~\\refMyPrereal.not_isPos_zero.","kind":"proof","summary":"Contrapositive of Lemma~\\refMyPrereal.not_isPos_zero.","labels":[],"detail_key":"p52"},{"id":"n42259","layer":"informal","project":"p52","title":"MyPrereal.isPos_quotient","kind":"lemma","summary":"If x \\approx x' and x is positive, then x' is positive.","labels":["MyPrereal.isPos_quotient"],"detail_key":"p52"},{"id":"n42260","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42261","layer":"informal","project":"p52","title":"MyPrereal.IsPos.add","kind":"lemma","summary":"Let x and y in MyPrereal both positive. Then x + y is positive.","labels":["MyPrereal.IsPos.add"],"detail_key":"p52"},{"id":"n42262","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42263","layer":"informal","project":"p52","title":"MyPrereal.IsPos.mul","kind":"lemma","summary":"Let x and y in MyPrereal both positive. Then x * y is positive.","labels":["MyPrereal.IsPos.mul"],"detail_key":"p52"},{"id":"n42264","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42265","layer":"informal","project":"p52","title":"MyPrereal.IsNonneg","kind":"definition","summary":"Given x in MyPrereal, we say that x is \\emphnonnegative if either x is positive or x \\approx 0.","labels":["MyPrereal.IsNonneg"],"detail_key":"p52"},{"id":"n42266","layer":"informal","project":"p52","title":"MyPrereal.IsNonneg_of_equiv_zero","kind":"lemma","summary":"If x \\approx 0, then x is nonnegative.","labels":["MyPrereal.IsNonneg_of_equiv_zero"],"detail_key":"p52"},{"id":"n42267","layer":"informal","project":"p52","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p52"},{"id":"n42268","layer":"informal","project":"p52","title":"MyPrereal.IsNonneg_of_nonneg","kind":"lemma","summary":"If there exists N such that 0 \\leq x_n for all n \\geq N, then x is nonnegative.","labels":["MyPrereal.IsNonneg_of_nonneg"],"detail_key":"p52"},{"id":"n42269","layer":"informal","project":"p52","title":"If x \\not\\approx 0 then, by an argument similar to Lemma~\\refMyPrereal.pos_of_not_equiv_z…","kind":"proof","summary":"If x \\not\\approx 0 then, by an argument similar to Lemma~\\refMyPrereal.pos_of_not_equiv_zero, x…","labels":[],"detail_key":"p52"},{"id":"n42270","layer":"informal","project":"p52","title":"MyPrereal.zero_nonneg","kind":"lemma","summary":"The zero of MyPrereal is nonnegative.","labels":["MyPrereal.zero_nonneg"],"detail_key":"p52"},{"id":"n42271","layer":"informal","project":"p52","title":"Clear, since 0 \\approx 0.","kind":"proof","summary":"Clear, since 0 \\approx 0.","labels":[],"detail_key":"p52"},{"id":"n42272","layer":"informal","project":"p52","title":"MyPrereal.one_nonneg","kind":"lemma","summary":"The one of MyPrereal is nonnegative.","labels":["MyPrereal.one_nonneg"],"detail_key":"p52"},{"id":"n42273","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyPrereal.one_pos.","kind":"proof","summary":"Clear because of Lemma~\\refMyPrereal.one_pos.","labels":[],"detail_key":"p52"},{"id":"n42274","layer":"informal","project":"p52","title":"MyPrereal.isNonneg_quotient","kind":"lemma","summary":"If x \\approx x' and x is nonnegative, then x' is nonnegative.","labels":["MyPrereal.isNonneg_quotient"],"detail_key":"p52"},{"id":"n42275","layer":"informal","project":"p52","title":"Follows from Lemma~\\refMyPrereal.isPos_quotient.","kind":"proof","summary":"Follows from Lemma~\\refMyPrereal.isPos_quotient.","labels":[],"detail_key":"p52"},{"id":"n42276","layer":"informal","project":"p52","title":"MyPrereal.eq_zero_of_isNonneg_of_isNonneg_neg","kind":"lemma","summary":"Let x be in MyPrereal such that both x and -x are nonnegative. Then x \\approx 0.","labels":["MyPrereal.eq_zero_of_isNonneg_of_isNonneg_neg"],"detail_key":"p52"},{"id":"n42277","layer":"informal","project":"p52","title":"Suppose x \\not\\approx 0. Since x is nonnegative, x is positive. Since -x is nonnegative,…","kind":"proof","summary":"Suppose x \\not\\approx 0. Since x is nonnegative, x is positive. Since -x is nonnegative, either…","labels":[],"detail_key":"p52"},{"id":"n42278","layer":"informal","project":"p52","title":"MyPrereal.isNonneg_neg_of_not_isNonneg","kind":"lemma","summary":"Let x be in MyPrereal such that x is not nonnegative. Then -x is nonnegative.","labels":["MyPrereal.isNonneg_neg_of_not_isNonneg"],"detail_key":"p52"},{"id":"n42279","layer":"informal","project":"p52","title":"If x is not nonnegative then it is in particular not eventually nonnegative, so -x is eve…","kind":"proof","summary":"If x is not nonnegative then it is in particular not eventually nonnegative, so -x is eventuall…","labels":[],"detail_key":"p52"},{"id":"n42280","layer":"informal","project":"p52","title":"MyReal.IsNonneg","kind":"definition","summary":"We say that ⟦ x ⟧ in MyReal is \\emphnonnegative if x is nonnegative. Thanks to Lemma~\\refMyPrer…","labels":["MyReal.IsNonneg"],"detail_key":"p52"},{"id":"n42281","layer":"informal","project":"p52","title":"MyReal.zero_nonneg","kind":"lemma","summary":"The zero of MyReal is nonnegative.","labels":["MyReal.zero_nonneg"],"detail_key":"p52"},{"id":"n42282","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyPrereal.zero_nonneg.","kind":"proof","summary":"Clear because of Lemma~\\refMyPrereal.zero_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42283","layer":"informal","project":"p52","title":"MyReal.eq_zero_of_isNonneg_of_isNonneg_neg","kind":"lemma","summary":"Let x be in MyReal such that both x and -x are nonnegative. Then x = 0.","labels":["MyReal.eq_zero_of_isNonneg_of_isNonneg_neg"],"detail_key":"p52"},{"id":"n42284","layer":"informal","project":"p52","title":"Follows from Lemma~\\refMyPrereal.eq_zero_of_isNonneg_of_isNonneg_neg.","kind":"proof","summary":"Follows from Lemma~\\refMyPrereal.eq_zero_of_isNonneg_of_isNonneg_neg.","labels":[],"detail_key":"p52"},{"id":"n42285","layer":"informal","project":"p52","title":"MyReal.IsNonneg.add","kind":"lemma","summary":"Let x and y in MyReal both nonnegative. Then x + y is nonnegative.","labels":["MyReal.IsNonneg.add"],"detail_key":"p52"},{"id":"n42286","layer":"informal","project":"p52","title":"Reduce to representatives and use Lemma~\\refMyPrereal.IsPos.add (treating the cases where…","kind":"proof","summary":"Reduce to representatives and use Lemma~\\refMyPrereal.IsPos.add (treating the cases where one o…","labels":[],"detail_key":"p52"},{"id":"n42287","layer":"informal","project":"p52","title":"MyReal.IsNonneg.mul","kind":"lemma","summary":"Let x and y in MyReal both nonnegative. Then x * y is nonnegative.","labels":["MyReal.IsNonneg.mul"],"detail_key":"p52"},{"id":"n42288","layer":"informal","project":"p52","title":"Reduce to representatives and use Lemma~\\refMyPrereal.IsPos.mul.","kind":"proof","summary":"Reduce to representatives and use Lemma~\\refMyPrereal.IsPos.mul.","labels":[],"detail_key":"p52"},{"id":"n42289","layer":"informal","project":"p52","title":"MyReal.le","kind":"definition","summary":"Let x and y in MyReal. We write x \\leq y if y - x is nonnegative.","labels":["MyReal.le"],"detail_key":"p52"},{"id":"n42290","layer":"informal","project":"p52","title":"MyReal.zero_le_iff_isNonneg","kind":"lemma","summary":"We have that 0 \\leq x if and only if x is nonnegative.","labels":["MyReal.zero_le_iff_isNonneg"],"detail_key":"p52"},{"id":"n42291","layer":"informal","project":"p52","title":"Clear.","kind":"proof","summary":"Clear.","labels":[],"detail_key":"p52"},{"id":"n42292","layer":"informal","project":"p52","title":"MyReal.zero_le_one","kind":"lemma","summary":"In MyReal we have that 0 \\leq 1.","labels":["MyReal.zero_le_one"],"detail_key":"p52"},{"id":"n42293","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyPrereal.one_nonneg.","kind":"proof","summary":"Clear because of Lemma~\\refMyPrereal.one_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42294","layer":"informal","project":"p52","title":"MyReal.le_refl","kind":"lemma","summary":"The relation \\leq on MyReal is reflexive.","labels":["MyReal.le_refl"],"detail_key":"p52"},{"id":"n42295","layer":"informal","project":"p52","title":"Clear because of Lemma~\\refMyReal.zero_nonneg.","kind":"proof","summary":"Clear because of Lemma~\\refMyReal.zero_nonneg.","labels":[],"detail_key":"p52"},{"id":"n42296","layer":"informal","project":"p52","title":"MyReal.le_trans","kind":"lemma","summary":"The relation \\leq on MyReal is transitive.","labels":["MyReal.le_trans"],"detail_key":"p52"},{"id":"n42297","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyReal.IsNonneg.add, since (z - y) + (y - x) = z - x.","kind":"proof","summary":"It follows from Lemma~\\refMyReal.IsNonneg.add, since (z - y) + (y - x) = z - x.","labels":[],"detail_key":"p52"},{"id":"n42298","layer":"informal","project":"p52","title":"MyReal.le_antisymm","kind":"lemma","summary":"The relation \\leq on MyReal is antisymmetric.","labels":["MyReal.le_antisymm"],"detail_key":"p52"},{"id":"n42299","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyReal.eq_zero_of_isNonneg_of_isNonneg_neg.","kind":"proof","summary":"It follows from Lemma~\\refMyReal.eq_zero_of_isNonneg_of_isNonneg_neg.","labels":[],"detail_key":"p52"},{"id":"n42300","layer":"informal","project":"p52","title":"MyReal.add_le_add_left","kind":"lemma","summary":"Let x, y and t in MyReal be such that x \\leq y. Then x + t \\leq y + t.","labels":["MyReal.add_le_add_left"],"detail_key":"p52"},{"id":"n42301","layer":"informal","project":"p52","title":"Clear from the definitions, since (y + t) - (x + t) = y - x.","kind":"proof","summary":"Clear from the definitions, since (y + t) - (x + t) = y - x.","labels":[],"detail_key":"p52"},{"id":"n42302","layer":"informal","project":"p52","title":"MyReal.mul_nonneg","kind":"lemma","summary":"Let x and y in MyReal be such that 0 \\leq x and 0 \\leq y. Then 0 \\leq x * y.","labels":["MyReal.mul_nonneg"],"detail_key":"p52"},{"id":"n42303","layer":"informal","project":"p52","title":"It follows from Lemma~\\refMyReal.IsNonneg.mul.","kind":"proof","summary":"It follows from Lemma~\\refMyReal.IsNonneg.mul.","labels":[],"detail_key":"p52"},{"id":"n42304","layer":"informal","project":"p52","title":"MyReal.k_le_iff","kind":"lemma","summary":"Let x and y in MyRat. We have that k(x) \\leq k(y) if and only if x \\leq y.","labels":["MyReal.k_le_iff"],"detail_key":"p52"},{"id":"n42305","layer":"informal","project":"p52","title":"Exercise.","kind":"proof","summary":"Exercise.","labels":[],"detail_key":"p52"},{"id":"n42306","layer":"informal","project":"p52","title":"MyReal.k_lt_iff","kind":"lemma","summary":"Let x and y in MyRat. We have that k(x) < k(y) if and only if x < y.","labels":["MyReal.k_lt_iff"],"detail_key":"p52"},{"id":"n42307","layer":"informal","project":"p52","title":"Immediate from Lemma~\\refMyReal.k_le_iff and Lemma~\\refMyReal.k_injective.","kind":"proof","summary":"Immediate from Lemma~\\refMyReal.k_le_iff and Lemma~\\refMyReal.k_injective.","labels":[],"detail_key":"p52"},{"id":"n42308","layer":"informal","project":"p52","title":"MyReal.le_total","kind":"lemma","summary":"The order \\leq on MyReal is a total order.","labels":["MyReal.le_total"],"detail_key":"p52"},{"id":"n42309","layer":"informal","project":"p52","title":"This follows by Lemma~\\refMyPrereal.isNonneg_neg_of_not_isNonneg.","kind":"proof","summary":"This follows by Lemma~\\refMyPrereal.isNonneg_neg_of_not_isNonneg.","labels":[],"detail_key":"p52"},{"id":"n42310","layer":"informal","project":"p52","title":"MyReal.linearOrder","kind":"lemma","summary":"We have that MyReal with \\leq is a \\emphlinear order.","labels":["MyReal.linearOrder"],"detail_key":"p52"},{"id":"n42311","layer":"informal","project":"p52","title":"Clear from the lemmas above.","kind":"proof","summary":"Clear from the lemmas above.","labels":[],"detail_key":"p52"},{"id":"n42312","layer":"informal","project":"p52","title":"MyReal.mul_pos","kind":"lemma","summary":"Let a and b in MyReal be such that 0 < a and 0 < b. Then 0 < a * b.","labels":["MyReal.mul_pos"],"detail_key":"p52"},{"id":"n42313","layer":"informal","project":"p52","title":"Exercise, using that a positive real is the class of a positive prereal and Lemma~\\refMyP…","kind":"proof","summary":"Exercise, using that a positive real is the class of a positive prereal and Lemma~\\refMyPrereal…","labels":[],"detail_key":"p52"},{"id":"n42314","layer":"informal","project":"p52","title":"MyReal.myRat_dense_rat'","kind":"lemma","summary":"Let x in MyReal and let \\varepsilon in MyRat with 0 < \\varepsilon. Then there exists r in MyRat…","labels":["MyReal.myRat_dense_rat'"],"detail_key":"p52"},{"id":"n42315","layer":"informal","project":"p52","title":"Pick a representative a of x; by the Cauchy property the rational r = a_N for N large eno…","kind":"proof","summary":"Pick a representative a of x; by the Cauchy property the rational r = a_N for N large enough wo…","labels":[],"detail_key":"p52"},{"id":"n42316","layer":"informal","project":"p52","title":"MyReal.myRat_dense_of_pos","kind":"lemma","summary":"Let x in MyReal with 0 < x. Then there exists r in MyRat with 0 < r and k(r) < x.","labels":["MyReal.myRat_dense_of_pos"],"detail_key":"p52"},{"id":"n42317","layer":"informal","project":"p52","title":"A positive real is the class of a positive prereal, which is eventually bounded below by…","kind":"proof","summary":"A positive real is the class of a positive prereal, which is eventually bounded below by a posi…","labels":[],"detail_key":"p52"},{"id":"n42318","layer":"informal","project":"p52","title":"MyReal.myRat_dense_rat","kind":"lemma","summary":"Let x in MyReal and let \\varepsilon in MyReal with 0 < \\varepsilon. Then there exists r in MyRa…","labels":["MyReal.myRat_dense_rat"],"detail_key":"p52"},{"id":"n42319","layer":"informal","project":"p52","title":"Combine Lemma~\\refMyReal.myRat_dense_of_pos (to find a rational below \\varepsilon) with L…","kind":"proof","summary":"Combine Lemma~\\refMyReal.myRat_dense_of_pos (to find a rational below \\varepsilon) with Lemma~\\…","labels":[],"detail_key":"p52"},{"id":"n42320","layer":"informal","project":"p52","title":"MyReal.TendsTo","kind":"definition","summary":"A sequence f \\colon MyNat\\to MyReal \\emphtends to x in MyReal if \\[ \\forall \\varepsilon > 0, \\…","labels":["MyReal.TendsTo"],"detail_key":"p52"},{"id":"n42321","layer":"informal","project":"p52","title":"MyReal.IsConvergent","kind":"definition","summary":"A sequence f \\colon MyNat\\to MyReal is \\emphconvergent if it tends to some x in MyReal.","labels":["MyReal.IsConvergent"],"detail_key":"p52"},{"id":"n42322","layer":"informal","project":"p52","title":"MyReal.IsCauchy","kind":"definition","summary":"A sequence f \\colon MyNat\\to MyReal is a \\emphCauchy sequence if \\[ \\forall \\varepsilon > 0, \\…","labels":["MyReal.IsCauchy"],"detail_key":"p52"},{"id":"n42323","layer":"informal","project":"p52","title":"MyReal.tendsTo_of_myRat_tendsTo","kind":"lemma","summary":"Let f \\colon MyNat\\to MyReal and x in MyReal. If for every rational \\varepsilon > 0 there exist…","labels":["MyReal.tendsTo_of_myRat_tendsTo"],"detail_key":"p52"},{"id":"n42324","layer":"informal","project":"p52","title":"It is enough to check the condition for \\varepsilon of the form k(\\delta), which is possi…","kind":"proof","summary":"It is enough to check the condition for \\varepsilon of the form k(\\delta), which is possible by…","labels":[],"detail_key":"p52"},{"id":"n42325","layer":"informal","project":"p52","title":"MyReal.tendsTo_myRat","kind":"lemma","summary":"Let x in MyPrereal. Then the sequence n \\mapsto k(x_n) tends to ⟦ x ⟧.","labels":["MyReal.tendsTo_myRat"],"detail_key":"p52"},{"id":"n42326","layer":"informal","project":"p52","title":"Immediate from the definition of \\approx and Lemma~\\refMyReal.tendsTo_of_myRat_tendsTo: t…","kind":"proof","summary":"Immediate from the definition of \\approx and Lemma~\\refMyReal.tendsTo_of_myRat_tendsTo: the Cau…","labels":[],"detail_key":"p52"},{"id":"n42327","layer":"informal","project":"p52","title":"MyReal.archimedean","kind":"lemma","summary":"Let x in MyReal. Then there exists a natural number n such that x \\leq k(i(n+1)), where i \\colo…","labels":["MyReal.archimedean"],"detail_key":"p52"},{"id":"n42328","layer":"informal","project":"p52","title":"Apply density with \\varepsilon = 1 to find r with |x-k(r)|<1, so x<k(r+1). Then use Lemma…","kind":"proof","summary":"Apply density with \\varepsilon = 1 to find r with |x-k(r)|<1, so x<k(r+1). 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Then there exists a rational r such that |x - k(r)| < k…","labels":["MyReal.ex_approx_punctual"],"detail_key":"p52"},{"id":"n42332","layer":"informal","project":"p52","title":"Apply density (Lemma~\\refMyReal.myRat_dense_rat') with \\varepsilon = i(n+1)^-1.","kind":"proof","summary":"Apply density (Lemma~\\refMyReal.myRat_dense_rat') with \\varepsilon = i(n+1)^-1.","labels":[],"detail_key":"p52"},{"id":"n42333","layer":"informal","project":"p52","title":"MyReal.ex_approx","kind":"lemma","summary":"Let f \\colon MyNat\\to MyReal. 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ra…","labels":["MyReal.IsCauchy.approx"],"detail_key":"p52"},{"id":"n42341","layer":"informal","project":"p52","title":"MyReal.complete","kind":"theorem","summary":"MyReal is complete: every Cauchy sequence f \\colon MyNat\\to MyReal is convergent.","labels":["MyReal.complete"],"detail_key":"p52"},{"id":"n42342","layer":"informal","project":"p52","title":"Let a = approx(f), seen as an element of MyPrereal (Definition~\\refMyReal.IsCauchy.approx…","kind":"proof","summary":"Let a = approx(f), seen as an element of MyPrereal (Definition~\\refMyReal.IsCauchy.approx), and…","labels":[],"detail_key":"p52"},{"id":"n42343","layer":"formal","project":"p52","title":"MyInt","kind":"def","summary":"Type","labels":[],"detail_key":"p52","name":"MyInt","module":"Numbers.integers"},{"id":"n42344","layer":"formal","project":"p52","title":"MyInt.add","kind":"def","summary":"MyInt → MyInt → 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(H…","labels":[],"detail_key":"p52","name":"MyRat.Quotient.mk_def","module":"Numbers.rationals"},{"id":"n42422","layer":"formal","project":"p52","title":"MyRat.add","kind":"def","summary":"MyRat → MyRat → MyRat","labels":[],"detail_key":"p52","name":"MyRat.add","module":"Numbers.rationals"},{"id":"n42423","layer":"formal","project":"p52","title":"MyRat.commRing","kind":"def","summary":"CommRing MyRat","labels":[],"detail_key":"p52","name":"MyRat.commRing","module":"Numbers.rationals"},{"id":"n42424","layer":"formal","project":"p52","title":"MyRat.field","kind":"def","summary":"Field MyRat","labels":[],"detail_key":"p52","name":"MyRat.field","module":"Numbers.rationals"},{"id":"n42425","layer":"formal","project":"p52","title":"MyRat.i","kind":"def","summary":"MyNat → MyRat","labels":[],"detail_key":"p52","name":"MyRat.i","module":"Numbers.rationals"},{"id":"n42426","layer":"formal","project":"p52","title":"MyRat.i_add","kind":"theorem","summary":"∀ (a b : MyNat), Eq (MyRat.i (HAdd.hAdd 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n)","labels":[],"detail_key":"p52","name":"MyRat.j_comp_eq_i","module":"Numbers.rationals"},{"id":"n42435","layer":"formal","project":"p52","title":"MyRat.j_injective","kind":"theorem","summary":"Function.Injective MyRat.j","labels":[],"detail_key":"p52","name":"MyRat.j_injective","module":"Numbers.rationals"},{"id":"n42436","layer":"formal","project":"p52","title":"MyRat.j_mul","kind":"theorem","summary":"∀ (a b : MyInt), Eq (MyRat.j (HMul.hMul a b)) (HMul.hMul (MyRat.j a) (MyRat.j b))","labels":[],"detail_key":"p52","name":"MyRat.j_mul","module":"Numbers.rationals"},{"id":"n42437","layer":"formal","project":"p52","title":"MyRat.j_one","kind":"theorem","summary":"Eq (MyRat.j 1) 1","labels":[],"detail_key":"p52","name":"MyRat.j_one","module":"Numbers.rationals"},{"id":"n42438","layer":"formal","project":"p52","title":"MyRat.j_zero","kind":"theorem","summary":"Eq (MyRat.j 0) 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MyRat","labels":[],"detail_key":"p52","name":"MyRat.one","module":"Numbers.rationals"},{"id":"n42443","layer":"formal","project":"p52","title":"MyRat.zero","kind":"def","summary":"Zero MyRat","labels":[],"detail_key":"p52","name":"MyRat.zero","module":"Numbers.rationals"},{"id":"n42444","layer":"formal","project":"p52","title":"MyRat.zero_ne_one","kind":"theorem","summary":"Ne 0 1","labels":[],"detail_key":"p52","name":"MyRat.zero_ne_one","module":"Numbers.rationals"},{"id":"n42445","layer":"formal","project":"p52","title":"MyRat.IsNonneg","kind":"def","summary":"MyRat → Prop","labels":[],"detail_key":"p52","name":"MyRat.IsNonneg","module":"Numbers.rationals_order"},{"id":"n42446","layer":"formal","project":"p52","title":"MyRat.add_le_add_left","kind":"theorem","summary":"∀ (x y : MyRat), LE.le x y → ∀ (t : MyRat), LE.le (HAdd.hAdd x t) (HAdd.hAdd y t)","labels":[],"detail_key":"p52","name":"MyRat.add_le_add_left","module":"Numbers.rationals_order"},{"id":"n42447","layer":"formal","project":"p52","title":"MyRat.archimedean","kind":"theorem","summary":"∀ (x : MyRat), Exists fun n => LE.le x (MyRat.i n)","labels":[],"detail_key":"p52","name":"MyRat.archimedean","module":"Numbers.rationals_order"},{"id":"n42448","layer":"formal","project":"p52","title":"MyRat.i_le_iff","kind":"theorem","summary":"∀ (a b : MyNat), Iff (LE.le (MyRat.i a) (MyRat.i b)) (LE.le a b)","labels":[],"detail_key":"p52","name":"MyRat.i_le_iff","module":"Numbers.rationals_order"},{"id":"n42449","layer":"formal","project":"p52","title":"MyRat.isNonneg_add_isNonneg","kind":"theorem","summary":"∀ x y : MyRat, x.IsNonneg → y.IsNonneg → (HAdd.hAdd x 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q)","labels":[],"detail_key":"p52","name":"MyRat.j_le_iff","module":"Numbers.rationals_order"},{"id":"n42453","layer":"formal","project":"p52","title":"MyRat.le","kind":"def","summary":"MyRat → MyRat → Prop","labels":[],"detail_key":"p52","name":"MyRat.le","module":"Numbers.rationals_order"},{"id":"n42454","layer":"formal","project":"p52","title":"MyRat.le_antisymm","kind":"theorem","summary":"∀ (x y : MyRat), x.le y → y.le x → Eq x y","labels":[],"detail_key":"p52","name":"MyRat.le_antisymm","module":"Numbers.rationals_order"},{"id":"n42455","layer":"formal","project":"p52","title":"MyRat.le_refl","kind":"theorem","summary":"∀ (x : MyRat), x.le x","labels":[],"detail_key":"p52","name":"MyRat.le_refl","module":"Numbers.rationals_order"},{"id":"n42456","layer":"formal","project":"p52","title":"MyRat.le_total","kind":"theorem","summary":"∀ (a b : MyRat), Or (LE.le a b) (LE.le b 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(x…","labels":[],"detail_key":"p52","name":"IsCauchy.bounded","module":"Numbers.reals"},{"id":"n42468","layer":"formal","project":"p52","title":"MyPrereal","kind":"def","summary":"Type","labels":[],"detail_key":"p52","name":"MyPrereal","module":"Numbers.reals"},{"id":"n42469","layer":"formal","project":"p52","title":"MyPrereal.IsCauchy.add","kind":"theorem","summary":"∀ x y : MyNat → MyRat, IsCauchy x → IsCauchy y → IsCauchy (HAdd.hAdd x y)","labels":[],"detail_key":"p52","name":"MyPrereal.IsCauchy.add","module":"Numbers.reals"},{"id":"n42470","layer":"formal","project":"p52","title":"MyPrereal.IsCauchy.const","kind":"theorem","summary":"∀ (x : MyRat), IsCauchy fun x_1 => x","labels":[],"detail_key":"p52","name":"MyPrereal.IsCauchy.const","module":"Numbers.reals"},{"id":"n42471","layer":"formal","project":"p52","title":"MyPrereal.IsCauchy.inv","kind":"theorem","summary":"∀ x : MyPrereal, Not (HasEquiv.Equiv x 0) → IsCauchy (Inv.inv 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x')","labels":[],"detail_key":"p52","name":"MyPrereal.neg_quotient","module":"Numbers.reals"},{"id":"n42494","layer":"formal","project":"p52","title":"MyPrereal.not_equiv_zero_of_isPos","kind":"theorem","summary":"∀ x : MyPrereal, x.IsPos → Not (HasEquiv.Equiv x 0)","labels":[],"detail_key":"p52","name":"MyPrereal.not_equiv_zero_of_isPos","module":"Numbers.reals"},{"id":"n42495","layer":"formal","project":"p52","title":"MyPrereal.not_isPos_zero","kind":"theorem","summary":"∀ x : MyPrereal, HasEquiv.Equiv x 0 → Not x.IsPos","labels":[],"detail_key":"p52","name":"MyPrereal.not_isPos_zero","module":"Numbers.reals"},{"id":"n42496","layer":"formal","project":"p52","title":"MyPrereal.one","kind":"def","summary":"One MyPrereal","labels":[],"detail_key":"p52","name":"MyPrereal.one","module":"Numbers.reals"},{"id":"n42497","layer":"formal","project":"p52","title":"MyPrereal.one_nonneg","kind":"theorem","summary":"MyPrereal.IsNonneg 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MyPrereal","labels":[],"detail_key":"p52","name":"MyPrereal.zero","module":"Numbers.reals"},{"id":"n42502","layer":"formal","project":"p52","title":"MyPrereal.zero_nonneg","kind":"theorem","summary":"MyPrereal.IsNonneg 0","labels":[],"detail_key":"p52","name":"MyPrereal.zero_nonneg","module":"Numbers.reals"},{"id":"n42503","layer":"formal","project":"p52","title":"MyReal","kind":"def","summary":"Type","labels":[],"detail_key":"p52","name":"MyReal","module":"Numbers.reals"},{"id":"n42504","layer":"formal","project":"p52","title":"MyReal.IsCauchy","kind":"def","summary":"(MyNat → MyReal) → Prop","labels":[],"detail_key":"p52","name":"MyReal.IsCauchy","module":"Numbers.reals"},{"id":"n42505","layer":"formal","project":"p52","title":"MyReal.IsCauchy.approx","kind":"def","summary":"f : MyNat → MyReal → MyReal.IsCauchy f → 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MyReal.k","labels":[],"detail_key":"p52","name":"MyReal.k_injective","module":"Numbers.reals"},{"id":"n42528","layer":"formal","project":"p52","title":"MyReal.k_inv","kind":"theorem","summary":"∀ (x : MyRat), Eq (MyReal.k (Inv.inv x)) (Inv.inv (MyReal.k x))","labels":[],"detail_key":"p52","name":"MyReal.k_inv","module":"Numbers.reals"},{"id":"n42529","layer":"formal","project":"p52","title":"MyReal.k_le_iff","kind":"theorem","summary":"∀ (x y : MyRat), Iff (LE.le (MyReal.k x) (MyReal.k y)) (LE.le x y)","labels":[],"detail_key":"p52","name":"MyReal.k_le_iff","module":"Numbers.reals"},{"id":"n42530","layer":"formal","project":"p52","title":"MyReal.k_lt_iff","kind":"theorem","summary":"∀ (x y : MyRat), Iff (LT.lt (MyReal.k x) (MyReal.k y)) (LT.lt x y)","labels":[],"detail_key":"p52","name":"MyReal.k_lt_iff","module":"Numbers.reals"},{"id":"n42531","layer":"formal","project":"p52","title":"MyReal.k_mul","kind":"theorem","summary":"∀ (x y : MyRat), Eq (MyReal.k (HMul.hMul x y)) (HMul.hMul (MyReal.k x) (MyReal.k y))","labels":[],"detail_key":"p52","name":"MyReal.k_mul","module":"Numbers.reals"},{"id":"n42532","layer":"formal","project":"p52","title":"MyReal.k_neg","kind":"theorem","summary":"∀ (x : MyRat), Eq (MyReal.k (Neg.neg x)) (Neg.neg (MyReal.k x))","labels":[],"detail_key":"p52","name":"MyReal.k_neg","module":"Numbers.reals"},{"id":"n42533","layer":"formal","project":"p52","title":"MyReal.k_one","kind":"theorem","summary":"Eq (MyReal.k 1) 1","labels":[],"detail_key":"p52","name":"MyReal.k_one","module":"Numbers.reals"},{"id":"n42534","layer":"formal","project":"p52","title":"MyReal.k_sub","kind":"theorem","summary":"∀ (x y : MyRat), Eq (MyReal.k (HSub.hSub x y)) (HSub.hSub (MyReal.k x) (MyReal.k y))","labels":[],"detail_key":"p52","name":"MyReal.k_sub","module":"Numbers.reals"},{"id":"n42535","layer":"formal","project":"p52","title":"MyReal.k_zero","kind":"theorem","summary":"Eq (MyReal.k 0) 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MyReal) ε : MyReal, LT.lt 0 ε → Exists fun r => LT.lt (abs (HSub.hSub x (MyReal.k r))) ε","labels":[],"detail_key":"p52","name":"MyReal.myRat_dense_rat","module":"Numbers.reals"},{"id":"n42548","layer":"formal","project":"p52","title":"MyReal.myRat_dense_rat'","kind":"theorem","summary":"∀ (x : MyReal) ε : MyRat, LT.lt 0 ε → Exists fun r => LE.le (abs (HSub.hSub x (MyReal.k r))) (M…","labels":[],"detail_key":"p52","name":"MyReal.myRat_dense_rat'","module":"Numbers.reals"},{"id":"n42549","layer":"formal","project":"p52","title":"MyReal.neg","kind":"def","summary":"MyReal → MyReal","labels":[],"detail_key":"p52","name":"MyReal.neg","module":"Numbers.reals"},{"id":"n42550","layer":"formal","project":"p52","title":"MyReal.one","kind":"def","summary":"One MyReal","labels":[],"detail_key":"p52","name":"MyReal.one","module":"Numbers.reals"},{"id":"n42551","layer":"formal","project":"p52","title":"MyReal.tendsTo_myRat","kind":"theorem","summary":"∀ (x : MyPrereal), MyReal.TendsTo (fun n => 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binary operation \\mathsfop satisfying \\mathsfop\\,(\\mathsfop\\…","labels":["def:xor-bij"],"detail_key":"p53"},{"id":"n42586","layer":"informal","project":"p53","title":"Sampling through a bijection is indistinguishable","kind":"lemma","summary":"[Sampling through a bijection is indistinguishable] For any bijection f : \\alpha \\simeq \\alpha…","labels":["lem:rhoare-sample-bij"],"detail_key":"p53"},{"id":"n42587","layer":"informal","project":"p53","title":"Set probability and conditioning","kind":"definition","summary":"[Set probability and conditioning] For a probability mass function p, the probability of a set…","labels":["def:pmf-condprob"],"detail_key":"p53"},{"id":"n42588","layer":"informal","project":"p53","title":"Bayes' theorem","kind":"theorem","summary":"[Bayes' theorem] For any events A, B under a mass function p, Pr_p[A \\mid B]\\cdotPr_p[B] = Pr_p…","labels":["thm:bayes"],"detail_key":"p53"},{"id":"n42589","layer":"informal","project":"p53","title":"Bayesian inversion 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of…","labels":["def:advantage"],"detail_key":"p53"},{"id":"n42593","layer":"informal","project":"p53","title":"Advantage against an adversary","kind":"definition","summary":"[Advantage against an adversary] Given games G_0, G_1 : \\mathsfSPComp\\,\\alpha and an adversary…","labels":["def:advantageA"],"detail_key":"p53"},{"id":"n42594","layer":"informal","project":"p53","title":"Triangle inequality for advantage","kind":"theorem","summary":"[Triangle inequality for advantage] For all games G_0, G_1, G_2 and every adversary A, \\[ \\math…","labels":["thm:advantageA-triangle"],"detail_key":"p53"},{"id":"n42595","layer":"informal","project":"p53","title":"Factoring a pure prefix","kind":"theorem","summary":"[Factoring a pure prefix] Let \\mathitpfx be a heap-independent (pure) computation. If for every…","labels":["thm:advantageA-isPure-bind"],"detail_key":"p53"},{"id":"n42596","layer":"informal","project":"p53","title":"Factoring a uniform-sampling prefix","kind":"theorem","summary":"[Factoring a uniform-sampling prefix] For a finite nonempty type \\alpha, if \\mathsfAdv^A(f\\,x,…","labels":["thm:advantageA-sample-bind"],"detail_key":"p53"},{"id":"n42597","layer":"informal","project":"p53","title":"Unsigned difference","kind":"definition","summary":"[Unsigned difference] The unsigned difference in R_\\ge 0^\\infty is \\mathsfabsDiff(a, b) = (a -…","labels":["def:absdiff"],"detail_key":"p53"},{"id":"n42598","layer":"informal","project":"p53","title":"Triangle inequality for the unsigned difference","kind":"lemma","summary":"[Triangle inequality for the unsigned difference] \\mathsfabsDiff(a, c) \\le \\mathsfabsDiff(a, b)…","labels":["lem:absdiff-triangle"],"detail_key":"p53"},{"id":"n42599","layer":"informal","project":"p53","title":"Statistical distance","kind":"definition","summary":"[Statistical distance] The \\emphstatistical distance between Kleisli morphisms f, g : \\alpha \\t…","labels":["def:sdist"],"detail_key":"p53"},{"id":"n42600","layer":"informal","project":"p53","title":"Symmetry of statistical distance","kind":"theorem","summary":"[Symmetry of statistical distance] \\mathsfsdist(f, g) = \\mathsfsdist(g, f).","labels":["thm:sdist-sym"],"detail_key":"p53"},{"id":"n42601","layer":"informal","project":"p53","title":"Kleisli morphisms as a pseudometric space","kind":"definition","summary":"[Kleisli morphisms as a pseudometric space] Kleisli morphisms \\alpha \\to \\mathsfSPComp\\,\\beta,…","labels":["def:klmorph"],"detail_key":"p53"},{"id":"n42602","layer":"informal","project":"p53","title":"Right post-processing lemma","kind":"theorem","summary":"[Right post-processing lemma] Post-composition with a shared continuation g does not increase d…","labels":["thm:sdist-comp-right"],"detail_key":"p53"},{"id":"n42603","layer":"informal","project":"p53","title":"Left post-processing lemma","kind":"theorem","summary":"[Left post-processing lemma] Pre-composition with a shared prefix f does not increase distance:…","labels":["thm:sdist-comp-left"],"detail_key":"p53"},{"id":"n42604","layer":"informal","project":"p53","title":"Composition of \\varepsilon-close morphisms","kind":"theorem","summary":"[Composition of \\varepsilon-close morphisms] If \\mathsfsdist(f_1, f_2) \\le \\varepsilon_1 and \\m…","labels":["thm:sdist-comp-add"],"detail_key":"p53"},{"id":"n42605","layer":"informal","project":"p53","title":"Advantage bounds statistical distance for pure games","kind":"theorem","summary":"[Advantage bounds statistical distance for pure games] If f and g are pointwise heap-independen…","labels":["thm:sdist-isPure-le"],"detail_key":"p53"},{"id":"n42606","layer":"informal","project":"p53","title":"Statistical distance from per-input advantage","kind":"theorem","summary":"[Statistical distance from per-input advantage] For pointwise-pure f, g: if \\mathsfAdv^D(f\\,a,…","labels":["thm:sdist-of-isPure-advantageA"],"detail_key":"p53"},{"id":"n42607","layer":"informal","project":"p53","title":"Statistical distance for oracle games","kind":"theorem","summary":"[Statistical distance for oracle games] Suppose G_real\\,A = A\\,o_real and G_ideal\\,A = A\\,o_ide…","labels":["thm:sdist-oracleGame-le"],"detail_key":"p53"},{"id":"n42608","layer":"informal","project":"p53","title":"Dependent hybrid bound","kind":"theorem","summary":"[Dependent hybrid bound] For a sequence of games G : N \\to \\mathsfSPComp\\,\\alpha and an adversa…","labels":["thm:advantage-hybrid-dep"],"detail_key":"p53"},{"id":"n42609","layer":"informal","project":"p53","title":"Uniform hybrid bound","kind":"theorem","summary":"[Uniform hybrid bound] If every step satisfies \\mathsfAdv^A(G_i, G_i+1) \\le \\varepsilon for i <…","labels":["thm:advantage-hybrid-uniform"],"detail_key":"p53"},{"id":"n42610","layer":"informal","project":"p53","title":"Hybrid with perfect steps","kind":"theorem","summary":"[Hybrid with perfect steps] If every adjacent pair G_i, G_i+1 is perfectly indistinguishable (r…","labels":["thm:advantage-hybrid-zero"],"detail_key":"p53"},{"id":"n42611","layer":"informal","project":"p53","title":"Factored hybrid bound","kind":"theorem","summary":"[Factored hybrid bound] When real and ideal games share a common post-processing f, and each ba…","labels":["thm:advantage-hybrid-factored"],"detail_key":"p53"},{"id":"n42612","layer":"informal","project":"p53","title":"Symmetric encryption scheme","kind":"definition","summary":"[Symmetric encryption scheme] An \\emphencryption scheme bundles key, plaintext, and ciphertext…","labels":["def:encscheme"],"detail_key":"p53"},{"id":"n42613","layer":"informal","project":"p53","title":"Correctness","kind":"definition","summary":"[Correctness] A scheme is \\emphcorrect if decryption recovers the message on every reachable pa…","labels":["def:enc-correct"],"detail_key":"p53"},{"id":"n42614","layer":"informal","project":"p53","title":"IND-CPA game","kind":"definition","summary":"[IND-CPA game] The IND-CPA game generates a fresh key and encrypts one of two adversary-chosen…","labels":["def:indcpa-game"],"detail_key":"p53"},{"id":"n42615","layer":"informal","project":"p53","title":"IND-CPA advantage","kind":"definition","summary":"[IND-CPA advantage] The IND-CPA advantage of A is the adversarial advantage between the two cha…","labels":["def:indcpa-adv"],"detail_key":"p53"},{"id":"n42616","layer":"informal","project":"p53","title":"Pseudorandom function family","kind":"definition","summary":"[Pseudorandom function family] A \\emphPRF family bundles finite nonempty key and output types,…","labels":["def:prfscheme"],"detail_key":"p53"},{"id":"n42617","layer":"informal","project":"p53","title":"PRF advantage","kind":"definition","summary":"[PRF advantage] The PRF advantage compares the real game (evaluate \\mathsfeval under a uniforml…","labels":["def:prf-adv"],"detail_key":"p53"},{"id":"n42618","layer":"informal","project":"p53","title":"Message authentication code","kind":"definition","summary":"[Message authentication code] A \\emphMAC scheme bundles finite nonempty key and tag types (tags…","labels":["def:macscheme"],"detail_key":"p53"},{"id":"n42619","layer":"informal","project":"p53","title":"EUF-CMA advantage","kind":"definition","summary":"[EUF-CMA advantage] In the single-query EUF-CMA game, the adversary sees a tag on a message m a…","labels":["def:eufcma-adv"],"detail_key":"p53"},{"id":"n42620","layer":"informal","project":"p53","title":"Monotonicity of lifting","kind":"lemma","summary":"[Monotonicity of lifting] If R \\subseteq S pointwise (i.e.\\ R\\,a\\,b \\to S\\,a\\,b for all a,b) an…","labels":["lem:liftr-mono"],"detail_key":"p53"},{"id":"n42621","layer":"informal","project":"p53","title":"Symmetry of lifting","kind":"lemma","summary":"[Symmetry of lifting] If d_1 \\mathrel\\simR d_2, then d_2 \\mathrel\\simR^-1 d_1, where R^-1\\,b\\,a…","labels":["lem:liftr-symm"],"detail_key":"p53"},{"id":"n42622","layer":"informal","project":"p53","title":"Equality coupling forces equality","kind":"lemma","summary":"[Equality coupling forces equality] If d_1 \\mathrel\\sim(=) d_2, then d_1 = d_2. An equality cou…","labels":["lem:liftr-eq-implies-eq"],"detail_key":"p53"},{"id":"n42623","layer":"informal","project":"p53","title":"Relational precondition","kind":"definition","summary":"[Relational precondition] A \\emphrelational precondition RPre is a predicate on pairs of heaps,…","labels":["def:rpre"],"detail_key":"p53"},{"id":"n42624","layer":"informal","project":"p53","title":"Relational postcondition","kind":"definition","summary":"[Relational postcondition] A \\emphrelational postcondition RPost\\,\\alpha\\,\\beta is a predicate…","labels":["def:rpost"],"detail_key":"p53"},{"id":"n42625","layer":"informal","project":"p53","title":"Equality precondition","kind":"definition","summary":"[Equality precondition] The equality precondition eqPre holds of (h_1,h_2) exactly when h_1 = h…","labels":["def:eqpre"],"detail_key":"p53"},{"id":"n42626","layer":"informal","project":"p53","title":"Equality postcondition","kind":"definition","summary":"[Equality postcondition] The equality postcondition eqPost relates (a_1,h_1) and (a_2,h_2) when…","labels":["def:eqpost"],"detail_key":"p53"},{"id":"n42627","layer":"informal","project":"p53","title":"The pRHL judgment","kind":"definition","summary":"[The pRHL judgment] For programs c_1 : SPComp\\,\\alpha, c_2 : SPComp\\,\\beta, a precondition \\Phi…","labels":["def:rhoare"],"detail_key":"p53"},{"id":"n42628","layer":"informal","project":"p53","title":"Consequence","kind":"theorem","summary":"[Consequence] If \\Phi' \\Rightarrow \\Phi (pointwise), \\Psi \\Rightarrow \\Psi' (pointwise), and \\\\…","labels":["thm:rhoare-conseq"],"detail_key":"p53"},{"id":"n42629","layer":"informal","project":"p53","title":"Symmetry","kind":"theorem","summary":"[Symmetry] The judgment is symmetric: from \\\\,\\\\;( \\sim \\lambda \\;\\h\\_1\\,h_2.\\ \\Phi\\,h_2\\,h_1)\\…","labels":["thm:rhoare-symm"],"detail_key":"p53"},{"id":"n42630","layer":"informal","project":"p53","title":"Sequential composition / bind","kind":"theorem","summary":"[Sequential composition / bind] Suppose \\\\,\\\\;\\Phi \\sim \\,\\;\\c\\_1\\,c_2\\,\\Psi and, for all a,b,…","labels":["thm:rhoare-bind"],"detail_key":"p53"},{"id":"n42631","layer":"informal","project":"p53","title":"Reflexivity","kind":"theorem","summary":"[Reflexivity] Every program is related to itself under equality: for all c, \\\\,\\\\;eqPre\\sim \\,\\…","labels":["thm:rhoare-refl"],"detail_key":"p53"},{"id":"n42632","layer":"informal","project":"p53","title":"Synchronized sampling, diagonal coupling","kind":"theorem","summary":"[Synchronized sampling, diagonal coupling] Sampling the same finite nonempty type on both sides…","labels":["thm:rhoare-sample-same"],"detail_key":"p53"},{"id":"n42633","layer":"informal","project":"p53","title":"Bijection sampling","kind":"theorem","summary":"[Bijection sampling] Sampling along a bijection f : \\alpha \\simeq \\beta relates the two samples…","labels":["thm:rhoare-sample-bij"],"detail_key":"p53"},{"id":"n42634","layer":"informal","project":"p53","title":"Frame","kind":"theorem","summary":"[Frame] If \\\\,\\\\;\\Phi \\sim \\,\\;\\c\\_1\\,c_2\\,\\Theta and the frame \\Psi follows from the postcondi…","labels":["thm:r-frame"],"detail_key":"p53"},{"id":"n42635","layer":"informal","project":"p53","title":"Assertion locality","kind":"definition","summary":"[Assertion locality] DependsOn\\,P\\,L holds when the assertion P only inspects locations in the…","labels":["def:depends-on"],"detail_key":"p53"},{"id":"n42636","layer":"informal","project":"p53","title":"Computation footprint","kind":"definition","summary":"[Computation footprint] PreservesOutside\\,c\\,L holds when c never modifies locations outside L:…","labels":["def:preserves-outside"],"detail_key":"p53"},{"id":"n42637","layer":"informal","project":"p53","title":"Location-based frame rule","kind":"theorem","summary":"[Location-based frame rule] Suppose both programs preserve everything outside L_c, the frame as…","labels":["thm:r-frame-local"],"detail_key":"p53"},{"id":"n42638","layer":"informal","project":"p53","title":"Reorder a pure prefix on the left","kind":"theorem","summary":"[Reorder a pure prefix on the left] If c is heap-independent (\\mathsfIsPure\\,c), the pure opera…","labels":["thm:rhoare-reorder-pure-l"],"detail_key":"p53"},{"id":"n42639","layer":"informal","project":"p53","title":"Swap two samples in opposite order","kind":"theorem","summary":"[Swap two samples in opposite order] When the left side samples \\alpha then \\beta and the right…","labels":["thm:rhoare-sample-comm"],"detail_key":"p53"},{"id":"n42640","layer":"informal","project":"p53","title":"Zero advantage from pRHL equality","kind":"theorem","summary":"[Zero advantage from pRHL equality] If \\\\,\\\\;eqPre\\sim \\,\\;\\G\\_0\\,G_1\\,eqPost, then no adversar…","labels":["thm:advantage-zero-of-rhoare"],"detail_key":"p53"},{"id":"n42641","layer":"informal","project":"p53","title":"Advantage factorization","kind":"theorem","summary":"[Advantage factorization] Post-composing both games with the same f moves f into the distinguis…","labels":["thm:advantage-factorization"],"detail_key":"p53"},{"id":"n42642","layer":"informal","project":"p53","title":"Factorization with a commuted pure prefix","kind":"theorem","summary":"[Factorization with a commuted pure prefix] When both games share a pure prefix \\mathitpfx, it…","labels":["thm:advantage-factorization-comm"],"detail_key":"p53"},{"id":"n42643","layer":"informal","project":"p53","title":"Symmetry of advantage","kind":"theorem","summary":"[Symmetry of advantage] The distinguishing advantage is symmetric in its two games: Adv\\,G_1\\,G…","labels":["thm:advantage-symm"],"detail_key":"p53"},{"id":"n42644","layer":"informal","project":"p53","title":"Operation signature","kind":"definition","summary":"[Operation signature] An \\emphoperation signature \\mathsfOpSig is a pair of types (\\mathsfsrc,…","labels":["def:opsig"],"detail_key":"p53"},{"id":"n42645","layer":"informal","project":"p53","title":"Interface","kind":"definition","summary":"[Interface] An \\emphinterface is a finite set of entries, each pairing an operation identifier…","labels":["def:interface"],"detail_key":"p53"},{"id":"n42646","layer":"informal","project":"p53","title":"Typed function","kind":"definition","summary":"[Typed function] A \\emphtyped function bundles an input type \\mathsfsrc, an output type \\mathsf…","labels":["def:typedfun"],"detail_key":"p53"},{"id":"n42647","layer":"informal","project":"p53","title":"Raw package","kind":"definition","summary":"[Raw package] A \\emphraw package is a finite set of package entries, each pairing an operation…","labels":["def:rawpackage"],"detail_key":"p53"},{"id":"n42648","layer":"informal","project":"p53","title":"Parallel composition of raw packages","kind":"definition","summary":"[Parallel composition of raw packages] For identifier-disjoint packages P_1, P_2, the parallel…","labels":["def:rawpackage-par"],"detail_key":"p53"},{"id":"n42649","layer":"informal","project":"p53","title":"Resolving an operation","kind":"definition","summary":"[Resolving an operation] \\mathsfresolve\\,P\\,id\\,x looks up the implementation of operation id,…","labels":["def:rawpackage-resolve"],"detail_key":"p53"},{"id":"n42650","layer":"informal","project":"p53","title":"Linking of raw packages is a stub","kind":"definition","summary":"[Linking of raw packages is a stub] \\mathsflink\\,P_1\\,P_2 = P_1 is a \\emphshallow-layer no-op s…","labels":["def:rawpackage-link"],"detail_key":"p53"},{"id":"n42651","layer":"informal","project":"p53","title":"Kleisli category of \\mathsfSPComp","kind":"definition","summary":"[Kleisli category of \\mathsfSPComp] \\mathsfKlSPComp is the type of objects of the Kleisli categ…","labels":["def:klspcomp"],"detail_key":"p53"},{"id":"n42652","layer":"informal","project":"p53","title":"Composition is bind","kind":"lemma","summary":"[Composition is bind] For f : \\alpha \\to \\beta and g : \\beta \\to \\gamma in \\mathsfKlSPComp, (f…","labels":["lem:klspcomp-comp"],"detail_key":"p53"},{"id":"n42653","layer":"informal","project":"p53","title":"Cocartesian monoidal category","kind":"definition","summary":"[Cocartesian monoidal category] A monoidal category is \\emphcocartesian when its unit is initia…","labels":["def:cocartesian-class"],"detail_key":"p53"},{"id":"n42654","layer":"informal","project":"p53","title":"Copairing in \\mathsfKlSPComp","kind":"definition","summary":"[Copairing in \\mathsfKlSPComp] Given f : \\alpha \\to \\gamma and g : \\beta \\to \\gamma in \\mathsfK…","labels":["def:klspcomp-codesc"],"detail_key":"p53"},{"id":"n42655","layer":"informal","project":"p53","title":"\\mathsfKlSPComp is cocartesian","kind":"proposition","summary":"[\\mathsfKlSPComp is cocartesian] The cofan BinaryCofan(\\mathsfklInl, \\mathsfklInr) is a colimit…","labels":["prop:klspcomp-cocartesian"],"detail_key":"p53"},{"id":"n42656","layer":"informal","project":"p53","title":"Affine monoidal category","kind":"definition","summary":"[Affine monoidal category] An \\emphaffine monoidal category equips every object with a natural…","labels":["def:affine-class"],"detail_key":"p53"},{"id":"n42657","layer":"informal","project":"p53","title":"Discard in \\mathsfKlSPComp","kind":"definition","summary":"[Discard in \\mathsfKlSPComp] The discard morphism \\mathsfklDel\\,\\alpha : \\alpha \\to Empty sends…","labels":["def:klspcomp-del"],"detail_key":"p53"},{"id":"n42658","layer":"informal","project":"p53","title":"Category of families","kind":"definition","summary":"[Category of families] For a category C, an object of \\mathsfFamObj\\,C is a pair (\\iota, \\maths…","labels":["def:famobj"],"detail_key":"p53"},{"id":"n42659","layer":"informal","project":"p53","title":"Package interface","kind":"definition","summary":"[Package interface] A \\emphpackage interface I is an index type \\iota together with domain and…","labels":["def:pkginterface"],"detail_key":"p53"},{"id":"n42660","layer":"informal","project":"p53","title":"Package implementation","kind":"definition","summary":"[Package implementation] A \\emphpackage implementation of interface I is a morphism \\mathsfdomF…","labels":["def:pkgimpl"],"detail_key":"p53"},{"id":"n42661","layer":"informal","project":"p53","title":"Kleisli linking of package implementations","kind":"definition","summary":"[Kleisli linking of package implementations] Linking \\mathsflink\\,p_1\\,p_2 = p_1 \\mathbin;\\!;p_…","labels":["def:pkgimpl-link"],"detail_key":"p53"},{"id":"n42662","layer":"informal","project":"p53","title":"Parallel composition of package implementations","kind":"definition","summary":"[Parallel composition of package implementations] \\mathsfpar\\,p_1\\,p_2 = p_1 \\otimes p_2 is the…","labels":["def:pkgimpl-par"],"detail_key":"p53"},{"id":"n42663","layer":"informal","project":"p53","title":"Bridge from deep interfaces to package interfaces","kind":"definition","summary":"[Bridge from deep interfaces to package interfaces] A list-based \\mathsfDeepInterface becomes a…","labels":["def:deepinterface-topkg"],"detail_key":"p53"},{"id":"n42664","layer":"informal","project":"p53","title":"Raw code: a free monad","kind":"definition","summary":"[Raw code: a free monad] \\mathsfRawCode\\,\\alpha is the free monad of computation syntax trees,…","labels":["def:rawcode"],"detail_key":"p53"},{"id":"n42665","layer":"informal","project":"p53","title":"Deep interface","kind":"definition","summary":"[Deep interface] A \\emphdeep interface is a list of operation entries (id, dom, codom) of match…","labels":["def:deepinterface"],"detail_key":"p53"},{"id":"n42666","layer":"informal","project":"p53","title":"Valid code bundle","kind":"definition","summary":"[Valid code bundle] Validity of code is the inductive predicate \\mathsfValidCode\\,L that every…","labels":["def:validcodebundle"],"detail_key":"p53"},{"id":"n42667","layer":"informal","project":"p53","title":"Deep package","kind":"definition","summary":"[Deep package] A \\emphdeep package bundles a finite location set \\mathsflocs, an import interfa…","labels":["def:deeppackage"],"detail_key":"p53"},{"id":"n42668","layer":"informal","project":"p53","title":"Separation","kind":"definition","summary":"[Separation] Two deep packages are \\emphseparated, \\mathsfsep\\,p_1\\,p_2, when their location se…","labels":["def:deeppackage-sep"],"detail_key":"p53"},{"id":"n42669","layer":"informal","project":"p53","title":"Oracle substitution","kind":"definition","summary":"[Oracle substitution] \\mathsfsubstOracle\\,c\\,\\mathsfenv recursively replaces every \\mathsforacl…","labels":["def:substoracle"],"detail_key":"p53"},{"id":"n42670","layer":"informal","project":"p53","title":"Substitution composes","kind":"lemma","summary":"[Substitution composes] (\\mathsfsubstOracle\\,c\\,\\mathsfenv_1)\\,\\mathsfsubstOracle\\,\\mathsfenv_2…","labels":["lem:substoracle-comp"],"detail_key":"p53"},{"id":"n42671","layer":"informal","project":"p53","title":"Substitution is a no-op on oracle-free code","kind":"lemma","summary":"[Substitution is a no-op on oracle-free code] If c contains no \\mathsforacleCall node (predicat…","labels":["lem:substoracle-self"],"detail_key":"p53"},{"id":"n42672","layer":"informal","project":"p53","title":"Deep linking","kind":"definition","summary":"[Deep linking] \\mathsflink\\,p_1\\,p_2 connects p_2's exports to p_1's imports. It uses the union…","labels":["def:deeppackage-link"],"detail_key":"p53"},{"id":"n42673","layer":"informal","project":"p53","title":"Deep parallel composition","kind":"definition","summary":"[Deep parallel composition] For separated p_1, p_2, \\mathsfpar\\,p_1\\,p_2 takes the union of loc…","labels":["def:deeppackage-par"],"detail_key":"p53"},{"id":"n42674","layer":"informal","project":"p53","title":"Identity package","kind":"definition","summary":"[Identity package] \\mathsfid\\,I is the stateless passthrough for interface I: it exports and im…","labels":["def:deeppackage-id"],"detail_key":"p53"},{"id":"n42675","layer":"informal","project":"p53","title":"Evaluation to \\mathsfSPComp","kind":"definition","summary":"[Evaluation to \\mathsfSPComp] \\mathsfRawCode.eval interprets a syntax tree as an \\mathsfSPComp…","labels":["def:eval"],"detail_key":"p53"},{"id":"n42676","layer":"informal","project":"p53","title":"Substitution commutes with evaluation","kind":"lemma","summary":"[Substitution commutes with evaluation] Evaluating \\mathsfsubstOracle\\,c\\,\\mathsfenv equals eva…","labels":["lem:eval-substoracle"],"detail_key":"p53"},{"id":"n42677","layer":"informal","project":"p53","title":"Running a deep package","kind":"definition","summary":"[Running a deep package] \\mathsfrunPkg\\,p evaluates the conventional main export (0, Unit, Bool…","labels":["def:runpkg"],"detail_key":"p53"},{"id":"n42678","layer":"informal","project":"p53","title":"Deep nominal advantage","kind":"definition","summary":"[Deep nominal advantage] For nominal packages G, G', A, \\[ \\mathsfDeepNomAdvantage\\,G\\,G'\\,A =…","labels":["def:deepnomadv"],"detail_key":"p53"},{"id":"n42679","layer":"informal","project":"p53","title":"Correctness of linking under evaluation","kind":"theorem","summary":"[Correctness of linking under evaluation] \\mathsfrunPkg(\\mathsflink\\,p_1\\,p_2) equals evaluatin…","labels":["thm:runpkg-link"],"detail_key":"p53"},{"id":"n42680","layer":"informal","project":"p53","title":"Associativity of linking under evaluation","kind":"theorem","summary":"[Associativity of linking under evaluation] \\mathsfrunPkg(\\mathsflink(\\mathsflink\\,p_1\\,p_2)\\,p…","labels":["thm:runpkg-link-assoc"],"detail_key":"p53"},{"id":"n42681","layer":"informal","project":"p53","title":"Interchange law","kind":"theorem","summary":"[Interchange law] Under an interface-matching condition on the oracle environments, \\[ \\mathsfr…","labels":["thm:runpkg-interchange"],"detail_key":"p53"},{"id":"n42682","layer":"informal","project":"p53","title":"Reduction lemma for advantage","kind":"theorem","summary":"[Reduction lemma for advantage] For compatible registries, \\[ \\mathsfDeepNomAdvantage\\,G_0\\,G_1…","labels":["thm:deepnomadv-link"],"detail_key":"p53"},{"id":"n42683","layer":"informal","project":"p53","title":"Game-based security for deep packages","kind":"definition","summary":"[Game-based security for deep packages] \\mathsfNomPkgSecure\\,G\\,G'\\,\\varepsilon holds when for…","labels":["def:nompkgsecure"],"detail_key":"p53"},{"id":"n42684","layer":"informal","project":"p53","title":"Forkable adversary","kind":"definition","summary":"[Forkable adversary] A \\emphforkable adversary for the single-query random-oracle model is a tw…","labels":["def:forkable-adversary"],"detail_key":"p53"},{"id":"n42685","layer":"informal","project":"p53","title":"Acceptance probability","kind":"definition","summary":"[Acceptance probability] The \\emphacceptance probability of A on input x under predicate \\maths…","labels":["def:fork-accept-prob"],"detail_key":"p53"},{"id":"n42686","layer":"informal","project":"p53","title":"Forking algorithm","kind":"definition","summary":"[Forking algorithm] The \\emphforking algorithm runs A once to obtain a commitment c, then sampl…","labels":["def:fork-alg"],"detail_key":"p53"},{"id":"n42687","layer":"informal","project":"p53","title":"Fork success probability","kind":"definition","summary":"[Fork success probability] The \\emphfork success probability is the probability that both repla…","labels":["def:fork-succ-prob"],"detail_key":"p53"},{"id":"n42688","layer":"informal","project":"p53","title":"AM--GM for extended non-negative reals","kind":"lemma","summary":"[AM--GM for extended non-negative reals] For all a, b \\in \\overlineR_\\geq 0, \\[ 2ab \\;\\leq\\; a^…","labels":["lem:ennreal-amgm"],"detail_key":"p53"},{"id":"n42689","layer":"informal","project":"p53","title":"Jensen's inequality under sub-PMF weights","kind":"lemma","summary":"[Jensen's inequality under sub-PMF weights] Let w : \\iota \\to \\overlineR_\\geq 0 with \\sum_i w_i…","labels":["lem:jensen-sub-pmf"],"detail_key":"p53"},{"id":"n42690","layer":"informal","project":"p53","title":"Conditional acceptance probability","kind":"definition","summary":"[Conditional acceptance probability] Fixing a commitment c, the \\emphconditional acceptance pro…","labels":["def:cond-acc-prob"],"detail_key":"p53"},{"id":"n42691","layer":"informal","project":"p53","title":"Conditional fork probability","kind":"definition","summary":"[Conditional fork probability] Fixing a commitment c, the \\emphconditional fork probability frk…","labels":["def:cond-fork-prob"],"detail_key":"p53"},{"id":"n42692","layer":"informal","project":"p53","title":"Exact conditional fork count","kind":"lemma","summary":"[Exact conditional fork count] For each fixed commitment c, exact counting over R \\times R give…","labels":["lem:cond-fork-prob-eq"],"detail_key":"p53"},{"id":"n42693","layer":"informal","project":"p53","title":"Decomposition over commitments","kind":"lemma","summary":"[Decomposition over commitments] Both acc(A,x) and frk(A,x) decompose as weighted sums over the…","labels":["lem:fork-decompose"],"detail_key":"p53"},{"id":"n42694","layer":"informal","project":"p53","title":"Forking lemma, q = 1","kind":"theorem","summary":"[Forking lemma, q = 1] For every forkable adversary A, input x, and acceptance predicate, \\[ fr…","labels":["thm:fork-success-ge"],"detail_key":"p53"},{"id":"n42695","layer":"informal","project":"p53","title":"q-query forkable adversary","kind":"definition","summary":"[q-query forkable adversary] A \\emphq-query forkable adversary has a probabilistic commit phase…","labels":["def:qforkable-adversary"],"detail_key":"p53"},{"id":"n42696","layer":"informal","project":"p53","title":"q-query acceptance probability","kind":"definition","summary":"[q-query acceptance probability] acc(A,x) for a q-query adversary is the probability that, over…","labels":["def:qaccept-prob"],"detail_key":"p53"},{"id":"n42697","layer":"informal","project":"p53","title":"Guess reduction to a single query","kind":"definition","summary":"[Guess reduction to a single query] For a guess I : Fin\\,q, the \\emphguess reduction builds a s…","labels":["def:guess-reduction"],"detail_key":"p53"},{"id":"n42698","layer":"informal","project":"p53","title":"Acceptance at a guess","kind":"definition","summary":"[Acceptance at a guess] The predicate \\mathsfacceptAtGuess(\\mathsfaccept, I) accepts a pair (j,…","labels":["def:accept-at-guess"],"detail_key":"p53"},{"id":"n42699","layer":"informal","project":"p53","title":"Guesses partition acceptance","kind":"lemma","summary":"[Guesses partition acceptance] Summing the reduced acceptance probabilities over all guesses re…","labels":["lem:sum-accept-at-guess"],"detail_key":"p53"},{"id":"n42700","layer":"informal","project":"p53","title":"Bellare--Neven aggregation","kind":"theorem","summary":"[Bellare--Neven aggregation] Let q > 0 and a, f : Fin\\,q \\to \\overlineR_\\geq 0 with f_j \\geq a_…","labels":["thm:bellare-neven-agg"],"detail_key":"p53"},{"id":"n42701","layer":"informal","project":"p53","title":"Indexed q-fork success probability","kind":"definition","summary":"[Indexed q-fork success probability] The \\emphindexed q-fork success probability sums the singl…","labels":["def:qfork-succ-prob"],"detail_key":"p53"},{"id":"n42702","layer":"informal","project":"p53","title":"General forking lemma, Bellare--Neven 2006","kind":"theorem","summary":"[General forking lemma, Bellare--Neven 2006] For every q-query forkable adversary A with q > 0,…","labels":["thm:qfork-success-ge"],"detail_key":"p53"},{"id":"n42703","layer":"informal","project":"p53","title":"Bilinear pairing group","kind":"definition","summary":"[Bilinear pairing group] A \\emphpairing group (Type~III) bundles three cyclic groups G_1, G_2,…","labels":["def:pairing-group"],"detail_key":"p53"},{"id":"n42704","layer":"informal","project":"p53","title":"Pairing bilinearity on exponents","kind":"lemma","summary":"[Pairing bilinearity on exponents] For integer exponents a, b, e(g_1^\\,a,\\, g_2^\\,b) = e(g_1, g…","labels":["def:pairing-bilinear"],"detail_key":"p53"},{"id":"n42705","layer":"informal","project":"p53","title":"KZG structured reference string","kind":"definition","summary":"[KZG structured reference string] For a secret \\alpha \\in Z_p and degree bound t, the KZG struc…","labels":["def:kzg-srs"],"detail_key":"p53"},{"id":"n42706","layer":"informal","project":"p53","title":"Discrete logarithm advantage","kind":"definition","summary":"[Discrete logarithm advantage] In the DL game the challenger samples x \\in Z_p, hands the adver…","labels":["def:dl"],"detail_key":"p53"},{"id":"n42707","layer":"informal","project":"p53","title":"Diffie--Hellman group","kind":"definition","summary":"[Diffie--Hellman group] Abstractly, a Diffie--Hellman group is a scalar-multiplication operatio…","labels":["def:ddh-def"],"detail_key":"p53"},{"id":"n42708","layer":"informal","project":"p53","title":"Decisional Diffie--Hellman advantage","kind":"definition","summary":"[Decisional Diffie--Hellman advantage] The DDH advantage is the distinguishing distance between…","labels":["def:ddh"],"detail_key":"p53"},{"id":"n42709","layer":"informal","project":"p53","title":"Computational Diffie--Hellman advantage","kind":"definition","summary":"[Computational Diffie--Hellman advantage] In the CDH game the adversary receives (g^a, g^b) and…","labels":["def:cdh"],"detail_key":"p53"},{"id":"n42710","layer":"informal","project":"p53","title":"co-CDH advantage","kind":"definition","summary":"[co-CDH advantage] In a Type~III pairing group the co-CDH game samples a, b \\in Z_p, gives the…","labels":["def:cocdh"],"detail_key":"p53"},{"id":"n42711","layer":"informal","project":"p53","title":"Gap Diffie--Hellman advantage","kind":"definition","summary":"[Gap Diffie--Hellman advantage] GapDH strengthens CDH by granting the adversary a DDH decision…","labels":["def:gapdh"],"detail_key":"p53"},{"id":"n42712","layer":"informal","project":"p53","title":"t-Strong Diffie--Hellman advantage","kind":"definition","summary":"[t-Strong Diffie--Hellman advantage] In the t-SDH game the challenger samples \\alpha and publis…","labels":["def:tsdh"],"detail_key":"p53"},{"id":"n42713","layer":"informal","project":"p53","title":"q-SDH group","kind":"definition","summary":"[q-SDH group] An abstract cyclic group of prime order carrying scalar arithmetic (addition, mul…","labels":["def:qsdh-group"],"detail_key":"p53"},{"id":"n42714","layer":"informal","project":"p53","title":"q-Strong Diffie--Hellman advantage","kind":"definition","summary":"[q-Strong Diffie--Hellman advantage] The challenger samples a secret x and hands the adversary…","labels":["def:qsdh"],"detail_key":"p53"},{"id":"n42715","layer":"informal","project":"p53","title":"Oracle Diffie--Hellman advantage","kind":"definition","summary":"[Oracle Diffie--Hellman advantage] ODH augments a DH group with a keyed hash H whose key is the…","labels":["def:odh"],"detail_key":"p53"},{"id":"n42716","layer":"informal","project":"p53","title":"DDH and PRF imply ODH","kind":"lemma","summary":"[DDH and PRF imply ODH] If the DH secret can be replaced by a random key at cost \\varepsilon_dd…","labels":["def:odh-reduction"],"detail_key":"p53"},{"id":"n42717","layer":"informal","project":"p53","title":"Collision-resistance advantage","kind":"definition","summary":"[Collision-resistance advantage] For a hash h, the search-CR game has the adversary output a pa…","labels":["def:cr-search"],"detail_key":"p53"},{"id":"n42718","layer":"informal","project":"p53","title":"Target collision resistance advantage","kind":"definition","summary":"[Target collision resistance advantage] In the TCR (second-preimage) game the adversary receive…","labels":["def:tcr"],"detail_key":"p53"},{"id":"n42719","layer":"informal","project":"p53","title":"Preimage-resistance advantage","kind":"definition","summary":"[Preimage-resistance advantage] In the preimage game the adversary receives y = h(x) for a rand…","labels":["def:pre"],"detail_key":"p53"},{"id":"n42720","layer":"informal","project":"p53","title":"One-wayness advantage","kind":"definition","summary":"[One-wayness advantage] For a function f, the inversion game samples x, hands the adversary f(x…","labels":["def:owf"],"detail_key":"p53"},{"id":"n42721","layer":"informal","project":"p53","title":"One-way permutation","kind":"definition","summary":"[One-way permutation] A one-way permutation is a one-way function that is additionally a biject…","labels":["def:owp"],"detail_key":"p53"},{"id":"n42722","layer":"informal","project":"p53","title":"Pseudorandom-generator advantage","kind":"definition","summary":"[Pseudorandom-generator advantage] For a stretch function G, the PRG advantage is the distingui…","labels":["def:prg"],"detail_key":"p53"},{"id":"n42723","layer":"informal","project":"p53","title":"RSA inversion advantage","kind":"definition","summary":"[RSA inversion advantage] The RSA game samples a key pair and a random preimage x, sets y = rsa…","labels":["def:rsa"],"detail_key":"p53"},{"id":"n42724","layer":"informal","project":"p53","title":"Pseudorandom-permutation advantage","kind":"definition","summary":"[Pseudorandom-permutation advantage] For a block cipher E, the PRP advantage is the distinguish…","labels":["def:prp"],"detail_key":"p53"},{"id":"n42725","layer":"informal","project":"p53","title":"Strong-PRP advantage","kind":"definition","summary":"[Strong-PRP advantage] The strong-PRP game grants the adversary both the forward oracle E(k, \\c…","labels":["def:sprp"],"detail_key":"p53"},{"id":"n42726","layer":"informal","project":"p53","title":"Pseudorandom-function advantage","kind":"definition","summary":"[Pseudorandom-function advantage] For a keyed function F, the (single-query) PRF advantage is t…","labels":["def:prf"],"detail_key":"p53"},{"id":"n42727","layer":"informal","project":"p53","title":"PRP/PRF switching bound","kind":"definition","summary":"[PRP/PRF switching bound] The birthday term q(q-1)/(2|W|) bounding the statistical gap between…","labels":["def:prp-prf-switch"],"detail_key":"p53"},{"id":"n42728","layer":"informal","project":"p53","title":"MAC forgery advantage","kind":"definition","summary":"[MAC forgery advantage] In the (indistinguishability form of) MAC game a random key is sampled…","labels":["def:mac"],"detail_key":"p53"},{"id":"n42729","layer":"informal","project":"p53","title":"AEAD confidentiality advantage","kind":"definition","summary":"[AEAD confidentiality advantage] For an authenticated-encryption scheme the (IND-CPA form) AEAD…","labels":["def:aead"],"detail_key":"p53"},{"id":"n42730","layer":"informal","project":"p53","title":"AEAD IND-CCA2 advantage","kind":"definition","summary":"[AEAD IND-CCA2 advantage] The IND-CCA2 game equips the adversary with both an encryption oracle…","labels":["def:aead-cca"],"detail_key":"p53"},{"id":"n42731","layer":"informal","project":"p53","title":"Projective x-only doubling","kind":"definition","summary":"[Projective x-only doubling] For a Montgomery curve constant a_24 = (A+2)/4, the projective dou…","labels":["def:xdbl"],"detail_key":"p53"},{"id":"n42732","layer":"informal","project":"p53","title":"Projective x-only differential addition","kind":"definition","summary":"[Projective x-only differential addition] Given projective x-coordinates of P, Q and their diff…","labels":["def:xdadd"],"detail_key":"p53"},{"id":"n42733","layer":"informal","project":"p53","title":"Symmetry of \\mathsfxdadd","kind":"lemma","summary":"[Symmetry of \\mathsfxdadd] \\mathsfxdadd\\,PZ\\,QZ\\,DZ = \\mathsfxdadd\\,QZ\\,PZ\\,DZ: the difference…","labels":["lem:xdadd-symm"],"detail_key":"p53"},{"id":"n42734","layer":"informal","project":"p53","title":"Montgomery curve parameters","kind":"definition","summary":"[Montgomery curve parameters] A parameter record over a commutative ring F carrying the curve c…","labels":["def:monty-params"],"detail_key":"p53"},{"id":"n42735","layer":"informal","project":"p53","title":"Curve25519 parameters","kind":"definition","summary":"[Curve25519 parameters] The Montgomery parameters of Curve25519: A = 486662 and a_24 = 121666 =…","labels":["def:curve25519-params"],"detail_key":"p53"},{"id":"n42736","layer":"informal","project":"p53","title":"Projective equivalence","kind":"definition","summary":"[Projective equivalence] Two pairs are x-equivalent, written (X_1,Z_1) \\sim (X_2,Z_2), iff X_1…","labels":["def:proj-equiv"],"detail_key":"p53"},{"id":"n42737","layer":"informal","project":"p53","title":"Restricted transitivity of \\sim","kind":"lemma","summary":"[Restricted transitivity of \\sim] Over a field, if p \\sim q, q \\sim r, and q is nondegenerate (…","labels":["lem:proj-equiv-trans"],"detail_key":"p53"},{"id":"n42738","layer":"informal","project":"p53","title":"\\mathsfxdbl respects \\sim","kind":"theorem","summary":"[\\mathsfxdbl respects \\sim] If p \\sim q then \\mathsfxdbl\\,a_24\\,p \\sim \\mathsfxdbl\\,a_24\\,q. Bo…","labels":["thm:xdbl-projequiv"],"detail_key":"p53"},{"id":"n42739","layer":"informal","project":"p53","title":"\\mathsfxdadd respects \\sim","kind":"theorem","summary":"[\\mathsfxdadd respects \\sim] If p \\sim p', q \\sim q', and d \\sim d' then \\mathsfxdadd\\,p\\,q\\,d…","labels":["thm:xdadd-projequiv"],"detail_key":"p53"},{"id":"n42740","layer":"informal","project":"p53","title":"\\mathsfMontyCurveGroup","kind":"definition","summary":"[\\mathsfMontyCurveGroup] A hypothesis class packaging the Costello--Smith~\\S4 correspondence: a…","labels":["def:monty-curve-group"],"detail_key":"p53"},{"id":"n42741","layer":"informal","project":"p53","title":"Bits to natural","kind":"definition","summary":"[Bits to natural] The natural number read from a bit list, MSB first: \\mathsfbitsToNat(b::bs) =…","labels":["def:bits-to-nat"],"detail_key":"p53"},{"id":"n42742","layer":"informal","project":"p53","title":"Ladder step","kind":"definition","summary":"[Ladder step] One double-and-add step on a state (R_0, R_1) of an abelian group, branching on b…","labels":["def:ladder-step"],"detail_key":"p53"},{"id":"n42743","layer":"informal","project":"p53","title":"Ladder fold","kind":"definition","summary":"[Ladder fold] Iterating \\mathsfladderStep over a bit list (head processed first, MSB first) fro…","labels":["def:ladder-fold"],"detail_key":"p53"},{"id":"n42744","layer":"informal","project":"p53","title":"Ladder invariant","kind":"definition","summary":"[Ladder invariant] The predicate \\mathsfLadderInv\\,P\\,a\\,(R_0,R_1) asserting R_0 = a \\cdot P an…","labels":["def:ladder-inv"],"detail_key":"p53"},{"id":"n42745","layer":"informal","project":"p53","title":"Montgomery ladder correctness (algebraic)","kind":"theorem","summary":"[Montgomery ladder correctness (algebraic)] For any abelian group G, any P : G and any bit list…","labels":["thm:ladder-fold-correct"],"detail_key":"p53"},{"id":"n42746","layer":"informal","project":"p53","title":"Field-level ladder step","kind":"definition","summary":"[Field-level ladder step] One step of the x-only ladder on projective state (R_0, R_1) \\in (F\\t…","labels":["def:xladder-step"],"detail_key":"p53"},{"id":"n42747","layer":"informal","project":"p53","title":"Field-level ladder fold","kind":"definition","summary":"[Field-level ladder fold] Iterating \\mathsfxladderStep over a bit list from the initial project…","labels":["def:xladder-fold"],"detail_key":"p53"},{"id":"n42748","layer":"informal","project":"p53","title":"Field-level ladder invariant","kind":"definition","summary":"[Field-level ladder invariant] The projective invariant tying the field state to the abstract g…","labels":["def:xladder-inv"],"detail_key":"p53"},{"id":"n42749","layer":"informal","project":"p53","title":"Field-level step preserves the invariant","kind":"theorem","summary":"[Field-level step preserves the invariant] Over a field, given negation-invariance of \\mathsfxP…","labels":["thm:xladder-step-preserves"],"detail_key":"p53"},{"id":"n42750","layer":"informal","project":"p53","title":"Montgomery curve as Weierstrass curve","kind":"definition","summary":"[Montgomery curve as Weierstrass curve] The equation y^2 = x^3 + A x^2 + x is literally the Wei…","labels":["def:montgomeryW"],"detail_key":"p53"},{"id":"n42751","layer":"informal","project":"p53","title":"Weierstrass equation is the Montgomery equation","kind":"lemma","summary":"[Weierstrass equation is the Montgomery equation] A point (x,y) lies on \\mathsfmontgomeryW\\,A i…","labels":["lem:montgomeryW-equation"],"detail_key":"p53"},{"id":"n42752","layer":"informal","project":"p53","title":"Projective x-coordinate map","kind":"definition","summary":"[Projective x-coordinate map] The map sending a Weierstrass affine point to its projective x-co…","labels":["def:xprojW"],"detail_key":"p53"},{"id":"n42753","layer":"informal","project":"p53","title":"\\mathsfxdbl doubling spec on Weierstrass points","kind":"theorem","summary":"[\\mathsfxdbl doubling spec on Weierstrass points] For 2 \\neq 0 and 4\\,a_24 = A + 2, and every p…","labels":["thm:xdbl-spec-w"],"detail_key":"p53"},{"id":"n42754","layer":"informal","project":"p53","title":"\\mathsfxdadd addition spec on Weierstrass points","kind":"theorem","summary":"[\\mathsfxdadd addition spec on Weierstrass points] For 2 \\neq 0 and all points P, Q on \\mathsfm…","labels":["thm:xdadd-spec-w"],"detail_key":"p53"},{"id":"n42755","layer":"informal","project":"p53","title":"Weierstrass transport into \\mathsfMontyCurveGroup","kind":"definition","summary":"[Weierstrass transport into \\mathsfMontyCurveGroup] Packaging: the Weierstrass group on (\\maths…","labels":["def:montgomeryW-mcg"],"detail_key":"p53"},{"id":"n42756","layer":"informal","project":"p53","title":"Curve25519 base-field prime","kind":"definition","summary":"[Curve25519 base-field prime] p = 2^255 - 19.","labels":["def:curve25519-prime"],"detail_key":"p53"},{"id":"n42757","layer":"informal","project":"p53","title":"Curve25519 base field","kind":"definition","summary":"[Curve25519 base field] The type alias F_p = Z/pZ for p = 2^255 - 19.","labels":["def:fp25519"],"detail_key":"p53"},{"id":"n42758","layer":"informal","project":"p53","title":"Base-field primality (AUCurves bridge, taken as an axiom)","kind":"definition","summary":"[Base-field primality (AUCurves bridge, taken as an axiom)] \\mathsfNat.Prime\\,(2^255 - 19). Pro…","labels":["def:curve25519-prime-prime"],"detail_key":"p53"},{"id":"n42759","layer":"informal","project":"p53","title":"Concrete Curve25519 point group","kind":"definition","summary":"[Concrete Curve25519 point group] The affine points of \\mathsfmontgomeryW\\,(486662) over F_p, a…","labels":["def:curve25519-point"],"detail_key":"p53"},{"id":"n42760","layer":"informal","project":"p53","title":"Curve25519 \\mathsfMontyCurveGroup instance","kind":"definition","summary":"[Curve25519 \\mathsfMontyCurveGroup instance] Specialising \\refdef:montgomeryW-mcg to A = 486662…","labels":["def:curve25519-mcg"],"detail_key":"p53"},{"id":"n42761","layer":"informal","project":"p53","title":"X25519 scalar multiplication","kind":"definition","summary":"[X25519 scalar multiplication] \\mathsfx25519\\,k\\,P = k \\cdot P in the group. For the Curve25519…","labels":["def:x25519"],"detail_key":"p53"},{"id":"n42762","layer":"informal","project":"p53","title":"Ladder computes X25519","kind":"theorem","summary":"[Ladder computes X25519] \\mathsfx25519\\,(\\mathsfbitsToNat\\,bits)\\,P = (\\mathsfladderFold\\,P\\,bi…","labels":["thm:x25519-eq-ladder"],"detail_key":"p53"},{"id":"n42763","layer":"informal","project":"p53","title":"Curve25519 subgroup order","kind":"definition","summary":"[Curve25519 subgroup order] \\ell = 2^252 + 27742317777372353535851937790883648493, the order of…","labels":["def:curve25519-suborder"],"detail_key":"p53"},{"id":"n42764","layer":"informal","project":"p53","title":"Subgroup order primality (AUCurves bridge, taken as an axiom)","kind":"definition","summary":"[Subgroup order primality (AUCurves bridge, taken as an axiom)] \\mathsfNat.Prime\\,\\ell. Bridged…","labels":["def:curve25519-suborder-prime"],"detail_key":"p53"},{"id":"n42765","layer":"informal","project":"p53","title":"Curve25519 base point (AUCurves bridge, taken as an axiom)","kind":"definition","summary":"[Curve25519 base point (AUCurves bridge, taken as an axiom)] A concrete \\mathsfCurve25519Point…","labels":["def:curve25519-basepoint"],"detail_key":"p53"},{"id":"n42766","layer":"informal","project":"p53","title":"Base-point order (AUCurves bridge, taken as an axiom)","kind":"definition","summary":"[Base-point order (AUCurves bridge, taken as an axiom)] \\mathsfaddOrderOf\\,B = \\ell: the base p…","labels":["def:curve25519-basepoint-addorder"],"detail_key":"p53"},{"id":"n42767","layer":"informal","project":"p53","title":"Ladder nondegeneracy","kind":"definition","summary":"[Ladder nondegeneracy] The predicate asserting that, at every scalar n \\le \\mathsfbitsToNat\\,bi…","labels":["def:ladder-nondeg"],"detail_key":"p53"},{"id":"n42768","layer":"informal","project":"p53","title":"Field-level ladder correctness for Curve25519","kind":"theorem","summary":"[Field-level ladder correctness for Curve25519] For a point P, a bit list, and \\mathsfLadderNon…","labels":["thm:xladder-curve25519-correct"],"detail_key":"p53"},{"id":"n42769","layer":"informal","project":"p53","title":"End-to-end X25519 correctness, conditional on nondegeneracy","kind":"theorem","summary":"[End-to-end X25519 correctness, conditional on nondegeneracy] Given \\mathsfLadderNondeg\\,P\\,bit…","labels":["thm:x25519-ladder-correct"],"detail_key":"p53"},{"id":"n42770","layer":"informal","project":"p53","title":"Base point discharges nondegeneracy","kind":"theorem","summary":"[Base point discharges nondegeneracy] For any bit list whose value is below the subgroup order…","labels":["thm:curve25519-basepoint-laddernondeg"],"detail_key":"p53"},{"id":"n42771","layer":"informal","project":"p53","title":"Unconditional X25519 ladder correctness for the base point","kind":"theorem","summary":"[Unconditional X25519 ladder correctness for the base point] Capstone. For every scalar below t…","labels":["thm:x25519-ladder-correct-basepoint"],"detail_key":"p53"},{"id":"n42772","layer":"informal","project":"p53","title":"One-time pad (over \\mathsfBool)","kind":"definition","summary":"[One-time pad (over \\mathsfBool)] The one-time pad \\mathsfBoolOTP is the encryption scheme on \\…","labels":["def:otp-scheme"],"detail_key":"p53"},{"id":"n42773","layer":"informal","project":"p53","title":"Correctness","kind":"theorem","summary":"[Correctness] Decryption inverts encryption: \\mathsfdec(k, \\mathsfenc(k, m)) = m.","labels":["thm:otp-correct"],"detail_key":"p53"},{"id":"n42774","layer":"informal","project":"p53","title":"Challenge games couple","kind":"lemma","summary":"[Challenge games couple] For all messages m_0, m_1, the two IND-CPA challenge games --- encrypt…","labels":["lem:otp-coupling"],"detail_key":"p53"},{"id":"n42775","layer":"informal","project":"p53","title":"Perfect IND-CPA security","kind":"theorem","summary":"[Perfect IND-CPA security] The one-time pad has \\emphperfect IND-CPA security: for all messages…","labels":["thm:otp-perfect"],"detail_key":"p53"},{"id":"n42776","layer":"informal","project":"p53","title":"Bijection PRF family","kind":"definition","summary":"[Bijection PRF family] A bijection PRF family has finite nonempty key, input, and output types…","labels":["def:bijprf"],"detail_key":"p53"},{"id":"n42777","layer":"informal","project":"p53","title":"Real and ideal oracles couple","kind":"lemma","summary":"[Real and ideal oracles couple] For every input x, the real oracle (a uniform key run through \\…","labels":["lem:bijprf-coupling"],"detail_key":"p53"},{"id":"n42778","layer":"informal","project":"p53","title":"Perfect PRF security","kind":"theorem","summary":"[Perfect PRF security] A bijection PRF family has perfect security: for every input x and adver…","labels":["thm:bijprf-perfect"],"detail_key":"p53"},{"id":"n42779","layer":"informal","project":"p53","title":"XOR PRF","kind":"corollary","summary":"[XOR PRF] The XOR pseudorandom function F(k, x) = k \\oplus x, instantiating the family by x \\ma…","labels":["cor:xorprf"],"detail_key":"p53"},{"id":"n42780","layer":"informal","project":"p53","title":"Cascade PRF security bound","kind":"theorem","summary":"[Cascade PRF security bound] The \\emphcomputational cascade result. For the double construction…","labels":["thm:cascade-prf"],"detail_key":"p53"},{"id":"n42781","layer":"informal","project":"p53","title":"Acceptance","kind":"definition","summary":"[Acceptance] A transcript (a, c, s) is accepting for public key \\mathitpk when g^s = a \\cdot \\m…","labels":["def:schnorr-accepts"],"detail_key":"p53"},{"id":"n42782","layer":"informal","project":"p53","title":"Completeness","kind":"theorem","summary":"[Completeness] The honest prover with witness w, commitment a = g^r, and response s = r + w c i…","labels":["thm:schnorr-complete"],"detail_key":"p53"},{"id":"n42783","layer":"informal","project":"p53","title":"Special soundness","kind":"theorem","summary":"[Special soundness] Two accepting transcripts (a, c_1, s_1) and (a, c_2, s_2) with the same com…","labels":["thm:schnorr-soundness"],"detail_key":"p53"},{"id":"n42784","layer":"informal","project":"p53","title":"Forking bound","kind":"theorem","summary":"[Forking bound] Instantiating the general forking lemma at a Schnorr forger A bounds the probab…","labels":["thm:schnorr-forking"],"detail_key":"p53"},{"id":"n42785","layer":"informal","project":"p53","title":"Discrete log from forking","kind":"theorem","summary":"[Discrete log from forking] Forking a Schnorr forger and applying special-soundness extraction…","labels":["thm:schnorr-dl"],"detail_key":"p53"},{"id":"n42786","layer":"informal","project":"p53","title":"Bijection generator","kind":"definition","summary":"[Bijection generator] A PRG scheme has a stretch map Seed \\to Output; a bijection generator is…","labels":["def:bijprg"],"detail_key":"p53"},{"id":"n42787","layer":"informal","project":"p53","title":"Real and ideal generators couple","kind":"lemma","summary":"[Real and ideal generators couple] Stretching a uniform seed through a bijection and sampling a…","labels":["lem:bijprg-coupling"],"detail_key":"p53"},{"id":"n42788","layer":"informal","project":"p53","title":"Perfect PRG security","kind":"theorem","summary":"[Perfect PRG security] A bijection generator has perfect security: for every adversary A, the P…","labels":["thm:bijprg-perfect"],"detail_key":"p53"},{"id":"n42789","layer":"informal","project":"p53","title":"Triple-from-double bound","kind":"theorem","summary":"[Triple-from-double bound] The \\emphcomputational length-tripling result. Extending a length-do…","labels":["thm:triple-prg"],"detail_key":"p53"},{"id":"n42790","layer":"informal","project":"p53","title":"Encrypt-then-MAC combinator","kind":"definition","summary":"[Encrypt-then-MAC combinator] Given an encryption scheme E and a MAC \\mathsfmac, the combinator…","labels":["def:etm"],"detail_key":"p53"},{"id":"n42791","layer":"informal","project":"p53","title":"Correctness is preserved","kind":"theorem","summary":"[Correctness is preserved] If the underlying encryption scheme is correct, so is the Encrypt-th…","labels":["thm:etm-correct"],"detail_key":"p53"},{"id":"n42792","layer":"informal","project":"p53","title":"Perfect IND-CPA of the XOR instantiation","kind":"theorem","summary":"[Perfect IND-CPA of the XOR instantiation] The one-time pad composed with the XOR MAC has perfe…","labels":["thm:etm-perfect"],"detail_key":"p53"},{"id":"n42793","layer":"informal","project":"p53","title":"Keyed hash tag","kind":"definition","summary":"[Keyed hash tag] A hash function keys a nonce to a tag; the reader accepts when the tag matches…","labels":["def:basichash"],"detail_key":"p53"},{"id":"n42794","layer":"informal","project":"p53","title":"Perfect authentication","kind":"theorem","summary":"[Perfect authentication] For any bijection-family hash, authentication is perfect: the real and…","labels":["thm:basichash-auth"],"detail_key":"p53"},{"id":"n42795","layer":"informal","project":"p53","title":"Unlinkability counterexample","kind":"theorem","summary":"[Unlinkability counterexample] The XOR hash leaks a key-independent correlation, \\,(k \\oplus n_…","labels":["thm:basichash-unlink"],"detail_key":"p53"},{"id":"n42796","layer":"informal","project":"p53","title":"Hybrid public-key scheme","kind":"definition","summary":"[Hybrid public-key scheme] From a KEM and a DEM, the hybrid scheme encapsulates a key, encrypts…","labels":["def:hybridpke"],"detail_key":"p53"},{"id":"n42797","layer":"informal","project":"p53","title":"Hybrid correctness","kind":"theorem","summary":"[Hybrid correctness] Component correctness of the KEM and DEM lifts to correctness of the hybri…","labels":["thm:hybridpke-correct"],"detail_key":"p53"},{"id":"n42798","layer":"informal","project":"p53","title":"Hybrid security bound","kind":"theorem","summary":"[Hybrid security bound] The hybrid scheme's IND-CCA advantage is bounded by the sum of the KEM,…","labels":["thm:kemdem-bound"],"detail_key":"p53"},{"id":"n42799","layer":"informal","project":"p53","title":"Perfect security of the XOR-DEM hybrid","kind":"theorem","summary":"[Perfect security of the XOR-DEM hybrid] A perfectly-secure KEM composed with the XOR one-time-…","labels":["thm:kemdem-perfect"],"detail_key":"p53"},{"id":"n42800","layer":"informal","project":"p53","title":"Commitment scheme","kind":"definition","summary":"[Commitment scheme] A commitment scheme has a \\mathsfcommit map (value and randomness to a comm…","labels":["def:commscheme"],"detail_key":"p53"},{"id":"n42801","layer":"informal","project":"p53","title":"Perfect hiding of the mask commitment","kind":"theorem","summary":"[Perfect hiding of the mask commitment] The mask commitment \\mathsfcommit(r, m) = r \\oplus m wi…","labels":["thm:comm-hiding"],"detail_key":"p53"},{"id":"n42802","layer":"informal","project":"p53","title":"Perfect binding of the identity commitment","kind":"theorem","summary":"[Perfect binding of the identity commitment] The identity commitment cannot be opened to two di…","labels":["thm:comm-binding"],"detail_key":"p53"},{"id":"n42803","layer":"informal","project":"p53","title":"Additive 2-of-2 sharing","kind":"definition","summary":"[Additive 2-of-2 sharing] The secret s is shared as (k, k \\oplus s) for a uniform key k; party…","labels":["def:ss-share"],"detail_key":"p53"},{"id":"n42804","layer":"informal","project":"p53","title":"Reconstruction","kind":"lemma","summary":"[Reconstruction] The two shares reconstruct the secret: k \\oplus (k \\oplus s) = s.","labels":["lem:ss-reconstruct"],"detail_key":"p53"},{"id":"n42805","layer":"informal","project":"p53","title":"Perfect privacy","kind":"theorem","summary":"[Perfect privacy] For either corrupted party, any pair of secrets, and any adversary, the priva…","labels":["thm:ss-privacy"],"detail_key":"p53"},{"id":"n42806","layer":"informal","project":"p53","title":"Bijection MAC family","kind":"definition","summary":"[Bijection MAC family] A bijection MAC family tags each message by a key-indexed bijection Key…","labels":["def:bijmac"],"detail_key":"p53"},{"id":"n42807","layer":"informal","project":"p53","title":"Optimal forgery probability","kind":"theorem","summary":"[Optimal forgery probability] A bijection MAC has single-query EUF-CMA forgery probability exac…","labels":["thm:bijmac-forgery"],"detail_key":"p53"},{"id":"n42808","layer":"informal","project":"p53","title":"XOR MAC","kind":"corollary","summary":"[XOR MAC] The XOR MAC \\mathsfmac(k, m) = k \\oplus m attains the optimal one-bit forgery probabi…","labels":["cor:xormac"],"detail_key":"p53"},{"id":"n42809","layer":"informal","project":"p53","title":"PRF-based encryption","kind":"definition","summary":"[PRF-based encryption] From a bijection PRF family, the scheme encrypts m as (r, F(k, r) \\oplus…","labels":["def:cpafromprf"],"detail_key":"p53"},{"id":"n42810","layer":"informal","project":"p53","title":"Correctness","kind":"theorem","summary":"[Correctness] Decryption recovers the message: F(k, r) \\oplus (F(k, r) \\oplus m) = m.","labels":["thm:cpafromprf-correct"],"detail_key":"p53"},{"id":"n42811","layer":"informal","project":"p53","title":"Perfect IND-CPA","kind":"theorem","summary":"[Perfect IND-CPA] Built on a perfect PRF, the scheme has perfect IND-CPA security: advantage ex…","labels":["thm:cpafromprf-perfect"],"detail_key":"p53"},{"id":"n42812","layer":"informal","project":"p53","title":"Computational IND-CPA from a PRF","kind":"theorem","summary":"[Computational IND-CPA from a PRF] For a general PRF F with advantage \\varepsilon, replacing F(…","labels":["thm:cpafromprf-bound"],"detail_key":"p53"},{"id":"n42813","layer":"informal","project":"p53","title":"Synthesized reduction","kind":"theorem","summary":"[Synthesized reduction] The PRF-swap hop is discharged, not assumed: an explicit distinguisher…","labels":["thm:cpafromprf-reduction"],"detail_key":"p53"},{"id":"n42814","layer":"informal","project":"p53","title":"ElGamal over a prime-order group","kind":"definition","summary":"[ElGamal over a prime-order group] With public key \\mathitpk = g^x, ElGamal encrypts m as (g^r,…","labels":["def:elgamal"],"detail_key":"p53"},{"id":"n42815","layer":"informal","project":"p53","title":"Correctness","kind":"theorem","summary":"[Correctness] Decryption inverts encryption: (m \\cdot \\mathitpk^r) \\cdot (g^r)^-x = m.","labels":["thm:elgamal-correct"],"detail_key":"p53"},{"id":"n42816","layer":"informal","project":"p53","title":"IND-CPA reduces to DDH","kind":"theorem","summary":"[IND-CPA reduces to DDH] ElGamal's IND-CPA advantage is bounded by the DDH advantage of two exp…","labels":["thm:elgamal-ddh"],"detail_key":"p53"},{"id":"n42817","layer":"informal","project":"p53","title":"Exact real-or-random bound","kind":"theorem","summary":"[Exact real-or-random bound] In the real-or-random framing (real encryption versus a random cip…","labels":["thm:elgamal-eq-ddh"],"detail_key":"p53"},{"id":"n42818","layer":"informal","project":"p53","title":"IND-CCA game","kind":"definition","summary":"[IND-CCA game] The adversary receives the challenge ciphertext and a decryption oracle that ans…","labels":["def:indcca"],"detail_key":"p53"},{"id":"n42819","layer":"informal","project":"p53","title":"Generic CCA reduction","kind":"theorem","summary":"[Generic CCA reduction] For any scheme, the IND-CCA advantage is bounded by its IND-CPA advanta…","labels":["thm:indcca-reduction"],"detail_key":"p53"},{"id":"n42820","layer":"informal","project":"p53","title":"Encrypt-then-MAC is authenticated encryption","kind":"theorem","summary":"[Encrypt-then-MAC is authenticated encryption] For the one-time-pad Encrypt-then-MAC scheme the…","labels":["thm:etm-cca"],"detail_key":"p53"},{"id":"n42821","layer":"informal","project":"p53","title":"Single-block CTR","kind":"definition","summary":"[Single-block CTR] A CTR block scheme is a keystream bijection family Block \\to (Key \\simeq Blo…","labels":["def:ctr"],"detail_key":"p53"},{"id":"n42822","layer":"informal","project":"p53","title":"Perfect IND-CPA","kind":"theorem","summary":"[Perfect IND-CPA] Single-block CTR from a bijection block cipher has perfect IND-CPA security:…","labels":["thm:ctr-perfect"],"detail_key":"p53"},{"id":"n42823","layer":"informal","project":"p53","title":"Computational bound from a PRF","kind":"theorem","summary":"[Computational bound from a PRF] For a general block cipher with PRF-advantage \\varepsilon, sin…","labels":["thm:ctr-bound"],"detail_key":"p53"},{"id":"n42824","layer":"informal","project":"p53","title":"Key-agreement distinguishing game","kind":"definition","summary":"[Key-agreement distinguishing game] The real game hands the distinguisher the transcript (g^a,…","labels":["def:ka-game"],"detail_key":"p53"},{"id":"n42825","layer":"informal","project":"p53","title":"Key indistinguishability equals DDH","kind":"theorem","summary":"[Key indistinguishability equals DDH] The key-agreement advantage is \\emphequal to the DDH adva…","labels":["thm:ka-ddh"],"detail_key":"p53"},{"id":"n42826","layer":"informal","project":"p53","title":"Security under DDH","kind":"corollary","summary":"[Security under DDH] If DDH is \\varepsilon-hard in the group, Diffie--Hellman key agreement has…","labels":["cor:ka-secure"],"detail_key":"p53"},{"id":"n42827","layer":"informal","project":"p53","title":"Single-block CBC","kind":"definition","summary":"[Single-block CBC] A CBC block cipher carries two coherent views --- a permutation per key (for…","labels":["def:cbc"],"detail_key":"p53"},{"id":"n42828","layer":"informal","project":"p53","title":"Perfect IND-CPA","kind":"theorem","summary":"[Perfect IND-CPA] Single-block CBC from a two-sided bijection block cipher has perfect IND-CPA…","labels":["thm:cbc-perfect"],"detail_key":"p53"},{"id":"n42829","layer":"informal","project":"p53","title":"Computational bound from a PRP","kind":"theorem","summary":"[Computational bound from a PRP] For a general block cipher with PRP-advantage \\varepsilon, sin…","labels":["thm:cbc-bound"],"detail_key":"p53"},{"id":"n42830","layer":"informal","project":"p53","title":"Polynomial share","kind":"definition","summary":"[Polynomial share] A degree-<t polynomial is its constant term s plus t-1 free coefficients; th…","labels":["def:shamir-share"],"detail_key":"p53"},{"id":"n42831","layer":"informal","project":"p53","title":"Reconstruction","kind":"theorem","summary":"[Reconstruction] Lagrange interpolation through any t distinct evaluation points recovers f(0)…","labels":["thm:shamir-reconstruct"],"detail_key":"p53"},{"id":"n42832","layer":"informal","project":"p53","title":"Perfect privacy","kind":"theorem","summary":"[Perfect privacy] For any t-1 nonzero evaluation points, any two secrets, and any adversary, th…","labels":["thm:shamir-privacy"],"detail_key":"p53"},{"id":"n42833","layer":"informal","project":"p53","title":"Completeness","kind":"theorem","summary":"[Completeness] Honest execution with a valid witness always verifies.","labels":["thm:sigma-complete"],"detail_key":"p53"},{"id":"n42834","layer":"informal","project":"p53","title":"Special honest-verifier zero knowledge","kind":"theorem","summary":"[Special honest-verifier zero knowledge] Real and simulated transcripts are identically distrib…","labels":["thm:sigma-shvzk"],"detail_key":"p53"},{"id":"n42835","layer":"informal","project":"p53","title":"Special soundness","kind":"theorem","summary":"[Special soundness] Two accepting transcripts sharing a commitment but differing in challenge y…","labels":["thm:sigma-sound"],"detail_key":"p53"},{"id":"n42836","layer":"informal","project":"p53","title":"Commitment hiding and binding","kind":"theorem","summary":"[Commitment hiding and binding] The associated commitment is perfectly hiding (a direct instanc…","labels":["thm:sigma-commitment"],"detail_key":"p53"},{"id":"n42837","layer":"formal","project":"p53","title":"CatCrypt.Core.KlSPComp.klDel","kind":"def","summary":"(α : CatCrypt.Core.KlSPComp) → Quiver.Hom α (have this := Empty; this)","labels":[],"detail_key":"p53","name":"CatCrypt.Core.KlSPComp.klDel","module":"CatCryptCore.Category.Affine"},{"id":"n42838","layer":"formal","project":"p53","title":"CategoryTheory.AffineMonoidalCategory","kind":"inductive","summary":"(C : Type u) → [inst : CategoryTheory.Category.v, u C] → [CategoryTheory.MonoidalCategory C] →…","labels":[],"detail_key":"p53","name":"CategoryTheory.AffineMonoidalCategory","module":"CatCryptCore.Category.Affine"},{"id":"n42839","layer":"formal","project":"p53","title":"CatCrypt.Core.KlSPComp.binaryCoproductIsColimit","kind":"def","summary":"(α β : CatCrypt.Core.KlSPComp) → CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryC…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.KlSPComp.binaryCoproductIsColimit","module":"CatCryptCore.Category.Cocartesian"},{"id":"n42840","layer":"formal","project":"p53","title":"CatCrypt.Core.KlSPComp.klDesc","kind":"def","summary":"α β γ : CatCrypt.Core.KlSPComp → Quiver.Hom α γ → Quiver.Hom β γ → Quiver.Hom (have this := Sum…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.KlSPComp.klDesc","module":"CatCryptCore.Category.Cocartesian"},{"id":"n42841","layer":"formal","project":"p53","title":"CategoryTheory.CocartesianMonoidalCategory","kind":"inductive","summary":"(C : Type u) → [CategoryTheory.Category.v, u C] → Type (max u v)","labels":[],"detail_key":"p53","name":"CategoryTheory.CocartesianMonoidalCategory","module":"CatCryptCore.Category.Cocartesian"},{"id":"n42842","layer":"formal","project":"p53","title":"CatCrypt.Category.FamObj","kind":"inductive","summary":"(C : Type u) → [CategoryTheory.Category.v, u C] → Type (max 1 u)","labels":[],"detail_key":"p53","name":"CatCrypt.Category.FamObj","module":"CatCryptCore.Category.Fam"},{"id":"n42843","layer":"formal","project":"p53","title":"CatCrypt.Core.KlSPComp","kind":"def","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Core.KlSPComp","module":"CatCryptCore.Category.KlSPComp"},{"id":"n42844","layer":"formal","project":"p53","title":"CatCrypt.Core.KlSPComp.comp_apply","kind":"theorem","summary":"∀ α β γ : CatCrypt.Core.KlSPComp (f : Quiver.Hom α β) (g : Quiver.Hom β γ) (x : α), Eq (Categor…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.KlSPComp.comp_apply","module":"CatCryptCore.Category.KlSPComp"},{"id":"n42845","layer":"formal","project":"p53","title":"CatCrypt.Category.DeepInterface.toPkgInterface","kind":"def","summary":"CatCrypt.Deep.DeepInterface → CatCrypt.Category.PkgInterface","labels":[],"detail_key":"p53","name":"CatCrypt.Category.DeepInterface.toPkgInterface","module":"CatCryptCore.Category.PkgFam"},{"id":"n42846","layer":"formal","project":"p53","title":"CatCrypt.Category.PkgImpl","kind":"def","summary":"CatCrypt.Category.PkgInterface → Type","labels":[],"detail_key":"p53","name":"CatCrypt.Category.PkgImpl","module":"CatCryptCore.Category.PkgFam"},{"id":"n42847","layer":"formal","project":"p53","title":"CatCrypt.Category.PkgImpl.link","kind":"def","summary":"D M C : CatCrypt.Category.FamObj CatCrypt.Core.KlSPComp → Quiver.Hom D M → Quiver.Hom M C → Qui…","labels":[],"detail_key":"p53","name":"CatCrypt.Category.PkgImpl.link","module":"CatCryptCore.Category.PkgFam"},{"id":"n42848","layer":"formal","project":"p53","title":"CatCrypt.Category.PkgImpl.par","kind":"def","summary":"I J : CatCrypt.Category.PkgInterface → CatCrypt.Category.PkgImpl I → CatCrypt.Category.PkgImpl…","labels":[],"detail_key":"p53","name":"CatCrypt.Category.PkgImpl.par","module":"CatCryptCore.Category.PkgFam"},{"id":"n42849","layer":"formal","project":"p53","title":"CatCrypt.Category.PkgInterface","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Category.PkgInterface","module":"CatCryptCore.Category.PkgFam"},{"id":"n42850","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp","kind":"def","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp","module":"CatCryptCore.Core.Code"},{"id":"n42851","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.bind","kind":"def","summary":"α : Type u_1 → β : Type u_2 → CatCrypt.Core.SPComp α → (α → CatCrypt.Core.SPComp β) → CatCrypt.…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.bind","module":"CatCryptCore.Core.Code"},{"id":"n42852","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.bind_assoc","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 γ : Type u_3 (c : CatCrypt.Core.SPComp α) (f : α → CatCrypt.Core.SP…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.bind_assoc","module":"CatCryptCore.Core.Code"},{"id":"n42853","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.bind_pure","kind":"theorem","summary":"∀ α : Type u_1 (c : CatCrypt.Core.SPComp α), Eq (c.bind CatCrypt.Core.SPComp.pure) c","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.bind_pure","module":"CatCryptCore.Core.Code"},{"id":"n42854","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.pure","kind":"def","summary":"α : Type u_1 → α → CatCrypt.Core.SPComp α","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.pure","module":"CatCryptCore.Core.Code"},{"id":"n42855","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.pure_bind","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 (a : α) (f : α → CatCrypt.Core.SPComp β), Eq ((CatCrypt.Core.SPComp…","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.pure_bind","module":"CatCryptCore.Core.Code"},{"id":"n42856","layer":"formal","project":"p53","title":"CatCrypt.Core.SPComp.sample","kind":"def","summary":"(α : Type u_4) → [Fintype α] → [Nonempty α] → CatCrypt.Core.SPComp α","labels":[],"detail_key":"p53","name":"CatCrypt.Core.SPComp.sample","module":"CatCryptCore.Core.Code"},{"id":"n42857","layer":"formal","project":"p53","title":"CatCrypt.Core.Heap","kind":"inductive","summary":"Type","labels":[],"detail_key":"p53","name":"CatCrypt.Core.Heap","module":"CatCryptCore.Core.Heap"},{"id":"n42858","layer":"formal","project":"p53","title":"CatCrypt.Crypto.Advantage","kind":"def","summary":"CatCrypt.Core.SPComp Bool → CatCrypt.Core.SPComp Bool → 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CatCrypt.Crypto.ECC.curve25519Prime","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519Prime_prime","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42921","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519SubgroupOrder","kind":"def","summary":"Nat","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519SubgroupOrder","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42922","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519SubgroupOrder_prime","kind":"axiom","summary":"Nat.Prime CatCrypt.Crypto.ECC.curve25519SubgroupOrder","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519SubgroupOrder_prime","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42923","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519_MontyCurveGroup","kind":"def","summary":"CatCrypt.Crypto.ECC.MontyCurveGroup CatCrypt.Crypto.ECC.Fp25519 (CatCrypt.Crypto.ECC.curve25519…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519_MontyCurveGroup","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42924","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519_basepoint","kind":"axiom","summary":"CatCrypt.Crypto.ECC.Curve25519Point","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519_basepoint","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42925","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519_basepoint_LadderNondeg","kind":"theorem","summary":"∀ (bits : List Bool), LT.lt (CatCrypt.Crypto.ECC.bitsToNat bits) CatCrypt.Crypto.ECC.curve25519…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519_basepoint_LadderNondeg","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42926","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519_basepoint_addOrder","kind":"theorem","summary":"Eq (addOrderOf CatCrypt.Crypto.ECC.curve25519_basepoint) CatCrypt.Crypto.ECC.curve25519Subgroup…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519_basepoint_addOrder","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42927","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.x25519","kind":"def","summary":"G : Type u_1 → [AddCommGroup G] → Nat → G → G","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.x25519","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42928","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.x25519_eq_ladder","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] (P : G) (bits : List Bool), Eq (CatCrypt.Crypto.ECC.x255…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.x25519_eq_ladder","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42929","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.x25519_ladder_correct","kind":"theorem","summary":"∀ (P : CatCrypt.Crypto.ECC.Curve25519Point) (bits : List Bool), CatCrypt.Crypto.ECC.LadderNonde…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.x25519_ladder_correct","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42930","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.x25519_ladder_correct_basepoint","kind":"theorem","summary":"∀ (bits : List Bool), LT.lt (CatCrypt.Crypto.ECC.bitsToNat bits) CatCrypt.Crypto.ECC.curve25519…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.x25519_ladder_correct_basepoint","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42931","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xladderFold_curve25519_correct","kind":"theorem","summary":"∀ (P : CatCrypt.Crypto.ECC.Curve25519Point) (bits : List Bool), CatCrypt.Crypto.ECC.LadderNonde…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xladderFold_curve25519_correct","module":"CatCryptCore.Crypto.KeyAgreement.Curve25519"},{"id":"n42932","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.montgomeryW","kind":"def","summary":"F : Type u_1 → [Field F] → F → WeierstrassCurve F","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.montgomeryW","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42933","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.montgomeryW_MontyCurveGroup","kind":"def","summary":"F : Type u_1 → [inst : Field F] → [inst_1 : DecidableEq F] → Ne 2 0 → (A a24 : F) → Eq (HMul.hM…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.montgomeryW_MontyCurveGroup","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42934","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.montgomeryW_equation","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] (A x y : F), Iff ((CatCrypt.Crypto.ECC.montgomeryW A).toAffine.…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.montgomeryW_equation","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42935","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xProjW","kind":"def","summary":"F : Type u_1 → [inst : Field F] → (A : F) → (CatCrypt.Crypto.ECC.montgomeryW A).toAffine.Point…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xProjW","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42936","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdadd_spec_W","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : DecidableEq F], Ne 2 0 → ∀ (A : F) (P Q : (CatCrypt.C…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdadd_spec_W","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42937","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdbl_spec_W","kind":"theorem","summary":"∀ F : Type u_1 [inst : Field F] [inst_1 : DecidableEq F], Ne 2 0 → ∀ (A a24 : F), Eq (HMul.hMul…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdbl_spec_W","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryAsWeierstrass"},{"id":"n42938","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.LadderInv","kind":"def","summary":"G : Type u_1 → [AddCommGroup G] → G → Nat → Prod G G → Prop","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.LadderInv","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryLadder"},{"id":"n42939","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.bitsToNat","kind":"def","summary":"List Bool → Nat","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.bitsToNat","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryLadder"},{"id":"n42940","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.ladderFold","kind":"def","summary":"G : Type u_1 → [AddCommGroup G] → G → List Bool → Prod G G","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.ladderFold","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryLadder"},{"id":"n42941","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.ladderFold_correct","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] (P : G) (bits : List Bool), CatCrypt.Crypto.ECC.LadderIn…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.ladderFold_correct","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryLadder"},{"id":"n42942","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.ladderStep","kind":"def","summary":"G : Type u_1 → [AddCommGroup G] → Prod G G → Bool → Prod G G","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.ladderStep","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryLadder"},{"id":"n42943","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.MontyCurveGroup","kind":"inductive","summary":"(F : Type u_2) → [inst : CommRing F] → CatCrypt.Crypto.ECC.MontyParams F → (G : Type u_3) → [Ad…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.MontyCurveGroup","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42944","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.MontyParams","kind":"inductive","summary":"(F : Type u_2) → [CommRing F] → Type u_2","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.MontyParams","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42945","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.ProjEquiv","kind":"def","summary":"F : Type u_1 → [CommRing F] → Prod F F → Prod F F → Prop","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.ProjEquiv","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42946","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.ProjEquiv.trans","kind":"theorem","summary":"∀ F : Type u_2 [inst : Field F] p q r : Prod F F, CatCrypt.Crypto.ECC.ProjEquiv p q → CatCrypt.…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.ProjEquiv.trans","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42947","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.XLadderInv","kind":"def","summary":"F : Type u_2 → [inst : CommRing F] → params : CatCrypt.Crypto.ECC.MontyParams F → G : Type u_3…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.XLadderInv","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42948","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.curve25519Params","kind":"def","summary":"(F : Type u_2) → [inst : CommRing F] → CatCrypt.Crypto.ECC.MontyParams F","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.curve25519Params","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42949","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdadd","kind":"def","summary":"F : Type u_1 → [CommRing F] → Prod F F → Prod F F → Prod F F → Prod F F","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdadd","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42950","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdadd_projEquiv","kind":"theorem","summary":"∀ F : Type u_1 [inst : CommRing F] p p' q q' d d' : Prod F F, CatCrypt.Crypto.ECC.ProjEquiv p p…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdadd_projEquiv","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42951","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdadd_symm","kind":"theorem","summary":"∀ F : Type u_1 [inst : CommRing F] (PZ QZ DZ : Prod F F), Eq (CatCrypt.Crypto.ECC.xdadd PZ QZ D…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdadd_symm","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42952","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdbl","kind":"def","summary":"F : Type u_1 → [CommRing F] → F → Prod F F → Prod F F","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdbl","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42953","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xdbl_projEquiv","kind":"theorem","summary":"∀ F : Type u_1 [inst : CommRing F] (a24 : F) p q : Prod F F, CatCrypt.Crypto.ECC.ProjEquiv p q…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xdbl_projEquiv","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42954","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xladderFold","kind":"def","summary":"F : Type u_1 → [CommRing F] → F → Prod F F → List Bool → Prod (Prod F F) (Prod F F)","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xladderFold","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42955","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xladderStep","kind":"def","summary":"F : Type u_1 → [CommRing F] → F → Prod F F → Prod (Prod F F) (Prod F F) → Bool → Prod (Prod F F…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xladderStep","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42956","layer":"formal","project":"p53","title":"CatCrypt.Crypto.ECC.xladderStep_preserves_invariant","kind":"theorem","summary":"∀ F : Type u_2 [inst : Field F] params : CatCrypt.Crypto.ECC.MontyParams F G : Type u_3 [inst_1…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.ECC.xladderStep_preserves_invariant","module":"CatCryptCore.Crypto.KeyAgreement.MontgomeryXOnly"},{"id":"n42957","layer":"formal","project":"p53","title":"CatCrypt.Crypto.DeepNomAdvantage","kind":"def","summary":"CatCrypt.Deep.NomPackage → CatCrypt.Deep.NomPackage → CatCrypt.Deep.NomPackage → ENNReal","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.DeepNomAdvantage","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42958","layer":"formal","project":"p53","title":"CatCrypt.Crypto.DeepNomAdvantage_link","kind":"theorem","summary":"∀ (G₀ G₁ A R : CatCrypt.Deep.NomPackage) (hregAR : Eq A.registry R.registry) (hregRG₀ : Eq R.re…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.DeepNomAdvantage_link","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42959","layer":"formal","project":"p53","title":"CatCrypt.Crypto.NomPkgSecure","kind":"def","summary":"CatCrypt.Deep.NomPackage → CatCrypt.Deep.NomPackage → (CatCrypt.Deep.NomPackage → ENNReal) → Pr…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.NomPkgSecure","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42960","layer":"formal","project":"p53","title":"CatCrypt.Crypto.runPkg","kind":"def","summary":"CatCrypt.Deep.DeepPackage → CatCrypt.Core.SPComp Bool","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.runPkg","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42961","layer":"formal","project":"p53","title":"CatCrypt.Crypto.runPkg_interchange","kind":"theorem","summary":"∀ (p₁ p₂ p₃ p₄ : CatCrypt.Deep.DeepPackage) (h₁₂ : p₁.sep p₂) (h₃₄ : p₃.sep p₄) (h_link_sep : (…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.runPkg_interchange","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42962","layer":"formal","project":"p53","title":"CatCrypt.Crypto.runPkg_link","kind":"theorem","summary":"∀ (p₁ p₂ : CatCrypt.Deep.DeepPackage), Eq (CatCrypt.Crypto.runPkg (p₁.link p₂)) (have this := C…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.runPkg_link","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42963","layer":"formal","project":"p53","title":"CatCrypt.Crypto.runPkg_link_assoc","kind":"theorem","summary":"∀ (p₁ p₂ p₃ : CatCrypt.Deep.DeepPackage), Eq (CatCrypt.Crypto.runPkg ((p₁.link p₂).link p₃)) (C…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.runPkg_link_assoc","module":"CatCryptCore.Crypto.NomAdvantage"},{"id":"n42964","layer":"formal","project":"p53","title":"CatCrypt.Crypto.PairingGroup","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.PairingGroup","module":"CatCryptCore.Crypto.PairingGroup"},{"id":"n42965","layer":"formal","project":"p53","title":"CatCrypt.Crypto.PairingGroup.e_bilinear","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (a b : Int), Eq (CatCrypt.Crypto.PairingGroup.e (HPow.hPow C…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.PairingGroup.e_bilinear","module":"CatCryptCore.Crypto.PairingGroup"},{"id":"n42966","layer":"formal","project":"p53","title":"CatCrypt.Crypto.PairingGroup.srs₁","kind":"def","summary":"P : CatCrypt.Crypto.PairingGroup → ZMod CatCrypt.Crypto.PairingGroup.p → (t : Nat) → Fin (HAdd.…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.PairingGroup.srs₁","module":"CatCryptCore.Crypto.PairingGroup"},{"id":"n42967","layer":"formal","project":"p53","title":"CatCrypt.Crypto.KlMorph","kind":"def","summary":"Type u_1 → Type u_2 → Type (max u_1 u_2)","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.KlMorph","module":"CatCryptCore.Crypto.SDist"},{"id":"n42968","layer":"formal","project":"p53","title":"CatCrypt.Crypto.absDiff","kind":"def","summary":"ENNReal → ENNReal → ENNReal","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.absDiff","module":"CatCryptCore.Crypto.SDist"},{"id":"n42969","layer":"formal","project":"p53","title":"CatCrypt.Crypto.absDiff_triangle","kind":"theorem","summary":"∀ (a b c : ENNReal), LE.le (CatCrypt.Crypto.absDiff a c) (HAdd.hAdd (CatCrypt.Crypto.absDiff a…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.absDiff_triangle","module":"CatCryptCore.Crypto.SDist"},{"id":"n42970","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist","kind":"def","summary":"α : Type u_1 → β : Type u_2 → (α → CatCrypt.Core.SPComp β) → (α → CatCrypt.Core.SPComp β) → ENN…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist","module":"CatCryptCore.Crypto.SDist"},{"id":"n42971","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_comp_add","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 γ : Type u_3 (f₁ f₂ : α → CatCrypt.Core.SPComp β) (g₁ g₂ : β → CatC…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_comp_add","module":"CatCryptCore.Crypto.SDist"},{"id":"n42972","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_comp_left","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 γ : Type u_3 (f : α → CatCrypt.Core.SPComp β) (g₁ g₂ : β → CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_comp_left","module":"CatCryptCore.Crypto.SDist"},{"id":"n42973","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_comp_right","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 γ : Type u_3 (f₁ f₂ : α → CatCrypt.Core.SPComp β) (g : β → CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_comp_right","module":"CatCryptCore.Crypto.SDist"},{"id":"n42974","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_isPure_le","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 f g : α → CatCrypt.Core.SPComp β ε : ENNReal, (∀ (a : α), (f a).IsP…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_isPure_le","module":"CatCryptCore.Crypto.SDist"},{"id":"n42975","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_of_isPure_advantageA","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 f g : α → CatCrypt.Core.SPComp β ε : ENNReal, (∀ (a : α), (f a).IsP…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_of_isPure_advantageA","module":"CatCryptCore.Crypto.SDist"},{"id":"n42976","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_oracleGame_le","kind":"theorem","summary":"∀ OracleIf : Type u_1 G_real G_ideal : (OracleIf → CatCrypt.Core.SPComp Bool) → CatCrypt.Core.S…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_oracleGame_le","module":"CatCryptCore.Crypto.SDist"},{"id":"n42977","layer":"formal","project":"p53","title":"CatCrypt.Crypto.sdist_sym","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 (f g : α → CatCrypt.Core.SPComp β), Eq (CatCrypt.Crypto.sdist f g)…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.sdist_sym","module":"CatCryptCore.Crypto.SDist"},{"id":"n42978","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.EUF_CMA_Adv","kind":"def","summary":"(M : CatCrypt.Crypto.SecurityDefs.MACScheme) → M.Message → M.Message → M.Tag → (Bool → CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.EUF_CMA_Adv","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42979","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.INDCCA_Game","kind":"def","summary":"(E : CatCrypt.Crypto.EncScheme) → E.Plaintext → E.Plaintext → Bool → ((E.Ciphertext → CatCrypt.…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.INDCCA_Game","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42980","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.INDCCA_reduces_to_INDCPA","kind":"theorem","summary":"∀ (E : CatCrypt.Crypto.EncScheme) (m₀ m₁ : E.Plaintext) (A : (E.Ciphertext → CatCrypt.Core.SPCo…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.INDCCA_reduces_to_INDCPA","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42981","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.INDCPA_Adv","kind":"def","summary":"(E : CatCrypt.Crypto.EncScheme) → E.Plaintext → E.Plaintext → (E.Ciphertext → CatCrypt.Core.SPC…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.INDCPA_Adv","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42982","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.INDCPA_Game","kind":"def","summary":"(E : CatCrypt.Crypto.EncScheme) → E.Plaintext → E.Plaintext → Bool → CatCrypt.Core.SPComp E.Cip…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.INDCPA_Game","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42983","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.MACScheme","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.MACScheme","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42984","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.PRFScheme","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.PRFScheme","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42985","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.PRF_Adv","kind":"def","summary":"(F : CatCrypt.Crypto.SecurityDefs.PRFScheme) → F.Input → (F.Output → CatCrypt.Core.SPComp Bool)…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.PRF_Adv","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42986","layer":"formal","project":"p53","title":"CatCrypt.Crypto.SecurityDefs.ccaDecOracle","kind":"def","summary":"(E : CatCrypt.Crypto.EncScheme) → E.Key → E.Ciphertext → E.Ciphertext → CatCrypt.Core.SPComp (O…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.SecurityDefs.ccaDecOracle","module":"CatCryptCore.Crypto.SecurityDefs"},{"id":"n42987","layer":"formal","project":"p53","title":"CatCrypt.Deep.RawCode.eval","kind":"def","summary":"α : Type u_4 → CatCrypt.Deep.RawCode α → CatCrypt.Core.SPComp α","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.RawCode.eval","module":"CatCryptCore.Deep.Eval"},{"id":"n42988","layer":"formal","project":"p53","title":"CatCrypt.Deep.eval_substOracle","kind":"theorem","summary":"∀ α : Type u (c : CatCrypt.Deep.RawCode α) (env : Nat → (dom codom : Type u) → dom → CatCrypt.D…","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.eval_substOracle","module":"CatCryptCore.Deep.Eval"},{"id":"n42989","layer":"formal","project":"p53","title":"CatCrypt.Deep.DeepInterface","kind":"inductive","summary":"Type (u + 1)","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.DeepInterface","module":"CatCryptCore.Deep.Package"},{"id":"n42990","layer":"formal","project":"p53","title":"CatCrypt.Deep.DeepPackage","kind":"inductive","summary":"Type (u + 1)","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.DeepPackage","module":"CatCryptCore.Deep.Package"},{"id":"n42991","layer":"formal","project":"p53","title":"CatCrypt.Deep.DeepPackage.id","kind":"def","summary":"CatCrypt.Deep.DeepInterface → CatCrypt.Deep.DeepPackage","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.DeepPackage.id","module":"CatCryptCore.Deep.Package"},{"id":"n42992","layer":"formal","project":"p53","title":"CatCrypt.Deep.DeepPackage.link","kind":"def","summary":"CatCrypt.Deep.DeepPackage 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1)","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.ValidCodeBundle","module":"CatCryptCore.Deep.Package"},{"id":"n42996","layer":"formal","project":"p53","title":"CatCrypt.Deep.RawCode.inductionOn","kind":"def","summary":"∀ α : Type u motive : α : Type u → CatCrypt.Deep.RawCode α → Prop (c : CatCrypt.Deep.RawCode α)…","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.RawCode.inductionOn","module":"CatCryptCore.Deep.RawCode"},{"id":"n42997","layer":"formal","project":"p53","title":"CatCrypt.Deep.RawCode.substOracle","kind":"def","summary":"α : Type u → CatCrypt.Deep.RawCode α → (Nat → (dom codom : Type u) → dom → CatCrypt.Deep.RawCod…","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.RawCode.substOracle","module":"CatCryptCore.Deep.RawCode"},{"id":"n42998","layer":"formal","project":"p53","title":"CatCrypt.Deep.RawCode.substOracle_comp","kind":"theorem","summary":"∀ α : Type u (c : CatCrypt.Deep.RawCode α) (env₁ env₂ : Nat → (dom codom : Type u) → dom → CatC…","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.RawCode.substOracle_comp","module":"CatCryptCore.Deep.RawCode"},{"id":"n42999","layer":"formal","project":"p53","title":"CatCrypt.Deep.RawCode.substOracle_eq_self","kind":"theorem","summary":"∀ α : Type u c : CatCrypt.Deep.RawCode α, c.NoOracleCall → ∀ (env : Nat → (dom codom : Type u)…","labels":[],"detail_key":"p53","name":"CatCrypt.Deep.RawCode.substOracle_eq_self","module":"CatCryptCore.Deep.RawCode"},{"id":"n43000","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.HashFunc","kind":"inductive","summary":"Type","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.HashFunc","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43001","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.auth_perfect_bij","kind":"theorem","summary":"∀ (H : CatCrypt.Examples.BasicHash.HashFunc), (∀ (n : CatCrypt.Examples.BasicHash.Word), Functi…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.auth_perfect_bij","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43002","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.auth_zero_advantage_xor","kind":"theorem","summary":"∀ (n : CatCrypt.Examples.BasicHash.Word) (A : CatCrypt.Examples.BasicHash.Word → CatCrypt.Core.…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.auth_zero_advantage_xor","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43003","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.unlink_ideal_not_always_true","kind":"theorem","summary":"∀ (n1 n2 : CatCrypt.Examples.BasicHash.Word), Ne n1 n2 → Exists fun k1 => Exists fun k2 => Eq (…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.unlink_ideal_not_always_true","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43004","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.unlink_real_always_true","kind":"theorem","summary":"∀ (n1 n2 : CatCrypt.Examples.BasicHash.Word), Eq ((CatCrypt.Examples.BasicHash.unlink_real CatC…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.unlink_real_always_true","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43005","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.xorHash","kind":"def","summary":"CatCrypt.Examples.BasicHash.HashFunc","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.xorHash","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43006","layer":"formal","project":"p53","title":"CatCrypt.Examples.BasicHash.xorHash_correlated","kind":"theorem","summary":"∀ (k n1 n2 : CatCrypt.Examples.BasicHash.Word), Eq ((xor k n1).xor (xor k n2)) (xor n1 n2)","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.BasicHash.xorHash_correlated","module":"CatCryptCore.Examples.BasicHash"},{"id":"n43007","layer":"formal","project":"p53","title":"CatCrypt.Examples.CBCMode.CBCBlockCipher","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CBCMode.CBCBlockCipher","module":"CatCryptCore.Examples.CBCMode"},{"id":"n43008","layer":"formal","project":"p53","title":"CatCrypt.Examples.CBCMode.boolCBC_perfect_indcpa","kind":"theorem","summary":"∀ (m₀ m₁ : Bool) (A : CatCrypt.Examples.CBCMode.boolCBC.toEncScheme.Ciphertext → CatCrypt.Core.…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CBCMode.boolCBC_perfect_indcpa","module":"CatCryptCore.Examples.CBCMode"},{"id":"n43009","layer":"formal","project":"p53","title":"CatCrypt.Examples.CBCMode.cbc_indcpa_bound","kind":"theorem","summary":"∀ (C : CatCrypt.Examples.CBCMode.CBCBlockCipher) (eval : C.Key → C.Block → C.Block) (m₀ m₁ : C.…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CBCMode.cbc_indcpa_bound","module":"CatCryptCore.Examples.CBCMode"},{"id":"n43010","layer":"formal","project":"p53","title":"CatCrypt.Examples.CBCMode.cbc_perfect_indcpa","kind":"theorem","summary":"∀ (C : CatCrypt.Examples.CBCMode.CBCBlockCipher) (m₀ m₁ : C.Block) (A : C.toEncScheme.Ciphertex…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CBCMode.cbc_perfect_indcpa","module":"CatCryptCore.Examples.CBCMode"},{"id":"n43011","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43012","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme.toBijPRFFamily","kind":"def","summary":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme → CatCrypt.Examples.PRF.BijPRFFamily","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme.toBijPRFFamily","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43013","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.bijCPA_correct","kind":"theorem","summary":"∀ (B : CatCrypt.Examples.CPAFromPRF.BijCPAScheme), B.toEncScheme.Correct","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.bijCPA_correct","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43014","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.bijCPA_perfect_indcpa","kind":"theorem","summary":"∀ (B : CatCrypt.Examples.CPAFromPRF.BijCPAScheme) (m₀ m₁ : B.Output) (A : B.toEncScheme.Ciphert…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.bijCPA_perfect_indcpa","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43015","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.boolXorCPA_perfect_indcpa","kind":"theorem","summary":"∀ (m₀ m₁ : Bool) (A : CatCrypt.Examples.CPAFromPRF.boolXorCPA.toEncScheme.Ciphertext → CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.boolXorCPA_perfect_indcpa","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43016","layer":"formal","project":"p53","title":"CatCrypt.Examples.CPAFromPRF.cpa_from_prf_bound","kind":"theorem","summary":"∀ (P : CatCrypt.Examples.CPAFromPRF.PRFEncScheme) (H : CatCrypt.Crypto.PRFAssumption P.F) (m₀ m…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CPAFromPRF.cpa_from_prf_bound","module":"CatCryptCore.Examples.CPAFromPRF"},{"id":"n43017","layer":"formal","project":"p53","title":"CatCrypt.Examples.CTRMode.CTRBlockScheme","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CTRMode.CTRBlockScheme","module":"CatCryptCore.Examples.CTRMode"},{"id":"n43018","layer":"formal","project":"p53","title":"CatCrypt.Examples.CTRMode.CTRBlockScheme.toBijCPAScheme","kind":"def","summary":"CatCrypt.Examples.CTRMode.CTRBlockScheme → CatCrypt.Examples.CPAFromPRF.BijCPAScheme","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CTRMode.CTRBlockScheme.toBijCPAScheme","module":"CatCryptCore.Examples.CTRMode"},{"id":"n43019","layer":"formal","project":"p53","title":"CatCrypt.Examples.CTRMode.boolCTR_perfect_indcpa","kind":"theorem","summary":"∀ (m₀ m₁ : CatCrypt.Examples.CTRMode.boolCTR.toEncScheme.Plaintext) (A : CatCrypt.Examples.CTRM…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CTRMode.boolCTR_perfect_indcpa","module":"CatCryptCore.Examples.CTRMode"},{"id":"n43020","layer":"formal","project":"p53","title":"CatCrypt.Examples.CTRMode.ctr_indcpa_bound","kind":"theorem","summary":"∀ (C : CatCrypt.Examples.CTRMode.CTRBlockScheme) F : CatCrypt.Crypto.SecurityDefs.PRFScheme (H…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CTRMode.ctr_indcpa_bound","module":"CatCryptCore.Examples.CTRMode"},{"id":"n43021","layer":"formal","project":"p53","title":"CatCrypt.Examples.CTRMode.ctr_perfect_indcpa","kind":"theorem","summary":"∀ (C : CatCrypt.Examples.CTRMode.CTRBlockScheme) (m₀ m₁ : C.toEncScheme.Plaintext) (A : C.toEnc…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.CTRMode.ctr_perfect_indcpa","module":"CatCryptCore.Examples.CTRMode"},{"id":"n43022","layer":"formal","project":"p53","title":"CatCrypt.Examples.Commitment.CommScheme","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.Commitment.CommScheme","module":"CatCryptCore.Examples.Commitment"},{"id":"n43023","layer":"formal","project":"p53","title":"CatCrypt.Examples.Commitment.idComm_perfectly_binding","kind":"theorem","summary":"CatCrypt.Examples.Commitment.IdComm.PerfectlyBinding","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.Commitment.idComm_perfectly_binding","module":"CatCryptCore.Examples.Commitment"},{"id":"n43024","layer":"formal","project":"p53","title":"CatCrypt.Examples.Commitment.maskComm_perfect_hiding","kind":"theorem","summary":"∀ (m₀ m₁ : Bool) (A : CatCrypt.Examples.Commitment.MaskComm.Commitment → CatCrypt.Core.SPComp B…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.Commitment.maskComm_perfect_hiding","module":"CatCryptCore.Examples.Commitment"},{"id":"n43025","layer":"formal","project":"p53","title":"CatCrypt.Examples.DiffieHellman.KA_Advantage","kind":"def","summary":"(P : CatCrypt.Crypto.PairingGroup) → CatCrypt.Examples.DiffieHellman.KA_Distinguisher CatCrypt.…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.DiffieHellman.KA_Advantage","module":"CatCryptCore.Examples.DiffieHellman"},{"id":"n43026","layer":"formal","project":"p53","title":"CatCrypt.Examples.DiffieHellman.KA_Game_Real","kind":"def","summary":"(P : CatCrypt.Crypto.PairingGroup) → CatCrypt.Examples.DiffieHellman.KA_Distinguisher CatCrypt.…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.DiffieHellman.KA_Game_Real","module":"CatCryptCore.Examples.DiffieHellman"},{"id":"n43027","layer":"formal","project":"p53","title":"CatCrypt.Examples.DiffieHellman.ka_advantage_eq_ddh","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (D : CatCrypt.Examples.DiffieHellman.KA_Distinguisher CatCry…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.DiffieHellman.ka_advantage_eq_ddh","module":"CatCryptCore.Examples.DiffieHellman"},{"id":"n43028","layer":"formal","project":"p53","title":"CatCrypt.Examples.DiffieHellman.ka_secure_under_ddh","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (ε : ENNReal), (∀ (A : CatCrypt.Crypto.Assumptions.DDH_Adver…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.DiffieHellman.ka_secure_under_ddh","module":"CatCryptCore.Examples.DiffieHellman"},{"id":"n43029","layer":"formal","project":"p53","title":"CatCrypt.Examples.ElGamal.elgamalDDH","kind":"def","summary":"(P : CatCrypt.Crypto.PairingGroup) → CatCrypt.Crypto.Assumptions.DDHDef CatCrypt.Crypto.Pairing…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.ElGamal.elgamalDDH","module":"CatCryptCore.Examples.ElGamal"},{"id":"n43030","layer":"formal","project":"p53","title":"CatCrypt.Examples.ElGamal.elgamalEnc","kind":"def","summary":"CatCrypt.Crypto.PairingGroup → CatCrypt.Crypto.EncScheme","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.ElGamal.elgamalEnc","module":"CatCryptCore.Examples.ElGamal"},{"id":"n43031","layer":"formal","project":"p53","title":"CatCrypt.Examples.ElGamal.elgamal_dec_enc","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (sk r : ZMod CatCrypt.Crypto.PairingGroup.p) (m : CatCrypt.C…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.ElGamal.elgamal_dec_enc","module":"CatCryptCore.Examples.ElGamal"},{"id":"n43032","layer":"formal","project":"p53","title":"CatCrypt.Examples.ElGamal.elgamal_indcpa_eq_ddh","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (m₀ m₁ : CatCrypt.Crypto.PairingGroup.G₁) (A : Prod CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.ElGamal.elgamal_indcpa_eq_ddh","module":"CatCryptCore.Examples.ElGamal"},{"id":"n43033","layer":"formal","project":"p53","title":"CatCrypt.Examples.ElGamal.elgamal_indcpa_le_ddh","kind":"theorem","summary":"∀ P : CatCrypt.Crypto.PairingGroup (m₀ m₁ : CatCrypt.Crypto.PairingGroup.G₁) (A : Prod CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.ElGamal.elgamal_indcpa_le_ddh","module":"CatCryptCore.Examples.ElGamal"},{"id":"n43034","layer":"formal","project":"p53","title":"CatCrypt.Examples.EncryptThenMAC.EtM","kind":"def","summary":"(E : CatCrypt.Crypto.EncScheme) → (MacKey Tag : Type) → [Fintype MacKey] → [Nonempty MacKey] →…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.EncryptThenMAC.EtM","module":"CatCryptCore.Examples.EncryptThenMAC"},{"id":"n43035","layer":"formal","project":"p53","title":"CatCrypt.Examples.EncryptThenMAC.EtM_correct","kind":"theorem","summary":"∀ (E : CatCrypt.Crypto.EncScheme) (MacKey Tag : Type) [inst : Fintype MacKey] [inst_1 : Nonempt…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.EncryptThenMAC.EtM_correct","module":"CatCryptCore.Examples.EncryptThenMAC"},{"id":"n43036","layer":"formal","project":"p53","title":"CatCrypt.Examples.EncryptThenMAC.boolEtM_perfect_indcpa","kind":"theorem","summary":"∀ (m₀ m₁ : Bool) (A : CatCrypt.Examples.EncryptThenMAC.BoolEtM.Ciphertext → CatCrypt.Core.SPCom…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.EncryptThenMAC.boolEtM_perfect_indcpa","module":"CatCryptCore.Examples.EncryptThenMAC"},{"id":"n43037","layer":"formal","project":"p53","title":"CatCrypt.Examples.EtMCCA.boolEtM_indcca_reduces","kind":"theorem","summary":"∀ (m₀ m₁ : Bool) (A : (CatCrypt.Examples.EncryptThenMAC.BoolEtM.Ciphertext → CatCrypt.Core.SPCo…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.EtMCCA.boolEtM_indcca_reduces","module":"CatCryptCore.Examples.EtMCCA"},{"id":"n43038","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.DEMScheme","kind":"inductive","summary":"Type → Type → Type → Type","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.DEMScheme","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43039","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.HybridPKE","kind":"def","summary":"PKey SKey Key EKey Plain Cipher : Type → CatCrypt.Examples.KEMDEM.KEMScheme PKey SKey Key EKey…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.HybridPKE","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43040","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.KEMScheme","kind":"inductive","summary":"Type → Type → Type → Type → Type","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.KEMScheme","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43041","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.hybridPKE_correct","kind":"theorem","summary":"∀ PKey SKey Key EKey Plain Cipher : Type (KEM : CatCrypt.Examples.KEMDEM.KEMScheme PKey SKey Ke…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.hybridPKE_correct","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43042","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.pke_perfect_security","kind":"theorem","summary":"∀ PKey SKey Key EKey Plain Cipher : Type [inst : Fintype Key] [inst_1 : Nonempty Key] [Fintype…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.pke_perfect_security","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43043","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.pke_security","kind":"theorem","summary":"∀ PKey SKey Key EKey Plain Cipher : Type [inst : Fintype Key] [inst_1 : Nonempty Key] [Fintype…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.pke_security","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43044","layer":"formal","project":"p53","title":"CatCrypt.Examples.KEMDEM.xorHybrid_advantage_zero","kind":"theorem","summary":"∀ PKey SKey EKey : Type [Fintype EKey] [Nonempty EKey] (KEM : CatCrypt.Examples.KEMDEM.KEMSchem…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.KEMDEM.xorHybrid_advantage_zero","module":"CatCryptCore.Examples.KEMDEM"},{"id":"n43045","layer":"formal","project":"p53","title":"CatCrypt.Examples.MAC.BijMACFamily","kind":"inductive","summary":"Type 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B.Tag), Eq (CatCryp…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.MAC.bijMAC_forgery_prob","module":"CatCryptCore.Examples.MAC"},{"id":"n43049","layer":"formal","project":"p53","title":"CatCrypt.Examples.MAC.boolXorMAC","kind":"def","summary":"CatCrypt.Examples.MAC.BijMACFamily","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.MAC.boolXorMAC","module":"CatCryptCore.Examples.MAC"},{"id":"n43050","layer":"formal","project":"p53","title":"CatCrypt.Examples.MAC.boolXorMAC_forgery_prob","kind":"theorem","summary":"∀ (m m_star t_star : Bool), Eq (CatCrypt.Crypto.prTrue 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ENNR…","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.PRF.cascade_prf_bound","module":"CatCryptCore.Examples.PRF"},{"id":"n43062","layer":"formal","project":"p53","title":"CatCrypt.Examples.PRG.BijPRG","kind":"inductive","summary":"Type 1","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.PRG.BijPRG","module":"CatCryptCore.Examples.PRG"},{"id":"n43063","layer":"formal","project":"p53","title":"CatCrypt.Examples.PRG.BijPRG.toPRGScheme","kind":"def","summary":"CatCrypt.Examples.PRG.BijPRG → CatCrypt.Examples.PRG.PRGScheme","labels":[],"detail_key":"p53","name":"CatCrypt.Examples.PRG.BijPRG.toPRGScheme","module":"CatCryptCore.Examples.PRG"},{"id":"n43064","layer":"formal","project":"p53","title":"CatCrypt.Examples.PRG.PRGScheme","kind":"inductive","summary":"Type 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CatCrypt.Prob.SD…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.bind_assoc","module":"CatCryptCore.Prob.SDistr"},{"id":"n43117","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.bind_congr_support","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 d : CatCrypt.Prob.SDistr α f g : α → CatCrypt.Prob.SDistr β, (∀ (a…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.bind_congr_support","module":"CatCryptCore.Prob.SDistr"},{"id":"n43118","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.bind_pure","kind":"theorem","summary":"∀ α : Type u_1 (d : CatCrypt.Prob.SDistr α), Eq (d.bind CatCrypt.Prob.SDistr.pure) d","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.bind_pure","module":"CatCryptCore.Prob.SDistr"},{"id":"n43119","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.fail","kind":"def","summary":"α : Type u_1 → CatCrypt.Prob.SDistr α","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.fail","module":"CatCryptCore.Prob.SDistr"},{"id":"n43120","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.mass","kind":"def","summary":"α : Type u_1 → CatCrypt.Prob.SDistr α → ENNReal","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.mass","module":"CatCryptCore.Prob.SDistr"},{"id":"n43121","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.pure","kind":"def","summary":"α : Type u_1 → α → CatCrypt.Prob.SDistr α","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.pure","module":"CatCryptCore.Prob.SDistr"},{"id":"n43122","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.pure_bind","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 (a : α) (f : α → CatCrypt.Prob.SDistr β), Eq ((CatCrypt.Prob.SDistr…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.pure_bind","module":"CatCryptCore.Prob.SDistr"},{"id":"n43123","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.support","kind":"def","summary":"α : Type u_1 → CatCrypt.Prob.SDistr α → Set α","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.support","module":"CatCryptCore.Prob.SDistr"},{"id":"n43124","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.uniform","kind":"def","summary":"(α : Type u_4) → [Fintype α] → [Nonempty α] → CatCrypt.Prob.SDistr α","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.uniform","module":"CatCryptCore.Prob.SDistr"},{"id":"n43125","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.uniform_bind_bij","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : Nonempty α] [inst…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.uniform_bind_bij","module":"CatCryptCore.Prob.SDistr"},{"id":"n43126","layer":"formal","project":"p53","title":"CatCrypt.Prob.poly_identity_test","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (φ ψ : Polynomial (ZMod p)), Ne φ ψ → LE.le (CatCrypt.C…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.poly_identity_test","module":"CatCryptCore.Prob.SchwartzZippel"},{"id":"n43127","layer":"formal","project":"p53","title":"CatCrypt.Prob.schwartz_zippel","kind":"theorem","summary":"∀ (p : Nat) [inst : Fact (Nat.Prime p)] (φ : Polynomial (ZMod p)), Ne φ 0 → LE.le (CatCrypt.Cry…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.schwartz_zippel","module":"CatCryptCore.Prob.SchwartzZippel"},{"id":"n43128","layer":"formal","project":"p53","title":"CatCrypt.Prob.SDistr.bind_support_witness","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 d : CatCrypt.Prob.SDistr α f : α → CatCrypt.Prob.SDistr β b : β, Ne…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.SDistr.bind_support_witness","module":"CatCryptCore.Prob.Support"},{"id":"n43129","layer":"formal","project":"p53","title":"CatCrypt.Prob.XorBij.boolXorBij","kind":"def","summary":"Bool → Equiv Bool Bool","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.XorBij.boolXorBij","module":"CatCryptCore.Prob.XorBij"},{"id":"n43130","layer":"formal","project":"p53","title":"CatCrypt.Prob.XorBij.boolXorBij_symm","kind":"theorem","summary":"∀ (x : Bool), Eq (CatCrypt.Prob.XorBij.boolXorBij x).symm (CatCrypt.Prob.XorBij.boolXorBij x)","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.XorBij.boolXorBij_symm","module":"CatCryptCore.Prob.XorBij"},{"id":"n43131","layer":"formal","project":"p53","title":"CatCrypt.Prob.XorBij.rHoare_sample_bij","kind":"theorem","summary":"∀ α : Type [inst : Fintype α] [inst_1 : Nonempty α] (f : Equiv α α), CatCrypt.Relational.rHoare…","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.XorBij.rHoare_sample_bij","module":"CatCryptCore.Prob.XorBij"},{"id":"n43132","layer":"formal","project":"p53","title":"CatCrypt.Prob.XorBij.xorEquiv","kind":"def","summary":"α : Type → (op : α → α → α) → (∀ (a b : α), Eq (op (op a b) b) a) → α → Equiv α α","labels":[],"detail_key":"p53","name":"CatCrypt.Prob.XorBij.xorEquiv","module":"CatCryptCore.Prob.XorBij"},{"id":"n43133","layer":"formal","project":"p53","title":"CatCrypt.Relational.RPost","kind":"def","summary":"Type u_1 → Type u_2 → Type (max u_1 u_2)","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.RPost","module":"CatCryptCore.Relational.Basic"},{"id":"n43134","layer":"formal","project":"p53","title":"CatCrypt.Relational.RPre","kind":"def","summary":"Type","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.RPre","module":"CatCryptCore.Relational.Basic"},{"id":"n43135","layer":"formal","project":"p53","title":"CatCrypt.Relational.eqPost","kind":"def","summary":"α : Type u_1 → CatCrypt.Relational.RPost α α","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.eqPost","module":"CatCryptCore.Relational.Basic"},{"id":"n43136","layer":"formal","project":"p53","title":"CatCrypt.Relational.eqPre","kind":"def","summary":"CatCrypt.Relational.RPre","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.eqPre","module":"CatCryptCore.Relational.Basic"},{"id":"n43137","layer":"formal","project":"p53","title":"CatCrypt.Relational.DependsOn","kind":"def","summary":"CatCrypt.Relational.RPre → CatCrypt.Core.LocSet → Prop","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.DependsOn","module":"CatCryptCore.Relational.Frame"},{"id":"n43138","layer":"formal","project":"p53","title":"CatCrypt.Relational.PreservesOutside","kind":"def","summary":"α : Type → CatCrypt.Core.SPComp α → CatCrypt.Core.LocSet → Prop","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.PreservesOutside","module":"CatCryptCore.Relational.Frame"},{"id":"n43139","layer":"formal","project":"p53","title":"CatCrypt.Relational.r_frame_local","kind":"theorem","summary":"∀ α β : Type Φ Frame : CatCrypt.Relational.RPre Ψ : CatCrypt.Relational.RPost α β c₁ : CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.r_frame_local","module":"CatCryptCore.Relational.Frame"},{"id":"n43140","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare","kind":"def","summary":"α : Type u_1 → β : Type u_2 → CatCrypt.Relational.RPre → CatCrypt.Core.SPComp α → CatCrypt.Core…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare","module":"CatCryptCore.Relational.Judgment"},{"id":"n43141","layer":"formal","project":"p53","title":"CatCrypt.Relational.advantage_factorization_comm","kind":"theorem","summary":"∀ α β γ : Type (pfx : CatCrypt.Core.SPComp α) (G₁ G₂ : CatCrypt.Core.SPComp β) (k : α → β → Cat…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.advantage_factorization_comm","module":"CatCryptCore.Relational.Reorder"},{"id":"n43142","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_reorder_pure_l","kind":"theorem","summary":"∀ α β γ δ : Type c : CatCrypt.Core.SPComp α d : CatCrypt.Core.SPComp β k : α → β → CatCrypt.Cor…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_reorder_pure_l","module":"CatCryptCore.Relational.Reorder"},{"id":"n43143","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_sample_comm","kind":"theorem","summary":"∀ α β γ : Type [inst : Fintype α] [inst_1 : Fintype β] [inst_2 : Nonempty α] [inst_3 : Nonempty…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_sample_comm","module":"CatCryptCore.Relational.Reorder"},{"id":"n43144","layer":"formal","project":"p53","title":"CatCrypt.Relational.advantage_factorization","kind":"theorem","summary":"∀ α β : Type (f : α → CatCrypt.Core.SPComp β) (G₁ G₂ : CatCrypt.Core.SPComp α) (A : β → CatCryp…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.advantage_factorization","module":"CatCryptCore.Relational.Rules"},{"id":"n43145","layer":"formal","project":"p53","title":"CatCrypt.Relational.advantage_symm","kind":"theorem","summary":"∀ (G₁ G₂ : CatCrypt.Core.SPComp Bool), Eq (CatCrypt.Crypto.Advantage G₁ G₂) (CatCrypt.Crypto.Ad…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.advantage_symm","module":"CatCryptCore.Relational.Rules"},{"id":"n43146","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_bind","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 γ : Type u_3 δ : Type u_4 Φ : CatCrypt.Relational.RPre Ψ : CatCrypt…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_bind","module":"CatCryptCore.Relational.Rules"},{"id":"n43147","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_conseq","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 Φ Φ' : CatCrypt.Relational.RPre Ψ Ψ' : CatCrypt.Relational.RPost α…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_conseq","module":"CatCryptCore.Relational.Rules"},{"id":"n43148","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_refl","kind":"theorem","summary":"∀ α : Type u_1 (c : CatCrypt.Core.SPComp α), CatCrypt.Relational.rHoare CatCrypt.Relational.eqP…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_refl","module":"CatCryptCore.Relational.Rules"},{"id":"n43149","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_sample_bij","kind":"theorem","summary":"∀ Φ : CatCrypt.Relational.RPre (α : Type u_5) (β : Type u_6) [inst : Fintype α] [inst_1 : Finty…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_sample_bij","module":"CatCryptCore.Relational.Rules"},{"id":"n43150","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_sample_same","kind":"theorem","summary":"∀ Φ : CatCrypt.Relational.RPre (α : Type u_5) [inst : Fintype α] [inst_1 : Nonempty α], CatCryp…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_sample_same","module":"CatCryptCore.Relational.Rules"},{"id":"n43151","layer":"formal","project":"p53","title":"CatCrypt.Relational.rHoare_symm","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 Φ : CatCrypt.Relational.RPre Ψ : CatCrypt.Relational.RPost α β c₁ :…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.rHoare_symm","module":"CatCryptCore.Relational.Rules"},{"id":"n43152","layer":"formal","project":"p53","title":"CatCrypt.Relational.r_frame","kind":"theorem","summary":"∀ α : Type u_1 β : Type u_2 Φ Ψ : CatCrypt.Relational.RPre Θ : CatCrypt.Relational.RPost α β c₁…","labels":[],"detail_key":"p53","name":"CatCrypt.Relational.r_frame","module":"CatCryptCore.Relational.Sync"},{"id":"n43153","layer":"formal","project":"p53","title":"CatCrypt.Crypto.advantageA_triangle","kind":"theorem","summary":"∀ α : Type (G₀ G₁ G₂ : CatCrypt.Core.SPComp α) (A : α → CatCrypt.Core.SPComp Bool), LE.le (CatC…","labels":[],"detail_key":"p53","name":"CatCrypt.Crypto.advantageA_triangle","module":"CatCryptCore.Tactics.Triangle"},{"id":"n43154","layer":"informal","project":"p54","title":"Magma","kind":"definition","summary":"[Magma] A \\emphmagma is a set G equipped with a binary operation \\diamond: G \\times G \\to G. A…","labels":["magma-def"],"detail_key":"p54"},{"id":"n43155","layer":"informal","project":"p54","title":"Free Magma","kind":"definition","summary":"[Free Magma] The \\emphfree magma M_X generated by a set X (which we call an \\emphalphabet) is t…","labels":["free-magma-def"],"detail_key":"p54"},{"id":"n43156","layer":"informal","project":"p54","title":"FreeMagma.elementsOfNumNodesEq_card_eq_catalan_mul_pow","kind":"lemma","summary":"For a finite alphabet X, the number of words of order n is C_n |X|^n+1, where C_n is the n^th C…","labels":[],"detail_key":"p54"},{"id":"n43157","layer":"informal","project":"p54","title":"Follows from standard properties of Catalan numbers.","kind":"proof","summary":"Follows from standard properties of Catalan numbers.","labels":[],"detail_key":"p54"},{"id":"n43158","layer":"informal","project":"p54","title":"Induced homomorphism","kind":"definition","summary":"[Induced homomorphism] Given a function f: X \\to G from an alphabet X to a magma G, the \\emphin…","labels":["induced-def"],"detail_key":"p54"},{"id":"n43159","layer":"informal","project":"p54","title":"Law","kind":"definition","summary":"[Law] Let X be a set. A \\emphlaw with alphabet X is a formal expression of the form w \\simeq w'…","labels":["law-def"],"detail_key":"p54"},{"id":"n43160","layer":"informal","project":"p54","title":"Models","kind":"definition","summary":"[Models] A \\emphtheory is a set \\Gamma of laws. Given a theory \\Gamma, a magma G is a \\emphmode…","labels":["models-def"],"detail_key":"p54"},{"id":"n43161","layer":"informal","project":"p54","title":"Derivation","kind":"definition","summary":"[Derivation] Given a theory \\Gamma and a law w\\simeq w' over a fixed alphabet X, we say that \\G…","labels":["derivation-def"],"detail_key":"p54"},{"id":"n43162","layer":"informal","project":"p54","title":"Birkhoff's completeness theorem","kind":"theorem","summary":"[Birkhoff's completeness theorem] For any theory \\Gamma and words w, w' over a fixed alphabet \\…","labels":["sound-complete"],"detail_key":"p54"},{"id":"n43163","layer":"informal","project":"p54","title":"(Sketch) The `only if' component is soundness, and follows from verifying that the rules…","kind":"proof","summary":"(Sketch) The `only if' component is soundness, and follows from verifying that the rules of inf…","labels":[],"detail_key":"p54"},{"id":"n43164","layer":"informal","project":"p54","title":"Compactness theorem","kind":"corollary","summary":"[Compactness theorem] Let \\Gamma be a theory, and let E be a law. Then \\Gamma \\models E if and…","labels":["compactness-thm"],"detail_key":"p54"},{"id":"n43165","layer":"informal","project":"p54","title":"The claim is obvious for \\models, and the claim then follows from \\Crefsound-complete.","kind":"proof","summary":"The claim is obvious for \\models, and the claim then follows from \\Crefsound-complete.","labels":[],"detail_key":"p54"},{"id":"n43166","layer":"informal","project":"p54","title":"Pushforward","kind":"lemma","summary":"[Pushforward] Let w \\simeq w' be a law with some alphabet X, G be a magma, and \\pi: X \\to Y be…","labels":["push"],"detail_key":"p54"},{"id":"n43167","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43168","layer":"informal","project":"p54","title":"Equivalence","kind":"lemma","summary":"[Equivalence] Let G be a magma and X be an alphabet. Then the relation G \\models w \\simeq w' is…","labels":["equiv"],"detail_key":"p54"},{"id":"n43169","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43170","layer":"informal","project":"p54","title":"Counting laws up to relabeling","kind":"lemma","summary":"[Counting laws up to relabeling] Up to relabeling, the number of laws w \\simeq w' of total orde…","labels":["law-count"],"detail_key":"p54"},{"id":"n43171","layer":"informal","project":"p54","title":"Follows from the properties of Catalan and Bell numbers.","kind":"proof","summary":"Follows from the properties of Catalan and Bell numbers.","labels":[],"detail_key":"p54"},{"id":"n43172","layer":"informal","project":"p54","title":"Counting laws up to relabeling and symmetry","kind":"lemma","summary":"[Counting laws up to relabeling and symmetry] Up to relabeling and symmetry, the number of laws…","labels":["law-count-sym"],"detail_key":"p54"},{"id":"n43173","layer":"informal","project":"p54","title":"Elementary counting.","kind":"proof","summary":"Elementary counting.","labels":[],"detail_key":"p54"},{"id":"n43174","layer":"informal","project":"p54","title":"Counting laws up to relabeling, symmetry, and triviality","kind":"lemma","summary":"[Counting laws up to relabeling, symmetry, and triviality] Up to relabeling, symmetry, and triv…","labels":["law-count-triv"],"detail_key":"p54"},{"id":"n43175","layer":"informal","project":"p54","title":"Routine counting.","kind":"proof","summary":"Routine counting.","labels":[],"detail_key":"p54"},{"id":"n43176","layer":"informal","project":"p54","title":"Equation 1","kind":"definition","summary":"[Equation 1] Equation 1 is the law 0 \\simeq0 (or the equation x=x). \\expandafter\\gdef\\csname Eq…","labels":["eq1"],"detail_key":"p54"},{"id":"n43177","layer":"informal","project":"p54","title":"Equation 2","kind":"definition","summary":"[Equation 2] Equation 2 is the law 0 \\simeq1 (or the equation x=y). \\expandafter\\gdef\\csname Eq…","labels":["eq2"],"detail_key":"p54"},{"id":"n43178","layer":"informal","project":"p54","title":"Equation 3","kind":"definition","summary":"[Equation 3] Equation 3 is the law 0 \\simeq0 \\diamond0 (or the equation x = x \\diamond x). \\exp…","labels":["eq3"],"detail_key":"p54"},{"id":"n43179","layer":"informal","project":"p54","title":"Equation 4","kind":"definition","summary":"[Equation 4] Equation 4 is the law 0 \\simeq0 \\diamond1 (or the equation x = x \\diamond y). \\exp…","labels":["eq4"],"detail_key":"p54"},{"id":"n43180","layer":"informal","project":"p54","title":"Equation 5","kind":"definition","summary":"[Equation 5] Equation 5 is the law 0 \\simeq1 \\diamond0 (or the equation x = y \\diamond x). \\exp…","labels":["eq5"],"detail_key":"p54"},{"id":"n43181","layer":"informal","project":"p54","title":"Equation 6","kind":"definition","summary":"[Equation 6] Equation 6 is the law 0 \\simeq1 \\diamond1 (or the equation x = y \\diamond y). \\exp…","labels":["eq6"],"detail_key":"p54"},{"id":"n43182","layer":"informal","project":"p54","title":"Equation 7","kind":"definition","summary":"[Equation 7] Equation 7 is the law 0 \\simeq1 \\diamond2 (or the equation x = y \\diamond z). \\exp…","labels":["eq7"],"detail_key":"p54"},{"id":"n43183","layer":"informal","project":"p54","title":"Equation 8","kind":"definition","summary":"[Equation 8] Equation 8 is the law 0 \\simeq0 \\diamond(0 \\diamond0) (or the equation x = x \\diam…","labels":["eq8"],"detail_key":"p54"},{"id":"n43184","layer":"informal","project":"p54","title":"Equation 14","kind":"definition","summary":"[Equation 14] Equation 14 is the law 0 \\simeq1 \\diamond(0 \\diamond1) (or the equation x = y \\di…","labels":["eq14"],"detail_key":"p54"},{"id":"n43185","layer":"informal","project":"p54","title":"Equation 16","kind":"definition","summary":"[Equation 16] Equation 16 is the law 0 \\simeq1 \\diamond(1 \\diamond0) (or the equation x = y \\di…","labels":["eq16"],"detail_key":"p54"},{"id":"n43186","layer":"informal","project":"p54","title":"Equation 23","kind":"definition","summary":"[Equation 23] Equation 23 is the law 0 \\simeq(0 \\diamond0) \\diamond0 (or the equation x = (x \\d…","labels":["eq23"],"detail_key":"p54"},{"id":"n43187","layer":"informal","project":"p54","title":"Equation 29","kind":"definition","summary":"[Equation 29] Equation 29 is the law 0 \\simeq(1 \\diamond0) \\diamond1 (or the equation x = (y \\d…","labels":["eq29"],"detail_key":"p54"},{"id":"n43188","layer":"informal","project":"p54","title":"Equation 38","kind":"definition","summary":"[Equation 38] Equation 38 is the law 0 \\diamond0 \\simeq0 \\diamond1 (or the equation x \\diamond…","labels":["eq38"],"detail_key":"p54"},{"id":"n43189","layer":"informal","project":"p54","title":"Equation 39","kind":"definition","summary":"[Equation 39] Equation 39 is the law 0 \\diamond0 \\simeq1 \\diamond0 (or the equation x \\diamond…","labels":["eq39"],"detail_key":"p54"},{"id":"n43190","layer":"informal","project":"p54","title":"Equation 40","kind":"definition","summary":"[Equation 40] Equation 40 is the law 0 \\diamond0 \\simeq1 \\diamond1 (or the equation x \\diamond…","labels":["eq40"],"detail_key":"p54"},{"id":"n43191","layer":"informal","project":"p54","title":"Equation 41","kind":"definition","summary":"[Equation 41] Equation 41 is the law 0 \\diamond0 \\simeq1 \\diamond2 (or the equation x \\diamond…","labels":["eq41"],"detail_key":"p54"},{"id":"n43192","layer":"informal","project":"p54","title":"Equation 42","kind":"definition","summary":"[Equation 42] Equation 42 is the law 0 \\diamond1 \\simeq0 \\diamond2 (or the equation x \\diamond…","labels":["eq42"],"detail_key":"p54"},{"id":"n43193","layer":"informal","project":"p54","title":"Equation 43","kind":"definition","summary":"[Equation 43] Equation 43 is the law 0 \\diamond1 \\simeq1 \\diamond0 (or the equation x \\diamond…","labels":["eq43"],"detail_key":"p54"},{"id":"n43194","layer":"informal","project":"p54","title":"Equation 45","kind":"definition","summary":"[Equation 45] Equation 45 is the law 0 \\diamond1 \\simeq2 \\diamond1 (or the equation x \\diamond…","labels":["eq45"],"detail_key":"p54"},{"id":"n43195","layer":"informal","project":"p54","title":"Equation 46","kind":"definition","summary":"[Equation 46] Equation 46 is the law 0 \\diamond1 \\simeq2 \\diamond3 (or the equation x \\diamond…","labels":["eq46"],"detail_key":"p54"},{"id":"n43196","layer":"informal","project":"p54","title":"Equation 63","kind":"definition","summary":"[Equation 63] Equation 63 is the law 0 \\simeq1 \\diamond(0 \\diamond(0 \\diamond1)) (or the equati…","labels":["eq63"],"detail_key":"p54"},{"id":"n43197","layer":"informal","project":"p54","title":"Equation 65","kind":"definition","summary":"[Equation 65] Equation 65 is the law 0 \\simeq1 \\diamond(0 \\diamond(1 \\diamond0)) (or the equati…","labels":["eq65"],"detail_key":"p54"},{"id":"n43198","layer":"informal","project":"p54","title":"Equation 168","kind":"definition","summary":"[Equation 168] Equation 168 is the law 0 \\simeq(1 \\diamond0) \\diamond(0 \\diamond2) (or the equa…","labels":["eq168"],"detail_key":"p54"},{"id":"n43199","layer":"informal","project":"p54","title":"Equation 206","kind":"definition","summary":"[Equation 206] Equation 206 is the law 0 \\simeq(0 \\diamond(0 \\diamond1)) \\diamond1 (or the equa…","labels":["eq206"],"detail_key":"p54"},{"id":"n43200","layer":"informal","project":"p54","title":"Equation 381","kind":"definition","summary":"[Equation 381] Equation 381 is the law 0 \\diamond1 \\simeq(0 \\diamond2) \\diamond1 (or the equati…","labels":["eq381"],"detail_key":"p54"},{"id":"n43201","layer":"informal","project":"p54","title":"Equation 387","kind":"definition","summary":"[Equation 387] Equation 387 is the law 0 \\diamond1 \\simeq(1 \\diamond1) \\diamond0 (or the equati…","labels":["eq387"],"detail_key":"p54"},{"id":"n43202","layer":"informal","project":"p54","title":"Equation 477","kind":"definition","summary":"[Equation 477] Equation 477 is the law 0 \\simeq1 \\diamond(0 \\diamond(1 \\diamond(1 \\diamond1)))…","labels":["eq477"],"detail_key":"p54"},{"id":"n43203","layer":"informal","project":"p54","title":"Equation 854","kind":"definition","summary":"[Equation 854] Equation 854 is the law 0 = 0 \\diamond((1 \\diamond2) \\diamond(0 \\diamond2)) (or…","labels":["eq854"],"detail_key":"p54"},{"id":"n43204","layer":"informal","project":"p54","title":"Equation 953","kind":"definition","summary":"[Equation 953] Equation 953 is the law 0 = 1 \\diamond((2 \\diamond0) \\diamond(2 \\diamond2)) (or…","labels":["eq953"],"detail_key":"p54"},{"id":"n43205","layer":"informal","project":"p54","title":"Equation 1485","kind":"definition","summary":"[Equation 1485] Equation 1485 is the law 0 \\simeq(1 \\diamond0) \\diamond(0 \\diamond(2 \\diamond1)…","labels":["eq1485"],"detail_key":"p54"},{"id":"n43206","layer":"informal","project":"p54","title":"Equation 1491","kind":"definition","summary":"[Equation 1491] Equation 1491 is the law 0 \\simeq(1 \\diamond0) \\diamond(1 \\diamond(1 \\diamond0)…","labels":["eq1491"],"detail_key":"p54"},{"id":"n43207","layer":"informal","project":"p54","title":"Equation 1571","kind":"definition","summary":"[Equation 1571] Equation 1571 is the law 0 \\simeq(1 \\diamond2) \\diamond(1 \\diamond(0 \\diamond2)…","labels":["eq1571"],"detail_key":"p54"},{"id":"n43208","layer":"informal","project":"p54","title":"Equation 1648","kind":"definition","summary":"[Equation 1648] Equation 1648 is the law 0 \\simeq(0 \\diamond1) \\diamond((0 \\diamond1) \\diamond1…","labels":["eq1648"],"detail_key":"p54"},{"id":"n43209","layer":"informal","project":"p54","title":"Equation 1657","kind":"definition","summary":"[Equation 1657] Equation 1657 is the law 0 \\simeq(0 \\diamond1) \\diamond((1 \\diamond1) \\diamond0…","labels":["eq1657"],"detail_key":"p54"},{"id":"n43210","layer":"informal","project":"p54","title":"Equation 1659","kind":"definition","summary":"[Equation 1659] Equation 1659 is the law 0 \\simeq(0 \\diamond1) \\diamond((1 \\diamond1) \\diamond2…","labels":["eq1659"],"detail_key":"p54"},{"id":"n43211","layer":"informal","project":"p54","title":"Equation 1661","kind":"definition","summary":"[Equation 1661] Equation 1661 is the law 0 \\simeq(0 \\diamond1) \\diamond((1 \\diamond2) \\diamond1…","labels":["eq1661"],"detail_key":"p54"},{"id":"n43212","layer":"informal","project":"p54","title":"Equation 1689","kind":"definition","summary":"[Equation 1689] Equation 1689 is the law 0 \\simeq(1 \\diamond0) \\diamond((0 \\diamond2) \\diamond2…","labels":["eq1689"],"detail_key":"p54"},{"id":"n43213","layer":"informal","project":"p54","title":"Equation 1701","kind":"definition","summary":"[Equation 1701] Equation 1701 is the law 0 \\simeq(1 \\diamond x) \\diamond((2 \\diamond0) \\diamond…","labels":["eq1701"],"detail_key":"p54"},{"id":"n43214","layer":"informal","project":"p54","title":"Equation 2662","kind":"definition","summary":"[Equation 2662] Equation 2662 is the law 0 \\simeq((0 \\diamond1) \\diamond(0 \\diamond1)) \\diamond…","labels":["eq2662"],"detail_key":"p54"},{"id":"n43215","layer":"informal","project":"p54","title":"Equation 3167","kind":"definition","summary":"[Equation 3167] Equation 3167 is the law 0 \\simeq(((1 \\diamond1) \\diamond2) \\diamond2) \\diamond…","labels":["eq3167"],"detail_key":"p54"},{"id":"n43216","layer":"informal","project":"p54","title":"Equation 3588","kind":"definition","summary":"[Equation 3588] Equation 3588 is the law 0 \\diamond1 \\simeq2 \\diamond((0 \\diamond1) \\diamond2)…","labels":["eq3588"],"detail_key":"p54"},{"id":"n43217","layer":"informal","project":"p54","title":"Equation 3722","kind":"definition","summary":"[Equation 3722] Equation 3722 is the law 0 \\diamond1 \\simeq(0 \\diamond1) \\diamond(0 \\diamond1)…","labels":["eq3722"],"detail_key":"p54"},{"id":"n43218","layer":"informal","project":"p54","title":"Equation 3744","kind":"definition","summary":"[Equation 3744] Equation 3744 is the law 0 \\diamond1 \\simeq(0 \\diamond2) \\diamond(3 \\diamond1)…","labels":["eq3744"],"detail_key":"p54"},{"id":"n43219","layer":"informal","project":"p54","title":"Equation 3994","kind":"definition","summary":"[Equation 3994] Equation 3994 is the law 0 \\diamond1 \\simeq(2 \\diamond(0 \\diamond1)) \\diamond2…","labels":["eq3994"],"detail_key":"p54"},{"id":"n43220","layer":"informal","project":"p54","title":"Equation 4315","kind":"definition","summary":"[Equation 4315] Equation 4315 is the law 0 \\diamond(1 \\diamond0) \\simeq0 \\diamond(1 \\diamond2)…","labels":["eq4315"],"detail_key":"p54"},{"id":"n43221","layer":"informal","project":"p54","title":"Equation 4512","kind":"definition","summary":"[Equation 4512] Equation 4512 is the law 0 \\diamond(1 \\diamond2) \\simeq(0 \\diamond1) \\diamond2…","labels":["eq4512"],"detail_key":"p54"},{"id":"n43222","layer":"informal","project":"p54","title":"Equation 4513","kind":"definition","summary":"[Equation 4513] Equation 4513 is the law 0 \\diamond(1 \\diamond2) \\simeq(0 \\diamond1) \\diamond3…","labels":["eq4513"],"detail_key":"p54"},{"id":"n43223","layer":"informal","project":"p54","title":"Equation 4522","kind":"definition","summary":"[Equation 4522] Equation 4522 is the law 0 \\diamond(1 \\diamond2) \\simeq(0 \\diamond3) \\diamond4…","labels":["eq4522"],"detail_key":"p54"},{"id":"n43224","layer":"informal","project":"p54","title":"Equation 4564","kind":"definition","summary":"[Equation 4564] Equation 4564 is the law 0 \\diamond(1 \\diamond2) \\simeq(3 \\diamond1) \\diamond2…","labels":["eq4564"],"detail_key":"p54"},{"id":"n43225","layer":"informal","project":"p54","title":"Equation 4579","kind":"definition","summary":"[Equation 4579] Equation 4579 is the law 0 \\diamond(1 \\diamond2) \\simeq(3 \\diamond4) \\diamond2…","labels":["eq4579"],"detail_key":"p54"},{"id":"n43226","layer":"informal","project":"p54","title":"Equation 4582","kind":"definition","summary":"[Equation 4582] Equation 4582 is the law 0 \\diamond(1 \\diamond2) \\simeq(3 \\diamond4) \\diamond5…","labels":["eq4582"],"detail_key":"p54"},{"id":"n43227","layer":"informal","project":"p54","title":"Equation 5093","kind":"definition","summary":"[Equation 5093] Equation 5093 is the law 0 \\simeq1 \\diamond(1 \\diamond(1 \\diamond(0 \\diamond(2…","labels":["eq5093"],"detail_key":"p54"},{"id":"n43228","layer":"informal","project":"p54","title":"Equation 26302","kind":"definition","summary":"[Equation 26302] Equation 26302 is the law 0 \\simeq(1 \\diamond((2 \\diamond0) \\diamond3)) \\diamo…","labels":["eq26302"],"detail_key":"p54"},{"id":"n43229","layer":"informal","project":"p54","title":"Equation 28770","kind":"definition","summary":"[Equation 28770] Equation 28770 is the law 0 \\simeq(((1 \\diamond1) \\diamond1) \\diamond0) \\diamo…","labels":["eq28770"],"detail_key":"p54"},{"id":"n43230","layer":"informal","project":"p54","title":"Equation 345169","kind":"definition","summary":"[Equation 345169] Equation 345169 is the law 0 \\simeq(1 \\diamond((0 \\diamond1) \\diamond1)) \\dia…","labels":["eq345169"],"detail_key":"p54"},{"id":"n43231","layer":"informal","project":"p54","title":"Equation 374794","kind":"definition","summary":"[Equation 374794] Equation 374794 is the law 0 \\simeq(((1 \\diamond1) \\diamond1) \\diamond0) \\dia…","labels":["eq374794"],"detail_key":"p54"},{"id":"n43232","layer":"informal","project":"p54","title":"Implication","kind":"definition","summary":"[Implication] A law E is said to \\emphimply another law E' if \\E\\ \\models E', or equivalently:…","labels":["impl"],"detail_key":"p54"},{"id":"n43233","layer":"informal","project":"p54","title":"Pre-order","kind":"lemma","summary":"[Pre-order] If we define E \\leq E' if E implies E', then this is a pre-order on the set of laws…","labels":["pre-order"],"detail_key":"p54"},{"id":"n43234","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43235","layer":"informal","project":"p54","title":"Maximal element","kind":"lemma","summary":"[Maximal element] The law 0 \\simeq0 is the maximal element in this pre-order.","labels":["maximal"],"detail_key":"p54"},{"id":"n43236","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43237","layer":"informal","project":"p54","title":"Minimal element","kind":"lemma","summary":"[Minimal element] The law 0 \\simeq1 is the minimal element in this pre-order.","labels":["minimal"],"detail_key":"p54"},{"id":"n43238","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43239","layer":"informal","project":"p54","title":"Duality of laws","kind":"lemma","summary":"[Duality of laws] The law w \\simeq w' implies w''\\simeq w''', if and only if w^op \\simeq(w')^op…","labels":["duality"],"detail_key":"p54"},{"id":"n43240","layer":"informal","project":"p54","title":"This follows from the fact that a magma G satisfies a law w \\simeq w' if and only if G^op…","kind":"proof","summary":"This follows from the fact that a magma G satisfies a law w \\simeq w' if and only if G^op satis…","labels":[],"detail_key":"p54"},{"id":"n43241","layer":"informal","project":"p54","title":"Diagonalization","kind":"theorem","summary":"[Diagonalization] An equational law of the form F(x_1,\\dots,x_n) = G(y_1,\\dots,y_m), where x_1,…","labels":["diag","prediag"],"detail_key":"p54"},{"id":"n43242","layer":"informal","project":"p54","title":"From two applications of \\Crefprediag one has F(x_1,\\dots,x_n) = G(y_1,\\dots,y_m) and F(x…","kind":"proof","summary":"From two applications of \\Crefprediag one has F(x_1,\\dots,x_n) = G(y_1,\\dots,y_m) and F(x'_1,\\d…","labels":[],"detail_key":"p54"},{"id":"n43243","layer":"informal","project":"p54","title":"Laws implied by the constant law","kind":"theorem","summary":"[Laws implied by the constant law] If w, w' each have order at least one, then the law w \\simeq…","labels":["constant-impl"],"detail_key":"p54"},{"id":"n43244","layer":"informal","project":"p54","title":"Routine.","kind":"proof","summary":"Routine.","labels":[],"detail_key":"p54"},{"id":"n43245","layer":"informal","project":"p54","title":"Criterion for implication","kind":"theorem","summary":"[Criterion for implication] If w \\simeq w' is such that every variable appears the same number…","labels":["variable-impl"],"detail_key":"p54"},{"id":"n43246","layer":"informal","project":"p54","title":"Consider the magma MS of multisets over an arbitrary set A (which can be seen as finitely…","kind":"proof","summary":"Consider the magma MS of multisets over an arbitrary set A (which can be seen as finitely suppo…","labels":[],"detail_key":"p54"},{"id":"n43247","layer":"informal","project":"p54","title":"387 implies 43","kind":"theorem","summary":"[387 implies 43] \\hrefhttps://teorth.github.io/equational_theories/implications/?387E387 (\\csna…","labels":["387_implies_43"],"detail_key":"p54"},{"id":"n43248","layer":"informal","project":"p54","title":"387-again","kind":"proof","summary":"(From \\hrefhttps://mathoverflow.net/a/450905/766MathOverflow). By \\hrefhttps://teorth.github.io…","labels":["387-again","idem","op-idem"],"detail_key":"p54"},{"id":"n43249","layer":"informal","project":"p54","title":"29 equivalent to 14","kind":"theorem","summary":"[29 equivalent to 14] \\hrefhttps://teorth.github.io/equational_theories/implications/?29E29 (\\c…","labels":["29_equiv_14"],"detail_key":"p54"},{"id":"n43250","layer":"informal","project":"p54","title":"By \\Crefduality it suffices to show that \\hrefhttps://teorth.github.io/equational_theorie…","kind":"proof","summary":"By \\Crefduality it suffices to show that \\hrefhttps://teorth.github.io/equational_theories/impl…","labels":[],"detail_key":"p54"},{"id":"n43251","layer":"informal","project":"p54","title":"14 implies 29","kind":"theorem","summary":"[14 implies 29] \\hrefhttps://teorth.github.io/equational_theories/implications/?14E14 implies \\…","labels":["14_implies_29"],"detail_key":"p54"},{"id":"n43252","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43253","layer":"informal","project":"p54","title":"3744 implies 3722, 381","kind":"theorem","summary":"","labels":["3744_implies_3722_381"],"detail_key":"p54"},{"id":"n43254","layer":"informal","project":"p54","title":"381-1","kind":"proof","summary":"By hypothesis, one has x \\diamond y = (x \\diamond z) \\diamond(w \\diamond y) for all x,y,z,w. Va…","labels":["381-1","381-2","381-3"],"detail_key":"p54"},{"id":"n43255","layer":"informal","project":"p54","title":"1689 is equivalent to 2","kind":"theorem","summary":"[1689 is equivalent to 2] \\hrefhttps://teorth.github.io/equational_theories/implications/?1689E…","labels":["1689_equiv_2"],"detail_key":"p54"},{"id":"n43256","layer":"informal","project":"p54","title":"Kisielewicz-tfftuz","kind":"proof","summary":"The implication of \\hrefhttps://teorth.github.io/equational_theories/implications/?1689E1689 fr…","labels":["Kisielewicz-tfftuz","Kisielewicz-ftg"],"detail_key":"p54"},{"id":"n43257","layer":"informal","project":"p54","title":"Consequences of 1571","kind":"theorem","summary":"","labels":["1571_impl"],"detail_key":"p54"},{"id":"n43258","layer":"informal","project":"p54","title":"1571-again","kind":"proof","summary":"Suppose that a magma G satisfies \\hrefhttps://teorth.github.io/equational_theories/implications…","labels":["1571-again","xxe","16-again","14-again"],"detail_key":"p54"},{"id":"n43259","layer":"informal","project":"p54","title":"953 is equivalent to 2","kind":"theorem","summary":"[953 is equivalent to 2] \\hrefhttps://teorth.github.io/equational_theories/implications/?953E95…","labels":["953_equiv_2"],"detail_key":"p54"},{"id":"n43260","layer":"informal","project":"p54","title":"It suffices to show that \\hrefhttps://teorth.github.io/equational_theories/implications/?…","kind":"proof","summary":"It suffices to show that \\hrefhttps://teorth.github.io/equational_theories/implications/?953E95…","labels":[],"detail_key":"p54"},{"id":"n43261","layer":"informal","project":"p54","title":"Sheffer stroke axiom","kind":"theorem","summary":"[Sheffer stroke axiom] Definition \\hrefhttps://teorth.github.io/equational_theories/implication…","labels":["sheffer"],"detail_key":"p54"},{"id":"n43262","layer":"informal","project":"p54","title":"See \\citemccune_et_al. In fact this is the shortest law with this property. A sketch of p…","kind":"proof","summary":"See \\citemccune_et_al. In fact this is the shortest law with this property. A sketch of proof f…","labels":[],"detail_key":"p54"},{"id":"n43263","layer":"informal","project":"p54","title":"Natural central groupoid axiom","kind":"theorem","summary":"[Natural central groupoid axiom] \\hrefhttps://teorth.github.io/equational_theories/implications…","labels":["natural-central-groupoid"],"detail_key":"p54"},{"id":"n43264","layer":"informal","project":"p54","title":"See \\cite[Theorem 5]knuth. The proof is quite lengthy; a sketch is as follows. It is easy…","kind":"proof","summary":"See \\cite[Theorem 5]knuth. The proof is quite lengthy; a sketch is as follows. It is easy to se…","labels":[],"detail_key":"p54"},{"id":"n43265","layer":"informal","project":"p54","title":"Kisielewicz's first Austin law","kind":"theorem","summary":"[Kisielewicz's first Austin law] \\hrefhttps://teorth.github.io/equational_theories/implications…","labels":["kis-thm"],"detail_key":"p54"},{"id":"n43266","layer":"informal","project":"p54","title":"First we show that every finite model of \\hrefhttps://teorth.github.io/equational_theorie…","kind":"proof","summary":"First we show that every finite model of \\hrefhttps://teorth.github.io/equational_theories/impl…","labels":[],"detail_key":"p54"},{"id":"n43267","layer":"informal","project":"p54","title":"Kisielewicz's second Austin law","kind":"theorem","summary":"[Kisielewicz's second Austin law] \\hrefhttps://teorth.github.io/equational_theories/implication…","labels":["kis-thm2"],"detail_key":"p54"},{"id":"n43268","layer":"informal","project":"p54","title":"kis2-law","kind":"proof","summary":"Using the y^2 and y^3 notation as before, the law reads x = (y^3 \\diamond x) \\diamond(y \\diamon…","labels":["kis2-law"],"detail_key":"p54"},{"id":"n43269","layer":"informal","project":"p54","title":"Equation 5093 has no non-trivial finite models","kind":"theorem","summary":"[Equation 5093 has no non-trivial finite models] \\hrefhttps://teorth.github.io/equational_theor…","labels":["5093-nontrivial"],"detail_key":"p54"},{"id":"n43270","layer":"informal","project":"p54","title":"From \\hrefhttps://teorth.github.io/equational_theories/implications/?5093E5093 we see tha…","kind":"proof","summary":"From \\hrefhttps://teorth.github.io/equational_theories/implications/?5093E5093 we see that the…","labels":[],"detail_key":"p54"},{"id":"n43271","layer":"informal","project":"p54","title":"Austin's finite model theorem","kind":"theorem","summary":"[Austin's finite model theorem] Any law with at most two variables has a non-trivial finite mod…","labels":["austin-two"],"detail_key":"p54"},{"id":"n43272","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43273","layer":"informal","project":"p54","title":"ffg","kind":"lemma","summary":"Let X be finite, and let f, g: X \\to X be such that f = f \\circ f \\circ g. Then f = f \\circ g \\…","labels":["ffg"],"detail_key":"p54"},{"id":"n43274","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43275","layer":"informal","project":"p54","title":"gff","kind":"lemma","summary":"Let X be finite, and let f, g: X \\to X be such that f = g \\circ f \\circ f. Then f = f \\circ g \\…","labels":["gff"],"detail_key":"p54"},{"id":"n43276","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43277","layer":"informal","project":"p54","title":"Eventual period","kind":"lemma","summary":"[Eventual period] Let X be finite and f: X \\to X. Then there exists n \\geq 1 such that f^2n = f…","labels":["period"],"detail_key":"p54"},{"id":"n43278","layer":"informal","project":"p54","title":"By the pigeonhole principle, there exists n \\geq 1, m \\geq 0 such that f^m+n = f^m, which…","kind":"proof","summary":"By the pigeonhole principle, there exists n \\geq 1, m \\geq 0 such that f^m+n = f^m, which impli…","labels":[],"detail_key":"p54"},{"id":"n43279","layer":"informal","project":"p54","title":"3994 implies 3588 for finite models","kind":"proposition","summary":"[3994 implies 3588 for finite models] All finite magmas which satisfy \\hrefhttps://teorth.githu…","labels":["finite_imp_3994_3588_thm"],"detail_key":"p54"},{"id":"n43280","layer":"informal","project":"p54","title":"finite_imp_3994_3588","kind":"proof","summary":"For a finite magma M, consider the set S = \\x \\diamond y | x, y \\in M\\. Now f_z : x \\mapsto z \\…","labels":["finite_imp_3994_3588"],"detail_key":"p54"},{"id":"n43281","layer":"informal","project":"p54","title":"3994 does not imply 3588 for infinite models","kind":"proposition","summary":"[3994 does not imply 3588 for infinite models] There exists a magma which satisfies \\hrefhttps:…","labels":["non_imp_3994_3588_thm"],"detail_key":"p54"},{"id":"n43282","layer":"informal","project":"p54","title":"Consider N, with x \\diamond y defined as x \\oplus y (bitwise XOR) if x and y are even, y+…","kind":"proof","summary":"Consider N, with x \\diamond y defined as x \\oplus y (bitwise XOR) if x and y are even, y+2 if o…","labels":[],"detail_key":"p54"},{"id":"n43283","layer":"informal","project":"p54","title":"3342","kind":"proposition","summary":"[3342] On a finite magma M, equation 3342, x \\diamond y= y \\diamond(x \\diamond(x \\diamond x)),…","labels":["3342"],"detail_key":"p54"},{"id":"n43284","layer":"informal","project":"p54","title":"Write Sx := x \\diamond, fx := x \\diamond Sx and Cx = Sx \\diamond x, then we have x \\diamo…","kind":"proof","summary":"Write Sx := x \\diamond, fx := x \\diamond Sx and Cx = Sx \\diamond x, then we have x \\diamond y=…","labels":[],"detail_key":"p54"},{"id":"n43285","layer":"informal","project":"p54","title":"1167 implies 1096","kind":"proposition","summary":"[1167 implies 1096] For finite magmas, Equation 1167, x = y \\diamond((z \\diamond(y \\diamond y))…","labels":["1167-1096"],"detail_key":"p54"},{"id":"n43286","layer":"informal","project":"p54","title":"We write 1167 as L_y L_z \\diamond Sy = I, hence L_y is invertible and L_z \\diamond Sy L_y…","kind":"proof","summary":"We write 1167 as L_y L_z \\diamond Sy = I, hence L_y is invertible and L_z \\diamond Sy L_y = I.…","labels":[],"detail_key":"p54"},{"id":"n43287","layer":"informal","project":"p54","title":"1133 implies 1167","kind":"proposition","summary":"[1133 implies 1167] For finite magmas, Equation 1133, x = y \\diamond((y \\diamond(z \\diamond y))…","labels":["1133-1167"],"detail_key":"p54"},{"id":"n43288","layer":"informal","project":"p54","title":"intermediate","kind":"proof","summary":"1133 asserts that L_y L_y \\diamond(z \\diamond y) = I, hence L_y is invertible and L_y \\diamond(…","labels":["intermediate"],"detail_key":"p54"},{"id":"n43289","layer":"informal","project":"p54","title":"1441 implies 4067, 1443 implies 3055","kind":"proposition","summary":"[1441 implies 4067, 1443 implies 3055] For finite magmas, Equation 1441, x = (x \\diamond y) \\di…","labels":["1441-4067-1443-3055"],"detail_key":"p54"},{"id":"n43290","layer":"informal","project":"p54","title":"Write \\tilde C x = x \\diamond Sx, then 1441 asserts that R_\\tilde Cx R_y x = x. Setting y…","kind":"proof","summary":"Write \\tilde C x = x \\diamond Sx, then 1441 asserts that R_\\tilde Cx R_y x = x. Setting y=Sx we…","labels":[],"detail_key":"p54"},{"id":"n43291","layer":"informal","project":"p54","title":"1681 implies 3877, 1701 implies 1035","kind":"proposition","summary":"[1681 implies 3877, 1701 implies 1035] For finite magmas, Equation 1681, x = (y \\diamond x) \\di…","labels":["1681-3877-1701-1035"],"detail_key":"p54"},{"id":"n43292","layer":"informal","project":"p54","title":"This is very similar to the previous proof. Write Cx = Sx \\diamond x, then 1681 asserts R…","kind":"proof","summary":"This is very similar to the previous proof. Write Cx = Sx \\diamond x, then 1681 asserts R_Cx L_…","labels":[],"detail_key":"p54"},{"id":"n43293","layer":"informal","project":"p54","title":"A magma G satisfying the left (\\hrefhttps://teorth.github.io/equational_theories/implicat…","kind":"example","summary":"A magma G satisfying the left (\\hrefhttps://teorth.github.io/equational_theories/implications/?…","labels":[],"detail_key":"p54"},{"id":"n43294","layer":"informal","project":"p54","title":"Linear magmas x\\diamond y = ax+by on a field (F,+,-,\\cdot,0,1) are translation-invariant…","kind":"example","summary":"Linear magmas x\\diamond y = ax+by on a field (F,+,-,\\cdot,0,1) are translation-invariant if a +…","labels":[],"detail_key":"p54"},{"id":"n43295","layer":"informal","project":"p54","title":"partial-solution","kind":"definition","summary":"A \\emphpartial solution (E_0, E_1, E_2, f) to \\Creffh consists of nested finite sets E_0 \\subse…","labels":["partial-solution"],"detail_key":"p54"},{"id":"n43296","layer":"informal","project":"p54","title":"Enlarging a partial solution","kind":"lemma","summary":"[Enlarging a partial solution] Let (E_0, E_1, E_2, f) be a partial solution to \\Creffh, and let…","labels":["iteration"],"detail_key":"p54"},{"id":"n43297","layer":"informal","project":"p54","title":"Because f maps E_1 \\backslash E_0 bijectively to E_2 \\backslash E_1, there are three case…","kind":"proof","summary":"Because f maps E_1 \\backslash E_0 bijectively to E_2 \\backslash E_1, there are three cases: \\it…","labels":[],"detail_key":"p54"},{"id":"n43298","layer":"informal","project":"p54","title":"extend","kind":"corollary","summary":"Every partial solution (E_0,E_1,E_2,f) to \\Creffh can be extended to a full solution \\tilde f:…","labels":["extend"],"detail_key":"p54"},{"id":"n43299","layer":"informal","project":"p54","title":"If we arb","kind":"proof","summary":"If we arb","labels":[],"detail_key":"p54"},{"id":"n43300","layer":"informal","project":"p54","title":"no-inject","kind":"corollary","summary":"There exists a solution f:Z\\to Z to \\Creffh such that the map h \\mapsto h + f(h) is not injecti…","labels":["no-inject"],"detail_key":"p54"},{"id":"n43301","layer":"informal","project":"p54","title":"Select integers h_0,h_1,h_2,h'_0,h'_1,h'_2 such that the quantities 0, h_0, h_1, h_2, h_0…","kind":"proof","summary":"Select integers h_0,h_1,h_2,h'_0,h'_1,h'_2 such that the quantities 0, h_0, h_1, h_2, h_0+h_1+h…","labels":[],"detail_key":"p54"},{"id":"n43302","layer":"informal","project":"p54","title":"Asterix does not imply Obelix","kind":"corollary","summary":"","labels":["asterix-obelix"],"detail_key":"p54"},{"id":"n43303","layer":"informal","project":"p54","title":"Note that L_y (y+h) = y + h + f(h), so the injectivity of the left-multiplication maps is…","kind":"proof","summary":"Note that L_y (y+h) = y + h + f(h), so the injectivity of the left-multiplication maps is equiv…","labels":[],"detail_key":"p54"},{"id":"n43304","layer":"informal","project":"p54","title":"Asterix implies Obelix for finite magmas","kind":"proposition","summary":"[Asterix implies Obelix for finite magmas] Any finite magma satisfying the Asterix law (\\hrefht…","labels":["asterix-obelix-finite"],"detail_key":"p54"},{"id":"n43305","layer":"informal","project":"p54","title":"From \\hrefhttps://teorth.github.io/equational_theories/implications/?65E65 we see the map…","kind":"proof","summary":"From \\hrefhttps://teorth.github.io/equational_theories/implications/?65E65 we see the map z \\ma…","labels":[],"detail_key":"p54"},{"id":"n43306","layer":"informal","project":"p54","title":"partial-solution2","kind":"definition","summary":"A \\emphpartial solution for an Obelix is a partial function f : G \\to G with the properties: \\i…","labels":["partial-solution2"],"detail_key":"p54"},{"id":"n43307","layer":"informal","project":"p54","title":"obelix-extend","kind":"lemma","summary":"For any E\\in \\mathscrE and any a\\in G, there is an extension E\\subseteq E'\\in \\mathscrE where t…","labels":["obelix-extend"],"detail_key":"p54"},{"id":"n43308","layer":"informal","project":"p54","title":"\\bf Case 1: Assume (a,b)\\in E for some b\\in G. If b\\in \\rm dom(E), then by condition (4)…","kind":"proof","summary":"\\bf Case 1: Assume (a,b)\\in E for some b\\in G. If b\\in \\rm dom(E), then by condition (4) we are…","labels":[],"detail_key":"p54"},{"id":"n43309","layer":"informal","project":"p54","title":"There is an Obelix magma that is not Asterix (equation 65).","kind":"corollary","summary":"There is an Obelix magma that is not Asterix (equation 65).","labels":[],"detail_key":"p54"},{"id":"n43310","layer":"informal","project":"p54","title":"This requires picking an initial set that still satisfies the correct initial closure pro…","kind":"proof","summary":"This requires picking an initial set that still satisfies the correct initial closure propertie…","labels":[],"detail_key":"p54"},{"id":"n43311","layer":"informal","project":"p54","title":"1722 extension","kind":"lemma","summary":"[1722 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently und…","labels":["1722-extension"],"detail_key":"p54"},{"id":"n43312","layer":"informal","project":"p54","title":"Suppose first that b=a, so a \\diamond a is undefined, then by Law 4 we have d \\diamond a…","kind":"proof","summary":"Suppose first that b=a, so a \\diamond a is undefined, then by Law 4 we have d \\diamond a \\neq a…","labels":[],"detail_key":"p54"},{"id":"n43313","layer":"informal","project":"p54","title":"713 extension","kind":"lemma","summary":"[713 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently unde…","labels":["713-extension"],"detail_key":"p54"},{"id":"n43314","layer":"informal","project":"p54","title":"First suppose that a \\diamond a is not defined. Set a_0 := a and select three new element…","kind":"proof","summary":"First suppose that a \\diamond a is not defined. Set a_0 := a and select three new elements a_1,…","labels":[],"detail_key":"p54"},{"id":"n43315","layer":"informal","project":"p54","title":"1289 extension","kind":"lemma","summary":"[1289 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently und…","labels":["1289-extension"],"detail_key":"p54"},{"id":"n43316","layer":"informal","project":"p54","title":"If a \\diamond b is currently undefined, introduce a new element c and define a \\diamond b…","kind":"proof","summary":"If a \\diamond b is currently undefined, introduce a new element c and define a \\diamond b = c a…","labels":[],"detail_key":"p54"},{"id":"n43317","layer":"informal","project":"p54","title":"73 extension","kind":"lemma","summary":"[73 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently undef…","labels":["73-extension"],"detail_key":"p54"},{"id":"n43318","layer":"informal","project":"p54","title":"If a \\diamond b is undefined, set it equal to a new element c. If we also have b = d \\dia…","kind":"proof","summary":"If a \\diamond b is undefined, set it equal to a new element c. If we also have b = d \\diamond a…","labels":[],"detail_key":"p54"},{"id":"n43319","layer":"informal","project":"p54","title":"63 extension","kind":"lemma","summary":"[63 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently undef…","labels":["63-extension"],"detail_key":"p54"},{"id":"n43320","layer":"informal","project":"p54","title":"If a = b, set a \\diamond' a = a. This clearly does not destroy Laws 3 or 4, and because o…","kind":"proof","summary":"If a = b, set a \\diamond' a = a. This clearly does not destroy Laws 3 or 4, and because of Law…","labels":[],"detail_key":"p54"},{"id":"n43321","layer":"informal","project":"p54","title":"1076 extension","kind":"lemma","summary":"[1076 extension] Suppose that \\diamond is a partial solution, and a \\diamond b is currently und…","labels":["1076-extension"],"detail_key":"p54"},{"id":"n43322","layer":"informal","project":"p54","title":"Set a \\diamond' b = c for some new element c. If d_1,d_2,...,d_n are the elements for whi…","kind":"proof","summary":"Set a \\diamond' b = c for some new element c. If d_1,d_2,...,d_n are the elements for which a \\…","labels":[],"detail_key":"p54"},{"id":"n43323","layer":"informal","project":"p54","title":"1648 does not imply 206","kind":"theorem","summary":"[1648 does not imply 206] There exists a magma which satisfies \\hrefhttps://teorth.github.io/eq…","labels":["non_imp_1648_206_thm"],"detail_key":"p54"},{"id":"n43324","layer":"informal","project":"p54","title":"1659 does not imply 4315","kind":"theorem","summary":"[1659 does not imply 4315] There exists a magma which satisfies \\hrefhttps://teorth.github.io/e…","labels":["non_imp_1659_4315_thm"],"detail_key":"p54"},{"id":"n43325","layer":"informal","project":"p54","title":"Promoting to E_1","kind":"lemma","summary":"[Promoting to E_1] If (E_1,E_2,f) is a partial solution and h \\in E_2 \\backslash E_1, then ther…","labels":["add-E1"],"detail_key":"p54"},{"id":"n43326","layer":"informal","project":"p54","title":"This will be a greedy construction, but we have to introduce a rather large number of add…","kind":"proof","summary":"This will be a greedy construction, but we have to introduce a rather large number of additiona…","labels":[],"detail_key":"p54"},{"id":"n43327","layer":"informal","project":"p54","title":"Promoting to E_2","kind":"lemma","summary":"[Promoting to E_2] If (E_1,E_2,f) is a partial solution and h \\in (Z^3 \\times Z) \\backslash E_2…","labels":["add-E2"],"detail_key":"p54"},{"id":"n43328","layer":"informal","project":"p54","title":"hp-form","kind":"proof","summary":"Let \\tilde E_2 denote the set E_2 together with all elements of the form -h', h' - f^2(h'), f^2…","labels":["hp-form"],"detail_key":"p54"},{"id":"n43329","layer":"informal","project":"p54","title":"Seed solution","kind":"lemma","summary":"[Seed solution] f_0: E_0 \\to Z is a partial solution.","labels":["finite-check"],"detail_key":"p54"},{"id":"n43330","layer":"informal","project":"p54","title":"Finite check.","kind":"proof","summary":"Finite check.","labels":[],"detail_key":"p54"},{"id":"n43331","layer":"informal","project":"p54","title":"Extension","kind":"lemma","summary":"[Extension] If f: E \\to Z is a partial solution and h_0 \\in Z, then there exists an extension f…","labels":["extension-lemma"],"detail_key":"p54"},{"id":"n43332","layer":"informal","project":"p54","title":"We divide into cases. \\bf Case 1: h_0 \\in E and f(h_0) \\in E. In this case we are already…","kind":"proof","summary":"We divide into cases. \\bf Case 1: h_0 \\in E and f(h_0) \\in E. In this case we are already done…","labels":[],"detail_key":"p54"},{"id":"n43333","layer":"informal","project":"p54","title":"Greedy completion of Dupont","kind":"corollary","summary":"[Greedy completion of Dupont] Every partial solution f: E \\to Z can be extended to a global sol…","labels":["dupont-iter"],"detail_key":"p54"},{"id":"n43334","layer":"informal","project":"p54","title":"Non-injective Dupont solution","kind":"corollary","summary":"[Non-injective Dupont solution] The Dupont equation admits non-injective solutions, and hence c…","labels":["non-inject"],"detail_key":"p54"},{"id":"n43335","layer":"informal","project":"p54","title":"It suffices to find a partial solution that violates injectivity. This can be done for in…","kind":"proof","summary":"It suffices to find a partial solution that violates injectivity. This can be done for instance…","labels":[],"detail_key":"p54"},{"id":"n43336","layer":"informal","project":"p54","title":"There exists a magma which satisfies Equation 3342, x \\diamond y = y \\diamond(x \\diamond(…","kind":"theorem","summary":"There exists a magma which satisfies Equation 3342, x \\diamond y = y \\diamond(x \\diamond(x \\dia…","labels":[],"detail_key":"p54"},{"id":"n43337","layer":"informal","project":"p54","title":"fdef","kind":"proof","summary":"We begin with some informal motivation. Writing f(x) := x \\diamond(x \\diamond x), we conclude t…","labels":["fdef","xop"],"detail_key":"p54"},{"id":"n43338","layer":"informal","project":"p54","title":"1437 example","kind":"theorem","summary":"[1437 example] There exists a magma that satisfies Equation 1437, x = (x \\diamond x) \\diamond(y…","labels":["1437-thm"],"detail_key":"p54"},{"id":"n43339","layer":"informal","project":"p54","title":"A first attempt would be the operation i \\diamond j := j+1 on Z/3Z; this satisfies 1437,…","kind":"proof","summary":"A first attempt would be the operation i \\diamond j := j+1 on Z/3Z; this satisfies 1437, but un…","labels":[],"detail_key":"p54"},{"id":"n43340","layer":"informal","project":"p54","title":"Note for category theorists","kind":"proof","summary":"[Note for category theorists] Let \\Pi denote the preorder of magma equations ordered by implica…","labels":[],"detail_key":"p54"},{"id":"n43341","layer":"informal","project":"p54","title":"Lifting Magma Family","kind":"definition","summary":"[Lifting Magma Family] A \\emphlifting magma family is a family of magmas \\G_\\alpha\\, one for ea…","labels":["lifting-magma-family"],"detail_key":"p54"},{"id":"n43342","layer":"informal","project":"p54","title":"The free abelian groups form a lifting magma family. When the underlying set is finite, t…","kind":"example","summary":"The free abelian groups form a lifting magma family. When the underlying set is finite, the gro…","labels":[],"detail_key":"p54"},{"id":"n43343","layer":"informal","project":"p54","title":"Lists form a lifting magma family.","kind":"example","summary":"Lists form a lifting magma family.","labels":[],"detail_key":"p54"},{"id":"n43344","layer":"informal","project":"p54","title":"Evaluation theorem for lifting magma families","kind":"theorem","summary":"[Evaluation theorem for lifting magma families] Suppose E is an equation involving a set of var…","labels":["lifting-magma-basis-evaluation"],"detail_key":"p54"},{"id":"n43345","layer":"informal","project":"p54","title":"For the forward direction, suppose E is satisfied by G_X. Then, by definition, any substi…","kind":"proof","summary":"For the forward direction, suppose E is satisfied by G_X. Then, by definition, any substitution…","labels":[],"detail_key":"p54"},{"id":"n43346","layer":"informal","project":"p54","title":"The fundamental property of invariants","kind":"theorem","summary":"[The fundamental property of invariants] Let E and E' be equations involving a set of variables…","labels":["fundamental-property-of-invariants"],"detail_key":"p54"},{"id":"n43347","layer":"informal","project":"p54","title":"Applying the evaluation \\Creflifting-magma-basis-evaluation, we see that E is satisfied b…","kind":"proof","summary":"Applying the evaluation \\Creflifting-magma-basis-evaluation, we see that E is satisfied by G_X.…","labels":[],"detail_key":"p54"},{"id":"n43348","layer":"informal","project":"p54","title":"The result of evaluating an expression along the function \\iota_X : X \\to G_X \\emphis the…","kind":"remark","summary":"The result of evaluating an expression along the function \\iota_X : X \\to G_X \\emphis the invar…","labels":[],"detail_key":"p54"},{"id":"n43349","layer":"informal","project":"p54","title":"Given an equation \\phi in the language of magmas (possibly involving logical operations o…","kind":"remark","summary":"Given an equation \\phi in the language of magmas (possibly involving logical operations other t…","labels":[],"detail_key":"p54"},{"id":"n43350","layer":"informal","project":"p54","title":"Suppose S is a finite set of equations in the language of magmas that is a confluent term…","kind":"remark","summary":"Suppose S is a finite set of equations in the language of magmas that is a confluent term rewri…","labels":[],"detail_key":"p54"},{"id":"n43351","layer":"informal","project":"p54","title":"compatibility-between-magma-laws","kind":"lemma","summary":"[Compatibility between magma laws over finite sets and the natural numbers] Let E be a magma la…","labels":["compatibility-between-magma-laws"],"detail_key":"p54"},{"id":"n43352","layer":"informal","project":"p54","title":"In the forward direction, suppose \\phi : N \\to M is a substitution. Then the restriction…","kind":"proof","summary":"In the forward direction, suppose \\phi : N \\to M is a substitution. Then the restriction of \\ph…","labels":[],"detail_key":"p54"},{"id":"n43353","layer":"informal","project":"p54","title":"Free magma relative to a theory","kind":"definition","summary":"[Free magma relative to a theory] Let \\Gamma be a theory with an alphabet X. A \\emphfree magma…","labels":["free-theory"],"detail_key":"p54"},{"id":"n43354","layer":"informal","project":"p54","title":"Existence and uniqueness of free magmas","kind":"theorem","summary":"[Existence and uniqueness of free magmas] Let \\Gamma be a theory with alphabet X. \\item[(i)] Th…","labels":["freemag-exist"],"detail_key":"p54"},{"id":"n43355","layer":"informal","project":"p54","title":"For (i), we define M_X,\\Gamma = M_X / \\sim, where the equivalence relation \\sim is define…","kind":"proof","summary":"For (i), we define M_X,\\Gamma = M_X / \\sim, where the equivalence relation \\sim is defined by r…","labels":[],"detail_key":"p54"},{"id":"n43356","layer":"informal","project":"p54","title":"Free associative magma","kind":"example","summary":"[Free associative magma] Let \\Gamma","labels":[],"detail_key":"p54"},{"id":"n43357","layer":"informal","project":"p54","title":"Free associative commutative magma","kind":"example","summary":"[Free associative commutative magma] Let \\Gamma","labels":["facm"],"detail_key":"p54"},{"id":"n43358","layer":"informal","project":"p54","title":"Free left absorptive magma","kind":"example","summary":"[Free left absorptive magma] Let \\Gamma consist of the left absorptive law (\\hrefhttps://teorth…","labels":["freeleft"],"detail_key":"p54"},{"id":"n43359","layer":"informal","project":"p54","title":"Free constant magma","kind":"example","summary":"[Free constant magma] Let \\Gamma consist of the constant law (\\hrefhttps://teorth.github.io/equ…","labels":["freeconst"],"detail_key":"p54"},{"id":"n43360","layer":"informal","project":"p54","title":"Canonical invariant","kind":"theorem","summary":"[Canonical invariant] Let \\Gamma be a theory with some alphabet X, and let M_X,\\Gamma be a free…","labels":["canonical-invariant"],"detail_key":"p54"},{"id":"n43361","layer":"informal","project":"p54","title":"By \\Creffreemag-exist we may take M_X,\\Gamma to be the canonical free magma constructed i…","kind":"proof","summary":"By \\Creffreemag-exist we may take M_X,\\Gamma to be the canonical free magma constructed in the…","labels":[],"detail_key":"p54"},{"id":"n43362","layer":"informal","project":"p54","title":"Criterion for anti-implication","kind":"corollary","summary":"[Criterion for anti-implication] Let \\Gamma be a theory with some alphabet X, and let M_X,\\Gamm…","labels":["anti-impl"],"detail_key":"p54"},{"id":"n43363","layer":"informal","project":"p54","title":"By \\Crefcanonical-invariant, the hypothesis \\iota_X,\\Gamma(w) = \\iota_X,\\Gamma(w') is equ…","kind":"proof","summary":"By \\Crefcanonical-invariant, the hypothesis \\iota_X,\\Gamma(w) = \\iota_X,\\Gamma(w') is equivalen…","labels":[],"detail_key":"p54"},{"id":"n43364","layer":"informal","project":"p54","title":"Let \\Gamma","kind":"example","summary":"Let \\Gamma","labels":[],"detail_key":"p54"},{"id":"n43365","layer":"informal","project":"p54","title":"Let \\Gamma consist of the left absorption law, so we can take M_X,\\Gamma = X as in \\Creff…","kind":"example","summary":"Let \\Gamma consist of the left absorption law, so we can take M_X,\\Gamma = X as in \\Creffreelef…","labels":[],"detail_key":"p54"},{"id":"n43366","layer":"informal","project":"p54","title":"Let \\Gamma consist of the constant law, so we can take M_X,\\Gamma = X \\uplus \\0\\ as in \\C…","kind":"example","summary":"Let \\Gamma consist of the constant law, so we can take M_X,\\Gamma = X \\uplus \\0\\ as in \\Creffre…","labels":[],"detail_key":"p54"},{"id":"n43367","layer":"informal","project":"p54","title":"Let \\Gamma","kind":"example","summary":"Let \\Gamma","labels":[],"detail_key":"p54"},{"id":"n43368","layer":"informal","project":"p54","title":"Confluent theory","kind":"definition","summary":"[Confluent theory] Let \\Gamma b","labels":["confluent-theory"],"detail_key":"p54"},{"id":"n43369","layer":"informal","project":"p54","title":"The associative law, \\hrefhttps://teorth.github.io/equational_theories/implications/?4512…","kind":"example","summary":"The associative law, \\hrefhttps://teorth.github.io/equational_theories/implications/?4512E4512,…","labels":[],"detail_key":"p54"},{"id":"n43370","layer":"informal","project":"p54","title":"The theory consisting of both the associative and commutative laws, \\hrefhttps://teorth.g…","kind":"example","summary":"The theory consisting of both the associative and commutative laws, \\hrefhttps://teorth.github.…","labels":[],"detail_key":"p54"},{"id":"n43371","layer":"informal","project":"p54","title":"The idempotent law, \\hrefhttps://teorth.github.io/equational_theories/implications/?3E3,…","kind":"example","summary":"The idempotent law, \\hrefhttps://teorth.github.io/equational_theories/implications/?3E3, appear…","labels":[],"detail_key":"p54"},{"id":"n43372","layer":"informal","project":"p54","title":"Free magma of a confluent theory","kind":"theorem","summary":"[Free magma of a confluent theory] Let \\Gamma be a confluent theory. Then the free magma M_X,\\G…","labels":["free-confluent"],"detail_key":"p54"},{"id":"n43373","layer":"informal","project":"p54","title":"Should just be a matter of expanding definitions properly.","kind":"proof","summary":"Should just be a matter of expanding definitions properly.","labels":[],"detail_key":"p54"},{"id":"n43374","layer":"informal","project":"p54","title":"Criterion for anti-implication","kind":"corollary","summary":"[Criterion for anti-implication] Let \\Gamma be a confluent theory. Then a law w \\simeq w' is a…","labels":["confluent-anti-impl"],"detail_key":"p54"},{"id":"n43375","layer":"informal","project":"p54","title":"Follows from \\Crefanti-impl.","kind":"proof","summary":"Follows from \\Crefanti-impl.","labels":[],"detail_key":"p54"},{"id":"n43376","layer":"informal","project":"p54","title":"477 confluent","kind":"theorem","summary":"[477 confluent] \\hrefhttps://teorth.github.io/equational_theories/implications/?477E477 (\\csnam…","labels":["477-confl"],"detail_key":"p54"},{"id":"n43377","layer":"informal","project":"p54","title":"See the notes \\hrefhttps://www.overleaf.com/project/66f847bb14d0d8f0b77f74e1here. A sketc…","kind":"proof","summary":"See the notes \\hrefhttps://www.overleaf.com/project/66f847bb14d0d8f0b77f74e1here. A sketch of p…","labels":[],"detail_key":"p54"},{"id":"n43378","layer":"informal","project":"p54","title":"477 lemma","kind":"lemma","summary":"[477 lemma] If Z and W are simple, then Z(W\\cdots(WW)) is simple.","labels":["477-lemma"],"detail_key":"p54"},{"id":"n43379","layer":"informal","project":"p54","title":"Assume the contrary. Then we have 2 cases. Case 1: W\\cdots(WW) matches the pattern y(x(y\\…","kind":"proof","summary":"Assume the contrary. Then we have 2 cases. Case 1: W\\cdots(WW) matches the pattern y(x(y\\cdots(…","labels":[],"detail_key":"p54"},{"id":"n43380","layer":"informal","project":"p54","title":"In the case the associative law (and taking a=b=1), the cocycle law becomes the familiar…","kind":"remark","summary":"In the case the associative law (and taking a=b=1), the cocycle law becomes the familiar f(x,y)…","labels":[],"detail_key":"p54"},{"id":"n43381","layer":"informal","project":"p54","title":"One can interpret the above cohomology group in terms of a partial chain complex 0 \\to C^…","kind":"remark","summary":"One can interpret the above cohomology group in terms of a partial chain complex 0 \\to C^0(G,M)…","labels":[],"detail_key":"p54"},{"id":"n43382","layer":"informal","project":"p54","title":"remark","kind":"remark","summary":"","labels":[],"detail_key":"p54"},{"id":"n43383","layer":"informal","project":"p54","title":"1485 equivalent to 2162","kind":"lemma","summary":"[1485 equivalent to 2162] \\hrefhttps://teorth.github.io/equational_theories/implications/?1485E…","labels":["1485-dual","2162"],"detail_key":"p54"},{"id":"n43384","layer":"informal","project":"p54","title":"It suffices to prove that \\Cref1485 implies \\Cref2162. Write w = y \\diamond z, then from…","kind":"proof","summary":"It suffices to prove that \\Cref1485 implies \\Cref2162. Write w = y \\diamond z, then from \\Cref1…","labels":[],"detail_key":"p54"},{"id":"n43385","layer":"informal","project":"p54","title":"Equivalent characterization of graph","kind":"lemma","summary":"[Equivalent characterization of graph] One has x \\to y if and only if x = w \\diamond y for some…","labels":["graph-dual"],"detail_key":"p54"},{"id":"n43386","layer":"informal","project":"p54","title":"If x \\to y then y = x \\diamond z, then writing z = z_1 \\diamond z_2 as before we obtain x…","kind":"proof","summary":"If x \\to y then y = x \\diamond z, then writing z = z_1 \\diamond z_2 as before we obtain x = (z_…","labels":[],"detail_key":"p54"},{"id":"n43387","layer":"informal","project":"p54","title":"Claim 4","kind":"lemma","summary":"[Claim 4] If a \\to b \\to c \\to d \\to e \\to a is a 5-cycle in the directed graph, and a \\to b \\t…","labels":["claim-4"],"detail_key":"p54"},{"id":"n43388","layer":"informal","project":"p54","title":"If a \\to b \\to c is good then b = a \\diamond c; if c \\to d \\to e is good then d = c \\diam…","kind":"proof","summary":"If a \\to b \\to c is good then b = a \\diamond c; if c \\to d \\to e is good then d = c \\diamond e;…","labels":[],"detail_key":"p54"},{"id":"n43389","layer":"informal","project":"p54","title":"Reversing the claims","kind":"lemma","summary":"[Reversing the claims] Let G be a directed graph, with some paths of length two in the graph de…","labels":["rev-claim"],"detail_key":"p54"},{"id":"n43390","layer":"informal","project":"p54","title":"Define an operation \\diamond: G \\times G \\to G by defining x \\diamond y to be the unique…","kind":"proof","summary":"Define an operation \\diamond: G \\times G \\to G by defining x \\diamond y to be the unique vertex…","labels":[],"detail_key":"p54"},{"id":"n43391","layer":"informal","project":"p54","title":"Completion property","kind":"proposition","summary":"[Completion property] Let G_0 be a directed graph satisfying claims 1', 2, 3, 4. Then any finit…","labels":["greedy-prop"],"detail_key":"p54"},{"id":"n43392","layer":"informal","project":"p54","title":"By the previous comments, we can ignore Claim 4 as it is automatic, and focus on completi…","kind":"proof","summary":"By the previous comments, we can ignore Claim 4 as it is automatic, and focus on completing the…","labels":[],"detail_key":"p54"},{"id":"n43393","layer":"informal","project":"p54","title":"1485 does not imply 1483","kind":"theorem","summary":"[1485 does not imply 1483] \\hrefhttps://teorth.github.io/equational_theories/implications/?1485…","labels":["1485-refutes"],"detail_key":"p54"},{"id":"n43394","layer":"informal","project":"p54","title":"Computer check reveals that the carrier G_0=\\0,1,2,3 ,4\\ with incidence matrix 1 & 1 & 0…","kind":"proof","summary":"Computer check reveals that the carrier G_0=\\0,1,2,3 ,4\\ with incidence matrix 1 & 1 & 0 & 0 &…","labels":[],"detail_key":"p54"},{"id":"n43395","layer":"informal","project":"p54","title":"Basic properties of 677 magma","kind":"lemma","summary":"[Basic properties of 677 magma] Let M be a finite magma satisfying \\eqref677. \\item (i) The lef…","labels":["677-basic"],"detail_key":"p54"},{"id":"n43396","layer":"informal","project":"p54","title":"From \\eqref677-alt we see that L_y is surjective, hence invertible on finite magmas, givi…","kind":"proof","summary":"From \\eqref677-alt we see that L_y is surjective, hence invertible on finite magmas, giving (i)…","labels":[],"detail_key":"p54"},{"id":"n43397","layer":"informal","project":"p54","title":"255-equiv","kind":"lemma","summary":"Let M be a finite magma satisfying \\eqref677, and let x \\in M. Then the following are equivalen…","labels":["255-equiv"],"detail_key":"p54"},{"id":"n43398","layer":"informal","project":"p54","title":"Clearly (ii) implies (i), which implies (iii). If (iii) holds, we apply \\eqref677-alt to…","kind":"proof","summary":"Clearly (ii) implies (i), which implies (iii). If (iii) holds, we apply \\eqref677-alt to conclu…","labels":[],"detail_key":"p54"},{"id":"n43399","layer":"informal","project":"p54","title":"No linear counterexamples","kind":"lemma","summary":"[No linear counterexamples] Suppose we have a finite magma M satisfying 677 which is linear in…","labels":["linear-obstruction"],"detail_key":"p54"},{"id":"n43400","layer":"informal","project":"p54","title":"By the previous lemma, it suffices to show that right multiplication R_x is surjective, o…","kind":"proof","summary":"By the previous lemma, it suffices to show that right multiplication R_x is surjective, or equi…","labels":[],"detail_key":"p54"},{"id":"n43401","layer":"informal","project":"p54","title":"No counterexamples via linear extension","kind":"lemma","summary":"[No counterexamples via linear extension] Suppose that we have a magma with carrier G \\times M…","labels":["linear-2"],"detail_key":"p54"},{"id":"n43402","layer":"informal","project":"p54","title":"By \\Cref677-basic, it suffices to show that for any (y,t), the equation (x,s) \\diamond(y,…","kind":"proof","summary":"By \\Cref677-basic, it suffices to show that for any (y,t), the equation (x,s) \\diamond(y,t) = (…","labels":[],"detail_key":"p54"},{"id":"n43403","layer":"informal","project":"p54","title":"Properties of operation","kind":"lemma","summary":"[Properties of operation] Let x,y \\in M_X be such that x \\diamond y = z. Then either x, y < z =…","labels":["op-prop"],"detail_key":"p54"},{"id":"n43404","layer":"informal","project":"p54","title":"Additional property","kind":"lemma","summary":"[Additional property] If x,y \\in M_X, then x \\diamond((y \\diamond x) \\diamond y) = (x, (y \\diam…","labels":["op-2-677"],"detail_key":"p54"},{"id":"n43405","layer":"informal","project":"p54","title":"Write z := y \\diamond x, u = z \\diamond y, v = x \\diamond u. Our task is to show that v =…","kind":"proof","summary":"Write z := y \\diamond x, u = z \\diamond y, v = x \\diamond u. Our task is to show that v = (x,u)…","labels":[],"detail_key":"p54"},{"id":"n43406","layer":"informal","project":"p54","title":"677-satisfy","kind":"corollary","summary":"The operation \\diamond satisfies \\eqref677.","labels":["677-satisfy"],"detail_key":"p54"},{"id":"n43407","layer":"informal","project":"p54","title":"By the previous lemma and definition of \\diamond, it suffices to show that y < (x, (y \\di…","kind":"proof","summary":"By the previous lemma and definition of \\diamond, it suffices to show that y < (x, (y \\diamond…","labels":[],"detail_key":"p54"},{"id":"n43408","layer":"informal","project":"p54","title":"Let M_X,677 be the magma generated by X with operation \\diamond. Then M_X,677 is the free…","kind":"corollary","summary":"Let M_X,677 be the magma generated by X with operation \\diamond. Then M_X,677 is the free magma…","labels":[],"detail_key":"p54"},{"id":"n43409","layer":"informal","project":"p54","title":"By the previous corollary, it suffices to show that every function f: X \\to M into a 677…","kind":"proof","summary":"By the previous corollary, it suffices to show that every function f: X \\to M into a 677 magma…","labels":[],"detail_key":"p54"},{"id":"n43410","layer":"informal","project":"p54","title":"Description of equivalence","kind":"theorem","summary":"[Description of equivalence] Let w be an irreducible word, and let w' be a word equivalent to w…","labels":["irred-desc"],"detail_key":"p54"},{"id":"n43411","layer":"informal","project":"p54","title":"We just verify claim (i), as claim (ii) is similar. The converse direction is clear from…","kind":"proof","summary":"We just verify claim (i), as claim (ii) is similar. The converse direction is clear from \\Cref8…","labels":[],"detail_key":"p54"},{"id":"n43412","layer":"informal","project":"p54","title":"Unique factorization","kind":"corollary","summary":"[Unique factorization] Two irreducible words w, w' are equivalent if and only if they are eithe…","labels":["unique-factorization"],"detail_key":"p54"},{"id":"n43413","layer":"informal","project":"p54","title":"Immediate from \\Crefirred-desc.","kind":"proof","summary":"Immediate from \\Crefirred-desc.","labels":[],"detail_key":"p54"},{"id":"n43414","layer":"informal","project":"p54","title":"Description of graph","kind":"corollary","summary":"[Description of graph] If w,w' are words, then w' \\to w holds if and only if w' \\sim (Y \\diamon…","labels":["graph-desc"],"detail_key":"p54"},{"id":"n43415","layer":"informal","project":"p54","title":"By replacing w,w' with irreducible equivalents, we may assume without loss of generality…","kind":"proof","summary":"By replacing w,w' with irreducible equivalents, we may assume without loss of generality that w…","labels":[],"detail_key":"p54"},{"id":"n43416","layer":"informal","project":"p54","title":"854 does not imply 3316, 3925","kind":"theorem","summary":"[854 does not imply 3316, 3925] The laws x \\diamond y = x \\diamond(y \\diamond(x \\diamond y)) an…","labels":["854-anti","3316","3925"],"detail_key":"p54"},{"id":"n43417","layer":"informal","project":"p54","title":"We work in the free group M_X on two generators X = \\x,y\\. It suffices to show that x \\di…","kind":"proof","summary":"We work in the free group M_X on two generators X = \\x,y\\. It suffices to show that x \\diamond…","labels":[],"detail_key":"p54"},{"id":"n43418","layer":"informal","project":"p54","title":"854 equivalences, I","kind":"lemma","summary":"[854 equivalences, I] For x, y in a 854 magma, the following are equivalent. \\item (i) y \\to x.…","labels":["854-equiv"],"detail_key":"p54"},{"id":"n43419","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43420","layer":"informal","project":"p54","title":"854 equivalences, II","kind":"lemma","summary":"[854 equivalences, II] For x,y in a 854 magma, the following are equivalent. \\item (i) y \\leq x…","labels":["854-equiv-2"],"detail_key":"p54"},{"id":"n43421","layer":"informal","project":"p54","title":"The equivalence of (i) and (ii) is by definition. If (ii) holds and y \\to z, then by \\Cre…","kind":"proof","summary":"The equivalence of (i) and (ii) is by definition. If (ii) holds and y \\to z, then by \\Cref854-e…","labels":[],"detail_key":"p54"},{"id":"n43422","layer":"informal","project":"p54","title":"The relation \\leq is a pre-order, and for each z, the sets \\ x: x \\to z \\ are upward clos…","kind":"corollary","summary":"The relation \\leq is a pre-order, and for each z, the sets \\ x: x \\to z \\ are upward closed in…","labels":[],"detail_key":"p54"},{"id":"n43423","layer":"informal","project":"p54","title":"Greedy construction","kind":"proposition","summary":"[Greedy construction] Suppose one has a partial 854 magma on N that is only finitely defined, a…","labels":["854-extend"],"detail_key":"p54"},{"id":"n43424","layer":"informal","project":"p54","title":"Define a directed graph by writing x \\to y if y \\diamond x is defined and equal to y. By…","kind":"proof","summary":"Define a directed graph by writing x \\to y if y \\diamond x is defined and equal to y. By Equati…","labels":[],"detail_key":"p54"},{"id":"n43425","layer":"informal","project":"p54","title":"854 extension","kind":"corollary","summary":"[854 extension] Suppose one has a partial 854 magma on N that is only finitely defined. Then it…","labels":["extend-854"],"detail_key":"p54"},{"id":"n43426","layer":"informal","project":"p54","title":"Apply the usual greedy algorithm.","kind":"proof","summary":"Apply the usual greedy algorithm.","labels":[],"detail_key":"p54"},{"id":"n43427","layer":"informal","project":"p54","title":"854 does not not imply 413","kind":"corollary","summary":"[854 does not not imply 413] There is an 854 magma which does not satisfy the 413 law x = x \\di…","labels":["854-413"],"detail_key":"p54"},{"id":"n43428","layer":"informal","project":"p54","title":"Create a partial magma by imposing the laws 1 \\diamond0 = 2, 0 \\diamond2 = 3, 2 \\diamond0…","kind":"proof","summary":"Create a partial magma by imposing the laws 1 \\diamond0 = 2, 0 \\diamond2 = 3, 2 \\diamond0 = 2,…","labels":[],"detail_key":"p54"},{"id":"n43429","layer":"informal","project":"p54","title":"854 does not not imply 1045","kind":"corollary","summary":"[854 does not not imply 1045] There is an 854 magma which does not satisfy the 1045 law x = x \\…","labels":["854-1045"],"detail_key":"p54"},{"id":"n43430","layer":"informal","project":"p54","title":"Similar to previous, but start with the seed 0 \\diamond0 = 2; 0 \\diamond1 = 0 \\diamond2 =…","kind":"proof","summary":"Similar to previous, but start with the seed 0 \\diamond0 = 2; 0 \\diamond1 = 0 \\diamond2 = 0; 1…","labels":[],"detail_key":"p54"},{"id":"n43431","layer":"informal","project":"p54","title":"The 854 relation","kind":"lemma","summary":"[The 854 relation] Let M be an 854 magma, and let \\to be the associated operation, thus x \\to y…","labels":["854-relation"],"detail_key":"p54"},{"id":"n43432","layer":"informal","project":"p54","title":"Case (i) follows from \\Cref378. For (ii), we observe y \\diamond x = y \\diamond((a \\diamon…","kind":"proof","summary":"Case (i) follows from \\Cref378. For (ii), we observe y \\diamond x = y \\diamond((a \\diamond b) \\…","labels":[],"detail_key":"p54"},{"id":"n43433","layer":"informal","project":"p54","title":"Relation on the free group","kind":"definition","summary":"[Relation on the free group] For x,y \\in M_X, we have x \\to y if and only if one of the followi…","labels":["free-relate"],"detail_key":"p54"},{"id":"n43434","layer":"informal","project":"p54","title":"Free 854 magma","kind":"theorem","summary":"[Free 854 magma] The magma M_X,854 is a free 854 magma on X.","labels":["free-854"],"detail_key":"p54"},{"id":"n43435","layer":"informal","project":"p54","title":"one-1","kind":"proof","summary":"Let f: X \\to M be a function from X to an 854 magma M, then \\varphi_f is a homomorphism from M_…","labels":["one-1","one-2","one-3","wz-1","wz"],"detail_key":"p54"},{"id":"n43436","layer":"informal","project":"p54","title":"Edge disjointness of left cycles","kind":"corollary","summary":"[Edge disjointness of left cycles] For any integer n, L_y x = L_z x \\implies L_y^n x = L_z^n x.","labels":["edge-disjoint"],"detail_key":"p54"},{"id":"n43437","layer":"informal","project":"p54","title":"This is trivial for n=0,1, and n=-1 follows from \\Cref906-b. Observe that if the claim ho…","kind":"proof","summary":"This is trivial for n=0,1, and n=-1 follows from \\Cref906-b. Observe that if the claim holds fo…","labels":[],"detail_key":"p54"},{"id":"n43438","layer":"informal","project":"p54","title":"906-3862","kind":"theorem","summary":"For finite magmas, equation 906 implies equation 3862, (x \\diamond(x \\diamond x)) \\diamond x =…","labels":["906-3862","3862"],"detail_key":"p54"},{"id":"n43439","layer":"informal","project":"p54","title":"Observe from \\Cref906-a that L_x S^2 x = L_x (L_x x \\diamond Sx) = x while from \\Cref906-…","kind":"proof","summary":"Observe from \\Cref906-a that L_x S^2 x = L_x (L_x x \\diamond Sx) = x while from \\Cref906-b we h…","labels":[],"detail_key":"p54"},{"id":"n43440","layer":"informal","project":"p54","title":"Construction of 1323 magmas","kind":"lemma","summary":"[Construction of 1323 magmas] Suppose that M is a magma such that R_Sy L_Sy = 1 and L_y R_y = R…","labels":["1323-construct","lr","lr-simp"],"detail_key":"p54"},{"id":"n43441","layer":"informal","project":"p54","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p54"},{"id":"n43442","layer":"informal","project":"p54","title":"Bijections","kind":"lemma","summary":"[Bijections] Let G be a countably infinite abelian torsion group of exponent 2. Then there exis…","labels":["bij"],"detail_key":"p54"},{"id":"n43443","layer":"informal","project":"p54","title":"Such a bijection can be easily constructed from the axiom of choice and a greedy algorith…","kind":"proof","summary":"Such a bijection can be easily constructed from the axiom of choice and a greedy algorithm, def…","labels":[],"detail_key":"p54"},{"id":"n43444","layer":"informal","project":"p54","title":"Building a magma","kind":"lemma","summary":"[Building a magma] Let G be a countably infinite abelian torsion group of exponent 2, and let \\…","labels":["build-magma","op-0","op-1","op-2","op-3"],"detail_key":"p54"},{"id":"n43445","layer":"informal","project":"p54","title":"With these rules, 0 is a unit, and the squaring operator is given by Sa = 0 and S(x,a) =…","kind":"proof","summary":"With these rules, 0 is a unit, and the squaring operator is given by Sa = 0 and S(x,a) = a, so…","labels":[],"detail_key":"p54"},{"id":"n43446","layer":"informal","project":"p54","title":"Partial solution","kind":"definition","summary":"[Partial solution] A \\emphpartial solution is a finite family F of tuples (x,y,z,a,b,c) \\in (Q^…","labels":["partial-1323"],"detail_key":"p54"},{"id":"n43447","layer":"informal","project":"p54","title":"Soundness","kind":"lemma","summary":"[Soundness] Let F be a partial solution. Then if one defines a partial operation \\diamond on M…","labels":["partial-1323-sound"],"detail_key":"p54"},{"id":"n43448","layer":"informal","project":"p54","title":"Routine.","kind":"proof","summary":"Routine.","labels":[],"detail_key":"p54"},{"id":"n43449","layer":"informal","project":"p54","title":"Greedy extension","kind":"lemma","summary":"[Greedy extension] If \\diamond is defined by a partial solution, and (x,a) \\diamond(y,b) is und…","labels":["greedy-1323"],"detail_key":"p54"},{"id":"n43450","layer":"informal","project":"p54","title":"We select a c \\in G \\backslash \\0\\ that has not previously been used by the partial solut…","kind":"proof","summary":"We select a c \\in G \\backslash \\0\\ that has not previously been used by the partial solution, l…","labels":[],"detail_key":"p54"},{"id":"n43451","layer":"informal","project":"p54","title":"Iterated greedy extension","kind":"corollary","summary":"[Iterated greedy extension] Every partial solution can be extended to a complete solution that…","labels":["greedy-iterate"],"detail_key":"p54"},{"id":"n43452","layer":"informal","project":"p54","title":"Apply the usual greedy algorithm.","kind":"proof","summary":"Apply the usual greedy algorithm.","labels":[],"detail_key":"p54"},{"id":"n43453","layer":"informal","project":"p54","title":"1323 does not imply 2744","kind":"corollary","summary":"[1323 does not imply 2744] There exists a 1323 magma which does not satisfy the 2744 equation R…","labels":["1323-refute-2744"],"detail_key":"p54"},{"id":"n43454","layer":"informal","project":"p54","title":"It suffices to produce a partial solution in which L_y is not injective. Pick distinct a,…","kind":"proof","summary":"It suffices to produce a partial solution in which L_y is not injective. Pick distinct a, b, b'…","labels":[],"detail_key":"p54"},{"id":"n43455","layer":"informal","project":"p54","title":"1516 seed","kind":"definition","summary":"[1516 seed] A \\emph1516 seed is a finite collection E of pairs (a,b) with a,b \\in Z satisfying…","labels":["1516-seed"],"detail_key":"p54"},{"id":"n43456","layer":"informal","project":"p54","title":"1516 extension","kind":"lemma","summary":"[1516 extension] Let E be a 1516 seed, and let a_0 \\in Z. Then there exists an extension E' of…","labels":["1516-ext"],"detail_key":"p54"},{"id":"n43457","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43458","layer":"informal","project":"p54","title":"1516 extension variant","kind":"lemma","summary":"[1516 extension variant] Let E be a 1516 seed, and let h \\in Z be non-zero. Then there exists a…","labels":["1516-ext-var"],"detail_key":"p54"},{"id":"n43459","layer":"informal","project":"p54","title":"If we choose a sufficiently large, and set a_i = i a (say) for i=1, 2, 3, 4, the claim si…","kind":"proof","summary":"If we choose a sufficiently large, and set a_i = i a (say) for i=1, 2, 3, 4, the claim simply f…","labels":[],"detail_key":"p54"},{"id":"n43460","layer":"informal","project":"p54","title":"Base magma","kind":"corollary","summary":"[Base magma] There exists a 1516 magma with carrier Z with the properties that \\item (i) Sa=a f…","labels":["1516-base"],"detail_key":"p54"},{"id":"n43461","layer":"informal","project":"p54","title":"By \\Cref1516-ext, \\Cref1516-ext-var and the greedy algorithm starting with the seed consi…","kind":"proof","summary":"By \\Cref1516-ext, \\Cref1516-ext-var and the greedy algorithm starting with the seed consisting…","labels":[],"detail_key":"p54"},{"id":"n43462","layer":"informal","project":"p54","title":"Useful elements","kind":"lemma","summary":"[Useful elements] One can assign c_y,b \\in Z for each y = (a,c,n) \\in G' and b \\in Z with the f…","labels":["aux"],"detail_key":"p54"},{"id":"n43463","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43464","layer":"informal","project":"p54","title":"Existence of partial solution","kind":"lemma","summary":"[Existence of partial solution] A partial solution exists.","labels":["part-exist"],"detail_key":"p54"},{"id":"n43465","layer":"informal","project":"p54","title":"We define L_c' y for y = (a,c,n) \\in G' as follows: \\item (i) If c' = a and n=0, then L_c…","kind":"proof","summary":"We define L_c' y for y = (a,c,n) \\in G' as follows: \\item (i) If c' = a and n=0, then L_c' y :=…","labels":[],"detail_key":"p54"},{"id":"n43466","layer":"informal","project":"p54","title":"First extension","kind":"lemma","summary":"[First extension] Suppose we have a partial solution for which L_c' y is currently undefined fo…","labels":["first-ext"],"detail_key":"p54"},{"id":"n43467","layer":"informal","project":"p54","title":"Write y = (a,c,n). Because L_a: Z\\to Z and L_c': Z\\to Z are surjective, we can find b \\in…","kind":"proof","summary":"Write y = (a,c,n). Because L_a: Z\\to Z and L_c': Z\\to Z are surjective, we can find b \\in Z suc…","labels":[],"detail_key":"p54"},{"id":"n43468","layer":"informal","project":"p54","title":"Second extension","kind":"lemma","summary":"[Second extension] Suppose we have a partial solution, and let c' \\in Z and y = (a,c,n) \\in G.…","labels":["second-ext"],"detail_key":"p54"},{"id":"n43469","layer":"informal","project":"p54","title":"Write y = (a,c,n). There are several cases. \\item Case 1: L_c' y = w for some w \\in G'. B…","kind":"proof","summary":"Write y = (a,c,n). There are several cases. \\item Case 1: L_c' y = w for some w \\in G'. By Lemm…","labels":[],"detail_key":"p54"},{"id":"n43470","layer":"informal","project":"p54","title":"Obtaining Axiom B","kind":"proposition","summary":"[Obtaining Axiom B] There exists a way to extend L_b: Z\\to Z to L_b: G \\to G for all b \\in Z, i…","labels":["axiom-b"],"detail_key":"p54"},{"id":"n43471","layer":"informal","project":"p54","title":"By iterating \\Creffirst-ext and \\Crefsecond-ext in alternation, we can find an increasing…","kind":"proof","summary":"By iterating \\Creffirst-ext and \\Crefsecond-ext in alternation, we can find an increasing chain…","labels":[],"detail_key":"p54"},{"id":"n43472","layer":"informal","project":"p54","title":"Obtaining Axioms A, C","kind":"proposition","summary":"[Obtaining Axioms A, C] One can find maps L_x: G \\to G for each ``non-square'' x \\in G', such t…","labels":["axiom-c"],"detail_key":"p54"},{"id":"n43473","layer":"informal","project":"p54","title":"axioma-again","kind":"proof","summary":"We can work with a single x \\in G'. Our task is to find a function L_x for which L_x x = Sx and…","labels":["axioma-again","axiomb-again"],"detail_key":"p54"},{"id":"n43474","layer":"informal","project":"p54","title":"1516-no-255","kind":"corollary","summary":"There exists a 1516 magma that does not satisfy the 255 equation x = ((x \\diamond x) \\diamond x…","labels":["1516-no-255"],"detail_key":"p54"},{"id":"n43475","layer":"informal","project":"p54","title":"Let us define the element x_0 := (0, 1, 0) \\in G'. Since S x_0 = 0, we know that 0 \\diamo…","kind":"proof","summary":"Let us define the element x_0 := (0, 1, 0) \\in G'. Since S x_0 = 0, we know that 0 \\diamond x_0…","labels":[],"detail_key":"p54"},{"id":"n43476","layer":"informal","project":"p54","title":"Extending a 1729 magma","kind":"theorem","summary":"[Extending a 1729 magma] Let SM be a magma satisfying 1729, and let N be another set disjoint f…","labels":["mag"],"detail_key":"p54"},{"id":"n43477","layer":"informal","project":"p54","title":"We need to show that \\diamond'' verifies the law \\Cref1729. In the case when x,y \\in SM,…","kind":"proof","summary":"We need to show that \\diamond'' verifies the law \\Cref1729. In the case when x,y \\in SM, then t…","labels":[],"detail_key":"p54"},{"id":"n43478","layer":"informal","project":"p54","title":"Definition of SM","kind":"definition","summary":"[Definition of SM] Take SM to be a countably infinite abelian group of exponent 4, generated by…","labels":["sm-def"],"detail_key":"p54"},{"id":"n43479","layer":"informal","project":"p54","title":"Basic properties of SM","kind":"lemma","summary":"[Basic properties of SM] SM is a 1729 magma, the squaring operation S: SM \\to SM is just the do…","labels":["sm-1729"],"detail_key":"p54"},{"id":"n43480","layer":"informal","project":"p54","title":"Routine verification.","kind":"proof","summary":"Routine verification.","labels":[],"detail_key":"p54"},{"id":"n43481","layer":"informal","project":"p54","title":"Definition of N","kind":"definition","summary":"[Definition of N] Take N to be the free non-abelian group with a generator e_a for each a \\in S…","labels":["n-def"],"detail_key":"p54"},{"id":"n43482","layer":"informal","project":"p54","title":"Basic properties of N","kind":"lemma","summary":"[Basic properties of N] N is countable, and \\leq is a partial ordering.","labels":["n-prop"],"detail_key":"p54"},{"id":"n43483","layer":"informal","project":"p54","title":"Routine verification.","kind":"proof","summary":"Routine verification.","labels":[],"detail_key":"p54"},{"id":"n43484","layer":"informal","project":"p54","title":"Definition of R'_a","kind":"definition","summary":"[Definition of R'_a] We set R'_a x := e_a x for all a \\in SM and x \\in N.","labels":["ra-defn","ra-def"],"detail_key":"p54"},{"id":"n43485","layer":"informal","project":"p54","title":"Basic properties of R'_a","kind":"lemma","summary":"[Basic properties of R'_a] The operators R'_a are bijective and satisfy axiom (ii).","labels":["ra-prop"],"detail_key":"p54"},{"id":"n43486","layer":"informal","project":"p54","title":"Routine verification.","kind":"proof","summary":"Routine verification.","labels":[],"detail_key":"p54"},{"id":"n43487","layer":"informal","project":"p54","title":"Using L'_0 to construct L'_a","kind":"lemma","summary":"[Using L'_0 to construct L'_a] Suppose we have a bijection L'_0: N \\to N that satisfies axiom (…","labels":["l0-la","la1"],"detail_key":"p54"},{"id":"n43488","layer":"informal","project":"p54","title":"Routine verification.","kind":"proof","summary":"Routine verification.","labels":[],"detail_key":"p54"},{"id":"n43489","layer":"informal","project":"p54","title":"Reduction to new axioms","kind":"lemma","summary":"[Reduction to new axioms] Suppose we can find a function S': N \\to SM, a bijection L'_0: N \\to…","labels":["axiom-reduce","Eq1729.reduce_to_new_axioms"],"detail_key":"p54"},{"id":"n43490","layer":"informal","project":"p54","title":"Construct the L'_a using \\Crefl0-la. By \\Crefra-prop and direct verification we can noe v…","kind":"proof","summary":"Construct the L'_a using \\Crefl0-la. By \\Crefra-prop and direct verification we can noe verify…","labels":[],"detail_key":"p54"},{"id":"n43491","layer":"informal","project":"p54","title":"Partial solution","kind":"definition","summary":"[Partial solution] A \\emphpartial solution (L'_0, \\diamond', S', I) is a collection of the foll…","labels":["part-sol"],"detail_key":"p54"},{"id":"n43492","layer":"informal","project":"p54","title":"Existence of partial solution","kind":"lemma","summary":"[Existence of partial solution] There exists a partial solution.","labels":["partial-exist"],"detail_key":"p54"},{"id":"n43493","layer":"informal","project":"p54","title":"Set L'_0, \\diamond', S' to be empty functions, and have the set of pending identities to…","kind":"proof","summary":"Set L'_0, \\diamond', S' to be empty functions, and have the set of pending identities to also b…","labels":[],"detail_key":"p54"},{"id":"n43494","layer":"informal","project":"p54","title":"Chain of partial solutions","kind":"lemma","summary":"[Chain of partial solutions] Suppose that one has a sequence (L'_0,n, \\diamond'_n, S'_n, I_n) o…","labels":["chain"],"detail_key":"p54"},{"id":"n43495","layer":"informal","project":"p54","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p54"},{"id":"n43496","layer":"informal","project":"p54","title":"Enlarging L'_0","kind":"proposition","summary":"[Enlarging L'_0] Suppose one has a partial solution in which L'_0 x is undefined for some x \\in…","labels":["enlarge-l0"],"detail_key":"p54"},{"id":"n43497","layer":"informal","project":"p54","title":"By axiom (i''), L'_0 (R'_0)^n x is undefined for every integer n. Let d = E_m be a genera…","kind":"proof","summary":"By axiom (i''), L'_0 (R'_0)^n x is undefined for every integer n. Let d = E_m be a generator of…","labels":[],"detail_key":"p54"},{"id":"n43498","layer":"informal","project":"p54","title":"Enlarging L'_0 many times","kind":"proposition","summary":"[Enlarging L'_0 many times] Suppose one has a partial solution. Let A be a finite subset of N.…","labels":["enlarge-l0-many"],"detail_key":"p54"},{"id":"n43499","layer":"informal","project":"p54","title":"Iterate Proposition \\refenlarge-l0 in the obvious fashion.","kind":"proof","summary":"Iterate Proposition \\refenlarge-l0 in the obvious fashion.","labels":[],"detail_key":"p54"},{"id":"n43500","layer":"informal","project":"p54","title":"Enlarging S' with induction hypothesis and axioms","kind":"proposition","summary":"[Enlarging S' with induction hypothesis and axioms] Suppose one has a partial solution in which…","labels":["enlarge-S-induct-axioms"],"detail_key":"p54"},{"id":"n43501","layer":"informal","project":"p54","title":"Let d_0, d_1 \\in SM be generators E_n_0 of SM that do not appear in the index a of any e_…","kind":"proof","summary":"Let d_0, d_1 \\in SM be generators E_n_0 of SM that do not appear in the index a of any e_a that…","labels":[],"detail_key":"p54"},{"id":"n43502","layer":"informal","project":"p54","title":"Enlarging S' with induction hypothesis","kind":"proposition","summary":"[Enlarging S' with induction hypothesis] Suppose one has a partial solution in which S'x is und…","labels":["enlarge-S-induct"],"detail_key":"p54"},{"id":"n43503","layer":"informal","project":"p54","title":"Let y_0 be the parent of x, that is to say the unique neighbor of x in the path to 1 (thi…","kind":"proof","summary":"Let y_0 be the parent of x, that is to say the unique neighbor of x in the path to 1 (this is o…","labels":[],"detail_key":"p54"},{"id":"n43504","layer":"informal","project":"p54","title":"Enlarging S'","kind":"proposition","summary":"[Enlarging S'] Suppose one has a partial solution in which S'x is undefined for some x \\in N. T…","labels":["enlarge-S"],"detail_key":"p54"},{"id":"n43505","layer":"informal","project":"p54","title":"Obtained by induction from Proposition \\refenlarge-S-induct, using the fact that there ar…","kind":"proof","summary":"Obtained by induction from Proposition \\refenlarge-S-induct, using the fact that there are no i…","labels":[],"detail_key":"p54"},{"id":"n43506","layer":"informal","project":"p54","title":"Enlarging \\diamond","kind":"proposition","summary":"[Enlarging \\diamond] Suppose one has a partial solution in which x \\diamond' y is undefined for…","labels":["enlarge-op"],"detail_key":"p54"},{"id":"n43507","layer":"informal","project":"p54","title":"By applying \\Crefenlarge-S as needed, we may assume without loss of generality that S'x a…","kind":"proof","summary":"By applying \\Crefenlarge-S as needed, we may assume without loss of generality that S'x and S'y…","labels":[],"detail_key":"p54"},{"id":"n43508","layer":"informal","project":"p54","title":"1729 does not imply 817","kind":"theorem","summary":"[1729 does not imply 817] There exists a magma that satisfies equation 1729 but not equation 81…","labels":["1729_refute_817"],"detail_key":"p54"},{"id":"n43509","layer":"informal","project":"p54","title":"Starting from \\Crefpartial-exist and applying \\Crefenlarge-l0, \\Crefenlarge-S, \\Crefenlar…","kind":"proof","summary":"Starting from \\Crefpartial-exist and applying \\Crefenlarge-l0, \\Crefenlarge-S, \\Crefenlarge-op…","labels":[],"detail_key":"p54"},{"id":"n43510","layer":"informal","project":"p54","title":"We write a \\rightarrow b if a\\ R\\ b holds in A, and say that a \\emphrewrites to (or \\emph…","kind":"definition","summary":"We write a \\rightarrow b if a\\ R\\ b holds in A, and say that a \\emphrewrites to (or \\emphreduce…","labels":[],"detail_key":"p54"},{"id":"n43511","layer":"informal","project":"p54","title":"We say that R is \\emphChurch-Rosser if whenever a\\leftrightarrow^* b, there exists some c…","kind":"definition","summary":"We say that R is \\emphChurch-Rosser if whenever a\\leftrightarrow^* b, there exists some c such…","labels":[],"detail_key":"p54"},{"id":"n43512","layer":"informal","project":"p54","title":"If R is Church-Rosser, then any normal form is \\emphunique.","kind":"lemma","summary":"If R is Church-Rosser, then any normal form is \\emphunique.","labels":[],"detail_key":"p54"},{"id":"n43513","layer":"informal","project":"p54","title":"\\item R is Church-Rosser iff it is confluent. \\item (Newman's lemma) if R is strongly nor…","kind":"theorem","summary":"\\item R is Church-Rosser iff it is confluent. \\item (Newman's lemma) if R is strongly normalizi…","labels":[],"detail_key":"p54"},{"id":"n43514","layer":"informal","project":"p54","title":"Given a rewrite system R and two rules \\rho_1: l_1\\rightarrow r_1 and \\rho_2: l_2\\rightar…","kind":"definition","summary":"Given a rewrite system R and two rules \\rho_1: l_1\\rightarrow r_1 and \\rho_2: l_2\\rightarrow r_…","labels":[],"detail_key":"p54"},{"id":"n43515","layer":"informal","project":"p54","title":"Given a rewrite system R, if for every critical pair (t, u) of R, there is a term v such…","kind":"theorem","summary":"Given a rewrite system R, if for every critical pair (t, u) of R, there is a term v such that t…","labels":[],"detail_key":"p54"},{"id":"n43516","layer":"informal","project":"p54","title":"14 implies 23","kind":"theorem","summary":"[14 implies 23] \\hrefhttps://teorth.github.io/equational_theories/implications/?14E14 (\\csname…","labels":["14_implies_23"],"detail_key":"p54"},{"id":"n43517","layer":"informal","project":"p54","title":"x = (x \\diamond x) \\diamond(x \\diamond(x \\diamond x)) = (x \\diamond x) \\diamond x","kind":"proof","summary":"x = (x \\diamond x) \\diamond(x \\diamond(x \\diamond x)) = (x \\diamond x) \\diamond x","labels":[],"detail_key":"p54"},{"id":"n43518","layer":"formal","project":"p54","title":"Completeness","kind":"theorem","summary":"∀ α : Type u_1 Γ : Ctx α E : Law.MagmaLaw α, models Γ E → Nonempty (derive Γ E)","labels":[],"detail_key":"p54","name":"Completeness","module":"equational_theories.Completeness"},{"id":"n43519","layer":"formal","project":"p54","title":"FreeMagma.EvalFreeMagmaWithLawsUniversalProperty","kind":"theorem","summary":"∀ α : Type u_1 G : Type Γ : Ctx α (φ : α → G) [ginst : Magma G] (modelsG : satisfiesSet G Γ) (ψ…","labels":[],"detail_key":"p54","name":"FreeMagma.EvalFreeMagmaWithLawsUniversalProperty","module":"equational_theories.Completeness"},{"id":"n43520","layer":"formal","project":"p54","title":"FreeMagmaWithLaws","kind":"def","summary":"α : Type u_1 → Type u → Ctx α → Type u","labels":[],"detail_key":"p54","name":"FreeMagmaWithLaws","module":"equational_theories.Completeness"},{"id":"n43521","layer":"formal","project":"p54","title":"FreeMagma.elementsOfNumNodesEq_card_eq_catalan_mul_pow","kind":"theorem","summary":"∀ (X : Type u_1) [inst : Fintype X] [inst_1 : DecidableEq X] (n : Nat), Eq (FreeMagma.elementsO…","labels":[],"detail_key":"p54","name":"FreeMagma.elementsOfNumNodesEq_card_eq_catalan_mul_pow","module":"equational_theories.Counting"},{"id":"n43522","layer":"formal","project":"p54","title":"Equation1","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1","module":"equational_theories.Equations.Basic"},{"id":"n43523","layer":"formal","project":"p54","title":"Equation14","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation14","module":"equational_theories.Equations.Basic"},{"id":"n43524","layer":"formal","project":"p54","title":"Equation1491","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1491","module":"equational_theories.Equations.Basic"},{"id":"n43525","layer":"formal","project":"p54","title":"Equation1571","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1571","module":"equational_theories.Equations.Basic"},{"id":"n43526","layer":"formal","project":"p54","title":"Equation16","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation16","module":"equational_theories.Equations.Basic"},{"id":"n43527","layer":"formal","project":"p54","title":"Equation1659","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1659","module":"equational_theories.Equations.Basic"},{"id":"n43528","layer":"formal","project":"p54","title":"Equation1661","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1661","module":"equational_theories.Equations.Basic"},{"id":"n43529","layer":"formal","project":"p54","title":"Equation168","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation168","module":"equational_theories.Equations.Basic"},{"id":"n43530","layer":"formal","project":"p54","title":"Equation1689","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1689","module":"equational_theories.Equations.Basic"},{"id":"n43531","layer":"formal","project":"p54","title":"Equation1701","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1701","module":"equational_theories.Equations.Basic"},{"id":"n43532","layer":"formal","project":"p54","title":"Equation2","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation2","module":"equational_theories.Equations.Basic"},{"id":"n43533","layer":"formal","project":"p54","title":"Equation23","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation23","module":"equational_theories.Equations.Basic"},{"id":"n43534","layer":"formal","project":"p54","title":"Equation2662","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation2662","module":"equational_theories.Equations.Basic"},{"id":"n43535","layer":"formal","project":"p54","title":"Equation28770","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation28770","module":"equational_theories.Equations.Basic"},{"id":"n43536","layer":"formal","project":"p54","title":"Equation29","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation29","module":"equational_theories.Equations.Basic"},{"id":"n43537","layer":"formal","project":"p54","title":"Equation3","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation3","module":"equational_theories.Equations.Basic"},{"id":"n43538","layer":"formal","project":"p54","title":"Equation3588","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation3588","module":"equational_theories.Equations.Basic"},{"id":"n43539","layer":"formal","project":"p54","title":"Equation3722","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation3722","module":"equational_theories.Equations.Basic"},{"id":"n43540","layer":"formal","project":"p54","title":"Equation3744","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation3744","module":"equational_theories.Equations.Basic"},{"id":"n43541","layer":"formal","project":"p54","title":"Equation374794","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation374794","module":"equational_theories.Equations.Basic"},{"id":"n43542","layer":"formal","project":"p54","title":"Equation38","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation38","module":"equational_theories.Equations.Basic"},{"id":"n43543","layer":"formal","project":"p54","title":"Equation381","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation381","module":"equational_theories.Equations.Basic"},{"id":"n43544","layer":"formal","project":"p54","title":"Equation387","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation387","module":"equational_theories.Equations.Basic"},{"id":"n43545","layer":"formal","project":"p54","title":"Equation39","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation39","module":"equational_theories.Equations.Basic"},{"id":"n43546","layer":"formal","project":"p54","title":"Equation4","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4","module":"equational_theories.Equations.Basic"},{"id":"n43547","layer":"formal","project":"p54","title":"Equation40","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation40","module":"equational_theories.Equations.Basic"},{"id":"n43548","layer":"formal","project":"p54","title":"Equation41","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation41","module":"equational_theories.Equations.Basic"},{"id":"n43549","layer":"formal","project":"p54","title":"Equation42","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation42","module":"equational_theories.Equations.Basic"},{"id":"n43550","layer":"formal","project":"p54","title":"Equation43","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation43","module":"equational_theories.Equations.Basic"},{"id":"n43551","layer":"formal","project":"p54","title":"Equation4315","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4315","module":"equational_theories.Equations.Basic"},{"id":"n43552","layer":"formal","project":"p54","title":"Equation45","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation45","module":"equational_theories.Equations.Basic"},{"id":"n43553","layer":"formal","project":"p54","title":"Equation4512","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4512","module":"equational_theories.Equations.Basic"},{"id":"n43554","layer":"formal","project":"p54","title":"Equation4513","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4513","module":"equational_theories.Equations.Basic"},{"id":"n43555","layer":"formal","project":"p54","title":"Equation4522","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4522","module":"equational_theories.Equations.Basic"},{"id":"n43556","layer":"formal","project":"p54","title":"Equation4564","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4564","module":"equational_theories.Equations.Basic"},{"id":"n43557","layer":"formal","project":"p54","title":"Equation4579","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4579","module":"equational_theories.Equations.Basic"},{"id":"n43558","layer":"formal","project":"p54","title":"Equation4582","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation4582","module":"equational_theories.Equations.Basic"},{"id":"n43559","layer":"formal","project":"p54","title":"Equation46","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation46","module":"equational_theories.Equations.Basic"},{"id":"n43560","layer":"formal","project":"p54","title":"Equation5","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation5","module":"equational_theories.Equations.Basic"},{"id":"n43561","layer":"formal","project":"p54","title":"Equation5093","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation5093","module":"equational_theories.Equations.Basic"},{"id":"n43562","layer":"formal","project":"p54","title":"Equation6","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation6","module":"equational_theories.Equations.Basic"},{"id":"n43563","layer":"formal","project":"p54","title":"Equation65","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation65","module":"equational_theories.Equations.Basic"},{"id":"n43564","layer":"formal","project":"p54","title":"Equation7","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation7","module":"equational_theories.Equations.Basic"},{"id":"n43565","layer":"formal","project":"p54","title":"Equation8","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation8","module":"equational_theories.Equations.Basic"},{"id":"n43566","layer":"formal","project":"p54","title":"Equation953","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation953","module":"equational_theories.Equations.Basic"},{"id":"n43567","layer":"formal","project":"p54","title":"Equation1648","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1648","module":"equational_theories.Equations.Eqns1000_1999"},{"id":"n43568","layer":"formal","project":"p54","title":"Equation1657","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation1657","module":"equational_theories.Equations.Eqns1000_1999"},{"id":"n43569","layer":"formal","project":"p54","title":"Equation206","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation206","module":"equational_theories.Equations.Eqns1_999"},{"id":"n43570","layer":"formal","project":"p54","title":"Equation477","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation477","module":"equational_theories.Equations.Eqns1_999"},{"id":"n43571","layer":"formal","project":"p54","title":"Equation63","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation63","module":"equational_theories.Equations.Eqns1_999"},{"id":"n43572","layer":"formal","project":"p54","title":"Equation3167","kind":"def","summary":"(G : Type u) → [a : Magma G] → Prop","labels":[],"detail_key":"p54","name":"Equation3167","module":"equational_theories.Equations.Eqns3000_3999"},{"id":"n43573","layer":"formal","project":"p54","title":"FiniteModel.Finite.f_ffg_implies_f_fgf","kind":"theorem","summary":"∀ G : Type u_1 [Finite G] (f g : G → G), Eq f (Function.comp f (Function.comp f g)) → Eq f (Fun…","labels":[],"detail_key":"p54","name":"FiniteModel.Finite.f_ffg_implies_f_fgf","module":"equational_theories.FiniteModel"},{"id":"n43574","layer":"formal","project":"p54","title":"FiniteModel.Finite.f_gff_implies_f_fgf","kind":"theorem","summary":"∀ G : Type u_1 [Finite G] (f g : G → G), Eq f (Function.comp g (Function.comp f f)) → Eq f (Fun…","labels":[],"detail_key":"p54","name":"FiniteModel.Finite.f_gff_implies_f_fgf","module":"equational_theories.FiniteModel"},{"id":"n43575","layer":"formal","project":"p54","title":"FiniteModel.Finite.fn_eventually_periodic'","kind":"theorem","summary":"∀ G : Type u_1 [Finite G] (f : G → G), Exists fun p => And (GT.gt p 0) (Eq (Nat.iterate f p) (N…","labels":[],"detail_key":"p54","name":"FiniteModel.Finite.fn_eventually_periodic'","module":"equational_theories.FiniteModel"},{"id":"n43576","layer":"formal","project":"p54","title":"Law.MagmaLaw.SameCount.derive","kind":"theorem","summary":"∀ α : Type u_2 [inst : DecidableEq α] Γ : Ctx α E : Law.MagmaLaw α (hE : derive Γ E), (∀ (E : L…","labels":[],"detail_key":"p54","name":"Law.MagmaLaw.SameCount.derive","module":"equational_theories.FreeComm"},{"id":"n43577","layer":"formal","project":"p54","title":"FreeMagma","kind":"inductive","summary":"Type u → Type u","labels":[],"detail_key":"p54","name":"FreeMagma","module":"equational_theories.FreeMagma"},{"id":"n43578","layer":"formal","project":"p54","title":"InfModel.Equation28770_not_implies_Equation2","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation28770 G) (Not (Equation2 G))","labels":[],"detail_key":"p54","name":"InfModel.Equation28770_not_implies_Equation2","module":"equational_theories.InfModel"},{"id":"n43579","layer":"formal","project":"p54","title":"InfModel.Equation374794_not_implies_Equation2","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation374794 G) (Not (Equation2 G))","labels":[],"detail_key":"p54","name":"InfModel.Equation374794_not_implies_Equation2","module":"equational_theories.InfModel"},{"id":"n43580","layer":"formal","project":"p54","title":"InfModel.Equation3994_not_implies_Equation3588","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation3994 G) (Not (Equation3588 G))","labels":[],"detail_key":"p54","name":"InfModel.Equation3994_not_implies_Equation3588","module":"equational_theories.InfModel"},{"id":"n43581","layer":"formal","project":"p54","title":"InfModel.Finite.Equation28770_implies_Equation2","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation28770 G → Equation2 G","labels":[],"detail_key":"p54","name":"InfModel.Finite.Equation28770_implies_Equation2","module":"equational_theories.InfModel"},{"id":"n43582","layer":"formal","project":"p54","title":"InfModel.Finite.Equation374794_implies_Equation2","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation374794 G → Equation2 G","labels":[],"detail_key":"p54","name":"InfModel.Finite.Equation374794_implies_Equation2","module":"equational_theories.InfModel"},{"id":"n43583","layer":"formal","project":"p54","title":"InfModel.Finite.Equation3994_implies_Equation3588","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation3994 G → Equation3588 G","labels":[],"detail_key":"p54","name":"InfModel.Finite.Equation3994_implies_Equation3588","module":"equational_theories.InfModel"},{"id":"n43584","layer":"formal","project":"p54","title":"InfModel.Finite.Equation5093_implies_Equation2","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation5093 G → Equation2 G","labels":[],"detail_key":"p54","name":"InfModel.Finite.Equation5093_implies_Equation2","module":"equational_theories.InfModel"},{"id":"n43585","layer":"formal","project":"p54","title":"InfModel.Finite.two_variable_laws","kind":"theorem","summary":"∀ α : Type [ht : Fintype α], Eq (Fintype.card α) 2 → ∀ (E : Law.MagmaLaw α) (z : α), FreeMagma.…","labels":[],"detail_key":"p54","name":"InfModel.Finite.two_variable_laws","module":"equational_theories.InfModel"},{"id":"n43586","layer":"formal","project":"p54","title":"LiftingMagmaFamily","kind":"inductive","summary":"(Type u_1 → Type u_2) → Type (max (u_1 + 1) u_2)","labels":[],"detail_key":"p54","name":"LiftingMagmaFamily","module":"equational_theories.LiftingMagmaFamilies"},{"id":"n43587","layer":"formal","project":"p54","title":"MagmaLaw.models_iff_satisfies_ι","kind":"theorem","summary":"∀ α : Type [inst : DecidableEq α] (G : Type → Type) [family : LiftingMagmaFamily G] (law : Law.…","labels":[],"detail_key":"p54","name":"MagmaLaw.models_iff_satisfies_ι","module":"equational_theories.LiftingMagmaFamilies"},{"id":"n43588","layer":"formal","project":"p54","title":"Law.MagmaLaw","kind":"inductive","summary":"Type u_1 → Type u_1","labels":[],"detail_key":"p54","name":"Law.MagmaLaw","module":"equational_theories.MagmaLaw"},{"id":"n43589","layer":"formal","project":"p54","title":"Law.satisfies_fin_satisfies_nat","kind":"theorem","summary":"∀ n : Nat (G : Type u_1) [inst : Magma G] (E : Law.MagmaLaw (Fin n)), Iff (satisfies G (Law.Mag…","labels":[],"detail_key":"p54","name":"Law.satisfies_fin_satisfies_nat","module":"equational_theories.MagmaLaw"},{"id":"n43590","layer":"formal","project":"p54","title":"models","kind":"def","summary":"α : Type u_1 → β : Type u_2 → Ctx α → Law.MagmaLaw β → Prop","labels":[],"detail_key":"p54","name":"models","module":"equational_theories.MagmaLaw"},{"id":"n43591","layer":"formal","project":"p54","title":"Law.MagmaLaw.implies_iff_dual","kind":"theorem","summary":"∀ α : Type u_2 l₁ l₂ : Law.MagmaLaw α, Iff (l₁.implies l₂) (l₁.dual.implies l₂.dual)","labels":[],"detail_key":"p54","name":"Law.MagmaLaw.implies_iff_dual","module":"equational_theories.MagmaOp"},{"id":"n43592","layer":"formal","project":"p54","title":"Eq1076.Greedy.lift","kind":"theorem","summary":"∀ (E : PartialMagma.Extension Nat) (a b : Nat), Exists fun E' => And (LE.le E E') (Membership.m…","labels":[],"detail_key":"p54","name":"Eq1076.Greedy.lift","module":"equational_theories.ManuallyProved.Equation1076"},{"id":"n43593","layer":"formal","project":"p54","title":"Eq1133.Finite.Equation1133_implies_Equation1167","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1133 G → Equation1167 G","labels":[],"detail_key":"p54","name":"Eq1133.Finite.Equation1133_implies_Equation1167","module":"equational_theories.ManuallyProved.Equation1133"},{"id":"n43594","layer":"formal","project":"p54","title":"Eq1133.Finite.Equation1167_implies_Equation1096","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1167 G → Equation1096 G","labels":[],"detail_key":"p54","name":"Eq1133.Finite.Equation1167_implies_Equation1096","module":"equational_theories.ManuallyProved.Equation1133"},{"id":"n43595","layer":"formal","project":"p54","title":"Eq1289.Greedy.lift","kind":"theorem","summary":"∀ (E : PartialMagma.Extension Nat) (a b : Nat), Exists fun E' => And (LE.le E E') (Membership.m…","labels":[],"detail_key":"p54","name":"Eq1289.Greedy.lift","module":"equational_theories.ManuallyProved.Equation1289"},{"id":"n43596","layer":"formal","project":"p54","title":"Eq1323.Equation1323_not_implies_Equation2744","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1323 G) (Not (Equation2744 G))","labels":[],"detail_key":"p54","name":"Eq1323.Equation1323_not_implies_Equation2744","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43597","layer":"formal","project":"p54","title":"Eq1323.eq1323_if_conditions","kind":"theorem","summary":"∀ (G : Type) (x : Magma G), (∀ (x_1 y : G), Eq (Magma.op (Magma.op (Magma.op y y) x_1) (Magma.o…","labels":[],"detail_key":"p54","name":"Eq1323.eq1323_if_conditions","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43598","layer":"formal","project":"p54","title":"Eq1323.exists_complete_function","kind":"theorem","summary":"∀ (seed : ↑Eq1323.PartialSolution), Exists fun f => And (Eq1323.Axiom3 f) (∀ (rel : Eq1323.Rela…","labels":[],"detail_key":"p54","name":"Eq1323.exists_complete_function","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43599","layer":"formal","project":"p54","title":"Eq1323.extend","kind":"theorem","summary":"∀ (S : ↑Eq1323.PartialSolution) (p : Eq1323.RelationLHS), Not (Eq1323.definedAt (Eq1323.closure…","labels":[],"detail_key":"p54","name":"Eq1323.extend","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43600","layer":"formal","project":"p54","title":"Eq1323.op","kind":"def","summary":"(Eq1323.RelationLHS → Eq1323.R) → Eq1323.G → Eq1323.G → Eq1323.G","labels":[],"detail_key":"p54","name":"Eq1323.op","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43601","layer":"formal","project":"p54","title":"Eq1323.op_Ly_Ry_eq_LSy","kind":"theorem","summary":"∀ (f : Eq1323.RelationLHS → Eq1323.R), Eq1323.Axiom3 f → ∀ (x y : Eq1323.G), Eq (Eq1323.op f y…","labels":[],"detail_key":"p54","name":"Eq1323.op_Ly_Ry_eq_LSy","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43602","layer":"formal","project":"p54","title":"Eq1323.op_RSy_LSy_eq_Id","kind":"theorem","summary":"∀ (f : Eq1323.RelationLHS → Eq1323.R) (x y : Eq1323.G), Eq (Eq1323.op f (Eq1323.op f (Eq1323.op…","labels":[],"detail_key":"p54","name":"Eq1323.op_RSy_LSy_eq_Id","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43603","layer":"formal","project":"p54","title":"Eq1323.ϕ","kind":"def","summary":"Eq1323.S' → Eq1323.S → Eq1323.A","labels":[],"detail_key":"p54","name":"Eq1323.ϕ","module":"equational_theories.ManuallyProved.Equation1323"},{"id":"n43604","layer":"formal","project":"p54","title":"Eq1441.Finite.Equation1441_implies_Equation4067","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1441 G → Equation4067 G","labels":[],"detail_key":"p54","name":"Eq1441.Finite.Equation1441_implies_Equation4067","module":"equational_theories.ManuallyProved.Equation1441"},{"id":"n43605","layer":"formal","project":"p54","title":"Eq1441.Finite.Equation1443_implies_Equation3055","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1443 G → Equation3055 G","labels":[],"detail_key":"p54","name":"Eq1441.Finite.Equation1443_implies_Equation3055","module":"equational_theories.ManuallyProved.Equation1441"},{"id":"n43606","layer":"formal","project":"p54","title":"Eq1441.Finite.Equation1681_implies_Equation3877","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1681 G → Equation3877 G","labels":[],"detail_key":"p54","name":"Eq1441.Finite.Equation1681_implies_Equation3877","module":"equational_theories.ManuallyProved.Equation1441"},{"id":"n43607","layer":"formal","project":"p54","title":"Eq1441.Finite.Equation1701_implies_Equation1035","kind":"theorem","summary":"∀ (G : Type) [inst : Magma G] [Finite G], Equation1701 G → Equation1035 G","labels":[],"detail_key":"p54","name":"Eq1441.Finite.Equation1701_implies_Equation1035","module":"equational_theories.ManuallyProved.Equation1441"},{"id":"n43608","layer":"formal","project":"p54","title":"Equation1648_not_implies_Equation206","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1648 G) (Not (Equation206 G))","labels":[],"detail_key":"p54","name":"Equation1648_not_implies_Equation206","module":"equational_theories.ManuallyProved.Equation1648"},{"id":"n43609","layer":"formal","project":"p54","title":"Equation1659_facts","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1659 G) (And (Not (Equation1631 G)) (And (Not (Equ…","labels":[],"detail_key":"p54","name":"Equation1659_facts","module":"equational_theories.ManuallyProved.Equation1659"},{"id":"n43610","layer":"formal","project":"p54","title":"Eq1722.Greedy.lift","kind":"theorem","summary":"∀ (E : PartialMagma.Extension Nat) (a b : Nat), Exists fun E' => And (LE.le E E') (Membership.m…","labels":[],"detail_key":"p54","name":"Eq1722.Greedy.lift","module":"equational_theories.ManuallyProved.Equation1722"},{"id":"n43611","layer":"formal","project":"p54","title":"Eq1729.PartialSolution","kind":"inductive","summary":"Type","labels":[],"detail_key":"p54","name":"Eq1729.PartialSolution","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43612","layer":"formal","project":"p54","title":"Eq1729.TrivialPartialSolution","kind":"def","summary":"Eq1729.PartialSolution","labels":[],"detail_key":"p54","name":"Eq1729.TrivialPartialSolution","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43613","layer":"formal","project":"p54","title":"Eq1729.enlarge_L₀'","kind":"theorem","summary":"∀ (sol : Eq1729.PartialSolution) (x : Eq1729.N), Exists fun sol' => And (LE.le sol sol') (And (…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_L₀'","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43614","layer":"formal","project":"p54","title":"Eq1729.enlarge_L₀'_multiple","kind":"theorem","summary":"∀ (sol : Eq1729.PartialSolution) (A : Finset Eq1729.N), Exists fun sol' => And (LE.le sol sol')…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_L₀'_multiple","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43615","layer":"formal","project":"p54","title":"Eq1729.enlarge_S'","kind":"theorem","summary":"∀ (sol : Eq1729.PartialSolution) (x : Eq1729.N), Exists fun sol' => And (LE.le sol sol') (Membe…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_S'","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43616","layer":"formal","project":"p54","title":"Eq1729.enlarge_S'_induction","kind":"theorem","summary":"∀ sol : Eq1729.PartialSolution x : Eq1729.N, (∀ (y : Eq1729.N), LT.lt y x → Membership.mem Eq17…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_S'_induction","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43617","layer":"formal","project":"p54","title":"Eq1729.enlarge_S'_induction_with_axioms","kind":"theorem","summary":"∀ (sol : Eq1729.PartialSolution_with_axioms), Exists fun sol' => And (LE.le sol.toPartialSoluti…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_S'_induction_with_axioms","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43618","layer":"formal","project":"p54","title":"Eq1729.enlarge_op","kind":"theorem","summary":"∀ (sol : Eq1729.PartialSolution) (x y : Eq1729.N), Exists fun sol' => And (LE.le sol sol') (Mem…","labels":[],"detail_key":"p54","name":"Eq1729.enlarge_op","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43619","layer":"formal","project":"p54","title":"Eq1729.use_chain","kind":"theorem","summary":"∀ sols : Set Eq1729.PartialSolution, IsChain (fun sol1 sol2 => LE.le sol1 sol2) sols → Nonempty…","labels":[],"detail_key":"p54","name":"Eq1729.use_chain","module":"equational_theories.ManuallyProved.Equation1729.MagmaConstruction"},{"id":"n43620","layer":"formal","project":"p54","title":"Eq1729.L'","kind":"def","summary":"L₀' : Eq1729.N → Eq1729.N → Eq1729.axiom_i' L₀' → Eq1729.SM → Equiv Eq1729.N Eq1729.N","labels":[],"detail_key":"p54","name":"Eq1729.L'","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43621","layer":"formal","project":"p54","title":"Eq1729.L'_0_eq_L₀'","kind":"theorem","summary":"∀ L₀' : Eq1729.N → Eq1729.N (h : Eq1729.axiom_i' L₀'), Eq (⇑(Eq1729.L' h 0)) L₀'","labels":[],"detail_key":"p54","name":"Eq1729.L'_0_eq_L₀'","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43622","layer":"formal","project":"p54","title":"Eq1729.N","kind":"def","summary":"Type","labels":[],"detail_key":"p54","name":"Eq1729.N","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43623","layer":"formal","project":"p54","title":"Eq1729.N_countable","kind":"theorem","summary":"Countable Eq1729.N","labels":[],"detail_key":"p54","name":"Eq1729.N_countable","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43624","layer":"formal","project":"p54","title":"Eq1729.N_order","kind":"def","summary":"PartialOrder Eq1729.N","labels":[],"detail_key":"p54","name":"Eq1729.N_order","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43625","layer":"formal","project":"p54","title":"Eq1729.R'","kind":"def","summary":"Eq1729.SM → Equiv Eq1729.N Eq1729.N","labels":[],"detail_key":"p54","name":"Eq1729.R'","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43626","layer":"formal","project":"p54","title":"Eq1729.R'_axiom_iia","kind":"theorem","summary":"∀ (a b : Eq1729.SM) (y : Eq1729.N), Ne a b → Ne ((Eq1729.R' a) y) ((Eq1729.R' b) y)","labels":[],"detail_key":"p54","name":"Eq1729.R'_axiom_iia","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43627","layer":"formal","project":"p54","title":"Eq1729.R'_axiom_iib","kind":"theorem","summary":"∀ (a : Eq1729.SM) (y : Eq1729.N), Ne ((Eq1729.R' a) y) y","labels":[],"detail_key":"p54","name":"Eq1729.R'_axiom_iib","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43628","layer":"formal","project":"p54","title":"Eq1729.SM","kind":"def","summary":"Type","labels":[],"detail_key":"p54","name":"Eq1729.SM","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43629","layer":"formal","project":"p54","title":"Eq1729.SM_satisfies_1729","kind":"theorem","summary":"Equation1729 Eq1729.SM","labels":[],"detail_key":"p54","name":"Eq1729.SM_satisfies_1729","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43630","layer":"formal","project":"p54","title":"Eq1729.SM_square_eq_double","kind":"theorem","summary":"∀ (a : Eq1729.SM), Eq (Eq1729.S a) (HAdd.hAdd a a)","labels":[],"detail_key":"p54","name":"Eq1729.SM_square_eq_double","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43631","layer":"formal","project":"p54","title":"Eq1729.SM_square_square_eq_zero","kind":"theorem","summary":"∀ (a : Eq1729.SM), Eq (Eq1729.S (Eq1729.S a)) 0","labels":[],"detail_key":"p54","name":"Eq1729.SM_square_square_eq_zero","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43632","layer":"formal","project":"p54","title":"Eq1729.parent","kind":"def","summary":"Eq1729.N → Eq1729.N","labels":[],"detail_key":"p54","name":"Eq1729.parent","module":"equational_theories.ManuallyProved.Equation1729.SmallMagma"},{"id":"n43633","layer":"formal","project":"p54","title":"Eq1729.not_817","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1729 G) (Not (Equation817 G))","labels":[],"detail_key":"p54","name":"Eq1729.not_817","module":"equational_theories.ManuallyProved.Equation1729"},{"id":"n43634","layer":"formal","project":"p54","title":"Eq3342.Finite.Equation3342_implies_Equation3522","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation3342 G → Equation3522 G","labels":[],"detail_key":"p54","name":"Eq3342.Finite.Equation3342_implies_Equation3522","module":"equational_theories.ManuallyProved.Equation3342"},{"id":"n43635","layer":"formal","project":"p54","title":"Eq3342.Finite.Equation3342_implies_Equation4118","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation3342 G → Equation4118 G","labels":[],"detail_key":"p54","name":"Eq3342.Finite.Equation3342_implies_Equation4118","module":"equational_theories.ManuallyProved.Equation3342"},{"id":"n43636","layer":"formal","project":"p54","title":"Eq63.Equation63_not_implies_Equation1692","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation63 G) (Not (Equation1692 G))","labels":[],"detail_key":"p54","name":"Eq63.Equation63_not_implies_Equation1692","module":"equational_theories.ManuallyProved.Equation63"},{"id":"n43637","layer":"formal","project":"p54","title":"Eq63.Greedy.Extension.Next","kind":"inductive","summary":"[Eq63.Greedy.Extension] → Eq63.G → Eq63.G → Prop","labels":[],"detail_key":"p54","name":"Eq63.Greedy.Extension.Next","module":"equational_theories.ManuallyProved.Equation63"},{"id":"n43638","layer":"formal","project":"p54","title":"Eq63.Greedy.Extension.next","kind":"def","summary":"[Eq63.Greedy.Extension] → Eq63.Greedy.PartialSolution","labels":[],"detail_key":"p54","name":"Eq63.Greedy.Extension.next","module":"equational_theories.ManuallyProved.Equation63"},{"id":"n43639","layer":"formal","project":"p54","title":"Eq63.Greedy.exists_extension","kind":"theorem","summary":"∀ (seed : Eq63.Greedy.PartialSolution), Exists fun f => And (Eq63.Greedy.thomson f) (∀ x y : Eq…","labels":[],"detail_key":"p54","name":"Eq63.Greedy.exists_extension","module":"equational_theories.ManuallyProved.Equation63"},{"id":"n43640","layer":"formal","project":"p54","title":"Refutation_854.Greedy.Extension1.next_ok","kind":"theorem","summary":"∀ [inst : Refutation_854.Greedy.Extension1], Refutation_854.Greedy.Extension1.next.OK","labels":[],"detail_key":"p54","name":"Refutation_854.Greedy.Extension1.next_ok","module":"equational_theories.ManuallyProved.Equation854"},{"id":"n43641","layer":"formal","project":"p54","title":"Refutation_854.Greedy.exists_extension","kind":"theorem","summary":"∀ (e₀ : Refutation_854.Greedy.Extension), Exists fun op => And (∀ (x y z : Nat), Eq x (op x (op…","labels":[],"detail_key":"p54","name":"Refutation_854.Greedy.exists_extension","module":"equational_theories.ManuallyProved.Equation854"},{"id":"n43642","layer":"formal","project":"p54","title":"Refutation_854.not_3316_3925","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation854 G) (And (Not (Equation3316 G)) (Not (Equation3…","labels":[],"detail_key":"p54","name":"Refutation_854.not_3316_3925","module":"equational_theories.ManuallyProved.Equation854"},{"id":"n43643","layer":"formal","project":"p54","title":"Refutation_854.not_413_1045","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation854 G) (And (Not (Equation413 G)) (Not (Equation10…","labels":[],"detail_key":"p54","name":"Refutation_854.not_413_1045","module":"equational_theories.ManuallyProved.Equation854"},{"id":"n43644","layer":"formal","project":"p54","title":"Refutation_854.unique_factorization","kind":"theorem","summary":"∀ a b c d : Refutation_854.G, Eq (Magma.op a b) (Magma.op c d) → Not (Refutation_854.Rel b a) →…","labels":[],"detail_key":"p54","name":"Refutation_854.unique_factorization","module":"equational_theories.ManuallyProved.Equation854"},{"id":"n43645","layer":"formal","project":"p54","title":"Eq906.Finite.Equation906_implies_Equation3862","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G] [Finite G], Equation906 G → Equation3862 G","labels":[],"detail_key":"p54","name":"Eq906.Finite.Equation906_implies_Equation3862","module":"equational_theories.ManuallyProved.Equation906"},{"id":"n43646","layer":"formal","project":"p54","title":"Law.MagmaLaw.Equation1_maximal","kind":"theorem","summary":"∀ (l : Law.MagmaLaw Nat), LE.le l (Law.MagmaLaw.mk 0 0)","labels":[],"detail_key":"p54","name":"Law.MagmaLaw.Equation1_maximal","module":"equational_theories.Preorder"},{"id":"n43647","layer":"formal","project":"p54","title":"Law.MagmaLaw.Equation2_minimal","kind":"theorem","summary":"∀ (l : Law.MagmaLaw Nat), LE.le (Law.MagmaLaw.mk 0 1) l","labels":[],"detail_key":"p54","name":"Law.MagmaLaw.Equation2_minimal","module":"equational_theories.Preorder"},{"id":"n43648","layer":"formal","project":"p54","title":"Sheffer.Equation345169_is_Boolean","kind":"def","summary":"(G : Type u_1) → [inst : Magma G] → Equation345169 G → [Inhabited G] → BooleanAlgebra G","labels":[],"detail_key":"p54","name":"Sheffer.Equation345169_is_Boolean","module":"equational_theories.Sheffer"},{"id":"n43649","layer":"formal","project":"p54","title":"Subgraph.Equation14_implies_Equation23","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation14 G → Equation23 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation14_implies_Equation23","module":"equational_theories.Subgraph"},{"id":"n43650","layer":"formal","project":"p54","title":"Subgraph.Equation14_implies_Equation29","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation14 G → Equation29 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation14_implies_Equation29","module":"equational_theories.Subgraph"},{"id":"n43651","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation16","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation16 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation16","module":"equational_theories.Subgraph"},{"id":"n43652","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation23","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation23 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation23","module":"equational_theories.Subgraph"},{"id":"n43653","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation2662","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation2662 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation2662","module":"equational_theories.Subgraph"},{"id":"n43654","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation40","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation40 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation40","module":"equational_theories.Subgraph"},{"id":"n43655","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation43","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation43 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation43","module":"equational_theories.Subgraph"},{"id":"n43656","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation4512","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation4512 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation4512","module":"equational_theories.Subgraph"},{"id":"n43657","layer":"formal","project":"p54","title":"Subgraph.Equation1571_implies_Equation8","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1571 G → Equation8 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1571_implies_Equation8","module":"equational_theories.Subgraph"},{"id":"n43658","layer":"formal","project":"p54","title":"Subgraph.Equation1689_implies_Equation2","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation1689 G → Equation2 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation1689_implies_Equation2","module":"equational_theories.Subgraph"},{"id":"n43659","layer":"formal","project":"p54","title":"Subgraph.Equation29_implies_Equation14","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation29 G → Equation14 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation29_implies_Equation14","module":"equational_theories.Subgraph"},{"id":"n43660","layer":"formal","project":"p54","title":"Subgraph.Equation2_implies_Equation1689","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation2 G → Equation1689 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation2_implies_Equation1689","module":"equational_theories.Subgraph"},{"id":"n43661","layer":"formal","project":"p54","title":"Subgraph.Equation3744_implies_Equation3722","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation3744 G → Equation3722 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation3744_implies_Equation3722","module":"equational_theories.Subgraph"},{"id":"n43662","layer":"formal","project":"p54","title":"Subgraph.Equation3744_implies_Equation381","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation3744 G → Equation381 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation3744_implies_Equation381","module":"equational_theories.Subgraph"},{"id":"n43663","layer":"formal","project":"p54","title":"Subgraph.Equation387_implies_Equation43","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation387 G → Equation43 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation387_implies_Equation43","module":"equational_theories.Subgraph"},{"id":"n43664","layer":"formal","project":"p54","title":"Subgraph.Equation953_implies_Equation2","kind":"theorem","summary":"∀ (G : Type u_1) [inst : Magma G], Equation953 G → Equation2 G","labels":[],"detail_key":"p54","name":"Subgraph.Equation953_implies_Equation2","module":"equational_theories.Subgraph"},{"id":"n43665","layer":"formal","project":"p54","title":"Refutation_1485.not_2087_2124","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1485 G) (And (Not (Equation2087 G)) (Not (Equation…","labels":[],"detail_key":"p54","name":"Refutation_1485.not_2087_2124","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43666","layer":"formal","project":"p54","title":"Refutation_1485.not_3457","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1485 G) (Not (Equation3457 G))","labels":[],"detail_key":"p54","name":"Refutation_1485.not_3457","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43667","layer":"formal","project":"p54","title":"Refutation_1485.not_3511","kind":"theorem","summary":"Exists fun G => Exists fun x => And (Equation1485 G) (Not (Equation3511 G))","labels":[],"detail_key":"p54","name":"Refutation_1485.not_3511","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43668","layer":"formal","project":"p54","title":"RelaxedVeryWeakCentralGroupoid.Greedy.exists_extension","kind":"theorem","summary":"∀ G : Type u [inst : RelaxedVeryWeakCentralGroupoid G] [Countable G] (e₀ : RelaxedVeryWeakCentr…","labels":[],"detail_key":"p54","name":"RelaxedVeryWeakCentralGroupoid.Greedy.exists_extension","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43669","layer":"formal","project":"p54","title":"RelaxedWeakCentralGroupoid.strictify","kind":"def","summary":"G : Type u_1 → [inst : RelaxedVeryWeakCentralGroupoid G] → [RelaxedVeryWeakCentralGroupoid.IsWe…","labels":[],"detail_key":"p54","name":"RelaxedWeakCentralGroupoid.strictify","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43670","layer":"formal","project":"p54","title":"WeakCentralGroupoid.Path.def'","kind":"theorem","summary":"∀ G : Type u_1 [inst : WeakCentralGroupoid G] x y : G, Iff (WeakCentralGroupoid.Path x y) (Exis…","labels":[],"detail_key":"p54","name":"WeakCentralGroupoid.Path.def'","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43671","layer":"formal","project":"p54","title":"WeakCentralGroupoid.dual_eqn","kind":"theorem","summary":"∀ G : Type u_1 [inst : WeakCentralGroupoid G] (x y z : G), Eq (Magma.op (Magma.op (Magma.op y z…","labels":[],"detail_key":"p54","name":"WeakCentralGroupoid.dual_eqn","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43672","layer":"formal","project":"p54","title":"WeakCentralGroupoid.isGood_five","kind":"theorem","summary":"∀ G : Type u_1 [inst : WeakCentralGroupoid G] a b c d e : G, WeakCentralGroupoid.IsGood a b c →…","labels":[],"detail_key":"p54","name":"WeakCentralGroupoid.isGood_five","module":"equational_theories.WeakCentralGroupoids"},{"id":"n43673","layer":"formal","project":"p55","title":"Expdb.automatic_uniformity_of_choicewise_bounded","kind":"theorem","summary":"∀ α : Type u_1 [inst : SeminormedAddCommGroup α] (E : Expdb.VariableObject (Set Real)), (∀ (i :…","labels":[],"detail_key":"p55","name":"Expdb.automatic_uniformity_of_choicewise_bounded","module":"Expdb.Basic.AutomaticUniformity"},{"id":"n43674","layer":"formal","project":"p55","title":"Expdb.automatic_uniformity_of_choicewise_infinitesimal","kind":"theorem","summary":"∀ α : Type u_1 [inst : SeminormedAddCommGroup α] (E : Expdb.VariableObject (Set Real)), (∀ (i :…","labels":[],"detail_key":"p55","name":"Expdb.automatic_uniformity_of_choicewise_infinitesimal","module":"Expdb.Basic.AutomaticUniformity"},{"id":"n43675","layer":"formal","project":"p55","title":"Expdb.IsExponentSumBound","kind":"def","summary":"NNReal → Real → Prop","labels":[],"detail_key":"p55","name":"Expdb.IsExponentSumBound","module":"Expdb.ExponentialSums.ExponentSumGrowth"},{"id":"n43676","layer":"formal","project":"p55","title":"Expdb.exponentSumGrowthExponent","kind":"def","summary":"NNReal → Real","labels":[],"detail_key":"p55","name":"Expdb.exponentSumGrowthExponent","module":"Expdb.ExponentialSums.ExponentSumGrowth"},{"id":"n43677","layer":"formal","project":"p55","title":"Expdb.IsExponentSumBoundNonAsymptotic","kind":"def","summary":"NNReal → Real → Prop","labels":[],"detail_key":"p55","name":"Expdb.IsExponentSumBoundNonAsymptotic","module":"Expdb.ExponentialSums.ExponentSumGrowthNonAsymptotic"},{"id":"n43678","layer":"formal","project":"p55","title":"Expdb.exponentSumGrowthExponent_le_iff_nonAsymptotic","kind":"theorem","summary":"∀ α : NNReal β : Real, Iff (LE.le (Expdb.exponentSumGrowthExponent α) β) (Expdb.IsExponentSumBo…","labels":[],"detail_key":"p55","name":"Expdb.exponentSumGrowthExponent_le_iff_nonAsymptotic","module":"Expdb.ExponentialSums.ExponentSumGrowthNonAsymptotic"},{"id":"n43679","layer":"formal","project":"p55","title":"Expdb.IsModelPhaseFunction","kind":"def","summary":"Expdb.VariableFunction (Expdb.VariableObject.fixed Real) Real → Prop","labels":[],"detail_key":"p55","name":"Expdb.IsModelPhaseFunction","module":"Expdb.ExponentialSums.PhaseFunctions"},{"id":"n43680","layer":"formal","project":"p55","title":"Expdb.IsPhaseFunction","kind":"def","summary":"Expdb.VariableFunction (Expdb.VariableObject.fixed Real) Real → Prop","labels":[],"detail_key":"p55","name":"Expdb.IsPhaseFunction","module":"Expdb.ExponentialSums.PhaseFunctions"},{"id":"n43681","layer":"formal","project":"p55","title":"Expdb.exponentSumGrowthExponent_eq_sub_one","kind":"theorem","summary":"∀ α : NNReal, LT.lt 1 α → Eq (Expdb.exponentSumGrowthExponent α) (HSub.hSub (↑α) 1)","labels":[],"detail_key":"p55","name":"Expdb.exponentSumGrowthExponent_eq_sub_one","module":"Expdb.ExponentialSums.TrivialBounds"},{"id":"n43682","layer":"formal","project":"p55","title":"Expdb.exponentSumGrowthExponent_mem_Icc","kind":"theorem","summary":"∀ α : NNReal, LE.le α 1 → Membership.mem (Set.Icc (HDiv.hDiv (↑α) 2) ↑α) (Expdb.exponentSumGrow…","labels":[],"detail_key":"p55","name":"Expdb.exponentSumGrowthExponent_mem_Icc","module":"Expdb.ExponentialSums.TrivialBounds"},{"id":"n43683","layer":"formal","project":"p55","title":"Expdb.exponentSumGrowthExponent_zero","kind":"theorem","summary":"Eq (Expdb.exponentSumGrowthExponent 0) 0","labels":[],"detail_key":"p55","name":"Expdb.exponentSumGrowthExponent_zero","module":"Expdb.ExponentialSums.TrivialBounds"},{"id":"n43684","layer":"formal","project":"p55","title":"Expdb.upperSemicontinuous_exponentSumGrowthExponent","kind":"theorem","summary":"UpperSemicontinuous Expdb.exponentSumGrowthExponent","labels":[],"detail_key":"p55","name":"Expdb.upperSemicontinuous_exponentSumGrowthExponent","module":"Expdb.ExponentialSums.UpperSemicontinuity"},{"id":"n43685","layer":"formal","project":"p55","title":"Expdb.l2_integral_estimate","kind":"theorem","summary":"Exists fun C => And (LT.lt 0 C) (∀ ι : Type u_1 [inst : Fintype ι] (a : ι → Complex) (ξ : ι → R…","labels":[],"detail_key":"p55","name":"Expdb.l2_integral_estimate","module":"Expdb.Fourier.L2Integral"},{"id":"n43686","layer":"formal","project":"p55","title":"Expdb.l2_integral_estimate_error","kind":"theorem","summary":"Exists fun C => And (LT.lt 0 C) (∀ ι : Type u_1 [inst : Fintype ι] (a : ι → Complex) (ξ : ι → R…","labels":[],"detail_key":"p55","name":"Expdb.l2_integral_estimate_error","module":"Expdb.Fourier.L2Integral"},{"id":"n43687","layer":"informal","project":"p55","title":"Automatic uniformity","kind":"proposition","summary":"[Automatic uniformity] Let E = E_i be a non-empty variable set, and let f = f_i: E \\to C be a v…","labels":["auto"],"detail_key":"p55"},{"id":"n43688","layer":"informal","project":"p55","title":"We begin with (i). Suppose that there is no uniform bound. Then for any fixed natural num…","kind":"proof","summary":"We begin with (i). Suppose that there is no uniform bound. Then for any fixed natural number n,…","labels":[],"detail_key":"p55"},{"id":"n43689","layer":"informal","project":"p55","title":"The proof gives slightly more as we only need to discard finitely many initial indices ra…","kind":"remark","summary":"The proof gives slightly more as we only need to discard finitely many initial indices rather t…","labels":[],"detail_key":"p55"},{"id":"n43690","layer":"informal","project":"p55","title":"remark","kind":"remark","summary":"","labels":[],"detail_key":"p55"},{"id":"n43691","layer":"informal","project":"p55","title":"remark","kind":"remark","summary":"","labels":[],"detail_key":"p55"},{"id":"n43692","layer":"informal","project":"p55","title":"L^2 integral estimate","kind":"lemma","summary":"[L^2 integral estimate] Let \\xi_1,\\dots,\\xi_R be real numbers that are 1/N-separated. Then for…","labels":["l2-int"],"detail_key":"p55"},{"id":"n43693","layer":"informal","project":"p55","title":"art","kind":"proof","summary":"We adapt the proof of \\cite[Theorem 9.1]ik. Without loss of generality we may normalize \\sum_r=…","labels":["art","trap"],"detail_key":"p55"},{"id":"n43694","layer":"informal","project":"p55","title":"Phase function","kind":"definition","summary":"[Phase function] A \\emphphase function is a (variable) smooth function F \\colon [1,2] \\to R. A…","labels":["phase-def","fpu"],"detail_key":"p55"},{"id":"n43695","layer":"informal","project":"p55","title":"Exponent sum growth exponent","kind":"definition","summary":"[Exponent sum growth exponent] For any fixed \\alpha \\geq 0, let \\beta(\\alpha) \\in R denote the…","labels":["beta-def"],"detail_key":"p55"},{"id":"n43696","layer":"informal","project":"p55","title":"Non-asymptotic definition of \\beta","kind":"lemma","summary":"[Non-asymptotic definition of \\beta] Let \\alpha \\geq 0 and \\overline\\beta \\in R be fixed. Then…","labels":["beta-asymp","fpu-bound"],"detail_key":"p55"},{"id":"n43697","layer":"informal","project":"p55","title":"It is easy to see that (ii) implies (i) by expanding out all the definitions (and using P…","kind":"proof","summary":"It is easy to see that (ii) implies (i) by expanding out all the definitions (and using Proposi…","labels":[],"detail_key":"p55"},{"id":"n43698","layer":"informal","project":"p55","title":"Trivial bounds on \\beta","kind":"lemma","summary":"[Trivial bounds on \\beta] For any fixed \\alpha > 1, we have \\beta(\\alpha) = \\alpha-1. For fixed…","labels":["beta-triv","beta-0"],"detail_key":"p55"},{"id":"n43699","layer":"informal","project":"p55","title":"nit","kind":"proof","summary":"Let T > 1 be unbounded, N = T^\\alpha+o(1), I \\subset [N,2N] an interval, and F a model phase fu…","labels":["nit"],"detail_key":"p55"},{"id":"n43700","layer":"informal","project":"p55","title":"Upper semicontinuity","kind":"lemma","summary":"[Upper semicontinuity] \\beta is an upper semicontinuous function.","labels":["beta-semicts"],"detail_key":"p55"},{"id":"n43701","layer":"informal","project":"p55","title":"Routine from the definition.","kind":"proof","summary":"Routine from the definition.","labels":[],"detail_key":"p55"},{"id":"n43702","layer":"informal","project":"p55","title":"Van der Corput A process for \\beta","kind":"lemma","summary":"[Van der Corput A process for \\beta] If 0 \\leq \\alpha \\leq 2/3 and h \\geq 0 then 2\\beta(\\alpha)…","labels":["vdca-beta"],"detail_key":"p55"},{"id":"n43703","layer":"informal","project":"p55","title":"By definition, there exists an unbounded T, N = T^\\alpha+o(1), F a model phase function,…","kind":"proof","summary":"By definition, there exists an unbounded T, N = T^\\alpha+o(1), F a model phase function, and I…","labels":[],"detail_key":"p55"},{"id":"n43704","layer":"informal","project":"p55","title":"Van der Corput inequality","kind":"proposition","summary":"[Van der Corput inequality] For any natural number k \\geq 2 and any \\alpha>0, one has \\beta(\\al…","labels":["beta-vdc","beta-1"],"detail_key":"p55"},{"id":"n43705","layer":"informal","project":"p55","title":"Follows from \\cite[Theorem 8.20]ik. It is also possible to prove this by induction on k u…","kind":"proof","summary":"Follows from \\cite[Theorem 8.20]ik. It is also possible to prove this by induction on k using L…","labels":[],"detail_key":"p55"},{"id":"n43706","layer":"informal","project":"p55","title":"Optimizing the van der Corput inequality","kind":"corollary","summary":"[Optimizing the van der Corput inequality] For any \\alpha > 0 one has \\beta(\\alpha) \\leq \\inf_k…","labels":["vdc-opt"],"detail_key":"p55"},{"id":"n43707","layer":"informal","project":"p55","title":"Let \\beta_k(\\alpha) = \\alpha + (1 - k\\alpha)/(2^k - 2) and \\[ \\alpha_k = \\frac2^k(k - 1)2…","kind":"proof","summary":"Let \\beta_k(\\alpha) = \\alpha + (1 - k\\alpha)/(2^k - 2) and \\[ \\alpha_k = \\frac2^k(k - 1)2^k + 2…","labels":[],"detail_key":"p55"},{"id":"n43708","layer":"informal","project":"p55","title":"interval-set","kind":"lemma","summary":"In Definition \\refbeta-def, one can take the interval I to be [N,2N].","labels":["interval-set"],"detail_key":"p55"},{"id":"n43709","layer":"informal","project":"p55","title":"three","kind":"proof","summary":"Suppose that \\alpha, \\overline\\beta are fixed quantities such that the bounds in Definition \\re…","labels":["three"],"detail_key":"p55"},{"id":"n43710","layer":"informal","project":"p55","title":"Reflection","kind":"lemma","summary":"[Reflection] For any 0 < \\alpha < 1, we have \\beta(\\alpha) - \\frac\\alpha2 = \\beta(1-\\alpha) - \\…","labels":["beta-reflect"],"detail_key":"p55"},{"id":"n43711","layer":"informal","project":"p55","title":"This is the van der Corput B-process. See e.g., \\cite[p 370]huxley_area_1996.","kind":"proof","summary":"This is the van der Corput B-process. See e.g., \\cite[p 370]huxley_area_1996.","labels":[],"detail_key":"p55"},{"id":"n43712","layer":"informal","project":"p55","title":"1989 Watt bound","kind":"theorem","summary":"[1989 Watt bound] For any 3/7 \\le \\alpha \\le 1/2, one has \\[ \\beta(\\alpha) \\le \\frac89560 + \\fr…","labels":["beta-Watt"],"detail_key":"p55"},{"id":"n43713","layer":"informal","project":"p55","title":"See \\cite[Theorem~5]watt_exponential_1989.","kind":"proof","summary":"See \\cite[Theorem~5]watt_exponential_1989.","labels":[],"detail_key":"p55"},{"id":"n43714","layer":"informal","project":"p55","title":"1991 Huxley--Kolesnik bound","kind":"theorem","summary":"[1991 Huxley--Kolesnik bound] For any 2/5 \\le \\alpha \\le 1/2 one has \\[ \\beta(\\alpha) \\le \\max\\…","labels":["beta-HK2"],"detail_key":"p55"},{"id":"n43715","layer":"informal","project":"p55","title":"See \\cite[Theorem~3]huxley_exponential_1991. Note that the paper contains an error, howev…","kind":"proof","summary":"See \\cite[Theorem~3]huxley_exponential_1991. Note that the paper contains an error, however thi…","labels":[],"detail_key":"p55"},{"id":"n43716","layer":"informal","project":"p55","title":"1993 Huxley bound","kind":"theorem","summary":"[1993 Huxley bound] For any 0 \\le \\alpha \\le 49/114, one has \\[ \\beta(\\alpha) \\le \\max\\left(\\fr…","labels":["beta-Huxley-4"],"detail_key":"p55"},{"id":"n43717","layer":"informal","project":"p55","title":"See \\cite[Theorem~1]huxley_exponential_1993.","kind":"proof","summary":"See \\cite[Theorem~1]huxley_exponential_1993.","labels":[],"detail_key":"p55"},{"id":"n43718","layer":"informal","project":"p55","title":"Second 1993 Huxley bound","kind":"theorem","summary":"[Second 1993 Huxley bound] If 0 \\leq \\alpha \\leq 1, then \\beta(\\alpha) is bounded by \\frac1146…","labels":["beta-Huxley-4a"],"detail_key":"p55"},{"id":"n43719","layer":"informal","project":"p55","title":"See \\cite[Theorem~3]huxley_exponential_1993.","kind":"proof","summary":"See \\cite[Theorem~3]huxley_exponential_1993.","labels":[],"detail_key":"p55"},{"id":"n43720","layer":"informal","project":"p55","title":"1995 Sargos bound","kind":"theorem","summary":"[1995 Sargos bound] \\cite[Th\\'eor\\`eme 2.4, Lemme 2.6]sargos_points_1995 For any 0 \\leq \\alpha…","labels":["sargos_1995"],"detail_key":"p55"},{"id":"n43721","layer":"informal","project":"p55","title":"1996 Huxley table","kind":"theorem","summary":"[1996 Huxley table] One can bound \\beta(\\alpha) by \\beta_0(\\alpha) for X \\leq \\alpha \\leq Y for…","labels":["huxley-table"],"detail_key":"p55"},{"id":"n43722","layer":"informal","project":"p55","title":"See \\cite[Table 17.1, Table 19.2]huxley_area_1996 (and also \\cite[\\S 3.0.2, 3.0.3]trudgia…","kind":"proof","summary":"See \\cite[Table 17.1, Table 19.2]huxley_area_1996 (and also \\cite[\\S 3.0.2, 3.0.3]trudgian-yang…","labels":[],"detail_key":"p55"},{"id":"n43723","layer":"informal","project":"p55","title":"2001 Huxley--Kolesnik bound","kind":"theorem","summary":"[2001 Huxley--Kolesnik bound] For any 2/5 \\le \\alpha \\le 1/2 one has \\beta(\\alpha) \\leq \\max\\le…","labels":["beta-HK"],"detail_key":"p55"},{"id":"n43724","layer":"informal","project":"p55","title":"See \\cite[Theorem~1]huxley_exponential_2001.","kind":"proof","summary":"See \\cite[Theorem~1]huxley_exponential_2001.","labels":[],"detail_key":"p55"},{"id":"n43725","layer":"informal","project":"p55","title":"2002 Robert--Sargos bound","kind":"theorem","summary":"[2002 Robert--Sargos bound] For any \\alpha > 0 one has \\beta(\\alpha) \\leq \\max\\left( \\alpha + \\…","labels":["beta-RS"],"detail_key":"p55"},{"id":"n43726","layer":"informal","project":"p55","title":"See \\cite[Theorem~1]robert_fourth_2002.","kind":"proof","summary":"See \\cite[Theorem~1]robert_fourth_2002.","labels":[],"detail_key":"p55"},{"id":"n43727","layer":"informal","project":"p55","title":"Sargos 2003 bound","kind":"theorem","summary":"[Sargos 2003 bound] For any \\alpha > 0 one has \\beta(\\alpha) \\leq \\max\\left( \\alpha + \\frac1-8\\…","labels":["sargos-bound"],"detail_key":"p55"},{"id":"n43728","layer":"informal","project":"p55","title":"See \\cite[Theorems~3, 4]sargos_analog_2003.","kind":"proof","summary":"See \\cite[Theorems~3, 4]sargos_analog_2003.","labels":[],"detail_key":"p55"},{"id":"n43729","layer":"informal","project":"p55","title":"Huxley bound","kind":"theorem","summary":"[Huxley bound] For any 1/3 \\le \\alpha \\le 1/2, one has \\[ \\beta(\\alpha) \\le \\max\\left(\\frac37 +…","labels":["beta-Huxley-5"],"detail_key":"p55"},{"id":"n43730","layer":"informal","project":"p55","title":"See \\cite[Proposition~1, Theorem~1]huxley_exponential_2005.","kind":"proof","summary":"See \\cite[Proposition~1, Theorem~1]huxley_exponential_2005.","labels":[],"detail_key":"p55"},{"id":"n43731","layer":"informal","project":"p55","title":"2016 Robert bound","kind":"theorem","summary":"[2016 Robert bound] For any 0 < \\alpha \\le 3/7 one has \\[ \\beta(\\alpha) \\le \\max\\left(\\alpha +…","labels":["beta-R"],"detail_key":"p55"},{"id":"n43732","layer":"informal","project":"p55","title":"See \\cite[Theorem~1]robert_fourth_2016.","kind":"proof","summary":"See \\cite[Theorem~1]robert_fourth_2016.","labels":[],"detail_key":"p55"},{"id":"n43733","layer":"informal","project":"p55","title":"Second 2016 Robert bound","kind":"theorem","summary":"[Second 2016 Robert bound] If k \\geq 4 and \\alpha \\geq -(1-k\\alpha) \\frack-12k-3 then \\beta(\\al…","labels":["beta-R2"],"detail_key":"p55"},{"id":"n43734","layer":"informal","project":"p55","title":"See \\cite[Theorem 10]robert_2016.","kind":"proof","summary":"See \\cite[Theorem 10]robert_2016.","labels":[],"detail_key":"p55"},{"id":"n43735","layer":"informal","project":"p55","title":"2017 Heath-Brown bound","kind":"theorem","summary":"[2017 Heath-Brown bound] For any \\alpha > 0 and any natural number k \\geq 3 one has \\beta(\\alph…","labels":["beta-HB"],"detail_key":"p55"},{"id":"n43736","layer":"informal","project":"p55","title":"See \\cite[Theorem~1]heathbrown_new_2017.","kind":"proof","summary":"See \\cite[Theorem~1]heathbrown_new_2017.","labels":[],"detail_key":"p55"},{"id":"n43737","layer":"informal","project":"p55","title":"2017 Bourgain bound","kind":"theorem","summary":"[2017 Bourgain bound] One has \\[ \\beta(\\alpha) \\le \\displaystyle\\frac29 + \\frac13\\alpha,&\\displ…","labels":["beta-Bourgain"],"detail_key":"p55"},{"id":"n43738","layer":"informal","project":"p55","title":"See \\cite[Equation~(3.18)]bourgain_decoupling_2017.","kind":"proof","summary":"See \\cite[Equation~(3.18)]bourgain_decoupling_2017.","labels":[],"detail_key":"p55"},{"id":"n43739","layer":"informal","project":"p55","title":"2020 Heath-Brown bound","kind":"theorem","summary":"[2020 Heath-Brown bound] If \\alpha is fixed with 1 \\leq 4\\alpha-1 \\leq 2 (i.e., 1/2 \\leq \\alpha…","labels":["beta-hb-2020"],"detail_key":"p55"},{"id":"n43740","layer":"informal","project":"p55","title":"See \\cite[Theorem 11.2]demeter_small_2020.","kind":"proof","summary":"See \\cite[Theorem 11.2]demeter_small_2020.","labels":[],"detail_key":"p55"},{"id":"n43741","layer":"informal","project":"p55","title":"Combined bound","kind":"theorem","summary":"[Combined bound] For X \\leq \\alpha \\leq Y, one has \\beta(\\alpha) \\leq \\beta_0(\\alpha), where \\b…","labels":["combined-bound"],"detail_key":"p55"},{"id":"n43742","layer":"informal","project":"p55","title":"See \\cite[Table 3]trudgian-yang.","kind":"proof","summary":"See \\cite[Table 3]trudgian-yang.","labels":[],"detail_key":"p55"},{"id":"n43743","layer":"informal","project":"p55","title":"Exponent pair","kind":"definition","summary":"[Exponent pair] An exponent pair is a (fixed) element (k,\\ell) of the triangle \\ (k,\\ell) \\in R…","labels":["exp-pair-def","exp-pair-triangle","ntf"],"detail_key":"p55"},{"id":"n43744","layer":"informal","project":"p55","title":"Non-asymptotic definition of exponent pair","kind":"lemma","summary":"[Non-asymptotic definition of exponent pair] Let (k,\\ell) be a fixed element of \\eqrefexp-pair-…","labels":[],"detail_key":"p55"},{"id":"n43745","layer":"informal","project":"p55","title":"Duality between exponent pairs and \\beta","kind":"lemma","summary":"[Duality between exponent pairs and \\beta] Let (k,\\ell) be in the triangle \\eqrefexp-pair-trian…","labels":["beta-duality"],"detail_key":"p55"},{"id":"n43746","layer":"informal","project":"p55","title":"If (i) holds, then for any 0 < \\alpha < 1, any unbounded T \\geq 1, any N = T^\\alpha+o(1),…","kind":"proof","summary":"If (i) holds, then for any 0 < \\alpha < 1, any unbounded T \\geq 1, any N = T^\\alpha+o(1), inter…","labels":[],"detail_key":"p55"},{"id":"n43747","layer":"informal","project":"p55","title":"Exponent pairs closed and convex","kind":"corollary","summary":"[Exponent pairs closed and convex] The set of exponent pairs is closed and convex.","labels":["exp-pair-closed"],"detail_key":"p55"},{"id":"n43748","layer":"informal","project":"p55","title":"Immediate from Lemma \\refbeta-duality.","kind":"proof","summary":"Immediate from Lemma \\refbeta-duality.","labels":[],"detail_key":"p55"},{"id":"n43749","layer":"informal","project":"p55","title":"Trivial exponent pairs","kind":"proposition","summary":"[Trivial exponent pairs] (0,1) and (1/2,1/2) are exponent pairs.","labels":["exp-pair-trivial"],"detail_key":"p55"},{"id":"n43750","layer":"informal","project":"p55","title":"Immediate from Lemma \\refbeta-duality and Lemma \\refbeta-triv.","kind":"proof","summary":"Immediate from Lemma \\refbeta-duality and Lemma \\refbeta-triv.","labels":[],"detail_key":"p55"},{"id":"n43751","layer":"informal","project":"p55","title":"Exponent pairs conjecture","kind":"conjecture","summary":"[Exponent pairs conjecture] (0,1/2) is an exponent pair. (Equivalently, by Corollary \\refexp-pa…","labels":["exp-pair-conj"],"detail_key":"p55"},{"id":"n43752","layer":"informal","project":"p55","title":"exp-pair-conj-beta","kind":"lemma","summary":"The exponent pair conjecture is equivalent to \\beta(\\alpha)=\\alpha/2 holding true for all 0 \\le…","labels":["exp-pair-conj-beta"],"detail_key":"p55"},{"id":"n43753","layer":"informal","project":"p55","title":"Clear from Lemma \\refbeta-duality and Lemma \\refbeta-triv.","kind":"proof","summary":"Clear from Lemma \\refbeta-duality and Lemma \\refbeta-triv.","labels":[],"detail_key":"p55"},{"id":"n43754","layer":"informal","project":"p55","title":"Van der Corput A-process","kind":"proposition","summary":"[Van der Corput A-process] If (k,\\ell) is an exponent pair, then so is A(k,\\ell) := \\left(\\frac…","labels":["vdc-a"],"detail_key":"p55"},{"id":"n43755","layer":"informal","project":"p55","title":"See \\cite[Lemma 2.8]ivic. It can also be deduced from Lemma \\refvdca-beta and Lemma \\refb…","kind":"proof","summary":"See \\cite[Lemma 2.8]ivic. It can also be deduced from Lemma \\refvdca-beta and Lemma \\refbeta-du…","labels":[],"detail_key":"p55"},{"id":"n43756","layer":"informal","project":"p55","title":"Van der Corput B-process","kind":"proposition","summary":"[Van der Corput B-process] If (k,\\ell) is an exponent pair, then so is B(k,\\ell) := \\left(\\ell-…","labels":["vdc-b"],"detail_key":"p55"},{"id":"n43757","layer":"informal","project":"p55","title":"See \\cite[Lemma 2.9]ivic. Alternatively, this can be derived from Lemma \\refbeta-reflect…","kind":"proof","summary":"See \\cite[Lemma 2.9]ivic. Alternatively, this can be derived from Lemma \\refbeta-reflect and Le…","labels":[],"detail_key":"p55"},{"id":"n43758","layer":"informal","project":"p55","title":"Classical van der Corput exponent pairs","kind":"proposition","summary":"[Classical van der Corput exponent pairs] For any natural number k \\geq 2, A^k-2 B(0,1) = \\left…","labels":["vdc-class"],"detail_key":"p55"},{"id":"n43759","layer":"informal","project":"p55","title":"Follows by induction from Proposition \\refvdc-a and Proposition \\refexp-pair-trivial; alt…","kind":"proof","summary":"Follows by induction from Proposition \\refvdc-a and Proposition \\refexp-pair-trivial; alternati…","labels":[],"detail_key":"p55"},{"id":"n43760","layer":"informal","project":"p55","title":"Additional exponent pairs","kind":"corollary","summary":"[Additional exponent pairs] The pairs \\left(\\frac1331, \\frac1631\\right), \\left(\\frac411,\\frac61…","labels":["add-exponential"],"detail_key":"p55"},{"id":"n43761","layer":"informal","project":"p55","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p55"},{"id":"n43762","layer":"informal","project":"p55","title":"Exponent pairs on the line of symmetry","kind":"theorem","summary":"[Exponent pairs on the line of symmetry] (k,k+1/2) is an exponent pair for \\item[(i)] k = 9/56…","labels":["line-sym"],"detail_key":"p55"},{"id":"n43763","layer":"informal","project":"p55","title":"Exponent pairs from the Bombieri--Iwaniec method","kind":"theorem","summary":"[Exponent pairs from the Bombieri--Iwaniec method] The following pairs are exponent pairs: \\ite…","labels":["exp_pair_bombieri-iwaniec"],"detail_key":"p55"},{"id":"n43764","layer":"informal","project":"p55","title":"Exponent pairs from derivative tests","kind":"theorem","summary":"[Exponent pairs from derivative tests] (k,1-mk) is an exponent pair when \\item[(i)] k=\\frac113…","labels":["exp_pair_deriv_test"],"detail_key":"p55"},{"id":"n43765","layer":"informal","project":"p55","title":"Huxley sequence","kind":"theorem","summary":"[Huxley sequence] \\cite[Table 17.3]huxley_area_1996 For any integer m \\geq 1, the pair \\left(\\f…","labels":["huxley_exp_pair"],"detail_key":"p55"},{"id":"n43766","layer":"informal","project":"p55","title":"1996 Heath--Brown sequence","kind":"theorem","summary":"[1996 Heath--Brown sequence] \\cite[(6.17.4)]titchmarsh_theory_1986 For any integer m \\geq 3, th…","labels":["heath-brown_exp_pair_1996"],"detail_key":"p55"},{"id":"n43767","layer":"informal","project":"p55","title":"2017 Heath--Brown sequence","kind":"theorem","summary":"[2017 Heath--Brown sequence] \\cite[Theorem 2]heathbrown_new_2017 For any integer m \\geq 3, the…","labels":["heath-brown_exp_pair_2017"],"detail_key":"p55"},{"id":"n43768","layer":"informal","project":"p55","title":"This follows from Theorem \\refbeta-HB and Lemma \\refbeta-duality, after some computation.","kind":"proof","summary":"This follows from Theorem \\refbeta-HB and Lemma \\refbeta-duality, after some computation.","labels":[],"detail_key":"p55"},{"id":"n43769","layer":"informal","project":"p55","title":"Sargos C-process","kind":"theorem","summary":"[Sargos C-process] \\cite[Theorem 5]sargos_analog_2003 If (k,\\ell) is an exponent pair, then so…","labels":["sargos_C"],"detail_key":"p55"},{"id":"n43770","layer":"informal","project":"p55","title":"Sargos D-process","kind":"theorem","summary":"[Sargos D-process] \\cite[Theorem 7.1]sargos_points_1995 If (k,\\ell) is an exponent pair, then o…","labels":["sargos_D"],"detail_key":"p55"},{"id":"n43771","layer":"informal","project":"p55","title":"trudgian_yang_eps","kind":"theorem","summary":"\\cite[Lemma 1.1]trudgian-yang The following are exponent pairs: (k_1,\\ell_1) &:= \\left(\\frac474…","labels":["trudgian_yang_eps"],"detail_key":"p55"},{"id":"n43772","layer":"informal","project":"p55","title":"For the pair (18/199, 593/796), apply Theorem \\refsargos_D with the pair (13/84, 55/84) f…","kind":"proof","summary":"For the pair (18/199, 593/796), apply Theorem \\refsargos_D with the pair (13/84, 55/84) from Th…","labels":[],"detail_key":"p55"},{"id":"n43773","layer":"informal","project":"p55","title":"Set of exponent pairs","kind":"corollary","summary":"","labels":["H-pairs"],"detail_key":"p55"},{"id":"n43774","layer":"informal","project":"p55","title":"Clear from Corollary \\refexp-pair-closed, Proposition \\refexp-pair-trivial, \\reftrudgian_…","kind":"proof","summary":"Clear from Corollary \\refexp-pair-closed, Proposition \\refexp-pair-trivial, \\reftrudgian_yang_e…","labels":[],"detail_key":"p55"},{"id":"n43775","layer":"informal","project":"p55","title":"New exponent pairs","kind":"theorem","summary":"[New exponent pairs] The following are exponent pairs: \\[ \\left(\\frac891282, \\frac9971282\\right…","labels":["new-exp-pair"],"detail_key":"p55"},{"id":"n43776","layer":"informal","project":"p55","title":"new-exp-pair-beta-bounds-table","kind":"proof","summary":"Using the bounds on \\beta(\\alpha) collected in Table \\refnew-exp-pair-beta-bounds-table, one ma…","labels":["new-exp-pair-beta-bounds-table"],"detail_key":"p55"},{"id":"n43777","layer":"informal","project":"p55","title":"Cushing (2025) exponent pairs","kind":"theorem","summary":"[Cushing (2025) exponent pairs] The following are exponent pairs: \\[ \\left(\\frac3114822, \\frac3…","labels":["new-exp-pairs-2"],"detail_key":"p55"},{"id":"n43778","layer":"informal","project":"p55","title":"Growth rate of zeta","kind":"definition","summary":"[Growth rate of zeta] For any fixed \\sigma \\in R, let \\mu(\\sigma) denote the least possible (fi…","labels":["zeta-grow-def"],"detail_key":"p55"},{"id":"n43779","layer":"informal","project":"p55","title":"Trivial bound","kind":"lemma","summary":"[Trivial bound] One has \\mu(\\sigma)=0 for all \\sigma \\geq 1.","labels":["zeta-grow-triv"],"detail_key":"p55"},{"id":"n43780","layer":"informal","project":"p55","title":"Immediate from the absolute convergence of the Dirichlet series for both \\zeta(s) and 1/\\…","kind":"proof","summary":"Immediate from the absolute convergence of the Dirichlet series for both \\zeta(s) and 1/\\zeta(s…","labels":[],"detail_key":"p55"},{"id":"n43781","layer":"informal","project":"p55","title":"Convexity","kind":"lemma","summary":"[Convexity] \\mu is convex.","labels":["zeta-convex"],"detail_key":"p55"},{"id":"n43782","layer":"informal","project":"p55","title":"Immediate from the Phragm\\'en--Lindel\\\"of principle; see e.g., \\cite[\\S A.8]ivic.","kind":"proof","summary":"Immediate from the Phragm\\'en--Lindel\\\"of principle; see e.g., \\cite[\\S A.8]ivic.","labels":[],"detail_key":"p55"},{"id":"n43783","layer":"informal","project":"p55","title":"Functional equation","kind":"lemma","summary":"[Functional equation] One has \\mu(1-\\sigma) = \\mu(\\sigma) + \\sigma - 1/2 for all 0 \\leq \\sigma…","labels":["zeta-functional"],"detail_key":"p55"},{"id":"n43784","layer":"informal","project":"p55","title":"Immediate from the functional equation for \\zeta and asymptotics of the Gamma function; s…","kind":"proof","summary":"Immediate from the functional equation for \\zeta and asymptotics of the Gamma function; see e.g…","labels":[],"detail_key":"p55"},{"id":"n43785","layer":"informal","project":"p55","title":"Left of critical strip","kind":"lemma","summary":"[Left of critical strip] One has \\mu(\\sigma)=1/2-\\sigma for \\sigma \\leq 0.","labels":["zeta-left"],"detail_key":"p55"},{"id":"n43786","layer":"informal","project":"p55","title":"Immediate from Lemmas \\refzeta-grow-triv, \\refzeta-functional.","kind":"proof","summary":"Immediate from Lemmas \\refzeta-grow-triv, \\refzeta-functional.","labels":[],"detail_key":"p55"},{"id":"n43787","layer":"informal","project":"p55","title":"Convexity bounds","kind":"lemma","summary":"[Convexity bounds] One has \\max(0, 1/2-\\sigma) \\leq \\mu(\\sigma) \\leq (1-\\sigma)/2 for 0 \\leq \\s…","labels":["zeta-convexity"],"detail_key":"p55"},{"id":"n43788","layer":"informal","project":"p55","title":"Immediate from Lemma \\refzeta-grow-triv, Lemma \\refzeta-left, and Lemma \\refzeta-convexit…","kind":"proof","summary":"Immediate from Lemma \\refzeta-grow-triv, Lemma \\refzeta-left, and Lemma \\refzeta-convexity.","labels":[],"detail_key":"p55"},{"id":"n43789","layer":"informal","project":"p55","title":"Connection with dual exponent pairs","kind":"lemma","summary":"[Connection with dual exponent pairs] For any 1/2 \\leq \\sigma \\leq 1, one has \\mu(\\sigma) \\leq…","labels":["mu-beta","{zeta-grow-def"],"detail_key":"p55"},{"id":"n43790","layer":"informal","project":"p55","title":"Let t be unbounded. From the Riemann--Siegel formula (see \\cite[Theorem 4.1]ivic) one has…","kind":"proof","summary":"Let t be unbounded. From the Riemann--Siegel formula (see \\cite[Theorem 4.1]ivic) one has \\zeta…","labels":[],"detail_key":"p55"},{"id":"n43791","layer":"informal","project":"p55","title":"Exponent pairs and \\mu","kind":"corollary","summary":"[Exponent pairs and \\mu] If (k,\\ell) is an exponent pair, then \\mu(\\ell-k) \\leq k.","labels":["exp-pair-mu"],"detail_key":"p55"},{"id":"n43792","layer":"informal","project":"p55","title":"Immediate from Lemma \\refmu-beta and Lemma \\refbeta-duality. See also \\cite[(7.57)]ivic.","kind":"proof","summary":"Immediate from Lemma \\refmu-beta and Lemma \\refbeta-duality. See also \\cite[(7.57)]ivic.","labels":[],"detail_key":"p55"},{"id":"n43793","layer":"informal","project":"p55","title":"Lindel\\\"of hypothesis","kind":"conjecture","summary":"[Lindel\\\"of hypothesis] One has \\mu(1/2)=0.","labels":["LH"],"detail_key":"p55"},{"id":"n43794","layer":"informal","project":"p55","title":"exp-pair_implies_lindelof","kind":"lemma","summary":"The exponent pair conjecture implies the Lindel\\\"of hypothesis.","labels":["exp-pair_implies_lindelof"],"detail_key":"p55"},{"id":"n43795","layer":"informal","project":"p55","title":"Immediate from Corollary \\refexp-pair-mu.","kind":"proof","summary":"Immediate from Corollary \\refexp-pair-mu.","labels":[],"detail_key":"p55"},{"id":"n43796","layer":"informal","project":"p55","title":"Conjectured value of \\mu","kind":"proposition","summary":"[Conjectured value of \\mu] We have the lower bound \\mu(\\sigma) \\geq \\max\\left(0, \\frac12-\\sigma…","labels":["mu-conj","muh"],"detail_key":"p55"},{"id":"n43797","layer":"informal","project":"p55","title":"Clearly equality in \\eqrefmuh implies the Lindel\\\"of hypothesis, while from the trivial b…","kind":"proof","summary":"Clearly equality in \\eqrefmuh implies the Lindel\\\"of hypothesis, while from the trivial bounds…","labels":[],"detail_key":"p55"},{"id":"n43798","layer":"informal","project":"p55","title":"Historical bounds","kind":"theorem","summary":"[Historical bounds] The upper bounds on \\mu(\\sigma) given by Table \\refmu-table are known.","labels":["mu-hist"],"detail_key":"p55"},{"id":"n43799","layer":"informal","project":"p55","title":"mu_est_thm","kind":"theorem","summary":"\\cite[Theorems 2.4-2.6]trudgian-yang We have \\[ \\mu(\\sigma) \\le (31 - 36\\sigma)/84 , & \\frac12…","labels":["mu_est_thm"],"detail_key":"p55"},{"id":"n43800","layer":"informal","project":"p55","title":"Heath-Brown \\citeheathbrown_new_2017 \\mu bounds","kind":"theorem","summary":"[Heath-Brown \\citeheathbrown_new_2017 \\mu bounds] For any integer k \\ge 3, one has \\[ \\mu\\left(…","labels":["hb_mu_bounds"],"detail_key":"p55"},{"id":"n43801","layer":"informal","project":"p55","title":"Follows from substituting \\Crefheath-brown_exp_pair_2017 into \\eqrefexp-pair-mu.","kind":"proof","summary":"Follows from substituting \\Crefheath-brown_exp_pair_2017 into \\eqrefexp-pair-mu.","labels":[],"detail_key":"p55"},{"id":"n43802","layer":"informal","project":"p55","title":"Growth exponent and zeroes","kind":"lemma","summary":"[Growth exponent and zeroes] Let 1/2 \\leq \\sigma_0 < 1 be fixed. Then the assertion \\mu(\\sigma_…","labels":[],"detail_key":"p55"},{"id":"n43803","layer":"informal","project":"p55","title":"This is a routine adaptation of Theorem 2 of \\urlhttps://terrytao.wordpress.com/2015/03/0…","kind":"proof","summary":"This is a routine adaptation of Theorem 2 of \\urlhttps://terrytao.wordpress.com/2015/03/01.","labels":[],"detail_key":"p55"},{"id":"n43804","layer":"informal","project":"p55","title":"Large value pattern","kind":"definition","summary":"[Large value pattern] A \\emphlarge value pattern is a tuple (N, T, V, (a_n)_n \\in [N,2N], J, W)…","labels":["large-pattern-def","V-large"],"detail_key":"p55"},{"id":"n43805","layer":"informal","project":"p55","title":"Large value exponent","kind":"definition","summary":"[Large value exponent] Let 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0 be fixed. We define LV(\\sigma…","labels":["lv-def"],"detail_key":"p55"},{"id":"n43806","layer":"informal","project":"p55","title":"Asymptotic form of large value exponent","kind":"lemma","summary":"[Asymptotic form of large value exponent] Let 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, and \\rho \\ge…","labels":["lv-asymp"],"detail_key":"p55"},{"id":"n43807","layer":"informal","project":"p55","title":"Basic properties","kind":"lemma","summary":"[Basic properties] \\item[(i)] (Monotonicity in \\sigma) For any \\tau \\geq 0, \\sigma \\mapsto LV(\\…","labels":["lv-basic"],"detail_key":"p55"},{"id":"n43808","layer":"informal","project":"p55","title":"All claims are clear except perhaps for the upper bound LV(\\sigma,\\tau') \\leq LV(\\sigma,\\…","kind":"proof","summary":"All claims are clear except perhaps for the upper bound LV(\\sigma,\\tau') \\leq LV(\\sigma,\\tau) +…","labels":[],"detail_key":"p55"},{"id":"n43809","layer":"informal","project":"p55","title":"Lower bound","kind":"lemma","summary":"[Lower bound] For any 1/2 < \\sigma \\leq 1 and \\tau \\geq 0, one has LV(\\sigma, \\tau) \\geq \\min(2…","labels":["lv-lower"],"detail_key":"p55"},{"id":"n43810","layer":"informal","project":"p55","title":"In view of Lemma \\reflv-basic(ii), it suffices to show that LV(\\sigma, 2-2\\sigma) \\geq 2-…","kind":"proof","summary":"In view of Lemma \\reflv-basic(ii), it suffices to show that LV(\\sigma, 2-2\\sigma) \\geq 2-2\\sigm…","labels":[],"detail_key":"p55"},{"id":"n43811","layer":"informal","project":"p55","title":"Montgomery conjecture","kind":"conjecture","summary":"[Montgomery conjecture] One has LV(\\sigma, \\tau) \\leq 2 - 2 \\sigma for all fixed 1/2 < \\sigma \\…","labels":["montgomery-conj","mont-conj"],"detail_key":"p55"},{"id":"n43812","layer":"informal","project":"p55","title":"Subdivision and the Montgomery conjecture","kind":"lemma","summary":"[Subdivision and the Montgomery conjecture] If \\sigma is fixed, and the Montgomery conjecture h…","labels":["montgomery-subdivide","lvt-opt"],"detail_key":"p55"},{"id":"n43813","layer":"informal","project":"p55","title":"Clear from Lemma \\reflv-basic(ii).","kind":"proof","summary":"Clear from Lemma \\reflv-basic(ii).","labels":[],"detail_key":"p55"},{"id":"n43814","layer":"informal","project":"p55","title":"Raising to a power","kind":"lemma","summary":"[Raising to a power] For any 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, and natural number k, one has…","labels":["power-lemma"],"detail_key":"p55"},{"id":"n43815","layer":"informal","project":"p55","title":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with T = N^k\\tau+o(1) and V =…","kind":"proof","summary":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with T = N^k\\tau+o(1) and V = N^\\si…","labels":[],"detail_key":"p55"},{"id":"n43816","layer":"informal","project":"p55","title":"L^2 mean value theorem","kind":"theorem","summary":"[L^2 mean value theorem] For any fixed 1/2 \\leq \\sigma \\leq 1 and \\tau\\geq 0 one has LV(\\sigma,…","labels":["l2-mvt"],"detail_key":"p55"},{"id":"n43817","layer":"informal","project":"p55","title":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with T = N^\\tau + o(1), V = N…","kind":"proof","summary":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with T = N^\\tau + o(1), V = N^\\sigm…","labels":[],"detail_key":"p55"},{"id":"n43818","layer":"informal","project":"p55","title":"Montgomery large values theorem","kind":"theorem","summary":"[Montgomery large values theorem] If 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0 is such that \\sup_1…","labels":["montgomery-lv","tab"],"detail_key":"p55"},{"id":"n43819","layer":"informal","project":"p55","title":"Set \\rho := LV(\\sigma,\\tau); we may assume without loss of generality that \\rho \\geq 0. T…","kind":"proof","summary":"Set \\rho := LV(\\sigma,\\tau); we may assume without loss of generality that \\rho \\geq 0. Then by…","labels":[],"detail_key":"p55"},{"id":"n43820","layer":"informal","project":"p55","title":"Converting an exponent pair to a large values theorem","kind":"corollary","summary":"[Converting an exponent pair to a large values theorem] If (k,\\ell) is an exponent pair, and 1/…","labels":["exp-lv"],"detail_key":"p55"},{"id":"n43821","layer":"informal","project":"p55","title":"By Lemma \\refmontgomery-subdivide it suffices to prove the latter claim. From Lemma \\refb…","kind":"proof","summary":"By Lemma \\refmontgomery-subdivide it suffices to prove the latter claim. From Lemma \\refbeta-du…","labels":[],"detail_key":"p55"},{"id":"n43822","layer":"informal","project":"p55","title":"Huxley large values theorem","kind":"theorem","summary":"[Huxley large values theorem] \\cite[Equation~(2.9)]Huxley Let 1/2 \\leq \\sigma \\leq 1 and \\tau \\…","labels":["huxley-lv"],"detail_key":"p55"},{"id":"n43823","layer":"informal","project":"p55","title":"Apply Corollary \\refexp-lv with the pair (k,\\ell) = (1/2,1/2) from Lemma \\refvdc-class.","kind":"proof","summary":"Apply Corollary \\refexp-lv with the pair (k,\\ell) = (1/2,1/2) from Lemma \\refvdc-class.","labels":[],"detail_key":"p55"},{"id":"n43824","layer":"informal","project":"p55","title":"Heath-Brown large values theorem, preliminary form","kind":"theorem","summary":"[Heath-Brown large values theorem, preliminary form] Let 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0…","labels":["heath_brown-lv-prelim"],"detail_key":"p55"},{"id":"n43825","layer":"informal","project":"p55","title":"Follows from \\cite[Lemma~1]heathbrown_zero_1979.","kind":"proof","summary":"Follows from \\cite[Lemma~1]heathbrown_zero_1979.","labels":[],"detail_key":"p55"},{"id":"n43826","layer":"informal","project":"p55","title":"Heath-Brown large values theorem, optimized","kind":"theorem","summary":"[Heath-Brown large values theorem, optimized] Let 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0 be fix…","labels":["hb-opt"],"detail_key":"p55"},{"id":"n43827","layer":"informal","project":"p55","title":"By Lemma \\refmontgomery-subdivide it suffices to show that LV(\\sigma,\\tau) \\leq 2-2\\sigma…","kind":"proof","summary":"By Lemma \\refmontgomery-subdivide it suffices to show that LV(\\sigma,\\tau) \\leq 2-2\\sigma for \\…","labels":[],"detail_key":"p55"},{"id":"n43828","layer":"informal","project":"p55","title":"Second Heath-Brown large values theorem","kind":"lemma","summary":"[Second Heath-Brown large values theorem] If 3/4 < \\sigma \\leq 1 and \\tau \\geq 0 are fixed, the…","labels":["hb-lvt-2"],"detail_key":"p55"},{"id":"n43829","layer":"informal","project":"p55","title":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with N \\geq 1 be unbounded, T…","kind":"proof","summary":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a large value pattern with N \\geq 1 be unbounded, T = N^\\…","labels":[],"detail_key":"p55"},{"id":"n43830","layer":"informal","project":"p55","title":"Jutila large values theorem","kind":"theorem","summary":"[Jutila large values theorem] For any integer k \\geq 1, one has LV(\\sigma,\\tau) \\leq \\max(2-2\\s…","labels":["jutila-lvt"],"detail_key":"p55"},{"id":"n43831","layer":"informal","project":"p55","title":"See \\cite[(1.4)]jutila_zero_density_1977 (setting V = N^\\sigma+o(1), T = N^\\tau+o(1), and…","kind":"proof","summary":"See \\cite[(1.4)]jutila_zero_density_1977 (setting V = N^\\sigma+o(1), T = N^\\tau+o(1), and G \\le…","labels":[],"detail_key":"p55"},{"id":"n43832","layer":"informal","project":"p55","title":"Large value zeta exponent","kind":"definition","summary":"","labels":["lvz-def"],"detail_key":"p55"},{"id":"n43833","layer":"informal","project":"p55","title":"Asymptotic form of large value exponent at zeta","kind":"lemma","summary":"[Asymptotic form of large value exponent at zeta] Let 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, and…","labels":["lvz-asymp"],"detail_key":"p55"},{"id":"n43834","layer":"informal","project":"p55","title":"Basic properties","kind":"lemma","summary":"[Basic properties] \\item[(i)] (Monotonicity in \\sigma) For any \\tau \\geq 0, \\sigma \\mapsto LV_\\…","labels":["lvz-basic"],"detail_key":"p55"},{"id":"n43835","layer":"informal","project":"p55","title":"r-targ","kind":"proof","summary":"The claims (i), (ii) are obvious. The claim (iii) is clear by setting a_n = 1_I in Definition \\…","labels":["r-targ"],"detail_key":"p55"},{"id":"n43836","layer":"informal","project":"p55","title":"Characterization of negative infinite value","kind":"lemma","summary":"[Characterization of negative infinite value] Let 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0 be fix…","labels":["lvz-infty"],"detail_key":"p55"},{"id":"n43837","layer":"informal","project":"p55","title":"Clearly (i) implies (ii). If (iii) holds, then in any zeta large value pattern (N,T,V,(a_…","kind":"proof","summary":"Clearly (i) implies (ii). If (iii) holds, then in any zeta large value pattern (N,T,V,(a_n)_n \\…","labels":[],"detail_key":"p55"},{"id":"n43838","layer":"informal","project":"p55","title":"beta-zeta-vanish","kind":"corollary","summary":"If \\tau \\geq 0 is fixed then LV_\\zeta(\\sigma,\\tau) = -\\infty whenever \\sigma > \\tau \\beta(1/\\ta…","labels":["beta-zeta-vanish"],"detail_key":"p55"},{"id":"n43839","layer":"informal","project":"p55","title":"Suppose one has data N, I obeying the hypotheses of Lemma \\reflvz-infty(iii), then by \\eq…","kind":"proof","summary":"Suppose one has data N, I obeying the hypotheses of Lemma \\reflvz-infty(iii), then by \\eqrefbet…","labels":[],"detail_key":"p55"},{"id":"n43840","layer":"informal","project":"p55","title":"lvz-mu","kind":"corollary","summary":"If \\tau > 0 and 1/2 \\leq \\sigma_0 \\leq 1 are fixed, then LV_\\zeta(\\sigma,\\tau) = -\\infty whenev…","labels":["lvz-mu"],"detail_key":"p55"},{"id":"n43841","layer":"informal","project":"p55","title":"From Definition \\refzeta-grow-def one has \\zeta(\\sigma_0 + it) \\ll |t|^\\mu(\\sigma_0) + o(…","kind":"proof","summary":"From Definition \\refzeta-grow-def one has \\zeta(\\sigma_0 + it) \\ll |t|^\\mu(\\sigma_0) + o(1) for…","labels":[],"detail_key":"p55"},{"id":"n43842","layer":"informal","project":"p55","title":"lvz-exp","kind":"corollary","summary":"If (k,\\ell) is an exponent pair, then LV_\\zeta(\\sigma,\\tau) = -\\infty whenever 1/2 \\leq \\sigma…","labels":["lvz-exp"],"detail_key":"p55"},{"id":"n43843","layer":"informal","project":"p55","title":"Immediate from Corollary \\refbeta-zeta-vanish and Lemma \\refbeta-duality; alternatively,…","kind":"proof","summary":"Immediate from Corollary \\refbeta-zeta-vanish and Lemma \\refbeta-duality; alternatively, one ca…","labels":[],"detail_key":"p55"},{"id":"n43844","layer":"informal","project":"p55","title":"lh-vanish","kind":"corollary","summary":"Assuming the Lindelof hypothesis, one has LV_\\zeta(\\sigma,\\tau) = -\\infty whenever \\sigma > 1/2…","labels":["lh-vanish"],"detail_key":"p55"},{"id":"n43845","layer":"informal","project":"p55","title":"Apply Corollary \\reflvz-mu with \\sigma_0=1/2, so that \\mu(\\sigma_0) vanishes from the Lin…","kind":"proof","summary":"Apply Corollary \\reflvz-mu with \\sigma_0=1/2, so that \\mu(\\sigma_0) vanishes from the Lindelof…","labels":[],"detail_key":"p55"},{"id":"n43846","layer":"informal","project":"p55","title":"Value at \\sigma=1/2","kind":"lemma","summary":"[Value at \\sigma=1/2] One has LV_\\zeta(1/2,\\tau) = \\tau for all \\tau \\geq 1.","labels":["lvz-2"],"detail_key":"p55"},{"id":"n43847","layer":"informal","project":"p55","title":"The upper bound LV_\\zeta(1/2,\\tau) \\leq \\tau follows from Lemma \\reflvz-basic(ii), so it…","kind":"proof","summary":"The upper bound LV_\\zeta(1/2,\\tau) \\leq \\tau follows from Lemma \\reflvz-basic(ii), so it suffic…","labels":[],"detail_key":"p55"},{"id":"n43848","layer":"informal","project":"p55","title":"Value at \\tau<1","kind":"lemma","summary":"[Value at \\tau<1] If 0 \\leq \\tau < 1, then LV_\\zeta(\\sigma,\\tau) is equal to -\\infty for \\sigma…","labels":["lvz-small-tau"],"detail_key":"p55"},{"id":"n43849","layer":"informal","project":"p55","title":"The first claim follows from Corollary \\refbeta-zeta-vanish and Lemma \\refbeta-triv. For…","kind":"proof","summary":"The first claim follows from Corollary \\refbeta-zeta-vanish and Lemma \\refbeta-triv. For the se…","labels":[],"detail_key":"p55"},{"id":"n43850","layer":"informal","project":"p55","title":"From exponent pairs to zeta large values estimate","kind":"lemma","summary":"[From exponent pairs to zeta large values estimate] \\cite[Theorem 8.2]ivic If (k,\\ell) is an ex…","labels":["zeta-from-exp"],"detail_key":"p55"},{"id":"n43851","layer":"informal","project":"p55","title":"Hal\\'asz--Montgomery inequality","kind":"lemma","summary":"[Hal\\'asz--Montgomery inequality] For any 1/2 \\leq \\sigma \\leq 1 and \\tau \\geq 0, we have LV(\\s…","labels":["hl-improv"],"detail_key":"p55"},{"id":"n43852","layer":"informal","project":"p55","title":"It suffices to show that LV(\\sigma,\\tau) \\leq \\max\\left(2-2\\sigma, 1 - 2\\sigma + \\sup_\\st…","kind":"proof","summary":"It suffices to show that LV(\\sigma,\\tau) \\leq \\max\\left(2-2\\sigma, 1 - 2\\sigma + \\sup_\\stackrel…","labels":[],"detail_key":"p55"},{"id":"n43853","layer":"informal","project":"p55","title":"Converting a bound on \\mu to a large values theorem","kind":"corollary","summary":"[Converting a bound on \\mu to a large values theorem] If 1/2 \\leq \\sigma \\leq 1, \\sigma' \\leq 1…","labels":["mu-lv"],"detail_key":"p55"},{"id":"n43854","layer":"informal","project":"p55","title":"By Lemma \\refmontgomery-subdivide it suffices to verify the claim for \\tau < \\frac2\\sigma…","kind":"proof","summary":"By Lemma \\refmontgomery-subdivide it suffices to verify the claim for \\tau < \\frac2\\sigma-1-\\si…","labels":[],"detail_key":"p55"},{"id":"n43855","layer":"informal","project":"p55","title":"Hal\\'asz-Tur\\'an large values theorem","kind":"theorem","summary":"[Hal\\'asz-Tur\\'an large values theorem] \\cite[Theorem 1]halasz_distribution_1969 On the Lindel\\…","labels":["htlv"],"detail_key":"p55"},{"id":"n43856","layer":"informal","project":"p55","title":"Immediate from Corollary \\refmu-lv, since \\mu(1/2)=0 in this case.","kind":"proof","summary":"Immediate from Corollary \\refmu-lv, since \\mu(1/2)=0 in this case.","labels":[],"detail_key":"p55"},{"id":"n43857","layer":"informal","project":"p55","title":"First Ivic large values theorem","kind":"theorem","summary":"[First Ivic large values theorem] \\cite[Lemma 8.2]ivic If \\tau \\geq 0 and 1/2 < \\sigma < \\sigma…","labels":["ivic-lvt-82"],"detail_key":"p55"},{"id":"n43858","layer":"informal","project":"p55","title":"We set \\theta to equal (3\\sigma-2)/(2\\sigma-1) \\hbox for 1/2 < \\sigma \\leq 2/3; (9\\sigma-…","kind":"proof","summary":"We set \\theta to equal (3\\sigma-2)/(2\\sigma-1) \\hbox for 1/2 < \\sigma \\leq 2/3; (9\\sigma-6)/(4\\…","labels":[],"detail_key":"p55"},{"id":"n43859","layer":"informal","project":"p55","title":"Second Ivic large values theorem","kind":"lemma","summary":"[Second Ivic large values theorem] \\cite[(11.40)]ivic For any 1/2 \\leq \\sigma \\leq 1 and \\tau \\…","labels":["ivic-lvt"],"detail_key":"p55"},{"id":"n43860","layer":"informal","project":"p55","title":"Write \\rho := LV(\\sigma,\\tau), and let \\varepsilon>0 be arbitrary. By Lemma \\refhl-improv…","kind":"proof","summary":"Write \\rho := LV(\\sigma,\\tau), and let \\varepsilon>0 be arbitrary. By Lemma \\refhl-improv, we m…","labels":[],"detail_key":"p55"},{"id":"n43861","layer":"informal","project":"p55","title":"Zeta moment exponents","kind":"definition","summary":"[Zeta moment exponents] For fixed \\sigma \\in R and A \\geq 0, we define M(\\sigma,A) to be the le…","labels":["zeta-moment-def"],"detail_key":"p55"},{"id":"n43862","layer":"informal","project":"p55","title":"Basic properties of M(\\sigma,A)","kind":"lemma","summary":"[Basic properties of M(\\sigma,A)] \\ \\item[(i)] M(\\sigma,A) is convex in \\sigma. \\item[(ii)] For…","labels":["zeta-moment-basic"],"detail_key":"p55"},{"id":"n43863","layer":"informal","project":"p55","title":"The claim (i) follows from the Phragmen-Lindel\\\"of principle. The claim (ii) follows from…","kind":"proof","summary":"The claim (i) follows from the Phragmen-Lindel\\\"of principle. The claim (ii) follows from H\\\"ol…","labels":[],"detail_key":"p55"},{"id":"n43864","layer":"informal","project":"p55","title":"Relationship with Lindel\\\"of hypothesis","kind":"corollary","summary":"[Relationship with Lindel\\\"of hypothesis] If the Lindel\\\"of hypothesis holds, then M(\\sigma,A)…","labels":["moment_from_lindelof"],"detail_key":"p55"},{"id":"n43865","layer":"informal","project":"p55","title":"mad_known_est","kind":"lemma","summary":"One has M(1/2,A)=1 for all 0 \\leq A \\leq 4.","labels":["mad_known_est"],"detail_key":"p55"},{"id":"n43866","layer":"informal","project":"p55","title":"Follows from H\\\"older's inequality and the standard estimates \\int_T^2T |\\zeta(1/2+it)|^2…","kind":"proof","summary":"Follows from H\\\"older's inequality and the standard estimates \\int_T^2T |\\zeta(1/2+it)|^2\\ dt =…","labels":[],"detail_key":"p55"},{"id":"n43867","layer":"informal","project":"p55","title":"mad","kind":"lemma","summary":"If 1/2 \\leq \\sigma_0 \\leq 1 and A \\geq 1, then M(\\sigma_0,A) = \\sup_\\tau \\geq 2; \\sigma \\geq 1/…","labels":["mad","M-form"],"detail_key":"p55"},{"id":"n43868","layer":"informal","project":"p55","title":"We first show the lower bound, or equivalently that A(\\sigma-\\sigma_0) + LV_\\zeta(\\sigma,…","kind":"proof","summary":"We first show the lower bound, or equivalently that A(\\sigma-\\sigma_0) + LV_\\zeta(\\sigma,\\tau)…","labels":[],"detail_key":"p55"},{"id":"n43869","layer":"informal","project":"p55","title":"Fourth moment bound","kind":"corollary","summary":"[Fourth moment bound] One has LV_\\zeta(\\sigma,\\tau) \\leq \\tau - 4 (\\sigma-1/2) for all 1/2 \\leq…","labels":["lvz-4"],"detail_key":"p55"},{"id":"n43870","layer":"informal","project":"p55","title":"Apply Lemma \\refmad with \\sigma_0 = 1/2 and A=4, using Lemma \\refzeta-moment-basic(iv).","kind":"proof","summary":"Apply Lemma \\refmad with \\sigma_0 = 1/2 and A=4, using Lemma \\refzeta-moment-basic(iv).","labels":[],"detail_key":"p55"},{"id":"n43871","layer":"informal","project":"p55","title":"Heath-Brown twelfth moment estimate","kind":"theorem","summary":"[Heath-Brown twelfth moment estimate] \\citeheathbrown_twelfth_1978 M(1/2,12) \\leq 2. Equivalent…","labels":["hb-12"],"detail_key":"p55"},{"id":"n43872","layer":"informal","project":"p55","title":"From Lemma \\refzeta-from-exp with the exponent pair (1/2,1/2) from Lemma \\refvdc-class we…","kind":"proof","summary":"From Lemma \\refzeta-from-exp with the exponent pair (1/2,1/2) from Lemma \\refvdc-class we have…","labels":[],"detail_key":"p55"},{"id":"n43873","layer":"informal","project":"p55","title":"Auxiliary Heath-Brown estimate","kind":"theorem","summary":"[Auxiliary Heath-Brown estimate] For \\tau \\geq 2 and 1/2 \\leq \\sigma \\leq 1, one has LV_\\zeta(\\…","labels":["hb-12-aux"],"detail_key":"p55"},{"id":"n43874","layer":"informal","project":"p55","title":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a zeta large value pattern with N, V = N^\\sigma+o(1…","kind":"proof","summary":"Let (N,T,V,(a_n)_n \\in [N,2N],J,W) be a zeta large value pattern with N, V = N^\\sigma+o(1), T =…","labels":[],"detail_key":"p55"},{"id":"n43875","layer":"informal","project":"p55","title":"Ivic's table of moment bounds","kind":"lemma","summary":"[Ivic's table of moment bounds] \\cite[Theorem 8.4]ivic We have M(\\sigma,A) = 1 when A is equal…","labels":["ivic-moment"],"detail_key":"p55"},{"id":"n43876","layer":"informal","project":"p55","title":"This is a computation using Lemma \\refzeta-from-exp, Theorem \\refivic-lvt-82, and Lemma \\…","kind":"proof","summary":"This is a computation using Lemma \\refzeta-from-exp, Theorem \\refivic-lvt-82, and Lemma \\refmad…","labels":[],"detail_key":"p55"},{"id":"n43877","layer":"informal","project":"p55","title":"Moment bounds for \\sigma=1/2","kind":"theorem","summary":"[Moment bounds for \\sigma=1/2] \\cite[Theorems 2.1, 2.2]trudgian-yang We have \\[ M(1/2, A) \\leq…","labels":["M_bound_larger_A"],"detail_key":"p55"},{"id":"n43878","layer":"informal","project":"p55","title":"Mixed moments","kind":"definition","summary":"[Mixed moments] For fixed 1/2 \\leq \\sigma \\leq 1, A \\geq 0, and h \\geq 0, let M(\\sigma,A, \\geq…","labels":["mixed-moment-def"],"detail_key":"p55"},{"id":"n43879","layer":"informal","project":"p55","title":"Mixed moments and large values of zeta","kind":"lemma","summary":"[Mixed moments and large values of zeta] If 1/2 \\leq \\sigma_0 \\leq 1, A \\geq 1, and h \\geq 0 ar…","labels":["mad-variant","msah","msah-2"],"detail_key":"p55"},{"id":"n43880","layer":"informal","project":"p55","title":"This is a routine modification of the proof of Lemma \\refmad.","kind":"proof","summary":"This is a routine modification of the proof of Lemma \\refmad.","labels":[],"detail_key":"p55"},{"id":"n43881","layer":"informal","project":"p55","title":"Mixed moments and exponent pairs","kind":"corollary","summary":"[Mixed moments and exponent pairs] If (k,\\ell) is an exponent pair with k > 0, then M(1/2,6, \\g…","labels":["ivic-split"],"detail_key":"p55"},{"id":"n43882","layer":"informal","project":"p55","title":"From Lemma \\refzeta-from-exp with the exponent pair (k,\\ell) we have LV_\\zeta(\\sigma,\\tau…","kind":"proof","summary":"From Lemma \\refzeta-from-exp with the exponent pair (k,\\ell) we have LV_\\zeta(\\sigma,\\tau) \\leq…","labels":[],"detail_key":"p55"},{"id":"n43883","layer":"informal","project":"p55","title":"Specific mixed moments","kind":"corollary","summary":"[Specific mixed moments] \\cite[(8.56)]ivic M(1/2, 6, \\geq 11/72) \\leq 1 and M(1/2,24, \\leq 11/7…","labels":["ivic-6-large"],"detail_key":"p55"},{"id":"n43884","layer":"informal","project":"p55","title":"Apply Corollary \\refivic-split with the exponent pair (4/18, 11/18) = BABA(1/6, 2/3) from…","kind":"proof","summary":"Apply Corollary \\refivic-split with the exponent pair (4/18, 11/18) = BABA(1/6, 2/3) from Corol…","labels":[],"detail_key":"p55"},{"id":"n43885","layer":"informal","project":"p55","title":"Large value theorems from mixed moment bounds","kind":"lemma","summary":"[Large value theorems from mixed moment bounds] \\cite[Proposition 2]bourgain_remarks_1995 Suppo…","labels":["bourgain-remark-1"],"detail_key":"p55"},{"id":"n43886","layer":"informal","project":"p55","title":"Zero density theorems from mixed moment bounds","kind":"lemma","summary":"[Zero density theorems from mixed moment bounds] \\cite[Proposition 5]bourgain_remarks_1995 Supp…","labels":["bourgain-remark-2"],"detail_key":"p55"},{"id":"n43887","layer":"informal","project":"p55","title":"Chen-Debruyne-Vidas large values theorem","kind":"lemma","summary":"[Chen-Debruyne-Vidas large values theorem] \\cite[Lemma A.1]chen_debruyne_vindas_density_2024 Le…","labels":["cdv-lv"],"detail_key":"p55"},{"id":"n43888","layer":"informal","project":"p55","title":"Additive energy","kind":"definition","summary":"[Additive energy] Let W be a finite set of real numbers. The \\emphadditive energy E_1(W) of suc…","labels":["energy-def"],"detail_key":"p55"},{"id":"n43889","layer":"informal","project":"p55","title":"Basic properties of additive energy","kind":"lemma","summary":"[Basic properties of additive energy] \\item[(i)] If W is a finite set of reals, then E_1(W) \\as…","labels":["add-energy"],"detail_key":"p55"},{"id":"n43890","layer":"informal","project":"p55","title":"For (i), we just prove the first estimate, as the second follows from the first by severa…","kind":"proof","summary":"For (i), we just prove the first estimate, as the second follows from the first by several appl…","labels":[],"detail_key":"p55"},{"id":"n43891","layer":"informal","project":"p55","title":"Cauchy--Schwarz and double \\zeta-sums","kind":"lemma","summary":"[Cauchy--Schwarz and double \\zeta-sums] \\cite[Lemma 3.4]bourgain_dirichlet_2000 If W,W' are fin…","labels":["cauchy-schwarz","ttww"],"detail_key":"p55"},{"id":"n43892","layer":"informal","project":"p55","title":"The left-hand side of \\eqrefttww can be rewritten as \\sum_n,m \\in [N,2N] a_n \\overlinea_m…","kind":"proof","summary":"The left-hand side of \\eqrefttww can be rewritten as \\sum_n,m \\in [N,2N] a_n \\overlinea_m \\over…","labels":[],"detail_key":"p55"},{"id":"n43893","layer":"informal","project":"p55","title":"Energy controlled by third moment","kind":"lemma","summary":"[Energy controlled by third moment] Suppose that (N,T,V,(a_n)_n \\in [N,2N],J,W) is a large valu…","labels":["energy-third"],"detail_key":"p55"},{"id":"n43894","layer":"informal","project":"p55","title":"By hypothesis, we have V^2 E_1(W) \\leq \\sum_t_1,t_2,t_3,t_4 \\in W: |t_1+t_2-t_3-t_4| \\leq…","kind":"proof","summary":"By hypothesis, we have V^2 E_1(W) \\leq \\sum_t_1,t_2,t_3,t_4 \\in W: |t_1+t_2-t_3-t_4| \\leq 1 \\le…","labels":[],"detail_key":"p55"},{"id":"n43895","layer":"informal","project":"p55","title":"wtu","kind":"lemma","summary":"If W \\subset [-T,T] is 1-separated and 1 \\leq N \\ll T^O(1), then one has S_4(N,W) \\ll T^o(1) u^…","labels":["wtu","v1","v2"],"detail_key":"p55"},{"id":"n43896","layer":"informal","project":"p55","title":"One can bound S_4(N,W) \\ll T^o(1) \\sum_t_1,t_2,t_3,t_4 \\in W \\int_t = t_1+t_2-t_3-t_4+O(T…","kind":"proof","summary":"One can bound S_4(N,W) \\ll T^o(1) \\sum_t_1,t_2,t_3,t_4 \\in W \\int_t = t_1+t_2-t_3-t_4+O(T^o(1))…","labels":[],"detail_key":"p55"},{"id":"n43897","layer":"informal","project":"p55","title":"Large value energy region","kind":"definition","summary":"[Large value energy region] The \\emphlarge value energy region E\\subset R^5 is defined to be th…","labels":["lv-edef"],"detail_key":"p55"},{"id":"n43898","layer":"informal","project":"p55","title":"Trivial containment","kind":"lemma","summary":"[Trivial containment] We have E_\\zeta \\subset E.","labels":["triv-contain"],"detail_key":"p55"},{"id":"n43899","layer":"informal","project":"p55","title":"energy-region-lv","kind":"lemma","summary":"For any fixed 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, we have LV(\\sigma,\\tau) = \\sup \\ \\rho: (\\sig…","labels":["energy-region-lv"],"detail_key":"p55"},{"id":"n43900","layer":"informal","project":"p55","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p55"},{"id":"n43901","layer":"informal","project":"p55","title":"lvze-def","kind":"definition","summary":"For any fixed 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, we define LV^*(\\sigma,\\tau) := \\sup \\ \\rho^*…","labels":["lvze-def"],"detail_key":"p55"},{"id":"n43902","layer":"informal","project":"p55","title":"Non-asymptotic form of large value energy region","kind":"lemma","summary":"[Non-asymptotic form of large value energy region] Let 1/2 \\leq \\sigma \\leq 1, \\tau \\geq 0, \\rh…","labels":["lve-asymp"],"detail_key":"p55"},{"id":"n43903","layer":"informal","project":"p55","title":"Basic properties","kind":"lemma","summary":"[Basic properties] \\ \\item[(i)] (Monotonicity in \\sigma) If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E,…","labels":["lve-basic"],"detail_key":"p55"},{"id":"n43904","layer":"informal","project":"p55","title":"The claim (i) is trivial, so we turn to (ii). By definition, there exists a large value p…","kind":"proof","summary":"The claim (i) is trivial, so we turn to (ii). By definition, there exists a large value pattern…","labels":[],"detail_key":"p55"},{"id":"n43905","layer":"informal","project":"p55","title":"Raising to a power","kind":"lemma","summary":"","labels":["power-energy"],"detail_key":"p55"},{"id":"n43906","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N >…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N > 1 unb…","labels":[],"detail_key":"p55"},{"id":"n43907","layer":"informal","project":"p55","title":"Monotonicity criterion","kind":"lemma","summary":"[Monotonicity criterion] Let E_1 be the intersection of sets E_i, each of the form \\(\\sigma, \\t…","labels":["lver_e_mono_crit"],"detail_key":"p55"},{"id":"n43908","layer":"informal","project":"p55","title":"Suppose tha","kind":"proof","summary":"Suppose tha","labels":[],"detail_key":"p55"},{"id":"n43909","layer":"informal","project":"p55","title":"Raising to a power, alternative formulation","kind":"lemma","summary":"[Raising to a power, alternative formulation] Let k be a positive integer, E_1 \\subseteq R^5 be…","labels":[],"detail_key":"p55"},{"id":"n43910","layer":"informal","project":"p55","title":"Suppose that (\\sigma, \\tau, \\rho, \\rho^*, s) \\in E\\subseteq E_1. By Lemma \\refpower-energ…","kind":"proof","summary":"Suppose that (\\sigma, \\tau, \\rho, \\rho^*, s) \\in E\\subseteq E_1. By Lemma \\refpower-energy and…","labels":[],"detail_key":"p55"},{"id":"n43911","layer":"informal","project":"p55","title":"Reflection principle","kind":"theorem","summary":"[Reflection principle] \\cite[\\S 11.5]ivic If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E with \\sigma \\geq…","labels":["reflect"],"detail_key":"p55"},{"id":"n43912","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43913","layer":"informal","project":"p55","title":"easy-double-zeta-bound","kind":"lemma","summary":"If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E with \\tau < 1, then s \\leq \\max(\\rho+1, 2\\rho)+1.","labels":["easy-double-zeta-bound"],"detail_key":"p55"},{"id":"n43914","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43915","layer":"informal","project":"p55","title":"double-zeta-from-exp-pair","kind":"lemma","summary":"\\cite[Lemma 11.2]ivic If (k,\\ell) is an exponent pair with k>0, and (\\sigma,\\tau,\\rho,\\rho^*,s)…","labels":["double-zeta-from-exp-pair"],"detail_key":"p55"},{"id":"n43916","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43917","layer":"informal","project":"p55","title":"Heath-Brown bound on double sums","kind":"lemma","summary":"[Heath-Brown bound on double sums] If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E, then s \\leq \\max( \\rho…","labels":["hb-double"],"detail_key":"p55"},{"id":"n43918","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43919","layer":"informal","project":"p55","title":"wtu-alt","kind":"lemma","summary":"If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E, then there exists (\\sigma,\\tau,\\rho',(\\rho')^*,s') \\in E…","labels":["wtu-alt"],"detail_key":"p55"},{"id":"n43920","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N \\…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N \\geq 1…","labels":[],"detail_key":"p55"},{"id":"n43921","layer":"informal","project":"p55","title":"Heath-Brown relation","kind":"theorem","summary":"[Heath-Brown relation] \\cite[(33)]heathbrown_zero_1979 If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E, th…","labels":["hbt"],"detail_key":"p55"},{"id":"n43922","layer":"informal","project":"p55","title":"By Lemma \\refwtu-alt we have \\rho^* + 2\\sigma \\leq \\kappa + ( \\max( \\rho+1, 2\\rho, 5\\rho/…","kind":"proof","summary":"By Lemma \\refwtu-alt we have \\rho^* + 2\\sigma \\leq \\kappa + ( \\max( \\rho+1, 2\\rho, 5\\rho/4+\\tau…","labels":[],"detail_key":"p55"},{"id":"n43923","layer":"informal","project":"p55","title":"Simplified Heath-Brown relation","kind":"corollary","summary":"[Simplified Heath-Brown relation] If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E and \\tau \\leq 3/2, then…","labels":["hb-energy-simp"],"detail_key":"p55"},{"id":"n43924","layer":"informal","project":"p55","title":"Apply the previous result. For \\tau \\leq 3/2 we observe that 5\\rho/4+\\tau/2 is less than…","kind":"proof","summary":"Apply the previous result. For \\tau \\leq 3/2 we observe that 5\\rho/4+\\tau/2 is less than 5\\rho/…","labels":[],"detail_key":"p55"},{"id":"n43925","layer":"informal","project":"p55","title":"If (k, \\ell) be an exponent pair with k > 0 and (\\sigma, \\tau, \\rho, \\rho^*, s)\\in E, the…","kind":"theorem","summary":"If (k, \\ell) be an exponent pair with k > 0 and (\\sigma, \\tau, \\rho, \\rho^*, s)\\in E, then \\rho…","labels":[],"detail_key":"p55"},{"id":"n43926","layer":"informal","project":"p55","title":"kappa-bounds","kind":"proof","summary":"By Lemma \\refwtu-alt and Lemma \\refdouble-zeta-from-exp-pair, there exists some (\\sigma, \\tau,…","labels":["kappa-bounds"],"detail_key":"p55"},{"id":"n43927","layer":"informal","project":"p55","title":"Second Heath-Brown relation","kind":"lemma","summary":"[Second Heath-Brown relation] If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E then \\rho \\leq \\max( 2-2\\sig…","labels":["hbt-2"],"detail_key":"p55"},{"id":"n43928","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43929","layer":"informal","project":"p55","title":"Guth-Maynard relation","kind":"lemma","summary":"[Guth-Maynard relation] If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E then \\rho \\leq \\max(2-2\\sigma, 1-2…","labels":["gm-1"],"detail_key":"p55"},{"id":"n43930","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43931","layer":"informal","project":"p55","title":"Second Guth-Maynard relation","kind":"lemma","summary":"[Second Guth-Maynard relation]\\cite[Lemma 1.7]guth-maynard If (\\sigma,\\tau,\\rho,\\rho^*,s) \\in E…","labels":[],"detail_key":"p55"},{"id":"n43932","layer":"informal","project":"p55","title":"By definition, we can find a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, we can find a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unboun…","labels":[],"detail_key":"p55"},{"id":"n43933","layer":"informal","project":"p55","title":"Third Guth-Maynard relation","kind":"lemma","summary":"[Third Guth-Maynard relation] If (\\sigma,\\tau,\\rho,\\rho^*, s) \\in E and 1 \\leq \\tau \\leq 4/3, t…","labels":["gm-3"],"detail_key":"p55"},{"id":"n43934","layer":"informal","project":"p55","title":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1…","kind":"proof","summary":"By definition, there exists a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,W) with N>1 unbou…","labels":[],"detail_key":"p55"},{"id":"n43935","layer":"informal","project":"p55","title":"Guth--Maynard large values theorem","kind":"theorem","summary":"[Guth--Maynard large values theorem] \\cite[Theorem~1.1]guth-maynard One has LV(\\sigma,\\tau) \\le…","labels":["guth-maynard-lvt"],"detail_key":"p55"},{"id":"n43936","layer":"informal","project":"p55","title":"For \\sigma \\leq 7/10 this follows from Lemma \\refl2-mvt, and for \\sigma \\geq 8/10 it foll…","kind":"proof","summary":"For \\sigma \\leq 7/10 this follows from Lemma \\refl2-mvt, and for \\sigma \\geq 8/10 it follows fr…","labels":[],"detail_key":"p55"},{"id":"n43937","layer":"informal","project":"p55","title":"Additional Guth--Maynard large values estimate","kind":"theorem","summary":"[Additional Guth--Maynard large values estimate] For any 1/2 < \\sigma < 1, \\tau \\geq 1, and nat…","labels":["guth-maynard-extra","general-tau","second-bound","third-bound"],"detail_key":"p55"},{"id":"n43938","layer":"informal","project":"p55","title":"(Sketch) The first bound \\eqrefgeneral-tau is \\cite[(12.1)]guth-maynard, and is proven by…","kind":"proof","summary":"(Sketch) The first bound \\eqrefgeneral-tau is \\cite[(12.1)]guth-maynard, and is proven by the s…","labels":[],"detail_key":"p55"},{"id":"n43939","layer":"informal","project":"p55","title":"Bourgain large values theorem","kind":"theorem","summary":"[Bourgain large values theorem] \\citebourgain_large_2000 Let 1/2 < \\sigma < 1 and \\tau > 0, and…","labels":["bourgain-lvt","rho1","rs"],"detail_key":"p55"},{"id":"n43940","layer":"informal","project":"p55","title":"rr","kind":"proof","summary":"By Definition \\reflv-def, we can find a large value pattern (N,T,V,(a_n)_n \\in [N,2N],J,R) with…","labels":["rr","Exp-1","Exp-2"],"detail_key":"p55"},{"id":"n43941","layer":"informal","project":"p55","title":"Bourgain large values theorem, simplified version","kind":"corollary","summary":"[Bourgain large values theorem, simplified version] \\cite[Lemma 4.60]bourgain_large_2000 Let th…","labels":["borg-lv-simp"],"detail_key":"p55"},{"id":"n43942","layer":"informal","project":"p55","title":"With \\rho \\leq \\min(1,4-2\\tau), 5\\rho/4+\\tau/2+1 and 2\\rho+1 are both bounded by \\rho+2,…","kind":"proof","summary":"With \\rho \\leq \\min(1,4-2\\tau), 5\\rho/4+\\tau/2+1 and 2\\rho+1 are both bounded by \\rho+2, hence…","labels":[],"detail_key":"p55"},{"id":"n43943","layer":"informal","project":"p55","title":"Bourgain large values theorem, optimized version","kind":"corollary","summary":"[Bourgain large values theorem, optimized version] For each row (\\rho_0, \\alpha_1, \\alpha_2, S)…","labels":["borg-lv-opt"],"detail_key":"p55"},{"id":"n43944","layer":"informal","project":"p55","title":"Follows from substituting the specified values of \\alpha_1 and \\alpha_2 and a routine cal…","kind":"proof","summary":"Follows from substituting the specified values of \\alpha_1 and \\alpha_2 and a routine calculati…","labels":[],"detail_key":"p55"},{"id":"n43945","layer":"informal","project":"p55","title":"Kerr large values theorem","kind":"lemma","summary":"[Kerr large values theorem] \\ \\item[(i)]\\cite[Theorem 2]kerr Let 3/4 < \\sigma \\leq 1, 0 \\leq \\t…","labels":["kerr-thm"],"detail_key":"p55"},{"id":"n43946","layer":"informal","project":"p55","title":"Zero density exponents","kind":"definition","summary":"[Zero density exponents] For \\sigma \\in R and T>0, let N(\\sigma,T) denote the number of zeroes…","labels":["zero-def"],"detail_key":"p55"},{"id":"n43947","layer":"informal","project":"p55","title":"Basic properties of A","kind":"lemma","summary":"[Basic properties of A] \\item[(i)] \\sigma \\mapsto (1-\\sigma) A(\\sigma) is non-increasing and le…","labels":["zero-basic"],"detail_key":"p55"},{"id":"n43948","layer":"informal","project":"p55","title":"The claim (i) is clear using the Riemann-von Mangoldt formula \\cite[Theorem 1.7]ivic and…","kind":"proof","summary":"The claim (i) is clear using the Riemann-von Mangoldt formula \\cite[Theorem 1.7]ivic and the fu…","labels":[],"detail_key":"p55"},{"id":"n43949","layer":"informal","project":"p55","title":"One can ask what happens if one omits the \\delta shift. Thus, define A_0(\\sigma) to be th…","kind":"remark","summary":"One can ask what happens if one omits the \\delta shift. Thus, define A_0(\\sigma) to be the infi…","labels":[],"detail_key":"p55"},{"id":"n43950","layer":"informal","project":"p55","title":"Density hypothesis","kind":"conjecture","summary":"[Density hypothesis] One has \\|A\\|_\\infty=2. Equivalently, A(\\sigma) \\leq 2 for all 1/2 \\leq \\s…","labels":["density-hypothesis"],"detail_key":"p55"},{"id":"n43951","layer":"informal","project":"p55","title":"Zero density from large values","kind":"lemma","summary":"[Zero density from large values] Let 1/2 < \\sigma < 1. Then A(\\sigma)(1-\\sigma) \\leq \\max( \\sup…","labels":["zero-from-large"],"detail_key":"p55"},{"id":"n43952","layer":"informal","project":"p55","title":"lvz-bound","kind":"proof","summary":"Write the right-hand side as B, then B \\geq 0 (from Lemma \\reflv-basic(iii)) and we have LV_\\ze…","labels":["lvz-bound","lv-bound","td"],"detail_key":"p55"},{"id":"n43953","layer":"informal","project":"p55","title":"Large values from zero density","kind":"lemma","summary":"[Large values from zero density] \\cite[Theorem 1.2]matomaki_teravainen_2024 If \\tau > 0 and 1/2…","labels":["zero-dens_implies_large"],"detail_key":"p55"},{"id":"n43954","layer":"informal","project":"p55","title":"Let N \\geq 1 be unbounded, T = N^\\tau+o(1), and I \\subset [N,2N] be an interval, and t_1,…","kind":"proof","summary":"Let N \\geq 1 be unbounded, T = N^\\tau+o(1), and I \\subset [N,2N] be an interval, and t_1,\\dots,…","labels":[],"detail_key":"p55"},{"id":"n43955","layer":"informal","project":"p55","title":"zero-large-cor-0","kind":"corollary","summary":"Let 1/2 < \\sigma < 1 and \\tau_0 > 0. Then A(\\sigma)(1-\\sigma) \\leq \\max \\left(\\sup_2 \\leq \\tau…","labels":["zero-large-cor-0"],"detail_key":"p55"},{"id":"n43956","layer":"informal","project":"p55","title":"lvz-b","kind":"proof","summary":"Denote the right-hand side by B, thus LV(\\sigma,\\tau) \\leq B\\tau for all \\tau_0 \\leq \\tau \\leq…","labels":["lvz-b","lvzo"],"detail_key":"p55"},{"id":"n43957","layer":"informal","project":"p55","title":"zero-large-cor","kind":"corollary","summary":"Let 1/2 < \\sigma < 1 and \\tau_0 > 0. Then A(\\sigma)(1-\\sigma) \\leq \\max \\left(\\sup_2 \\leq \\tau…","labels":["zero-large-cor"],"detail_key":"p55"},{"id":"n43958","layer":"informal","project":"p55","title":"Applying Corollary \\refzero-large-cor-0 with \\tau replaced by 4\\tau_0/3, it suffices to s…","kind":"proof","summary":"Applying Corollary \\refzero-large-cor-0 with \\tau replaced by 4\\tau_0/3, it suffices to show th…","labels":[],"detail_key":"p55"},{"id":"n43959","layer":"informal","project":"p55","title":"zero-large-cor2","kind":"corollary","summary":"Let 1/2 < \\sigma < 1 and \\tau_0 > 0. Suppose that one has the bounds LV(\\sigma,\\tau) \\leq (3-3\\…","labels":["zero-large-cor2","lvo","lvoz"],"detail_key":"p55"},{"id":"n43960","layer":"informal","project":"p55","title":"zero-large-cor3","kind":"corollary","summary":"Let 1/2 < \\sigma < 1 and \\tau_0 > 0. Suppose that one has the bound \\eqreflvoz for 2 \\leq \\tau…","labels":["zero-large-cor3"],"detail_key":"p55"},{"id":"n43961","layer":"informal","project":"p55","title":"We may assume that \\tau_0 \\geq 3-3\\sigma, since otherwise the claim follows from the Riem…","kind":"proof","summary":"We may assume that \\tau_0 \\geq 3-3\\sigma, since otherwise the claim follows from the Riemann--v…","labels":[],"detail_key":"p55"},{"id":"n43962","layer":"informal","project":"p55","title":"montgomery_implies_density","kind":"theorem","summary":"The Montgomery conjecture implies the density hypothesis.","labels":["montgomery_implies_density"],"detail_key":"p55"},{"id":"n43963","layer":"informal","project":"p55","title":"Apply Corollary \\refzero-large-cor2 with \\tau_0=3/2 (so that \\eqreflvoz is vacuously true…","kind":"proof","summary":"Apply Corollary \\refzero-large-cor2 with \\tau_0=3/2 (so that \\eqreflvoz is vacuously true).","labels":[],"detail_key":"p55"},{"id":"n43964","layer":"informal","project":"p55","title":"lindelof_implies_density","kind":"theorem","summary":"The Lindelof hypothesis implies the density hypothesis, and also that A(\\sigma) \\leq 0 for 3/4…","labels":["lindelof_implies_density"],"detail_key":"p55"},{"id":"n43965","layer":"informal","project":"p55","title":"ap","kind":"proof","summary":"The first result is proved in \\citeingham_estimation_1940, and the second result is due to \\cit…","labels":["ap"],"detail_key":"p55"},{"id":"n43966","layer":"informal","project":"p55","title":"Ingham's first bound","kind":"theorem","summary":"[Ingham's first bound] \\citeingham_difference_1937 (See also \\citetitchmarsh_theory_1986) For a…","labels":["thm:ingham-first"],"detail_key":"p55"},{"id":"n43967","layer":"informal","project":"p55","title":"We give here a proof (somewhat different from the original proof) that passes through Cor…","kind":"proof","summary":"We give here a proof (somewhat different from the original proof) that passes through Corollary…","labels":[],"detail_key":"p55"},{"id":"n43968","layer":"informal","project":"p55","title":"Ingham's second bound","kind":"theorem","summary":"[Ingham's second bound] \\citeingham_estimation_1940 For any 1/2 < \\sigma < 1, one has A(\\sigma)…","labels":["thm:ingham_zero_density2"],"detail_key":"p55"},{"id":"n43969","layer":"informal","project":"p55","title":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 2-\\sigma. Here we have 4\\tau_0/3 <…","kind":"proof","summary":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 2-\\sigma. Here we have 4\\tau_0/3 < 2 sinc…","labels":[],"detail_key":"p55"},{"id":"n43970","layer":"informal","project":"p55","title":"Huxley bound","kind":"theorem","summary":"[Huxley bound] \\citeHuxley For any 1/2 < \\sigma < 1, one has A(\\sigma) \\leq \\frac33\\sigma-1. (I…","labels":["huxley-bound"],"detail_key":"p55"},{"id":"n43971","layer":"informal","project":"p55","title":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 3\\sigma-1. The Montgomery conjectur…","kind":"proof","summary":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 3\\sigma-1. The Montgomery conjecture hypo…","labels":[],"detail_key":"p55"},{"id":"n43972","layer":"informal","project":"p55","title":"Guth--Maynard bound","kind":"theorem","summary":"[Guth--Maynard bound] For any 1/2 < \\sigma < 1, one has A(\\sigma) \\leq \\frac153+5\\sigma.","labels":["guth-maynard-density"],"detail_key":"p55"},{"id":"n43973","layer":"informal","project":"p55","title":"18-5","kind":"proof","summary":"We may assume that 7/10 < \\sigma < 8/10, since the bound follows from the Ingham and Huxley bou…","labels":["18-5"],"detail_key":"p55"},{"id":"n43974","layer":"informal","project":"p55","title":"Jutila zero density theorem","kind":"theorem","summary":"[Jutila zero density theorem] \\citejutila_zero_density_1977 The zero density hypothesis is true…","labels":["jutila-density"],"detail_key":"p55"},{"id":"n43975","layer":"informal","project":"p55","title":"We apply Corollary \\refzero-large-cor with \\tau_0 := 3/2, then it suffices to show that L…","kind":"proof","summary":"We apply Corollary \\refzero-large-cor with \\tau_0 := 3/2, then it suffices to show that LV(\\sig…","labels":[],"detail_key":"p55"},{"id":"n43976","layer":"informal","project":"p55","title":"Heath-Brown zero density theorem","kind":"theorem","summary":"","labels":["hb-density"],"detail_key":"p55"},{"id":"n43977","layer":"informal","project":"p55","title":"For the first estimate, we apply Corollary \\refzero-large-cor2 with \\tau_0 := \\frac7\\sigm…","kind":"proof","summary":"For the first estimate, we apply Corollary \\refzero-large-cor2 with \\tau_0 := \\frac7\\sigma-13.…","labels":[],"detail_key":"p55"},{"id":"n43978","layer":"informal","project":"p55","title":"3-40","kind":"lemma","summary":"(3/40, 31/40) is an exponent pair. In particular, by Corollary \\refexp-pair-mu, \\mu(7/10) \\leq…","labels":["3-40"],"detail_key":"p55"},{"id":"n43979","layer":"informal","project":"p55","title":"This can be derived from the Watt exponent pair W := (89/560, 1/2 + 89/560) from Theorem…","kind":"proof","summary":"This can be derived from the Watt exponent pair W := (89/560, 1/2 + 89/560) from Theorem \\refli…","labels":[],"detail_key":"p55"},{"id":"n43980","layer":"informal","project":"p55","title":"Improved Heath-Brown zero density theorem","kind":"theorem","summary":"[Improved Heath-Brown zero density theorem] For any 7/10 < \\sigma \\leq 1, one has A(\\sigma) \\le…","labels":["hb-density2"],"detail_key":"p55"},{"id":"n43981","layer":"informal","project":"p55","title":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 10\\sigma-7. The claim \\eqreflvo aga…","kind":"proof","summary":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := 10\\sigma-7. The claim \\eqreflvo again fol…","labels":[],"detail_key":"p55"},{"id":"n43982","layer":"informal","project":"p55","title":"Bourgain result on density hypothesis","kind":"theorem","summary":"[Bourgain result on density hypothesis] The density hypothesis holds for \\sigma > 25/32.","labels":["bourgain-density"],"detail_key":"p55"},{"id":"n43983","layer":"informal","project":"p55","title":"rot","kind":"proof","summary":"The arguments below are a translation of the original arguments of Bourgain \\citebourgain_large…","labels":["rot","rhomax","r0-start","tau-check","40"],"detail_key":"p55"},{"id":"n43984","layer":"informal","project":"p55","title":"Improved Bourgain density hypothesis bound","kind":"theorem","summary":"[Improved Bourgain density hypothesis bound] For 17/22 \\leq \\sigma \\leq 4/5, one has A(\\sigma)…","labels":["bourgain-density-improved"],"detail_key":"p55"},{"id":"n43985","layer":"informal","project":"p55","title":"slope","kind":"proof","summary":"We apply Corollary \\refzero-large-cor3 with \\tau_0 := \\min( \\frac 9(3\\sigma-2)2, \\frac8(2\\sigma…","labels":["slope","lv-st","lvz-st","tau-lower","40-alt","bound-1","taut","tar"],"detail_key":"p55"},{"id":"n43986","layer":"informal","project":"p55","title":"Bourgain zero density theorem","kind":"theorem","summary":"[Bourgain zero density theorem] \\cite[Proposition 3]bourgain_remarks_1995 Let (k,\\ell) be an ex…","labels":["bourgain-zd"],"detail_key":"p55"},{"id":"n43987","layer":"informal","project":"p55","title":"Special case of Bourgain's zero density theorem","kind":"corollary","summary":"[Special case of Bourgain's zero density theorem] \\cite[Corollary 4]bourgain_remarks_1995 One h…","labels":["bourgain-zero-density"],"detail_key":"p55"},{"id":"n43988","layer":"informal","project":"p55","title":"Apply Theorem \\refbourgain-zd with the classical pairs (\\frac114,\\frac1114) and (\\frac16,…","kind":"proof","summary":"Apply Theorem \\refbourgain-zd with the classical pairs (\\frac114,\\frac1114) and (\\frac16, \\frac…","labels":[],"detail_key":"p55"},{"id":"n43989","layer":"informal","project":"p55","title":"Optimized Bourgain zero density bound","kind":"corollary","summary":"[Optimized Bourgain zero density bound] One has \\[ A(\\sigma) \\leq \\dfrac1112(4 \\sigma - 3) & \\d…","labels":["bourgain-zero-density-optimized"],"detail_key":"p55"},{"id":"n43990","layer":"informal","project":"p55","title":"Let S(\\sigma) denote the closure of the region \\Bigg\\(k, \\ell) : 0 < k < \\frac15, \\frac35…","kind":"proof","summary":"Let S(\\sigma) denote the closure of the region \\Bigg\\(k, \\ell) : 0 < k < \\frac15, \\frac35 < \\el…","labels":[],"detail_key":"p55"},{"id":"n43991","layer":"informal","project":"p55","title":"1980 Ivic zero density bound","kind":"lemma","summary":"[1980 Ivic zero density bound] \\citeivic_exponent_pairs, \\cite[Theorem 11.2]ivic We have A(\\sig…","labels":["ivic-zero-density"],"detail_key":"p55"},{"id":"n43992","layer":"informal","project":"p55","title":"From Lemma \\refivic-lvt we have LV(\\sigma,\\tau) \\leq \\max( 2-2\\sigma, \\tau + 9-12\\sigma,…","kind":"proof","summary":"From Lemma \\refivic-lvt we have LV(\\sigma,\\tau) \\leq \\max( 2-2\\sigma, \\tau + 9-12\\sigma, \\tau -…","labels":[],"detail_key":"p55"},{"id":"n43993","layer":"informal","project":"p55","title":"Zero density from \\mu bound","kind":"lemma","summary":"[Zero density from \\mu bound] \\cite[Theorem 12.3]montgomery_topics_1971 If 1/2 \\leq \\alpha \\leq…","labels":["zero_from_mu"],"detail_key":"p55"},{"id":"n43994","layer":"informal","project":"p55","title":"1971 Montgomery zero density bound","kind":"corollary","summary":"[1971 Montgomery zero density bound] \\citemontgomery_topics_1971, \\cite[Theorem 11.3]ivic For a…","labels":["ivic-zero-density-large"],"detail_key":"p55"},{"id":"n43995","layer":"informal","project":"p55","title":"Apply the previous lemma with \\alpha = 5\\sigma-4.","kind":"proof","summary":"Apply the previous lemma with \\alpha = 5\\sigma-4.","labels":[],"detail_key":"p55"},{"id":"n43996","layer":"informal","project":"p55","title":"Preliminary large values estimate","kind":"lemma","summary":"[Preliminary large values estimate] If m \\geq 2 is an integer, 3/4 < \\sigma \\leq 1, and (k,\\ell…","labels":["a-ivt-1"],"detail_key":"p55"},{"id":"n43997","layer":"informal","project":"p55","title":"See \\cite[(11.74)]ivic.","kind":"proof","summary":"See \\cite[(11.74)]ivic.","labels":[],"detail_key":"p55"},{"id":"n43998","layer":"informal","project":"p55","title":"General zero density estimate","kind":"lemma","summary":"[General zero density estimate] \\cite[(11.76), (11.77)]ivic If (k,\\ell) is an exponent pair, an…","labels":["gzd"],"detail_key":"p55"},{"id":"n43999","layer":"informal","project":"p55","title":"With the hypothesis on \\sigma, one sees from Lemma \\refa-ivt-1 that LV(\\sigma,\\tau) \\leq…","kind":"proof","summary":"With the hypothesis on \\sigma, one sees from Lemma \\refa-ivt-1 that LV(\\sigma,\\tau) \\leq \\max(…","labels":[],"detail_key":"p55"},{"id":"n44000","layer":"informal","project":"p55","title":"1980-1984 Ivic zero density bound","kind":"corollary","summary":"[1980-1984 Ivic zero density bound] \\citeivic_exponent_pairs, \\cite[Theorem 11.4]ivic One can b…","labels":["further_ivic_zero"],"detail_key":"p55"},{"id":"n44001","layer":"informal","project":"p55","title":"Apply Lemma \\refgzd with m=2 and (k,\\ell) = (\\frac97251, \\frac132251) for the first claim…","kind":"proof","summary":"Apply Lemma \\refgzd with m=2 and (k,\\ell) = (\\frac97251, \\frac132251) for the first claim; the…","labels":[],"detail_key":"p55"},{"id":"n44002","layer":"informal","project":"p55","title":"2000 Bourgain zero density theorem","kind":"theorem","summary":"[2000 Bourgain zero density theorem] \\citebourgain_dirichlet_2000 One has A(\\sigma) \\leq 3/2\\si…","labels":["bourgain-zero-density-2000"],"detail_key":"p55"},{"id":"n44003","layer":"informal","project":"p55","title":"Preliminary large values theorem","kind":"lemma","summary":"[Preliminary large values theorem] If 1/2 \\leq \\sigma \\leq 1 and \\tau < 8\\sigma-5, then LV(\\sig…","labels":["a-ivt"],"detail_key":"p55"},{"id":"n44004","layer":"informal","project":"p55","title":"See \\cite[(11.95)]ivic.","kind":"proof","summary":"See \\cite[(11.95)]ivic.","labels":[],"detail_key":"p55"},{"id":"n44005","layer":"informal","project":"p55","title":"Zero density estimates for \\sigma close to 3/4","kind":"corollary","summary":"[Zero density estimates for \\sigma close to 3/4] \\cite[Theorem 11.5]ivic One has A(\\sigma) \\leq…","labels":["ivic-near-34"],"detail_key":"p55"},{"id":"n44006","layer":"informal","project":"p55","title":"For 3/4 \\leq \\sigma \\leq 10/13, we see from Lemma \\refa-ivt that the bound LV(\\sigma,\\tau…","kind":"proof","summary":"For 3/4 \\leq \\sigma \\leq 10/13, we see from Lemma \\refa-ivt that the bound LV(\\sigma,\\tau) \\leq…","labels":[],"detail_key":"p55"},{"id":"n44007","layer":"informal","project":"p55","title":"Pintz zero density theorem","kind":"theorem","summary":"[Pintz zero density theorem] \\cite[Theorem 1]pintz_density_2023 If k \\geq 4, \\ell \\geq 3 are in…","labels":["pintz-density","eta-b","eta-l"],"detail_key":"p55"},{"id":"n44008","layer":"informal","project":"p55","title":"tau0-def","kind":"proof","summary":"We apply Corollary \\refzero-large-cor2 with \\tau_0 := \\min( \\ell (1 - 2(\\ell-1) \\eta), \\frac34…","labels":["tau0-def","taub"],"detail_key":"p55"},{"id":"n44009","layer":"informal","project":"p55","title":"Chen-Debruyne-Vidas density theorem","kind":"theorem","summary":"[Chen-Debruyne-Vidas density theorem] \\citechen_debruyne_vindas_density_2024 For any 279/314 \\l…","labels":["cdv-density"],"detail_key":"p55"},{"id":"n44010","layer":"informal","project":"p55","title":"kerr-prop","kind":"proposition","summary":"\\cite[Theorems 6, 7]kerr One has A(\\sigma) \\leq \\frac32\\sigma for \\sigma \\geq 23/29, and A(\\sig…","labels":["kerr-prop"],"detail_key":"p55"},{"id":"n44011","layer":"informal","project":"p55","title":"\\citesimonic","kind":"theorem","summary":"[\\citesimonic] For T \\ge 3 and 1/2 \\le \\sigma \\le 0.778, one has N(\\sigma, 2 T)-N(\\sigma, T) \\l…","labels":[],"detail_key":"p55"},{"id":"n44012","layer":"informal","project":"p55","title":"\\citechourasiya_explicit_2025","kind":"theorem","summary":"[\\citechourasiya_explicit_2025]For every T\\ge 3 and 1/2\\le \\sigma\\le 5/8 one has \\[ N(\\sigma, T…","labels":[],"detail_key":"p55"},{"id":"n44013","layer":"informal","project":"p55","title":"\\citechourasiya_explicit_2024","kind":"theorem","summary":"[\\citechourasiya_explicit_2024]For every T\\ge 3 and \\sigma\\ge 3/5, one has N(\\sigma, T) \\leq 0.…","labels":[],"detail_key":"p55"},{"id":"n44014","layer":"informal","project":"p55","title":"\\citeramare_explicit_2016","kind":"theorem","summary":"[\\citeramare_explicit_2016] For every T\\ge 3 and \\sigma\\ge 0.52 one has N(\\sigma, T) \\leq 965(3…","labels":["th:ramareexpl"],"detail_key":"p55"},{"id":"n44015","layer":"informal","project":"p55","title":"\\citeKadiri_explicit_2018","kind":"theorem","summary":"[\\citeKadiri_explicit_2018] For each tuple (\\sigma_0, A, B) of Table \\refzerodensity_kadiri, on…","labels":["th:explicitkln"],"detail_key":"p55"},{"id":"n44016","layer":"informal","project":"p55","title":"\\citebellotti_2024","kind":"theorem","summary":"[\\citebellotti_2024] For every T\\ge 3 and \\sigma\\in[0.9927,1], one has N(\\sigma, T) \\leq 4.45 \\…","labels":["th:explbellogfree"],"detail_key":"p55"},{"id":"n44017","layer":"informal","project":"p55","title":"\\citebellotti","kind":"theorem","summary":"[\\citebellotti] For every \\sigma\\in[0.98,1] and T\\ge 3, one has: N(\\sigma,T)\\le2.15\\cdot 10^23\\…","labels":["th:explicitkvbel"],"detail_key":"p55"},{"id":"n44018","layer":"informal","project":"p55","title":"\\citebellotti_2024","kind":"corollary","summary":"[\\citebellotti_2024] For every T\\ge \\exp(6.7\\cdot 10^12) and \\sigma\\in[0.98,1], one hasN(\\sigma…","labels":["th:kvaslokgfreebel"],"detail_key":"p55"},{"id":"n44019","layer":"informal","project":"p55","title":"Zero density exponents","kind":"definition","summary":"[Zero density exponents] For 1/2 \\leq \\sigma \\leq 1 and T>0, let N^*(\\sigma,T) denote the addit…","labels":["zeroe-def"],"detail_key":"p55"},{"id":"n44020","layer":"informal","project":"p55","title":"Basic properties of A^*","kind":"lemma","summary":"[Basic properties of A^*] \\item[(i)] We have the trivial bounds 2A(\\sigma), 4A(\\sigma)-\\frac11-…","labels":["zeroe-basic"],"detail_key":"p55"},{"id":"n44021","layer":"informal","project":"p55","title":"The claim (i) follows from Lemma \\refadd-energy(iv), and the remaining claims then follow…","kind":"proof","summary":"The claim (i) follows from Lemma \\refadd-energy(iv), and the remaining claims then follow from…","labels":[],"detail_key":"p55"},{"id":"n44022","layer":"informal","project":"p55","title":"Zero density energy from large values energy","kind":"lemma","summary":"[Zero density energy from large values energy] Let 1/2 < \\sigma < 1. Then A^*(\\sigma)(1-\\sigma)…","labels":["zeroe-from-large"],"detail_key":"p55"},{"id":"n44023","layer":"informal","project":"p55","title":"lvze-bound","kind":"proof","summary":"Write the right-hand side as B, then B \\geq 0 (from Lemma \\reflve-basic(iii)) and we have LV^*_…","labels":["lvze-bound","lve-bound"],"detail_key":"p55"},{"id":"n44024","layer":"informal","project":"p55","title":"zeroe-large-cor-0","kind":"corollary","summary":"Let 1/2 < \\sigma < 1 and \\tau_0 > 0 be fixed. Then A^*(\\sigma)(1-\\sigma) \\leq \\max \\left(\\sup_2…","labels":["zeroe-large-cor-0"],"detail_key":"p55"},{"id":"n44025","layer":"informal","project":"p55","title":"Repeat the proof of Corollary \\refzero-large-cor-0.","kind":"proof","summary":"Repeat the proof of Corollary \\refzero-large-cor-0.","labels":[],"detail_key":"p55"},{"id":"n44026","layer":"informal","project":"p55","title":"Additive energy under the Lindelof hypothesis","kind":"proposition","summary":"[Additive energy under the Lindelof hypothesis] Let 1/2 \\leq \\sigma \\leq 1 be fixed. Then one h…","labels":["zeroe-lindelof"],"detail_key":"p55"},{"id":"n44027","layer":"informal","project":"p55","title":"See \\cite[Lemma 4]heath_brown_consecutive_II.","kind":"proof","summary":"See \\cite[Lemma 4]heath_brown_consecutive_II.","labels":[],"detail_key":"p55"},{"id":"n44028","layer":"informal","project":"p55","title":"Heath-Brown's additive energy bound","kind":"theorem","summary":"[Heath-Brown's additive energy bound] \\cite[Theorem 2]heathbrown_zero_1979 Let 1/2 \\leq \\sigma…","labels":["hb-energy-bound"],"detail_key":"p55"},{"id":"n44029","layer":"informal","project":"p55","title":"second-claim","kind":"proof","summary":"We first suppose that \\sigma \\leq 3/4. Here we apply Corollary \\refzeroe-large-cor-0 with \\tau_…","labels":["second-claim","smak","rho-k","first-claim","second-claim'","rhok","rhok-star","rhost","4s","rhok-simp"],"detail_key":"p55"},{"id":"n44030","layer":"informal","project":"p55","title":"imp-hb-energy-bound","kind":"theorem","summary":"For 3/4 \\le \\sigma \\le 5/6 one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac18 - 19\\sigma2(3\\sigma - 1…","labels":["imp-hb-energy-bound"],"detail_key":"p55"},{"id":"n44031","layer":"informal","project":"p55","title":"imphb-lver-ineq","kind":"proof","summary":"Throughout assume that 3/4 \\le \\sigma \\le 5/6. Choose \\[ \\tau_0 = 8\\sigma - 4. \\] We will show…","labels":["imphb-lver-ineq","imphb-zlver-ineq","hb-lv-rho-form","huxley-lv-rho-form2","zlver:tau-gradient-1","ze-ihb-rho-bound-k","ze-ihb-rho-1","ze-ihb-rho-2"],"detail_key":"p55"},{"id":"n44032","layer":"informal","project":"p55","title":"imp-energy-bound1","kind":"theorem","summary":"For 2/3 \\le \\sigma \\le 3/4, one has A^*(\\sigma) &\\le \\frac10 - 11\\sigma(2 - \\sigma)(1 - \\sigma)…","labels":["imp-energy-bound1"],"detail_key":"p55"},{"id":"n44033","layer":"informal","project":"p55","title":"Take \\tau_0=2 in Corollary \\refzeroe-large-cor-0. The LV^*_\\zeta supremum is now trivial,…","kind":"proof","summary":"Take \\tau_0=2 in Corollary \\refzeroe-large-cor-0. The LV^*_\\zeta supremum is now trivial, so it…","labels":[],"detail_key":"p55"},{"id":"n44034","layer":"informal","project":"p55","title":"imp-energy-bound2","kind":"theorem","summary":"For 7/10 \\le \\sigma \\le 3/4, one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac5(18 - 19\\sigma)2(5\\sigm…","labels":["imp-energy-bound2"],"detail_key":"p55"},{"id":"n44035","layer":"informal","project":"p55","title":"ze-bound2-tau-factor-bounds","kind":"proof","summary":"Throughout assume 7/10 \\le \\sigma \\le 3/4 and take \\tau_0 = 2 in Corollary \\refzeroe-large-cor-…","labels":["ze-bound2-tau-factor-bounds","ze-bound2-rho1","ze-bound2-rho2","ze-bound2-rhok-bound","ze-bound2-rhostar-boundk","ze-bound2-rho-case1-1","ze-bound2-rho-star-bound","ze-bound2-rhok-temp1","ze-bound2-rhostar-k1","ze-bound2-rhostar-k1-case1","ze-bound2-rhostar-k1-case2","ze-bound2-rhostar-k1-case3","ze-bound2-rhok-temp2"],"detail_key":"p55"},{"id":"n44036","layer":"informal","project":"p55","title":"imp-energy-bound3","kind":"theorem","summary":"For 3/4 \\le \\sigma \\le 4/5, one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac197 - 220\\sigma8(5\\sigma…","labels":["imp-energy-bound3"],"detail_key":"p55"},{"id":"n44037","layer":"informal","project":"p55","title":"ze-bound3-energy-req","kind":"proof","summary":"Throughout assume that 3/4 \\le \\sigma \\le 4/5 and take \\tau_0 := 8\\sigma - 4 in Corollary \\refz…","labels":["ze-bound3-energy-req","ze-bound3-energyzeta-req","ze-bound3-kdefn","ze-bound3-rhok","ze-bound3-rhostar2"],"detail_key":"p55"},{"id":"n44038","layer":"informal","project":"p55","title":"imp-energy-bound4","kind":"theorem","summary":"For 664/877 \\le \\sigma \\le 31/40, one has \\[ A^*(\\sigma)\\le \\max\\left(\\frac72 - 91\\sigma7(11\\si…","labels":["imp-energy-bound4"],"detail_key":"p55"},{"id":"n44039","layer":"informal","project":"p55","title":"ze-bound5-rhostar1","kind":"proof","summary":"Fix 664/877 \\le \\sigma \\le 31/40 and take \\tau_0 = 2. It suffices to show that \\[ \\rho^* \\le \\m…","labels":["ze-bound5-rhostar1","ze-bound5-rhostar2","ze-bound5-rho-bound","ze-bound5-julita-lvt","ze-bound5-julita-lvt1","ze-bound5-rhostar-bound","ze-bound5-rhostar-bound3","ze-bound5-julita-lvt-2"],"detail_key":"p55"},{"id":"n44040","layer":"informal","project":"p55","title":"imp-energy-bound6","kind":"theorem","summary":"For 42/55 \\le \\sigma \\le 79/103, one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac18 - 19\\sigma6(15\\si…","labels":["imp-energy-bound6"],"detail_key":"p55"},{"id":"n44041","layer":"informal","project":"p55","title":"ze-bound6-tuples","kind":"proof","summary":"Fix 42/55 \\le \\sigma \\le 79/103 and take \\tau_0 = 2. It suffices to show that \\[ \\rho^* \\le \\ma…","labels":["ze-bound6-tuples","ze-bound6-rho-bound","ze-bound6-rho","ze-bound6-rhostar-bound-case01","ze-bound6-rhostar-bound-case02","ze-bound6-rhostar-bound-case1","ze-bound6-rhostar-bound","ze-bound6-rhostar-bound3","ze-bound6-rho-case2","ze-bound6-rho-final"],"detail_key":"p55"},{"id":"n44042","layer":"informal","project":"p55","title":"imp-energy-bound7","kind":"theorem","summary":"For 79/103 \\le \\sigma \\le 84/109, one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac18 - 19\\sigma2(37\\s…","labels":["imp-energy-bound7"],"detail_key":"p55"},{"id":"n44043","layer":"informal","project":"p55","title":"ze-bound7-rhostar-bound","kind":"proof","summary":"Fix 79/103 \\le \\sigma \\le 84/109 and take \\[ \\tau_0 = (36\\sigma - 16)/5,& 79/103 \\le \\sigma < 3…","labels":["ze-bound7-rhostar-bound","ze-bound7-tuples","ze-bound7-rhostar-bound-case01","ze-bound7-rhostar-bound-case02","ze-bound7-rhostar-final1","ze-bound7-rhostar-zeta-bound"],"detail_key":"p55"},{"id":"n44044","layer":"informal","project":"p55","title":"imp-energy-bound8","kind":"theorem","summary":"For 84/109 \\le \\sigma \\le 5/6, one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac18 - 19\\sigma9(3\\sigma…","labels":["imp-energy-bound8"],"detail_key":"p55"},{"id":"n44045","layer":"informal","project":"p55","title":"ze-guthmaynard-thm2","kind":"theorem","summary":"For 165/226 \\le \\sigma \\le 42/55 one has \\[ A^*(\\sigma) \\le \\max\\left(\\frac457 - 546\\sigma2(61…","labels":["ze-guthmaynard-thm2"],"detail_key":"p55"},{"id":"n44046","layer":"informal","project":"p55","title":"Zero free region of \\zeta(s)","kind":"definition","summary":"[Zero free region of \\zeta(s)] A zero-free region of the Riemann zeta function is a set D \\subs…","labels":["zeta-zero-free-def"],"detail_key":"p55"},{"id":"n44047","layer":"informal","project":"p55","title":"Basic properties of zero free regions","kind":"lemma","summary":"[Basic properties of zero free regions] The following properties hold: \\item[(i)] (Symmetry abo…","labels":["zero-free-basic-lem"],"detail_key":"p55"},{"id":"n44048","layer":"informal","project":"p55","title":"Claim (i) follows directly from the property \\overline\\zeta(s) = \\zeta(\\overlines). Claim…","kind":"proof","summary":"Claim (i) follows directly from the property \\overline\\zeta(s) = \\zeta(\\overlines). Claim (ii)…","labels":[],"detail_key":"p55"},{"id":"n44049","layer":"informal","project":"p55","title":"Riemann hypothesis","kind":"conjecture","summary":"[Riemann hypothesis] If \\rho is a non-trivial zero of the Riemann zeta function, then \\Re \\rho…","labels":["rh"],"detail_key":"p55"},{"id":"n44050","layer":"informal","project":"p55","title":"Non-vanishing on the 1-line","kind":"theorem","summary":"[Non-vanishing on the 1-line] One has \\zeta(1 + it) \\ne 0 for any real t.","labels":[],"detail_key":"p55"},{"id":"n44051","layer":"informal","project":"p55","title":"For \\Re s > 1, one has \\[ \\Re \\log \\zeta(s) = \\sum_p\\sum_m = 1^\\infty\\frac\\cos (t \\log p^…","kind":"proof","summary":"For \\Re s > 1, one has \\[ \\Re \\log \\zeta(s) = \\sum_p\\sum_m = 1^\\infty\\frac\\cos (t \\log p^m)mp^m…","labels":[],"detail_key":"p55"},{"id":"n44052","layer":"informal","project":"p55","title":"Relation to growth exponents of zeta","kind":"lemma","summary":"[Relation to growth exponents of zeta] Suppose and 0 < g(t) \\le 1 < f(t) are real-valued functi…","labels":["mu_to_zero_free"],"detail_key":"p55"},{"id":"n44053","layer":"informal","project":"p55","title":"See \\cite[Theorem 3.10]titchmarsh_theory_1986.","kind":"proof","summary":"See \\cite[Theorem 3.10]titchmarsh_theory_1986.","labels":[],"detail_key":"p55"},{"id":"n44054","layer":"informal","project":"p55","title":"Classical zero free region","kind":"theorem","summary":"[Classical zero free region] One has \\zeta(\\sigma + it) \\ne 0 if \\[ \\sigma \\ge 1 - \\fracA\\log t…","labels":["zfr-classical"],"detail_key":"p55"},{"id":"n44055","layer":"informal","project":"p55","title":"Thanks to the convexity bound \\mu(\\sigma) \\le (1-\\sigma)/2, one may take g(t) = 1/2, f(t)…","kind":"proof","summary":"Thanks to the convexity bound \\mu(\\sigma) \\le (1-\\sigma)/2, one may take g(t) = 1/2, f(t) = t^1…","labels":[],"detail_key":"p55"},{"id":"n44056","layer":"informal","project":"p55","title":"Littlewood zero free region","kind":"theorem","summary":"[Littlewood zero free region] One has \\zeta(\\sigma + it) \\ne 0 if \\[ \\sigma \\ge 1 - \\fracA \\log…","labels":["zfr-littlewood"],"detail_key":"p55"},{"id":"n44057","layer":"informal","project":"p55","title":"Follows from the zeta bound corresponding to \\[ \\mu\\left(1 - \\frack2^k - 2\\right) \\le \\fr…","kind":"proof","summary":"Follows from the zeta bound corresponding to \\[ \\mu\\left(1 - \\frack2^k - 2\\right) \\le \\frac12^k…","labels":[],"detail_key":"p55"},{"id":"n44058","layer":"informal","project":"p55","title":"Chudakov zero free region","kind":"theorem","summary":"[Chudakov zero free region] One has \\zeta(\\sigma + it) \\ne 0 if \\[ \\sigma \\ge 1 - \\frac1(\\log t…","labels":["zfr-chudakov"],"detail_key":"p55"},{"id":"n44059","layer":"informal","project":"p55","title":"Korobov-Vinogradov zero free region","kind":"theorem","summary":"[Korobov-Vinogradov zero free region] One has \\zeta(\\sigma + it) \\ne 0 if \\[ \\sigma \\ge 1 - \\fr…","labels":["zfr-vk"],"detail_key":"p55"},{"id":"n44060","layer":"informal","project":"p55","title":"Via estimates of Vinogradov's integral, one may obtain an estimate of the form (see e.g.…","kind":"proof","summary":"Via estimates of Vinogradov's integral, one may obtain an estimate of the form (see e.g. Richer…","labels":[],"detail_key":"p55"},{"id":"n44061","layer":"informal","project":"p55","title":"For all x \\ge 1 define the Chebyshev prime counting functions \\psi(x), \\theta(x) and \\pi(…","kind":"definition","summary":"For all x \\ge 1 define the Chebyshev prime counting functions \\psi(x), \\theta(x) and \\pi(x) as…","labels":[],"detail_key":"p55"},{"id":"n44062","layer":"informal","project":"p55","title":"Prime number theorem","kind":"theorem","summary":"[Prime number theorem] As x \\to \\infty, \\[ \\pi(x) \\sim \\fracx\\log x \\sim li(x) := \\int_2^x\\frac…","labels":[],"detail_key":"p55"},{"id":"n44063","layer":"informal","project":"p55","title":"Prime number theorem, alternative formulations","kind":"theorem","summary":"[Prime number theorem, alternative formulations] As x \\to \\infty, one has \\psi(x) \\sim x and \\t…","labels":[],"detail_key":"p55"},{"id":"n44064","layer":"informal","project":"p55","title":"Korobov--Vinogradov estimate","kind":"theorem","summary":"[Korobov--Vinogradov estimate] There exists a positive constant A, such that \\psi(x) - x, \\; \\t…","labels":[],"detail_key":"p55"},{"id":"n44065","layer":"informal","project":"p55","title":"\\citekoch_sur_1901","kind":"theorem","summary":"[\\citekoch_sur_1901] If the Riemann hypothesis is true, then \\[ \\psi(x) - x,\\; \\theta(x) - x \\l…","labels":[],"detail_key":"p55"},{"id":"n44066","layer":"informal","project":"p55","title":"Heath-Brown \\citeheathbrown_gaps_1982","kind":"theorem","summary":"[Heath-Brown \\citeheathbrown_gaps_1982] Assume that the Riemann hypothesis is true. Furthermore…","labels":[],"detail_key":"p55"},{"id":"n44067","layer":"informal","project":"p55","title":"Relation to zero free regions","kind":"lemma","summary":"[Relation to zero free regions] \\citeingham_distribution_1990 Suppose \\zeta(\\sigma + it) \\ne 0…","labels":["zero_free_to_pnt"],"detail_key":"p55"},{"id":"n44068","layer":"informal","project":"p55","title":"\\citeturan_new_1984 Theorem 40.1","kind":"theorem","summary":"[\\citeturan_new_1984 Theorem 40.1] If for some 0 < \\alpha \\le 1 one has \\[ \\psi(x) - x \\ll x \\e…","labels":[],"detail_key":"p55"},{"id":"n44069","layer":"informal","project":"p55","title":"Schmidt \\citeschmidt_uber_1903","kind":"theorem","summary":"[Schmidt \\citeschmidt_uber_1903] As x \\to \\infty, \\[ \\psi(x) = x + \\Omega(x^1/2). \\]","labels":[],"detail_key":"p55"},{"id":"n44070","layer":"informal","project":"p55","title":"Littlewood \\citelittlewood_sur_1914","kind":"theorem","summary":"[Littlewood \\citelittlewood_sur_1914] If the Riemann hypothesis is true, then as x \\to \\infty,…","labels":[],"detail_key":"p55"},{"id":"n44071","layer":"informal","project":"p55","title":"Grosswald \\citegrosswald_sur_1965","kind":"theorem","summary":"[Grosswald \\citegrosswald_sur_1965] If \\[ \\theta = \\sup_\\rho: \\zeta(\\rho) = 0\\Re \\rho > 1/2 \\]…","labels":[],"detail_key":"p55"},{"id":"n44072","layer":"informal","project":"p55","title":"Prime number theorem in short interval exponents","kind":"definition","summary":"[Prime number theorem in short interval exponents] \\item[(i)] We let \\theta_PNT denote the leas…","labels":["pnt-ap"],"detail_key":"p55"},{"id":"n44073","layer":"informal","project":"p55","title":"Trivial bounds","kind":"lemma","summary":"[Trivial bounds] We have 0 \\leq \\theta_gap-AA\\leq \\theta_PNT-AA, \\theta_gap\\leq \\theta_PNT\\leq…","labels":["pnt-triv"],"detail_key":"p55"},{"id":"n44074","layer":"informal","project":"p55","title":"These are all immediate, after noting from the prime number theorem that \\sum_p_n \\leq x…","kind":"proof","summary":"These are all immediate, after noting from the prime number theorem that \\sum_p_n \\leq x p_n+1…","labels":[],"detail_key":"p55"},{"id":"n44075","layer":"informal","project":"p55","title":"Prime gap conjecture","kind":"conjecture","summary":"[Prime gap conjecture] \\theta_PNT= 0, and hence (by Lemma \\refpnt-triv) \\theta_gap-AA= \\theta_P…","labels":[],"detail_key":"p55"},{"id":"n44076","layer":"informal","project":"p55","title":"Zero density theorems and prime gaps","kind":"proposition","summary":"[Zero density theorems and prime gaps] Let \\|A\\|_\\infty := \\sup_1/2 \\leq \\sigma \\leq 1 A(\\sigma…","labels":["prime-gap","A-def"],"detail_key":"p55"},{"id":"n44077","layer":"informal","project":"p55","title":"See for instance \\cite[\\S 13.2]guth-maynard.","kind":"proof","summary":"See for instance \\cite[\\S 13.2]guth-maynard.","labels":[],"detail_key":"p55"},{"id":"n44078","layer":"informal","project":"p55","title":"Ingham-Huxley bound","kind":"corollary","summary":"[Ingham-Huxley bound] We have \\theta_PNT\\leq \\frac712 and \\theta_PNT-AA\\leq \\frac16.","labels":[],"detail_key":"p55"},{"id":"n44079","layer":"informal","project":"p55","title":"From Theorem \\refthm:ingham_zero_density2 and Theorem \\refhuxley-bound one as \\|A\\|_\\inft…","kind":"proof","summary":"From Theorem \\refthm:ingham_zero_density2 and Theorem \\refhuxley-bound one as \\|A\\|_\\infty \\leq…","labels":[],"detail_key":"p55"},{"id":"n44080","layer":"informal","project":"p55","title":"Ingham-Guth-Maynard bound","kind":"corollary","summary":"[Ingham-Guth-Maynard bound]\\citeguth-maynard We have \\theta_PNT\\leq \\frac1730 and \\theta_PNT-AA…","labels":[],"detail_key":"p55"},{"id":"n44081","layer":"informal","project":"p55","title":"From Theorem \\refthm:ingham_zero_density2 and Theorem \\refguth-maynard-density one as \\|A…","kind":"proof","summary":"From Theorem \\refthm:ingham_zero_density2 and Theorem \\refguth-maynard-density one as \\|A\\|_\\in…","labels":[],"detail_key":"p55"},{"id":"n44082","layer":"informal","project":"p55","title":"The density hypothesis implies that \\theta_PNT\\leq 1/2 and \\theta_PNT-AA= 0.","kind":"corollary","summary":"The density hypothesis implies that \\theta_PNT\\leq 1/2 and \\theta_PNT-AA= 0.","labels":[],"detail_key":"p55"},{"id":"n44083","layer":"informal","project":"p55","title":"bhp-thm","kind":"theorem","summary":"\\citeli_number_2025 We have \\theta_gap\\leq 13/25 = 0.52.","labels":["bhp-thm"],"detail_key":"p55"},{"id":"n44084","layer":"informal","project":"p55","title":"gapsquare-from-a","kind":"proposition","summary":"We have \\theta_gap,2\\leq \\max\\left( 2-\\frac2\\|A\\|_\\infty, \\sup_1/2 \\leq \\sigma \\leq 1 \\max(\\alp…","labels":["gapsquare-from-a"],"detail_key":"p55"},{"id":"n44085","layer":"informal","project":"p55","title":"See \\cite[Lemma 2]heath_brown_consecutive_II. We remark that this lemma allows \\sigma to…","kind":"proof","summary":"See \\cite[Lemma 2]heath_brown_consecutive_II. We remark that this lemma allows \\sigma to range…","labels":[],"detail_key":"p55"},{"id":"n44086","layer":"informal","project":"p55","title":"\\ \\item[(i)] Assuming the Riemann hypothesis, \\theta_gap,2= 1. (Selberg, 1943 \\citeselber…","kind":"corollary","summary":"\\ \\item[(i)] Assuming the Riemann hypothesis, \\theta_gap,2= 1. (Selberg, 1943 \\citeselberg_1943…","labels":[],"detail_key":"p55"},{"id":"n44087","layer":"informal","project":"p55","title":"For (i), we observe that \\|A\\|_\\infty=2 and that one can take A(\\sigma)=B(\\sigma)=\\vareps…","kind":"proof","summary":"For (i), we observe that \\|A\\|_\\infty=2 and that one can take A(\\sigma)=B(\\sigma)=\\varepsilon f…","labels":[],"detail_key":"p55"},{"id":"n44088","layer":"informal","project":"p55","title":"Trivial bounds on large gaps","kind":"proposition","summary":"[Trivial bounds on large gaps] One has \\theta_gap,>\\le \\theta_gap,\\geq. If \\theta_gap< 1/2, the…","labels":["trivial-large-gap"],"detail_key":"p55"},{"id":"n44089","layer":"informal","project":"p55","title":"Bounds on \\mu","kind":"lemma","summary":"[Bounds on \\mu] \\ \\item[(i)] \\cite[Theorem 2(i)]bazzanella-perelli For sufficiently small \\Delt…","labels":["baz-bound"],"detail_key":"p55"},{"id":"n44090","layer":"informal","project":"p55","title":"\\citeFI2004 Let \\chi=\\chi_D denotes the real primitive character of conductor D, x \\geqsl…","kind":"theorem","summary":"\\citeFI2004 Let \\chi=\\chi_D denotes the real primitive character of conductor D, x \\geqslant D^…","labels":[],"detail_key":"p55"},{"id":"n44091","layer":"informal","project":"p55","title":"Consequences of the prime number theorem","kind":"theorem","summary":"[Consequences of the prime number theorem] One has \\[ \\liminf_n\\to\\infty\\fracp_n + 1 - p_n\\log…","labels":[],"detail_key":"p55"},{"id":"n44092","layer":"informal","project":"p55","title":"Twin prime conjecture","kind":"conjecture","summary":"[Twin prime conjecture] One has \\[ \\liminf_n\\to\\infty(p_n + 1 - p_n) = 2. \\]","labels":[],"detail_key":"p55"},{"id":"n44093","layer":"informal","project":"p55","title":"Cram\\'er \\citecramer","kind":"conjecture","summary":"[Cram\\'er \\citecramer] One has \\[ \\limsup_X \\to \\infty\\fracG(X)(\\log X)^2 = 1. \\]","labels":[],"detail_key":"p55"},{"id":"n44094","layer":"informal","project":"p55","title":"Polymath 8b \\citepolymath_variants_2014","kind":"theorem","summary":"[Polymath 8b \\citepolymath_variants_2014] One has \\[ \\liminf_n\\to\\infty(p_n + 1 - p_n) \\le 246.…","labels":[],"detail_key":"p55"},{"id":"n44095","layer":"informal","project":"p55","title":"Maynard \\citemaynard_small_2015","kind":"theorem","summary":"[Maynard \\citemaynard_small_2015] Assuming the Elliott-Halberstam conjecture (EH), one has \\[ \\…","labels":[],"detail_key":"p55"},{"id":"n44096","layer":"informal","project":"p55","title":"Polymath 8b \\citepolymath_variants_2014","kind":"theorem","summary":"[Polymath 8b \\citepolymath_variants_2014] Assuming the Generalized Elliott-Halberstam conjectur…","labels":[],"detail_key":"p55"},{"id":"n44097","layer":"informal","project":"p55","title":"Ford--Green--Konyagin--Maynard--Tao (2017) \\citeford_long_2017","kind":"theorem","summary":"[Ford--Green--Konyagin--Maynard--Tao (2017) \\citeford_long_2017] For unbounded X, one has \\[ G(…","labels":[],"detail_key":"p55"},{"id":"n44098","layer":"informal","project":"p55","title":"Divisor sum exponents","kind":"definition","summary":"[Divisor sum exponents] Let k \\geq 1 be a fixed integer. Then, \\alpha_k is the least (fixed) ex…","labels":["divisor-def"],"detail_key":"p55"},{"id":"n44099","layer":"informal","project":"p55","title":"d_1 exponent","kind":"lemma","summary":"[d_1 exponent] One has \\alpha_1=\\beta_1=0.","labels":["divisor-1"],"detail_key":"p55"},{"id":"n44100","layer":"informal","project":"p55","title":"Follows from \\sum_n \\le x1 = x + O(1).","kind":"proof","summary":"Follows from \\sum_n \\le x1 = x + O(1).","labels":[],"detail_key":"p55"},{"id":"n44101","layer":"informal","project":"p55","title":"Hardy \\citehardy_average_1917","kind":"theorem","summary":"[Hardy \\citehardy_average_1917] One has \\beta_2 = 1/4.","labels":["avg-divisor-2"],"detail_key":"p55"},{"id":"n44102","layer":"informal","project":"p55","title":"Cram\\'er \\citecramer_uber_1922","kind":"theorem","summary":"[Cram\\'er \\citecramer_uber_1922] One has \\beta_3 = 1/3.","labels":["avg-divisor-3"],"detail_key":"p55"},{"id":"n44103","layer":"informal","project":"p55","title":"Lower bound on \\alpha_k and \\beta_k","kind":"lemma","summary":"[Lower bound on \\alpha_k and \\beta_k] For all k \\geq 1, one has \\[ \\alpha_k \\geq \\beta_k \\geq \\…","labels":["divisor-lower"],"detail_key":"p55"},{"id":"n44104","layer":"informal","project":"p55","title":"The first inequality follows from inserting the bound \\Delta_k(x) \\ll x^\\alpha_k + o(1) i…","kind":"proof","summary":"The first inequality follows from inserting the bound \\Delta_k(x) \\ll x^\\alpha_k + o(1) into th…","labels":[],"detail_key":"p55"},{"id":"n44105","layer":"informal","project":"p55","title":"Generalised Dirichlet divisor problem conjecture","kind":"conjecture","summary":"[Generalised Dirichlet divisor problem conjecture] For all k \\geq 1, one has \\[ \\alpha_k = \\bet…","labels":[],"detail_key":"p55"},{"id":"n44106","layer":"informal","project":"p55","title":"divisor-2-bound","kind":"theorem","summary":"\\cite[Theorem 1.2]li_yang_gauss_2024 One has \\alpha_2 \\leq \\alpha^* = 0.314483\\ldots, where \\al…","labels":["divisor-2-bound"],"detail_key":"p55"},{"id":"n44107","layer":"informal","project":"p55","title":"divisor-kolesnik","kind":"theorem","summary":"\\citekolesnik One has \\alpha_3 \\leq 43/96.","labels":["divisor-kolesnik"],"detail_key":"p55"},{"id":"n44108","layer":"informal","project":"p55","title":"mas","kind":"lemma","summary":"Let k \\geq 2 be an integer. If M(\\sigma,k) = 1 then \\alpha_k \\leq \\sigma.","labels":["mas"],"detail_key":"p55"},{"id":"n44109","layer":"informal","project":"p55","title":"See \\cite[\\S 13.3]ivic.","kind":"proof","summary":"See \\cite[\\S 13.3]ivic.","labels":[],"detail_key":"p55"},{"id":"n44110","layer":"informal","project":"p55","title":"Piltz bound","kind":"lemma","summary":"[Piltz bound] For k \\ge 2, one has \\[ \\alpha_k \\le 1 - \\frac1k. \\]","labels":["piltz-alpha"],"detail_key":"p55"},{"id":"n44111","layer":"informal","project":"p55","title":"Voronoi, Landau bound","kind":"lemma","summary":"[Voronoi, Landau bound] For k \\ge 2, one has \\[ \\alpha_k \\leq 1 - \\frac2k + 1. \\]","labels":["voronoi-alpha"],"detail_key":"p55"},{"id":"n44112","layer":"informal","project":"p55","title":"See Voronoi \\citevoronoi_sur_1903 for k = 2 and Landau \\citelandau_uber_1912 for k \\ge 3.","kind":"proof","summary":"See Voronoi \\citevoronoi_sur_1903 for k = 2 and Landau \\citelandau_uber_1912 for k \\ge 3.","labels":[],"detail_key":"p55"},{"id":"n44113","layer":"informal","project":"p55","title":"Hardy--Littlewood bound for k \\ge 4","kind":"lemma","summary":"[Hardy--Littlewood bound for k \\ge 4] For k \\ge 4, one has \\[ \\alpha_k \\leq 1 - \\frac3k + 2. \\]","labels":["hl-alpha"],"detail_key":"p55"},{"id":"n44114","layer":"informal","project":"p55","title":"See \\citehardy_littlewood_approximate_1923. The original proof relied on the assumption t…","kind":"proof","summary":"See \\citehardy_littlewood_approximate_1923. The original proof relied on the assumption that \\m…","labels":[],"detail_key":"p55"},{"id":"n44115","layer":"informal","project":"p55","title":"Tong bound for 4 \\le k \\le 11","kind":"lemma","summary":"[Tong bound for 4 \\le k \\le 11] One has 4 \\alpha_4 &\\le 1/2,\\qquad &&\\alpha_5 \\le 4/7,\\qquad &&…","labels":[],"detail_key":"p55"},{"id":"n44116","layer":"informal","project":"p55","title":"See Tong \\citeTong_divisor.","kind":"proof","summary":"See Tong \\citeTong_divisor.","labels":[],"detail_key":"p55"},{"id":"n44117","layer":"informal","project":"p55","title":"\\citeheathbrown_mean_1981 For 4 \\le k \\le 8, one has \\[ \\alpha_k \\leq \\frac3k-44k. \\]","kind":"theorem","summary":"\\citeheathbrown_mean_1981 For 4 \\le k \\le 8, one has \\[ \\alpha_k \\leq \\frac3k-44k. \\]","labels":[],"detail_key":"p55"},{"id":"n44118","layer":"informal","project":"p55","title":"Ivi\\'c--Ouellet bound for large k","kind":"theorem","summary":"[Ivi\\'c--Ouellet bound for large k]\\citeivic_ouellet_1989 One has \\alpha_10 \\le 27/40,\\qquad \\a…","labels":[],"detail_key":"p55"},{"id":"n44119","layer":"informal","project":"p55","title":"\\cite[Theorem 13.12]ivic One can bound \\alpha_k by (3k-4)/4k & \\hbox for 4 \\leq k \\leq 8…","kind":"theorem","summary":"\\cite[Theorem 13.12]ivic One can bound \\alpha_k by (3k-4)/4k & \\hbox for 4 \\leq k \\leq 8 \\\\ 35/…","labels":[],"detail_key":"p55"},{"id":"n44120","layer":"informal","project":"p55","title":"Heath-Brown bound for large k","kind":"lemma","summary":"[Heath-Brown bound for large k] For any k \\ge 2, one has \\[ \\alpha_k \\le 1 - 0.849k^-2/3. \\]","labels":["hb-alpha-large"],"detail_key":"p55"},{"id":"n44121","layer":"informal","project":"p55","title":"See Heath-Brown \\citeheathbrown_new_2017.","kind":"proof","summary":"See Heath-Brown \\citeheathbrown_new_2017.","labels":[],"detail_key":"p55"},{"id":"n44122","layer":"informal","project":"p55","title":"\\citebellotti_generalised_2023","kind":"theorem","summary":"[\\citebellotti_generalised_2023]For integer k \\ge 30, one has \\[ \\alpha_k \\leq 1 - 1.421(k - 1.…","labels":[],"detail_key":"p55"},{"id":"n44123","layer":"informal","project":"p55","title":"Trudgian--Yang bound for large k","kind":"theorem","summary":"[Trudgian--Yang bound for large k][\\citetrudgian-yang, Theorem 2.9]One has \\alpha_9 \\le 0.64720…","labels":[],"detail_key":"p55"},{"id":"n44124","layer":"informal","project":"p55","title":"Li bound for large k","kind":"theorem","summary":"[Li bound for large k][\\citeli_Dirichlet_2025, Theorem 2]One has \\alpha_9 \\le 0.638889,\\qquad \\…","labels":[],"detail_key":"p55"},{"id":"n44125","layer":"informal","project":"p55","title":"Pythagorean triple exponent","kind":"definition","summary":"[Pythagorean triple exponent] Let \\theta_Pythag be the least exponent for which one has P(N) =…","labels":["pythag-def"],"detail_key":"p55"},{"id":"n44126","layer":"informal","project":"p55","title":"pythag-14","kind":"lemma","summary":"One has \\theta_Pythag\\leq 1/4.","labels":["pythag-14"],"detail_key":"p55"},{"id":"n44127","layer":"informal","project":"p55","title":"See \\citewild_1955, duttlinger_schwarz. The previous bound \\theta_Pythag\\leq 1/3 was obta…","kind":"proof","summary":"See \\citewild_1955, duttlinger_schwarz. The previous bound \\theta_Pythag\\leq 1/3 was obtained i…","labels":[],"detail_key":"p55"},{"id":"n44128","layer":"informal","project":"p55","title":"exp_pair_to_pythag","kind":"lemma","summary":"If (k,\\ell) is an exponent pair, and RH holds, then \\theta_Pythag\\leq \\max( \\frac13 - \\frac56 \\…","labels":["exp_pair_to_pythag"],"detail_key":"p55"},{"id":"n44129","layer":"informal","project":"p55","title":"See \\citemenzer and \\cite[Section 5.10]trudgian-yang.","kind":"proof","summary":"See \\citemenzer and \\cite[Section 5.10]trudgian-yang.","labels":[],"detail_key":"p55"},{"id":"n44130","layer":"informal","project":"p55","title":"pythag-71-316","kind":"lemma","summary":"Assuming RH, one has \\theta_Pythag\\leq 71/316.","labels":["pythag-71-316"],"detail_key":"p55"},{"id":"n44131","layer":"informal","project":"p55","title":"See \\cite[Section 5.10]trudgian-yang.","kind":"proof","summary":"See \\cite[Section 5.10]trudgian-yang.","labels":[],"detail_key":"p55"},{"id":"n44132","layer":"informal","project":"p55","title":"Prime counting function on arithmetic progressions","kind":"definition","summary":"[Prime counting function on arithmetic progressions] Suppose a,q\\in Z with \\gcd(a,q)=1. For eac…","labels":[],"detail_key":"p55"},{"id":"n44133","layer":"informal","project":"p55","title":"Logarithmic integral function","kind":"definition","summary":"[Logarithmic integral function] Define the offset logarithmic integral function for x\\geq2 by L…","labels":[],"detail_key":"p55"},{"id":"n44134","layer":"informal","project":"p55","title":"Brun-Titchmarsh theorem under GRH (1929) \\citetitchmarsh_divisor_1930","kind":"theorem","summary":"[Brun-Titchmarsh theorem under GRH (1929) \\citetitchmarsh_divisor_1930] Under the Generalized R…","labels":["titchmarsh-GRH-asymptotic"],"detail_key":"p55"},{"id":"n44135","layer":"informal","project":"p55","title":"Walfisz (1936) \\citewalfisz_1936","kind":"theorem","summary":"[Walfisz (1936) \\citewalfisz_1936] Fix B\\geq0 and suppose q\\leq(\\log x)^B. Then there exists A=…","labels":["walfisz-small-q-asymptotic"],"detail_key":"p55"},{"id":"n44136","layer":"informal","project":"p55","title":"Brun-Titchmarsh theorem (1929) \\citetitchmarsh_divisor_1930","kind":"theorem","summary":"[Brun-Titchmarsh theorem (1929) \\citetitchmarsh_divisor_1930] If 0<\\theta<1 and q\\leq x^\\theta,…","labels":[],"detail_key":"p55"},{"id":"n44137","layer":"informal","project":"p55","title":"Lint, Richert (1965) \\citelint_richert_1965","kind":"theorem","summary":"[Lint, Richert (1965) \\citelint_richert_1965] If y>q, then \\pi(x+y;q,a)-\\pi(x;q,a)<\\frac2y\\varp…","labels":[],"detail_key":"p55"},{"id":"n44138","layer":"informal","project":"p55","title":"Montgomery, Vaughan (1973) \\citemontgomery_vaughan_1973","kind":"theorem","summary":"[Montgomery, Vaughan (1973) \\citemontgomery_vaughan_1973] If y>q, then \\pi(x+y;q,a)-\\pi(x;q,a)<…","labels":[],"detail_key":"p55"},{"id":"n44139","layer":"informal","project":"p55","title":"\\theta and C_\\theta","kind":"definition","summary":"[\\theta and C_\\theta] Suppose x>0 and q\\in Z. Define \\theta:=\\frac\\log q\\log x, and let C_\\thet…","labels":[],"detail_key":"p55"},{"id":"n44140","layer":"informal","project":"p55","title":"From exponent pairs to Brun--Titchmarsh","kind":"theorem","summary":"[From exponent pairs to Brun--Titchmarsh] \\cite[Theorem 1.4]xi-zheng If (k,\\ell) is an exponent…","labels":["convert"],"detail_key":"p55"},{"id":"n44141","layer":"informal","project":"p55","title":"Linnik's constant L","kind":"definition","summary":"[Linnik's constant L] Define L to be the infimum over all L'>0 where there exists q_0(L')>0 suc…","labels":[],"detail_key":"p55"},{"id":"n44142","layer":"informal","project":"p55","title":"Maynard (2013) \\citemaynard_2013","kind":"theorem","summary":"[Maynard (2013) \\citemaynard_2013] For sufficiently large q and x>q^8, we have \\frac\\log qq^1/2…","labels":[],"detail_key":"p55"},{"id":"n44143","layer":"informal","project":"p55","title":"Maynard (2013) \\citemaynard_2013","kind":"theorem","summary":"[Maynard (2013) \\citemaynard_2013] Let \\epsilon>0. There exists q_0(\\epsilon)>0 such that for a…","labels":[],"detail_key":"p55"},{"id":"n44144","layer":"informal","project":"p55","title":"addbasis","kind":"definition","summary":"Let A\\subsetN be such that there exists k for which \\underbraceA+A+\\cdots+A_k\\text times=N Then…","labels":["addbasis"],"detail_key":"p55"},{"id":"n44145","layer":"informal","project":"p55","title":"g","kind":"definition","summary":"For any k\\ge1 let A_k=\\n^k:n\\inN\\cup\\0\\\\. Let g(k) be the order of A_k when it exists. That is,…","labels":["{g"],"detail_key":"p55"},{"id":"n44146","layer":"informal","project":"p55","title":"G","kind":"definition","summary":"For any k\\ge1, let G(k) be the minimum m such that there exists N\\ge1 for which \\underbraceA_k+…","labels":["{G"],"detail_key":"p55"},{"id":"n44147","layer":"informal","project":"p55","title":"Lagrange's Four Square Theorem","kind":"theorem","summary":"[Lagrange's Four Square Theorem] We have g(2)=4; that is every natural number may be written as…","labels":[],"detail_key":"p55"},{"id":"n44148","layer":"informal","project":"p55","title":"Linnik \\citeLinnik_Линник1943","kind":"theorem","summary":"[Linnik \\citeLinnik_Линник1943] g(k) exists for all k\\ge1.","labels":["HilbertExistence"],"detail_key":"p55"},{"id":"n44149","layer":"informal","project":"p55","title":"Let G_1(k) be the smallest number m such that d(\\underbraceA_k+\\cdots+A_k_m\\text times)=1…","kind":"definition","summary":"Let G_1(k) be the smallest number m such that d(\\underbraceA_k+\\cdots+A_k_m\\text times)=1 where…","labels":[],"detail_key":"p55"},{"id":"n44150","layer":"informal","project":"p55","title":"Brudern and Wooley 2022 \\citebruedern2022waringsproblemlargerpowers","kind":"theorem","summary":"[Brudern and Wooley 2022 \\citebruedern2022waringsproblemlargerpowers] For all k\\ge1, G(k)<k(\\lo…","labels":["BW2022"],"detail_key":"p55"},{"id":"n44151","layer":"informal","project":"p55","title":"G(3)\\ge4","kind":"lemma","summary":"G(3)\\ge4","labels":[],"detail_key":"p55"},{"id":"n44152","layer":"informal","project":"p55","title":"Note cubes are congruent 1,-1,0 modulo 9. Thus, numbers congruent 4,5 modulo 9 may not be…","kind":"proof","summary":"Note cubes are congruent 1,-1,0 modulo 9. Thus, numbers congruent 4,5 modulo 9 may not be expre…","labels":[],"detail_key":"p55"},{"id":"n44153","layer":"informal","project":"p55","title":"Linnik \\citeLinnik_1943_sum_cubes","kind":"theorem","summary":"[Linnik \\citeLinnik_1943_sum_cubes] G(3)\\le7","labels":[],"detail_key":"p55"},{"id":"n44154","layer":"informal","project":"p55","title":"Kamke","kind":"theorem","summary":"[Kamke] Let f(x) be an integer valued polynomial such that there does not exist d\\inN such that…","labels":[],"detail_key":"p55"},{"id":"n44155","layer":"informal","project":"p55","title":"Wooley","kind":"theorem","summary":"[Wooley] Assuming GRH, then x_1^2+x_2^2+x_3^3+x_4^3+x_5^6+x_6^6=n is solvable for sufficiently…","labels":["WooleyThm"],"detail_key":"p55"},{"id":"n44156","layer":"informal","project":"p55","title":"Liu, Wooley, Yu \\citeLIU2004298","kind":"theorem","summary":"[Liu, Wooley, Yu \\citeLIU2004298] Let E(N) be the number of integers n\\equiv 4\\;(mod\\; 24) for…","labels":[],"detail_key":"p55"},{"id":"n44157","layer":"informal","project":"p55","title":"For k\\inN, let H(k) be the minimum integer s such that p_1^k+p_2^k+\\cdots+p_s^k=n is solv…","kind":"definition","summary":"For k\\inN, let H(k) be the minimum integer s such that p_1^k+p_2^k+\\cdots+p_s^k=n is solvable f…","labels":[],"detail_key":"p55"},{"id":"n44158","layer":"informal","project":"p55","title":"Wooley, Kawada 2001 \\citekawada_koichi_wooley_trevor_2001","kind":"theorem","summary":"[Wooley, Kawada 2001 \\citekawada_koichi_wooley_trevor_2001] We have \\item H(4)\\le14 \\item For a…","labels":[],"detail_key":"p55"},{"id":"n44159","layer":"informal","project":"p55","title":"Kumchev, Wooley 2016 \\citeKumchev2016","kind":"theorem","summary":"[Kumchev, Wooley 2016 \\citeKumchev2016] For large values of k, H(k)\\le(4k-2)\\log k-(2\\log 2-1)k…","labels":["tab:Hkbounds"],"detail_key":"p55"},{"id":"n44160","layer":"informal","project":"p55","title":"Define the Schnirelmann density of A\\subset N as \\sigma A=\\inf_n\\ge1\\frac\\#(A\\cap J_n)n","kind":"definition","summary":"Define the Schnirelmann density of A\\subset N as \\sigma A=\\inf_n\\ge1\\frac\\#(A\\cap J_n)n","labels":[],"detail_key":"p55"},{"id":"n44161","layer":"informal","project":"p55","title":"Define the lower asymptotic density of A\\subset N as \\delta A=\\liminf_n\\to\\infty\\frac\\#(A…","kind":"definition","summary":"Define the lower asymptotic density of A\\subset N as \\delta A=\\liminf_n\\to\\infty\\frac\\#(A\\cap J…","labels":[],"detail_key":"p55"},{"id":"n44162","layer":"informal","project":"p55","title":"Schnirelmann \\citeSchnirelmann1933","kind":"theorem","summary":"[Schnirelmann \\citeSchnirelmann1933] Suppose \\sigma A>0. Then A is an additive basis for N.","labels":["addbasisthm"],"detail_key":"p55"},{"id":"n44163","layer":"informal","project":"p55","title":"Schnirelmann \\citeSchnirelmann1933","kind":"theorem","summary":"[Schnirelmann \\citeSchnirelmann1933] Let P denote the set of primes. Then, \\delta( P+ P)>0. The…","labels":[],"detail_key":"p55"},{"id":"n44164","layer":"informal","project":"p55","title":"Romanoff \\citeASNSP_1995_4_22_4_645_0","kind":"theorem","summary":"[Romanoff \\citeASNSP_1995_4_22_4_645_0] Let \\mathfrak S_a=\\p+a^k:p\\in P, k\\inN\\. Then, \\sigma\\m…","labels":[],"detail_key":"p55"},{"id":"n44165","layer":"informal","project":"p55","title":"B\\subset N is called an essential component if \\sigma(A+B)>\\sigma(A) for any A\\subsetN wi…","kind":"definition","summary":"B\\subset N is called an essential component if \\sigma(A+B)>\\sigma(A) for any A\\subsetN with 0<\\…","labels":[],"detail_key":"p55"},{"id":"n44166","layer":"informal","project":"p55","title":"For fixed integer k \\ge 2, define \\theta^Gauss_k as the least (fixed) exponent for which…","kind":"definition","summary":"For fixed integer k \\ge 2, define \\theta^Gauss_k as the least (fixed) exponent for which \\[ S_k…","labels":[],"detail_key":"p55"},{"id":"n44167","layer":"informal","project":"p55","title":"gauss-circle-conj","kind":"conjecture","summary":"One has \\[ \\theta^Gauss_k = 1/2,& k = 2,\\\\ k - 2,& k \\ge 3. \\]","labels":["gauss-circle-conj"],"detail_key":"p55"},{"id":"n44168","layer":"informal","project":"p55","title":"For integer k \\ge 4, one has \\theta^Gauss_k = k - 2.","kind":"theorem","summary":"For integer k \\ge 4, one has \\theta^Gauss_k = k - 2.","labels":[],"detail_key":"p55"},{"id":"n44169","layer":"informal","project":"p55","title":"gauss-circle-lower-23","kind":"theorem","summary":"One has \\theta^Gauss_2 \\ge 1/2 and \\theta^Gauss_3 \\ge 1.","labels":["gauss-circle-lower-23"],"detail_key":"p55"},{"id":"n44170","layer":"informal","project":"p55","title":"Li--Yang (2023) \\citeli_yang_gauss_2024","kind":"theorem","summary":"[Li--Yang (2023) \\citeli_yang_gauss_2024] One has \\theta_2^Gauss \\le 2\\alpha, where \\alpha = 0.…","labels":[],"detail_key":"p55"},{"id":"n44171","layer":"informal","project":"p55","title":"The value of","kind":"remark","summary":"The value of","labels":[],"detail_key":"p55"},{"id":"n44172","layer":"informal","project":"p56","title":"Concavity","kind":"lemma","summary":"[Concavity] h is strictly concave on [0,\\infty).","labels":["concave"],"detail_key":"p56"},{"id":"n44173","layer":"informal","project":"p56","title":"Check that h' is strictly monotone decreasing.","kind":"proof","summary":"Check that h' is strictly monotone decreasing.","labels":[],"detail_key":"p56"},{"id":"n44174","layer":"informal","project":"p56","title":"log sum inequality","kind":"lemma","summary":"[log sum inequality] If S is a finite set, and a_s,b_s are non-negative for s\\in S, then \\sum_s…","labels":["log-sum"],"detail_key":"p56"},{"id":"n44175","layer":"informal","project":"p56","title":"Let B:=\\sum_s\\in S b_s. Apply Jensen and \\Crefconcave to show that \\sum_s\\in S \\fracb_sB…","kind":"proof","summary":"Let B:=\\sum_s\\in S b_s. Apply Jensen and \\Crefconcave to show that \\sum_s\\in S \\fracb_sB h(\\fra…","labels":[],"detail_key":"p56"},{"id":"n44176","layer":"informal","project":"p56","title":"converse log sum","kind":"lemma","summary":"[converse log sum] If equality holds in \\Creflog-sum, then a_s=r\\cdot b_s for every s\\in S, for…","labels":["converse-log-sum"],"detail_key":"p56"},{"id":"n44177","layer":"informal","project":"p56","title":"By the fact that h is strictly concave and the equality condition of Jensen.","kind":"proof","summary":"By the fact that h is strictly concave and the equality condition of Jensen.","labels":[],"detail_key":"p56"},{"id":"n44178","layer":"informal","project":"p56","title":"Entropy","kind":"definition","summary":"[Entropy] If X is an S-valued random variable, the entropy H[X] of X is defined H[X] := \\sum_s…","labels":["entropy-def"],"detail_key":"p56"},{"id":"n44179","layer":"informal","project":"p56","title":"Entropy and relabeling","kind":"lemma","summary":"[Entropy and relabeling] \\item[(i)] If X: \\Omega \\to S and Y: \\Omega \\to T are random variables…","labels":["relabeled-entropy"],"detail_key":"p56"},{"id":"n44180","layer":"informal","project":"p56","title":"Expand out both entropies and rearrange.","kind":"proof","summary":"Expand out both entropies and rearrange.","labels":[],"detail_key":"p56"},{"id":"n44181","layer":"informal","project":"p56","title":"Jensen bound","kind":"lemma","summary":"[Jensen bound] If X is an S-valued random variable, then H[X] \\leq \\log |S|.","labels":["jensen-bound"],"detail_key":"p56"},{"id":"n44182","layer":"informal","project":"p56","title":"This is a direct consequence of \\Crefconcave and Jensen's inequality.","kind":"proof","summary":"This is a direct consequence of \\Crefconcave and Jensen's inequality.","labels":[],"detail_key":"p56"},{"id":"n44183","layer":"informal","project":"p56","title":"Uniform distribution","kind":"definition","summary":"[Uniform distribution] If H is a subset of S, an S-random variable X is said to be uniformly di…","labels":["uniform-def"],"detail_key":"p56"},{"id":"n44184","layer":"informal","project":"p56","title":"Uniform distributions exist","kind":"lemma","summary":"[Uniform distributions exist] Given a finite non-empty subset H of a set S, there exists a rand…","labels":["unif-exist"],"detail_key":"p56"},{"id":"n44185","layer":"informal","project":"p56","title":"Direct construction.","kind":"proof","summary":"Direct construction.","labels":[],"detail_key":"p56"},{"id":"n44186","layer":"informal","project":"p56","title":"Entropy of uniform random variable","kind":"lemma","summary":"[Entropy of uniform random variable] If X is S-valued random variable, then H[X] = \\log |S| if…","labels":["uniform-entropy"],"detail_key":"p56"},{"id":"n44187","layer":"informal","project":"p56","title":"Direct computation in one direction. Converse direction needs the strict Jensen inequalit…","kind":"proof","summary":"Direct computation in one direction. Converse direction needs the strict Jensen inequality and…","labels":[],"detail_key":"p56"},{"id":"n44188","layer":"informal","project":"p56","title":"Entropy of uniform random variable, II","kind":"lemma","summary":"[Entropy of uniform random variable, II] If X is uniformly distributed on H, then, then H[X] =…","labels":["uniform-entropy-II"],"detail_key":"p56"},{"id":"n44189","layer":"informal","project":"p56","title":"Direct computation.","kind":"proof","summary":"Direct computation.","labels":[],"detail_key":"p56"},{"id":"n44190","layer":"informal","project":"p56","title":"Bounded entropy implies concentration","kind":"lemma","summary":"[Bounded entropy implies concentration] If X is an S-valued random variable, then there exists…","labels":["bound-conc"],"detail_key":"p56"},{"id":"n44191","layer":"informal","project":"p56","title":"We have H[X] = \\sum_s \\in S P[X=s] \\log \\frac1P[X=s] \\geq \\min_s \\in S \\log \\frac1P[X=s]…","kind":"proof","summary":"We have H[X] = \\sum_s \\in S P[X=s] \\log \\frac1P[X=s] \\geq \\min_s \\in S \\log \\frac1P[X=s] and th…","labels":[],"detail_key":"p56"},{"id":"n44192","layer":"informal","project":"p56","title":"Commutativity and associativity of joint entropy","kind":"lemma","summary":"[Commutativity and associativity of joint entropy] If X: \\Omega \\to S, Y: \\Omega \\to T, and Z:…","labels":["entropy-comm"],"detail_key":"p56"},{"id":"n44193","layer":"informal","project":"p56","title":"Set up an injection from (X,Y) to (Y,X) and use \\Crefrelabeled-entropy for the first clai…","kind":"proof","summary":"Set up an injection from (X,Y) to (Y,X) and use \\Crefrelabeled-entropy for the first claim. Sim…","labels":[],"detail_key":"p56"},{"id":"n44194","layer":"informal","project":"p56","title":"Conditioned event","kind":"definition","summary":"[Conditioned event] If X: \\Omega \\to S is an S-valued random variable and E is an event in \\Ome…","labels":["condition-event-def"],"detail_key":"p56"},{"id":"n44195","layer":"informal","project":"p56","title":"Conditional entropy","kind":"definition","summary":"[Conditional entropy] If X: \\Omega \\to S and Y: \\Omega \\to T are random variables, the conditio…","labels":["conditional-entropy-def"],"detail_key":"p56"},{"id":"n44196","layer":"informal","project":"p56","title":"Conditional entropy and relabeling","kind":"lemma","summary":"[Conditional entropy and relabeling] If X: \\Omega \\to S, Y: \\Omega \\to T, and Z: \\Omega \\to U a…","labels":["relabeled-entropy-cond"],"detail_key":"p56"},{"id":"n44197","layer":"informal","project":"p56","title":"For the first part, use \\Crefconditional-entropy-def and then \\Crefrelabeled-entropy. The…","kind":"proof","summary":"For the first part, use \\Crefconditional-entropy-def and then \\Crefrelabeled-entropy. The secon…","labels":[],"detail_key":"p56"},{"id":"n44198","layer":"informal","project":"p56","title":"Chain rule","kind":"lemma","summary":"[Chain rule] If X: \\Omega \\to S and Y: \\Omega \\to T are random variables, then H[X, Y] = H[Y] +…","labels":["chain-rule"],"detail_key":"p56"},{"id":"n44199","layer":"informal","project":"p56","title":"Direct computation.","kind":"proof","summary":"Direct computation.","labels":[],"detail_key":"p56"},{"id":"n44200","layer":"informal","project":"p56","title":"Conditional chain rule","kind":"lemma","summary":"[Conditional chain rule] If X: \\Omega \\to S, Y: \\Omega \\to T, Z: \\Omega \\to U are random variab…","labels":["conditional-chain-rule"],"detail_key":"p56"},{"id":"n44201","layer":"informal","project":"p56","title":"For each z \\in U, we can apply \\Crefchain-rule to the random variables (X|Z=z) and (Y|Z=z…","kind":"proof","summary":"For each z \\in U, we can apply \\Crefchain-rule to the random variables (X|Z=z) and (Y|Z=z) to o…","labels":[],"detail_key":"p56"},{"id":"n44202","layer":"informal","project":"p56","title":"Mutual information","kind":"definition","summary":"[Mutual information] If X: \\Omega \\to S, Y: \\Omega \\to T are random variables, then I[X : Y] :=…","labels":["information-def"],"detail_key":"p56"},{"id":"n44203","layer":"informal","project":"p56","title":"Alternative formulae for mutual information","kind":"lemma","summary":"[Alternative formulae for mutual information] With notation as above, we have I[X : Y] = I[Y:X]…","labels":["alternative-mutual"],"detail_key":"p56"},{"id":"n44204","layer":"informal","project":"p56","title":"Immediate from Lemmas \\refentropy-comm, \\refchain-rule.","kind":"proof","summary":"Immediate from Lemmas \\refentropy-comm, \\refchain-rule.","labels":[],"detail_key":"p56"},{"id":"n44205","layer":"informal","project":"p56","title":"Nonnegativity of mutual information","kind":"lemma","summary":"[Nonnegativity of mutual information] We have I[X:Y] \\geq 0.","labels":["mutual-nonneg"],"detail_key":"p56"},{"id":"n44206","layer":"informal","project":"p56","title":"An application of jensen's inequality and \\Crefconcave,alternative-mutual.","kind":"proof","summary":"An application of jensen's inequality and \\Crefconcave,alternative-mutual.","labels":[],"detail_key":"p56"},{"id":"n44207","layer":"informal","project":"p56","title":"Subadditivity","kind":"corollary","summary":"[Subadditivity] With notation as above, we have H[X,Y] \\leq H[X] + H[Y].","labels":["subadditive"],"detail_key":"p56"},{"id":"n44208","layer":"informal","project":"p56","title":"Use \\Crefmutual-nonneg.","kind":"proof","summary":"Use \\Crefmutual-nonneg.","labels":[],"detail_key":"p56"},{"id":"n44209","layer":"informal","project":"p56","title":"Conditioning reduces entropy","kind":"corollary","summary":"[Conditioning reduces entropy] With notation as above, we have H[X|Y] \\leq H[X].","labels":["cond-reduce"],"detail_key":"p56"},{"id":"n44210","layer":"informal","project":"p56","title":"Combine \\Crefmutual-nonneg with \\Crefalternative-mutual.","kind":"proof","summary":"Combine \\Crefmutual-nonneg with \\Crefalternative-mutual.","labels":[],"detail_key":"p56"},{"id":"n44211","layer":"informal","project":"p56","title":"Submodularity","kind":"corollary","summary":"[Submodularity] With three random variables X,Y,Z, one has H[X|Y,Z] \\leq H[X|Z].","labels":["submodularity"],"detail_key":"p56"},{"id":"n44212","layer":"informal","project":"p56","title":"Apply the ``averaging over conditioning'' argument to \\Crefcond-reduce.","kind":"proof","summary":"Apply the ``averaging over conditioning'' argument to \\Crefcond-reduce.","labels":[],"detail_key":"p56"},{"id":"n44213","layer":"informal","project":"p56","title":"Alternate form of submodularity","kind":"corollary","summary":"[Alternate form of submodularity] With three random variables X,Y,Z, one has H[X,Y,Z] + H[Z] \\l…","labels":["alt-submodularity"],"detail_key":"p56"},{"id":"n44214","layer":"informal","project":"p56","title":"Apply \\Crefsubmodularity and \\Crefchain-rule.","kind":"proof","summary":"Apply \\Crefsubmodularity and \\Crefchain-rule.","labels":[],"detail_key":"p56"},{"id":"n44215","layer":"informal","project":"p56","title":"Independent random variables","kind":"definition","summary":"[Independent random variables] Two random variables X: \\Omega \\to S and Y: \\Omega \\to T are ind…","labels":["independent-def"],"detail_key":"p56"},{"id":"n44216","layer":"informal","project":"p56","title":"Vanishing of mutual information","kind":"lemma","summary":"[Vanishing of mutual information] If X,Y are random variables, then I[X:Y] = 0 if and only if X…","labels":["vanish-entropy"],"detail_key":"p56"},{"id":"n44217","layer":"informal","project":"p56","title":"An application of the equality case of Jensen's inequality and \\Crefconcave.","kind":"proof","summary":"An application of the equality case of Jensen's inequality and \\Crefconcave.","labels":[],"detail_key":"p56"},{"id":"n44218","layer":"informal","project":"p56","title":"Additivity of entropy","kind":"corollary","summary":"[Additivity of entropy] If X,Y are random variables, then H[X,Y] = H[X] + H[Y] if and only if X…","labels":["add-entropy"],"detail_key":"p56"},{"id":"n44219","layer":"informal","project":"p56","title":"Direct from \\Crefvanish-entropy.","kind":"proof","summary":"Direct from \\Crefvanish-entropy.","labels":[],"detail_key":"p56"},{"id":"n44220","layer":"informal","project":"p56","title":"Conditional mutual information","kind":"definition","summary":"[Conditional mutual information] If X,Y,Z are random variables, with Z U-valued, then I[X:Y|Z]…","labels":["conditional-mutual-def"],"detail_key":"p56"},{"id":"n44221","layer":"informal","project":"p56","title":"Alternate formula for conditional mutual information","kind":"lemma","summary":"[Alternate formula for conditional mutual information] We have I[X:Y|Z] := H[X|Z] + H[Y|Z] - H[…","labels":["conditional-mutual-alt"],"detail_key":"p56"},{"id":"n44222","layer":"informal","project":"p56","title":"Routine computation.","kind":"proof","summary":"Routine computation.","labels":[],"detail_key":"p56"},{"id":"n44223","layer":"informal","project":"p56","title":"Nonnegativity of conditional mutual information","kind":"lemma","summary":"[Nonnegativity of conditional mutual information] If X,Y,Z are random variables, then I[X:Y|Z]…","labels":["conditional-nonneg"],"detail_key":"p56"},{"id":"n44224","layer":"informal","project":"p56","title":"Use \\Crefconditional-mutual-def and \\Crefsubmodularity.","kind":"proof","summary":"Use \\Crefconditional-mutual-def and \\Crefsubmodularity.","labels":[],"detail_key":"p56"},{"id":"n44225","layer":"informal","project":"p56","title":"Conditionally independent random variables","kind":"definition","summary":"[Conditionally independent random variables] Two random variables X: \\Omega \\to S and Y: \\Omega…","labels":["conditional-independent-def"],"detail_key":"p56"},{"id":"n44226","layer":"informal","project":"p56","title":"Vanishing conditional mutual information","kind":"lemma","summary":"[Vanishing conditional mutual information] If X,Y,Z are random variables, then I[X:Y|Z] = 0 iff…","labels":["conditional-vanish"],"detail_key":"p56"},{"id":"n44227","layer":"informal","project":"p56","title":"Immediate from \\Crefvanish-entropy and \\Crefconditional-independent-def.","kind":"proof","summary":"Immediate from \\Crefvanish-entropy and \\Crefconditional-independent-def.","labels":[],"detail_key":"p56"},{"id":"n44228","layer":"informal","project":"p56","title":"Entropy of conditionally independent variables","kind":"corollary","summary":"[Entropy of conditionally independent variables] If X, Y are conditionally independent over Z,…","labels":["cond-trial-ent"],"detail_key":"p56"},{"id":"n44229","layer":"informal","project":"p56","title":"Immediate from \\Crefconditional-vanish and \\Crefconditional-mutual-alt.","kind":"proof","summary":"Immediate from \\Crefconditional-vanish and \\Crefconditional-mutual-alt.","labels":[],"detail_key":"p56"},{"id":"n44230","layer":"informal","project":"p56","title":"Negation preserves entropy","kind":"lemma","summary":"[Negation preserves entropy] If X is G-valued, then H[-X]=H[X].","labels":["neg-ent"],"detail_key":"p56"},{"id":"n44231","layer":"informal","project":"p56","title":"Immediate from \\Crefrelabeled-entropy.","kind":"proof","summary":"Immediate from \\Crefrelabeled-entropy.","labels":[],"detail_key":"p56"},{"id":"n44232","layer":"informal","project":"p56","title":"Shearing preserves entropy","kind":"lemma","summary":"[Shearing preserves entropy] If X,Y are G-valued, then H[X \\pm Y | Y]=H[X|Y] and H[X \\pm Y, Y]…","labels":["shear-ent"],"detail_key":"p56"},{"id":"n44233","layer":"informal","project":"p56","title":"Immediate from \\Crefrelabeled-entropy-cond and \\Crefchain-rule.","kind":"proof","summary":"Immediate from \\Crefrelabeled-entropy-cond and \\Crefchain-rule.","labels":[],"detail_key":"p56"},{"id":"n44234","layer":"informal","project":"p56","title":"Lower bound of sumset","kind":"lemma","summary":"[Lower bound of sumset] If X,Y are G-valued random variables on \\Omega, we have \\max(H[X], H[Y]…","labels":["sumset-lower-gen"],"detail_key":"p56"},{"id":"n44235","layer":"informal","project":"p56","title":"By \\Crefcond-reduce, \\refshear-ent, \\refalternative-mutual, \\refneg-ent we have H[X\\pm Y]…","kind":"proof","summary":"By \\Crefcond-reduce, \\refshear-ent, \\refalternative-mutual, \\refneg-ent we have H[X\\pm Y] \\geq…","labels":[],"detail_key":"p56"},{"id":"n44236","layer":"informal","project":"p56","title":"Conditional lower bound on sumset","kind":"corollary","summary":"[Conditional lower bound on sumset] If X,Y are G-valued random variables on \\Omega and Z is ano…","labels":["sumset-lower-gen-cond"],"detail_key":"p56"},{"id":"n44237","layer":"informal","project":"p56","title":"This follows from \\Crefsumset-lower-gen by conditioning to Z = z and summing over z (weig…","kind":"proof","summary":"This follows from \\Crefsumset-lower-gen by conditioning to Z = z and summing over z (weighted b…","labels":[],"detail_key":"p56"},{"id":"n44238","layer":"informal","project":"p56","title":"Independent lower bound on sumset","kind":"corollary","summary":"[Independent lower bound on sumset] If X,Y are independent G-valued random variables, then \\max…","labels":["sumset-lower"],"detail_key":"p56"},{"id":"n44239","layer":"informal","project":"p56","title":"Combine \\Crefsumset-lower-gen with \\Crefvanish-entropy.","kind":"proof","summary":"Combine \\Crefsumset-lower-gen with \\Crefvanish-entropy.","labels":[],"detail_key":"p56"},{"id":"n44240","layer":"informal","project":"p56","title":"Copy preserves entropy","kind":"lemma","summary":"[Copy preserves entropy] If X' is a copy of X then H[X'] = H[X].","labels":["copy-ent"],"detail_key":"p56"},{"id":"n44241","layer":"informal","project":"p56","title":"Immediate from \\Crefentropy-def.","kind":"proof","summary":"Immediate from \\Crefentropy-def.","labels":[],"detail_key":"p56"},{"id":"n44242","layer":"informal","project":"p56","title":"Existence of independent copies","kind":"lemma","summary":"[Existence of independent copies] Let X_i : \\Omega_i \\to S_i be random variables for i=1,\\dots,…","labels":["independent-exist"],"detail_key":"p56"},{"id":"n44243","layer":"informal","project":"p56","title":"Explicit computation.","kind":"proof","summary":"Explicit computation.","labels":[],"detail_key":"p56"},{"id":"n44244","layer":"informal","project":"p56","title":"Ruzsa distance","kind":"definition","summary":"[Ruzsa distance] Let X,Y be G-valued random variables (not necessarily on the same sample space…","labels":["ruz-dist-def"],"detail_key":"p56"},{"id":"n44245","layer":"informal","project":"p56","title":"Distance from zero","kind":"lemma","summary":"[Distance from zero] If X is a G-valued random variable and 0 is the random variable taking the…","labels":["dist-zero"],"detail_key":"p56"},{"id":"n44246","layer":"informal","project":"p56","title":"This is an immediate consequence of the definitions and X-0\\equiv X and H(0)=0.","kind":"proof","summary":"This is an immediate consequence of the definitions and X-0\\equiv X and H(0)=0.","labels":[],"detail_key":"p56"},{"id":"n44247","layer":"informal","project":"p56","title":"Copy preserves Ruzsa distance","kind":"lemma","summary":"[Copy preserves Ruzsa distance] If X',Y' are copies of X,Y respectively then d[X';Y']=d[X ;Y].","labels":["ruz-copy"],"detail_key":"p56"},{"id":"n44248","layer":"informal","project":"p56","title":"Immediate from Definitions \\refruz-dist-def and \\Crefcopy-ent.","kind":"proof","summary":"Immediate from Definitions \\refruz-dist-def and \\Crefcopy-ent.","labels":[],"detail_key":"p56"},{"id":"n44249","layer":"informal","project":"p56","title":"Ruzsa distance in independent case","kind":"lemma","summary":"[Ruzsa distance in independent case] If X,Y are independent G-random variables then d[X ;Y] :=…","labels":["ruz-indep"],"detail_key":"p56"},{"id":"n44250","layer":"informal","project":"p56","title":"Immediate from \\Crefruz-dist-def and Lemmas \\refrelabeled-entropy, \\refcopy-ent.","kind":"proof","summary":"Immediate from \\Crefruz-dist-def and Lemmas \\refrelabeled-entropy, \\refcopy-ent.","labels":[],"detail_key":"p56"},{"id":"n44251","layer":"informal","project":"p56","title":"Distance symmetric","kind":"lemma","summary":"[Distance symmetric] If X,Y are G-valued random variables, then d[X ;Y] = d[Y;X].","labels":["ruzsa-symm"],"detail_key":"p56"},{"id":"n44252","layer":"informal","project":"p56","title":"Immediate from \\Crefneg-ent and \\Crefruz-dist-def.","kind":"proof","summary":"Immediate from \\Crefneg-ent and \\Crefruz-dist-def.","labels":[],"detail_key":"p56"},{"id":"n44253","layer":"informal","project":"p56","title":"Distance controls entropy difference","kind":"lemma","summary":"[Distance controls entropy difference] If X,Y are G-valued random variables, then |H[X]-H[Y]| \\…","labels":["ruzsa-diff"],"detail_key":"p56"},{"id":"n44254","layer":"informal","project":"p56","title":"Immediate from \\Crefsumset-lower and \\Crefruz-dist-def, and also \\Crefneg-ent.","kind":"proof","summary":"Immediate from \\Crefsumset-lower and \\Crefruz-dist-def, and also \\Crefneg-ent.","labels":[],"detail_key":"p56"},{"id":"n44255","layer":"informal","project":"p56","title":"Distance controls entropy growth","kind":"lemma","summary":"[Distance controls entropy growth] If X,Y are independent G-valued random variables, then H[X-Y…","labels":["ruzsa-growth"],"detail_key":"p56"},{"id":"n44256","layer":"informal","project":"p56","title":"Immediate from \\Crefsumset-lower and \\Crefruz-dist-def, and also \\Crefneg-ent.","kind":"proof","summary":"Immediate from \\Crefsumset-lower and \\Crefruz-dist-def, and also \\Crefneg-ent.","labels":[],"detail_key":"p56"},{"id":"n44257","layer":"informal","project":"p56","title":"Distance nonnegative","kind":"lemma","summary":"[Distance nonnegative] If X,Y are G-valued random variables, then d[X ;Y] \\geq 0.","labels":["ruzsa-nonneg"],"detail_key":"p56"},{"id":"n44258","layer":"informal","project":"p56","title":"Immediate from \\Crefruzsa-diff.","kind":"proof","summary":"Immediate from \\Crefruzsa-diff.","labels":[],"detail_key":"p56"},{"id":"n44259","layer":"informal","project":"p56","title":"Projection entropy and distance","kind":"lemma","summary":"[Projection entropy and distance] If G is an additive group and X is a G-valued random variable…","labels":["dist-projection"],"detail_key":"p56"},{"id":"n44260","layer":"informal","project":"p56","title":"WLOG, we make X, U_H independent (\\Crefindependent-exist). Now by Lemmas \\refsubmodularit…","kind":"proof","summary":"WLOG, we make X, U_H independent (\\Crefindependent-exist). Now by Lemmas \\refsubmodularity, \\re…","labels":[],"detail_key":"p56"},{"id":"n44261","layer":"informal","project":"p56","title":"Improved Ruzsa triangle inequality","kind":"lemma","summary":"[Improved Ruzsa triangle inequality] If X,Y,Z are G-valued random variables on \\Omega with (X,Y…","labels":["ruzsa-triangle-improved","submod-explicit"],"detail_key":"p56"},{"id":"n44262","layer":"informal","project":"p56","title":"Apply \\Crefalt-submodularity to obtain \\[H[X - Z, X - Y] + H[Y, X - Y] \\geq H[X - Z, Y, X…","kind":"proof","summary":"Apply \\Crefalt-submodularity to obtain \\[H[X - Z, X - Y] + H[Y, X - Y] \\geq H[X - Z, Y, X - Y]…","labels":[],"detail_key":"p56"},{"id":"n44263","layer":"informal","project":"p56","title":"Ruzsa triangle inequality","kind":"lemma","summary":"[Ruzsa triangle inequality] If X,Y,Z are G-valued random variables, then d[X ;Y] \\leq d[X ;Z] +…","labels":["ruzsa-triangle"],"detail_key":"p56"},{"id":"n44264","layer":"informal","project":"p56","title":"By \\Crefruz-copy and Lemmas \\refindependent-exist, \\refruz-indep, it suffices to prove th…","kind":"proof","summary":"By \\Crefruz-copy and Lemmas \\refindependent-exist, \\refruz-indep, it suffices to prove this ine…","labels":[],"detail_key":"p56"},{"id":"n44265","layer":"informal","project":"p56","title":"Conditioned Ruzsa distance","kind":"definition","summary":"[Conditioned Ruzsa distance] If (X, Z) and (Y, W) are random variables (where X and Y are G-val…","labels":["cond-dist-def"],"detail_key":"p56"},{"id":"n44266","layer":"informal","project":"p56","title":"Alternate form of distance","kind":"lemma","summary":"[Alternate form of distance] The expression d[X | Z;Y | W] is unchanged if (X,Z) or (Y,W) is re…","labels":["cond-dist-alt"],"detail_key":"p56"},{"id":"n44267","layer":"informal","project":"p56","title":"Straightforward thanks to \\Crefcopy-ent, \\Crefruz-copy, \\Crefruz-indep, \\Crefcond-dist-de…","kind":"proof","summary":"Straightforward thanks to \\Crefcopy-ent, \\Crefruz-copy, \\Crefruz-indep, \\Crefcond-dist-def, \\Cr…","labels":[],"detail_key":"p56"},{"id":"n44268","layer":"informal","project":"p56","title":"Kaimanovich-Vershik-Madiman inequality","kind":"lemma","summary":"[Kaimanovich-Vershik-Madiman inequality] Suppose that X, Y, Z are independent G-valued random v…","labels":["kv"],"detail_key":"p56"},{"id":"n44269","layer":"informal","project":"p56","title":"From \\Crefsubmodularity we have H[X, X + Y+ Z] + H[Z, X + Y+ Z] \\geq H[X, Z, X + Y+ Z] +…","kind":"proof","summary":"From \\Crefsubmodularity we have H[X, X + Y+ Z] + H[Z, X + Y+ Z] \\geq H[X, Z, X + Y+ Z] + H[X +…","labels":[],"detail_key":"p56"},{"id":"n44270","layer":"informal","project":"p56","title":"Existence of conditional independent trials","kind":"lemma","summary":"[Existence of conditional independent trials] For X,Y random variables, there exist random vari…","labels":["cond-indep-exist"],"detail_key":"p56"},{"id":"n44271","layer":"informal","project":"p56","title":"Explicit construction.","kind":"proof","summary":"Explicit construction.","labels":[],"detail_key":"p56"},{"id":"n44272","layer":"informal","project":"p56","title":"Balog-Szemer\\'edi-Gowers","kind":"lemma","summary":"[Balog-Szemer\\'edi-Gowers] Let A,B be G-valued random variables on \\Omega, and set Z := A+B. Th…","labels":["entropic-bsg","2-bsg-takeaway"],"detail_key":"p56"},{"id":"n44273","layer":"informal","project":"p56","title":"lhs-to-bound","kind":"proof","summary":"Let (A_1, B_1) and (A_2, B_2) (and Z', which by abuse of notation we call Z) be conditionally i…","labels":["lhs-to-bound","bsg-31","bsg-24","bsg-23","bsg-25"],"detail_key":"p56"},{"id":"n44274","layer":"informal","project":"p56","title":"Upper bound on conditioned Ruzsa distance","kind":"lemma","summary":"[Upper bound on conditioned Ruzsa distance] Suppose that (X, Z) and (Y, W) are random variables…","labels":["cond-dist-fact"],"detail_key":"p56"},{"id":"n44275","layer":"informal","project":"p56","title":"Using \\Crefcond-dist-alt and \\Crefindependent-exist, if (X',Z'), (Y',W') are independent…","kind":"proof","summary":"Using \\Crefcond-dist-alt and \\Crefindependent-exist, if (X',Z'), (Y',W') are independent copies…","labels":[],"detail_key":"p56"},{"id":"n44276","layer":"informal","project":"p56","title":"Comparison of Ruzsa distances, I","kind":"lemma","summary":"[Comparison of Ruzsa distances, I] Let X, Y, Z be random variables taking values in some abelia…","labels":["first-useful","lem51-a","ruzsa-3"],"detail_key":"p56"},{"id":"n44277","layer":"informal","project":"p56","title":"We first prove~\\eqreflem51-a. We may assume (taking an independent copy, using \\Crefindep…","kind":"proof","summary":"We first prove~\\eqreflem51-a. We may assume (taking an independent copy, using \\Crefindependent…","labels":[],"detail_key":"p56"},{"id":"n44278","layer":"informal","project":"p56","title":"Comparison of Ruzsa distances, II","kind":"lemma","summary":"[Comparison of Ruzsa distances, II] Let X, Y, Z, Z' be random variables taking values in some a…","labels":["second-useful","7111"],"detail_key":"p56"},{"id":"n44279","layer":"informal","project":"p56","title":"By \\Creffirst-useful (with a change of variables) we have \\[d[X ; Y + Z | Y + Z + Z'] - d…","kind":"proof","summary":"By \\Creffirst-useful (with a change of variables) we have \\[d[X ; Y + Z | Y + Z + Z'] - d[X ; Y…","labels":[],"detail_key":"p56"},{"id":"n44280","layer":"informal","project":"p56","title":"Symmetry group","kind":"definition","summary":"[Symmetry group] If X is a G-valued random variable, then the symmetry group Sym[X] is the set…","labels":["sym-group-def"],"detail_key":"p56"},{"id":"n44281","layer":"informal","project":"p56","title":"Symmetry group is a group","kind":"lemma","summary":"[Symmetry group is a group] If X is a G-valued random variable, then Sym[X] is a subgroup of G.","labels":["sym-group"],"detail_key":"p56"},{"id":"n44282","layer":"informal","project":"p56","title":"Direct verification of the group axioms.","kind":"proof","summary":"Direct verification of the group axioms.","labels":[],"detail_key":"p56"},{"id":"n44283","layer":"informal","project":"p56","title":"Zero Ruzsa distance implies large symmetry group","kind":"lemma","summary":"[Zero Ruzsa distance implies large symmetry group] If X is a G-valued random variable such that…","labels":["zero-large"],"detail_key":"p56"},{"id":"n44284","layer":"informal","project":"p56","title":"Let X_1,X_2 be independent copies of X (from \\Crefindependent-exist). Let A denote the ra…","kind":"proof","summary":"Let X_1,X_2 be independent copies of X (from \\Crefindependent-exist). Let A denote the range of…","labels":[],"detail_key":"p56"},{"id":"n44285","layer":"informal","project":"p56","title":"Translate is uniform on symmetry group","kind":"lemma","summary":"[Translate is uniform on symmetry group] If X is a G-valued random variable with d[X ;X]=0, and…","labels":["sym-zero"],"detail_key":"p56"},{"id":"n44286","layer":"informal","project":"p56","title":"The law of X-x_0 is invariant under Sym[X], non-zero at the origin, and supported on Sym[…","kind":"proof","summary":"The law of X-x_0 is invariant under Sym[X], non-zero at the origin, and supported on Sym[X], gi…","labels":[],"detail_key":"p56"},{"id":"n44287","layer":"informal","project":"p56","title":"Symmetric 100\\% inverse theorem","kind":"lemma","summary":"[Symmetric 100\\% inverse theorem] Suppose that X is a G-valued random variable such that d[X ;X…","labels":["lem:100pc-self"],"detail_key":"p56"},{"id":"n44288","layer":"informal","project":"p56","title":"Take H to be the symmetry group of X, which is a group by \\Crefsym-group. From \\Crefsym-z…","kind":"proof","summary":"Take H to be the symmetry group of X, which is a group by \\Crefsym-group. From \\Crefsym-zero, X…","labels":[],"detail_key":"p56"},{"id":"n44289","layer":"informal","project":"p56","title":"General 100\\% inverse theorem","kind":"corollary","summary":"[General 100\\% inverse theorem] Suppose that X_1,X_2 are G-valued random variables such that d[…","labels":["lem:100pc"],"detail_key":"p56"},{"id":"n44290","layer":"informal","project":"p56","title":"Using \\Crefruzsa-triangle and \\Crefruzsa-nonneg we have d[X_1;X_1]=0, hence by \\Creflem:1…","kind":"proof","summary":"Using \\Crefruzsa-triangle and \\Crefruzsa-nonneg we have d[X_1;X_1]=0, hence by \\Creflem:100pc-s…","labels":[],"detail_key":"p56"},{"id":"n44291","layer":"informal","project":"p56","title":"General fibring identity","kind":"proposition","summary":"[General fibring identity] Let \\pi : H \\to H' be a homomorphism additive groups, and let Z_1,Z_…","labels":["fibring-ident"],"detail_key":"p56"},{"id":"n44292","layer":"informal","project":"p56","title":"Let Z_1,Z_2 be independent throughout (this is possible by \\Crefruz-copy and \\Crefindepen…","kind":"proof","summary":"Let Z_1,Z_2 be independent throughout (this is possible by \\Crefruz-copy and \\Crefindependent-e…","labels":[],"detail_key":"p56"},{"id":"n44293","layer":"informal","project":"p56","title":"fibring-ineq","kind":"corollary","summary":"If \\pi:G\\to H is a homomorphism of additive groups and X,Y are G-valued random variables then \\…","labels":["fibring-ineq"],"detail_key":"p56"},{"id":"n44294","layer":"informal","project":"p56","title":"By \\Creffibring-ident and the nonnegativity of conditional Ruzsa distance (from \\Crefruzs…","kind":"proof","summary":"By \\Creffibring-ident and the nonnegativity of conditional Ruzsa distance (from \\Crefruzsa-nonn…","labels":[],"detail_key":"p56"},{"id":"n44295","layer":"informal","project":"p56","title":"Specific fibring identity","kind":"corollary","summary":"[Specific fibring identity] Let Y_1,Y_2,Y_3 and Y_4 be independent G-valued random variables. T…","labels":["cor-fibre"],"detail_key":"p56"},{"id":"n44296","layer":"informal","project":"p56","title":"We apply \\Creffibring-ident with H := G \\times G, H' := G, \\pi the addition homomorphism…","kind":"proof","summary":"We apply \\Creffibring-ident with H := G \\times G, H' := G, \\pi the addition homomorphism \\pi(x,…","labels":[],"detail_key":"p56"},{"id":"n44297","layer":"informal","project":"p56","title":"eta-def","kind":"definition","summary":"\\eta := 1/9.","labels":["eta-def"],"detail_key":"p56"},{"id":"n44298","layer":"informal","project":"p56","title":"\\tau functional","kind":"definition","summary":"[\\tau functional] If X_1,X_2 are two G-valued random variables, then \\tau[X_1; X_2] := d[X_1; X…","labels":["tau-def"],"detail_key":"p56"},{"id":"n44299","layer":"informal","project":"p56","title":"\\tau depends only on distribution","kind":"lemma","summary":"[\\tau depends only on distribution] If X'_1, X'_2 are copies of X_1,X_2, then \\tau[X'_1;X'_2] =…","labels":["tau-copy"],"detail_key":"p56"},{"id":"n44300","layer":"informal","project":"p56","title":"Immediate from \\Crefcopy-ent.","kind":"proof","summary":"Immediate from \\Crefcopy-ent.","labels":[],"detail_key":"p56"},{"id":"n44301","layer":"informal","project":"p56","title":"\\tau-minimizer","kind":"definition","summary":"[\\tau-minimizer] A pair of G-valued random variables X_1, X_2 are said to be a \\tau-minimizer i…","labels":["tau-min-def"],"detail_key":"p56"},{"id":"n44302","layer":"informal","project":"p56","title":"\\tau has minimum","kind":"proposition","summary":"[\\tau has minimum] A pair X_1, X_2 of \\tau-minimizers exist.","labels":["tau-min"],"detail_key":"p56"},{"id":"n44303","layer":"informal","project":"p56","title":"By \\Creftau-copy, \\tau only depends on the probability distributions of X_1, X_2. This ra…","kind":"proof","summary":"By \\Creftau-copy, \\tau only depends on the probability distributions of X_1, X_2. This ranges o…","labels":[],"detail_key":"p56"},{"id":"n44304","layer":"informal","project":"p56","title":"Distance lower bound","kind":"lemma","summary":"[Distance lower bound] For any G-valued random variables X'_1,X'_2, one has d[X'_1;X'_2] \\geq k…","labels":["distance-lower"],"detail_key":"p56"},{"id":"n44305","layer":"informal","project":"p56","title":"Immediate from \\Creftau-def and \\Creftau-min.","kind":"proof","summary":"Immediate from \\Creftau-def and \\Creftau-min.","labels":[],"detail_key":"p56"},{"id":"n44306","layer":"informal","project":"p56","title":"Conditional distance lower bound","kind":"lemma","summary":"[Conditional distance lower bound] For any G-valued random variables X'_1,X'_2 and random varia…","labels":["cond-distance-lower"],"detail_key":"p56"},{"id":"n44307","layer":"informal","project":"p56","title":"Apply \\Crefdistance-lower to conditioned random variables and then average.","kind":"proof","summary":"Apply \\Crefdistance-lower to conditioned random variables and then average.","labels":[],"detail_key":"p56"},{"id":"n44308","layer":"informal","project":"p56","title":"Fibring identity for first estimate","kind":"lemma","summary":"[Fibring identity for first estimate] We have & d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\t…","labels":["first-fibre"],"detail_key":"p56"},{"id":"n44309","layer":"informal","project":"p56","title":"Immediate from \\Crefcor-fibre.","kind":"proof","summary":"Immediate from \\Crefcor-fibre.","labels":[],"detail_key":"p56"},{"id":"n44310","layer":"informal","project":"p56","title":"Lower bound on distances","kind":"lemma","summary":"[Lower bound on distances] We have d[X_1+\\tilde X_2; X_2+\\tilde X_1] \\geq k &- \\eta (d[X^0_1; X…","labels":["first-dist-sum"],"detail_key":"p56"},{"id":"n44311","layer":"informal","project":"p56","title":"Immediate from \\Crefdistance-lower.","kind":"proof","summary":"Immediate from \\Crefdistance-lower.","labels":[],"detail_key":"p56"},{"id":"n44312","layer":"informal","project":"p56","title":"Lower bound on conditional distances","kind":"lemma","summary":"[Lower bound on conditional distances] We have & d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] \\\\ &…","labels":["first-cond"],"detail_key":"p56"},{"id":"n44313","layer":"informal","project":"p56","title":"Immediate from \\Crefcond-distance-lower.","kind":"proof","summary":"Immediate from \\Crefcond-distance-lower.","labels":[],"detail_key":"p56"},{"id":"n44314","layer":"informal","project":"p56","title":"Upper bound on distance differences","kind":"lemma","summary":"[Upper bound on distance differences] We have d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] &\\leq \\t…","labels":["first-upper"],"detail_key":"p56"},{"id":"n44315","layer":"informal","project":"p56","title":"Immediate from \\Creffirst-useful (and recalling that k is defined to be d[X_1;X_2]).","kind":"proof","summary":"Immediate from \\Creffirst-useful (and recalling that k is defined to be d[X_1;X_2]).","labels":[],"detail_key":"p56"},{"id":"n44316","layer":"informal","project":"p56","title":"First estimate","kind":"lemma","summary":"[First estimate] We have I_1 \\leq 2 \\eta k.","labels":["first-estimate"],"detail_key":"p56"},{"id":"n44317","layer":"informal","project":"p56","title":"Take a suitable linear combination of \\Creffirst-fibre, \\Creffirst-dist-sum, \\Creffirst-c…","kind":"proof","summary":"Take a suitable linear combination of \\Creffirst-fibre, \\Creffirst-dist-sum, \\Creffirst-cond, a…","labels":[],"detail_key":"p56"},{"id":"n44318","layer":"informal","project":"p56","title":"Entropy bound on quadruple sum","kind":"lemma","summary":"[Entropy bound on quadruple sum] With the same notation, we have H[X_1+X_2+\\tilde X_1+\\tilde X_…","labels":["foursum-bound","HS-bound"],"detail_key":"p56"},{"id":"n44319","layer":"informal","project":"p56","title":"Subtracting \\Creffirst-cond from \\Creffirst-fibre, and combining the resulting inequality…","kind":"proof","summary":"Subtracting \\Creffirst-cond from \\Creffirst-fibre, and combining the resulting inequality with…","labels":[],"detail_key":"p56"},{"id":"n44320","layer":"informal","project":"p56","title":"Distance between sums","kind":"lemma","summary":"[Distance between sums] We have d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k - \\frac\\eta2 ( d[X_1;…","labels":["dist-sums"],"detail_key":"p56"},{"id":"n44321","layer":"informal","project":"p56","title":"From \\Crefdistance-lower one has d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k &- \\eta(d[X^0_1…","kind":"proof","summary":"From \\Crefdistance-lower one has d[X_1+\\tilde X_1; X_2+\\tilde X_2] \\geq k &- \\eta(d[X^0_1;X_1]…","labels":[],"detail_key":"p56"},{"id":"n44322","layer":"informal","project":"p56","title":"second-estimate-aux","kind":"lemma","summary":"We have \\[d[X_1;X_1] + d[X_2;X_2] \\leq 2 k + \\frac2(2 \\eta k - I_1)1-\\eta. \\]","labels":["second-estimate-aux"],"detail_key":"p56"},{"id":"n44323","layer":"informal","project":"p56","title":"We may use \\Crefruz-indep to expand & d[X_1+\\tilde X_1;X_2+\\tilde X_2] \\\\ &= H[X_1+\\tilde…","kind":"proof","summary":"We may use \\Crefruz-indep to expand & d[X_1+\\tilde X_1;X_2+\\tilde X_2] \\\\ &= H[X_1+\\tilde X_1 +…","labels":[],"detail_key":"p56"},{"id":"n44324","layer":"informal","project":"p56","title":"Second estimate","kind":"lemma","summary":"[Second estimate] We have I_2 \\leq 2 \\eta k + \\frac2 \\eta (2 \\eta k - I_1)1 - \\eta.","labels":["second-estimate"],"detail_key":"p56"},{"id":"n44325","layer":"informal","project":"p56","title":"combined","kind":"proof","summary":"We apply \\Crefcor-fibre, but now with the choice \\[ (Y_1,Y_2,Y_3,Y_4) := (X_2, X_1, \\tilde X_2,…","labels":["combined"],"detail_key":"p56"},{"id":"n44326","layer":"informal","project":"p56","title":"Symmetry identity","kind":"lemma","summary":"[Symmetry identity] We have I(U:W | S) = I(V:W | S).","labels":["symm-lemma"],"detail_key":"p56"},{"id":"n44327","layer":"informal","project":"p56","title":"This should follow from \\Crefcopy-ent, \\Crefconditional-mutual-alt, and \\Crefchain-rule.","kind":"proof","summary":"This should follow from \\Crefcopy-ent, \\Crefconditional-mutual-alt, and \\Crefchain-rule.","labels":[],"detail_key":"p56"},{"id":"n44328","layer":"informal","project":"p56","title":"Bound on conditional mutual informations","kind":"lemma","summary":"[Bound on conditional mutual informations] We have I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W…","labels":["uvw-s"],"detail_key":"p56"},{"id":"n44329","layer":"informal","project":"p56","title":"From the definitions of I_1,I_2 and \\Crefsymm-lemma, we see that \\[ I_1 = I(U : V \\, | \\,…","kind":"proof","summary":"From the definitions of I_1,I_2 and \\Crefsymm-lemma, we see that \\[ I_1 = I(U : V \\, | \\, S), \\…","labels":[],"detail_key":"p56"},{"id":"n44330","layer":"informal","project":"p56","title":"Bound on distance increments","kind":"lemma","summary":"[Bound on distance increments] We have \\sum_i=1^2 \\sum_A\\in\\U,V,W\\ \\big(d[X^0_i;A|S] & - d[X^0_…","labels":["total-dist"],"detail_key":"p56"},{"id":"n44331","layer":"informal","project":"p56","title":"By \\Crefsecond-useful (taking X = X_1^0, Y = X_1, Z = X_2 and Z' = \\tilde X_1 + \\tilde X_…","kind":"proof","summary":"By \\Crefsecond-useful (taking X = X_1^0, Y = X_1, Z = X_2 and Z' = \\tilde X_1 + \\tilde X_2, so…","labels":[],"detail_key":"p56"},{"id":"n44332","layer":"informal","project":"p56","title":"Key identity","kind":"lemma","summary":"[Key identity] We have U+V+W=0.","labels":["key-ident"],"detail_key":"p56"},{"id":"n44333","layer":"informal","project":"p56","title":"Obvious because we are in characteristic two.","kind":"proof","summary":"Obvious because we are in characteristic two.","labels":[],"detail_key":"p56"},{"id":"n44334","layer":"informal","project":"p56","title":"Constructing good variables, I","kind":"lemma","summary":"[Constructing good variables, I] One has k \\leq \\delta + \\eta (& d[X^0_1;T_1]-d[X^0_1;X_1]) + \\…","labels":["construct-good-prelim"],"detail_key":"p56"},{"id":"n44335","layer":"informal","project":"p56","title":"bsg-t1t2","kind":"proof","summary":"We apply \\Crefentropic-bsg with (A,B) = (T_1, T_2) there. Since T_1 + T_2 = T_3, the conclusion…","labels":["bsg-t1t2"],"detail_key":"p56"},{"id":"n44336","layer":"informal","project":"p56","title":"Constructing good variables, II","kind":"lemma","summary":"[Constructing good variables, II] One has k & \\leq \\delta + \\frac\\eta3 \\biggl( \\delta + \\sum_i=…","labels":["construct-good"],"detail_key":"p56"},{"id":"n44337","layer":"informal","project":"p56","title":"Average \\Crefconstruct-good-prelim over all six permutations of T_1,T_2,T_3.","kind":"proof","summary":"Average \\Crefconstruct-good-prelim over all six permutations of T_1,T_2,T_3.","labels":[],"detail_key":"p56"},{"id":"n44338","layer":"informal","project":"p56","title":"\\tau-decrement","kind":"theorem","summary":"[\\tau-decrement] Let X_1, X_2 be tau-minimizers. Then d[X_1;X_2] = 0.","labels":["de-prop"],"detail_key":"p56"},{"id":"n44339","layer":"informal","project":"p56","title":"Set k := d[X_1;X_2]. Applying \\Crefconstruct-good with any random variables (T_1,T_2,T_3)…","kind":"proof","summary":"Set k := d[X_1;X_2]. Applying \\Crefconstruct-good with any random variables (T_1,T_2,T_3) such…","labels":[],"detail_key":"p56"},{"id":"n44340","layer":"informal","project":"p56","title":"Entropy version of PFR","kind":"theorem","summary":"[Entropy version of PFR] Let G = F_2^n, and suppose that X^0_1, X^0_2 are G-valued random varia…","labels":["entropy-pfr"],"detail_key":"p56"},{"id":"n44341","layer":"informal","project":"p56","title":"Let X_1, X_2 be the \\tau-minimizer from \\Creftau-min. From \\Crefde-prop, d[X_1;X_2]=0. Fr…","kind":"proof","summary":"Let X_1, X_2 be the \\tau-minimizer from \\Creftau-min. From \\Crefde-prop, d[X_1;X_2]=0. From \\Cr…","labels":[],"detail_key":"p56"},{"id":"n44342","layer":"informal","project":"p56","title":"Ruzsa covering lemma","kind":"lemma","summary":"[Ruzsa covering lemma] If A,B are finite non-empty subsets of a group G, then A can be covered…","labels":["ruz-cov"],"detail_key":"p56"},{"id":"n44343","layer":"informal","project":"p56","title":"Cover A greedily by disjoint translates of B.","kind":"proof","summary":"Cover A greedily by disjoint translates of B.","labels":[],"detail_key":"p56"},{"id":"n44344","layer":"informal","project":"p56","title":"pfr_aux","kind":"lemma","summary":"If A \\subset \\bf F_2^n is non-empty and |A+A| \\leq K|A|, then A can be covered by at most K ^ 1…","labels":["pfr_aux","ah"],"detail_key":"p56"},{"id":"n44345","layer":"informal","project":"p56","title":"uauh","kind":"proof","summary":"Let U_A be the uniform distribution on A (which exists by \\Crefunif-exist), thus H[U_A] = \\log…","labels":["uauh"],"detail_key":"p56"},{"id":"n44346","layer":"informal","project":"p56","title":"PFR","kind":"theorem","summary":"[PFR] If A \\subset \\bf F_2^n is non-empty and |A+A| \\leq K|A|, then A can be covered by most 2K…","labels":["pfr"],"detail_key":"p56"},{"id":"n44347","layer":"informal","project":"p56","title":"Let H be given by \\Crefpfr_aux. If |H| \\leq |A| then we are already done thanks to~\\eqref…","kind":"proof","summary":"Let H be given by \\Crefpfr_aux. If |H| \\leq |A| then we are already done thanks to~\\eqrefah. If…","labels":[],"detail_key":"p56"},{"id":"n44348","layer":"informal","project":"p56","title":"PFR in infinite groups","kind":"corollary","summary":"[PFR in infinite groups] If G is an abelian 2-torsion group, A \\subset G is non-empty finite, a…","labels":["pfr-cor"],"detail_key":"p56"},{"id":"n44349","layer":"informal","project":"p56","title":"Apply \\Crefpfr to the group generated by A, which is isomorphic to F_2^n for some n.","kind":"proof","summary":"Apply \\Crefpfr to the group generated by A, which is isomorphic to F_2^n for some n.","labels":[],"detail_key":"p56"},{"id":"n44350","layer":"informal","project":"p56","title":"New definition of \\eta","kind":"definition","summary":"[New definition of \\eta] \\eta is a real parameter with \\eta > 0.","labels":["eta-def-new"],"detail_key":"p56"},{"id":"n44351","layer":"informal","project":"p56","title":"Constructing good variables, I'","kind":"lemma","summary":"[Constructing good variables, I'] One has k \\leq \\delta + \\eta (& d[X^0_1;T_1|T_3]-d[X^0_1;X_1]…","labels":["construct-good-prelim-improv"],"detail_key":"p56"},{"id":"n44352","layer":"informal","project":"p56","title":"bsg-t1t2'","kind":"proof","summary":"We apply \\Crefentropic-bsg with (A,B) = (T_1, T_2) there. Since T_1 + T_2 = T_3, the conclusion…","labels":["bsg-t1t2'"],"detail_key":"p56"},{"id":"n44353","layer":"informal","project":"p56","title":"Constructing good variables, II'","kind":"lemma","summary":"[Constructing good variables, II'] One has k & \\leq \\delta + \\frac\\eta6 \\sum_i=1^2 \\sum_1 \\leq…","labels":["construct-good-improv"],"detail_key":"p56"},{"id":"n44354","layer":"informal","project":"p56","title":"Average \\Crefconstruct-good-prelim-improv over all six permutations of T_1,T_2,T_3.","kind":"proof","summary":"Average \\Crefconstruct-good-prelim-improv over all six permutations of T_1,T_2,T_3.","labels":[],"detail_key":"p56"},{"id":"n44355","layer":"informal","project":"p56","title":"Constructing good variables, III'","kind":"lemma","summary":"[Constructing good variables, III'] One has k & \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I…","labels":["averaged-construct-good"],"detail_key":"p56"},{"id":"n44356","layer":"informal","project":"p56","title":"For each s in the range of S, apply \\Crefconstruct-good-improv with T_1,T_2,T_3 equal to…","kind":"proof","summary":"For each s in the range of S, apply \\Crefconstruct-good-improv with T_1,T_2,T_3 equal to (U|S=s…","labels":[],"detail_key":"p56"},{"id":"n44357","layer":"informal","project":"p56","title":"General inequality","kind":"lemma","summary":"[General inequality] Let X_1, X_2, X_3, X_4 be independent G-valued random variables, and let Y…","labels":["gen-ineq"],"detail_key":"p56"},{"id":"n44358","layer":"informal","project":"p56","title":"On the one hand, by \\Crefcond-dist-fact and two applications of \\Creffirst-useful we have…","kind":"proof","summary":"On the one hand, by \\Crefcond-dist-fact and two applications of \\Creffirst-useful we have &d[Y;…","labels":[],"detail_key":"p56"},{"id":"n44359","layer":"informal","project":"p56","title":"Bound on distance differences","kind":"lemma","summary":"[Bound on distance differences] We have &\\sum_i=1^2 \\sum_A,B \\in \\U,V,W\\: A \\neq B d[X_i^0;A|B,…","labels":["dist-diff-bound"],"detail_key":"p56"},{"id":"n44360","layer":"informal","project":"p56","title":"If we apply \\Crefgen-ineq with X_1:=X_1, Y:=X_1^0 and (X_2,X_3,X_4) equal to the 3! permu…","kind":"proof","summary":"If we apply \\Crefgen-ineq with X_1:=X_1, Y:=X_1^0 and (X_2,X_3,X_4) equal to the 3! permutation…","labels":[],"detail_key":"p56"},{"id":"n44361","layer":"informal","project":"p56","title":"Improved \\tau-decrement","kind":"theorem","summary":"[Improved \\tau-decrement] Suppose 0 < \\eta < 1/8. Let X_1, X_2 be tau-minimizers. Then d[X_1;X_…","labels":["de-prop-improv"],"detail_key":"p56"},{"id":"n44362","layer":"informal","project":"p56","title":"From \\Crefaveraged-construct-good, \\Crefdist-diff-bound, and \\Crefuvw-s one has \\[ k \\leq…","kind":"proof","summary":"From \\Crefaveraged-construct-good, \\Crefdist-diff-bound, and \\Crefuvw-s one has \\[ k \\leq 8\\eta…","labels":[],"detail_key":"p56"},{"id":"n44363","layer":"informal","project":"p56","title":"Limiting improved \\tau-decrement","kind":"theorem","summary":"[Limiting improved \\tau-decrement] For \\eta = 1/8, there exist tau-minimizers X_1, X_2 satisfyi…","labels":["de-prop-lim-improv"],"detail_key":"p56"},{"id":"n44364","layer":"informal","project":"p56","title":"For each \\eta<1/8, consider minimizers X_1^\\eta and X_2^\\eta from \\Creftau-min. By \\Crefd…","kind":"proof","summary":"For each \\eta<1/8, consider minimizers X_1^\\eta and X_2^\\eta from \\Creftau-min. By \\Crefde-prop…","labels":[],"detail_key":"p56"},{"id":"n44365","layer":"informal","project":"p56","title":"Improved entropy version of PFR","kind":"theorem","summary":"[Improved entropy version of PFR] Let G = F_2^n, and suppose that X^0_1, X^0_2 are G-valued ran…","labels":["entropy-pfr-improv"],"detail_key":"p56"},{"id":"n44366","layer":"informal","project":"p56","title":"Let X_1, X_2 be the good \\tau-minimizer from \\Crefde-prop-lim-improv. By construction, d[…","kind":"proof","summary":"Let X_1, X_2 be the good \\tau-minimizer from \\Crefde-prop-lim-improv. By construction, d[X_1;X_…","labels":[],"detail_key":"p56"},{"id":"n44367","layer":"informal","project":"p56","title":"pfr_aux-improv","kind":"lemma","summary":"If A \\subset \\bf F_2^n is non-empty and |A+A| \\leq K|A|, then A can be covered by at most K^6 |…","labels":["pfr_aux-improv"],"detail_key":"p56"},{"id":"n44368","layer":"informal","project":"p56","title":"By repeating the proof of \\Crefpfr_aux and using \\Crefentropy-pfr-improv one can obtain t…","kind":"proof","summary":"By repeating the proof of \\Crefpfr_aux and using \\Crefentropy-pfr-improv one can obtain the cla…","labels":[],"detail_key":"p56"},{"id":"n44369","layer":"informal","project":"p56","title":"Improved PFR","kind":"theorem","summary":"[Improved PFR] If A \\subset \\bf F_2^n is non-empty and |A+A| \\leq K|A|, then A can be covered b…","labels":["pfr-improv"],"detail_key":"p56"},{"id":"n44370","layer":"informal","project":"p56","title":"By repeating the proof of \\Crefpfr and using \\Crefpfr_aux-improv one can obtain the claim…","kind":"proof","summary":"By repeating the proof of \\Crefpfr and using \\Crefpfr_aux-improv one can obtain the claim with…","labels":[],"detail_key":"p56"},{"id":"n44371","layer":"informal","project":"p56","title":"Hahn-Banach type theorem","kind":"lemma","summary":"[Hahn-Banach type theorem] Let H_0 be a subgroup of G. Then every homomorphism \\phi: H_0 \\to G'…","labels":["hb-thm"],"detail_key":"p56"},{"id":"n44372","layer":"informal","project":"p56","title":"By induction it suffices to treat the case where H_0 has index 2 in G, but then the exten…","kind":"proof","summary":"By induction it suffices to treat the case where H_0 has index 2 in G, but then the extension c…","labels":[],"detail_key":"p56"},{"id":"n44373","layer":"informal","project":"p56","title":"Goursat type theorem","kind":"lemma","summary":"[Goursat type theorem] Let H be a subgroup of G \\times G'. Then there exists a subgroup H_0 of…","labels":["goursat"],"detail_key":"p56"},{"id":"n44374","layer":"informal","project":"p56","title":"We can take H_0 to be the projection of H to G, and H_1 to be the slice H_1 := \\ y: (0,y)…","kind":"proof","summary":"We can take H_0 to be the projection of H to G, and H_1 to be the slice H_1 := \\ y: (0,y) \\in H…","labels":[],"detail_key":"p56"},{"id":"n44375","layer":"informal","project":"p56","title":"Homomorphism form of PFR","kind":"theorem","summary":"[Homomorphism form of PFR] Let f: G \\to G' be a function, and let S denote the set S := \\ f(x+y…","labels":["hom-pfr"],"detail_key":"p56"},{"id":"n44376","layer":"informal","project":"p56","title":"Consider the graph A \\subset G \\times G' defined by A := \\ (x,f(x)): x \\in G \\. Clearly,…","kind":"proof","summary":"Consider the graph A \\subset G \\times G' defined by A := \\ (x,f(x)): x \\in G \\. Clearly, |A| =…","labels":[],"detail_key":"p56"},{"id":"n44377","layer":"informal","project":"p56","title":"Additive energy","kind":"definition","summary":"[Additive energy] If G is a finite additive group and A is a subset of G, the \\emphadditive ene…","labels":["energy-def"],"detail_key":"p56"},{"id":"n44378","layer":"informal","project":"p56","title":"Cauchy--Schwarz bound","kind":"lemma","summary":"[Cauchy--Schwarz bound] If G is a finite additive group and A,B are subsets of G, then \\left( \\…","labels":["cs-bound"],"detail_key":"p56"},{"id":"n44379","layer":"informal","project":"p56","title":"If B is empty then the claim is trivial (with the Lean convention 0/0), so without loss o…","kind":"proof","summary":"If B is empty then the claim is trivial (with the Lean convention 0/0), so without loss of gene…","labels":[],"detail_key":"p56"},{"id":"n44380","layer":"informal","project":"p56","title":"Balog--Szemer\\'edi--Gowers lemma","kind":"lemma","summary":"[Balog--Szemer\\'edi--Gowers lemma] Let G be a finite abelian group, and let A be a non-empty su…","labels":["bsg"],"detail_key":"p56"},{"id":"n44381","layer":"informal","project":"p56","title":"See \\urlhttps://terrytao.files.wordpress.com/2024/01/simplebsg.pdf.","kind":"proof","summary":"See \\urlhttps://terrytao.files.wordpress.com/2024/01/simplebsg.pdf.","labels":[],"detail_key":"p56"},{"id":"n44382","layer":"informal","project":"p56","title":"Approximate homomorphism form of PFR","kind":"theorem","summary":"[Approximate homomorphism form of PFR] Let G,G' be finite abelian 2-groups. Let f: G \\to G' be…","labels":["approx-hom-pfr"],"detail_key":"p56"},{"id":"n44383","layer":"informal","project":"p56","title":"Consider the graph A \\subset G \\times G' defined by A := \\ (x,f(x)): x \\in G \\. Clearly,…","kind":"proof","summary":"Consider the graph A \\subset G \\times G' defined by A := \\ (x,f(x)): x \\in G \\. Clearly, |A| =…","labels":[],"detail_key":"p56"},{"id":"n44384","layer":"informal","project":"p56","title":"Duality","kind":"lemma","summary":"[Duality] Let G be a finite abelian 2-group. Then the finite abelian 2-group Hom(G,Z/2Z) of hom…","labels":["gdual"],"detail_key":"p56"},{"id":"n44385","layer":"informal","project":"p56","title":"By the classification of finite abelian groups, G is isomorphic to (Z/2Z)^n. Then Hom(G,Z…","kind":"proof","summary":"By the classification of finite abelian groups, G is isomorphic to (Z/2Z)^n. Then Hom(G,Z/2Z) i…","labels":[],"detail_key":"p56"},{"id":"n44386","layer":"informal","project":"p56","title":"Counting","kind":"lemma","summary":"[Counting] Let G be a finite abelian 2-group, and let x \\in G be non-zero. Then there are |G|/2…","labels":["gcount"],"detail_key":"p56"},{"id":"n44387","layer":"informal","project":"p56","title":"The map \\phi \\mapsto \\phi(x) is a homomorphism from Hom(G,Z/2Z) to Z/2Z, and by Lemma \\re…","kind":"proof","summary":"The map \\phi \\mapsto \\phi(x) is a homomorphism from Hom(G,Z/2Z) to Z/2Z, and by Lemma \\refgdual…","labels":[],"detail_key":"p56"},{"id":"n44388","layer":"informal","project":"p56","title":"Slicing","kind":"lemma","summary":"[Slicing] Let G be a finite abelian 2-group, and let A be a subset of G. Then there exists a ho…","labels":["gslice"],"detail_key":"p56"},{"id":"n44389","layer":"informal","project":"p56","title":"We have \\sum_\\phi \\in Hom(G,Z/2Z) |A \\cap \\phi^-1(1)| = \\sum_x \\in A |\\ \\phi \\in Hom(G,Z/…","kind":"proof","summary":"We have \\sum_\\phi \\in Hom(G,Z/2Z) |A \\cap \\phi^-1(1)| = \\sum_x \\in A |\\ \\phi \\in Hom(G,Z/2Z): \\…","labels":[],"detail_key":"p56"},{"id":"n44390","layer":"informal","project":"p56","title":"Approximate homomorphism form of PFR, no constant term","kind":"corollary","summary":"[Approximate homomorphism form of PFR, no constant term] Let G,G' be finite abelian 2-groups. L…","labels":["approx-hom-pfr-no-const"],"detail_key":"p56"},{"id":"n44391","layer":"informal","project":"p56","title":"By Theorem \\refapprox-hom-pfr, there exists a homomorphism \\phi: G \\to G' and a constant…","kind":"proof","summary":"By Theorem \\refapprox-hom-pfr, there exists a homomorphism \\phi: G \\to G' and a constant c \\in…","labels":[],"detail_key":"p56"},{"id":"n44392","layer":"informal","project":"p56","title":"torsion-free-doubling","kind":"lemma","summary":"If G is torsion-free and X,Y are G-valued random variables then d[X;2Y]\\leq 5d[X;Y].","labels":["torsion-free-doubling"],"detail_key":"p56"},{"id":"n44393","layer":"informal","project":"p56","title":"Let Y_1,Y_2 be independent copies of Y (also independent of X). Since G is torsion-free w…","kind":"proof","summary":"Let Y_1,Y_2 be independent copies of Y (also independent of X). Since G is torsion-free we know…","labels":[],"detail_key":"p56"},{"id":"n44394","layer":"informal","project":"p56","title":"torsion-dist-shrinking","kind":"lemma","summary":"If G is a torsion-free group and X,Y are G-valued random variables and \\phi:G\\to F_2^d is a hom…","labels":["torsion-dist-shrinking"],"detail_key":"p56"},{"id":"n44395","layer":"informal","project":"p56","title":"By \\Creffibring-ineq and \\Creftorsion-free-doubling we have \\[d[\\phi(X);\\phi(2Y)]\\leq d[X…","kind":"proof","summary":"By \\Creffibring-ineq and \\Creftorsion-free-doubling we have \\[d[\\phi(X);\\phi(2Y)]\\leq d[X;2Y]\\l…","labels":[],"detail_key":"p56"},{"id":"n44396","layer":"informal","project":"p56","title":"app-ent-pfr","kind":"lemma","summary":"Let G=F_2^n and \\alpha\\in (0,1) and let X,Y be G-valued random variables such that \\[H(X)+H(Y)>…","labels":["app-ent-pfr"],"detail_key":"p56"},{"id":"n44397","layer":"informal","project":"p56","title":"By \\Crefentropy-pfr-improv there exists a subgroup H such that d[X;U_H] + d[Y;U_H] \\leq 1…","kind":"proof","summary":"By \\Crefentropy-pfr-improv there exists a subgroup H such that d[X;U_H] + d[Y;U_H] \\leq 10 d[X;…","labels":[],"detail_key":"p56"},{"id":"n44398","layer":"informal","project":"p56","title":"pfr-projection'","kind":"lemma","summary":"If G=F_2^d and \\alpha\\in (0,1) and X,Y are G-valued random variables then there is a subgroup H…","labels":["pfr-projection'"],"detail_key":"p56"},{"id":"n44399","layer":"informal","project":"p56","title":"Let H\\leq F_2^d be a maximal subgroup such that \\[H(\\psi(X))+H(\\psi(Y))> \\frac20\\alpha d[…","kind":"proof","summary":"Let H\\leq F_2^d be a maximal subgroup such that \\[H(\\psi(X))+H(\\psi(Y))> \\frac20\\alpha d[\\psi(X…","labels":[],"detail_key":"p56"},{"id":"n44400","layer":"informal","project":"p56","title":"pfr-projection","kind":"lemma","summary":"If G=F_2^d and \\alpha\\in (0,1) and X,Y are G-valued random variables then there is a subgroup H…","labels":["pfr-projection"],"detail_key":"p56"},{"id":"n44401","layer":"informal","project":"p56","title":"Specialize \\Crefpfr-projection' to \\alpha=3/5. In the second inequality, it gives a bound…","kind":"proof","summary":"Specialize \\Crefpfr-projection' to \\alpha=3/5. In the second inequality, it gives a bound 100/3…","labels":[],"detail_key":"p56"},{"id":"n44402","layer":"informal","project":"p56","title":"single-fibres","kind":"lemma","summary":"Let \\phi:G\\to H be a homomorphism and A,B\\subseteq G be finite subsets. If x,y\\in H then let A_…","labels":["single-fibres"],"detail_key":"p56"},{"id":"n44403","layer":"informal","project":"p56","title":"The random variables (U_A\\mid \\phi(U_A)=x) and (U_B\\mid \\phi(U_B)=y) are equal in distrib…","kind":"proof","summary":"The random variables (U_A\\mid \\phi(U_A)=x) and (U_B\\mid \\phi(U_B)=y) are equal in distribution…","labels":[],"detail_key":"p56"},{"id":"n44404","layer":"informal","project":"p56","title":"dimension-def","kind":"definition","summary":"If A\\subseteq Z^d then by \\dim(A) we mean the dimension of the span of A-A over the reals -- eq…","labels":["dimension-def"],"detail_key":"p56"},{"id":"n44405","layer":"informal","project":"p56","title":"weak-pfr-asymm","kind":"theorem","summary":"If A,B\\subseteq Z^d are finite non-empty sets then there exist non-empty A'\\subseteq A and B'\\s…","labels":["weak-pfr-asymm"],"detail_key":"p56"},{"id":"n44406","layer":"informal","project":"p56","title":"Without loss of generality we can assume that A and B are not both inside (possibly disti…","kind":"proof","summary":"Without loss of generality we can assume that A and B are not both inside (possibly distinct) c…","labels":[],"detail_key":"p56"},{"id":"n44407","layer":"informal","project":"p56","title":"weak-pfr-symm","kind":"theorem","summary":"If A\\subseteq Z^d is a finite non-empty set with d[U_A;U_A]\\leq \\log K then there exists a non-…","labels":["weak-pfr-symm"],"detail_key":"p56"},{"id":"n44408","layer":"informal","project":"p56","title":"Immediate from \\Crefweak-pfr-asymm and rearranging.","kind":"proof","summary":"Immediate from \\Crefweak-pfr-asymm and rearranging.","labels":[],"detail_key":"p56"},{"id":"n44409","layer":"informal","project":"p56","title":"weak-pfr-int","kind":"theorem","summary":"Let A\\subseteq Z^d and \\lvert A-A\\rvert\\leq K\\lvert A\\rvert. There exists A'\\subseteq A such th…","labels":["weak-pfr-int"],"detail_key":"p56"},{"id":"n44410","layer":"informal","project":"p56","title":"As in the beginning of \\Crefpfr the doubling condition forces d[U_A;U_A]\\leq \\log K, and…","kind":"proof","summary":"As in the beginning of \\Crefpfr the doubling condition forces d[U_A;U_A]\\leq \\log K, and then w…","labels":[],"detail_key":"p56"},{"id":"n44411","layer":"informal","project":"p56","title":"Data processing for a single variable","kind":"lemma","summary":"[Data processing for a single variable] Let X be a random variable. Then for any function f on…","labels":["data-process-single"],"detail_key":"p56"},{"id":"n44412","layer":"informal","project":"p56","title":"We have H[X] = H[X,f(X)] = H[f(X)] + H[X|f(X)] thanks to \\Crefrelabeled-entropy and \\Cref…","kind":"proof","summary":"We have H[X] = H[X,f(X)] = H[f(X)] + H[X|f(X)] thanks to \\Crefrelabeled-entropy and \\Crefchain-…","labels":[],"detail_key":"p56"},{"id":"n44413","layer":"informal","project":"p56","title":"One-sided unconditional data processing inequality","kind":"lemma","summary":"[One-sided unconditional data processing inequality] Let X,Y be random variables. For any funct…","labels":["data-process-unc-one"],"detail_key":"p56"},{"id":"n44414","layer":"informal","project":"p56","title":"By \\Crefalternative-mutual it suffices to show that H[Y|X] \\leq H[Y|f(X)]. But this follo…","kind":"proof","summary":"By \\Crefalternative-mutual it suffices to show that H[Y|X] \\leq H[Y|f(X)]. But this follows fro…","labels":[],"detail_key":"p56"},{"id":"n44415","layer":"informal","project":"p56","title":"Unconditional data processing inequality","kind":"lemma","summary":"[Unconditional data processing inequality] Let X,Y be random variables. For any functions f, g…","labels":["data-process-unc"],"detail_key":"p56"},{"id":"n44416","layer":"informal","project":"p56","title":"From \\Crefdata-process-unc-one, \\Crefentropy-comm we have I[f(X) : Y] \\leq I[X:Y] and I[f…","kind":"proof","summary":"From \\Crefdata-process-unc-one, \\Crefentropy-comm we have I[f(X) : Y] \\leq I[X:Y] and I[f(X): g…","labels":[],"detail_key":"p56"},{"id":"n44417","layer":"informal","project":"p56","title":"Data processing inequality","kind":"lemma","summary":"[Data processing inequality] Let X,Y,Z. For any functions f, g on the ranges of X, Y respective…","labels":["data-process"],"detail_key":"p56"},{"id":"n44418","layer":"informal","project":"p56","title":"Apply \\Crefdata-process-unc to X,Y conditioned to the event Z=z, multiply by \\bf P[Z=z],…","kind":"proof","summary":"Apply \\Crefdata-process-unc to X,Y conditioned to the event Z=z, multiply by \\bf P[Z=z], and su…","labels":[],"detail_key":"p56"},{"id":"n44419","layer":"informal","project":"p56","title":"Flipping a sign","kind":"lemma","summary":"[Flipping a sign] If X,Y are G-valued, then d[X ; -Y] \\leq 3 d[X;Y].","labels":["sign-flip"],"detail_key":"p56"},{"id":"n44420","layer":"informal","project":"p56","title":"Without loss of generality (using \\Crefruz-copy and \\Crefindependent-exist) we may take X…","kind":"proof","summary":"Without loss of generality (using \\Crefruz-copy and \\Crefindependent-exist) we may take X,Y to…","labels":[],"detail_key":"p56"},{"id":"n44421","layer":"informal","project":"p56","title":"Kaimonovich--Vershik--Madiman inequality","kind":"lemma","summary":"[Kaimonovich--Vershik--Madiman inequality] If n \\geq 0 and X, Y_1, \\dots, Y_n are jointly indep…","labels":["klm-1"],"detail_key":"p56"},{"id":"n44422","layer":"informal","project":"p56","title":"This is trivial for n=0,1, while the n=2 case is \\Crefkv. Now suppose inductively that n…","kind":"proof","summary":"This is trivial for n=0,1, while the n=2 case is \\Crefkv. Now suppose inductively that n > 2, a…","labels":[],"detail_key":"p56"},{"id":"n44423","layer":"informal","project":"p56","title":"Kaimonovich--Vershik--Madiman inequality, II","kind":"lemma","summary":"[Kaimonovich--Vershik--Madiman inequality, II] If n \\geq 1 and X, Y_1, \\dots, Y_n are jointly i…","labels":["klm-2"],"detail_key":"p56"},{"id":"n44424","layer":"informal","project":"p56","title":"Applying \\Crefklm-1 with all the Y_i replaced by -Y_i, and using \\Crefneg-ent and \\Crefru…","kind":"proof","summary":"Applying \\Crefklm-1 with all the Y_i replaced by -Y_i, and using \\Crefneg-ent and \\Crefruz-inde…","labels":[],"detail_key":"p56"},{"id":"n44425","layer":"informal","project":"p56","title":"Kaimonovich--Vershik--Madiman inequality, III","kind":"lemma","summary":"[Kaimonovich--Vershik--Madiman inequality, III] If n \\geq 1 and X, Y_1, \\dots, Y_n are jointly…","labels":["klm-3"],"detail_key":"p56"},{"id":"n44426","layer":"informal","project":"p56","title":"From \\Crefkv one has H\\left[-X + \\sum_i=1^n Y_i\\right] \\leq H[ - X + Y_1 ] + H\\left[ \\sum…","kind":"proof","summary":"From \\Crefkv one has H\\left[-X + \\sum_i=1^n Y_i\\right] \\leq H[ - X + Y_1 ] + H\\left[ \\sum_i=1^n…","labels":[],"detail_key":"p56"},{"id":"n44427","layer":"informal","project":"p56","title":"Comparing sums","kind":"lemma","summary":"[Comparing sums] Let (X_i)_1 \\leq i \\leq m and (Y_j)_1 \\leq j \\leq l be tuples of jointly indep…","labels":["compare-sums"],"detail_key":"p56"},{"id":"n44428","layer":"informal","project":"p56","title":"Write W := \\sum_i=1^m X_i. From \\Crefsumset-lower we have H[\\sum_j=1^l Y_j] \\leq H[-W + \\…","kind":"proof","summary":"Write W := \\sum_i=1^m X_i. From \\Crefsumset-lower we have H[\\sum_j=1^l Y_j] \\leq H[-W + \\sum_j=…","labels":[],"detail_key":"p56"},{"id":"n44429","layer":"informal","project":"p56","title":"Sums of dilates I","kind":"lemma","summary":"[Sums of dilates I] Let X,Y,X' be independent G-valued random variables, with X' a copy of X, a…","labels":["sum-dilate-I"],"detail_key":"p56"},{"id":"n44430","layer":"informal","project":"p56","title":"From \\Crefruzsa-triangle-improved we have H[(X-Y)-aY] \\leq H[(X-Y) - X'] + H[X'-aY] - H[X…","kind":"proof","summary":"From \\Crefruzsa-triangle-improved we have H[(X-Y)-aY] \\leq H[(X-Y) - X'] + H[X'-aY] - H[X'] whi…","labels":[],"detail_key":"p56"},{"id":"n44431","layer":"informal","project":"p56","title":"Sums of dilates II","kind":"lemma","summary":"[Sums of dilates II] Let X,Y be independent G-valued random variables, and let a be an integer.…","labels":["sum-dilate-II"],"detail_key":"p56"},{"id":"n44432","layer":"informal","project":"p56","title":"From \\Crefkv one has H[Y-X+X'] - H[Y-X] \\leq H[Y+X'] - H[Y] = H[Y+X] - H[Y] which by \\Cre…","kind":"proof","summary":"From \\Crefkv one has H[Y-X+X'] - H[Y-X] \\leq H[Y+X'] - H[Y] = H[Y+X] - H[Y] which by \\Crefruz-i…","labels":[],"detail_key":"p56"},{"id":"n44433","layer":"informal","project":"p56","title":"Multidistance","kind":"definition","summary":"[Multidistance] Let m be a positive integer, and let X_[m] = (X_i)_1 \\leq i \\leq m be an m-tupl…","labels":["multidist-def"],"detail_key":"p56"},{"id":"n44434","layer":"informal","project":"p56","title":"Multidistance of copy","kind":"lemma","summary":"[Multidistance of copy] If X_[m] = (X_i)_1 \\leq i \\leq m and Y_[m] = (Y_i)_1 \\leq i \\leq m are…","labels":["multidist-copy"],"detail_key":"p56"},{"id":"n44435","layer":"informal","project":"p56","title":"Clear from Lemma \\refcopy-ent.","kind":"proof","summary":"Clear from Lemma \\refcopy-ent.","labels":[],"detail_key":"p56"},{"id":"n44436","layer":"informal","project":"p56","title":"Multidistance of independent variables","kind":"lemma","summary":"[Multidistance of independent variables] If X_[m] = (X_i)_1 \\leq i \\leq m are jointly independe…","labels":["multidist-indep"],"detail_key":"p56"},{"id":"n44437","layer":"informal","project":"p56","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p56"},{"id":"n44438","layer":"informal","project":"p56","title":"Nonnegativity","kind":"lemma","summary":"[Nonnegativity] For any such tuple, we have D[X_[m]] \\geq 0.","labels":["multidist-nonneg"],"detail_key":"p56"},{"id":"n44439","layer":"informal","project":"p56","title":"From \\Crefsumset-lower one has H[\\sum_i =1^m \\tilde X_i] \\geq H[\\tilde X_i] for each 1 \\l…","kind":"proof","summary":"From \\Crefsumset-lower one has H[\\sum_i =1^m \\tilde X_i] \\geq H[\\tilde X_i] for each 1 \\leq i \\…","labels":[],"detail_key":"p56"},{"id":"n44440","layer":"informal","project":"p56","title":"Relabeling","kind":"lemma","summary":"[Relabeling] If \\phi: \\1,\\dots,m\\ \\to \\1,\\dots,m\\ is a bijection, then D[X_[m]] = D[(X_\\phi(j))…","labels":["multidist-perm"],"detail_key":"p56"},{"id":"n44441","layer":"informal","project":"p56","title":"Trivial.","kind":"proof","summary":"Trivial.","labels":[],"detail_key":"p56"},{"id":"n44442","layer":"informal","project":"p56","title":"Multidistance and Ruzsa distance, I","kind":"lemma","summary":"[Multidistance and Ruzsa distance, I] Let m \\ge 2, and let X_[m] be a tuple of G-valued random…","labels":["multidist-ruzsa-I"],"detail_key":"p56"},{"id":"n44443","layer":"informal","project":"p56","title":"By \\Crefruz-copy, \\Crefmultidist-copy we may take the X_i to be jointly independent. From…","kind":"proof","summary":"By \\Crefruz-copy, \\Crefmultidist-copy we may take the X_i to be jointly independent. From \\Cref…","labels":[],"detail_key":"p56"},{"id":"n44444","layer":"informal","project":"p56","title":"Multidistance and Ruzsa distance, II","kind":"lemma","summary":"[Multidistance and Ruzsa distance, II] Let m \\ge 2, and let X_[m] be a tuple of G-valued random…","labels":["multidist-ruzsa-II"],"detail_key":"p56"},{"id":"n44445","layer":"informal","project":"p56","title":"From \\Crefruzsa-triangle we have d[X_j;X_j] \\leq 2 d[X_j;-X_k], and applying this to ever…","kind":"proof","summary":"From \\Crefruzsa-triangle we have d[X_j;X_j] \\leq 2 d[X_j;-X_k], and applying this to every summ…","labels":[],"detail_key":"p56"},{"id":"n44446","layer":"informal","project":"p56","title":"Multidistance and Ruzsa distance, III","kind":"lemma","summary":"[Multidistance and Ruzsa distance, III] Let m \\ge 2, and let X_[m] be a tuple of G-valued rando…","labels":["multidist-ruzsa-III"],"detail_key":"p56"},{"id":"n44447","layer":"informal","project":"p56","title":"By \\Crefruz-copy, \\Crefmultidist-copy we may take the X_i to be jointly independent. Let…","kind":"proof","summary":"By \\Crefruz-copy, \\Crefmultidist-copy we may take the X_i to be jointly independent. Let X_0 be…","labels":[],"detail_key":"p56"},{"id":"n44448","layer":"informal","project":"p56","title":"Multidistance and Ruzsa distance, IV","kind":"lemma","summary":"[Multidistance and Ruzsa distance, IV] Let m \\ge 2, and let X_[m] be a tuple of independent G-v…","labels":["multidist-ruzsa-IV"],"detail_key":"p56"},{"id":"n44449","layer":"informal","project":"p56","title":"7922","kind":"proof","summary":"Take (X'_i)_1 \\leq i \\leq m to be further independent copies of (X_i)_1 \\leq i \\leq m (which ex…","labels":["7922"],"detail_key":"p56"},{"id":"n44450","layer":"informal","project":"p56","title":"Vanishing","kind":"proposition","summary":"[Vanishing] If D[X_[m]]=0, then for each 1 \\leq i \\leq m there is a finite subgroup H_i \\leq G…","labels":["multi-zero"],"detail_key":"p56"},{"id":"n44451","layer":"informal","project":"p56","title":"From \\Crefmultidist-ruzsa-II and \\Crefruzsa-nonneg we have d[X_j; X_j]=0 for all 1 \\leq j…","kind":"proof","summary":"From \\Crefmultidist-ruzsa-II and \\Crefruzsa-nonneg we have d[X_j; X_j]=0 for all 1 \\leq j \\leq…","labels":[],"detail_key":"p56"},{"id":"n44452","layer":"informal","project":"p56","title":"\\eta","kind":"definition","summary":"[\\eta] We set \\eta := \\frac132m^3.","labels":["eta-def-multi"],"detail_key":"p56"},{"id":"n44453","layer":"informal","project":"p56","title":"\\tau-functional","kind":"definition","summary":"[\\tau-functional] If (X_i)_1 \\leq i \\leq m is a tuple, we define its \\tau-functional \\tau[ (X_i…","labels":["tau-def-multi"],"detail_key":"p56"},{"id":"n44454","layer":"informal","project":"p56","title":"\\tau-minimizer","kind":"definition","summary":"[\\tau-minimizer] A \\tau-minimizer is a tuple (X_i)_1 \\leq i \\leq m that minimizes the \\tau-func…","labels":["tau-min-multi"],"detail_key":"p56"},{"id":"n44455","layer":"informal","project":"p56","title":"Existence of \\tau-minimizer","kind":"proposition","summary":"[Existence of \\tau-minimizer] If G is finite, then a \\tau-minimizer exists.","labels":["tau-min-exist-multi"],"detail_key":"p56"},{"id":"n44456","layer":"informal","project":"p56","title":"This is similar to the proof of \\Creftau-min.","kind":"proof","summary":"This is similar to the proof of \\Creftau-min.","labels":[],"detail_key":"p56"},{"id":"n44457","layer":"informal","project":"p56","title":"Minimizer close to reference variables","kind":"proposition","summary":"[Minimizer close to reference variables] If (X_i)_1 \\leq i \\leq m is a \\tau-minimizer, then \\su…","labels":["tau-ref"],"detail_key":"p56"},{"id":"n44458","layer":"informal","project":"p56","title":"By \\Creftau-min-multi we have \\tau[ (X_i)_1 \\leq i \\leq m] \\leq \\tau[ (X^0)_1 \\leq i \\leq…","kind":"proof","summary":"By \\Creftau-min-multi we have \\tau[ (X_i)_1 \\leq i \\leq m] \\leq \\tau[ (X^0)_1 \\leq i \\leq m] an…","labels":[],"detail_key":"p56"},{"id":"n44459","layer":"informal","project":"p56","title":"Lower bound on multidistance","kind":"lemma","summary":"[Lower bound on multidistance] If (X_i)_1 \\leq i \\leq m is a \\tau-minimizer, and k := D[(X_i)_1…","labels":["multidist-lower"],"detail_key":"p56"},{"id":"n44460","layer":"informal","project":"p56","title":"By \\Creftau-min-multi we have \\tau[ (X_i)_1 \\leq i \\leq m] \\leq \\tau[ (X'_i)_1 \\leq i \\le…","kind":"proof","summary":"By \\Creftau-min-multi we have \\tau[ (X_i)_1 \\leq i \\leq m] \\leq \\tau[ (X'_i)_1 \\leq i \\leq m] a…","labels":[],"detail_key":"p56"},{"id":"n44461","layer":"informal","project":"p56","title":"Conditional multidistance","kind":"definition","summary":"[Conditional multidistance] If X_[m] = (X_i)_1 \\leq i \\leq m and Y_[m] = (Y_i)_1 \\leq i \\leq m…","labels":["cond-multidist-def","multi-def-cond-alt"],"detail_key":"p56"},{"id":"n44462","layer":"informal","project":"p56","title":"Alternate form of conditional multidistance","kind":"lemma","summary":"[Alternate form of conditional multidistance] If the (X_i,Y_i) are independent, D[ X_[m] | Y_[m…","labels":["cond-multidist-alt","multi-def-cond"],"detail_key":"p56"},{"id":"n44463","layer":"informal","project":"p56","title":"This is routine from \\Crefconditional-entropy-def and Definitions \\refmultidist-def and \\…","kind":"proof","summary":"This is routine from \\Crefconditional-entropy-def and Definitions \\refmultidist-def and \\refcon…","labels":[],"detail_key":"p56"},{"id":"n44464","layer":"informal","project":"p56","title":"Conditional multidistance nonnegative","kind":"lemma","summary":"[Conditional multidistance nonnegative] If X_[m] = (X_i)_1 \\leq i \\leq m and Y_[m] = (Y_i)_1 \\l…","labels":["cond-multidist-nonneg"],"detail_key":"p56"},{"id":"n44465","layer":"informal","project":"p56","title":"Clear from \\Crefmultidist-nonneg and \\Crefcond-multidist-def, except that some care may n…","kind":"proof","summary":"Clear from \\Crefmultidist-nonneg and \\Crefcond-multidist-def, except that some care may need to…","labels":[],"detail_key":"p56"},{"id":"n44466","layer":"informal","project":"p56","title":"Lower bound on conditional multidistance","kind":"lemma","summary":"[Lower bound on conditional multidistance] If (X_i)_1 \\leq i \\leq m is a \\tau-minimizer, and k…","labels":["cond-multidist-lower"],"detail_key":"p56"},{"id":"n44467","layer":"informal","project":"p56","title":"Immediate from \\Crefmultidist-lower, \\Crefcond-multidist-alt, and \\Crefcond-dist-def.","kind":"proof","summary":"Immediate from \\Crefmultidist-lower, \\Crefcond-multidist-alt, and \\Crefcond-dist-def.","labels":[],"detail_key":"p56"},{"id":"n44468","layer":"informal","project":"p56","title":"Lower bound on conditional multidistance, II","kind":"corollary","summary":"[Lower bound on conditional multidistance, II] With the notation of the previous lemma, we have…","labels":["cond-multidist-lower-II","5.3-conv"],"detail_key":"p56"},{"id":"n44469","layer":"informal","project":"p56","title":"This follows from \\Crefcond-multidist-lower and \\Crefmultidist-perm.","kind":"proof","summary":"This follows from \\Crefcond-multidist-lower and \\Crefmultidist-perm.","labels":[],"detail_key":"p56"},{"id":"n44470","layer":"informal","project":"p56","title":"Multidistance chain rule","kind":"lemma","summary":"[Multidistance chain rule] Let \\pi \\colon G \\to H be a homomorphism of abelian groups and let X…","labels":["multidist-chain-rule","chain-eq"],"detail_key":"p56"},{"id":"n44471","layer":"informal","project":"p56","title":"chain-1","kind":"proof","summary":"For notational brevity during this proof, write S := \\sum_i=1^m X_i. From \\Crefconditional-mutu…","labels":["chain-1","chain-2"],"detail_key":"p56"},{"id":"n44472","layer":"informal","project":"p56","title":"Conditional multidistance chain rule","kind":"lemma","summary":"[Conditional multidistance chain rule] Let \\pi \\colon G \\to H be a homomorphism of abelian grou…","labels":["multidist-chain-rule-cond","chain-eq-cond"],"detail_key":"p56"},{"id":"n44473","layer":"informal","project":"p56","title":"For each y_i in the support of p_Y_i, apply \\Crefmultidist-chain-rule with X_i replaced b…","kind":"proof","summary":"For each y_i in the support of p_Y_i, apply \\Crefmultidist-chain-rule with X_i replaced by the…","labels":[],"detail_key":"p56"},{"id":"n44474","layer":"informal","project":"p56","title":"multidist-chain-rule-iter","kind":"lemma","summary":"Let m be a positive integer. Suppose one has a sequence G_m \\to G_m-1 \\to \\dots \\to G_1 \\to G_0…","labels":["multidist-chain-rule-iter","g-seq","chain-eq-cond'","chain-eq-cond''"],"detail_key":"p56"},{"id":"n44475","layer":"informal","project":"p56","title":"From \\Crefmultidist-chain-rule-cond (taking Y_[m] = \\pi_d-1(X_[m]) and \\pi = \\pi_d there,…","kind":"proof","summary":"From \\Crefmultidist-chain-rule-cond (taking Y_[m] = \\pi_d-1(X_[m]) and \\pi = \\pi_d there, and n…","labels":[],"detail_key":"p56"},{"id":"n44476","layer":"informal","project":"p56","title":"cor-multid","kind":"corollary","summary":"Let G be an abelian group and let m \\geq 2. Suppose that X_i,j, 1 \\leq i, j \\leq m, are indepen…","labels":["cor-multid"],"detail_key":"p56"},{"id":"n44477","layer":"informal","project":"p56","title":"In \\Crefmultidist-chain-rule-iter we take G_d := G^d with the maps \\pi_d \\colon G^m \\to G…","kind":"proof","summary":"In \\Crefmultidist-chain-rule-iter we take G_d := G^d with the maps \\pi_d \\colon G^m \\to G^d for…","labels":[],"detail_key":"p56"},{"id":"n44478","layer":"informal","project":"p56","title":"Bounding mutual information","kind":"proposition","summary":"[Bounding mutual information] Suppose that X_i,j, 1 \\leq i,j \\leq m, are jointly independent G-…","labels":["key","I-ineq"],"detail_key":"p56"},{"id":"n44479","layer":"informal","project":"p56","title":"441","kind":"proof","summary":"For each j \\in \\1,\\dots,m\\ we call the tuple (X_i,j)_i = 1^m a \\emphcolumn and for each i \\in \\…","labels":["441","54a","55a","eq:distbnd1","eq:distbnd2","eq:distbnd3"],"detail_key":"p56"},{"id":"n44480","layer":"informal","project":"p56","title":"Additional random variables","kind":"definition","summary":"[Additional random variables] By a slight abuse of notation, we identify Z/mZ and \\1,\\dots,m\\ i…","labels":["more-random","pqr-defs"],"detail_key":"p56"},{"id":"n44481","layer":"informal","project":"p56","title":"Zero-sum","kind":"lemma","summary":"[Zero-sum] We have Z_1+Z_2+Z_3= 0","labels":["Zero-sum","eq:sum-zero"],"detail_key":"p56"},{"id":"n44482","layer":"informal","project":"p56","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p56"},{"id":"n44483","layer":"informal","project":"p56","title":"Mutual information bound","kind":"proposition","summary":"[Mutual information bound] We have \\[ I[Z_1 : Z_2\\, |\\, W],\\ I[Z_2 : Z_3\\, |\\, W],\\ I[Z_1 : Z_3…","labels":["prop:52","t-def"],"detail_key":"p56"},{"id":"n44484","layer":"informal","project":"p56","title":"We analyze these variables by \\Crefkey in several different ways. In the first applicatio…","kind":"proof","summary":"We analyze these variables by \\Crefkey in several different ways. In the first application, tak…","labels":[],"detail_key":"p56"},{"id":"n44485","layer":"informal","project":"p56","title":"Entropy of W","kind":"lemma","summary":"[Entropy of W] We have H[W] \\leq (2m-1)k + \\frac1m \\sum_i=1^m H[X_i].","labels":["ent-w"],"detail_key":"p56"},{"id":"n44486","layer":"informal","project":"p56","title":"eq:s-bound","kind":"proof","summary":"Without loss of generality, we may take X_1,\\dots,X_m to be independent. Write S = \\sum_i=1^m X…","labels":["eq:s-bound","eq:ent-s"],"detail_key":"p56"},{"id":"n44487","layer":"informal","project":"p56","title":"Entropy of Z_2","kind":"lemma","summary":"[Entropy of Z_2] We have H[Z_2] \\leq (8m^2-16m+1) k + \\frac1m \\sum_i=1^m H[X_i].","labels":["ent-z2"],"detail_key":"p56"},{"id":"n44488","layer":"informal","project":"p56","title":"We observe \\[ H[Z_2] = H[\\sum_j \\in Z/mZ j Q_j]. \\] Applying \\Crefklm-1 one has H[Z_2] &\\…","kind":"proof","summary":"We observe \\[ H[Z_2] = H[\\sum_j \\in Z/mZ j Q_j]. \\] Applying \\Crefklm-1 one has H[Z_2] &\\leq \\s…","labels":[],"detail_key":"p56"},{"id":"n44489","layer":"informal","project":"p56","title":"Mutual information bound","kind":"lemma","summary":"[Mutual information bound] We have I[W : Z_2] \\leq 2 (m-1) k.","labels":["mutual-w-z2"],"detail_key":"p56"},{"id":"n44490","layer":"informal","project":"p56","title":"From \\Crefalternative-mutual we have I[W : Z_2] = H[W] - H[W | Z_2], and since Z_2 = \\sum…","kind":"proof","summary":"From \\Crefalternative-mutual we have I[W : Z_2] = H[W] - H[W | Z_2], and since Z_2 = \\sum_j=1^m…","labels":[],"detail_key":"p56"},{"id":"n44491","layer":"informal","project":"p56","title":"Distance bound","kind":"lemma","summary":"[Distance bound] We have \\sum_i=1^m d[X_i;Z_2|W] \\leq 4(m^3-m^2) k.","labels":["xi-z2-w-dist"],"detail_key":"p56"},{"id":"n44492","layer":"informal","project":"p56","title":"in-a-bit-6","kind":"proof","summary":"For each i \\in \\1,\\dots, m\\, using \\Crefklm-3 (noting the sum Z_2 contains X_i as a summand) we…","labels":["in-a-bit-6"],"detail_key":"p56"},{"id":"n44493","layer":"informal","project":"p56","title":"Application of BSG","kind":"lemma","summary":"[Application of BSG] Let G be an abelian group, let (T_1,T_2,T_3) be a G^3-valued random variab…","labels":["lem:get-better","eq:get-better"],"detail_key":"p56"},{"id":"n44494","layer":"informal","project":"p56","title":"514a","kind":"proof","summary":"We apply \\Crefentropic-bsg with X=T_1 and Y=T_2. Since T_1+T_2=-T_3, we find that \\sum_z p_T_3(…","labels":["514a"],"detail_key":"p56"},{"id":"n44495","layer":"informal","project":"p56","title":"Vanishing entropy","kind":"proposition","summary":"[Vanishing entropy] We have k = 0.","labels":["k-vanish"],"detail_key":"p56"},{"id":"n44496","layer":"informal","project":"p56","title":"eq:delta-w","kind":"proof","summary":"For each value W=w, apply \\Creflem:get-better (and \\CrefZero-sum) to \\[ T_1 = (Z_1 \\,|\\, W \\mat…","labels":["eq:delta-w","eq:uw1","eq:uw2","eq:uw3"],"detail_key":"p56"},{"id":"n44497","layer":"informal","project":"p56","title":"Entropy form of PFR","kind":"theorem","summary":"[Entropy form of PFR] Suppose that G is a finite abelian group of torsion m. Suppose that X is…","labels":["main-entropy"],"detail_key":"p56"},{"id":"n44498","layer":"informal","project":"p56","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p56"},{"id":"n44499","layer":"informal","project":"p56","title":"pfr_aux_torsion","kind":"lemma","summary":"Suppose that G is a finite abelian group of torsion m. If A \\subset G is non-empty and |A+A| \\l…","labels":["pfr_aux_torsion","ah2"],"detail_key":"p56"},{"id":"n44500","layer":"informal","project":"p56","title":"Repeat the proof of \\Crefpfr_aux, but with \\Crefmain-entropy in place of \\Crefentropy-pfr…","kind":"proof","summary":"Repeat the proof of \\Crefpfr_aux, but with \\Crefmain-entropy in place of \\Crefentropy-pfr. Beca…","labels":[],"detail_key":"p56"},{"id":"n44501","layer":"informal","project":"p56","title":"PFR","kind":"theorem","summary":"[PFR] Suppose that G is a finite abelian group of torsion m. If A \\subset G is non-empty and |A…","labels":["pfr-torsion"],"detail_key":"p56"},{"id":"n44502","layer":"informal","project":"p56","title":"Repeat the proof of \\Crefpfr, but with \\Crefpfr_aux_torsion in place of \\Crefpfr_aux.","kind":"proof","summary":"Repeat the proof of \\Crefpfr, but with \\Crefpfr_aux_torsion in place of \\Crefpfr_aux.","labels":[],"detail_key":"p56"},{"id":"n44503","layer":"informal","project":"p56","title":"Kullback--Leibler divergence","kind":"definition","summary":"[Kullback--Leibler divergence] If X,Y are two G-valued random variables, the Kullback--Leibler…","labels":["kl-div"],"detail_key":"p56"},{"id":"n44504","layer":"informal","project":"p56","title":"Kullback--Leibler divergence of copy","kind":"lemma","summary":"[Kullback--Leibler divergence of copy] If X' is a copy of X, and Y' is a copy of Y, then D_KL(X…","labels":["kl-div-copy"],"detail_key":"p56"},{"id":"n44505","layer":"informal","project":"p56","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p56"},{"id":"n44506","layer":"informal","project":"p56","title":"Gibbs inequality","kind":"lemma","summary":"[Gibbs inequality] D_KL(X\\Vert Y) \\geq 0.","labels":["Gibbs"],"detail_key":"p56"},{"id":"n44507","layer":"informal","project":"p56","title":"Apply \\Creflog-sum on the definition.","kind":"proof","summary":"Apply \\Creflog-sum on the definition.","labels":[],"detail_key":"p56"},{"id":"n44508","layer":"informal","project":"p56","title":"Converse Gibbs inequality","kind":"lemma","summary":"[Converse Gibbs inequality] If D_KL(X\\Vert Y) = 0, then Y is a copy of X.","labels":["Gibbs-converse"],"detail_key":"p56"},{"id":"n44509","layer":"informal","project":"p56","title":"Apply \\Crefconverse-log-sum.","kind":"proof","summary":"Apply \\Crefconverse-log-sum.","labels":[],"detail_key":"p56"},{"id":"n44510","layer":"informal","project":"p56","title":"Convexity of Kullback--Leibler","kind":"lemma","summary":"[Convexity of Kullback--Leibler] If S is a finite set, \\sum_s \\in S w_s = 1 for some non-negati…","labels":["kl-div-convex"],"detail_key":"p56"},{"id":"n44511","layer":"informal","project":"p56","title":"For each x, replace \\log \\fracP(X_s=x)P(Y_s=x) in the definition with \\log \\fracw_sP(X_s=…","kind":"proof","summary":"For each x, replace \\log \\fracP(X_s=x)P(Y_s=x) in the definition with \\log \\fracw_sP(X_s=x)w_sP…","labels":[],"detail_key":"p56"},{"id":"n44512","layer":"informal","project":"p56","title":"Kullback--Leibler and injections","kind":"lemma","summary":"[Kullback--Leibler and injections] If f:G \\to H is an injection, then D_KL(f(X)\\Vert f(Y)) = D_…","labels":["kl-div-inj"],"detail_key":"p56"},{"id":"n44513","layer":"informal","project":"p56","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p56"},{"id":"n44514","layer":"informal","project":"p56","title":"Kullback--Leibler and sums","kind":"lemma","summary":"[Kullback--Leibler and sums] If X, Y, Z are independent G-valued random variables, then D_KL(X+…","labels":["kl-sums"],"detail_key":"p56"},{"id":"n44515","layer":"informal","project":"p56","title":"For each z, D_KL(X+z\\Vert Y+z)=D_KL(X\\Vert Y) by \\Crefkl-div-inj. Then apply \\Crefkl-div-…","kind":"proof","summary":"For each z, D_KL(X+z\\Vert Y+z)=D_KL(X\\Vert Y) by \\Crefkl-div-inj. Then apply \\Crefkl-div-convex…","labels":[],"detail_key":"p56"},{"id":"n44516","layer":"informal","project":"p56","title":"Conditional Kullback--Leibler divergence","kind":"definition","summary":"[Conditional Kullback--Leibler divergence] If X,Y,Z are random variables, with X,Z defined on t…","labels":["ckl-div"],"detail_key":"p56"},{"id":"n44517","layer":"informal","project":"p56","title":"Kullback--Leibler and conditioning","kind":"lemma","summary":"[Kullback--Leibler and conditioning] If X, Y are independent G-valued random variables, and Z i…","labels":["kl-cond"],"detail_key":"p56"},{"id":"n44518","layer":"informal","project":"p56","title":"Compare the terms correspond to each x\\in G on both sides.","kind":"proof","summary":"Compare the terms correspond to each x\\in G on both sides.","labels":[],"detail_key":"p56"},{"id":"n44519","layer":"informal","project":"p56","title":"Conditional Gibbs inequality","kind":"lemma","summary":"[Conditional Gibbs inequality] D_KL((X|W)\\Vert Y) \\geq 0.","labels":["Conditional-Gibbs"],"detail_key":"p56"},{"id":"n44520","layer":"informal","project":"p56","title":"Clear from Definition \\refckl-div and Lemma \\refGibbs.","kind":"proof","summary":"Clear from Definition \\refckl-div and Lemma \\refGibbs.","labels":[],"detail_key":"p56"},{"id":"n44521","layer":"informal","project":"p56","title":"Rho minus","kind":"definition","summary":"[Rho minus] For any G-valued random variable X, we define \\rho^-(X) to be the infimum of D_KL(X…","labels":["rhominus-def"],"detail_key":"p56"},{"id":"n44522","layer":"informal","project":"p56","title":"Rho plus","kind":"definition","summary":"[Rho plus] For any G-valued random variable X, we define \\rho^+(X) := \\rho^-(X) + H(X) - H(U_A).","labels":["rhoplus-def"],"detail_key":"p56"},{"id":"n44523","layer":"informal","project":"p56","title":"Rho minus non-negative","kind":"lemma","summary":"[Rho minus non-negative] We have \\rho^-(X) \\geq 0.","labels":["rhominus-nonneg","rhoMinus_nonneg"],"detail_key":"p56"},{"id":"n44524","layer":"informal","project":"p56","title":"Clear from Lemma \\refConditional-Gibbs.","kind":"proof","summary":"Clear from Lemma \\refConditional-Gibbs.","labels":[],"detail_key":"p56"},{"id":"n44525","layer":"informal","project":"p56","title":"Rho minus of subgroup","kind":"lemma","summary":"[Rho minus of subgroup] If H is a finite subgroup of G, then \\rho^-(U_H) = \\log |A| - \\log \\max…","labels":["rhominus-subgroup"],"detail_key":"p56"},{"id":"n44526","layer":"informal","project":"p56","title":"For every G-valued random variable T that is independent of Y, D_KL(U_H \\Vert U_A+T) = \\s…","kind":"proof","summary":"For every G-valued random variable T that is independent of Y, D_KL(U_H \\Vert U_A+T) = \\sum_h\\i…","labels":[],"detail_key":"p56"},{"id":"n44527","layer":"informal","project":"p56","title":"Rho plus of subgroup","kind":"corollary","summary":"[Rho plus of subgroup] If H is a finite subgroup of G, then \\rho^+(U_H) = \\log |H| - \\log \\max_…","labels":["rhoplus-subgroup"],"detail_key":"p56"},{"id":"n44528","layer":"informal","project":"p56","title":"Straightforward by definition and \\Crefrhominus-subgroup.","kind":"proof","summary":"Straightforward by definition and \\Crefrhominus-subgroup.","labels":[],"detail_key":"p56"},{"id":"n44529","layer":"informal","project":"p56","title":"Rho functional","kind":"definition","summary":"[Rho functional] We define \\rho(X) := (\\rho^+(X) + \\rho^-(X))/2.","labels":["rho-def"],"detail_key":"p56"},{"id":"n44530","layer":"informal","project":"p56","title":"rho-init","kind":"lemma","summary":"We have \\rho(U_A) = 0.","labels":["rho-init","rho_of_uniform"],"detail_key":"p56"},{"id":"n44531","layer":"informal","project":"p56","title":"\\rho^-(U_A)\\le 0 by the choice T=0. The claim then follows from \\Crefrhominus-nonneg.","kind":"proof","summary":"\\rho^-(U_A)\\le 0 by the choice T=0. The claim then follows from \\Crefrhominus-nonneg.","labels":[],"detail_key":"p56"},{"id":"n44532","layer":"informal","project":"p56","title":"Rho of subgroup","kind":"lemma","summary":"[Rho of subgroup] If H is a finite subgroup of G, and \\rho(U_H) \\leq r, then there exists t suc…","labels":["rho-subgroup"],"detail_key":"p56"},{"id":"n44533","layer":"informal","project":"p56","title":"The first claim is a direct corollary of \\Crefrhominus-subgroup and \\Crefrhoplus-subgroup…","kind":"proof","summary":"The first claim is a direct corollary of \\Crefrhominus-subgroup and \\Crefrhoplus-subgroup. To s…","labels":[],"detail_key":"p56"},{"id":"n44534","layer":"informal","project":"p56","title":"Rho invariant","kind":"lemma","summary":"[Rho invariant] For any s \\in G, \\rho(X+s) = \\rho(X).","labels":["rho-invariant"],"detail_key":"p56"},{"id":"n44535","layer":"informal","project":"p56","title":"Observe that by \\Crefkl-div-inj, \\inf_T D_KL(X\\Vert U_A+T)=\\inf_T D_KL(X+s\\Vert U_A+T+s)=…","kind":"proof","summary":"Observe that by \\Crefkl-div-inj, \\inf_T D_KL(X\\Vert U_A+T)=\\inf_T D_KL(X+s\\Vert U_A+T+s)=\\inf_T…","labels":[],"detail_key":"p56"},{"id":"n44536","layer":"informal","project":"p56","title":"Rho continuous","kind":"lemma","summary":"[Rho continuous] \\rho(X) depends continuously on the distribution of X.","labels":["rho-cts"],"detail_key":"p56"},{"id":"n44537","layer":"informal","project":"p56","title":"Clear from definition.","kind":"proof","summary":"Clear from definition.","labels":[],"detail_key":"p56"},{"id":"n44538","layer":"informal","project":"p56","title":"Rho and sums","kind":"lemma","summary":"[Rho and sums] If X,Y are independent, one has \\rho^-(X+Y) \\leq \\rho^-(X) \\rho^+(X+Y) \\leq \\rho…","labels":["rho-sums"],"detail_key":"p56"},{"id":"n44539","layer":"informal","project":"p56","title":"The first inequality follows from \\Crefkl-sums. The second and third inequalities are dir…","kind":"proof","summary":"The first inequality follows from \\Crefkl-sums. The second and third inequalities are direct co…","labels":[],"detail_key":"p56"},{"id":"n44540","layer":"informal","project":"p56","title":"Conditional Rho functional","kind":"definition","summary":"[Conditional Rho functional] We define \\rho(X|Y) := \\sum_y \\bf P(Y=y) \\rho(X|Y=y).","labels":["rho-cond-def"],"detail_key":"p56"},{"id":"n44541","layer":"informal","project":"p56","title":"Conditional rho and translation","kind":"lemma","summary":"[Conditional rho and translation] For any s\\in G, \\rho(X+s|Y)=\\rho(X|Y).","labels":["rho-cond-invariant"],"detail_key":"p56"},{"id":"n44542","layer":"informal","project":"p56","title":"Direct corollary of \\Crefrho-invariant.","kind":"proof","summary":"Direct corollary of \\Crefrho-invariant.","labels":[],"detail_key":"p56"},{"id":"n44543","layer":"informal","project":"p56","title":"Conditional rho and relabeling","kind":"lemma","summary":"[Conditional rho and relabeling] If f is injective, then \\rho(X|f(Y))=\\rho(X|Y).","labels":["rho-cond-relabeled"],"detail_key":"p56"},{"id":"n44544","layer":"informal","project":"p56","title":"Clear from the definition.","kind":"proof","summary":"Clear from the definition.","labels":[],"detail_key":"p56"},{"id":"n44545","layer":"informal","project":"p56","title":"Rho and conditioning","kind":"lemma","summary":"[Rho and conditioning] If X,Z are defined on the same space, one has \\rho^-(X|Z) \\leq \\rho^-(X)…","labels":["rho-cond"],"detail_key":"p56"},{"id":"n44546","layer":"informal","project":"p56","title":"The first inequality follows from \\Crefkl-cond. The second and third inequalities are dir…","kind":"proof","summary":"The first inequality follows from \\Crefkl-cond. The second and third inequalities are direct co…","labels":[],"detail_key":"p56"},{"id":"n44547","layer":"informal","project":"p56","title":"Rho and sums, symmetrized","kind":"lemma","summary":"[Rho and sums, symmetrized] If X,Y are independent, then \\rho(X+Y) \\leq \\frac12(\\rho(X)+\\rho(Y)…","labels":["rho-sums-sym"],"detail_key":"p56"},{"id":"n44548","layer":"informal","project":"p56","title":"Apply \\Crefrho-sums for (X,Y) and (Y,X) and take their average.","kind":"proof","summary":"Apply \\Crefrho-sums for (X,Y) and (Y,X) and take their average.","labels":[],"detail_key":"p56"},{"id":"n44549","layer":"informal","project":"p56","title":"Rho and conditioning, symmetrized","kind":"lemma","summary":"[Rho and conditioning, symmetrized] If X,Y are independent, then \\rho(X | X+Y) \\leq \\frac12(\\rh…","labels":["rho-cond-sym"],"detail_key":"p56"},{"id":"n44550","layer":"informal","project":"p56","title":"First apply \\Crefrho-cond to get \\rho(X|X+Y)\\le \\rho(X) + \\frac12(H[X+Y]-H[Y]), and \\rho(…","kind":"proof","summary":"First apply \\Crefrho-cond to get \\rho(X|X+Y)\\le \\rho(X) + \\frac12(H[X+Y]-H[Y]), and \\rho(Y|X+Y)…","labels":[],"detail_key":"p56"},{"id":"n44551","layer":"informal","project":"p56","title":"phi-min-def","kind":"definition","summary":"Given G-valued random variables X,Y, define \\phi[X;Y] := d[X;Y] + \\eta(\\rho(X) + \\rho(Y)) and d…","labels":["phi-min-def"],"detail_key":"p56"},{"id":"n44552","layer":"informal","project":"p56","title":"\\phi-minimizers exist","kind":"lemma","summary":"[\\phi-minimizers exist] There exists a \\phi-minimizer.","labels":["phi-min-exist"],"detail_key":"p56"},{"id":"n44553","layer":"informal","project":"p56","title":"Clear from compactness.","kind":"proof","summary":"Clear from compactness.","labels":[],"detail_key":"p56"},{"id":"n44554","layer":"informal","project":"p56","title":"phi-first-estimate","kind":"lemma","summary":"I_1\\le 2\\eta d[X_1;X_2]","labels":["phi-first-estimate"],"detail_key":"p56"},{"id":"n44555","layer":"informal","project":"p56","title":"Similar to \\Creffirst-estimate: get upper bounds for d[X_1;X_2] by \\phi[X_1;X_2]\\le \\phi[…","kind":"proof","summary":"Similar to \\Creffirst-estimate: get upper bounds for d[X_1;X_2] by \\phi[X_1;X_2]\\le \\phi[X_1+X_…","labels":[],"detail_key":"p56"},{"id":"n44556","layer":"informal","project":"p56","title":"I1-I2-diff","kind":"lemma","summary":"d[X_1;X_1]+d[X_2;X_2]= 2d[X_1;X_2]+(I_2-I_1).","labels":["I1-I2-diff"],"detail_key":"p56"},{"id":"n44557","layer":"informal","project":"p56","title":"Compare \\Creffirst-fibre with the identity obtained from applying \\Crefcor-fibre on (X_1,…","kind":"proof","summary":"Compare \\Creffirst-fibre with the identity obtained from applying \\Crefcor-fibre on (X_1,\\tilde…","labels":[],"detail_key":"p56"},{"id":"n44558","layer":"informal","project":"p56","title":"phi-second-estimate","kind":"lemma","summary":"I_2\\le 2\\eta d[X_1;X_2] + \\frac\\eta1-\\eta(2\\eta d[X_1;X_2]-I_1).","labels":["phi-second-estimate"],"detail_key":"p56"},{"id":"n44559","layer":"informal","project":"p56","title":"First of all, by \\phi[X_1;X_2]\\le \\phi[X_1+\\tilde X_1;X_2+\\tilde X_2], \\phi[X_1;X_2]\\le \\…","kind":"proof","summary":"First of all, by \\phi[X_1;X_2]\\le \\phi[X_1+\\tilde X_1;X_2+\\tilde X_2], \\phi[X_1;X_2]\\le \\phi[X_…","labels":[],"detail_key":"p56"},{"id":"n44560","layer":"informal","project":"p56","title":"rho-BSG-triplet","kind":"lemma","summary":"If G-valued random variables T_1,T_2,T_3 satisfy T_1+T_2+T_3=0, then d[X_1;X_2]\\le 3I[T_1:T_2]…","labels":["rho-BSG-triplet"],"detail_key":"p56"},{"id":"n44561","layer":"informal","project":"p56","title":"Conditioned on every T_3=t, d[X_1;X_2]\\le d[T_1|T_3=t;T_2|T_3=t]+\\eta(\\rho(T_1|T_3=t)+\\rh…","kind":"proof","summary":"Conditioned on every T_3=t, d[X_1;X_2]\\le d[T_1|T_3=t;T_2|T_3=t]+\\eta(\\rho(T_1|T_3=t)+\\rho(T_2|…","labels":[],"detail_key":"p56"},{"id":"n44562","layer":"informal","project":"p56","title":"rho-BSG-triplet-symmetrized","kind":"lemma","summary":"If G-valued random variables T_1,T_2,T_3 satisfy T_1+T_2+T_3=0, then d[X_1;X_2] \\leq \\sum_1 \\le…","labels":["rho-BSG-triplet-symmetrized"],"detail_key":"p56"},{"id":"n44563","layer":"informal","project":"p56","title":"Take the average of \\Crefrho-BSG-triplet over all 6 permutations of T_1,T_2,T_3.","kind":"proof","summary":"Take the average of \\Crefrho-BSG-triplet over all 6 permutations of T_1,T_2,T_3.","labels":[],"detail_key":"p56"},{"id":"n44564","layer":"informal","project":"p56","title":"rho-increase","kind":"lemma","summary":"For independent random variables Y_1,Y_2,Y_3,Y_4 over G, define S:=Y_1+Y_2+Y_3+Y_4, T_1:=Y_1+Y_…","labels":["rho-increase"],"detail_key":"p56"},{"id":"n44565","layer":"informal","project":"p56","title":"Let T_1':=Y_3+Y_4, T_2':=Y_2+Y_4. First note that \\rho(T_1|T_2,S) &\\le \\rho(T_1|S) + \\fra…","kind":"proof","summary":"Let T_1':=Y_3+Y_4, T_2':=Y_2+Y_4. First note that \\rho(T_1|T_2,S) &\\le \\rho(T_1|S) + \\frac12I(T…","labels":[],"detail_key":"p56"},{"id":"n44566","layer":"informal","project":"p56","title":"rho-increase-symmetrized","kind":"lemma","summary":"For independent random variables Y_1,Y_2,Y_3,Y_4 over G, define T_1:=Y_1+Y_2,T_2:=Y_1+Y_3,T_3:=…","labels":["rho-increase-symmetrized"],"detail_key":"p56"},{"id":"n44567","layer":"informal","project":"p56","title":"Apply Lemma \\refrho-increase on (Y_i,Y_j,Y_k,Y_4) for (i,j,k)=(1,2,3),(2,3,1),(1,3,2), an…","kind":"proof","summary":"Apply Lemma \\refrho-increase on (Y_i,Y_j,Y_k,Y_4) for (i,j,k)=(1,2,3),(2,3,1),(1,3,2), and take…","labels":[],"detail_key":"p56"},{"id":"n44568","layer":"informal","project":"p56","title":"phi-minimizer-zero-distance","kind":"proposition","summary":"If X_1,X_2 is a \\phi-minimizer, then d[X_1;X_2] = 0.","labels":["phi-minimizer-zero-distance"],"detail_key":"p56"},{"id":"n44569","layer":"informal","project":"p56","title":"eq:further-bsg","kind":"proof","summary":"Consider T_1:=X_1+X_2,T_2:=X_1+\\tilde X_1, T_3:=\\tilde X_1 + X_2, and S=X_1+X_2+\\tilde X_1 + \\t…","labels":["eq:further-bsg"],"detail_key":"p56"},{"id":"n44570","layer":"informal","project":"p56","title":"pfr-rho","kind":"proposition","summary":"For any random variables Y_1,Y_2, there exist a subgroup H such that 2\\rho(U_H) \\leq \\rho(Y_1)…","labels":["pfr-rho"],"detail_key":"p56"},{"id":"n44571","layer":"informal","project":"p56","title":"Let X_1,X_2 be a \\phi-minimizer. By \\Crefphi-minimizer-zero-distance d[X_1;X_2]=0, which…","kind":"proof","summary":"Let X_1,X_2 be a \\phi-minimizer. By \\Crefphi-minimizer-zero-distance d[X_1;X_2]=0, which by \\Cr…","labels":[],"detail_key":"p56"},{"id":"n44572","layer":"informal","project":"p56","title":"pfr-9-aux","kind":"corollary","summary":"If |A+A| \\leq K|A|, then there exists a subgroup H and t\\in G such that |A \\cap (H+t)| \\geq K^-…","labels":["pfr-9-aux"],"detail_key":"p56"},{"id":"n44573","layer":"informal","project":"p56","title":"Apply \\Crefpfr-rho on U_A,U_A to get a subspace such that 2\\rho(U_H)\\le 2\\rho(U_A)+8d[U_A…","kind":"proof","summary":"Apply \\Crefpfr-rho on U_A,U_A to get a subspace such that 2\\rho(U_H)\\le 2\\rho(U_A)+8d[U_A;U_A].…","labels":[],"detail_key":"p56"},{"id":"n44574","layer":"informal","project":"p56","title":"pfr-9-aux'","kind":"corollary","summary":"If |A+A| \\leq K|A|, then there exist a subgroup H and a subset c of G with A \\subseteq c + H, s…","labels":["pfr-9-aux'"],"detail_key":"p56"},{"id":"n44575","layer":"informal","project":"p56","title":"Apply \\Crefpfr-9-aux and \\Crefruz-cov to get the result, as in the proof of \\Crefpfr_aux.","kind":"proof","summary":"Apply \\Crefpfr-9-aux and \\Crefruz-cov to get the result, as in the proof of \\Crefpfr_aux.","labels":[],"detail_key":"p56"},{"id":"n44576","layer":"informal","project":"p56","title":"PFR with \\texorpdfstringC=9C=9","kind":"theorem","summary":"[PFR with \\texorpdfstringC=9C=9] If A \\subset \\bf F_2^n is finite non-empty with |A+A| \\leq K|A…","labels":["pfr-9"],"detail_key":"p56"},{"id":"n44577","layer":"informal","project":"p56","title":"Given \\Crefpfr-9-aux', the proof is the same as that of \\Crefpfr.","kind":"proof","summary":"Given \\Crefpfr-9-aux', the proof is the same as that of \\Crefpfr.","labels":[],"detail_key":"p56"},{"id":"n44578","layer":"formal","project":"p56","title":"approx_hom_pfr'","kind":"theorem","summary":"∀ G : Type u_1 G' : Type u_2 [inst : AddCommGroup G] [inst_1 : Fintype G] [inst_2 : 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[DiscreteMeasurableS…","labels":[],"detail_key":"p56","name":"rho_of_subgroup","module":"PFR.RhoFunctional"},{"id":"n44783","layer":"formal","project":"p56","title":"rho_of_sum","kind":"theorem","summary":"∀ G : Type uG [inst : AddCommGroup G] [Finite G] [hGm : MeasurableSpace G] [DiscreteMeasurableS…","labels":[],"detail_key":"p56","name":"rho_of_sum","module":"PFR.RhoFunctional"},{"id":"n44784","layer":"formal","project":"p56","title":"rho_of_sum_le","kind":"theorem","summary":"∀ G : Type uG [inst : AddCommGroup G] [Finite G] [hGm : MeasurableSpace G] [DiscreteMeasurableS…","labels":[],"detail_key":"p56","name":"rho_of_sum_le","module":"PFR.RhoFunctional"},{"id":"n44785","layer":"formal","project":"p56","title":"rho_of_translate","kind":"theorem","summary":"∀ G : Type uG [inst : AddCommGroup G] [Finite G] [hGm : MeasurableSpace G] [DiscreteMeasurableS…","labels":[],"detail_key":"p56","name":"rho_of_translate","module":"PFR.RhoFunctional"},{"id":"n44786","layer":"formal","project":"p56","title":"rdist_of_sums_ge'","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Finite G] [hG : MeasurableSpace G] [MeasurableSingleton…","labels":[],"detail_key":"p56","name":"rdist_of_sums_ge'","module":"PFR.SecondEstimate"},{"id":"n44787","layer":"formal","project":"p56","title":"second_estimate","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Finite G] [hG : MeasurableSpace G] [MeasurableSingleton…","labels":[],"detail_key":"p56","name":"second_estimate","module":"PFR.SecondEstimate"},{"id":"n44788","layer":"formal","project":"p56","title":"second_estimate_aux","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Finite G] [hG : MeasurableSpace G] [MeasurableSingleton…","labels":[],"detail_key":"p56","name":"second_estimate_aux","module":"PFR.SecondEstimate"},{"id":"n44789","layer":"formal","project":"p56","title":"ProbabilityTheory.IdentDistrib.tau_eq","kind":"theorem","summary":"∀ Ω₀₁ : Type u_1 Ω₀₂ : Type u_2 [inst : MeasureTheory.MeasureSpace Ω₀₁] [inst_1 : MeasureTheory…","labels":[],"detail_key":"p56","name":"ProbabilityTheory.IdentDistrib.tau_eq","module":"PFR.TauFunctional"},{"id":"n44790","layer":"formal","project":"p56","title":"TauMinimizes","kind":"def","summary":"Ω₀₁ : Type u_1 → Ω₀₂ : Type u_2 → [inst : MeasureTheory.MeasureSpace Ω₀₁] → [inst_1 : MeasureTh…","labels":[],"detail_key":"p56","name":"TauMinimizes","module":"PFR.TauFunctional"},{"id":"n44791","layer":"formal","project":"p56","title":"condRuzsaDistance_ge_of_min","kind":"theorem","summary":"∀ Ω₀₁ : Type u_1 Ω₀₂ : Type u_2 [inst : MeasureTheory.MeasureSpace Ω₀₁] [inst_1 : MeasureTheory…","labels":[],"detail_key":"p56","name":"condRuzsaDistance_ge_of_min","module":"PFR.TauFunctional"},{"id":"n44792","layer":"formal","project":"p56","title":"distance_ge_of_min","kind":"theorem","summary":"∀ Ω₀₁ : Type u_1 Ω₀₂ : Type u_2 [inst : MeasureTheory.MeasureSpace Ω₀₁] [inst_1 : MeasureTheory…","labels":[],"detail_key":"p56","name":"distance_ge_of_min","module":"PFR.TauFunctional"},{"id":"n44793","layer":"formal","project":"p56","title":"tau","kind":"def","summary":"Ω₀₁ : Type u_1 → Ω₀₂ : Type u_2 → [inst : MeasureTheory.MeasureSpace Ω₀₁] → [inst_1 : MeasureTh…","labels":[],"detail_key":"p56","name":"tau","module":"PFR.TauFunctional"},{"id":"n44794","layer":"formal","project":"p56","title":"tau_minimizer_exists","kind":"theorem","summary":"∀ Ω₀₁ : Type u_1 Ω₀₂ : Type u_2 [inst : MeasureTheory.MeasureSpace Ω₀₁] [inst_1 : MeasureTheory…","labels":[],"detail_key":"p56","name":"tau_minimizer_exists","module":"PFR.TauFunctional"},{"id":"n44795","layer":"formal","project":"p56","title":"dist_of_U_add_le","kind":"theorem","summary":"∀ G : Type u_1 [inst : MeasurableFinGroup G] Ω : Type u [hΩ : MeasureTheory.MeasureSpace Ω] [Me…","labels":[],"detail_key":"p56","name":"dist_of_U_add_le","module":"PFR.TorsionEndgame"},{"id":"n44796","layer":"formal","project":"p56","title":"dist_of_X_U_H_le","kind":"theorem","summary":"∀ G : Type u [inst : AddCommGroup G] [Finite G] [inst_2 : MeasurableSpace G] [MeasurableSinglet…","labels":[],"detail_key":"p56","name":"dist_of_X_U_H_le","module":"PFR.TorsionEndgame"},{"id":"n44797","layer":"formal","project":"p56","title":"entropy_of_W_le","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"entropy_of_W_le","module":"PFR.TorsionEndgame"},{"id":"n44798","layer":"formal","project":"p56","title":"entropy_of_Z_two_le","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"entropy_of_Z_two_le","module":"PFR.TorsionEndgame"},{"id":"n44799","layer":"formal","project":"p56","title":"k_eq_zero","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"k_eq_zero","module":"PFR.TorsionEndgame"},{"id":"n44800","layer":"formal","project":"p56","title":"mutual_information_le_t_12","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"mutual_information_le_t_12","module":"PFR.TorsionEndgame"},{"id":"n44801","layer":"formal","project":"p56","title":"mutual_information_le_t_13","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"mutual_information_le_t_13","module":"PFR.TorsionEndgame"},{"id":"n44802","layer":"formal","project":"p56","title":"mutual_information_le_t_23","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"mutual_information_le_t_23","module":"PFR.TorsionEndgame"},{"id":"n44803","layer":"formal","project":"p56","title":"mutual_of_W_Z_two_le","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"mutual_of_W_Z_two_le","module":"PFR.TorsionEndgame"},{"id":"n44804","layer":"formal","project":"p56","title":"sum_of_conditional_distance_le","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"sum_of_conditional_distance_le","module":"PFR.TorsionEndgame"},{"id":"n44805","layer":"formal","project":"p56","title":"sum_of_z_eq_zero","kind":"theorem","summary":"∀ G Ωₒ : Type u [inst : MeasurableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ωₒ] p : multiRe…","labels":[],"detail_key":"p56","name":"sum_of_z_eq_zero","module":"PFR.TorsionEndgame"},{"id":"n44806","layer":"formal","project":"p56","title":"torsion_PFR","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Finite G] m : Nat, GE.ge m 2 → (∀ (x : G), Eq (HSMul.hS…","labels":[],"detail_key":"p56","name":"torsion_PFR","module":"PFR.TorsionEndgame"},{"id":"n44807","layer":"formal","project":"p56","title":"torsion_PFR_conjecture_aux","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Finite G] m : Nat, GE.ge m 2 → (∀ (x : G), Eq (HSMul.hS…","labels":[],"detail_key":"p56","name":"torsion_PFR_conjecture_aux","module":"PFR.TorsionEndgame"},{"id":"n44808","layer":"formal","project":"p56","title":"PFR_projection","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [inst_1 : Module (ZMod 2) G] [Finite G] [inst_3 : Measur…","labels":[],"detail_key":"p56","name":"PFR_projection","module":"PFR.WeakPFR"},{"id":"n44809","layer":"formal","project":"p56","title":"PFR_projection'","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [inst_1 : Module (ZMod 2) G] [Finite G] [inst_3 : Measur…","labels":[],"detail_key":"p56","name":"PFR_projection'","module":"PFR.WeakPFR"},{"id":"n44810","layer":"formal","project":"p56","title":"app_ent_PFR","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [inst_1 : Module (ZMod 2) G] [Finite G] [inst_3 : Measur…","labels":[],"detail_key":"p56","name":"app_ent_PFR","module":"PFR.WeakPFR"},{"id":"n44811","layer":"formal","project":"p56","title":"single_fibres","kind":"theorem","summary":"∀ G : Type u_1 H : Type u_2 Ω : Type u_3 Ω' : Type u_4 [inst : AddCommGroup G] [Countable G] [i…","labels":[],"detail_key":"p56","name":"single_fibres","module":"PFR.WeakPFR"},{"id":"n44812","layer":"formal","project":"p56","title":"torsion_dist_shrinking","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [inst_1 : MeasurableSpace G] [MeasurableSingletonClass G…","labels":[],"detail_key":"p56","name":"torsion_dist_shrinking","module":"PFR.WeakPFR"},{"id":"n44813","layer":"formal","project":"p56","title":"torsion_free_doubling","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [inst_1 : MeasurableSpace G] [MeasurableSingletonClass G…","labels":[],"detail_key":"p56","name":"torsion_free_doubling","module":"PFR.WeakPFR"},{"id":"n44814","layer":"formal","project":"p56","title":"weak_PFR","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Module.Free Int G] [Module.Finite Int G] [Countable G]…","labels":[],"detail_key":"p56","name":"weak_PFR","module":"PFR.WeakPFR"},{"id":"n44815","layer":"formal","project":"p56","title":"weak_PFR_asymm","kind":"theorem","summary":"∀ G : Type u_1 [inst : AddCommGroup G] [Module.Free Int G] [Module.Finite Int G] [Countable G]…","labels":[],"detail_key":"p56","name":"weak_PFR_asymm","module":"PFR.WeakPFR"},{"id":"n44816","layer":"formal","project":"p56","title":"weak_PFR_int","kind":"theorem","summary":"∀ G : Type u_2 [inst : AddCommGroup G] [Module.Free Int G] [Module.Finite Int G] A : Set G [A_f…","labels":[],"detail_key":"p56","name":"weak_PFR_int","module":"PFR.WeakPFR"},{"id":"n44817","layer":"informal","project":"p57","title":"SpherePacking.balls","kind":"definition","summary":"Given a set X \\subset R^d and a real number r > 0 (known as the \\emphseparation radius) such th…","labels":["SpherePacking.balls"],"detail_key":"p57"},{"id":"n44818","layer":"informal","project":"p57","title":"SpherePacking","kind":"remark","summary":"Note that","labels":["SpherePacking"],"detail_key":"p57"},{"id":"n44819","layer":"informal","project":"p57","title":"SpherePacking.finiteDensity","kind":"definition","summary":"The \\emphfinite density of a packing P is defined as \\[ \\Delta_P(R):=\\fracVol(P\\cap B_d(0,R))Vo…","labels":["SpherePacking.finiteDensity"],"detail_key":"p57"},{"id":"n44820","layer":"informal","project":"p57","title":"SpherePacking.density","kind":"definition","summary":"We define the \\emphdensity of a packing P as the limit superior \\[ \\Delta_P:=\\limsup\\limits_R\\t…","labels":["SpherePacking.density"],"detail_key":"p57"},{"id":"n44821","layer":"informal","project":"p57","title":"SpherePackingConstant","kind":"definition","summary":"The \\emphsphere packing constant is defined as supremum of packing densities over all possible…","labels":["SpherePackingConstant"],"detail_key":"p57"},{"id":"n44822","layer":"informal","project":"p57","title":"SpherePacking.scale","kind":"definition","summary":"Given a sphere packing P(X) with separation radius r, we defined the \\emphscaled packing with r…","labels":["SpherePacking.scale"],"detail_key":"p57"},{"id":"n44823","layer":"informal","project":"p57","title":"SpherePacking.scale_finiteDensity","kind":"lemma","summary":"Let P(X) be a sphere packing and c a positive real number. Then, for all R > 0, \\[ \\Delta_P(cX)…","labels":["SpherePacking.scale_finiteDensity"],"detail_key":"p57"},{"id":"n44824","layer":"informal","project":"p57","title":"The proof follows by direct computation: \\[ \\Delta_P(cX)(cR) = \\fracVol\\!\\left(P(cX) \\cap…","kind":"proof","summary":"The proof follows by direct computation: \\[ \\Delta_P(cX)(cR) = \\fracVol\\!\\left(P(cX) \\cap B_d(0…","labels":[],"detail_key":"p57"},{"id":"n44825","layer":"informal","project":"p57","title":"SpherePacking.scale_density","kind":"lemma","summary":"Let P(X) be a sphere packing and c a positive real number. Then, the density of the scaled pack…","labels":["SpherePacking.scale_density"],"detail_key":"p57"},{"id":"n44826","layer":"informal","project":"p57","title":"One can show, using relatively unsophisticated real analysis, that \\[ \\limsup_R \\to \\inft…","kind":"proof","summary":"One can show, using relatively unsophisticated real analysis, that \\[ \\limsup_R \\to \\infty \\Del…","labels":[],"detail_key":"p57"},{"id":"n44827","layer":"informal","project":"p57","title":"SpherePacking.constant_eq_constant_normalized","kind":"lemma","summary":"\\[ \\Delta_d = \\sup\\limits_\\substackP\\subset R^d \\\\ \\textsphere packing \\\\ \\textsep.~rad. = 1 \\D…","labels":["SpherePacking.constant_eq_constant_normalized"],"detail_key":"p57"},{"id":"n44828","layer":"informal","project":"p57","title":"That the","kind":"proof","summary":"That the","labels":[],"detail_key":"p57"},{"id":"n44829","layer":"informal","project":"p57","title":"IsZLattice","kind":"definition","summary":"We say that an additive subgroup \\Lambda \\leq R^d is a \\emphlattice if it is discrete and its R…","labels":["IsZLattice"],"detail_key":"p57"},{"id":"n44830","layer":"informal","project":"p57","title":"def:dual-lattice","kind":"definition","summary":"The \\emphdual lattice of a lattice \\Lambda is the set \\[ \\Lambda^* := \\left\\ v \\in R^d \\; \\midd…","labels":["def:dual-lattice"],"detail_key":"p57"},{"id":"n44831","layer":"informal","project":"p57","title":"PeriodicSpherePacking","kind":"definition","summary":"We say that a sphere packing P(X) is (\\Lambda-)\\emphperiodic if there exists a lattice \\Lambda…","labels":["PeriodicSpherePacking"],"detail_key":"p57"},{"id":"n44832","layer":"informal","project":"p57","title":"def:Periodic-sphere-packing-constant","kind":"definition","summary":"The periodic sphere packing constant is defined to be \\Delta_d^\\textperiodic := \\sup_\\substackP…","labels":["def:Periodic-sphere-packing-constant"],"detail_key":"p57"},{"id":"n44833","layer":"informal","project":"p57","title":"thm:periodic-packing-optimal","kind":"theorem","summary":"For all d, the periodic sphere packing constant in R^d is equal to the sphere packing constant…","labels":["thm:periodic-packing-optimal"],"detail_key":"p57"},{"id":"n44834","layer":"informal","project":"p57","title":"The following proof was written by Junyan Xu in the thread for \\hrefhttps://github.com/th…","kind":"proof","summary":"The following proof was written by Junyan Xu in the thread for \\hrefhttps://github.com/thefunda…","labels":[],"detail_key":"p57"},{"id":"n44835","layer":"informal","project":"p57","title":"theorem:CE_Main","kind":"theorem","summary":"All \\emphperiodic packing P \\subseteq R^8 has density satisfying \\Delta_P \\leq \\Delta_E_8 = \\fr…","labels":["theorem:CE_Main"],"detail_key":"p57"},{"id":"n44836","layer":"informal","project":"p57","title":"Directly follows from \\Crefthm:Cohn-Elkies-general applied to the function f(x)=g(x/\\sqrt…","kind":"proof","summary":"Directly follows from \\Crefthm:Cohn-Elkies-general applied to the function f(x)=g(x/\\sqrt2) of…","labels":[],"detail_key":"p57"},{"id":"n44837","layer":"informal","project":"p57","title":"corollary:upper-bound-E8","kind":"corollary","summary":"All packing P \\subseteq R^8 has density satisfying \\Delta_P \\leq \\Delta_E_8.","labels":["corollary:upper-bound-E8"],"detail_key":"p57"},{"id":"n44838","layer":"informal","project":"p57","title":"This is a direct consequence of Theorem \\crefthm:periodic-packing-optimal and \\creftheore…","kind":"proof","summary":"This is a direct consequence of Theorem \\crefthm:periodic-packing-optimal and \\creftheorem:CE_M…","labels":[],"detail_key":"p57"},{"id":"n44839","layer":"informal","project":"p57","title":"MainTheorem","kind":"corollary","summary":"\\Delta_8 = \\Delta_E_8.","labels":["MainTheorem"],"detail_key":"p57"},{"id":"n44840","layer":"informal","project":"p57","title":"By definition, \\Delta_E_8 \\leq \\Delta_8, while \\crefcorollary:upper-bound-E8 shows \\Delta…","kind":"proof","summary":"By definition, \\Delta_E_8 \\leq \\Delta_8, while \\crefcorollary:upper-bound-E8 shows \\Delta_8 = \\…","labels":[],"detail_key":"p57"},{"id":"n44841","layer":"informal","project":"p57","title":"lemma:sp-finite-density-bound","kind":"lemma","summary":"For any R > 0, \\[ \\left|X \\cap B_d\\left(R - \\fracr2\\right)\\right| \\cdot \\fracVol\\left(B_d\\left(…","labels":["lemma:sp-finite-density-bound"],"detail_key":"p57"},{"id":"n44842","layer":"informal","project":"p57","title":"The high level idea is to prove that P \\cap B_d(R) = \\left(\\bigcup_x \\in X B_d\\left(x, \\f…","kind":"proof","summary":"The high level idea is to prove that P \\cap B_d(R) = \\left(\\bigcup_x \\in X B_d\\left(x, \\fracr2\\…","labels":[],"detail_key":"p57"},{"id":"n44843","layer":"informal","project":"p57","title":"lemma:lattice-points-bound","kind":"lemma","summary":"For all R, we have the following inequality relating the number of lattice points from \\Lambda…","labels":["lemma:lattice-points-bound"],"detail_key":"p57"},{"id":"n44844","layer":"informal","project":"p57","title":"For the first inequality, it suffices to prove that B_d(R - L) \\subseteq \\bigcup_x \\in \\L…","kind":"proof","summary":"For the first inequality, it suffices to prove that B_d(R - L) \\subseteq \\bigcup_x \\in \\Lambda…","labels":[],"detail_key":"p57"},{"id":"n44845","layer":"informal","project":"p57","title":"lemma:periodic-points-bounds","kind":"lemma","summary":"For all R, we have the following inequality relating the number of points from X (periodic w.r.…","labels":["lemma:periodic-points-bounds"],"detail_key":"p57"},{"id":"n44846","layer":"informal","project":"p57","title":"For the first inequality, we notice that \\bigcup_x \\in \\Lambda \\cap B_d(R - L) (x + D) \\s…","kind":"proof","summary":"For the first inequality, we notice that \\bigcup_x \\in \\Lambda \\cap B_d(R - L) (x + D) \\subsete…","labels":[],"detail_key":"p57"},{"id":"n44847","layer":"informal","project":"p57","title":"lemma:volume-ball-ratio-limit","kind":"lemma","summary":"For any constant C > 0, we have \\[ \\lim_R \\to \\infty \\fracVol(B_d(R))Vol(B_d(R + C)) = 1 \\]","labels":["lemma:volume-ball-ratio-limit"],"detail_key":"p57"},{"id":"n44848","layer":"informal","project":"p57","title":"Write out the formula for volume of a ball and simplify. More specifically, we have Vol(B…","kind":"proof","summary":"Write out the formula for volume of a ball and simplify. More specifically, we have Vol(B_d(R))…","labels":[],"detail_key":"p57"},{"id":"n44849","layer":"informal","project":"p57","title":"theorem:psp-density","kind":"theorem","summary":"For a periodic sphere packing P = P(X) with centers X periodic to the lattice \\Lambda and separ…","labels":["theorem:psp-density"],"detail_key":"p57"},{"id":"n44850","layer":"informal","project":"p57","title":"Fix any fundamental domain D (induced by any basis) of the lattice \\Lambda. Combining \\cr…","kind":"proof","summary":"Fix any fundamental domain D (induced by any basis) of the lattice \\Lambda. Combining \\creflemm…","labels":[],"detail_key":"p57"},{"id":"n44851","layer":"informal","project":"p57","title":"E8-Set","kind":"definition","summary":"(E_8-lattice, Definition 1) We define the \\emphE_8-lattice (as a subset of R^8) to be \\Lambda_8…","labels":["E8-Set"],"detail_key":"p57"},{"id":"n44852","layer":"informal","project":"p57","title":"E8-Matrix","kind":"definition","summary":"(E_8-lattice, Definition 2) We define the \\emphE_8 basis vectors to be the set of vectors \\[ B_…","labels":["E8-Matrix"],"detail_key":"p57"},{"id":"n44853","layer":"informal","project":"p57","title":"E8-defs-equivalent","kind":"theorem","summary":"The two definitions above coincide, i.e. \\Lambda_8 = span_Z(B_8).","labels":["E8-defs-equivalent"],"detail_key":"p57"},{"id":"n44854","layer":"informal","project":"p57","title":"We prove each side contains the other side. For a vector \\vecv \\in \\Lambda_8 \\subseteq R^…","kind":"proof","summary":"We prove each side contains the other side. For a vector \\vecv \\in \\Lambda_8 \\subseteq R^8, we…","labels":[],"detail_key":"p57"},{"id":"n44855","layer":"informal","project":"p57","title":"E8-is-basis","kind":"lemma","summary":"B_8 is a R-basis of R^8.","labels":["E8-is-basis"],"detail_key":"p57"},{"id":"n44856","layer":"informal","project":"p57","title":"It suffices to prove that B_8 \\in GL_8(R). We prove this by explicitly defining the inver…","kind":"proof","summary":"It suffices to prove that B_8 \\in GL_8(R). We prove this by explicitly defining the inverse mat…","labels":[],"detail_key":"p57"},{"id":"n44857","layer":"informal","project":"p57","title":"E8-Lattice","kind":"lemma","summary":"\\Lambda_8 is an additive subgroup of R^8.","labels":["E8-Lattice"],"detail_key":"p57"},{"id":"n44858","layer":"informal","project":"p57","title":"Trivially follows from that \\Lambda_8 \\subseteq R^8 is the Z-span of B_8 and hence an add…","kind":"proof","summary":"Trivially follows from that \\Lambda_8 \\subseteq R^8 is the Z-span of B_8 and hence an additive…","labels":[],"detail_key":"p57"},{"id":"n44859","layer":"informal","project":"p57","title":"E8-vector-norms","kind":"lemma","summary":"All vectors in \\Lambda_8 have norm of the form \\sqrt2n, where n is a nonnegative integer.","labels":["E8-vector-norms"],"detail_key":"p57"},{"id":"n44860","layer":"informal","project":"p57","title":"Writing \\vecv = \\sum_i c_iB_8^i, we have \\|v\\|^2 = \\sum_i \\sum_j c_ic_j (B_8^i \\cdot B_8^…","kind":"proof","summary":"Writing \\vecv = \\sum_i c_iB_8^i, we have \\|v\\|^2 = \\sum_i \\sum_j c_ic_j (B_8^i \\cdot B_8^j). Co…","labels":[],"detail_key":"p57"},{"id":"n44861","layer":"informal","project":"p57","title":"instDiscreteE8Lattice","kind":"lemma","summary":"c\\Lambda_8 is discrete, i.e. that the subspace topology induced by its inclusion into R^8 is th…","labels":["instDiscreteE8Lattice"],"detail_key":"p57"},{"id":"n44862","layer":"informal","project":"p57","title":"Since \\Lambda_8 is a topological group and + is continuous, it suffices to prove that \\0\\…","kind":"proof","summary":"Since \\Lambda_8 is a topological group and + is continuous, it suffices to prove that \\0\\ is op…","labels":[],"detail_key":"p57"},{"id":"n44863","layer":"informal","project":"p57","title":"instLatticeE8","kind":"lemma","summary":"c\\Lambda_8 is a Z-lattice, i.e. it is discrete and spans R^8 over R.","labels":["instLatticeE8"],"detail_key":"p57"},{"id":"n44864","layer":"informal","project":"p57","title":"The first part is by \\crefinstDiscreteE8Lattice, and the second part follows from that B_…","kind":"proof","summary":"The first part is by \\crefinstDiscreteE8Lattice, and the second part follows from that B_8 is a…","labels":[],"detail_key":"p57"},{"id":"n44865","layer":"informal","project":"p57","title":"E8Packing","kind":"definition","summary":"The \\emphE_8 sphere packing is the (periodic) sphere packing with separation \\sqrt2, whose set…","labels":["E8Packing"],"detail_key":"p57"},{"id":"n44866","layer":"informal","project":"p57","title":"E8Packing-covol","kind":"lemma","summary":"Vol\\!\\left(\\Lambda_8\\right) = Covol(R^8 / \\Lambda_8) = 1.","labels":["E8Packing-covol"],"detail_key":"p57"},{"id":"n44867","layer":"informal","project":"p57","title":"\\colorredIn theory this should follow directly from \\det(\\Lambda_8) = 1, but Lean hates m…","kind":"proof","summary":"\\colorredIn theory this should follow directly from \\det(\\Lambda_8) = 1, but Lean hates me and…","labels":[],"detail_key":"p57"},{"id":"n44868","layer":"informal","project":"p57","title":"E8Packing-density","kind":"theorem","summary":"We have \\Delta_P(E_8) = \\frac\\pi^4384.","labels":["E8Packing-density"],"detail_key":"p57"},{"id":"n44869","layer":"informal","project":"p57","title":"By \\creftheorem:psp-density, we have \\Delta_P(E_8) = |E_8 / E_8| \\cdot \\fracVol\\!\\left(B_…","kind":"proof","summary":"By \\creftheorem:psp-density, we have \\Delta_P(E_8) = |E_8 / E_8| \\cdot \\fracVol\\!\\left(B_8(\\sqr…","labels":[],"detail_key":"p57"},{"id":"n44870","layer":"informal","project":"p57","title":"def:Fourier-Transform","kind":"definition","summary":"The Fourier transform of an L^1-function f:R^d\\toC is defined as \\[ F(f)(y) = \\widehatf(y) := \\…","labels":["def:Fourier-Transform"],"detail_key":"p57"},{"id":"n44871","layer":"informal","project":"p57","title":"lemma:Gaussian-Fourier","kind":"lemma","summary":"F(e^\\pi i \\|x\\|^2 z)(y) = z^-4\\,e^\\pi i \\|y\\|^2 \\,(\\frac-1z) .","labels":["lemma:Gaussian-Fourier"],"detail_key":"p57"},{"id":"n44872","layer":"informal","project":"p57","title":"\\colorredFill in proof.","kind":"proof","summary":"\\colorredFill in proof.","labels":[],"detail_key":"p57"},{"id":"n44873","layer":"informal","project":"p57","title":"def:Schwartz-Space","kind":"definition","summary":"A C^\\infty~fun","labels":["def:Schwartz-Space"],"detail_key":"p57"},{"id":"n44874","layer":"informal","project":"p57","title":"lemma:Fourier-transform-is-automorphism","kind":"lemma","summary":"The Fourier transform is a continuous, linear automorphism of the space of Schwartz functions.","labels":["lemma:Fourier-transform-is-automorphism"],"detail_key":"p57"},{"id":"n44875","layer":"informal","project":"p57","title":"We do not","kind":"proof","summary":"We do not","labels":[],"detail_key":"p57"},{"id":"n44876","layer":"informal","project":"p57","title":"lemma:inv-power-summable","kind":"lemma","summary":"Let X \\subset R^d be a set of sphere packing centres of separation 1 that is periodic with some…","labels":["lemma:inv-power-summable"],"detail_key":"p57"},{"id":"n44877","layer":"informal","project":"p57","title":"First, note that it does not matter how we number the (countably many) elements of the di…","kind":"proof","summary":"First, note that it does not matter how we number the (countably many) elements of the discrete…","labels":[],"detail_key":"p57"},{"id":"n44878","layer":"informal","project":"p57","title":"lemma:Schwartz-summable","kind":"lemma","summary":"Let f : R^d \\to C be a Schwartz function and let X \\subset R^d be periodic with respect to some…","labels":["lemma:Schwartz-summable"],"detail_key":"p57"},{"id":"n44879","layer":"informal","project":"p57","title":"Without loss of generality, assume that 0 \\notin X: if 0 \\in X, then we can add the f(0)…","kind":"proof","summary":"Without loss of generality, assume that 0 \\notin X: if 0 \\in X, then we can add the f(0) term t…","labels":[],"detail_key":"p57"},{"id":"n44880","layer":"informal","project":"p57","title":"Poisson summation formula","kind":"theorem","summary":"[Poisson summation formula] Let \\Lambda be a lattice in R^d, and let f:R^d\\toR be a Schwartz fu…","labels":["thm:Poisson-summation-formula"],"detail_key":"p57"},{"id":"n44881","layer":"informal","project":"p57","title":"One possible proof would be by induction on d. However, there are numerous nuances involv…","kind":"proof","summary":"One possible proof would be by induction on d. However, there are numerous nuances involved, pa…","labels":[],"detail_key":"p57"},{"id":"n44882","layer":"informal","project":"p57","title":"thm:smooth-fast-decay-schwartz","kind":"theorem","summary":"Assume f : R\\to C is smooth on [0, \\infty) and for all k, n \\in N, there exists C \\in R such th…","labels":["thm:smooth-fast-decay-schwartz"],"detail_key":"p57"},{"id":"n44883","layer":"informal","project":"p57","title":"Cohn--Elkies \\citeElkiesCohn","kind":"theorem","summary":"[Cohn--Elkies \\citeElkiesCohn] Let X\\subsetR^d be a discrete subset such that \\|x-y\\|\\geq 1 for…","labels":["thm:Cohn-Elkies-periodic","eqn:Cohn-Elkies-condition-1","eqn:Cohn-Elkies-condition-2"],"detail_key":"p57"},{"id":"n44884","layer":"informal","project":"p57","title":"eqn:_sharp_X_1","kind":"proof","summary":"Here we reproduce the proof given in \\citeElkiesCohn. The inequality \\sharp (X/\\Lambda)\\cdot f(…","labels":["eqn:_sharp_X_1","eqn:_sharp_X_2"],"detail_key":"p57"},{"id":"n44885","layer":"informal","project":"p57","title":"Cohn--Elkies \\citeElkiesCohn","kind":"theorem","summary":"[Cohn--Elkies \\citeElkiesCohn] Let f:R^d\\toR be a Schwartz function that is not identically zer…","labels":["thm:Cohn-Elkies-general"],"detail_key":"p57"},{"id":"n44886","layer":"informal","project":"p57","title":"periodic_constant_eq_constant","kind":"proof","summary":"The result follows immediately from Theorem~\\refthm:periodic-packing-optimal and \\Crefthm:Cohn-…","labels":[],"detail_key":"p57"},{"id":"n44887","layer":"informal","project":"p57","title":"thm:g","kind":"theorem","summary":"There exists a radial Schwartz function g:R^8\\toR which satisfies: g(x)&\\leq 0\\mbox for \\|x\\|\\g…","labels":["thm:g","eqn:g1","eqn:g2","eqn:g3"],"detail_key":"p57"},{"id":"n44888","layer":"informal","project":"p57","title":"def:Gamma-1-Action","kind":"lemma","summary":"The modular group \\Gamma_1:=SL_2(Z) acts on \\mathfrakH by linear fractional transformations \\le…","labels":["def:Gamma-1-Action"],"detail_key":"p57"},{"id":"n44889","layer":"informal","project":"p57","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p57"},{"id":"n44890","layer":"informal","project":"p57","title":"def:level-N-princ-cong-subgp","kind":"definition","summary":"The \\emphlevel N principal congruence subgroup of \\Gamma_1 is \\Gamma(N):=\\left\\\\left.\\left( a&b…","labels":["def:level-N-princ-cong-subgp"],"detail_key":"p57"},{"id":"n44891","layer":"informal","project":"p57","title":"def:congruence-subgroup","kind":"definition","summary":"A subgroup \\Gamma\\subset\\Gamma_1 is called a \\emphcongruence subgroup if \\Gamma(N)\\subset\\Gamma…","labels":["def:congruence-subgroup"],"detail_key":"p57"},{"id":"n44892","layer":"informal","project":"p57","title":"def:Gamma-generators","kind":"definition","summary":"Define the matrices \\[ S = 0 & -1 \\\\ 1 & 0 \\in \\Gamma_1, T = 1 & 1 \\\\ 0 & 1 \\in \\Gamma_1, \\alph…","labels":["def:Gamma-generators"],"detail_key":"p57"},{"id":"n44893","layer":"informal","project":"p57","title":"lemma:Gamma-1-generators","kind":"lemma","summary":"We have \\Gamma(1) = \\langle S, T, -I \\rangle.","labels":["lemma:Gamma-1-generators"],"detail_key":"p57"},{"id":"n44894","layer":"informal","project":"p57","title":"See~\\cite[Exercise 1.1.1]first course.","kind":"proof","summary":"See~\\cite[Exercise 1.1.1]first course.","labels":[],"detail_key":"p57"},{"id":"n44895","layer":"informal","project":"p57","title":"lemma:Gamma-2-generators","kind":"lemma","summary":"We have \\Gamma(2) = \\langle \\alpha, \\beta, -I \\rangle.","labels":["lemma:Gamma-2-generators"],"detail_key":"p57"},{"id":"n44896","layer":"informal","project":"p57","title":"See~\\cite[Exercise 1.2.4]first course.","kind":"proof","summary":"See~\\cite[Exercise 1.2.4]first course.","labels":[],"detail_key":"p57"},{"id":"n44897","layer":"informal","project":"p57","title":"def:automorphy-factor","kind":"definition","summary":"The \\emphautomorphy factor of weight k is defined as j_k(z,\\left( a&b\\\\c&d \\right)):=(cz+d)^-k.","labels":["def:automorphy-factor"],"detail_key":"p57"},{"id":"n44898","layer":"informal","project":"p57","title":"lemma:automorphy-factor-chain-rule","kind":"lemma","summary":"The automorphy factor satisfies the \\emphchain rule j_k(z,\\gamma_1\\gamma_2)=j_k(z,\\gamma_1)\\,j_…","labels":["lemma:automorphy-factor-chain-rule"],"detail_key":"p57"},{"id":"n44899","layer":"informal","project":"p57","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p57"},{"id":"n44900","layer":"informal","project":"p57","title":"def:slash-operator","kind":"definition","summary":"Let F be a function on \\mathfrakH and \\gamma\\inSL_2(Z). Then the \\emphslash operator acts on F…","labels":["def:slash-operator"],"detail_key":"p57"},{"id":"n44901","layer":"informal","project":"p57","title":"lemma:slash-operator-chain-rule","kind":"lemma","summary":"The chain rule implies F|_k\\gamma_1\\gamma_2=(F|_k\\gamma_1)|_k\\gamma_2.","labels":["lemma:slash-operator-chain-rule"],"detail_key":"p57"},{"id":"n44902","layer":"informal","project":"p57","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p57"},{"id":"n44903","layer":"informal","project":"p57","title":"lemma:slash-negI-even-weight","kind":"lemma","summary":"For even k, F|_k(-I) = F.","labels":["lemma:slash-negI-even-weight"],"detail_key":"p57"},{"id":"n44904","layer":"informal","project":"p57","title":"Follows from the definition of the slash operator: (F|_k(-I))(z) = (-1)^-kF((-I)z) = F(z).","kind":"proof","summary":"Follows from the definition of the slash operator: (F|_k(-I))(z) = (-1)^-kF((-I)z) = F(z).","labels":[],"detail_key":"p57"},{"id":"n44905","layer":"informal","project":"p57","title":"def:Mk","kind":"definition","summary":"Let \\Gamma denote a subgroup of SL_2(Z), then a modular form of level \\Gamma and weight k \\in Z…","labels":["def:Mk"],"detail_key":"p57"},{"id":"n44906","layer":"informal","project":"p57","title":"def:Ek","kind":"definition","summary":"For an even integer k\\geq 4 we define the \\emphweight k Eisenstein series as E_k(z):=\\frac12\\su…","labels":["def:Ek","eqn:Ek-definition"],"detail_key":"p57"},{"id":"n44907","layer":"informal","project":"p57","title":"lemma:Ek-is-modular-form","kind":"lemma","summary":"For all k, E_k\\in M_k(\\Gamma_1). Especially, we have E_k \\left(-\\frac1z\\right) = z^k E_k(z).","labels":["lemma:Ek-is-modular-form","eqn:Ek-trans-S"],"detail_key":"p57"},{"id":"n44908","layer":"informal","project":"p57","title":"This follows from the fact that the sum converges absolutely. Now apply slash operator wi…","kind":"proof","summary":"This follows from the fact that the sum converges absolutely. Now apply slash operator with \\ga…","labels":[],"detail_key":"p57"},{"id":"n44909","layer":"informal","project":"p57","title":"lemma:mod_form_poly_growth","kind":"lemma","summary":": Let \\Gamma be a finite index subgroup of SL_2(Z) and f \\in M_k(\\Gamma) be a modular form of w…","labels":["lemma:mod_form_poly_growth"],"detail_key":"p57"},{"id":"n44910","layer":"informal","project":"p57","title":"Note that the assumption on the polynomial growth holds when f is a holomorphic modular f…","kind":"proof","summary":"Note that the assumption on the polynomial growth holds when f is a holomorphic modular form, w…","labels":[],"detail_key":"p57"},{"id":"n44911","layer":"informal","project":"p57","title":"lemma:Ek-Fourier","kind":"lemma","summary":"The Eisenstein series possesses the Fourier expansion E_k(z)=1+\\frac2\\zeta(1-k)\\sum_n=1^\\infty…","labels":["lemma:Ek-Fourier","eqn:Ek-Fourier"],"detail_key":"p57"},{"id":"n44912","layer":"informal","project":"p57","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p57"},{"id":"n44913","layer":"informal","project":"p57","title":"def:disc-definition","kind":"definition","summary":"The \\emphdiscriminant form \\Delta(z) is given by \\Delta(z) = e^2 \\pi i z \\prod_n \\ge 1 (1 - e^2…","labels":["def:disc-definition","eqn:disc-definition"],"detail_key":"p57"},{"id":"n44914","layer":"informal","project":"p57","title":"def:E2","kind":"definition","summary":"We set E_2(z):= 1-24\\sum_n=1^\\infty \\sigma_1(n)\\,e^2\\pi i n z.","labels":["def:E2","eqn:E2"],"detail_key":"p57"},{"id":"n44915","layer":"informal","project":"p57","title":"lemma:E2-transform-S","kind":"lemma","summary":"This function is not modular, however it satisfies z^-2\\,E_2\\left(-\\frac1z\\right) = E_2(z) -\\fr…","labels":["lemma:E2-transform-S","eqn:E2-S-transform"],"detail_key":"p57"},{"id":"n44916","layer":"informal","project":"p57","title":"This is exercise 1.2.8 of \\citefirst course.","kind":"proof","summary":"This is exercise 1.2.8 of \\citefirst course.","labels":[],"detail_key":"p57"},{"id":"n44917","layer":"informal","project":"p57","title":"lemma:E2-transform-general","kind":"lemma","summary":"(cz + d)^-2 E_2\\left(\\fracaz + bcx + d\\right) = E_2(z) - \\frac6ic\\pi (cz + d), \\quad a & b \\\\ c…","labels":["lemma:E2-transform-general","eqn:E2-transform-general"],"detail_key":"p57"},{"id":"n44918","layer":"informal","project":"p57","title":"Use the fact that SL_2(Z) is generated by S and T. Then \\eqrefeqn:E2-transform-general fo…","kind":"proof","summary":"Use the fact that SL_2(Z) is generated by S and T. Then \\eqrefeqn:E2-transform-general follows…","labels":[],"detail_key":"p57"},{"id":"n44919","layer":"informal","project":"p57","title":"def:dedekind_eta","kind":"definition","summary":"The Dedekind eta function is defined as \\eta(z) = q^1/24 \\prod_n \\ge 1 (1 - q^n) where q = e^2\\…","labels":["def:dedekind_eta"],"detail_key":"p57"},{"id":"n44920","layer":"informal","project":"p57","title":"lemma:dedekind_eta_transformation","kind":"lemma","summary":"The Dedekind eta function transforms as \\eta\\left(-\\frac1z\\right) = \\sqrt-iz \\eta(z).","labels":["lemma:dedekind_eta_transformation"],"detail_key":"p57"},{"id":"n44921","layer":"informal","project":"p57","title":"Consider the logarithmic derivative of \\eta, which one can easily see is equal to \\frac\\p…","kind":"proof","summary":"Consider the logarithmic derivative of \\eta, which one can easily see is equal to \\frac\\pi i12…","labels":[],"detail_key":"p57"},{"id":"n44922","layer":"informal","project":"p57","title":"lemma:disc-cuspform","kind":"lemma","summary":"\\Delta(z) \\in M_12(\\Gamma_1). Especially, we have \\Delta\\left(-\\frac1z\\right) = z^12 \\Delta(z).…","labels":["lemma:disc-cuspform","eqn:disc-trans-S"],"detail_key":"p57"},{"id":"n44923","layer":"informal","project":"p57","title":"The fact that it is invariant under translation is clear from the definition, so we only…","kind":"proof","summary":"The fact that it is invariant under translation is clear from the definition, so we only need t…","labels":[],"detail_key":"p57"},{"id":"n44924","layer":"informal","project":"p57","title":"lemma:disc-E4E6","kind":"lemma","summary":"We have \\Delta(z) = (E_4^3-E_6^2)/1728.","labels":["lemma:disc-E4E6"],"detail_key":"p57"},{"id":"n44925","layer":"informal","project":"p57","title":"We only need to show its a cuspform, since once we have this, dividing the rhs by \\Delta…","kind":"proof","summary":"We only need to show its a cuspform, since once we have this, dividing the rhs by \\Delta would…","labels":[],"detail_key":"p57"},{"id":"n44926","layer":"informal","project":"p57","title":"cor:disc-pos","kind":"corollary","summary":"\\Delta(it) > 0 for all t > 0.","labels":["cor:disc-pos"],"detail_key":"p57"},{"id":"n44927","layer":"informal","project":"p57","title":"By \\refdef:disc-definition, we have \\Delta(it) = e^-2 \\pi t \\prod_n \\ge 1 (1 - e^-2 \\pi n…","kind":"proof","summary":"By \\refdef:disc-definition, we have \\Delta(it) = e^-2 \\pi t \\prod_n \\ge 1 (1 - e^-2 \\pi n t)^24…","labels":[],"detail_key":"p57"},{"id":"n44928","layer":"informal","project":"p57","title":"cor:disc-nonvanishing","kind":"corollary","summary":"\\Delta(z) \\neq 0 for all z \\in \\mathfrakH.","labels":["cor:disc-nonvanishing"],"detail_key":"p57"},{"id":"n44929","layer":"informal","project":"p57","title":"This follows from the product formula.","kind":"proof","summary":"This follows from the product formula.","labels":[],"detail_key":"p57"},{"id":"n44930","layer":"informal","project":"p57","title":"thm:nonpos_wt","kind":"theorem","summary":"Let k \\in Z with k < 0. Then M_k(\\Gamma_1) = \\0\\ and moreover \\dim M_0(\\Gamma(1)) = 1.","labels":["thm:nonpos_wt"],"detail_key":"p57"},{"id":"n44931","layer":"informal","project":"p57","title":"The proof makes use of the maximum modulus principle, as its already been formalised we s…","kind":"proof","summary":"The proof makes use of the maximum modulus principle, as its already been formalised we skip th…","labels":[],"detail_key":"p57"},{"id":"n44932","layer":"informal","project":"p57","title":"thm:lvl1_dims","kind":"theorem","summary":"Let k \\in Z with k \\ge 0 and even. Then \\dim M_k(\\Gamma_1) = \\lfloor k / 12 \\rfloor if k \\equiv…","labels":["thm:lvl1_dims"],"detail_key":"p57"},{"id":"n44933","layer":"informal","project":"p57","title":"First we note that for 2 < k we have \\dim(M_k(\\Gamma_1)) = 1 + \\dim S_k(\\Gamma_1). This f…","kind":"proof","summary":"First we note that for 2 < k we have \\dim(M_k(\\Gamma_1)) = 1 + \\dim S_k(\\Gamma_1). This follows…","labels":[],"detail_key":"p57"},{"id":"n44934","layer":"informal","project":"p57","title":"thm:dim-mf-general-level","kind":"theorem","summary":"Let \\Gamma be a congruence subgroup. Then M_k(\\Gamma) is finite-dimensional.","labels":["thm:dim-mf-general-level"],"detail_key":"p57"},{"id":"n44935","layer":"informal","project":"p57","title":"We know tha","kind":"proof","summary":"We know tha","labels":[],"detail_key":"p57"},{"id":"n44936","layer":"informal","project":"p57","title":"cor:dim-mf","kind":"corollary","summary":"We have \\dim M_2(SL_2(Z)) &= 0, \\\\ \\dim M_4(SL_2(Z)) &= 1, \\\\ \\dim M_6(SL_2(Z)) &= 1, \\\\ \\dim M…","labels":["cor:dim-mf","eqn:dimM2","eqn:dimM4","eqn:dimM6","eqn:dimM8","eqn:dimS4","eqn:dimS6","eqn:dimS8"],"detail_key":"p57"},{"id":"n44937","layer":"informal","project":"p57","title":"proof","kind":"proof","summary":"","labels":[],"detail_key":"p57"},{"id":"n44938","layer":"informal","project":"p57","title":"def:th00-th01-th10","kind":"definition","summary":"We define three different theta functions (so called ``Thetanullwerte'') as \\Theta_2(z) = \\thet…","labels":["def:th00-th01-th10"],"detail_key":"p57"},{"id":"n44939","layer":"informal","project":"p57","title":"def:H2-H3-H4","kind":"definition","summary":"Define H_2 = \\Theta_2^4, \\quad H_3 = \\Theta_3^4, \\quad H_4 = \\Theta_4^4.","labels":["def:H2-H3-H4","eqn:H2-H3-H4"],"detail_key":"p57"},{"id":"n44940","layer":"informal","project":"p57","title":"lemma:theta-transform-S-T","kind":"lemma","summary":"These elements act on the theta functions in the following way H_2 | S &= -H_4 \\\\ H_3 | S &= -H…","labels":["lemma:theta-transform-S-T","eqn:H2-transform-S","eqn:H3-transform-S","eqn:H4-transform-S","eqn:H2-transform-T","eqn:H3-transform-T","eqn:H4-transform-T"],"detail_key":"p57"},{"id":"n44941","layer":"informal","project":"p57","title":"eqn:jacobi2","kind":"proof","summary":"The last three identities easily follow from the definition. For example, \\eqrefeqn:H2-transfor…","labels":["eqn:jacobi2","eqn:Th2-as-jacobi2","eqn:Th3-as-jacobi2","eqn:Th4-as-jacobi2","eqn:jacobi2transform"],"detail_key":"p57"},{"id":"n44942","layer":"informal","project":"p57","title":"lemma:theta-slash-invariant","kind":"lemma","summary":"H_2, H_3, and H_4 are slash invariant under \\Gamma(2), i.e. for all \\gamma \\in \\Gamma(2) and i…","labels":["lemma:theta-slash-invariant"],"detail_key":"p57"},{"id":"n44943","layer":"informal","project":"p57","title":"eqn:matrix","kind":"proof","summary":"By \\creflemma:Gamma-2-generators and \\creflemma:slash-operator-chain-rule, it suffices to show…","labels":["eqn:matrix"],"detail_key":"p57"},{"id":"n44944","layer":"informal","project":"p57","title":"lemma:theta-bounded-im-infty","kind":"lemma","summary":"For all \\gamma \\in \\Gamma_1, H_2|_2 \\gamma, H_3|_2 \\gamma, and H_4|_2 \\gamma are holomorphic at…","labels":["lemma:theta-bounded-im-infty"],"detail_key":"p57"},{"id":"n44945","layer":"informal","project":"p57","title":"We want to show that for \\gamma \\in \\Gamma_1, \\|H_2|_2\\gamma(z)\\| is bounded as z \\in H \\…","kind":"proof","summary":"We want to show that for \\gamma \\in \\Gamma_1, \\|H_2|_2\\gamma(z)\\| is bounded as z \\in H \\to i\\i…","labels":[],"detail_key":"p57"},{"id":"n44946","layer":"informal","project":"p57","title":"lemma:theta-modular","kind":"lemma","summary":"H_2, H_3, and H_4 belong to M_2(\\Gamma(2)).","labels":["lemma:theta-modular"],"detail_key":"p57"},{"id":"n44947","layer":"informal","project":"p57","title":"From \\creflemma:theta-slash-invariant and \\creflemma:theta-bounded-im-infty, it remains o…","kind":"proof","summary":"From \\creflemma:theta-slash-invariant and \\creflemma:theta-bounded-im-infty, it remains ot prov…","labels":[],"detail_key":"p57"},{"id":"n44948","layer":"informal","project":"p57","title":"prop:H2-fourier","kind":"proposition","summary":"H_2 admits a Fourier series of the form H_2(z) = \\sum_n \\ge 1 c_H_2(n) e^\\pi i n z for some c_H…","labels":["prop:H2-fourier"],"detail_key":"p57"},{"id":"n44949","layer":"informal","project":"p57","title":"We have H_2(z) &= \\Theta_2(z)^4 \\\\ &= \\left(\\sum_n \\in Z e^\\pi i (n + \\frac12)^2 z\\right)…","kind":"proof","summary":"We have H_2(z) &= \\Theta_2(z)^4 \\\\ &= \\left(\\sum_n \\in Z e^\\pi i (n + \\frac12)^2 z\\right)^4 \\\\…","labels":[],"detail_key":"p57"},{"id":"n44950","layer":"informal","project":"p57","title":"prop:H3-fourier","kind":"proposition","summary":"H_3 admits a Fourier series of the form H_3(z) = \\sum_n \\ge 0 c_H_3(n) e^\\pi i n z for some c_H…","labels":["prop:H3-fourier"],"detail_key":"p57"},{"id":"n44951","layer":"informal","project":"p57","title":"We have H_3(z) = \\Theta_3(z)^4 = \\left(\\sum_n \\in Z e^\\pi i n^2 z\\right)^4 = \\left(1 + 2…","kind":"proof","summary":"We have H_3(z) = \\Theta_3(z)^4 = \\left(\\sum_n \\in Z e^\\pi i n^2 z\\right)^4 = \\left(1 + 2 \\sum_n…","labels":[],"detail_key":"p57"},{"id":"n44952","layer":"informal","project":"p57","title":"prop:H4-fourier","kind":"proposition","summary":"H_4 admits a Fourier series of the form H_4(z) = \\sum_n \\ge 0 c_H_4(n) e^\\pi i n z for some c_H…","labels":["prop:H4-fourier"],"detail_key":"p57"},{"id":"n44953","layer":"informal","project":"p57","title":"lemma:jacobi-identity","kind":"lemma","summary":"These three theta functions satisfy the \\emphJacobi identity H_2 + H_4 = H_3 \\Leftrightarrow \\T…","labels":["lemma:jacobi-identity","eqn:jacobi-identity"],"detail_key":"p57"},{"id":"n44954","layer":"informal","project":"p57","title":"Let f = (H_2 + H_4 - H_3)^2. Obviously, f is a modular form of weight 4 and level \\Gamma(…","kind":"proof","summary":"Let f = (H_2 + H_4 - H_3)^2. Obviously, f is a modular form of weight 4 and level \\Gamma(2). Ho…","labels":[],"detail_key":"p57"},{"id":"n44955","layer":"informal","project":"p57","title":"lemma:lv1-lv2-identities","kind":"lemma","summary":"We have E_4 &= \\frac12(H_2^2 + H_3^2 + H_4^2) = H_2^2 + H_2H_4 + H_4^2 \\\\ E_6 &= \\frac12 (H_2 +…","labels":["lemma:lv1-lv2-identities","eqn:e4theta","eqn:e6theta","eqn:disctheta"],"detail_key":"p57"},{"id":"n44956","layer":"informal","project":"p57","title":"We can prove these similarly as Lemma \\reflemma:jacobi-identity. Right hand sides of \\eqr…","kind":"proof","summary":"We can prove these similarly as Lemma \\reflemma:jacobi-identity. Right hand sides of \\eqrefeqn:…","labels":[],"detail_key":"p57"},{"id":"n44957","layer":"informal","project":"p57","title":"cor:theta-pos","kind":"corollary","summary":"H_2(it) and H_4(it) are positive for t > 0.","labels":["cor:theta-pos"],"detail_key":"p57"},{"id":"n44958","layer":"informal","project":"p57","title":"By the transformation law \\eqrefeqn:H2-transform-S, it is enough to prove the positivity…","kind":"proof","summary":"By the transformation law \\eqrefeqn:H2-transform-S, it is enough to prove the positivity for \\T…","labels":[],"detail_key":"p57"},{"id":"n44959","layer":"informal","project":"p57","title":"def:derivative","kind":"definition","summary":"Let F be a quasimodular form. We define the (normalized) derivative of F as F' = DF := \\frac12\\…","labels":["def:derivative","eqn:derivative"],"detail_key":"p57"},{"id":"n44960","layer":"informal","project":"p57","title":"lemma:der-q-series","kind":"lemma","summary":"We have an equality of operators D = q \\fracddq. In particular, the q-series of the derivative…","labels":["lemma:der-q-series"],"detail_key":"p57"},{"id":"n44961","layer":"informal","project":"p57","title":"Directly follows from the definition \\eqrefdef:derivative, where \\frac12 \\pi i\\fracddze^2…","kind":"proof","summary":"Directly follows from the definition \\eqrefdef:derivative, where \\frac12 \\pi i\\fracddze^2\\pi i…","labels":[],"detail_key":"p57"},{"id":"n44962","layer":"informal","project":"p57","title":"def:serre-der","kind":"definition","summary":"For k \\in R, define the weight k Serre derivative \\partial_k of a modular form F as \\partial_kF…","labels":["def:serre-der","eqn:serre-der"],"detail_key":"p57"},{"id":"n44963","layer":"informal","project":"p57","title":"thm:serre-der-equiv-action","kind":"theorem","summary":"Serre derivative \\partial_k is equivariant with the slash action of SL_2(Z) in the following se…","labels":["thm:serre-der-equiv-action"],"detail_key":"p57"},{"id":"n44964","layer":"informal","project":"p57","title":"Let G = \\partial_kF = F' - \\frack12E_2 F. From F \\in M_k(\\Gamma), we have (F|_k\\gamma)(z)…","kind":"proof","summary":"Let G = \\partial_kF = F' - \\frack12E_2 F. From F \\in M_k(\\Gamma), we have (F|_k\\gamma)(z) := (c…","labels":[],"detail_key":"p57"},{"id":"n44965","layer":"informal","project":"p57","title":"thm:serre-der-modularity","kind":"theorem","summary":"Let F be a modular form of weight k and level \\Gamma. Then, \\partial_kF is a modular form of we…","labels":["thm:serre-der-modularity"],"detail_key":"p57"},{"id":"n44966","layer":"informal","project":"p57","title":"Immediate from Theorem \\refthm:serre-der-equiv-action since F|_k\\gamma = F for all \\gamma…","kind":"proof","summary":"Immediate from Theorem \\refthm:serre-der-equiv-action since F|_k\\gamma = F for all \\gamma \\in \\…","labels":[],"detail_key":"p57"},{"id":"n44967","layer":"informal","project":"p57","title":"More generally, the following theorem holds: if F is a quasimodular form of weight k and…","kind":"remark","summary":"More generally, the following theorem holds: if F is a quasimodular form of weight k and depth…","labels":[],"detail_key":"p57"},{"id":"n44968","layer":"informal","project":"p57","title":"thm:ramanujan-formula","kind":"theorem","summary":"We have E_2' &= \\fracE_2^2 - E_412 \\\\ E_4' &= \\fracE_2 E_4 - E_63 \\\\ E_6' &= \\fracE_2 E_6 - E_4…","labels":["thm:ramanujan-formula","eqn:DE2","eqn:DE4","eqn:DE6"],"detail_key":"p57"},{"id":"n44969","layer":"informal","project":"p57","title":"eqn:SE2","kind":"proof","summary":"In terms of Serre derivatives, these are equivalent to \\partial_1E_2 &= -\\frac112 E_4 \\\\ \\parti…","labels":["eqn:SE2","eqn:SE4","eqn:SE6","eqn:DE2-transform","eqn:E2sq-transform"],"detail_key":"p57"},{"id":"n44970","layer":"informal","project":"p57","title":"cor:logder-disc-E2","kind":"corollary","summary":"\\Delta' = E_2 \\Delta.","labels":["cor:logder-disc-E2","eqn:logder-disc-E2"],"detail_key":"p57"},{"id":"n44971","layer":"informal","project":"p57","title":"By Ramanujan's formula \\eqrefeqn:DE4 and \\eqrefeqn:DE6, \\Delta' = \\frac3 E_4^2 E_4' - 2 E…","kind":"proof","summary":"By Ramanujan's formula \\eqrefeqn:DE4 and \\eqrefeqn:DE6, \\Delta' = \\frac3 E_4^2 E_4' - 2 E_6 E_6…","labels":[],"detail_key":"p57"},{"id":"n44972","layer":"informal","project":"p57","title":"prop:theta-der","kind":"proposition","summary":"We have H_2' &= \\frac16 (H_2^2 + 2 H_2 H_4 + E_2 H_2) \\\\ H_3' &= \\frac16 (H_2^2 - H_4^2 + E_2 H…","labels":["prop:theta-der","eqn:H2-der","eqn:H3-der","eqn:H4-der","eqn:H2-serre-der","eqn:H3-serre-der","eqn:H4-serre-der"],"detail_key":"p57"},{"id":"n44973","layer":"informal","project":"p57","title":"Equivalences are obvious from the definition of the Serre derivative. Define f_2, f_3, f_…","kind":"proof","summary":"Equivalences are obvious from the definition of the Serre derivative. Define f_2, f_3, f_4 be t…","labels":[],"detail_key":"p57"},{"id":"n44974","layer":"informal","project":"p57","title":"thm:serre-der-prod-rule","kind":"theorem","summary":"The Serre derivative satisfies the following product rule: for any quasimodular forms F and G,…","labels":["thm:serre-der-prod-rule"],"detail_key":"p57"},{"id":"n44975","layer":"informal","project":"p57","title":"It follows from the definition: \\partial_w_1 + w_2 (FG) &= (FG)' - \\fracw_1 + w_212 E_2 (…","kind":"proof","summary":"It follows from the definition: \\partial_w_1 + w_2 (FG) &= (FG)' - \\fracw_1 + w_212 E_2 (FG) \\\\…","labels":[],"detail_key":"p57"},{"id":"n44976","layer":"informal","project":"p57","title":"thm:anti-serre-der-pos","kind":"theorem","summary":"Let F be a holomorphic quasimodular cusp form with real Fourier coefficients. Assume that there…","labels":["thm:anti-serre-der-pos"],"detail_key":"p57"},{"id":"n44977","layer":"informal","project":"p57","title":"By \\eqrefeqn:logder-disc-E2, we have \\fracddt \\left( \\fracF(it)\\Delta(it)^\\frack12\\right)…","kind":"proof","summary":"By \\eqrefeqn:logder-disc-E2, we have \\fracddt \\left( \\fracF(it)\\Delta(it)^\\frack12\\right) &= (-…","labels":[],"detail_key":"p57"},{"id":"n44978","layer":"informal","project":"p57","title":"def:phi4-phi2-phi0","kind":"definition","summary":"\\phi_-4 &:= \\fracE_4^2\\Delta \\\\ \\phi_-2 &:= \\fracE_4(E_2 E_4 - E_6)\\Delta \\\\ \\phi_0 &:= \\frac(E…","labels":["def:phi4-phi2-phi0","eqn:_def_phi4","eqn:_def_phi2","eqn:_def_phi0"],"detail_key":"p57"},{"id":"n44979","layer":"informal","project":"p57","title":"lemma:phi0-transform","kind":"lemma","summary":"We have \\phi_0(z + 1) &= \\phi_0(z) \\\\ \\phi_0\\left(-\\frac1z\\right) &= \\phi_0(z)-\\frac12i\\pi\\,\\fr…","labels":["lemma:phi0-transform","eqn:phi0-trans-T","eqn:phi0-trans-S"],"detail_key":"p57"},{"id":"n44980","layer":"informal","project":"p57","title":"\\eqrefeqn:phi0-trans-T easily follows from periodicity of Eisenstein series and \\Delta(z)…","kind":"proof","summary":"\\eqrefeqn:phi0-trans-T easily follows from periodicity of Eisenstein series and \\Delta(z). For…","labels":[],"detail_key":"p57"},{"id":"n44981","layer":"informal","project":"p57","title":"def:a-definition","kind":"definition","summary":"Define a_rad : R\\to C by a_rad(r) := I_1(r) + I_2(r) + I_3(r) + I_4(r) + I_5(r) + I_6(r) where…","labels":["def:a-definition","eqn:a-rad-definition","eqn:a-I1","eqn:a-I2","eqn:a-I3","eqn:a-I4","eqn:a-I5","eqn:a-I6"],"detail_key":"p57"},{"id":"n44982","layer":"informal","project":"p57","title":"lemma:mod-div-disc-bound","kind":"lemma","summary":"Let f(z) be a holomorphic function with a Fourier expansion f(z) = \\sum_n \\ge n_0 c_f(n) e^\\pi…","labels":["lemma:mod-div-disc-bound"],"detail_key":"p57"},{"id":"n44983","layer":"informal","project":"p57","title":"By the product formula \\eqrefeqn:disc-definition, \\left|\\fracf(z)\\Delta(z)\\right| &= \\lef…","kind":"proof","summary":"By the product formula \\eqrefeqn:disc-definition, \\left|\\fracf(z)\\Delta(z)\\right| &= \\left|\\fra…","labels":[],"detail_key":"p57"},{"id":"n44984","layer":"informal","project":"p57","title":"cor:phi0-bound","kind":"corollary","summary":"There exists a constant C_0 > 0 such that |\\phi_0(z)| \\le C_0 e^-2 \\pi \\Im z for all z with \\Im…","labels":["cor:phi0-bound","eqn:phi0-bound"],"detail_key":"p57"},{"id":"n44985","layer":"informal","project":"p57","title":"By Ramanujan's formula, E_2 E_4 - E_6 = 3E_4' = 720 \\sum_n \\ge 1 n \\sigma_3(n) e^2 \\pi i…","kind":"proof","summary":"By Ramanujan's formula, E_2 E_4 - E_6 = 3E_4' = 720 \\sum_n \\ge 1 n \\sigma_3(n) e^2 \\pi i n z an…","labels":[],"detail_key":"p57"},{"id":"n44986","layer":"informal","project":"p57","title":"cor:phi2-bound","kind":"corollary","summary":"There exists a constant C_-2 > 0 such that |\\phi_-2(z)| \\le C_-2 for all z with \\Im z > 1/2.","labels":["cor:phi2-bound","eqn:phi2-bound"],"detail_key":"p57"},{"id":"n44987","layer":"informal","project":"p57","title":"cor:phi4-bound","kind":"corollary","summary":"There exists a constant C_-4 > 0 such that |\\phi_-4(z)| \\le C_-4 e^2 \\pi \\Im z for all z with \\…","labels":["cor:phi4-bound","eqn:phi4-bound"],"detail_key":"p57"},{"id":"n44988","layer":"informal","project":"p57","title":"lem:integral-bound","kind":"lemma","summary":"For all n \\in N, there exists a constant C' such that for all r \\geq 0, r^n \\cdot \\int_1^\\infty…","labels":["lem:integral-bound"],"detail_key":"p57"},{"id":"n44989","layer":"informal","project":"p57","title":"Fix n \\in N. We know there exists a constant C such that for all x \\geq 0, \\left\\lvert x…","kind":"proof","summary":"Fix n \\in N. We know there exists a constant C such that for all x \\geq 0, \\left\\lvert x \\right…","labels":[],"detail_key":"p57"},{"id":"n44990","layer":"informal","project":"p57","title":"lem:bound-I1-I3-I5","kind":"lemma","summary":"There exists C > 0 such that for all r \\geq 0, |I_1(r)|, |I_3(r)|, |I_5(r)| \\leq C \\int_1^\\inft…","labels":["lem:bound-I1-I3-I5"],"detail_key":"p57"},{"id":"n44991","layer":"informal","project":"p57","title":"We only prove the bound for I_1(r), as the other two are similar. By the change of variab…","kind":"proof","summary":"We only prove the bound for I_1(r), as the other two are similar. By the change of variable z =…","labels":[],"detail_key":"p57"},{"id":"n44992","layer":"informal","project":"p57","title":"lem:bound-I2-I4-I6","kind":"lemma","summary":"There exist C_1, C_2 > 0 such that for all r \\geq 0, |I_2(r)|, |I_4(r)| \\leq C_1 e^-\\pi r and |…","labels":["lem:bound-I2-I4-I6"],"detail_key":"p57"},{"id":"n44993","layer":"informal","project":"p57","title":"For I_2(r), parametrize z as z = t + i for t \\in [-1,0], and we have I_2(r) = \\int_-1^0 \\…","kind":"proof","summary":"For I_2(r), parametrize z as z = t + i for t \\in [-1,0], and we have I_2(r) = \\int_-1^0 \\phi_0\\…","labels":[],"detail_key":"p57"},{"id":"n44994","layer":"informal","project":"p57","title":"prop:a-schwartz","kind":"proposition","summary":"a(x) is a Schwartz function.","labels":["prop:a-schwartz"],"detail_key":"p57"},{"id":"n44995","layer":"informal","project":"p57","title":"By Theorem \\refthm:smooth-fast-decay-schwartz, it suffices to show that the function is s…","kind":"proof","summary":"By Theorem \\refthm:smooth-fast-decay-schwartz, it suffices to show that the function is smooth…","labels":[],"detail_key":"p57"},{"id":"n44996","layer":"informal","project":"p57","title":"prop:a-fourier","kind":"proposition","summary":"a(x) satisfies \\eqrefeqn:a-fourier.","labels":["prop:a-fourier"],"detail_key":"p57"},{"id":"n44997","layer":"informal","project":"p57","title":"eqn:gaussian_Fourier","kind":"proof","summary":"We recall that the Fourier transform of a Gaussian function is F(e^\\pi i \\|x\\|^2 z)(y)=z^-4\\,e^…","labels":["eqn:gaussian_Fourier"],"detail_key":"p57"},{"id":"n44998","layer":"informal","project":"p57","title":"cor:phi0-near-0-infty","kind":"corollary","summary":"We have \\phi_0\\left(\\fracit\\right) &= O(e^-2 \\pi / t) \\quad \\textas t \\to 0 \\\\ \\phi_0\\left(\\fra…","labels":["cor:phi0-near-0-infty","eqn:phi0-near-0","eqn:phi0-near-infty"],"detail_key":"p57"},{"id":"n44999","layer":"informal","project":"p57","title":"The first estimate follows from \\eqrefeqn:phi0-bound with z = i/t. For the second estimat…","kind":"proof","summary":"The first estimate follows from \\eqrefeqn:phi0-bound with z = i/t. For the second estimate, by…","labels":[],"detail_key":"p57"},{"id":"n45000","layer":"informal","project":"p57","title":"prop:a-double-zeros","kind":"proposition","summary":"For r>\\sqrt2 we can express a(r) in the following form a(r)=-4\\sin(\\pi r^2/2)^2\\,\\int\\limits_0^…","labels":["prop:a-double-zeros","eqn:_a_double_zeroes"],"detail_key":"p57"},{"id":"n45001","layer":"informal","project":"p57","title":"We denote the right hand side of \\eqrefeqn:_a_double_zeroes by d(r). Convergence of the i…","kind":"proof","summary":"We denote the right hand side of \\eqrefeqn:_a_double_zeroes by d(r). Convergence of the integra…","labels":[],"detail_key":"p57"},{"id":"n45002","layer":"informal","project":"p57","title":"prop:a-another-integral","kind":"proposition","summary":"For r\\geq0 we have a(r)=&4i\\,\\sin(\\pi r^2/2)^2\\,\\Bigg(\\frac36\\pi^3\\,(r^2-2)-\\frac8640\\pi^3\\,r^4…","labels":["prop:a-another-integral","eqn:a-another-integral"],"detail_key":"p57"},{"id":"n45003","layer":"informal","project":"p57","title":"eqn:_phi_asymptotic","kind":"proof","summary":"Suppose that r>\\sqrt2. Then by Proposition~\\refprop:a-double-zeros a(r)=4i\\,\\sin(\\pi r^2/2)^2\\,…","labels":["eqn:_phi_asymptotic"],"detail_key":"p57"},{"id":"n45004","layer":"informal","project":"p57","title":"prop:a0","kind":"proposition","summary":"We have a(0) = -\\fraci8640.","labels":["prop:a0"],"detail_key":"p57"},{"id":"n45005","layer":"informal","project":"p57","title":"These identities follow immediately from the previous proposition.","kind":"proof","summary":"These identities follow immediately from the previous proposition.","labels":[],"detail_key":"p57"},{"id":"n45006","layer":"informal","project":"p57","title":"def:_h","kind":"definition","summary":"h(z) := 128 \\fracH_3(z) + H_4(z)H_2(z)^2.","labels":["def:_h","eqn:_h_define__dup2"],"detail_key":"p57"},{"id":"n45007","layer":"informal","project":"p57","title":"def:psiI-psiT-psiS","kind":"definition","summary":"We define the following three functions \\psi_I\\,:=\\,&h-h|_-2ST \\\\ \\psi_T\\,:=\\,&\\psi_I|_-2T \\\\ \\…","labels":["def:psiI-psiT-psiS","eqn:psiI-define","eqn:psiT-define","eqn:psiS-define"],"detail_key":"p57"},{"id":"n45008","layer":"informal","project":"p57","title":"lemma:psi-new","kind":"lemma","summary":"\\psi_I(z), \\psi_S(z), \\psi_T(z) can be written as \\psi_I(z) &= \\fracH_4^3 (5 H_2^2 + 5 H_2 H_4…","labels":["lemma:psi-new","eqn:psiI-new","eqn:psiS-new","eqn:psiT-new"],"detail_key":"p57"},{"id":"n45009","layer":"informal","project":"p57","title":"By Lemma \\reflemma:theta-transform-S-T, we have H_2|_-2ST = (-H_4)|_-2T = -H_3, \\\\ H_3|_-…","kind":"proof","summary":"By Lemma \\reflemma:theta-transform-S-T, we have H_2|_-2ST = (-H_4)|_-2T = -H_3, \\\\ H_3|_-2ST =…","labels":[],"detail_key":"p57"},{"id":"n45010","layer":"informal","project":"p57","title":"lemma:psiI-psiT-psiS-fourier","kind":"lemma","summary":"The Fourier expansions of these functions are \\psi_I(z)\\,=\\,&q^-1 + 144 + O(q^1/2) \\\\ \\psi_T(z)…","labels":["lemma:psiI-psiT-psiS-fourier","eqn:_psi_fourier_I","eqn:_psi_fourier_T"],"detail_key":"p57"},{"id":"n45011","layer":"informal","project":"p57","title":"def:b-definition","kind":"definition","summary":"Define b_rad: R\\to C by b_rad(r) := J_1(r) + J_2(r) + J_3(r) + J_4(r) + J_5(r) + J_6(r) where f…","labels":["def:b-definition","eqn:b-definition","eqn:J1","eqn:J2","eqn:J3","eqn:J4","eqn:J5","eqn:J6"],"detail_key":"p57"},{"id":"n45012","layer":"informal","project":"p57","title":"lemma:psi-bound","kind":"lemma","summary":"There exist constants C_I, C_S, C_T > 0 such that |\\psi_I(z)| &\\le C_I e^2\\pi \\Im z, \\\\ |\\psi_T…","labels":["lemma:psi-bound","eqn:psiI-bound","eqn:psiT-bound","eqn:psiS-bound"],"detail_key":"p57"},{"id":"n45013","layer":"informal","project":"p57","title":"The proof is similar to that of Lemma \\refcor:phi0-bound, follows from Lemma \\reflemma:mo…","kind":"proof","summary":"The proof is similar to that of Lemma \\refcor:phi0-bound, follows from Lemma \\reflemma:mod-div-…","labels":[],"detail_key":"p57"},{"id":"n45014","layer":"informal","project":"p57","title":"lemma:bound-J1-J3-J5","kind":"lemma","summary":"There exist a constant C > 0 such that |J_1(r)|, |J_3(r)|, |J_5(r)| &\\le C \\int_1^\\infty e^-\\pi…","labels":["lemma:bound-J1-J3-J5"],"detail_key":"p57"},{"id":"n45015","layer":"informal","project":"p57","title":"lemma:bound-J2-J4-J6","kind":"lemma","summary":"There exist C_1, C_2 > 0 such that |J_2(r)|, |J_4(r)| &\\le C_1 e^-\\pi r \\\\ |J_6(r)| &\\le C_2 \\f…","labels":["lemma:bound-J2-J4-J6"],"detail_key":"p57"},{"id":"n45016","layer":"informal","project":"p57","title":"prop:b-schwartz","kind":"proposition","summary":"b(x) is a Schwartz function.","labels":["prop:b-schwartz"],"detail_key":"p57"},{"id":"n45017","layer":"informal","project":"p57","title":"Similar to the proof of \\refprop:a-schwartz.","kind":"proof","summary":"Similar to the proof of \\refprop:a-schwartz.","labels":[],"detail_key":"p57"},{"id":"n45018","layer":"informal","project":"p57","title":"prop:b-fourier","kind":"proposition","summary":"b(x) satisfies \\eqrefeqn:b-fourier.","labels":["prop:b-fourier"],"detail_key":"p57"},{"id":"n45019","layer":"informal","project":"p57","title":"Here, we repeat the arguments used in the proof of Proposition~\\refprop:a-fourier. We use…","kind":"proof","summary":"Here, we repeat the arguments used in the proof of Proposition~\\refprop:a-fourier. We use ident…","labels":[],"detail_key":"p57"},{"id":"n45020","layer":"informal","project":"p57","title":"cor:psiI-near-0-infty","kind":"corollary","summary":"We have \\psi_I(it) &= O(t^2 e^\\pi/t) \\quad \\textas t \\to 0 \\\\ \\psi_I(it) &= O(e^2 \\pi t) \\quad…","labels":["cor:psiI-near-0-infty","eqn:psiI-near-0","eqn:psiI-near-infty"],"detail_key":"p57"},{"id":"n45021","layer":"informal","project":"p57","title":"By \\eqrefeqn:psiS-define, we have \\psi_I(it) = (it)^-2 \\psi_S\\left(\\frac-1it\\right) = -t^…","kind":"proof","summary":"By \\eqrefeqn:psiS-define, we have \\psi_I(it) = (it)^-2 \\psi_S\\left(\\frac-1it\\right) = -t^-2 \\ps…","labels":[],"detail_key":"p57"},{"id":"n45022","layer":"informal","project":"p57","title":"prop:b-double-zeros","kind":"proposition","summary":"For r>\\sqrt2 function b(r) can be expressed as b(r)=-4\\sin(\\pi r^2/2)^2\\,\\int\\limits_0^i\\infty\\…","labels":["prop:b-double-zeros","eqn:_b_double_zeroes"],"detail_key":"p57"},{"id":"n45023","layer":"informal","project":"p57","title":"eqn:_inside_proof_1","kind":"proof","summary":"We denote the right hand side of~\\eqrefeqn:_b_double_zeroes by c(r). By Corollary \\refcor:psiI-…","labels":["eqn:_inside_proof_1","eqn:_c1","eqn:_c2"],"detail_key":"p57"},{"id":"n45024","layer":"informal","project":"p57","title":"prop:b-another-integral","kind":"proposition","summary":"For r\\geq0 we have b(r)=4i\\,\\sin(\\pi r^2/2)^2\\,\\left(\\frac144\\pi\\,r^2+\\frac1\\pi\\,(r^2-2)+\\int\\l…","labels":["prop:b-another-integral","eqn:b-another-integral"],"detail_key":"p57"},{"id":"n45025","layer":"informal","project":"p57","title":"eqn:_psi_asymptotic","kind":"proof","summary":"The proof is analogous to the proof of Proposition~\\refprop:a-another-integral. First, suppose…","labels":["eqn:_psi_asymptotic"],"detail_key":"p57"},{"id":"n45026","layer":"informal","project":"p57","title":"prop:b0","kind":"proposition","summary":"We have b(0) = 0.","labels":["prop:b0"],"detail_key":"p57"},{"id":"n45027","layer":"informal","project":"p57","title":"These identities follow immediately from the previous proposition.","kind":"proof","summary":"These identities follow immediately from the previous proposition.","labels":[],"detail_key":"p57"},{"id":"n45028","layer":"informal","project":"p57","title":"prop:ineqA","kind":"proposition","summary":"Consider the function A:(0,\\infty)\\toC defined as A(t):=-t^2\\phi_0(i/t)-\\frac36\\pi^2\\,\\psi_I(it…","labels":["prop:ineqA","eqn:defA","eqn:ineqA"],"detail_key":"p57"},{"id":"n45029","layer":"informal","project":"p57","title":"prop:ineqB","kind":"proposition","summary":"Consider the function B:(0,\\infty)\\toC defined as B(t) := -t^2\\phi_0(i/t)+\\frac36\\pi^2\\,\\psi_I(…","labels":["prop:ineqB","eqn:defB","eqn:ineqB"],"detail_key":"p57"},{"id":"n45030","layer":"informal","project":"p57","title":"def:FG-definition","kind":"definition","summary":"Define two (quasi) modular forms as F(z) &= (E_2(z) E_4(z) - E_6(z))^2 \\\\ G(z) &= H_2(z)^3 (2 H…","labels":["def:FG-definition","eqn:defF","eqn:defG"],"detail_key":"p57"},{"id":"n45031","layer":"informal","project":"p57","title":"lemma:F-G-phi-psi-identities","kind":"lemma","summary":"We have \\phi_0 &= \\fracF\\Delta \\\\ \\psi_S &= -\\frac12 \\fracG\\Delta","labels":["lemma:F-G-phi-psi-identities","eqn:phi0-F","eqn:psiS-G"],"detail_key":"p57"},{"id":"n45032","layer":"informal","project":"p57","title":"\\eqrefeqn:phi0-F is clear. \\eqrefeqn:psiS-G is already shown in Lemma \\reflemma:psi-new.","kind":"proof","summary":"\\eqrefeqn:phi0-F is clear. \\eqrefeqn:psiS-G is already shown in Lemma \\reflemma:psi-new.","labels":[],"detail_key":"p57"},{"id":"n45033","layer":"informal","project":"p57","title":"lemma:ineqABnew-equiv","kind":"lemma","summary":"Inequality \\eqrefeqn:ineqA and \\eqrefeqn:ineqB are equivalent to F(it) + \\frac18\\pi^2 G(it) > 0…","labels":["lemma:ineqABnew-equiv","eqn:ineqAnew","eqn:ineqBnew"],"detail_key":"p57"},{"id":"n45034","layer":"informal","project":"p57","title":"By \\eqrefeqn:psiS-define, \\psi_I(it) = (\\psi_S|_-2S)(it) = (it)^2\\psi_S\\left(-\\frac1it\\ri…","kind":"proof","summary":"By \\eqrefeqn:psiS-define, \\psi_I(it) = (\\psi_S|_-2S)(it) = (it)^2\\psi_S\\left(-\\frac1it\\right) =…","labels":[],"detail_key":"p57"},{"id":"n45035","layer":"informal","project":"p57","title":"lemma:F-G-pos","kind":"lemma","summary":"For all t > 0, we have F(it) > 0 and G(it) > 0.","labels":["lemma:F-G-pos"],"detail_key":"p57"},{"id":"n45036","layer":"informal","project":"p57","title":"By Ramanujan's identity \\eqrefeqn:DE4, we have F(z) = 9 E_4'(z)^2 and F(it) = 9E_4'(it)^2…","kind":"proof","summary":"By Ramanujan's identity \\eqrefeqn:DE4, we have F(z) = 9 E_4'(z)^2 and F(it) = 9E_4'(it)^2 = 9 \\…","labels":[],"detail_key":"p57"},{"id":"n45037","layer":"informal","project":"p57","title":"cor:ineqAnew","kind":"corollary","summary":"\\eqrefeqn:ineqAnew holds.","labels":["cor:ineqAnew"],"detail_key":"p57"},{"id":"n45038","layer":"informal","project":"p57","title":"This directly follows from Lemma \\reflemma:F-G-pos.","kind":"proof","summary":"This directly follows from Lemma \\reflemma:F-G-pos.","labels":[],"detail_key":"p57"},{"id":"n45039","layer":"informal","project":"p57","title":"lemma:FG-de","kind":"lemma","summary":"F and G satisfy the following differential equations: \\partial_12\\partial_10 F - \\frac56 E_4 F…","labels":["lemma:FG-de","eqn:ddf","eqn:ddg"],"detail_key":"p57"},{"id":"n45040","layer":"informal","project":"p57","title":"eqn:S5","kind":"proof","summary":"Both can be shown by direct computations. By Ramanujan's identities (Theorem \\refthm:ramanujan-…","labels":["eqn:S5","eqn:S7"],"detail_key":"p57"},{"id":"n45041","layer":"informal","project":"p57","title":"cor:MLDE-pos","kind":"corollary","summary":"\\eqrefeqn:ddf (resp. \\eqrefeqn:ddg) is positive (resp. negative) on the (positive) imaginary ax…","labels":["cor:MLDE-pos"],"detail_key":"p57"},{"id":"n45042","layer":"informal","project":"p57","title":"From \\eqrefeqn:E2 and Lemma \\refcor:disc-pos, \\ifplastex 7200 (-E_2'(it)) \\Delta(it) = 72…","kind":"proof","summary":"From \\eqrefeqn:E2 and Lemma \\refcor:disc-pos, \\ifplastex 7200 (-E_2'(it)) \\Delta(it) = 7200 \\cd…","labels":[],"detail_key":"p57"},{"id":"n45043","layer":"informal","project":"p57","title":"lemma:Qlim","kind":"lemma","summary":"We have \\lim_t \\to 0^+ Q(t) = \\frac18\\pi^2.","labels":["lemma:Qlim"],"detail_key":"p57"},{"id":"n45044","layer":"informal","project":"p57","title":"We have \\lim_t \\to 0^+ Q(t) = \\lim_t \\to 0^+ \\fracF(it)G(it) = \\lim_t \\to \\infty \\fracF(i…","kind":"proof","summary":"We have \\lim_t \\to 0^+ Q(t) = \\lim_t \\to 0^+ \\fracF(it)G(it) = \\lim_t \\to \\infty \\fracF(i/t)G(i…","labels":[],"detail_key":"p57"},{"id":"n45045","layer":"informal","project":"p57","title":"lemma:log-der-inf","kind":"lemma","summary":"Let F be a quasimodular form where the vanishing order of F at infinity is n_0 > 0, i.e. F(z) =…","labels":["lemma:log-der-inf"],"detail_key":"p57"},{"id":"n45046","layer":"informal","project":"p57","title":"By Lemma \\reflemma:der-q-series, \\lim_t \\to \\infty \\fracF'(it)F(it) = \\lim_t \\to \\infty \\…","kind":"proof","summary":"By Lemma \\reflemma:der-q-series, \\lim_t \\to \\infty \\fracF'(it)F(it) = \\lim_t \\to \\infty \\frac\\s…","labels":[],"detail_key":"p57"},{"id":"n45047","layer":"informal","project":"p57","title":"prop:Qdec","kind":"proposition","summary":"The function t \\mapsto Q(t) is strictly decreasing.","labels":["prop:Qdec"],"detail_key":"p57"},{"id":"n45048","layer":"informal","project":"p57","title":"It is enough to show that \\fracddt \\left(\\fracF(it)G(it)\\right) < 0 &\\Leftrightarrow (- 2…","kind":"proof","summary":"It is enough to show that \\fracddt \\left(\\fracF(it)G(it)\\right) < 0 &\\Leftrightarrow (- 2\\pi) \\…","labels":[],"detail_key":"p57"},{"id":"n45049","layer":"informal","project":"p57","title":"cor:ineqBnew","kind":"corollary","summary":"\\eqrefeqn:ineqBnew holds.","labels":["cor:ineqBnew"],"detail_key":"p57"},{"id":"n45050","layer":"informal","project":"p57","title":"\\fracF(it)G(it) = Q(t) < \\lim_u \\to 0^+ Q(u) = \\frac18\\pi^2 and by Lemma \\reflemma:F-G-po…","kind":"proof","summary":"\\fracF(it)G(it) = Q(t) < \\lim_u \\to 0^+ Q(u) = \\frac18\\pi^2 and by Lemma \\reflemma:F-G-pos, \\eq…","labels":[],"detail_key":"p57"},{"id":"n45051","layer":"informal","project":"p57","title":"thm:g1","kind":"theorem","summary":"The function g(x):=\\frac\\pi\\,i8640a(x)+\\fraci240\\pi\\,b(x) satisfies conditions \\eqrefeqn:g1--\\e…","labels":["thm:g1"],"detail_key":"p57"},{"id":"n45052","layer":"informal","project":"p57","title":"eqn:g_A","kind":"proof","summary":"First, we prove that \\eqrefeqn:g1 holds. By Propositions~\\refprop:a-double-zeros and \\refprop:b…","labels":["eqn:g_A","eqn:g_B"],"detail_key":"p57"},{"id":"n45053","layer":"formal","project":"p57","title":"E8Basis_volume","kind":"theorem","summary":"Eq (MeasureTheory.volume (ZSpan.fundamentalDomain (E8Basis Real))) 1","labels":[],"detail_key":"p57","name":"E8Basis_volume","module":"SpherePacking.Basic.E8"},{"id":"n45054","layer":"formal","project":"p57","title":"E8Lattice","kind":"def","summary":"Submodule Int (EuclideanSpace Real (Fin 8))","labels":[],"detail_key":"p57","name":"E8Lattice","module":"SpherePacking.Basic.E8"},{"id":"n45055","layer":"formal","project":"p57","title":"E8Matrix","kind":"def","summary":"(R : Type u_2) → [Field R] → Matrix (Fin 8) (Fin 8) R","labels":[],"detail_key":"p57","name":"E8Matrix","module":"SpherePacking.Basic.E8"},{"id":"n45056","layer":"formal","project":"p57","title":"E8Packing","kind":"def","summary":"PeriodicSpherePacking 8","labels":[],"detail_key":"p57","name":"E8Packing","module":"SpherePacking.Basic.E8"},{"id":"n45057","layer":"formal","project":"p57","title":"E8Packing_density","kind":"theorem","summary":"Eq E8Packing.density (HDiv.hDiv (HPow.hPow (ENNReal.ofReal Real.pi) 4) 384)","labels":[],"detail_key":"p57","name":"E8Packing_density","module":"SpherePacking.Basic.E8"},{"id":"n45058","layer":"formal","project":"p57","title":"E8_norm_eq_sqrt_even","kind":"theorem","summary":"∀ (v : Fin 8 → Real), Membership.mem (Submodule.E8 Real) v → Exists fun n => And (Even n) (Eq (…","labels":[],"detail_key":"p57","name":"E8_norm_eq_sqrt_even","module":"SpherePacking.Basic.E8"},{"id":"n45059","layer":"formal","project":"p57","title":"Submodule.E8","kind":"def","summary":"(R : Type u_2) → [inst : Field R] → [NeZero 2] → Submodule Int (Fin 8 → R)","labels":[],"detail_key":"p57","name":"Submodule.E8","module":"SpherePacking.Basic.E8"},{"id":"n45060","layer":"formal","project":"p57","title":"instDiscreteE8Lattice","kind":"theorem","summary":"DiscreteTopology (Subtype fun x => Membership.mem E8Lattice x)","labels":[],"detail_key":"p57","name":"instDiscreteE8Lattice","module":"SpherePacking.Basic.E8"},{"id":"n45061","layer":"formal","project":"p57","title":"instIsZLatticeE8Lattice","kind":"theorem","summary":"IsZLattice Real E8Lattice","labels":[],"detail_key":"p57","name":"instIsZLatticeE8Lattice","module":"SpherePacking.Basic.E8"},{"id":"n45062","layer":"formal","project":"p57","title":"span_E8Matrix","kind":"theorem","summary":"∀ (R : Type u_2) [inst : Field R] [inst_1 : CharZero R], Eq (Submodule.span Int (Set.range (E8M…","labels":[],"detail_key":"p57","name":"span_E8Matrix","module":"SpherePacking.Basic.E8"},{"id":"n45063","layer":"formal","project":"p57","title":"span_E8Matrix_eq_top","kind":"theorem","summary":"∀ (R : Type u_2) [inst : Field R] [NeZero 2], Eq (Submodule.span R (Set.range (E8Matrix R).row)…","labels":[],"detail_key":"p57","name":"span_E8Matrix_eq_top","module":"SpherePacking.Basic.E8"},{"id":"n45064","layer":"formal","project":"p57","title":"PeriodicSpherePacking.aux2_ge'","kind":"theorem","summary":"∀ d : Nat (S : PeriodicSpherePacking d) ι : Type u_1 [Finite ι] L : Real (R : Real) (b : Module…","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking.aux2_ge'","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45065","layer":"formal","project":"p57","title":"PeriodicSpherePacking.aux2_le'","kind":"theorem","summary":"∀ d : Nat (S : PeriodicSpherePacking d) ι : Type u_1 [Finite ι] L : Real (R : Real) (b : Module…","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking.aux2_le'","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45066","layer":"formal","project":"p57","title":"PeriodicSpherePacking.aux_ge","kind":"theorem","summary":"∀ d : Nat (S : PeriodicSpherePacking d), LT.lt 0 d → ∀ ι : Type u_1 [Finite ι] (b : Module.Basi…","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking.aux_ge","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45067","layer":"formal","project":"p57","title":"PeriodicSpherePacking.aux_le","kind":"theorem","summary":"∀ d : Nat (S : PeriodicSpherePacking d), LT.lt 0 d → ∀ ι : Type u_1 [Finite ι] (b : Module.Basi…","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking.aux_le","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45068","layer":"formal","project":"p57","title":"PeriodicSpherePacking.density_eq","kind":"theorem","summary":"∀ d : Nat S : PeriodicSpherePacking d ι : Type u_3 [Finite ι] (b : Module.Basis ι Int (Subtype…","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking.density_eq","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45069","layer":"formal","project":"p57","title":"periodic_constant_eq_constant","kind":"theorem","summary":"∀ d : Nat, LT.lt 0 d → Eq (PeriodicSpherePackingConstant d) (SpherePackingConstant d)","labels":[],"detail_key":"p57","name":"periodic_constant_eq_constant","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45070","layer":"formal","project":"p57","title":"periodic_constant_eq_periodic_constant_normalized","kind":"theorem","summary":"∀ d : Nat, LT.lt 0 d → Eq (PeriodicSpherePackingConstant d) (iSup fun S => iSup fun x => S.dens…","labels":[],"detail_key":"p57","name":"periodic_constant_eq_periodic_constant_normalized","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45071","layer":"formal","project":"p57","title":"volume_ball_ratio_tendsto_nhds_one''","kind":"theorem","summary":"∀ d : Nat C C' : Real, LT.lt 0 d → Filter.Tendsto (fun R => HDiv.hDiv (MeasureTheory.volume (Me…","labels":[],"detail_key":"p57","name":"volume_ball_ratio_tendsto_nhds_one''","module":"SpherePacking.Basic.PeriodicPacking"},{"id":"n45072","layer":"formal","project":"p57","title":"PeriodicSpherePacking","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p57","name":"PeriodicSpherePacking","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45073","layer":"formal","project":"p57","title":"PeriodicSpherePackingConstant","kind":"def","summary":"Nat → ENNReal","labels":[],"detail_key":"p57","name":"PeriodicSpherePackingConstant","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45074","layer":"formal","project":"p57","title":"SpherePacking","kind":"inductive","summary":"Nat → Type","labels":[],"detail_key":"p57","name":"SpherePacking","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45075","layer":"formal","project":"p57","title":"SpherePacking.balls","kind":"def","summary":"d : Nat → SpherePacking d → Set (EuclideanSpace Real (Fin d))","labels":[],"detail_key":"p57","name":"SpherePacking.balls","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45076","layer":"formal","project":"p57","title":"SpherePacking.constant_eq_constant_normalized","kind":"theorem","summary":"∀ d : Nat, LT.lt 0 d → Eq (SpherePackingConstant d) (iSup fun S => iSup fun x => S.density)","labels":[],"detail_key":"p57","name":"SpherePacking.constant_eq_constant_normalized","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45077","layer":"formal","project":"p57","title":"SpherePacking.density","kind":"def","summary":"d : Nat → SpherePacking d → ENNReal","labels":[],"detail_key":"p57","name":"SpherePacking.density","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45078","layer":"formal","project":"p57","title":"SpherePacking.finiteDensity","kind":"def","summary":"d : Nat → SpherePacking d → Real → ENNReal","labels":[],"detail_key":"p57","name":"SpherePacking.finiteDensity","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45079","layer":"formal","project":"p57","title":"SpherePacking.finiteDensity_ge","kind":"theorem","summary":"∀ d : Nat (S : SpherePacking d), LT.lt 0 d → ∀ (R : Real), GE.ge (S.finiteDensity R) (HDiv.hDiv…","labels":[],"detail_key":"p57","name":"SpherePacking.finiteDensity_ge","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45080","layer":"formal","project":"p57","title":"SpherePacking.finiteDensity_le","kind":"theorem","summary":"∀ d : Nat (S : SpherePacking d), LT.lt 0 d → ∀ (R : Real), LE.le (S.finiteDensity R) (HDiv.hDiv…","labels":[],"detail_key":"p57","name":"SpherePacking.finiteDensity_le","module":"SpherePacking.Basic.SpherePacking"},{"id":"n45081","layer":"formal","project":"p57","title":"SpherePacking.scale","kind":"def","summary":"d : Nat → SpherePacking d → c : Real → LT.lt 0 c → SpherePacking 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Discrete…","labels":[],"detail_key":"p57","name":"SchwartzMap.PoissonSummation_Lattices","module":"SpherePacking.CohnElkies.Prereqs"},{"id":"n45088","layer":"formal","project":"p57","title":"MagicFunction.PolyFourierCoeffBound.DivDiscBoundOfPolyFourierCoeff","kind":"theorem","summary":"∀ (z : UpperHalfPlane), LT.lt (1 / 2) z.im → ∀ (c : Int → Complex) (n₀ : Int), (Summable fun i…","labels":[],"detail_key":"p57","name":"MagicFunction.PolyFourierCoeffBound.DivDiscBoundOfPolyFourierCoeff","module":"SpherePacking.MagicFunction.PolyFourierCoeffBound"},{"id":"n45089","layer":"formal","project":"p57","title":"MagicFunction.a.RadialFunctions.a","kind":"def","summary":"EuclideanSpace Real (Fin 8) → Complex","labels":[],"detail_key":"p57","name":"MagicFunction.a.RadialFunctions.a","module":"SpherePacking.MagicFunction.a.Basic"},{"id":"n45090","layer":"formal","project":"p57","title":"MagicFunction.a.RealIntegrals.a'","kind":"def","summary":"Real → 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ModularForm.discriminant","labels":[],"detail_key":"p57","name":"Δ_imag_axis_pos","module":"SpherePacking.ModularForms.Delta"},{"id":"n45100","layer":"formal","project":"p57","title":"D","kind":"def","summary":"(UpperHalfPlane → Complex) → UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"D","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45101","layer":"formal","project":"p57","title":"D_qexp_tsum_pnat","kind":"theorem","summary":"∀ (a : PNat → Complex) (z : UpperHalfPlane), (Summable fun n => HMul.hMul (a n) (Complex.exp (H…","labels":[],"detail_key":"p57","name":"D_qexp_tsum_pnat","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45102","layer":"formal","project":"p57","title":"serre_D","kind":"def","summary":"Complex → (UpperHalfPlane → Complex) → UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"serre_D","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45103","layer":"formal","project":"p57","title":"serre_D_mul","kind":"theorem","summary":"∀ (k₁ k₂ : Int) (F G : UpperHalfPlane → Complex), MDifferentiable (modelWithCornersSelf Complex…","labels":[],"detail_key":"p57","name":"serre_D_mul","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45104","layer":"formal","project":"p57","title":"serre_D_slash_equivariant","kind":"theorem","summary":"∀ (k : Int) (F : UpperHalfPlane → Complex), MDifferentiable (modelWithCornersSelf Complex Compl…","labels":[],"detail_key":"p57","name":"serre_D_slash_equivariant","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45105","layer":"formal","project":"p57","title":"serre_D_slash_invariant","kind":"theorem","summary":"∀ (k : Int) (F : UpperHalfPlane → Complex), MDifferentiable (modelWithCornersSelf Complex Compl…","labels":[],"detail_key":"p57","name":"serre_D_slash_invariant","module":"SpherePacking.ModularForms.Derivative"},{"id":"n45106","layer":"formal","project":"p57","title":"dim_gen_cong_levels","kind":"theorem","summary":"∀ (k : Int) (Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) Int)), Ne Γ.index 0 → FiniteDimens…","labels":[],"detail_key":"p57","name":"dim_gen_cong_levels","module":"SpherePacking.ModularForms.DimensionFormulas"},{"id":"n45107","layer":"formal","project":"p57","title":"E₂_eq","kind":"theorem","summary":"∀ (z : UpperHalfPlane), Eq (E₂ z) (HSub.hSub 1 (HMul.hMul 24 (tsum fun n => HDiv.hDiv (HMul.hMu…","labels":[],"detail_key":"p57","name":"E₂_eq","module":"SpherePacking.ModularForms.E2"},{"id":"n45108","layer":"formal","project":"p57","title":"E₂_slash_transform","kind":"theorem","summary":"∀ (γ : Matrix.SpecialLinearGroup (Fin 2) Int), Eq (SlashAction.map 2 γ E₂) (HSub.hSub E₂ (HSMul…","labels":[],"detail_key":"p57","name":"E₂_slash_transform","module":"SpherePacking.ModularForms.E2"},{"id":"n45109","layer":"formal","project":"p57","title":"E₂_transform","kind":"theorem","summary":"∀ (z : UpperHalfPlane), Eq (SlashAction.map 2 ModularGroup.S E₂ z) (HAdd.hAdd (E₂ z) (HDiv.hDiv…","labels":[],"detail_key":"p57","name":"E₂_transform","module":"SpherePacking.ModularForms.E2"},{"id":"n45110","layer":"formal","project":"p57","title":"E","kind":"def","summary":"(k : Int) → LE.le 3 k → ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL Real) (Congr…","labels":[],"detail_key":"p57","name":"E","module":"SpherePacking.ModularForms.Eisenstein"},{"id":"n45111","layer":"formal","project":"p57","title":"F","kind":"def","summary":"UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"F","module":"SpherePacking.ModularForms.FG"},{"id":"n45112","layer":"formal","project":"p57","title":"FG_inequality_1","kind":"theorem","summary":"∀ t : Real, LT.lt 0 t → GT.gt (HAdd.hAdd (FReal t) (HMul.hMul (HMul.hMul 18 (HPow.hPow Real.pi…","labels":[],"detail_key":"p57","name":"FG_inequality_1","module":"SpherePacking.ModularForms.FG"},{"id":"n45113","layer":"formal","project":"p57","title":"FG_inequality_2","kind":"theorem","summary":"∀ t : Real, LT.lt 0 t → LT.lt (HSub.hSub (FReal t) (HMul.hMul (HMul.hMul 18 (HPow.hPow Real.pi…","labels":[],"detail_key":"p57","name":"FG_inequality_2","module":"SpherePacking.ModularForms.FG"},{"id":"n45114","layer":"formal","project":"p57","title":"F_imag_axis_pos","kind":"theorem","summary":"ResToImagAxis.Pos F","labels":[],"detail_key":"p57","name":"F_imag_axis_pos","module":"SpherePacking.ModularForms.FG"},{"id":"n45115","layer":"formal","project":"p57","title":"FmodG_rightLimitAt_zero","kind":"theorem","summary":"Filter.Tendsto FmodGReal (nhdsWithin 0 (Set.Ioi 0)) (nhds (HMul.hMul 18 (HPow.hPow Real.pi (-2)…","labels":[],"detail_key":"p57","name":"FmodG_rightLimitAt_zero","module":"SpherePacking.ModularForms.FG"},{"id":"n45116","layer":"formal","project":"p57","title":"FmodG_strictAntiOn","kind":"theorem","summary":"StrictAntiOn FmodGReal (Set.Ioi 0)","labels":[],"detail_key":"p57","name":"FmodG_strictAntiOn","module":"SpherePacking.ModularForms.FG"},{"id":"n45117","layer":"formal","project":"p57","title":"G","kind":"def","summary":"UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"G","module":"SpherePacking.ModularForms.FG"},{"id":"n45118","layer":"formal","project":"p57","title":"G_imag_axis_pos","kind":"theorem","summary":"ResToImagAxis.Pos G","labels":[],"detail_key":"p57","name":"G_imag_axis_pos","module":"SpherePacking.ModularForms.FG"},{"id":"n45119","layer":"formal","project":"p57","title":"MLDE_F","kind":"theorem","summary":"Eq (serre_D 12 (serre_D 10 F)) (HAdd.hAdd (HMul.hMul (HMul.hMul (HMul.hMul 5 (Inv.inv 6)) E₄.to…","labels":[],"detail_key":"p57","name":"MLDE_F","module":"SpherePacking.ModularForms.FG"},{"id":"n45120","layer":"formal","project":"p57","title":"MLDE_G","kind":"theorem","summary":"Eq (serre_D 12 (serre_D 10 G)) (HSub.hSub (HMul.hMul (HMul.hMul (HMul.hMul 5 (Inv.inv 6)) E₄.to…","labels":[],"detail_key":"p57","name":"MLDE_G","module":"SpherePacking.ModularForms.FG"},{"id":"n45121","layer":"formal","project":"p57","title":"H₂_SIF","kind":"def","summary":"SlashInvariantForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL Real) (CongruenceSubgroup.Gam…","labels":[],"detail_key":"p57","name":"H₂_SIF","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45122","layer":"formal","project":"p57","title":"H₂_S_action","kind":"theorem","summary":"Eq (SlashAction.map 2 ModularGroup.S H₂) (Neg.neg 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(CongruenceSubgroup.Gam…","labels":[],"detail_key":"p57","name":"H₃_SIF","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45126","layer":"formal","project":"p57","title":"H₃_S_action","kind":"theorem","summary":"Eq (SlashAction.map 2 ModularGroup.S H₃) (Neg.neg H₃)","labels":[],"detail_key":"p57","name":"H₃_S_action","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45127","layer":"formal","project":"p57","title":"H₃_T_action","kind":"theorem","summary":"Eq (SlashAction.map 2 ModularGroup.T H₃) H₄","labels":[],"detail_key":"p57","name":"H₃_T_action","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45128","layer":"formal","project":"p57","title":"H₄_SIF","kind":"def","summary":"SlashInvariantForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL Real) (CongruenceSubgroup.Gam…","labels":[],"detail_key":"p57","name":"H₄_SIF","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45129","layer":"formal","project":"p57","title":"H₄_S_action","kind":"theorem","summary":"Eq (SlashAction.map 2 ModularGroup.S H₄) (Neg.neg H₂)","labels":[],"detail_key":"p57","name":"H₄_S_action","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45130","layer":"formal","project":"p57","title":"H₄_T_action","kind":"theorem","summary":"Eq (SlashAction.map 2 ModularGroup.T H₄) H₃","labels":[],"detail_key":"p57","name":"H₄_T_action","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45131","layer":"formal","project":"p57","title":"H₄_imag_axis_pos","kind":"theorem","summary":"ResToImagAxis.Pos H₄","labels":[],"detail_key":"p57","name":"H₄_imag_axis_pos","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45132","layer":"formal","project":"p57","title":"isBoundedAtImInfty_H_slash","kind":"theorem","summary":"∀ (γ : Matrix.SpecialLinearGroup (Fin 2) Int), And (UpperHalfPlane.IsBoundedAtImInfty (SlashAct…","labels":[],"detail_key":"p57","name":"isBoundedAtImInfty_H_slash","module":"SpherePacking.ModularForms.JacobiTheta.Basic"},{"id":"n45133","layer":"formal","project":"p57","title":"H₂","kind":"def","summary":"UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"H₂","module":"SpherePacking.ModularForms.JacobiTheta.Defs"},{"id":"n45134","layer":"formal","project":"p57","title":"H₃","kind":"def","summary":"UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"H₃","module":"SpherePacking.ModularForms.JacobiTheta.Defs"},{"id":"n45135","layer":"formal","project":"p57","title":"H₄","kind":"def","summary":"UpperHalfPlane → Complex","labels":[],"detail_key":"p57","name":"H₄","module":"SpherePacking.ModularForms.JacobiTheta.Defs"},{"id":"n45136","layer":"formal","project":"p57","title":"Θ₂","kind":"def","summary":"UpperHalfPlane → 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2","labels":[],"detail_key":"p57","name":"H₂_MF","module":"SpherePacking.ModularForms.JacobiTheta.MDifferentiable"},{"id":"n45141","layer":"formal","project":"p57","title":"H₃_MF","kind":"def","summary":"ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL Real) (CongruenceSubgroup.Gamma 2)) 2","labels":[],"detail_key":"p57","name":"H₃_MF","module":"SpherePacking.ModularForms.JacobiTheta.MDifferentiable"},{"id":"n45142","layer":"formal","project":"p57","title":"H₄_MF","kind":"def","summary":"ModularForm (Subgroup.map (Matrix.SpecialLinearGroup.mapGL Real) (CongruenceSubgroup.Gamma 2)) 2","labels":[],"detail_key":"p57","name":"H₄_MF","module":"SpherePacking.ModularForms.JacobiTheta.MDifferentiable"},{"id":"n45143","layer":"formal","project":"p57","title":"ramanujan_E₂","kind":"theorem","summary":"Eq (D E₂) (HMul.hMul (Inv.inv 12) (HSub.hSub (HMul.hMul E₂ E₂) E₄.toFun))","labels":[],"detail_key":"p57","name":"ramanujan_E₂","module":"SpherePacking.ModularForms.RamanujanIdentities"},{"id":"n45144","layer":"formal","project":"p57","title":"ramanujan_E₂'","kind":"theorem","summary":"Eq (serre_D 1 E₂) (HMul.hMul (Neg.neg (Inv.inv 12)) E₄.toFun)","labels":[],"detail_key":"p57","name":"ramanujan_E₂'","module":"SpherePacking.ModularForms.RamanujanIdentities"},{"id":"n45145","layer":"formal","project":"p57","title":"ramanujan_E₄","kind":"theorem","summary":"Eq (D E₄.toFun) (HMul.hMul (Inv.inv 3) (HSub.hSub (HMul.hMul E₂ E₄.toFun) E₆.toFun))","labels":[],"detail_key":"p57","name":"ramanujan_E₄","module":"SpherePacking.ModularForms.RamanujanIdentities"},{"id":"n45146","layer":"formal","project":"p57","title":"ramanujan_E₄'","kind":"theorem","summary":"Eq (serre_D 4 E₄.toFun) (HMul.hMul (Neg.neg (Inv.inv 3)) 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ModularGroup.T)))","labels":[],"detail_key":"p57","name":"SL2Z_generate","module":"SpherePacking.ModularForms.SlashActionAuxil"},{"id":"n45150","layer":"formal","project":"p57","title":"modular_slash_negI_of_even","kind":"theorem","summary":"∀ (f : UpperHalfPlane → Complex) (k : Int), Even k → Eq (SlashAction.map k (↑negI) f) f","labels":[],"detail_key":"p57","name":"modular_slash_negI_of_even","module":"SpherePacking.ModularForms.SlashActionAuxil"},{"id":"n45151","layer":"formal","project":"p57","title":"Γ2_generate","kind":"theorem","summary":"Eq Top.top (Subgroup.closure (insert α (insert β (singleton negI))))","labels":[],"detail_key":"p57","name":"Γ2_generate","module":"SpherePacking.ModularForms.SlashActionAuxil"},{"id":"n45152","layer":"formal","project":"p57","title":"α","kind":"def","summary":"Subtype fun x => Membership.mem (CongruenceSubgroup.Gamma 2) 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[inst : PseudoMetricSpace X] x : MeasurableSpace X μ : MeasureTheory.Measure X,…","labels":[],"detail_key":"p59","name":"lowerSemicontinuous_measure_ball","module":"SardMoreira.MeasureBallSemicontinuous"},{"id":"n45441","layer":"formal","project":"p59","title":"lowerSemicontinuous_measure_ball_toUpper_symm","kind":"theorem","summary":"∀ X : Type u_1 [inst : PseudoMetricSpace X] x : MeasurableSpace X μ : MeasureTheory.Measure X,…","labels":[],"detail_key":"p59","name":"lowerSemicontinuous_measure_ball_toUpper_symm","module":"SardMoreira.MeasureBallSemicontinuous"},{"id":"n45442","layer":"informal","project":"p59","title":"lem:measurable-meas-ball","kind":"lemma","summary":"Let \\(E\\) and \\(F\\) be finite dimensional real normed spaces. Let \\(s \\subset E \\times F\\) be a…","labels":["lem:measurable-meas-ball"],"detail_key":"p59"},{"id":"n45443","layer":"informal","project":"p59","title":"Consider the set \\(t \\subset E \\times E \\times F \\times R\\) given by \\(t = \\(z, x, y, r)…","kind":"proof","summary":"Consider the set \\(t \\subset E \\times E \\times F \\times R\\) given by \\(t = \\(z, x, y, r) \\mid \\…","labels":[],"detail_key":"p59"},{"id":"n45444","layer":"informal","project":"p59","title":"cor:measurable-density","kind":"corollary","summary":"Let \\(E\\), \\(F\\), \\(s\\), and \\(\\mu\\) be as in \\autoreflem:measurable-meas-ball. Then the set of…","labels":["cor:measurable-density"],"detail_key":"p59"},{"id":"n45445","layer":"informal","project":"p59","title":"The theorem immediately follows from the non-parametrized version and measurability of th…","kind":"proof","summary":"The theorem immediately follows from the non-parametrized version and measurability of the set…","labels":[],"detail_key":"p59"},{"id":"n45446","layer":"informal","project":"p59","title":"lem:meas-ball-lowersemicont","kind":"lemma","summary":"Let \\(X\\) be a (pseudo) metric space with a measure \\(\\mu\\). Consider the measure of an open ba…","labels":["lem:meas-ball-lowersemicont"],"detail_key":"p59"},{"id":"n45447","layer":"informal","project":"p59","title":"Consider a point \\((x, r)\\) and an extended nonnegative real number \\(m < \\mu(B_r(x))\\) S…","kind":"proof","summary":"Consider a point \\((x, r)\\) and an extended nonnegative real number \\(m < \\mu(B_r(x))\\) Since t…","labels":[],"detail_key":"p59"},{"id":"n45448","layer":"informal","project":"p59","title":"cor:meas-ball-lowersemicont","kind":"corollary","summary":"Let \\(X\\) be a (pseudo) metric space with a measure \\(\\mu\\). Then \\(\\mu(B_r(x))\\) is lower semi…","labels":["cor:meas-ball-lowersemicont"],"detail_key":"p59"},{"id":"n45449","layer":"informal","project":"p59","title":"This lemma immediately follows from \\autoreflem:meas-ball-lowersemicont and the fact that…","kind":"proof","summary":"This lemma immediately follows from \\autoreflem:meas-ball-lowersemicont and the fact that the u…","labels":[],"detail_key":"p59"},{"id":"n45450","layer":"informal","project":"p59","title":"cor:meas-ball-mesaurable","kind":"corollary","summary":"The measure of an open ball in a (pseudo) metric space is measurable in \\((x, r)\\in X\\times R\\).","labels":["cor:meas-ball-mesaurable"],"detail_key":"p59"},{"id":"n45451","layer":"informal","project":"p59","title":"This statement immediately follows from \\autorefcor:meas-ball-lowersemicont and the fact…","kind":"proof","summary":"This statement immediately follows from \\autorefcor:meas-ball-lowersemicont and the fact that a…","labels":[],"detail_key":"p59"},{"id":"n45452","layer":"informal","project":"p59","title":"cor:meas-ball-min","kind":"corollary","summary":"If \\(X\\) is a (pseudo) metric space with a measure \\(\\mu\\) and \\(s\\) is a nonempty compact set…","labels":["cor:meas-ball-min"],"detail_key":"p59"},{"id":"n45453","layer":"informal","project":"p59","title":"This statement immediately follows from \\autorefcor:meas-ball-lowersemicont and the fact…","kind":"proof","summary":"This statement immediately follows from \\autorefcor:meas-ball-lowersemicont and the fact that a…","labels":[],"detail_key":"p59"},{"id":"n45454","layer":"informal","project":"p59","title":"cor:meas-ball-gt-pos","kind":"corollary","summary":"If \\(X\\) is a compact (pseudo) metric space and \\(\\mu\\) is a measure on \\(X\\) that is positive…","labels":["cor:meas-ball-gt-pos"],"detail_key":"p59"},{"id":"n45455","layer":"informal","project":"p59","title":"If \\(X\\) is nonempty, then we choose \\(x\\) as in \\autorefcor:meas-ball-min, then choose \\…","kind":"proof","summary":"If \\(X\\) is nonempty, then we choose \\(x\\) as in \\autorefcor:meas-ball-min, then choose \\(0 < \\…","labels":[],"detail_key":"p59"},{"id":"n45456","layer":"informal","project":"p59","title":"lem:Ck-differentiable-iteratedFDeriv","kind":"lemma","summary":"If \\(f \\colon E \\to F\\) is \\(C^k\\) at \\(a\\) and \\(l < k\\), then \\(D^lf\\) is differentiable at \\…","labels":["lem:Ck-differentiable-iteratedFDeriv"],"detail_key":"p59"},{"id":"n45457","layer":"informal","project":"p59","title":"Since \\(f\\) is \\(C^k\\) at \\(a\\) and \\(l + 1 \\le k\\), \\(D^lf\\) is \\(C^1\\) at \\(a\\), thus i…","kind":"proof","summary":"Since \\(f\\) is \\(C^k\\) at \\(a\\) and \\(l + 1 \\le k\\), \\(D^lf\\) is \\(C^1\\) at \\(a\\), thus it's di…","labels":[],"detail_key":"p59"},{"id":"n45458","layer":"informal","project":"p59","title":"lem:iteratedFDeriv-prod","kind":"lemma","summary":"If \\(f\\colon E \\to F\\) and \\(g\\colon E \\to G\\) are \\(C^k\\) at \\(a\\), then \\(D^k(x \\mapsto (f(x)…","labels":["lem:iteratedFDeriv-prod"],"detail_key":"p59"},{"id":"n45459","layer":"informal","project":"p59","title":"This lemma immediately follows from uniqueness of the iterated derivative and lemmas in t…","kind":"proof","summary":"This lemma immediately follows from uniqueness of the iterated derivative and lemmas in the lib…","labels":[],"detail_key":"p59"},{"id":"n45460","layer":"informal","project":"p59","title":"def:cdh-at","kind":"definition","summary":"We say that a map \\(f\\colon E\\to F\\) is \\emph\\(C^k+(\\alpha)\\) at a point \\(a\\), if it is \\(C^k\\…","labels":["def:cdh-at"],"detail_key":"p59"},{"id":"n45461","layer":"informal","project":"p59","title":"def:contdiffholder-imp-cdh-at","kind":"lemma","summary":"If \\(f\\colon E \\to F\\) is \\(C^k+\\alpha\\) on an open set \\(U\\), i.e., \\(f\\) is \\(C^k\\) on \\(U\\)…","labels":["def:contdiffholder-imp-cdh-at"],"detail_key":"p59"},{"id":"n45462","layer":"informal","project":"p59","title":"The proof follows immediately from definitions.","kind":"proof","summary":"The proof follows immediately from definitions.","labels":[],"detail_key":"p59"},{"id":"n45463","layer":"informal","project":"p59","title":"lem:cdh-at-zero","kind":"lemma","summary":"A map \\(f\\colon E\\to F\\) is \\(C^k+(0)\\) at \\(a\\) iff it is \\(C^k\\) at \\(a\\).","labels":["lem:cdh-at-zero"],"detail_key":"p59"},{"id":"n45464","layer":"informal","project":"p59","title":"The forward implication follows from the definition. For the backward implication, we nee…","kind":"proof","summary":"The forward implication follows from the definition. For the backward implication, we need to s…","labels":[],"detail_key":"p59"},{"id":"n45465","layer":"informal","project":"p59","title":"lem:cdh-at-of-contDiffAt","kind":"lemma","summary":"If \\(f\\colon E\\to F\\) is \\(C^l\\) at \\(a\\) with \\(l > k\\), then it is \\(C^k+(\\alpha)\\) at \\(a\\).","labels":["lem:cdh-at-of-contDiffAt"],"detail_key":"p59"},{"id":"n45466","layer":"informal","project":"p59","title":"Since \\(D^kf\\) is differentiable at \\(a\\), we have \\(D^kf(x)-D^kf(a)=O(x - a)=O(\\|x - a\\|…","kind":"proof","summary":"Since \\(D^kf\\) is differentiable at \\(a\\), we have \\(D^kf(x)-D^kf(a)=O(x - a)=O(\\|x - a\\|^\\alph…","labels":[],"detail_key":"p59"},{"id":"n45467","layer":"informal","project":"p59","title":"lem:cdh-at-mono","kind":"lemma","summary":"Let \\(f\\colon E\\to F\\) be a map which is \\(C^k+(\\alpha)\\) at \\(a\\). Let \\(l\\) be a natural numb…","labels":["lem:cdh-at-mono"],"detail_key":"p59"},{"id":"n45468","layer":"informal","project":"p59","title":"Note that \\(l\\le k\\), hence \\(f\\) is \\(C^l\\) at \\(a\\). In order to show \\(D^lf(x) - D^lf(…","kind":"proof","summary":"Note that \\(l\\le k\\), hence \\(f\\) is \\(C^l\\) at \\(a\\). In order to show \\(D^lf(x) - D^lf(a) = O…","labels":[],"detail_key":"p59"},{"id":"n45469","layer":"informal","project":"p59","title":"lem:cdh-at-prodMk","kind":"lemma","summary":"If \\(f \\colon E \\to F\\) and \\(g \\colon E \\to G\\) are \\(C^k+(\\alpha)\\) at \\(a \\in E\\), then \\(x…","labels":["lem:cdh-at-prodMk"],"detail_key":"p59"},{"id":"n45470","layer":"informal","project":"p59","title":"The lemma immediately follows from \\autoreflem:iteratedFDeriv-prod.","kind":"proof","summary":"The lemma immediately follows from \\autoreflem:iteratedFDeriv-prod.","labels":[],"detail_key":"p59"},{"id":"n45471","layer":"informal","project":"p59","title":"lem:cdh-at-clm-comp","kind":"lemma","summary":"If \\(f \\colon E \\to F\\) is \\(C^k+(\\alpha)\\) at \\(a \\in E\\) and \\(g \\colon F \\to G\\) is a contin…","labels":["lem:cdh-at-clm-comp"],"detail_key":"p59"},{"id":"n45472","layer":"informal","project":"p59","title":"Immediately follows from \\(D^k(g \\circ f) = g\\circ D^kf\\).","kind":"proof","summary":"Immediately follows from \\(D^k(g \\circ f) = g\\circ D^kf\\).","labels":[],"detail_key":"p59"},{"id":"n45473","layer":"informal","project":"p59","title":"\\autoreflem:cdh-at-clm-comp immediately follows from a more general \\autoreflem:cdh-at-co…","kind":"remark","summary":"\\autoreflem:cdh-at-clm-comp immediately follows from a more general \\autoreflem:cdh-at-comp bel…","labels":[],"detail_key":"p59"},{"id":"n45474","layer":"informal","project":"p59","title":"lem:cdh-at-fderiv","kind":"lemma","summary":"If \\(f \\colon E \\to F\\) is \\(C^k+1+(\\alpha)\\) at \\(a \\in E\\), then \\(Df\\) is \\(C^k+(\\alpha)\\) a…","labels":["lem:cdh-at-fderiv"],"detail_key":"p59"},{"id":"n45475","layer":"informal","project":"p59","title":"The lemma immediately follows from \\(D^k+1f = D(D^kf)\\).","kind":"proof","summary":"The lemma immediately follows from \\(D^k+1f = D(D^kf)\\).","labels":[],"detail_key":"p59"},{"id":"n45476","layer":"informal","project":"p59","title":"lem:cdh-at-iteratedFDeriv","kind":"corollary","summary":"If \\(f \\colon E \\to F\\) is \\(C^k+l+(\\alpha)\\) at \\(a \\in E\\), then \\(D^kf\\) is \\(C^l+(\\alpha)\\)…","labels":["lem:cdh-at-iteratedFDeriv"],"detail_key":"p59"},{"id":"n45477","layer":"informal","project":"p59","title":"The proof by induction on \\(k\\) using \\autoreflem:cdh-at-fderiv and \\autoreflem:cdh-at-cl…","kind":"proof","summary":"The proof by induction on \\(k\\) using \\autoreflem:cdh-at-fderiv and \\autoreflem:cdh-at-clm-comp…","labels":[],"detail_key":"p59"},{"id":"n45478","layer":"informal","project":"p59","title":"lem:cdh-at-comp","kind":"lemma","summary":"Consider \\(g\\colon F \\to G\\), \\(f\\colon E \\to F\\), and \\(a \\in E\\) such that \\(g\\) is \\(C^k+(\\a…","labels":["lem:cdh-at-comp"],"detail_key":"p59"},{"id":"n45479","layer":"informal","project":"p59","title":"Since \\(g\\) is \\(C^k\\) at \\(f(a)\\) and \\(f\\) is \\(C^k\\) at \\(a\\), the composition \\(g\\cir…","kind":"proof","summary":"Since \\(g\\) is \\(C^k\\) at \\(f(a)\\) and \\(f\\) is \\(C^k\\) at \\(a\\), the composition \\(g\\circ f\\)…","labels":[],"detail_key":"p59"},{"id":"n45480","layer":"informal","project":"p59","title":"cor:cdh-at-arith","kind":"corollary","summary":"Arithmetic operations (addition, subtraction, scalar multiplication, multiplication) of \\(C^k+(…","labels":["cor:cdh-at-arith"],"detail_key":"p59"},{"id":"n45481","layer":"informal","project":"p59","title":"This fact immediately follows from \\autoreflem:cdh-at-comp and \\autoreflem:cdh-at-of-cont…","kind":"proof","summary":"This fact immediately follows from \\autoreflem:cdh-at-comp and \\autoreflem:cdh-at-of-contDiffAt.","labels":[],"detail_key":"p59"},{"id":"n45482","layer":"informal","project":"p59","title":"It would be nice to formalize the proofs of \\autoreflem:cdh-at-comp above and \\autorefthm…","kind":"remark","summary":"It would be nice to formalize the proofs of \\autoreflem:cdh-at-comp above and \\autorefthm:cdh-a…","labels":[],"detail_key":"p59"},{"id":"n45483","layer":"informal","project":"p59","title":"def:cdh-on","kind":"definition","summary":"We say that a map \\(f\\colon E\\to F\\) is \\emph\\(C^k+(\\alpha)\\) on a pair of sets \\(K\\), \\(U\\), i…","labels":["def:cdh-on"],"detail_key":"p59"},{"id":"n45484","layer":"informal","project":"p59","title":"lem:exists-cdh-on","kind":"lemma","summary":"Given a map \\(f\\colon E \\to F\\), a set \\(K\\), a natural number \\(k\\), and a real number \\(\\alph…","labels":["lem:exists-cdh-on"],"detail_key":"p59"},{"id":"n45485","layer":"informal","project":"p59","title":"Consider two cases. \\paragraphCase 1: \\(f\\) is \\(C^k+(\\alpha)\\) at each \\(a \\in K\\). In t…","kind":"proof","summary":"Consider two cases. \\paragraphCase 1: \\(f\\) is \\(C^k+(\\alpha)\\) at each \\(a \\in K\\). In this ca…","labels":[],"detail_key":"p59"},{"id":"n45486","layer":"informal","project":"p59","title":"lem:cdh-on-comp","kind":"lemma","summary":"Suppose that \\item \\(k > 0\\), \\(\\alpha \\in [0, 1]\\); \\item \\(g\\colon F\\to G\\) is \\(C^k+(\\alpha)…","labels":["lem:cdh-on-comp"],"detail_key":"p59"},{"id":"n45487","layer":"informal","project":"p59","title":"This statement immediately follows from \\autoreflem:cdh-at-comp.","kind":"proof","summary":"This statement immediately follows from \\autoreflem:cdh-at-comp.","labels":[],"detail_key":"p59"},{"id":"n45488","layer":"informal","project":"p59","title":"thm:cdh-at-inverse","kind":"theorem","summary":"Consider a map \\(f\\colon E\\to F\\) between two finite dimensional real normed spaces. Suppose th…","labels":["thm:cdh-at-inverse"],"detail_key":"p59"},{"id":"n45489","layer":"informal","project":"p59","title":"From the usual inverse function theorem for \\(C^k\\) maps, we know that \\(g=f^-1\\) is \\(C^…","kind":"proof","summary":"From the usual inverse function theorem for \\(C^k\\) maps, we know that \\(g=f^-1\\) is \\(C^k\\) in…","labels":[],"detail_key":"p59"},{"id":"n45490","layer":"informal","project":"p59","title":"thm:cdh-at-implicit-ker","kind":"lemma","summary":"Let \\(E\\), \\(F\\), and \\(G\\) be finite dimensional real normed spaces. Consider a set \\(s \\subse…","labels":["thm:cdh-at-implicit-ker"],"detail_key":"p59"},{"id":"n45491","layer":"informal","project":"p59","title":"We only need this lemma for \\(G= R\\). If the proof in this case is a bit easier, then it'…","kind":"remark","summary":"We only need this lemma for \\(G= R\\). If the proof in this case is a bit easier, then it's OK t…","labels":[],"detail_key":"p59"},{"id":"n45492","layer":"informal","project":"p59","title":"Let \\(\\pi\\colon F\\to S\\) be a continuous linear projection, \\(\\pi \\circ \\pi = \\pi\\). Cons…","kind":"proof","summary":"Let \\(\\pi\\colon F\\to S\\) be a continuous linear projection, \\(\\pi \\circ \\pi = \\pi\\). Consider t…","labels":[],"detail_key":"p59"},{"id":"n45493","layer":"informal","project":"p59","title":"thm:cdh-at-implicit-dichotomy","kind":"theorem","summary":"Let \\(E\\) and \\(F\\) be finite dimensional real normed spaces. Consider a set \\(s \\subset E \\tim…","labels":["thm:cdh-at-implicit-dichotomy"],"detail_key":"p59"},{"id":"n45494","layer":"informal","project":"p59","title":"This theorem is a special case of \\autorefthm:cdh-at-implicit-ker for \\(G= R\\).","kind":"proof","summary":"This theorem is a special case of \\autorefthm:cdh-at-implicit-ker for \\(G= R\\).","labels":[],"detail_key":"p59"},{"id":"n45495","layer":"informal","project":"p59","title":"def:moreira-chart","kind":"definition","summary":"Consider a natural number \\(k\\), a real number \\(\\alpha\\in[0, 1]\\), a set \\(s \\subset E \\times…","labels":["def:moreira-chart"],"detail_key":"p59"},{"id":"n45496","layer":"informal","project":"p59","title":"def:moreira-chart-map","kind":"definition","summary":"Given a Moreira chart, for each \\(i \\le l\\) we define the map \\(\\Psi_i\\colon E\\times R^d_i\\to R…","labels":["def:moreira-chart-map"],"detail_key":"p59"},{"id":"n45497","layer":"informal","project":"p59","title":"lem:moreira-chart-map-props","kind":"lemma","summary":"Given a \\(C^k+(\\alpha)\\) Moreira chart, \\item each map \\(\\Psi_i\\) is \\(C^k-i+(\\alpha)\\) on \\((t…","labels":["lem:moreira-chart-map-props"],"detail_key":"p59"},{"id":"n45498","layer":"informal","project":"p59","title":"lem:cdh-at-sub-affine-le-of-meas","kind":"lemma","summary":"Let \\(E\\) and \\(F\\) be a real normed spaces. Consider a function \\(f\\colon E\\to F\\), points \\(a…","labels":["lem:cdh-at-sub-affine-le-of-meas"],"detail_key":"p59"},{"id":"n45499","layer":"informal","project":"p59","title":"Let \\(s\\) be the set of \\(t \\in [0, 1]\\) such that \\(D_b - af(a + t(b - a)) \\ne 0\\). We h…","kind":"proof","summary":"Let \\(s\\) be the set of \\(t \\in [0, 1]\\) such that \\(D_b - af(a + t(b - a)) \\ne 0\\). We have \\|…","labels":[],"detail_key":"p59"},{"id":"n45500","layer":"informal","project":"p59","title":"The constant can be improved, but probably we don't need this. In order to do this, we ne…","kind":"remark","summary":"The constant can be improved, but probably we don't need this. In order to do this, we need to…","labels":[],"detail_key":"p59"},{"id":"n45501","layer":"informal","project":"p59","title":"cor:cdh-at-sub-le-of-meas-fderiv","kind":"corollary","summary":"Let \\(E\\) and \\(F\\) be a real normed spaces. 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_norms_one_bit","candidate_count":1},{"source":"n2114","target":"n14974","status":"resolved","lean_ref":"TNLean.PEPS.InjectiveUnionContractionChain.union_contraction_injective","candidate_count":1},{"source":"n2114","target":"n14975","status":"resolved","lean_ref":"TNLean.PEPS.InjectiveUnionContractionChain.union_contraction_ker_eq_bot","candidate_count":1},{"source":"n21140","target":"n22311","status":"resolved","lean_ref":"GeneralHypercontractivity.low_norms_hypercontractivity","candidate_count":1},{"source":"n21141","target":"n22306","status":"resolved","lean_ref":"GeneralHypercontractivity.high_norms_hypercontractivity","candidate_count":1},{"source":"n21142","target":"n22303","status":"resolved","lean_ref":"GeneralHypercontractivity.general_one_function_hypercontractivity","candidate_count":1},{"source":"n21143","target":"n22304","status":"resolved","lean_ref":"GeneralHypercontractivity.general_two_function_hypercontractivity","candidate_count":1},{"source":"n21144","target":"n27822","status":"ambiguous","lean_ref":"KKL.noisyInfluence","candidate_count":2},{"source":"n21145","target":"n27816","status":"ambiguous","lean_ref":"KKL.lowDegreePart","candidate_count":2},{"source":"n21146","target":"n27811","status":"ambiguous","lean_ref":"KKL.highDegreePart","candidate_count":2},{"source":"n21147","target":"n27813","status":"ambiguous","lean_ref":"KKL.influentialCoords","candidate_count":2},{"source":"n21148","target":"n27803","status":"ambiguous","lean_ref":"KKL.IsJunta","candidate_count":2},{"source":"n21149","target":"n27815","status":"ambiguous","lean_ref":"KKL.l2DistSq","candidate_count":2},{"source":"n21150","target":"n27824","status":"ambiguous","lean_ref":"KKL.noisyInfluence_one","candidate_count":2},{"source":"n21151","target":"n27828","status":"ambiguous","lean_ref":"KKL.sum_noisyInfluence","candidate_count":2},{"source":"n21152","target":"n27830","status":"ambiguous","lean_ref":"KKL.totalInfluence_eq_sum_influences","candidate_count":2},{"source":"n21153","target":"n27808","status":"ambiguous","lean_ref":"KKL.expect_sq_pm_one","candidate_count":2},{"source":"n21154","target":"n27827","status":"ambiguous","lean_ref":"KKL.sum_fourier_sq_eq_expect_sq","candidate_count":2},{"source":"n21155","target":"n27820","status":"ambiguous","lean_ref":"KKL.low_plus_high_eq","candidate_count":2},{"source":"n21156","target":"n27809","status":"ambiguous","lean_ref":"KKL.fourierCoeff_lowDegreePart","candidate_count":2},{"source":"n21157","target":"n27819","status":"ambiguous","lean_ref":"KKL.lowDegree_l2_error","candidate_count":2},{"source":"n21158","target":"n27829","status":"ambiguous","lean_ref":"KKL.tail_fourier_weight_bound","candidate_count":2},{"source":"n21159","target":"n27818","status":"ambiguous","lean_ref":"KKL.lowDegree_approx","candidate_count":2},{"source":"n21160","target":"n27814","status":"ambiguous","lean_ref":"KKL.influential_coords_card","candidate_count":2},{"source":"n21161","target":"n27817","status":"ambiguous","lean_ref":"KKL.lowDegreePart_depends_on_influential","candidate_count":2},{"source":"n21162","target":"n27823","status":"ambiguous","lean_ref":"KKL.noisyInfluence_le_influence","candidate_count":2},{"source":"n21163","target":"n27825","status":"ambiguous","lean_ref":"KKL.noisyInfluence_power_bound","candidate_count":2},{"source":"n21164","target":"n27807","status":"ambiguous","lean_ref":"KKL.cauchy_schwarz_influences","candidate_count":2},{"source":"n21165","target":"n27821","status":"ambiguous","lean_ref":"KKL.max_influence_from_sum_sq","candidate_count":2},{"source":"n21166","target":"n27805","status":"ambiguous","lean_ref":"KKL.KKL_trivial","candidate_count":2},{"source":"n21167","target":"n27826","status":"ambiguous","lean_ref":"KKL.noisyTotalInfluence","candidate_count":2},{"source":"n21168","target":"n27804","status":"ambiguous","lean_ref":"KKL.KKL_balanced","candidate_count":2},{"source":"n21169","target":"n27810","status":"ambiguous","lean_ref":"KKL.friedgut_junta","candidate_count":2},{"source":"n2117","target":"n8784","status":"resolved","lean_ref":"TNLean.Algebra.ProjectivelyEquivalent","candidate_count":1},{"source":"n21170","target":"n27806","status":"ambiguous","lean_ref":"KKL.balanced_totalInfluence_ge_one","candidate_count":2},{"source":"n21171","target":"n27812","status":"ambiguous","lean_ref":"KKL.influence_entropy_nonneg","candidate_count":2},{"source":"n21172","target":"n22479","status":"resolved","lean_ref":"DecisionTree.size","candidate_count":1},{"source":"n21173","target":"n22477","status":"resolved","lean_ref":"DecisionTree.signEval","candidate_count":1},{"source":"n21174","target":"n22467","status":"resolved","lean_ref":"DecisionTree.coeffs","candidate_count":1},{"source":"n21175","target":"n26417","status":"ambiguous","lean_ref":"DecisionTree.symmDiff_singleton_invol","candidate_count":2},{"source":"n21176","target":"n26416","status":"ambiguous","lean_ref":"DecisionTree.sum_symmDiff_reindex","candidate_count":2},{"source":"n21177","target":"n26408","status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orusAtSite_basisVec","candidate_count":1},{"source":"n38256","target":"n40360","status":"resolved","lean_ref":"Pphi2.torusEmbeddedTwoPoint_eq_lattice_cross_moment","candidate_count":1},{"source":"n38257","target":"n40362","status":"resolved","lean_ref":"Pphi2.torusEmbeddedTwoPoint_eq_spectral_sum","candidate_count":1},{"source":"n38258","target":"n40350","status":"resolved","lean_ref":"Pphi2.latticeTestFn_norm_sq_bounded","candidate_count":1},{"source":"n38259","target":"n40347","status":"resolved","lean_ref":"Pphi2.latticeTestFn","candidate_count":1},{"source":"n3826","target":"n13775","status":"resolved","lean_ref":"FrustrationFree.norm_sq_apply_projection_le_of_norm_comp_le","candidate_count":1},{"source":"n38260","target":"n40351","status":"resolved","lean_ref":"Pphi2.lattice_second_moment_eq_inner","candidate_count":1},{"source":"n38261","target":"n40355","status":"resolved","lean_ref":"Pphi2.torusContinuumGreen_nonneg","candidate_count":1},{"source":"n38262","target":"n40364","status":"resolved","lean_ref":"Pphi2.torusEmbeddedTwoPoint_uniform_bound","candidate_count":1},{"source":"n38263","target":"n40356","status":"resolved","lean_ref":"Pphi2.torusContinuumGreen_pos","candidate_count":1},{"source":"n38264","target":"n40357","status":"resolved","lean_ref":"Pphi2.torusEmbedLift.congr_simp","candidate_count":1},{"source":"n38265","target":"n40358","status":"resolved","lean_ref":"Pphi2.torusEmbedLift_eval_eq","candidate_count":1},{"source":"n38266","target":"n40365","status":"resolved","lean_ref":"Pphi2.torus_propagator_convergence","candidate_count":1},{"source":"n38267","target":"n40349","status":"resolved","lean_ref":"Pphi2.latticeTestFn_expand","candidate_count":1},{"source":"n38268","target":"n40345","status":"resolved","lean_ref":"Pphi2.dm_basis_eq_fourierBasis","candidate_count":1},{"source":"n38269","target":"n40359","status":"resolved","lean_ref":"Pphi2.torusEmbeddedTwoPoint.congr_simp","candidate_count":1},{"source":"n38270","target":"n39598","status":"resolved","lean_ref":"Pphi2.latticeConfigEuclideanTimeShift","candidate_count":1},{"source":"n38271","target":"n39606","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeShift_mod","candidate_count":1},{"source":"n38272","target":"n39599","status":"resolved","lean_ref":"Pphi2.latticeConfigEuclideanTimeShift_mod","candidate_count":1},{"source":"n38273","target":"n39604","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeShift","candidate_count":1},{"source":"n38274","target":"n39600","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeSeparation","candidate_count":1},{"source":"n38275","target":"n39601","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeSeparation_eq_min","candidate_count":1},{"source":"n38276","target":"n39603","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeSeparation_val","candidate_count":1},{"source":"n38277","target":"n39605","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeShift.congr_simp","candidate_count":1},{"source":"n38278","target":"n39602","status":"resolved","lean_ref":"Pphi2.latticeEuclideanTimeSeparation_mod","candidate_count":1},{"source":"n38279","target":"n40430","status":"resolved","lean_ref":"Pphi2.timeCoupling_eq_zero_iff","candidate_count":1},{"source":"n3828","target":"n13851","status":"resolved","lean_ref":"FrustrationFree.movingWindow_sum_Ico_le","candidate_count":1},{"source":"n3828","target":"n13852","status":"resolved","lean_ref":"FrustrationFree.movingWindow_sum_le","candidate_count":1},{"source":"n3828","target":"n13853","status":"resolved","lean_ref":"FrustrationFree.movingWindow_sum_range_le","candidate_count":1},{"source":"n38280","target":"n40426","status":"resolved","lean_ref":"Pphi2.spatialAction.congr_simp","candidate_count":1},{"source":"n38281","target":"n40424","status":"resolved","lean_ref":"Pphi2.SpatialField","candidate_count":1},{"source":"n38282","target":"n40431","status":"resolved","lean_ref":"Pphi2.timeCoupling_nonneg","candidate_count":1},{"source":"n38283","target":"n40429","status":"resolved","lean_ref":"Pphi2.timeCoupling","candidate_count":1},{"source":"n38284","target":"n40434","status":"resolved","lean_ref":"Pphi2.transferKernel_symmetric","candidate_count":1},{"source":"n38285","target":"n40427","status":"resolved","lean_ref":"Pphi2.spatialKinetic","candidate_count":1},{"source":"n38286","target":"n40432","status":"resolved","lean_ref":"Pphi2.transferKernel","candidate_count":1},{"source":"n38287","target":"n40428","status":"resolved","lean_ref":"Pphi2.spatialPotential","candidate_count":1},{"source":"n38288","target":"n40433","status":"resolved","lean_ref":"Pphi2.transferKernel_pos","candidate_count":1},{"source":"n38289","target":"n40425","status":"resolved","lean_ref":"Pphi2.spatialAction","candidate_count":1},{"source":"n38290","target":"n40155","status":"resolved","lean_ref":"Pphi2.wickMonomial_add_binomial","candidate_count":1},{"source":"n38291","target":"n40156","status":"resolved","lean_ref":"Pphi2.wickMonomial_add_sub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ommitment.maskComm_perfect_hiding","candidate_count":1},{"source":"n42802","target":"n43023","status":"resolved","lean_ref":"CatCrypt.Examples.Commitment.idComm_perfectly_binding","candidate_count":1},{"source":"n42803","target":"n43075","status":"resolved","lean_ref":"CatCrypt.Examples.SecretSharing.revealShare","candidate_count":1},{"source":"n42804","target":"n43076","status":"resolved","lean_ref":"CatCrypt.Examples.SecretSharing.shareXor_reconstruct","candidate_count":1},{"source":"n42805","target":"n43077","status":"resolved","lean_ref":"CatCrypt.Examples.SecretSharing.ss_perfect_privacy","candidate_count":1},{"source":"n42806","target":"n43045","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.BijMACFamily","candidate_count":1},{"source":"n42806","target":"n43046","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.BijMACFamily.toMACScheme","candidate_count":1},{"source":"n42807","target":"n43047","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.bijMAC_euf_cma_adv","candidate_count":1},{"source":"n42807","target":"n43048","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.bijMAC_forgery_prob","candidate_count":1},{"source":"n42808","target":"n43049","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.boolXorMAC","candidate_count":1},{"source":"n42808","target":"n43050","status":"resolved","lean_ref":"CatCrypt.Examples.MAC.boolXorMAC_forgery_prob","candidate_count":1},{"source":"n42809","target":"n43011","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme","candidate_count":1},{"source":"n42809","target":"n43012","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.BijCPAScheme.toBijPRFFamily","candidate_count":1},{"source":"n4281","target":"n13222","status":"resolved","lean_ref":"MPSTensor.chainGroundSpace_toTensorFromBlocks_le_iSup_of_blockDiagonal_boundary_groundSpaceMap","candidate_count":1},{"source":"n42810","target":"n43013","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.bijCPA_correct","candidate_count":1},{"source":"n42811","target":"n43014","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.bijCPA_perfect_indcpa","candidate_count":1},{"source":"n42811","target":"n43015","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.boolXorCPA_perfect_indcpa","candidate_count":1},{"source":"n42812","target":"n43016","status":"resolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.cpa_from_prf_bound","candidate_count":1},{"source":"n42813","target":"n25934","status":"unresolved","lean_ref":"CatCrypt.Examples.CPAFromPRF.cpa_from_prf_reduction","candidate_count":0},{"source":"n42814","target":"n43029","status":"resolved","lean_ref":"CatCrypt.Examples.ElGamal.elgamalDDH","candidate_count":1},{"source":"n42814","target":"n43030","status":"resolved","lean_ref":"CatCrypt.Examples.ElGamal.elgamalEnc","candidate_count":1},{"source":"n42815","target":"n43031","status":"resolved","lean_ref":"CatCrypt.Examples.ElGamal.elgamal_dec_enc","candidate_count":1},{"source":"n42816","target":"n43033","status":"resolved","lean_ref":"CatCrypt.Examples.ElGamal.elgamal_indcpa_le_ddh","candidate_count":1},{"source":"n42817","target":"n43032","status":"resolved","lean_ref":"CatCrypt.Examples.ElGamal.elgamal_indcpa_eq_ddh","candidate_count":1},{"source":"n42818","target":"n42979","status":"resolved","lean_ref":"CatCrypt.Crypto.SecurityDefs.INDCCA_Game","candidate_count":1},{"source":"n42818","target":"n42986","status":"resolved","lean_ref":"CatCrypt.Crypto.SecurityDefs.ccaDecOracle","candidate_count":1},{"source":"n42819","target":"n42980","status":"resolved","lean_ref":"CatCrypt.Crypto.SecurityDefs.INDCCA_reduces_to_INDCPA","candidate_count":1},{"source":"n42820","target":"n43037","status":"resolved","lean_ref":"CatCrypt.Examples.EtMCCA.boolEtM_indcca_reduces","candidate_count":1},{"source":"n42821","target":"n43017","status":"resolved","lean_ref":"CatCrypt.Examples.CTRMode.CTRBlockScheme","candidate_count":1},{"source":"n42821","target":"n43018","status":"resolved","lean_ref":"CatCrypt.Examples.CTRMode.CTRBlockScheme.toBijCPAScheme","candidate_count":1},{"source":"n42822","target":"n43019","status":"resolved","lean_ref":"CatCrypt.Examples.CTRMode.boolCTR_perfect_indcpa","candidate_count":1},{"source":"n42822","target":"n43021","status":"resolved","lean_ref":"CatCrypt.Examples.CTRMode.ctr_perfect_indcpa","candidate_count":1},{"source":"n42823","target":"n43020","status":"resolved","lean_ref":"CatCrypt.Examples.CTRMode.ctr_indcpa_bound","candidate_count":1},{"source":"n42824","target":"n43025","status":"resolved","lean_ref":"CatCrypt.Examples.DiffieHellman.KA_Advantage","candidate_count":1},{"source":"n42824","target":"n43026","status":"resolved","lean_ref":"CatCrypt.Examples.DiffieHellman.KA_Game_Real","candidate_count":1},{"source":"n42825","target":"n43027","status":"resolved","lean_ref":"CatCrypt.Examples.DiffieHellman.ka_advantage_eq_ddh","candidate_count":1},{"source":"n42826","target":"n43028","status":"resolved","lean_ref":"CatCrypt.Examples.DiffieHe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